This is beta site. https://beta.mathcity.org/ 2026-06-15T20:56:53+00:00 https://beta.mathcity.org/ https://beta.mathcity.org/_media/logo.png text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/quote-of-the-day/mar?rev=1737476042&do=diff Quotes for the March “”“”“” “” “” --- “”“”“” --- “”“”“” --- “”“”“” --- “”“”“” --- “”“”“” --- “”“”“” --- “”“”“” --- “” text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/quote-of-the-day/may?rev=1737476042&do=diff Quotes for the May “”“”“” مورس کلائن (1908-1992) ---Morris Kline (1908-1992) --- “”“”“” ڈی آرسی تھامسن (1860-1948) ---D'Arcy Thompson (1860-1948) --- “”“”“” Vito Volterra (1860-1940) ---Vito Volterra (1860-1940) text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/quote-of-the-day/apr?rev=1737476042&do=diff Quotes for the April “”“”“” لیو لینڈاؤ (1908-1968) ---Lev Landau (1908-1968) --- “”“”“” Hannes Alfvén (1908-1995) ---Hannes Alfvén (1908-1995) --- “”“”“” جوزف برٹرینڈ (1822-1900) ---Joseph Bertrand (1822-1900) text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/quote-of-the-day/feb?rev=1737476042&do=diff Quotes for the February “”“”“” --- “”“”“” --- “”“”“” --- “”“”“” --- “”“”“” --- “”“”“” --- “”“”“” --- “”“”“” --- “”“”“ text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-3-p2?rev=1737476037&do=diff Question 2, Exercise 2.3 Solutions of Question 2 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2(i) $$\begin{bmatrix}4 & -2 & 5 \\ 2 & 1 & 0 \\ -1 & 2 & 3 \end{bmatrix}$$$$A=\begin{bmatrix} 4 & -2 & 5 \\ 2 & 1 & 0 \\ -1 & 2 & 3 \end{bmatrix}.$$\begin{align}|A|&=\begin{vmatrix}4 & -2 & 5 \\ 2 & 1 & 0 \\ -1 & 2 & 3 \end{vmatrix}\\ &=… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-5-p1?rev=1737476039&do=diff Question 1, Exercise 2.5 Solutions of Question 1 of Exercise 2.5 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left[\begin{array}{ccc}1 & 3 & 5 \\ -6 & 8 & 3 \\ -4 & 6 & 5\end{array}\right]$\begin{align*} & \quad \left[\begin{array}{ccc}1 & 3 & 5 \\ -6 & 8 & 3 \\ -4 & 6 & 5\end{array}\right]\\ \sim & \text{R} \left[\begin{array}{ccc} 1 & 3 & 5 \\ 0 & … text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/playground/wrap?rev=1737476042&do=diff Examples for the Wrap Plugin Basic syntax An uppercase <WRAP> (or alternatively <block> or <div>) creates a div and should be used for “big” containers, surrounding paragraphs, lists, tables, etc. <WRAP classes #id width :language> "big" content </WRAP> or <block classes #id width :language> "big" content </block> or <div classes #id width :language> "big" content </div> text/html 2024-08-05T06:27:23+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/wiki/syntax?rev=1722839243&do=diff Formatting Syntax DokuWiki supports some simple markup language, which tries to make the datafiles to be as readable as possible. This page contains all possible syntax you may use when editing the pages. Simply have a look at the source of this page by pressing text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-5-p2?rev=1737476039&do=diff Question 2, Exercise 2.5 Solutions of Question 2 of Exercise 2.5 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left[\begin{array}{ccc}5 & 9 & 3 \\ 3 & -5 & 6 \\ 2 & 10 & 6\end{array}\right]$\begin{align*}&\quad\left[ \begin{array}{ccc} 5 & 9 & 3 \\ 3 & -5 & 6 \\ 2 & 10 & 6 \end{array} \right]\\ \sim & \text{R}\left[ \begin{array}{ccc} 1 & \frac{9}{… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-3-p1?rev=1737476037&do=diff Question 1, Exercise 2.3 Solutions of Question 1 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) $\begin{bmatrix}1 & 3 & -1 \\2 & 1 & 4 \\3 & 4 & -5\end{bmatrix}$\begin{align}&\begin{bmatrix} 1 & 3 & -1 \\ 2 & 1 & 4 \\ 3 & 4 & -5 \end{bmatrix}\\ \underset{\sim}{R}&\begin{bmatrix} 1 & 3 & -1 \\ 0 & -5 & 6 \\ 0 & -5 & -2 \end{bmat… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p4?rev=1737476039&do=diff Question 4, Exercise 2.6 Solutions of Question 4 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 x_{1}-x_{2}-x_{3}=2$$3 x_{1}-4 x_{2}+3 x_{3}=7$$4 x_{1}+2 x_{2}-5 x_{3}=10$\begin{align*} 2x_1 - x_2 - x_3 &= 2, \\ 3x_1 - 4x_2 + 3x_3 &= 7, \\ 4x_1 + 2x_2 - 5x_3 &= 10, \end{align*}\begin{align*} A_b &= \begin{bmatrix} 2 & -1 & -1 & : & 2 … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-5-p6?rev=1737476038&do=diff Question 9 Exercise 6.5 Solutions of Question 9 of Exercise 6.5 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$2$$\dfrac{1}{7}$$\dfrac{1}{5}$\begin{align} P(\text { Ajmal scicction })&=\dfrac{1}{7} \\ \Rightarrow P(\text { Ajmal not selected })&=\dfrac{6}{7} \\ P(\text { Bushra selection })&=\dfrac{1}{5} \\ \Rightarrow P(\text { Bushra not selected }… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-3-p6?rev=1737476038&do=diff Question 9 & 10 Exercise 4.3 Solutions of Question 9 & 10 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9 $$306,315,324,333, \ldots, 693$$$a=306$$$d=(315-306) = 9 \text { and } a_n=693 .$$$n$\begin{align}a_n&=a_1+(n-1) d \text { becomes } \\ \Rightarrow a_1+(n-1) d&=693 \\ \Rightarrow 306+(n-1) \cdot 9&=693 \\ \Rightarrow 9 n&=396 \\ \Rightarr… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p9?rev=1737476038&do=diff Question 13 Exercise 6.2 Solutions of Question 13 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\mathrm{E}$$n=10$$m_1=4$$E, m_2=2$$L$$m_3=2$$C$\begin{align}\text{total number of permutations are} &=\left(\begin{array}{c} n \\ m_1, m_2, m_3 \end{array}\right)\\&=\left(\begin{array}{c} 10 \\ 4,2,2 \end{array}\right) \\ & =\dfrac{10 !}… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p3?rev=1737476040&do=diff Question 3, Exercise 9.1 Solutions of Question 3 of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y=7 \cos 4x$\begin{align*} & -1\leq \cos 4x \leq 1 \,\, \forall \,\, x\in \mathbb{R} \\ \implies & -7\leq 7 \cos 4x \leq 7 \\ \end{align*}$= ]-\infty, \infty[ = \mathbb{R}$$=[-7,7]$$y=\cos \frac{x}{3}$\begin{align*} & -1\leq \cos \frac{x}{3} \… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p7?rev=1737476039&do=diff Question 7 and 8, Exercise 2.6 Solutions of Question 7 and 8 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{ccc}3 & 2 & 1 \\ 4 & -1 & 2 \\ 7 & 3 & -3\end{array}\right]$$A^{-1}$$3 x+4 y+7 z=14 ; 2 x-y+3 z=4 ; \quad x+2 y-3 z=0$\begin{align*} A &= \begin{bmatrix} 3 & 2 & 1 \\ 4 & -1 & 2 \\ 7 & 3 & -3 \end{bmatrix}\\ |… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-ptb/sol/ch01/ex1-1?rev=1737476037&do=diff Exercise 1.1 (Solutions) <lead>Notes (Solutions) of Exercise 1.1: Textbook of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board (PTB) Lahore.</lead> The main topics of this exercise are properties of real numbers, binary operation, addition and multiplication law, properties of equality, properties of inequality (order properties), field, rule of fractions. These notes are based on the new Student Learning Outcomes (SLOs). Version: 4.0, Available at Ma… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p9?rev=1737476039&do=diff Question 9, Exercise 1.2 Solutions of Question 9 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 9(i) $(2+4 i)^{-1}$$z=2+4i$\begin{align} Re(2+4i)^{-1} & = Re(z^{-1}) = \dfrac{Re(z)}{|z|^2} \\ & =\dfrac{2}{2^2+4^2} = \dfrac{2}{20}\\ &= \dfrac{1}{10}. \end{align}\begin{align} Im(2+4i)^{-1} & = Im(z^{-1}) = -\dfrac{Im(z)}{|z|^2} \\ & =-\df… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-5-p3?rev=1737476039&do=diff Question 3, Exercise 2.5 Solutions of Question 3 of Exercise 2.5 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left[\begin{array}{ccc}0 & -1 & -1 \\ -1 & 3 & 0 \\ 1 & -1 & 4\end{array}\right]$$A A^{-1}=A^{-1} A=I$\begin{align*} A&=\left[ \begin{array}{ccc} 0 & -1 & -1 \\ -1 & 3 & 0 \\ 1 & -1 & 4 \end{array} \right]\\ |A|&=0+1(-4)-1(1-3)\\ &=-4+3\… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-8-p3?rev=1737476040&do=diff Question 5 and 6, Exercise 4.8 Solutions of Question 5 and 6 of Exercise 4.8 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $3+4+6+10+18+34+66+\dots$$n$$$ S_{n}=3+4+6+10+18+\ldots +T_{n} $$$$ S_{n}=3+4+6+10+\ldots +T_{n-1}+T_{n}. $$\begin{align*} S_{n}-S_{n}& =3+4+6+10+18+\ldots +T_{n} \\ & -\left(3+4+6+10+\ldots +T_{n-1}+T_{n}\right) \end{align*}\begin{align… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p2?rev=1737476039&do=diff Question 2, Exercise 1.2 Solutions of Question 2 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 2 $$ \left(z_{1} z_{2}\right)\left(z_{3} z_{4}\right)=\left(z_{1} z_{3}\right)\left(z_{2} z_{4}\right)=z_{3}\left(z_{1} z_{2}\right) z_{4} $$\begin{align} &(z_1 z_2)(z_3 z_4) \\ =&(z_1 z_2)z_5 \quad \text {Let }z_5=z_3 z_4 \\ =&z_1 (z_2 z_5) \… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p2?rev=1737476039&do=diff Question 2, Exercise 2.6 Solutions of Question 2 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\lambda$$\lambda$$2 x_{1}-\lambda x_{2}+x_{3}=0$$2 x_{1}+3 x_{2}-x_{3}=0$$3 x_{1}-2 x_{2}+4 x_{3}=0$\begin{align*} &2 x_{1}-\lambda x_{2}+x_{3}=0 \cdots(i)\\ &2 x_{1}+3 x_{2}-x_{3}=0\cdots(ii)\\ &3 x_{1}-2 x_{2}+4 x_{3}=0\cdots(iii)\\ \end{ali… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p3?rev=1737476039&do=diff Question 3, Exercise 2.6 Solutions of Question 3 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 x+3 y+4 z=2$$2 x+y+z=5$$3 x-2 y+z=-3$\begin{align*} \begin{aligned} 2x + 3y + 4z &= 2 \\ 2x + y + z &= 5 \\ 3x - 2y + z &= -3 \end{aligned}\end{align*}\begin{align*} A_{b} &=\quad \left[\begin{array}{cccc} 2 & 3 & 4 & 2 \\ 2 & 1 & 1 & 5 \\ 3… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-4-p5?rev=1737476037&do=diff Question 5 Exercise 3.4 Solutions of Question 5 of Exercise 3.4 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) Use the vector product to compute the area of the triangle with the given vertices $P(-2,-3), \quad Q(3,2)\quad$$\quad R(-1,-8)$$P Q$$\bar{P} R$\begin{align}\text{Area of triangle}&=\dfrac{1}{2}|\overrightarrow{P Q} \times \overrightarrow{P R}| \\ \text { S… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-4-p2?rev=1737476038&do=diff Question 2 & 3 Exercise 4.4 Solutions of Question 2 & 3 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2 $27$$243$$$a_3=27 \quad\text{and}\quad a_5=243$$\begin{align}a_3&=a_1 r^2=27\\ a_5&=a_1 r^4=243.\end{align}\begin{align}\dfrac{a_1 r^4}{a_1 r^2}&=\dfrac{243}{27}=9 \\ \Rightarrow r^2&=9 \text { or } r= \pm 3 .\end{align}$$a_1(9)=27 \quad \te… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/mathcraft/sample-01-latex?rev=1737476040&do=diff MathCraft: PDF to LaTeX file: Sample-01 If the PDF file provided by you as follows: Then the output LaTeX file is as follows: \documentclass[10pt]{amsart} %\usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} %\usepackage[version=4]{mhchem} \usepackage{stmaryrd} \usepackage{bbold} \usepackage{hyperref} \usepackage{enumerate} \hypersetup{colorlinks=true, linkcolor=blue, filecolor=magenta, urlcolor=cyan,} \urlstyle{same} \title{… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-3-p2?rev=1737476037&do=diff Question 2 and 3 Exercise 3.3 Solutions of Question 2 and 3 of Exercise 3.3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2 $$\vec{a}=2 \hat{i} + 2 \hat{j}-5 \hat{k}, \quad \vec{b}=2 \hat{i}+\hat{j}-7 \hat{k}$$\begin{align}\vec{a}+\vec{b}&=(2 \hat{i}+2 \hat{j}-5 \hat{k})+(2 \hat{i}+\hat{j}-7 \hat{k}) \\ \Rightarrow &=4 \hat{i}+3 \hat{j}-12 \hat{k}\\ \Rightarrow|\vec{a}+\… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-3-p4?rev=1737476038&do=diff Question 5 & 6 Exercise 4.3 Solutions of Question 5 & 6 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5 $20$$120$$$a-2 d, a-d, a+d, a+2 d,$$$Condition-1$$20$\begin{align}a-3 d+a-d+a+d+a+3 d&=20 \\ \Rightarrow 4 a&=20\\ \Rightarrow a&=5 .\end{align}$Condition-2$$120$\begin{align}(a-3 d)^2+(a-d)^2+(a+d)^2+(a+2 d)^2&=120 \\ \Rightarrow a^2-6 a d+… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-5-p5?rev=1737476038&do=diff Question 5 & 6 Exercise 4.5 Solutions of Question 5 & 6 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5 $r$$S_{10}=244 S_5$$$S_n=\dfrac{a_1(r^n-1)}{r-1}$$$$S_{10}=\dfrac{a_1(r^{10}-1)}{r-1} \quad \text{and}\quad S_5=\dfrac{a_1(r^5-1)}{r-1}$$$S_{10}$$S_S$\begin{align}\dfrac{a_1(r^{10}-1)}{r-1}&=244 \dfrac{a_1(r^5-1)}{r-1} \\ \Rightarrow r^{10}-… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-8-p1?rev=1737476040&do=diff Question 1 and 2, Exercise 4.8 Solutions of Question 1 and 2 of Exercise 4.8 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $3+7+13+21+\ldots$$n$$$ S_{n}=3+7+13+21+31+\ldots +T_{n} $$$$ S_{n}=3+7+13+21+\ldots +T_{n-1}+T_{n}.$$\begin{align*} S_{n}-S_{n}& =3+7+13+21+31+\ldots +T_{n} \\ & -\left(3+7+13+21+\ldots +T_{n-1}+T_{n}\right) \end{align*}\begin{align*} \… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-8-p2?rev=1737476040&do=diff Question 3 and 4, Exercise 4.8 Solutions of Question 3 and 4 of Exercise 4.8 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $1+4+13+40+121+ \ldots$$n$$$ S_{n}=1+4+13+40+121+\ldots +T_{n} $$$$ S_{n}=1+4+13+40+\ldots +T_{n-1}+T_{n}. $$\begin{align*} S_{n}-S_{n}& =1+4+13+40+121+\ldots +T_{n} \\ & -\left(1+4+13+40+\ldots +T_{n-1}+T_{n}\right) \end{align*}\begin… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p5?rev=1737476037&do=diff Question 5, Exercise 2.2 Solutions of Question 5 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) $\begin{vmatrix}a & b & c\\l & m & n\\x & y & z \end{vmatrix}=\begin{vmatrix}a & l & x\\b & m & y\\c & n & z \end{vmatrix}$\begin{align}L.H.S.&=\begin{vmatrix} a & b & c \\ l & m & n \\ x & y & z \end{vmatrix}\\ &=\begin{vmatrix} a & b &… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-5-p4?rev=1737476038&do=diff Question 4 Exercise 4.5 Solutions of Question 4 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4(i) $0 . \overline{8}$$$0 . \overline{8}=0.888888 \ldots$$\begin{align}0 . \overline{8}&=0.8+0.08+0.008 \div 0.0008+ \ldots\\ \text { or } 0 . \overline{8}&=0.8+(0.1)(0.8) +(0.1)^2(0.8)+\ldots \ldots \ldots \ldots .(\mathrm{i})\end{align}$$a_1=0.8, \… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/definitions?rev=1737476039&do=diff Definitions: Mathematics 11 NBF Model Textbook of Mathematics for Class XI is published by National Book Foundation (NBF), Islamabad, Pakistan. NBF can be considered as Federal Textbook Board Islamabad. The book has total of nine (9) chapters. Definition of the book provide the quick overview of the book.$x+iy$$x,y\in\mathbb{R}$$i^2=1$$\mathbb{C}$$x+i y$$x$$y$$x$$y$$Re(z)=x$$Im(z)=y$$z=x+i y$$\bar{z}$$\bar{z}=x-i y$$z=x+i y$$|z|$$|z|=\sqrt{x^{2}+y^{2}}$$z=x+iy$$Re(z)=x$$Im(z)=y$$Re(z^{-1})= \d… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-1-p11?rev=1737476036&do=diff Question 13, Exercise 10.1 Solutions of Question 13 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$r\,\,\sin \left( \theta +\phi \right)$$\theta$$\phi$$4\sin \theta +3\cos \theta .$$4\sin \theta +3\cos \theta$$r\sin(\theta + \varphi)$$$4\sin \theta +3\cos \theta=r\cos\varphi\sin\theta+r\sin\varphi\cos\theta --- (1)$… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-3-p3?rev=1737476037&do=diff Question 3, Exercise 2.3 Solutions of Question 3 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3(i) $$\left[ \begin{matrix} 1 & 0 & -2 \\ 2 & 2 & 1 \\ -1 & 2 & 3 \\ \end{matrix} \right]$$\begin{align}&\begin{bmatrix} 1 & 0 & -2 \\ 2 & 2 & 1 \\ -1 & 2 & 3 \end{bmatrix}\\ \underset{\sim}{R}& \begin{bmatrix} 1 & 0 & -2 \\ 0 &… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-5-p1?rev=1737476038&do=diff Question 1 Exercise 4.5 Solutions of Question 1 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) $3+6+12+\ldots+3.2^9$$a_1=3, \quad r=\dfrac{6}{3}=2$$a_n=3.2^9$$n$$$a_n=a_1 r^{n-1}$$\begin{align}3.2^9&=3(2)^{n-1} \text { or }(2)^{n-1}=\dfrac{3.2^9}{3} \\ \Rightarrow(2)^{n-1}&=2^9 \\ \Rightarrow n-1&=9 \text { or } n=10 \\ \text {. Now }\qua… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-5-p6?rev=1737476038&do=diff Question 7 & 8 Exercise 4.5 Solutions of Question 7 & 8 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7 $\operatorname{sum} S_n$$n$$\{(\dfrac{1}{2})^n\}$$$\{(\dfrac{1}{2})^n\}=\dfrac{1}{2}, \dfrac{1}{2^2}, \dfrac{1}{2^3}, \ldots$$$$a_1=\dfrac{1}{2}$$$$r=\dfrac{\dfrac{1}{2^2}}{\dfrac{1}{2}}=\dfrac{1}{2}$$\begin{align}S_n&=\dfrac{a_1(1-r^n)}{1-r… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p11?rev=1737476039&do=diff Question 13, Exercise 10.1 Solutions of Question 13 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$r\,\,\sin \left( \theta +\phi \right)$$\theta$$\phi$$4\sin \theta +3\cos \theta .$$4\sin \theta +3\cos \theta$$r\sin(\theta + \varphi)$$$4\sin \theta +3\cos \theta=r\cos\varphi\sin\theta+r\sin\varphi\cos\theta --- (1)$… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-ptb/sol/ch02/ex2-8?rev=1737476037&do=diff Exercise 2.8 (Solutions) <lead>Notes (Solutions) of Exercise 2.8: Textbook of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board (PTB) Lahore.</lead> The main topic of this exercise are binary operation, semi-group, monoid, groups and abelian groups. These notes are based on the new Student Learning Outcomes (SLOs). Version: 4.1, Available at MathCity.org $\oplus$$G=\{0,1\}$\[ \begin{array}{|c|c|c|} \hline \oplus & 0 & 1 \\ \hline 0 & 1 & 1 \\ \hl… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-3-p4?rev=1737476037&do=diff Question 4, Exercise 2.3 Solutions of Question 4 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4 $\begin{bmatrix}2 & 3 & 4 & 5 \\3 & 4 & 5 & 6 \\4 & 5 & 6 & 7 \\9 & 10 & 11 & 12\end{bmatrix}$\begin{align}&\begin{bmatrix} 2 & 3 & 4 & 5 \\ 3 & 4 & 5 & 6 \\ 4 & 5 & 6 & 7 \\ 9 & 10 & 11 & 12 \end{bmatrix}\\ \underset{\sim}{R}&\begin{bm… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-3-p5?rev=1737476037&do=diff Question 7 & 8 Exercise 3.3 Solutions of Question 7 & 8 of Exercise 3.3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7(i) $\vec{a}$$\vec{b}$$\vec{a}=-\dfrac{3}{2} \hat{j}+\dfrac{4}{5} \hat{k} \cdot \vec{b}=\hat{i}-2 \hat{j}-2 \hat{k}$$\vec{a}$$\vec{b}$$\vec{b}$$\vec{a}$$\vec{a}=-\dfrac{3}{2} \hat{j}+\dfrac{4}{5} \hat{k}\quad$$\vec{b}=\hat{i}-2 \hat{j}-2 \hat{k}$\begin{a… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/review-ex3-p3?rev=1737476038&do=diff Question 4 & 5 Review Exercise 3 Solutions of Question 4 & 5 of Review Exercise 3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4 $\vec{r}=x \hat{i}+y \hat{j}+z \hat{k}$$(\vec{r} \times \hat{i}) \cdot(\bar{r} \times \hat{j})+x y$$$(\vec{r} \times \hat{i}) \cdot(\vec{r} \times \hat{j})+x y $$\begin{align}\text { Now } \vec{r} \times \hat{i}&=\left|\begin{array}{ccc} \hat{… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p5?rev=1737476038&do=diff Question 8 Exercise 4.2 Solutions of Question 8 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 8 $\dfrac{b+c-a}{a}, \dfrac{c+a-b}{b}, \dfrac{a+b-c}{c}$$\dfrac{1}{a}, \dfrac{1}{b}, \dfrac{1}{c}$$\dfrac{b+c-a}{a}, \dfrac{c+a-b}{b}, \dfrac{a+b-c}{c}$\begin{align}\therefore \dfrac{c+a-b}{b}-\dfrac{b+c-a}{a}&=\dfrac{a+b-c}{c}-\dfrac{c+a-b}{b} \\ \te… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-3-p3?rev=1737476038&do=diff Question 3 & 4 Exercise 4.3 Solutions of Question 3 & 4 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 $5$$25$$350$$5$$25$$350$$$25,30,35, \ldots, 350.$$$a_1=25, d=5$$a_n=350$$n$\begin{align}a_n&=a_1+(n-1) d\end{align}\begin{align} 350&=25+(n-1)(5) \\ \Rightarrow 5 n-5+25&=350 \\ \Rightarrow 5 n&=350-20=330 \\ \Rightarrow n&=66, \text { now f… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-3-p7?rev=1737476038&do=diff Question 11 & 12 Exercise 4.3 Solutions of Question 11 & 12 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$16 \mathrm{ft}$$48 \mathrm{ft}$$80 \mathrm{ft}$$a_1=16 \mathrm{ft}$$2^{\text {nd }}$$a_2=48 \mathrm{ft}$$a_3=80 \mathrm{ft}$$16,48,80, \ldots \quad$$d=48-16=32$$S_6$\begin{align}S_n&=\dfrac{n}{2}[2 a_1+(n-1) d] \\ \therefore S_6&=\dfrac{6}{2}(2.16+5… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-3-p3?rev=1737476038&do=diff Question 3 Exercise 5.3 Solutions of Question 3 of Exercise 5.3 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 $n$$n$$4+10+18+28+40+\ldots$\begin{align} & a_2-a_1=10-4=6 \\ & a_3-a_2=18-10=8 \\ & a_4-a_3=28-18=10 \\ & \text {... ... ... } \\ & \text {... ... ... } \\ & a_n-a_{n \quad 1}=(\mathrm{n}-1) \text { term of the sequence } \end{align}$6,10,8, \ldot… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-3-p4?rev=1737476038&do=diff Question 4 Exercise 5.3 Solutions of Question 4 of Exercise 5.3 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4 $n$$n$$3+5+11+29+83+245+\ldots$\begin{align} & a_2-a_1=5-3=2 \\ & a_3-a_2=11-5=6 \\ & a_4-a_3=29-11=18 \\ & \text {... ... ... } \\ & \text {... ... ... } \\ & a_n-a_{n-1}=(\mathrm{n}-1) \text { term ofthe sequence }\end{align}$6,10,18, \ldots$\beg… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-3-p5?rev=1737476038&do=diff Question 5 Exercise 5.3 Solutions of Question 5 of Exercise 5.3 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5 $n$$n$$3+9+21+45+93+189+\ldots$\begin{align} & a_2-a_1=9-3=6 \\ & a_3-a_2=21-9=12 \\ & a_4-a_3=45-21=24\\ & \text {... ... ... } \\ & \text {... ... ... } \\ &a_n-a_{n-1}=(\mathrm{n}-1)\quad \text{ term of the sequence}\quad 6,12,24, \ldots\end{ali… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-3-p2?rev=1737476038&do=diff Question 2 Exercise 6.3 Solutions of Question 2 of Exercise 6.3 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$n$$r$${ }^n P_r=840$${ }^n C_r=35$\begin{align} &^n P_r=\dfrac{n !}{(n-r) !}=840 ....(i)\\ &^n C_r=\dfrac{n !}{(n-r) ! r !}=35....(ii)\end{align}\begin{align}\dfrac{n !}{(n-r) !} \cdot \dfrac{(n-r) ! r !}{n !}&=\dfrac{840}{35}\\ r!&=24\\ \te… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p2?rev=1737476038&do=diff Question 2 Exercise 7.2 Solutions of Question 2 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$4^{th}$$(2+a)^7$$\ln$$n=7$$a=2$$b=a$$$T_{r+1}=\frac{7 !}{(7-r) ! r !}(2)^{7-r } a^r $$$4^{\text {th }}$$r=3$\begin{align} & T_{3+1}=\dfrac{7 !}{(7-3) ! 3 !} 2^{7-3} a^3 \\ & \Rightarrow T_4=\dfrac{7 !}{4 ! 3 !} \cdot 2^4 a^3 \\ & \Rightarrow… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p9?rev=1737476039&do=diff Question 11 Exercise 7.3 Solutions of Question 11 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$y=\frac{1}{2^2}+\frac{1.3}{2 !} \cdot \frac{1}{2^4}+\frac{1 \cdot 3 \cdot 5}{3 !} \cdot \frac{1}{2^6}+\ldots$$y^2+2 y-1=0$$y=\frac{1}{2^2}+\frac{1.3}{2 !} \cdot \frac{1}{2^4}+\frac{1.3 \cdot 5}{3 !} \cdot \frac{1}{2^6}+\ldots$$$ S=y+1=1+\f… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p3?rev=1737476039&do=diff Question 3, Exercise 1.2 Solutions of Question 3 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 3(i) $z \in \mathbb{C}$$z$$z=\bar{z}$$$z=a+ib\quad \text{where}\quad a,b\in \mathbb{R}\, ... (1)$$$z$$\overline{z}=z$$z$$z$$b=0$\begin{align} &z=a \\ \implies &\bar{z}=a \end{align}$z=\bar{z}$$\overline{z}=z$$z$\begin{align}& z=\bar{z}\\ \Righ… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p3?rev=1737476039&do=diff Question 3, Exercise 1.4 Solutions of Question 3 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 3(i) $\left(x_{1}+i y_{1}\right)\left(x_{2}+i y_{2}\right)\left(x_{3}+i y_{3}\right) \ldots\left(x_{n}+i y_{n}\right)=a+i b$$\left(x_{1}^{2}+y_{1}^{2}\right)\left(x_{2}^{2}+y_{2}^{2}\right)\left(x_{3}^{2}+y_{3}^{2}\right) \ldots\left(x_{n}^{2}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p6?rev=1737476039&do=diff Question 6, Exercise 2.6 Solutions of Question 6 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $5 x+3 y+z=6$$2 x+y+3 z=19$$x+2 y+4 z=25$\begin{align*} A &= \begin{bmatrix} 5 & 3 & 1 \\ 2 & 1 & 3 \\ 1 & 2 & 4 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad B = \begin{bmatrix} 6 \\ 19 \\ 25 \end{bmatrix} \end{alig… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-3-p4?rev=1737476037&do=diff Question 5, Exercise 10.3 Solutions of Question 5 of Exercise 10.3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$\cos {{20}^{\circ }}\cos {{40}^{\circ }}\cos {{60}^{\circ }}\cos {{80}^{\circ }}=\dfrac{1}{16}.$$$2\cos \alpha \cos \beta =\cos \left( \alpha +\beta \right)+\cos \left( \alpha -\beta \right)$\begin{align}L.H.S.&=\cos {… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/review-ex3-p2?rev=1737476038&do=diff Question 2 & 3 Review Exercise 3 Solutions of Question 2 & 3 of Review Exercise 3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2 $\lambda$$\mu$$$(\hat{i}+3 \hat{j}+9 \hat{k}) \times(3 \hat{i}-\lambda \hat{j}+\mu \hat{k})=\overrightarrow{0} \text {. }$$\begin{align}(\hat{i}+3 \hat{j}+9 \hat{k}) \times(3 \hat{i}-\lambda \hat{j}+\mu \hat{k})&=\vec{O} \\ \Rightarrow\left|\b… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/review-ex3-p5?rev=1737476038&do=diff Question 8 & 9 Review Exercise 3 Solutions of Question 8 & 9 of Review Exercise 3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 8 $(0,0,2),(-1,3,2),(1,0,4)$$A(0,0,2)$$B(-1,3,2)$$C(1,0,4)$$\vec{a}=\overrightarrow{A B}=(-1,3,2)-(0,0,2)$$\Rightarrow \vec{a}=(-1,3,0)$$\vec{b}=\overrightarrow{B C}=(1,0,4)-(-1,3,2)$$\Rightarrow \vec{b}=(2,-3,2)$$$ \text{Area of triangle} =\dfr… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-1-p4?rev=1737476038&do=diff Question 6 Exercise 4.1 Solutions of Question 6 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Note The general recursive definition formula defined for Pascal sequences is $$P_0=1, P_{r+1}=\dfrac{n-r}{r+1} P_r, \text{ where } r=0,1,2,3,\ldots.$$$n=5$$n=5$$$P_0=1, P_{r+1}=\dfrac{5-r}{r+1} P_r, \text{ where } r=0,1,2,3,\ldots.$$$r=0$\begin{align}&P_{0+1… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-3-p8?rev=1737476038&do=diff Question 13 & 14 Exercise 4.3 Solutions of Question 13 & 14 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.\begin{align}\text{Total number of rows}& n=40,\\ \text{Seats in a first row} a_1&=20\\ \text{Seat in a second row} a_2&=23\\ \text{Seats in third row} a_3&=26\end{align}$20,23,26, \ldots$$S_{40}$$$S_n=\dfrac{n}{2} [{2} a_1+(n-1) d] \text {.}$$\begin… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-5-p3?rev=1737476038&do=diff Question 3 Exercise 4.5 Solutions of Question 3 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 $a_2=2$$a_3=1$$a_1$$r$$$a_n=a_1 r^{n-1}$$$$a_2=a_1 r=2....(i)$$$$a_3=a_1 r^2=1...(ii)$$\begin{align}\dfrac{a_1 r^2}{a_1 r}&=\dfrac{1}{2}\\ \Rightarrow r&=\dfrac{1}{2} \text {, }\end{align}\begin{align}\dfrac{a_1}{2}&=2\\ \Rightarrow a_1&=4 \text {. … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-5-p7?rev=1737476038&do=diff Question 9 & 10 Exercise 4.5 Solutions of Question 9 & 10 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9 $9$$n$$r$$a_1$$$S_n=\dfrac{a_1[r^n-1]}{r-1}$$$$S_6=\dfrac{a_1(r^5-1)}{r-1}$$$$S_3=\dfrac{a_1(r^3-1)}{r-1} \text {. }$$$3$$9$$6$\begin{align} \dfrac{a_1(r^6-1)}{r-1}&=9 \dfrac{a_1(r^3-1)}{r-1} \\ \Rightarrow r^6-1-9(r^3-1) \\ \Rightarrow r^… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-5-p8?rev=1737476038&do=diff Question 11 & 12 Exercise 4.5 Solutions of Question 11 & 12 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$p^{t h}, q^{t h}$$r^{t h}$$a, b, c$$a^{q-r} b^{r-p} c^{p-q}=1$$a_n=a_1 r^{n-1}$$a_p=a_1 r^{p-1}=a \quad a_q=a_1 r^{q-1}=b$$a_r=a_1 r^{r-1}$\begin{align}a^{q-r}&=(a_1 r^{p-1})^{q-r} . \\ b^{r-p}&=(a_1 r^{q-1})^{r-p}, \text { and } \\ c^{p-q}&=(a_1 r^… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-4-p1?rev=1737476038&do=diff Question 1 Exercise 5.3 Solutions of Question 1 of Exercise 5.4 of Unit 05: Mascellaneous series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) $\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+\ldots$$n$$$T_n=\dfrac{1}{n(n+1)}$$$T_n$$$\dfrac{1}{n(n+1)}=\dfrac{A}{n}+\dfrac{B}{(n+1)}$$$n(n+1)$$$1=A(n+1)+B n=(A+B) n+A$$$n$$$A+B=0 \text{and} A=1$$$A=1$\begin{align}1+B&=0\\ B&=-1\end{align}\beg… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-4-p2?rev=1737476038&do=diff Question 2 & 3 Exercise 5.4 Solutions of Question 2 & 3 of Exercise 5.4 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2 $\sum_{k=1}^n \dfrac{1}{9 k^2+3 k-2}$\begin{align}\text { Let } S_n&=\sum_{k=1}^n \dfrac{1}{9 k^2+3 k-2} \\ S_n&=\sum_{k=1}^n \dfrac{1}{9 k^2+6 k-3 k-2} \\ & =\sum_{k=1}^n \dfrac{1}{3 k(3 k+2)-1(3 k+2)} \\ S_n&=\sum_{k=1}^n \dfrac{1}{(3 k-1… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-3-p1?rev=1737476038&do=diff Question 1 Exercise 6.3 Solutions of Question 1 of Exercise 6.3 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$^n C_2=36$$n$\begin{align}&^n C_2=36\\ & \Rightarrow \dfrac{n !}{(n-2) ! 2 !}=36 \\ & \Rightarrow \dfrac{n(n-1)(n-2) !}{(n-2) ! \cdot 2}=36 \\ & \Rightarrow n(n-1)=72 \\ & \Rightarrow n^2-n-72=0 \\ & \Rightarrow n^2-9 n+8 n-72=0\\ & \Rightar… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-3-p4?rev=1737476039&do=diff Question 5, Exercise 10.3 Solutions of Question 5 of Exercise 10.3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$\cos {{20}^{\circ }}\cos {{40}^{\circ }}\cos {{60}^{\circ }}\cos {{80}^{\circ }}=\dfrac{1}{16}.$$$2\cos \alpha \cos \beta =\cos \left( \alpha +\beta \right)+\cos \left( \alpha -\beta \right)$\begin{align}L.H.S.&=\cos {… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-1-p1?rev=1737476039&do=diff Question 1, Exercise 2.1 Solutions of Question 1 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{lll}1 & 3 & 0 \\ 2 & 0 & 1\end{array}\right]$\begin{align}\text{Order of A}&= 2\times 3\end{align}$B=\left[\begin{array}{ll}1 & 2 \\ 2 & 3 \\ 3 & 4\end{array}\right]$\begin{align}\text{Order of B}&= 3\times 2\end{align}$C… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p8?rev=1737476039&do=diff Question 13, Exercise 4.2 Solutions of Question 13 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $7$$17$$a=7$$b=17$\begin{align*} \text{A.M.} &= \frac{a + b}{2}\\ &= \frac{7 + 17}{2} \\ &= \frac{24}{2} = 12. \end{align*}$12$$3+3 \sqrt{2}$$7-3 \sqrt{2}$$a=3+3\sqrt{2}$$b=7-3\sqrt{2}$\begin{align*} \text{A.M.} &= \frac{a + b}{2}\\ &= \frac{(3 + 3… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-6-p2?rev=1737476040&do=diff Question 3 & 4, Exercise 4.6 Solutions of Question 3 & 4 of Exercise 4.6 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\frac{1}{18}, \frac{1}{13}, \frac{1}{8}, \ldots \quad 20$\begin{align*} &\frac{1}{18}, \frac{1}{13}, \frac{1}{8}, \ldots \quad \text{ is in H.P.} \\ &18, 13, 8, \ldots \quad \text{ is in A.P.} \end{align*}$a_1 = 18$$d = 13 - 18 = -5$$a_{20}.… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-6-p3?rev=1737476040&do=diff Question 5 & 6, Exercise 4.6 Solutions of Question 5 & 6 of Exercise 4.6 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\dfrac{1}{27}, \dfrac{1}{20}, \dfrac{1}{13}, \ldots \quad$\begin{align*} &\frac{1}{27}, \frac{1}{20}, \frac{1}{13}, \ldots \quad \text{ is in H.P.} \\ &27, 20, 13, \ldots \quad \text{ is in A.P.} \end{align*}$a_1 = 27$$d = 20 - 27 = -7$$a_n=… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p1?rev=1737476040&do=diff Question 1, Exercise 9.1 Solutions of Question 1 of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y=2-2 \operatorname{Cos} \theta$\begin{align*} -1 \leq \operatorname{Cos} \theta \leq 1 \end{align*}$-2$\begin{align*} & 2 \geq -2 \operatorname{Cos} \theta \geq -2 \end{align*}$2$\begin{align*} & 4 \geq 2-2 \operatorname{Cos} \theta \geq 0 \\ … text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/junaid?rev=1737476042&do=diff Dr. Junaid Alam Khan <callout type=“info” icon=“true”> This is a personal web page of Dr. Junaid Alam Khan Associate Professor Institute of Business Administration, Karachi - PAKISTAN. IBA Profile: <https://oric.iba.edu.pk/profile.php?id=jakhan> ResearchGate Profile: <https://www.researchgate.net/scientific-contributions/2141923316_Junaid_Alam_Khan> Facebook: text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-3-p6?rev=1737476037&do=diff Question 9 & 10, Exercise 3.3 Solutions of Question 9 & 10 of Exercise 3.3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9 $\vec{k}-2 \hat{i}+3 \hat{j}+\hat{k}$$\vec{S}=2 \hat{i}+\hat{j}-\hat{k}$\begin{align}W &=\vec{F} \cdot s \\ \Rightarrow W &=(2 \hat{i}+3 \hat{j}+\hat{k}) \cdot(2 \hat{i}+\hat{j}-\hat{k}) \\ \Rightarrow W &=2(2) \div 3(1)+1(-1) \\ \Rightarrow W &=4+3 … text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-3-p7?rev=1737476037&do=diff Question 11, Exercise 3.3 Solutions of Question 11 of Exercise 3.3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 11 (i) Show that the vectors $3 \hat{i}-2 \hat{j}+$$\hat{k} . \quad \hat{i}-3 \hat{j}-5 \hat{k}$$2 \hat{i}+\hat{j}-4 \hat{k}$$\vec{a}=3 \hat{i}-2 \hat{j}+\hat{k}$$\vec{b}=\hat{i}-3 \hat{j}+5 \hat{k}$$\vec{c}=2 \hat{i}+\hat{j}-4 \hat{k}$\begin{align}|\vec{a}|&… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-4-p7?rev=1737476037&do=diff Question 7 & 8 Exercise 3.4 Solutions of Question 7 & 8 of Exercise 3.4 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7 If $\vec{A}+\vec{B}+\vec{C}=\vec{O}$$$\vec{A} \times \vec{B}=\vec{B} \times \vec{C}=\vec{C} \times \vec{A}.$$$$\vec{A}+\vec{B}+\vec{C}=\vec{O} \text {. }$$$\vec{A}$$$\vec{A} \times(\vec{A}+\vec{B}+\vec{C})=0$$\begin{align}\Rightarrow \vec{A} \times \ve… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-5-p1?rev=1737476038&do=diff Question 1 & 2 Exercise 3.5 Solutions of Question 1 & 2 of Exercise 3.5 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1 Find $\vec{a} \cdot \vec{b} \times \vec{c}$$\vec{a}=2 \hat{i}+\hat{j}+3 \hat{k}$$\vec{b}=-\hat{i}+2 \hat{j}+\hat{k} \quad \text { and }\quad \vec{c}=3 \hat{i}+\hat{j}+2 \hat{k} \text {. }$\begin{align}V&=\vec{a} \cdot \vec{b} \times \vec{c}\\ &=\left|\… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-3-p5?rev=1737476038&do=diff Question 7 & 8 Exercise 4.3 Solutions of Question 7 & 8 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7 $1+3-5+7+9-11+13+15-$$17+\ldots$$3 n$\begin{align}&(1+7+13+\ldots)+(3+9+15+\ldots)- \\ & (5+11+17+\ldots) \ldots \ldots \ldots . . .(1)\end{align}$\mathrm{n}$$n$$3 n$$$1+7+13+\ldots$$$$a_1=1, d=7-1=6$$$n$\begin{align}S_n&=\dfrac{n}{2}[2 a_1+… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-4-p3?rev=1737476038&do=diff Question 4 & 5 Exercise 4.4 Solutions of Question 4 & 5 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4 $\dfrac{1}{64}$$r=\dfrac{1}{2}$$a_1=16$$a_n=\dfrac{1}{64}$$r=\dfrac{1}{2}$$n$$$a_n=a_1 r^{n-1} \quad \text{then}$$\begin{align}\dfrac{1}{64}&=16(\dfrac{1}{2})^{n-1} \\ \Rightarrow(\dfrac{1}{2})^{n-1}&=\dfrac{1}{64 \times 16}=\dfrac{1}{1024} … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-4-p4?rev=1737476038&do=diff Question 6 & 7 Exercise 4.4 Solutions of Question 6 & 7 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6 $a_{10}=l, a_{13}=m$$a_{16}=n;\quad$$\ln =m^2$$a_n=a_1 r^{n-1}$\begin{align}a_{10}&=a_1 r^9=l \\ a_{13}&=a_1 r^{12}=m\\ \text{and} \quad a_{16}&=a_1 e^{\mathbf{A 5}}=n\end{align}\begin{align}a_{10} \cdot a_{16}&=\ln =(a_1 r^9)(a_1 r^{15})\\ … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-4-p5?rev=1737476038&do=diff Question 8 Exercise 4.4 Solutions of Question 8 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 8(i) $3.14$$2.71$$a=3.14$$b=2.71$$$G= \pm \sqrt{(3.14)(2.71)}= \pm 2.94$$$$G=2.94 \quad \text{or} \quad -2.94$$$-6$$-216$$a=-6$$b=-216$\begin{align}G&= \pm \sqrt{(-6)(-216)}= \pm \sqrt{1296} \\ \Rightarrow G&= \pm 36\end{align}$$G=36 \quad \text{or} \… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-5-p2?rev=1737476038&do=diff Question 2 Exercise 4.5 Solutions of Question 2 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2(i) $a_1, a_n, n_2 r$$S_n$$a_1=1, \quad r=-2, \quad a_n=64$$n$$S_n$$a_n=a_1 r^{n-1}$\begin{align}64&=(-2)^{n-1}\\ \Rightarrow(-2)^{n-1}&=(-2)^6 \\ \Rightarrow n-1&=6 \\ \Rightarrow n&=7\\ S_7&=\dfrac{a_1[r^{\prime \prime}-1]}{r-1}\\ \text{then}\\ S_7… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-2-p2?rev=1737476038&do=diff Question 2 & 3 Exercise 5.2 Solutions of Question 2 & 3 of Exercise 5.2 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2 $1+3^2 x+5^2 x^2+7^2 x^3+\ldots, x<1$\begin{align} & S_{\infty}=1+3^2 x+5^2 x^2+7^2 x^3+\ldots ..(1)\\ & x S_{\infty}=x+3^2 x^2+5^2 x^3+7^2 x^4+\ldots..(2)\end{align}\begin{align}& (1-x) S_{\infty}=1^2+(3^2-1^2) x+(5^2-3^2) x^2+(7^2-5^2) x^… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-3-p2?rev=1737476038&do=diff Question 2 Exercise 5.3 Solutions of Question 2 of Exercise 5.3 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2 $n$$n$$4+14+30+52+80+114+\ldots$\begin{align} & a_2-a_1=14-4=10 \\ & a_3-a_2=30-14=16 \\ & a_4-a_3=52-30=22 \\ & \cdots \quad \cdots \quad \cdots \\ & \cdots \quad \cdots \quad \cdots \\ & a_n-a_{n-1}=(\mathrm{n}-1)\text{ term of the sequence} 10,1… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-5-p2?rev=1737476038&do=diff Question 3 and 4 Exercise 6.5 Solutions of Question 3 and 4 of Exercise 6.5 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$P(A)=0.5$$P(A \cup B)=0.6$$P(B)$$A$$B$$\mathrm{A}$$B$$A \cap B=\emptyset$\begin{align}P(A \cup B)&=P(A)+P(B)\\ \Rightarrow P(B)&=P(A \cup B)-P(A)\\ &=0.6-.0 .5=0.1 \end{align}$30$$1$$30.$\begin{align}S&=\{1,2,3, \ldots, 50\} \tex… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p7?rev=1737476038&do=diff Question 11 Review Exercise 6 Solutions of Question 11 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$n(S)=4$$$\dfrac{1}{4}$$$\quad P( orange )=\dfrac{1}{4}$$$\dfrac{1}{4}$$\dfrac{1}{4}$\begin{align}P(\operatorname{Red})&=\dfrac{1}{4}\\ P( Green )&=\dfrac{1}{4}\end{align}$P(R \cap G)=\phi$$R$$G$\begin{align}\boldsymbol{P}( Red o… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p5?rev=1737476038&do=diff Question 5 Exercise 7.2 Solutions of Question 5 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$(\dfrac{a}{x}+b x)^8$$a=\dfrac{a}{x}$$b=b x$$n=8$$n-8$$8+1=9$$$(\dfrac{8+2}{2})^{t h}=5^{t h}$$T_{r+1}$$$T_{r+1}=\dfrac{8 !}{(8-r) ! r !}(\dfrac{a}{x})^{8-r}(b x)^r$$$T_5$$r=4$\begin{align}T_5&=\dfrac{8 !}{(8-4) ! 4 !}(\dfrac{a}{x})^{8-4}(b … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-6-p1?rev=1737476040&do=diff Question 1 and 2, Exercise 4.6 Solutions of Question 1 and 2 of Exercise 4.6 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\frac{1}{9}, \frac{1}{12}, \frac{1}{15}, \cdots \quad 7$$$\frac{1}{9}, \frac{1}{12}, \frac{1}{15}, \cdots \text{ is in H.P.}$$$$9, 12, 15, ... \text{ is in A.P.}$$$a_1=9$$d=12-9=3$$a_7=?$$$ a_n=a_1+(n-1)d. $$\begin{align*} a_7&=9+(6)(3) … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-6-p5?rev=1737476040&do=diff Question 9 & 10, Exercise 4.6 Solutions of Question 9 & 10 of Exercise 4.6 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\frac{1}{7}, \frac{1}{6},-1,-\frac{1}{3}, \ldots$$$\frac{1}{7}, \frac{1}{6}, -1, -\frac{1}{3}, \ldots \text{ is in H.P.}$$$$7, 6, -1, -3, \ldots \text{ is in A.P.}$$$a_1 = 7$$d = 6 - 7 = -1$$a_8=?$$$ a_n = a_1 + (n-1)d. $$\begin{align*} a_… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-2-p6?rev=1737476036&do=diff Question 7, Exercise 1.2 Solutions of Question 7 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7(i) $\dfrac{2+3i}{5-2i}$\begin{align}&\dfrac{2+3i}{5-2i} \\ =&\dfrac{2+3i}{5-2i}\times \dfrac{5+2i}{5+2i} \quad \text{by rationalizing} \\ =&\dfrac{10-6+15i+4i}{25+4}\\ =&\dfrac{4+19i}{29}\\ =&\dfrac{4}{29}+\dfrac{19}{29}i \end{align}$=\dfrac{4}{29}$$=\… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-2-p6?rev=1737476037&do=diff Question 7, Exercise 1.2 Solutions of Question 7 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7(i) $\dfrac{2+3i}{5-2i}$\begin{align}&\dfrac{2+3i}{5-2i} \\ =&\dfrac{2+3i}{5-2i}\times \dfrac{5+2i}{5+2i} \quad \text{by rationalizing} \\ =&\dfrac{10-6+15i+4i}{25+4}\\ =&\dfrac{4+19i}{29}\\ =&\dfrac{4}{29}+\dfrac{19}{29}i \end{align}$=\dfrac{4}{29}$$=\… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-3-p4?rev=1737476037&do=diff Question 5, Exercise 1.3 Solutions of Question 5 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) ${{z}^{2}}+z+3=0$$${{z}^{2}}+z+3=0.$$$a=1$$b=1$$c=3$\begin{align}z&=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\\ &=\dfrac{-1\pm \sqrt{{{\left( 1 \right)}^{2}}-4\left( 1 \right)\left( 3 \right)}}{2\left( 1 \right)}\\ &=\dfrac{-1\pm \sqrt{1-12}}{2}\\ &=\… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p5?rev=1737476037&do=diff Question 5 & 6, Exercise 2.1 Solutions of Question 5 & 6 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$A= \begin{bmatrix} 0 & 2b & -2 \\ 3 & 1 & 3 \\ 3a & 3 & -1 \end{bmatrix}$$a$$b$$A=\begin{bmatrix} 0 & 2b & -2 \\ 3 & 1 & 3 \\ 3a & 3 & -1 \end{bmatrix}$$$A^t=\left[ \begin{matrix} 0 & 3 & 3a \\ 2b & 1 & 3 \\ -2 & 3 & -1 \\ \end{ma… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-2-p4?rev=1737476037&do=diff Question 5 & 6, Exercise 3.2 Solutions of Question 5 & 6 of Exercise 3.2 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5 Find the length of the vector $\overrightarrow{AB}$$\vec{A}(-3,5)$$\vec{B}(7,9)$$\overrightarrow{AB}$$\vec{A}$$\vec{B}$$$\overrightarrow{OA}=-3\hat{i}+5\hat{j},$$$$\overrightarrow{OB}=7\hat{i}+9\hat{j}.$$\begin{align}\overrightarrow{AB}&=\overrightarr… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-3-p8?rev=1737476037&do=diff Question 12 & 13, Exercise 3.3 Solutions of Question 12 & 13 of Exercise 3.3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 12 $\overrightarrow{B A} \cdot \overrightarrow{A C}=0$$|\vec{a}|=\vec{b}|=| \vec{c} \mid=$$\vec{b}=-\vec{c}$$\triangle A B O$\begin{align}\overrightarrow{O B}+\overrightarrow{A B}&=\overrightarrow{O A}\\ \Rightarrow \overrightarrow{B A}&=\overrightar… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-4-p3?rev=1737476037&do=diff Question 3 Exercise 3.4 Solutions of Question 3 of Exercise 3.4 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3(i) Find a unit vector that is orthogonal to the given vector $\vec{a}=\hat{i}- 2 \hat{j}+3 \hat{k}, \quad \vec{b}=2 \hat{i}+\hat{j}-\hat{k}$$\hat{n}$$\vec{a}$$\vec{b}$\begin{align}\hat{n}&=\dfrac{\vec{a} \times \vec{b}}{\mid \vec{a} \times \vec{b}} \\ \text { … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-4-p8?rev=1737476038&do=diff Question 9 Exercise 3.4 Solutions of Question 9 of Exercise 3.4 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9(i) Find the area of parallelogram whose diagonals are $\vec{a}=4 \hat{i}+\hat{j}-2 \hat{k}\quad$$\quad\vec{b}=-2 \hat{i}+3 \hat{j}+4 \hat{k}$$\vec{c}$$\vec{d}$$E$$E$\begin{align}\overrightarrow{A E}&=\overrightarrow{E C}\\ &=\dfrac{1}{2} \vec{a}\\ &=2 \hat{i}+… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-5-p3?rev=1737476038&do=diff Question 5(i) & 5(ii) Exercise 3.5 Solutions of Question 5(i) & 5(ii) of Exercise 3.5 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) $\vec{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}\quad$$\quad\vec{b}=b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k}\quad$$\vec{a} \times \vec{b}\quad$$\vec{a} \times \vec{b}$$\vec{a}$$\vec{b}$$\vec{a} \times \vec{b}$$\vec{a}$$\vec{b}$$\vec{a} \times \v… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-5-p4?rev=1737476038&do=diff Question 5(iii) & 5(iv) Exercise 3.5 Solutions of Question 5(iii) & 5(iv) of Exercise 3.5 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\vec{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}\quad$$\quad\vec{b}=b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k}\quad$$(\vec{a}. \vec{b})^2,\quad|a|^2,\quad|b|^2$\begin{align}\vec{a} \cdot \vec{b}&=(a_1 \hat{i}+a_2 \hat{j} + a_3 \hat{k}) \cdot(b_1 \hat{i}+b_2 \… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-5-p7?rev=1737476038&do=diff Question 8 Exercise 3.5 Solutions of Question 8 of Exercise 3.5 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 8(i) Find the volume of tetrahedron with the Vectors as coterminous edges \begin{align}\vec{a}&=\hat{i}+2 \hat{j}+3 \hat{k},\\ \vec{b}&=4 \hat{i}+5 \hat{j}+6 \hat{k}, \\ \vec{c}&=7 \hat{j}+8 \hat{k}\end{align}\begin{align}V&=\dfrac{1}{6}[\vec{u} \cdot \vec{v} \… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/review-ex3-p4?rev=1737476038&do=diff Question 6 & 7 Review Exercise 3 Solutions of Question 6 & 7 of Review Exercise 3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6 $\lambda$$\vec{a}=\hat{i}+3 \hat{j}+\hat{k}$$\bar{b}=2 \hat{i}-\hat{j}-\hat{k}$$\vec{c}=\lambda \hat{j}+3 \hat{k}$\begin{align}\vec{a} \cdot \vec{b} \times \vec{c}&=0 \\ \Rightarrow\left|\begin{array}{ccc} 1 & 3 & 1 \\ 2 & -1 & -1 \\ 0 & \lamb… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p10?rev=1737476038&do=diff Question 14 Exercise 4.2 Solutions of Question 14 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 14(i) $A_1, A_2, A_3$$6, A_1, A_2, A_3, 41$$$a_1=6 \text{ and } a_6=41.$$\begin{align}& a_5=11\\ \Rightarrow &a_1+4 d=41 \\ \Rightarrow &6+4 d=41 \\ \Rightarrow &d=\dfrac{41-6}{4}\\ &=\dfrac{35}{4}.\end{align}\begin{align} A_1&=a+d=6+\dfrac{35}{4} \… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-3-p2?rev=1737476038&do=diff Question 2 Exercise 4.3 Solutions of Question 2 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2(i) $a_1, a_n, n, d$$S_n$$a_1=2, n=17, d=3$$a_1=2, n=17, d=3$$a_{17}$$S_{17}$$$a_{n}=a_1+(n-1)d.$$$$a_{17}=2+(17-1)(3)=50.$$$$S_n=\dfrac{n}{2}[a_1+a_n]$$\begin{align}S_{17}&=\dfrac{17}{2}(a_1+a_17) \\ &=\dfrac{17}{2}(2+50)=442.\end{align}$a_{17}=50$$… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-4-p6?rev=1737476038&do=diff Question 9 Exercise 4.4 Solutions of Question 9 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9(i) $3 \dfrac{5}{9}=\dfrac{32}{9}\quad$$\quad40 \dfrac{1}{2}=\dfrac{81}{2}$$G_1, G_2, G_3, G_4$$G_5$$\dfrac{32}{9}$$\dfrac{81}{2}$$\dfrac{32}{9}, G_1, G_2, G_3, G_4, G_5, \dfrac{81}{2}$$a_7=\dfrac{81}{2}$$a_1=\dfrac{32}{9}$\begin{align}a_1 r^6&=\dfra… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-4-p7?rev=1737476038&do=diff Question 10 Exercise 4.4 Solutions of Question 10 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 10 $48$$18$$a$$b$$1$$48$$$\quad a-b=48....(i)$$$a$$b$$$G=\sqrt{a b}$$$a$$b$$$A=\dfrac{a+b}{2}$$$2$$A \cdot M=G \cdot M+18$$A \cdot M-G \cdot M=18$$$\Rightarrow \dfrac{a+b}{2}-\sqrt{a b}=18$$$$(a+b)-2 \sqrt{a b}=36 \text {. }$$$a=b+48$\begin{align}(b… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-5-p9?rev=1737476038&do=diff Question 13 & 14 Exercise 4.5 Solutions of Question 13 & 14 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$y=\dfrac{x}{3}+\dfrac{x^2}{3^2}+\dfrac{x^3}{3^3}+\ldots$$0<x<3$$x=\dfrac{3 y}{1+y}$$$1+y=1+\dfrac{x}{3}+\dfrac{x^2}{3^2}+\dfrac{x^3}{3^3}$$$a_1=1$$r=\dfrac{x}{3}$$|r|=\dfrac{x}{3}<1$$0<x<3$$S_{\infty}=\dfrac{a_1}{1-r}$$a_1, \quad r$$$S_{\infty}=\dfr… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-1-p2?rev=1737476038&do=diff Question 2 & 3 Exercise 5.1 Solutions of Question 2 & 3 of Exercise 5.1 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Q2 Find the sum $1.2+2.3+3.4+\ldots+99.100$$1+2+3+\ldots+99$$2+3+4+\ldots+100$$n^{\text {th }}$$n(n+1)$$n^{\text {th }}$$\quad T_j=j(j+1)=j^2+j$$j=1$$j=99$$$ \begin{aligned} & \sum_{j=1}^{99} \tau_j=\sum_{j=1}^{99} j^2+\sum_{j=1}^{99} j \\ & =\frac{99… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/re-ex5-p5?rev=1737476038&do=diff Question 7 Review Exercise Solutions of Question 7 of Review Exercise of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7(i) $1.2^2+3.3^2+5.4^2+\ldots$$n$$1,3,5, \ldots,(2 n-1)$$2^2, 3^2, 4^2, \ldots,(n+1)^2$\begin{align} & a_n=(2 n-1)(n+1)^2 \\ & a_n=(2 n-1)(n^2+2 n+1) \\ & a_n=2 n^3+3 n^2-1\end{align}\begin{align} \sum_{r=1}^n a_r&=2 \sum_{r=1}^n r^3+\sum_{r=1… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-1-p2?rev=1737476038&do=diff Question 3 & 4 Exercise 6.1 Solutions of Question 3 & 4 of Exercise 6.1 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{1}{6 !}+\dfrac{2}{7 !}+\dfrac{3}{8 !}=\dfrac{75}{8 !}$\begin{align}\dfrac{1}{6 !}+\dfrac{2}{7 !}+\dfrac{3}{8 !}&=\dfrac{1}{6 !}+\dfrac{2}{7.6 !}+\dfrac{3}{8.7 .6 !} \\ & =\dfrac{56+16+3}{8 !}\\ &=\dfrac{75}{8 !}\end{align}$\df… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p1?rev=1737476038&do=diff Question 1 and 2 Exercise 6.2 Solutions of Question 1 and 2 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$^6 P_6$\begin{align}^6 P_6&=\dfrac{6 !}{(6-6) !}\\ &=6 !=720\end{align}$^{20} P_2$\begin{align}^{20} P_2&=\dfrac{20 !}{(20-2) !}\\ &=\dfrac{20.19 .18 !}{18 !}\\ &=20 \times 19=380\end{align}$^{16} P_3$\begin{align}^{16} P_3&=\dfr… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-4-p1?rev=1737476038&do=diff Question 1 Exercise 6.4 Solutions of Question 1 of Exercise 6.4 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$S=\{1,2,3,4,5,6\}$$5$$5$\begin{align}A&=\{5\}\\ P(A)&=\dfrac{n(A)}{n(S)}\\ &=\dfrac{1}{6} \end{align}$S=\{1,2,3,4,5,6\}$$1$$1$\begin{align}B&=\{\}\\ &=\phi \text{then}\\ P(B)&=\dfrac{n(B)}{n(S)}\\ &=\dfrac{0}{6}\\ &=0\end{align}$S=\{1,2,3,4,… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-4-p4?rev=1737476038&do=diff Question 4 Exercise 6.4 Solutions of Question 4 of Exercise 6.4 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.\begin{align}S&=(HHII,HHT.HTH.HTT.THII.THT.TTH,TT7),\\ \text{then} n(S)&=2^3=8\end{align}$$A=\{H H H\}$$$$n(A)=1$$$P(A)=\dfrac{n(A)}{n(S)}=\dfrac{1}{8}$\begin{align}S&=(HHII,HHT.HTH.HTT.THII.THT.TTH,TT7),\\ \text{then} n(S)&=2^3=8\end{align}… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p13?rev=1737476038&do=diff Question 13 Exercise 7.1 Solutions of Question 13 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$2^n>n \forall n \in \mathbf{N}$$n=1$$2^n=2^1=2$$n=1$$2>1$$n=1$$n=l>I$$2^k>k\cdots(i)$$n=k+1$\begin{align} & 2^{k+1}=2^k \cdot 2>k \cdot 2 \quad \text { by (i) } \\ & \Rightarrow 2^{k+1}>2 k=k+k \\ &\Rightarrow 2^{k+1}>k+1 \text {. as } k>1… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p7?rev=1737476038&do=diff Question 7 Exercise 7.2 Solutions of Question 7 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$(2+\sqrt{3})^5+(2-\sqrt{3})^5$\begin{align}(2+\sqrt{3})^5+(2 \cdot \sqrt{3})^5& =[(2)^5+{ }^5 C_1 \cdot 2^4 \cdot \sqrt{3}+{ }^5 C_2 \cdot 2^3 \cdot(\sqrt{3})^2 \\ & +^5 C_3 \cdot 2^2 \cdot(\sqrt{3})^4+{ }^5 C_4 \cdot 2 \cdot(\sqrt{3})^4 \\ … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p2?rev=1737476039&do=diff Question 2 Exercise 7.3 Solutions of Question 2 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sqrt{26}$$$ \begin{aligned} & \sqrt{26}=\sqrt{25+1} \\ & =\sqrt{25} \sqrt{1+\frac{1}{25}}=5\left[1+\frac{1}{25}\right]^{\frac{1}{2}} \end{aligned} $$$$ \begin{aligned} & \sqrt{26}=5\left[1+\frac{1}{25}\right]^{\frac{1}{2}} \\ & =5\left[1+\f… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p5?rev=1737476039&do=diff Question 5 and 6 Exercise 7.3 Solutions of Question 5 and 6 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$x$$x^2$$x$$$ \frac{(8+3 x)^{\frac{2}{3}}}{(2+3 x) \sqrt{4-5 x}}=1-\frac{5 x}{8} $$$$ \frac{\sqrt[4]{3}-3 x j^{\frac{2}{3}}}{2 \cdot 3 x+4-5 x} $$$$ \begin{aligned} & =\frac{8^{\frac{2}{3}}\left(1+\frac{3 x}{8}\right)^{\frac{2}{3}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p8?rev=1737476039&do=diff Question 10 Exercise 7.3 Solutions of Question 10 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$1-\frac{1}{2^2}+\frac{1.3}{2 !} \cdot \frac{1}{2^4}+\ldots$$(1+x)^n$$$ \begin{aligned} & 1+n x+\frac{n(n-1)}{2 !} x^2 \\ & +\frac{n(n-1(n-2))}{3 !} x^3+\ldots \end{aligned} $$$n x=-\frac{1}{4}$$\frac{n(n-1)}{2 !} x^2=\frac{1.3}{2 !} \cdot … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p10?rev=1737476039&do=diff Question 12 Exercise 7.3 Solutions of Question 12 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$2 y=\frac{1}{2^2}+\frac{1.3}{2 !} \cdot \frac{1}{2^4}+\frac{1.3 \cdot 5}{3 !} \cdot \frac{1}{2^6}+\ldots$$4 y^2+4 y-1=0$$$ 2 y=\frac{1}{2^2}+\frac{1.3}{2 !} \cdot \frac{1}{2^4}-\frac{1.3 \cdot 5}{3 !} \cdot \frac{1}{2^6}+\ldots $$$S=2 y+1=… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p11?rev=1737476039&do=diff Question 13 Exercise 7.3 Solutions of Question 13 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$x$$x^3$$x$$n^{\text {th }}$$1+x$$\frac{2 n+(n+1) x}{2 n+(n-1) x}$$$ \begin{aligned} & (1+x)^{\frac{1}{n}}=\frac{2 n+(n+1) x}{2 n+(n-1) x} \\ & \frac{2 n+(n+1) x}{2 n+(n-1) x} \\ & =1+\frac{1}{n} x+\frac{\frac{1}{n}\left(\frac{1}{n}-1\right… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-1-p5?rev=1737476039&do=diff Question 5, Exercise 1.1 Solutions of Question 5 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 5 $z$$4z-3\bar{z}=\dfrac{1-18i}{2-i}$$z=x+iy$$\bar{z}=x-iy$\begin{align}&4z-3\bar{z}=\dfrac{1-18i}{2-i}\\ \implies &4(x+iy)-3(x-iy)=\dfrac{1-18i}{2-i}\times \dfrac{2+i}{2+i}\\ \implies &4x+4iy-3x+3iy=\dfrac{(1-18i)(2+i)}{2^2-i^2} \end{align}\b… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-3-p6?rev=1737476039&do=diff Question 6, Exercise 2.3 Solutions of Question 6 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{ccc}2 & 1 & -3 \\ 0 & 1 & 0 \\ 2 & 1 & 6\end{array}\right]$$A^{-1}$$A A^{-1}=A^{-1} A=I_{3}$\begin{align*} A &= \begin{bmatrix} 2 & 1 & -3 \\ 0 & 1 & 0 \\ 2 & 1 & 6 \end{bmatrix} \end{align*}$ A^{-1} $$ A $\begin{align*} … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p1?rev=1737476039&do=diff Question 1, Exercise 2.6 Solutions of Question 1 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $ 2 x_{1}-3 x_{2}+4 x_{3}=0$$x_{1}-2 x_{2}+3 x_{3}=0$$4 x_{1}+x_{2}-6 x_{3}=0$\begin{align*} &2 x_{1}-3 x_{2}+4 x_{3}=0\cdots (i)\\ &x_{1}-2 x_{2}+3 x_{3}=0\cdots (ii)\\ &4 x_{1}+x_{2}-6 x_{3}=0\cdots (iii)\\ \end{align*}\begin{align*} A &= \le… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-6-p4?rev=1737476040&do=diff Question 7 & 8, Exercise 4.6 Solutions of Question 7 & 8 of Exercise 4.6 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\frac{1}{4}, \frac{1}{7}, \frac{1}{10}, \frac{1}{13}, \ldots$$ \frac{1}{4}, \frac{1}{7}, \frac{1}{10}, \frac{1}{13}, \ldots $$ a_1 = \frac{1}{4} $$d = \frac{1}{7} - \frac{1}{4} = -\frac{3}{28},$$ n = 14$$$a_n = a_1 + (n-1)d.$$\begin{align*} … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p9?rev=1737476040&do=diff Question 10, Exercise 8.1 Solutions of Question 10 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin \left(\dfrac{\pi}{2}-\alpha\right)=\cos \alpha$\begin{align*} L.H.S & = \sin \left(\frac{\pi}{2}-\alpha\right) \\ & =\sin\frac{\pi}{2} \cos \alpha - \cos \frac{\pi}{2} \sin\alpha \\ & = 1\times \cos \alpha - 0 \times \sin\alpha \\ & =… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-2-p4?rev=1737476036&do=diff Question 5, Exercise 1.2 Solutions of Question 5 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) ${{z}_{1}}=2+4i$${{z}_{2}}=1-3i$$\overline{{{z}_{1}}+{{z}_{2}}}=\overline{{{z}_{1}}}+\overline{{{z}_{2}}}$${{z}_{1}}=2+4i$${{z}_{2}}=1-3i$$\overline{{{z}_{1}}}=2-4i$$\overline{{{z}_{2}}}=1+3i$\begin{align}z_1+z_2&=2+4i+1-3i\\ &=3+i \end{align}\begin… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-2-p5?rev=1737476036&do=diff Question 6, Exercise 1.2 Solutions of Question 6 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6(i) ${{z}_{1}}$${{z}_{2}}$$|{{z}_{1}}{{z}_{2}}|=|{{z}_{1}}||{{z}_{2}}|$${{z}_{1}}=a+bi$${{z}_{2}}=c+di$$|z_1=\sqrt{a^2+b^2}|$$|z_2=\sqrt{c^2+d^2}|$\begin{align} L.H.S.&=|{{z}_{1}}{{z}_{2}}|\\ &=|(a+bi)(c+di)|\\ &=|ac-bd+(ad+bc)i|\\ &=\sqrt{{{(ac-bd)}^{… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-2-p7?rev=1737476036&do=diff Question 8, Exercise 1.2 Solutions of Question 8 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 8(i) $z+\overline{z}=2\operatorname{Re}\left( z \right)$$z=a+ib$$\overline{z}=a-ib$\begin{align}z+\overline{z}&=\left( a+ib \right)+\left( a-ib \right)\\ &=a+ib+a-ib\\ &=2a\\ z+\overline{z}&=2\operatorname{Re}\left( z \right)\end{align}$z-\overline{z}=2i… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-2-p6?rev=1737476036&do=diff Question 7, Exercise 10.2 Solutions of Question 7 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${{\cos }^{4}}\theta -{{\sin }^{4}}\theta =\dfrac{1}{\sec 2\theta }$\begin{align}L.H.S&={{\cos }^{4}}\theta -{{\sin }^{4}}\theta \\ &=\left( {{\cos }^{2}}\theta -{{\sin }^{2}}\theta \right)\left( {{\cos }^{2}}\theta +{{\… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-2-p4?rev=1737476037&do=diff Question 5, Exercise 1.2 Solutions of Question 5 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) ${{z}_{1}}=2+4i$${{z}_{2}}=1-3i$$\overline{{{z}_{1}}+{{z}_{2}}}=\overline{{{z}_{1}}}+\overline{{{z}_{2}}}$${{z}_{1}}=2+4i$${{z}_{2}}=1-3i$$\overline{{{z}_{1}}}=2-4i$$\overline{{{z}_{2}}}=1+3i$\begin{align}z_1+z_2&=2+4i+1-3i\\ &=3+i \end{align}\begin… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-2-p5?rev=1737476037&do=diff Question 6, Exercise 1.2 Solutions of Question 6 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6(i) ${{z}_{1}}$${{z}_{2}}$$|{{z}_{1}}{{z}_{2}}|=|{{z}_{1}}||{{z}_{2}}|$${{z}_{1}}=a+bi$${{z}_{2}}=c+di$$|z_1=\sqrt{a^2+b^2}|$$|z_2=\sqrt{c^2+d^2}|$\begin{align} L.H.S.&=|{{z}_{1}}{{z}_{2}}|\\ &=|(a+bi)(c+di)|\\ &=|ac-bd+(ad+bc)i|\\ &=\sqrt{{{(ac-bd)}^{… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-2-p7?rev=1737476037&do=diff Question 8, Exercise 1.2 Solutions of Question 8 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 8(i) $z+\overline{z}=2\operatorname{Re}\left( z \right)$$z=a+ib$$\overline{z}=a-ib$\begin{align}z+\overline{z}&=\left( a+ib \right)+\left( a-ib \right)\\ &=a+ib+a-ib\\ &=2a\\ z+\overline{z}&=2\operatorname{Re}\left( z \right)\end{align}$z-\overline{z}=2i… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-2-p7?rev=1737476037&do=diff Question 9 & 10, Exercise 3.2 Solutions of Question 9 & 10 of Exercise 3.2 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9 $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}, $$$ and $$, find a vector of magnitude of $$ unit which is parallel to the vector $\begin{align}2\overrightarrow{a}-\overrightarrow{b}+3\overrightarrow{c}&=2(\hat{i}+\hat{j}+\hat{k})-(4\hat{i}-2\hat{j}+3\h… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-3-p3?rev=1737476037&do=diff Question 4 and 5 Exercise 3.3 Solutions of Question 4 and 5 of Exercise 3.3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4 $\hat{i}+7 \hat{j} + 3 \hat{k}$$\hat{i}-\hat{j}+2 \hat{k}$$2 \hat{i}-$$\hat{j}+3 \hat{k}$$\vec{a}=\hat{i}+7 \hat{j}+3 \hat{k}$$\vec{b}=\hat{i}-\hat{j}+2 \hat{k}$$\vec{c} = 2 \hat{i}-\hat{j}-3 \hat{k}$\begin{align}\vec{a} \cdot \vec{b}&=(\hat{i}+7 \h… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-4-p2?rev=1737476037&do=diff Question 2 Exercise 3.4 Solutions of Question 2 of Exercise 3.4 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2(i) Show in two different ways that the vectors $\vec{a}$$\vec{b}$$\vec{a}=-\hat{i}+2 \hat{j}-3 \hat{k}, \quad \vec{b}=2 \hat{i}-4 \hat{j}+$$6 \hat{k}$\begin{align}\vec{a} \times \vec{b}&=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ -1 & 2 & -3 \\ 2 … text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-4-p6?rev=1737476037&do=diff Question 6 Exercise 3.4 Solutions of Question 6 of Exercise 3.4 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6(i) A force $\vec{F}=3 \hat{i}-2 \hat{j}+5 \hat{k}$$(1,-2,2)$$\vec{r}$$P(1,-2.2)$$O(0,0,0)$\begin{align}\vec{r}&=\overrightarrow{O P}\\ &=(1,-2,2)-(0,0,0) \\ \Rightarrow \vec{r}&=(1,-2,2).\\ \text { Hence } \vec{M}-\vec{r} \times \vec{F}&=\left|\begin{array}{cc… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-5-p2?rev=1737476038&do=diff Question 3 & 4 Exercise 3.5 Solutions of Question 3 & 4 of Exercise 3.5 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 For the vectors $\vec{a}=3 \hat{i}+2 \hat{k}$$\vec{b}=\hat{i}+2 \hat{j}+\hat{k}\quad$$\quad\vec{c}=-\hat{j}+4 \hat{k}$$\vec{a} \cdot \vec{b} \times \vec{c}=\vec{b} \cdot \vec{c} \times \vec{a}=\vec{c} \cdot \vec{a} \times \vec{b}$$\vec{a} \cdot \vec{b}… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p12?rev=1737476038&do=diff Question 16 Exercise 4.2 Solutions of Question 16 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 16 $5$$8$$5$$8$$A_1, A_2, A_3, A_4, A_5$$5$$8$$5, A_1, A_2, A_3, A_4, A_5, 8$$$a_1=5 \text{ and } a_7=8.$$\begin{align}&a_7=a+6d\\ \implies &8=5+6d\\ \implies &6d=8-5\\ \implies &d=\dfrac{3}{6}=\dfrac{1}{2}. \end{align}\begin{align} A_1&=a+d=5+\dfra… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-5-p10?rev=1737476038&do=diff Question 15 & 16 Exercise 4.5 Solutions of Question 15 & 16 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$2$$4$$15^{\text {th }}$$a_1=R s .1$$a_2=R s .2$$a_3=R s .4$$1,2,4,8, \ldots$$a_1=1 . \quad r=2 . \quad n=15$$a_n=a_1 r^{n-1}$$15^{1 / 2}$$$a_{15}=a_1 r^{14} $$$$a_{15}=1 .(2)^{1 4}=R s .16384 $$$$S_{30}=\dfrac{a_1(r^{30}-1)}{r-1} $$$r-2$$a_1=1$\begi… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-3-p6?rev=1737476038&do=diff Question 6 Exercise 5.3 Solutions of Question 6 of Exercise 5.3 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6 $n$$n$$28+32+52+152+652+\ldots$\begin{align} & a_2-a_1=32-28=4 \\ & a_3-a_2=52-32=20 \\ & a_4-a_3=152-52=100 \\ & \ldots \quad \cdots \quad \cdots \\ & \cdots \quad \cdots \quad \cdots \\ & a_n-a_{n-1}=(\mathrm{n}-1) \text { term ofthe sequence } 4… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/re-ex5-p2?rev=1737476038&do=diff Question 2 & 3 Review Exercise Solutions of Question 2 & 3 of Review Exercise of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$n$$1.2+2.3+3.4+\ldots$$n^{\text {th }}$$$a_n=n(n+1)=n^2+n$$\begin{align} \sum_{r=1}^n a_r&=\sum_{r=1}^n r^2+\sum_{r=1}^n r \\ & =\dfrac{n(n+1)(2 n+1)}{6}+\dfrac{n(n+1)}{2} \\ & =\dfrac{n(n+1)}{2}[\dfrac{2 n+1}{3}+1] \\ & =\dfrac{n(n+1)}{2} \cdot … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/re-ex5-p4?rev=1737476038&do=diff Question 5 & 6 Review Exercise Solutions of Question 5 & 6 of Review Exercise of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$5+12 x+19 x^2+26 x^3+\ldots$$n$\begin{align}S_n&=5+12 x+19 x^2+26 x^3+\cdots+(7 n-2) x^{n-1}...(i)\\ x S_n&=5 x+12 x^2+19 x^3+\cdots+(7 n-9) x^{n-1}+(7 n-1) x^n....(ii)\end{align}\begin{align}(1-x) S_n&=5+(12-5) x+(19-12) x^2+\cdots\\ &+[7 n-2-(… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/re-ex5-p6?rev=1737476038&do=diff Question 8 Review Exercise Solutions of Question 8 of Review Exercise of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 8(i) $n$$n^{t h}$$n^3+3^n.$$n^h$$$a_n=n^3+3^n$$\begin{align}\sum_{r=1}^n a_r&=\sum_{r=1}^n r^3+\sum_{r=1}^n 3^r \\ & =[\dfrac{n(n+1)}{2}]^2+\dfrac{3(3^n-1)}{3-1} \\ & =\dfrac{n^2(n+1)^2}{4}+\dfrac{3}{2}(3^n-1) \end{align}$n$$$S_n=\dfrac{n^2(n+1… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/re-ex5-p7?rev=1737476038&do=diff Question 9 Review Exercise Solutions of Question 9 of Review Exercise of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9(i) $n$$3+7+13+21+31+\ldots$\begin{align} & a_2-a_1=7-3=4 \\ & a_3-a_2=13-7=6 \\ & a_4-a_3=21-13=8 \\ & \ldots \quad \ldots \quad \ldots \\ & \ldots \quad \cdots \quad \ldots \\ & a_n-a_{n-1}=(n-1) \text { term of the series } \\ & 4,6,8, \ldo… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p3?rev=1737476038&do=diff Question 5 and 6 Exercise 6.2 Solutions of Question 5 and 6 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$7.$$7$$7$\begin{align}^7 P_7&=\dfrac{7 !}{(7-7) !}\\ & =7 !\\ &=5,040 \end{align}$2,4,5,7,9$$2,4,5,7,9$$\mathrm{n} . \mathrm{m}$$e$$$=5.4 .3 .2=120\quad \text{or}$$$$^5 P_4=\dfrac{5 !}{5-4} !=120$$$2$$4$$3$$E_1$$m_1=2$$E_2$$m_2=3… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p6?rev=1737476038&do=diff Question 10 Exercise 6.2 Solutions of Question 10 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$n=8$$r=5$\begin{align}^8 P_5&=\dfrac{8 !}{(8-5) !}\\ &=\dfrac{8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 !}{3 !}\\ &=6720\end{align}\begin{align}^2 P_2 \times^7 P_4&=2 \times \dfrac{7 !}{(7-4) !}\\ &=2 \times\dfrac{7.6 .5 .4 .3 !}{3 !}\\ &=… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-4-p3?rev=1737476038&do=diff Question 3 Exercise 6.4 Solutions of Question 3 of Exercise 6.4 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$8$$8$$2^8$$$n(S)=256$$$$\dfrac{1}{256}$$$8$$$A=\{8\}$$$${ }^8 C_8=\dfrac{8 !}{(8-8) ! 8 !}=1$$$8$$$P(A)=\dfrac{1}{256}$$$7$$8$$2^8$$$n(S)=256$$$$\dfrac{1}{256}$$$7$$$B=\{7\}$$$7$$8$$$n(B)={ }^8 C_7=\dfrac{8 !}{(8-7) ! 7 !}=8$$$7$$8$$$P(B)=\d… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-5-p3?rev=1737476038&do=diff Question 5 and 6 Exercise 6.5 Solutions of Question 5 and 6 of Exercise 6.5 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{8}{9}$$$E=\{ event\, passing\, the\, test \}$$$$E^{\prime}=\{ event\, failing\, the\, test \}$$$E$$E^{\prime}$$P(E)=\dfrac{8}{9}$\begin{align}P(E^{\prime})&=1-P(E)=1-\dfrac{8}{9}=\dfrac{1}{9}\end{align}$4$$4$\begin{align}S… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p2?rev=1737476038&do=diff Question 2 Review Exercise 6 Solutions of Question 2 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${ }^{2 n} C_r={ }^{2 n} C_{r+2}$$r$\begin{align} { }^{2 n} C_r&={ }^{2 n} C_{r+2} \\ \Rightarrow \dfrac{(2 n) !}{(2 n-r) ! r !}&=\dfrac{(2 n) !}{(2 n-(r+2)) !(r+2) !}\end{align}$(2 n)$\begin{align} \Rightarrow \dfrac{1}{(2 n-r) ! r… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p6?rev=1737476038&do=diff Question 6 Exercise 7.1 Solutions of Question 6 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$1(1 !)+2(2 !)+3(3 !)+\ldots+n(n !)= -(n+1) !-1$$n=1$$$1(1 !)=1=(1+1) !-1=2 !-1=1 $$$n=1$$n=k$\begin{align}1(1 !)+2(2 !)+3(3 !)+\ldots+k(k !)& =(k+1) !-1 \ldots . .(i)\end{align}$n=k+1$$(k+1)^{t h}$$a_{k+1}=(k+1)[(k+1) !]$$a_{k-1}$\begin{ali… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p7?rev=1737476038&do=diff Question 7 Exercise 7.1 Solutions of Question 7 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$1.2+2.3+3.4+\ldots+n(n+1)=\dfrac{n(n+1)(n+2)}{3}$$n=1$$$1.2=2=\dfrac{1(1+1)(1+2)}{3}=2 $$$n=1$$n=k$\begin{align}1.2+2.3+3.4+\ldots+k(k+1)& =\dfrac{k(k+1)(k+2)}{3}....(i)\end{align}$n=k+1$$(k-1)^{t h}$$a_{k+1}=(k+1)(k+ 2)$$(k+1)^{\text {th }}… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p14?rev=1737476038&do=diff Question 14 Exercise 7.1 Solutions of Question 14 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$5$$3^{2 n-1}+2^{2 n-1}$$n$$n=1$$$3^{2 n-1}+2^{2 n-1}=3^{2.1-1}+2^{2.1-1}=5 \text {. }$$$5$$5$$5$$5.$$n=1$$n=k>1$$54$$3^{2 k} 1+2^{2 k} \quad 1$$$3^{2 k-1}+2^{2 k-1}=5 Q$$$Q$$n=k+1$\begin{align} 3^{2(k+1)-1}+2^{2(k+1)-1} & =3^{2 k+2-1}+2^{2… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p4?rev=1737476038&do=diff Question 4 Exercise 7.2 Solutions of Question 4 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$x^{23}$$(x^2-x)^{20}$$n=20, \quad a=x^2$$b=-x$$T_{r, 1}$$x^{23}$\begin{align}T_{r-1}&=\dfrac{20 !}{(20-r) ! r !}(x^2)^{20 r}(-x)^r \\ & =\dfrac{20 !}{(20-r) ! r !}(-1)^r \cdot x^{40-2 r+r} \\ & =\dfrac{20 !}{(20-r) ! r !}(-1)^r x^{40-r}\end{… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p8?rev=1737476038&do=diff Question 8 Exercise 7.2 Solutions of Question 8 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$(3-2 x)^{10}$$x=\frac{3}{4}$$\left(3-2,1^{10}=3^{10}\left(1-\frac{3 x}{2}\right)^{10}\right.$$\left(1-\frac{3 x}{2}\right)^{10}$$p+1$$: 3-\mathbf{2}_1 1^{10}$$T_{5} !=\left(\begin{array}{c}10 \\ 5\end{array}\right) 3^{10} 5-2 \gamma^{15}$$x=… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p9?rev=1737476039&do=diff Question 9 Exercise 7.2 Solutions of Question 9 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$(x-y)="$$x=12$$y-4$$x=12$$$ \begin{aligned} & \left(x \quad y=20(12-y)^{20}\right. \\ & =12^{2 n}\left(\begin{array}{ll} 1 & \frac{y}{12} \end{array}\right)^{31} \end{aligned} $$$\frac{(n+1) \cdot x}{1+|x|}$$\left(\frac{1}{12}\right)^2 \cdot… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p3?rev=1737476039&do=diff Question 3 Exercise 7.3 Solutions of Question 3 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sqrt{\frac{1-x}{1+x}}$$x^3$$\sqrt{\frac{1-x}{1+x}}$$$ =(1-x)^{\frac{1}{2}}(1+x)^{-\frac{1}{2}} \text {. } $$$$ \begin{aligned} & (1-x)^{\frac{1}{2}}(1+x)^{\frac{1}{2}} \\ & =\left[1-\frac{x}{2}+\frac{\frac{1}{2}\left(\frac{1}{2}-1\right)}{2… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p6?rev=1737476039&do=diff Question 7 and 8 Exercise 7.3 Solutions of Question 7 and 8 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$x^4$$(1-x)^{\frac{1}{4}}+(1-x)^{\frac{1}{4}}=a-b x^2$$a$$b$$$ \begin{aligned} & (1+x)^{\frac{1}{4}}+(1-x)^{\frac{1}{4}} \\ & =\left[1+\frac{x}{4}+\frac{\frac{1}{4}\left(\frac{1}{4}-1\right)}{2 !} x^2+\right. \\ & \left.\frac{\fra… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/re-ex7-p3?rev=1737476039&do=diff Question 3 & 4 Review Exercise 7 Solutions of Question 3 & 4 of Review Exercise 7 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$(2 x-4 y)^7$$n=7, a=2 x$$b=-4 y$$$ \begin{aligned} & T_{3+1}=\frac{7 !}{(7-3) ! 3 !}(2 x)^{7 \cdot 3}(-4 y)^3 \\ & =\frac{7 !}{(7-3) ! 3 !} \cdot\left(2^4\right) \cdot(-4)^3 \cdot x^4 y^3 \\ & \Rightarrow T_4=-35840 x^4 y^3… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/re-ex7-p4?rev=1737476039&do=diff Question 5 & 6 Review Exercise 7 Solutions of Question 5 & 6 of Review Exercise 7 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\left(\frac{2}{x^2}+\frac{x^2}{2}\right)^{10}$$n=10, a^{\prime}=\frac{2}{x^2}$$b=\frac{x^2}{2}$$T_{r+1}$$x$$$ \begin{aligned} & T_{r+1}=\frac{10 !}{(10-r) ! r !}\left(\frac{2}{x^2}\right)^{10 r}\left(\frac{x^2}{2}\right)^r … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-2-p6?rev=1737476039&do=diff Question 7, Exercise 10.2 Solutions of Question 7 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${{\cos }^{4}}\theta -{{\sin }^{4}}\theta =\dfrac{1}{\sec 2\theta }$\begin{align}L.H.S&={{\cos }^{4}}\theta -{{\sin }^{4}}\theta \\ &=\left( {{\cos }^{2}}\theta -{{\sin }^{2}}\theta \right)\left( {{\cos }^{2}}\theta +{{\… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p5?rev=1737476039&do=diff Question 5, Exercise 1.2 Solutions of Question 5 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 5 $z_1$$z_2$$|z_1+z_2|^2-|z_1-z_2|^2=4Re(z_1)Re(z_2)$\begin{align}z_1&=x_1+iy_1 \text{ and } z_2&=x_2+iy_2\end{align}\begin{align}z_1+z_2&=x_1+iy_1+x_2+iy_2\\ &=x_1+x_2+i(y_1+y_2)\\ |z_1+z_2|^2&=(x_1+x_2)^2+(y_1+y_2)^2\\ &=x^2_1+x^2_2+2x_1x_… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p6?rev=1737476039&do=diff Question 6, Exercise 1.2 Solutions of Question 6 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 6 $\lambda$$\left|\dfrac{z_{1}}{z_{2}}+\lambda\right|=\sqrt{\lambda+2}$$z_{1}=3+i$$z_{2}=1+i$\begin{align} &z_{1}=3+i\text{ and } z_{2}=1+i.\end{align}\begin{align} \dfrac{z_1}{z_2} &= \dfrac{3+i}{1+i}\\ &=\dfrac{(3+i)(1-i)}{(1+i)(1-i)} \\ &=\… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-3-p3?rev=1737476039&do=diff Question 3, Exercise 1.3 Solutions of Question 3 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 3(i) $\dfrac{1}{3} z^{2}+2 z-16=0$\begin{align}&\dfrac{1}{3}z^{2}+2 z-16=0\\ \implies &z^{2} + 6z - 48 = 0 \end{align}$$ z = \dfrac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a},$$$$a = 1,\quad b = 6,\quad \text{and}\quad c = -48.$$\begin{align} z& = \d… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p8?rev=1737476039&do=diff Question 7, Exercise 1.4 Solutions of Question 7 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 7(i) $\arg (z-1)=-\dfrac{\pi}{4}$$z=x+iy$\begin{align*} &\arg (z-1)=-\dfrac{\pi}{4} \\ \implies & \arg(x+iy-1) = -\dfrac{\pi}{4} \\ \implies & \arg(x-1+iy) = -\dfrac{\pi}{4} \\ \implies & \tan^{-1}\left(\dfrac{y}{x-1}\right) = -\dfrac{\pi}{4} … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p9?rev=1737476039&do=diff Question 9, Exercise 2.2 Solutions of Question 9 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A$$B$$3 \times 3$$(A+B)^{t}=A^{t}+B^{t}$\begin{align*} A &= \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} \\ B &= \begin{pmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p7?rev=1737476039&do=diff Question 11 and 12, Exercise 4.2 Solutions of Question 11 and 12 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $1000$$3000$$2$$5000$$3$$20$$$1000, 3000, 5000, \dots, \text{ upto 20 terms}.$$$a_1 = 1000$$d=3000-1000=2000$$S_20=?$$$S_n =\frac{n}{2}[2a_1+(n-1)d],$$\begin{align*} S_{20} &= \frac{20}{2}[2(1000)+(20-1)2000]\\ &= 10 [2000+(19)2000] \… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p9?rev=1737476039&do=diff Question 14 and 15, Exercise 4.2 Solutions of Question 14 and 15 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $b$$10$$b$$20$$a= b$$b=20$\begin{align*} &\text{A.M.} = \frac{a + b}{2} \\ \implies & 10 = \frac{b + 20}{2} \\ \implies & 20 = b + 20 \\ \implies & b = 20 - 20 \\ \implies & b = 0 \end{align*}$b = 0$$b$$25$$b$$20$$b$$10$$b$$-10$$x$$y$… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p4?rev=1737476039&do=diff Question 7 and 8, Exercise 4.3 Solutions of Question 7 and 8 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $9+11+13+15+\cdots$$n=12$$a_1=9$$d=11-9=2$$n=12$$S_n$\begin{align} S_n&=\frac{n}{2}[2a_1+(n-1)d] \\ \implies S_{12}&=\frac{12}{2}[2(9)+(12-1)(2)]\\ &=6\times [18+22]\\ &=240. \end{align}$S_{12}=240$$2$$100$$2$$100$$$2+4+6+...+100 (50 \tex… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p5?rev=1737476039&do=diff Question 9 and 10, Exercise 4.3 Solutions of Question 9 and 10 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $1$$99$$1$$99$$$1+3+5+...+99 (50 \text{ terms}).$$$a_{1}=1$$n=50$$d=3-1=2$$S_n$\begin{align} S_n&=\frac{n}{2}[2a_1+(n-1)d] \\ \implies S_{50}&=\frac{50}{2}[2(1)+(50-1)(2)]\\ &=25\times [2+98]\\ &=2500. \end{align}$1$$99$$2500$$14$$523$$… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p13?rev=1737476039&do=diff Question 26 and 27, Exercise 4.4 Solutions of Question 26 and 27 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $16\,\, ft$$6$$16\,\,ft$$a_1$$a_2$$a_3,...$$$a_1 = 16\times \dfrac{1}{4} = 4\,\, ft.$$$r=\dfrac{1}{4}$$a_6$$$a_{n}=a_{1} r^{n-1}.$$\begin{align*} a_{6}&=a_{1} r^5 \\ &=(4)\left(\dfrac{1}{4} \right)^5 \\ & = \dfrac{1}{256} \end{align*}… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-6-p6?rev=1737476040&do=diff Question 11, Exercise 4.6 Solutions of Question 11 of Exercise 4.6 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\dfrac{2}{3}$$\dfrac{4}{7}$$a=\dfrac{2}{3}$$b=\dfrac{4}{7}$\begin{align*} \text{H.M.}&=\frac{2ab}{a+b} \\ &=\frac{2\times\frac{2}{3}\times\frac{4}{7}}{\frac{2}{3}+\frac{4}{7}} \\ &=\frac{16/21}{26/21} \\ &=\frac{8}{13} \\ \end{align*}$\dfrac{8}{13… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-8-p5?rev=1737476040&do=diff Question 9 and 10, Exercise 4.8 Solutions of Question 9 and 10 of Exercise 4.8 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$\frac{1}{1 \cdot 3}+\frac{1}{2 \cdot 5}+\frac{1}{3 \cdot 7}+\ldots \ldots \text{ up to } \infty$$$\sum_{k=3}^{n} \dfrac{1}{(k+1)(k+2)}$\begin{align*} T_k &= \frac{1}{(k+1)(k+2)}. \end{align*}\begin{align*} \frac{1}{(k+1)(k+2)} = \frac… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-1-p4?rev=1737476040&do=diff Question 6 and 7, Exercise 5.1 Solutions of Question 6 and 7 of Exercise 5.1 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $m$$2 x^{3}+3 x^{2}-3 x-m$$x-2$$p(x)=2 x^{3}+3 x^{2}-3 x-m$$x-c=x-2$$\implies c=2$\begin{align*} \text{Remainder} & = p(c) = p(2) \\ & = 2(2)^{3} + 3(2)^{2} - 3(2) - m \\ & = 2(8) + 3(4) - 3(2) - m \\ & = 16 + 12 - 6 - m \\ & = 22 - m. \end{align… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p5?rev=1737476040&do=diff Question 5 and 6, Exercise 8.1 Solutions of Question 5 and 6 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin \alpha=\dfrac{4}{5}, \tan \beta=-\dfrac{5}{12}$$\cos (\alpha+\beta)$$\cos (\alpha-\beta)$$\sin \alpha=\dfrac{4}{5}$$\alpha$$\tan \beta=-\dfrac{5}{12}$$\beta$$$\cos \alpha=\pm \sqrt{1-\sin^2\alpha}.$$$\alpha$$\cos$\begin{alig… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p10?rev=1737476040&do=diff Question 11, Exercise 8.1 Solutions of Question 11 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\dfrac{\sin \left(180^{\circ}+\lambda\right) \cos \left(270^{\circ}+\lambda\right)}{\sin \left(180^{\circ}-\lambda\right) \cos \left(270^{\circ}-\lambda\right)}=1$\begin{align*} L.H.S & = \dfrac{\sin \left(180^{\circ}+\lambda\right) \cos \… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p6?rev=1737476040&do=diff Question 8(i, ii & iii) Exercise 8.2 Solutions of Question 8(i, ii & iii) of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $(\sin \theta+\cos \theta)^{2}=1+\sin 2 \theta$\begin{align*} LHS & = (\sin \theta+\cos \theta)^{2} \\ &=\sin^2\theta + \cos^2\theta +2\sin \theta \cos\theta\\ &= 1+2\sin \theta \cos\theta \quad (\because \sin^2\theta… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p11?rev=1737476040&do=diff Question 8(xvi, xvii & xviii) Exercise 8.2 Solutions of Question 8(xvi, xvii & xviii) of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\dfrac{1-\cos ^{2} \beta}{2-2 \cos \beta}=\cos ^{2} \dfrac{\beta}{2}$\begin{align*} LHS &= \dfrac{1-\cos ^{2} \beta}{2-2 \cos \beta}\\ &= \dfrac{\sin ^{2} \beta}{2-2 \cos \beta}\\ &=\dfrac{4\sin ^{2} \fr… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-3-p5?rev=1737476040&do=diff Question 3(i, ii, iii, iv & v) Exercise 8.3 Solutions of Question 3(i, ii, iii, iv & v) of Exercise 8.3 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\dfrac{\cos (\alpha + \beta)}{\cos(\alpha - \beta)}=\dfrac{1- \tan \alpha \tan \beta}{1+ \tan \alpha \tan \beta}$\begin{align*} RHS & = \dfrac{1- \tan \alpha \tan \beta}{1+ \tan \alpha \tan \beta} \\ & … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/re-ex-p2?rev=1737476040&do=diff Question 2 and 3, Review Exercise Solutions of Question 2 and 3 of Review Exercise of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\cos \theta -\sin \theta=\sqrt{2}\sin \theta,$$\cos \theta+ \sin \theta=\sqrt{2} \cos \theta$$$\cos \theta -\sin \theta=\sqrt{2}\sin \theta$$\begin{align*} & \cos \theta=\sqrt{2}\sin \theta + \sin \theta \\ \implies & \cos \the… text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/quote-of-the-day?rev=1737476042&do=diff Quote of the Day [Quote of the Day] The “quote of the day” is a useful tool for inspiration, motivation, and self-examination It offers a daily serving of knowledge from reputable and accomplished people in a range of sectors, including literature, politics, science, and entertainment. Everyday reading and reflection on a thought-provoking quote can help people learn more, see things from other people's perspectives, and advance their personal development. Sharing the text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_mcqs?rev=1737476035&do=diff MCQs/Objective: HSSC-I <well> Short Questions by Mr. Akhtar Abbas NEW Short Questions without answers by Mr. Akhtar Abbas for FSc Part 1. </well><well> MCQs-Short Questions by Mr Parvez Khan MCQs and Short Question by Mr. Parvez Khan composed by Momin Ali: Text Book of Algebra and Trigonometry Class XI (Punjab Textbook Board, Lahore) </well><well> text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/group-theory-important-definitions-and-results?rev=1737476041&do=diff Group Theory: Important Definitions and Results [Group Theory: Important Definitions and Results] These notes are made and shared by Mr. Akhtar Abbas. We are really very thankful to him for providing these notes and appreciates his effort to publish these notes on MathCity.org. These notes contains important definitions with examples and related theorem, which might be helpful to prepare interviews or any other written test after graduation like PPSC, FPSC or etc.$G$$a\in G$$\exists$$e\in G$$a\… text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/ppsc/ppsc-maths-2021?rev=1737476042&do=diff PPSC Paper 2021 (Lecturer in Mathematics) [PPSC Paper 2011 (Lecturer in Mathematics)] On this page, we have given question from old (past) paper of Lecturer in Mathematics conducted in year 2021. This is a MCQs paper and answers are given at the end of the paper. At the end of the PDF is also given to download. This paper is provided by Ms. \(2018\)$4$\(6\)$8$$10$\(X\)\(Y\)\(X\times Y\)\(\parallel (x,y) \parallel=\parallel x\parallel+\parallel y\parallel, \,\forall \, (x,y)\in X \times Y\)\(f(… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-1-p2?rev=1737476036&do=diff Question 2 & 3, Exercise 1.1 Solutions of Question 2 & 3 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2 ${{i}^{107}}+{{i}^{112}}+{{i}^{122}}+{{i}^{153}}=0$\begin{align}L.H.S.&={{i}^{107}}+{{i}^{112}}+{{i}^{122}}+{{i}^{153}}\\ &=i\cdot i^{106}+i^{112}+i^{122}+i\cdot i^{152}\\ &=i.{{\left( {{i}^{2}} \right)}^{53}}+{{\left( {{i}^{2}} \right)}^{56}}+… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-1-p3?rev=1737476036&do=diff Question 4, Exercise 1.1 Solutions of Question 4 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4(i) $\left( a,0 \right)\left( 2,-b \right)$\begin{align}&\left( a,0 \right)-\left( 2,-b \right)\\ &=\left( a+0i \right)-\left( 2-bi \right)\\ &=\left( a-2 \right)+\left( 0+b \right)i\\ &=\left( a-2 \right)+bi\end{align}$\left( -3,\dfrac{1}{2} \right)\le… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-1-p4?rev=1737476036&do=diff Question 5, Exercise 1.1 Solutions of Question 5 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) $8i+11,-7+5i$\begin{align}&(8i+11)\times (-7+5i)\\ &=\left( 11+8i \right)\times \left( -7+5i \right)\\ &=\left( -77+40{{i}^{2}} \right)+\left( 55-56 \right)i\\ &=\left( -77+40\left( -1 \right) \right)+\left( 55-56 \right)i\\ &=\left( -77-40 \right)+… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-1-p5?rev=1737476036&do=diff Question 6, Exercise 1.1 Solutions of Question 6 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6(i) $\dfrac{4+i}{3+5i}$$a+ib$\begin{align}\dfrac{4+i}{3+5i}&=\dfrac{4+i}{3+5i}\times \dfrac{3-5i}{3-5i}\\ &=\dfrac{\left( 12+5 \right)+\left( 3-20 \right)i}{9-25{{i}^{2}}}\\ &=\dfrac{17-17i}{9+25}\\ &=\dfrac{17}{34}-\dfrac{17}{34}i\\ &=\dfrac{1}{2}-\dfr… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-1-p6?rev=1737476036&do=diff Question 7, Exercise 1.1 Solutions of Question 7 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7(i) ${{z}_{1}}=1+2i$${{z}_{2}}=2+3i$$|{{z}_{1}}+{{z}_{2}}|$$z_1=1+2i$$z_2=2+3i$\begin{align} {{z}_{1}}+{{z}_{2}}&=1+2i+2+3i\\ &=1+2+2i+3i\\ &=3+5i \end{align}\begin{align} |z_1+z_2|&=\sqrt{3^2+5^2}\\ &=\sqrt{9+25}\\ &=\sqrt{34}\end{align}${{z}_{1}}=1+2… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-1-p7?rev=1737476036&do=diff Question 8, Exercise 1.1 Solutions of Question 8 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 8(i) $\dfrac{1-2i}{2+i}+\dfrac{4-i}{3+2i}$$a+ib.$\begin{align}&\dfrac{1-2i}{2+i}+\dfrac{4-i}{3+2i}\\ &=\dfrac{\left( 3+2i \right)\left( 1-2i \right)+\left( 2+i \right)\left( 4-i \right)}{\left( 2+i \right)\left( 3+2i \right)}\\ &=\dfrac{\left( 3+4+2i-6i … text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-1-p8?rev=1737476036&do=diff Question 9 & 10, Exercise 1.1 Solutions of Question 9 & 10 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9 $\dfrac{\left( 3-2i \right)\left( 2+3i \right)}{\left( 1+2i \right)\left( 2-i \right)}$\begin{align}z&=\dfrac{\left( 3-2i \right)\left( 2+3i \right)}{\left( 1+2i \right)\left( 2-i \right)}\\ &=\dfrac{6+6+9i-4i}{2+2+4i-i}\\ &=\dfrac{12+5i}{4+3… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-2-p2?rev=1737476036&do=diff Question 2, Exercise 1.2 Solutions of Question 2 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2 $z_1=-1+i$, $z_2=3-2i$${{z}_{3}}=2-2i$${{z}_{1}}=-1+i$${{z}_{2}}=3-2i$${{z}_{3}}=2-2i$$$(z_1+z_2)+z_3=z_1+(z_2+z_3).$$\begin{align} {{z}_{1}}+{{z}_{2}}&=\left( -1+i \right)+\left( 3-2i \right)\\ &=2-i\end{align}\begin{align} \left( {{z}_{1}}+{{z}_{2}… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-2-p3?rev=1737476036&do=diff Question 3 & 4, Exercise 1.2 Solutions of Question 3 & 4 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 ${{z}_{1}}=\sqrt{3}+\sqrt{2}i$${{z}_{2}}=\sqrt{2}-\sqrt{3}i$${{z}_{3}}=2+3i$${{z}_{1}}=\sqrt{3}+\sqrt{2}i$${{z}_{2}}=\sqrt{2}-\sqrt{3}i$${{z}_{3}}=2+3i$\begin{align}{{z}_{1}}\left( {{z}_{2}}+{{z}_{3}} \right)&={{z}_{1}}{{z}_{2}}+{{z}_{1}}{{z}_{… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-3-p2?rev=1737476036&do=diff Question 2, Exercise 1.3 Solutions of Question 2 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2(i) $P(z)$$$P\left( z \right)={{z}^{3}}+6z+20$$$$p\left( z \right)={{z}^{3}}+6z+20$$$(z-a)$$P(z)$$P(a)=0$$z=-2$\begin{align} P(-2)&=(-2)^3+6(-2)+20\\ &=-8-12+20\\ &=0\end{align}$z+2$${{z}^{3}}+6z+20$$$\begin{array}{c|cccc} -2 & 1 & 0 & 6 & 20 \\ & \d… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-3-p3?rev=1737476036&do=diff Question 3 & 4, Exercise 1.3 Solutions of Question 3 & 4 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 ${{z}_{1}}=-1+i$${{z}_{2}}=-1-i$${{z}^{2}}+2z+2=0$$$z^2+2z_1+2=0\quad \ldots (i)$$$z_1=-1+i$\begin{align}L.H.S &= (-1+i)^2+2(-1+i)+2\\ &=1-2i-1-2+2i+2\\ &=0=R.H.S\end{align}$z_1=-1+i$$z_2=-1-i$\begin{align} L.H.S&=(-1-i)^2+2(-1-i)+2\\ &=1+2i-1-… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-3-p4?rev=1737476036&do=diff Question 5, Exercise 1.3 Solutions of Question 5 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) ${{z}^{2}}+z+3=0$${{z}^{2}}+z+3=0$$a=1,\,\,\,b=1$$c=3$\begin{align}z&=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\\ z&=\dfrac{-\left( 1 \right)\pm \sqrt{{{\left( 1 \right)}^{2}}-4\left( 1 \right)\left( 3 \right)}}{2\left( 1 \right)}\\ z&=\dfrac{-1\pm \s… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/review-ex-1-p2?rev=1737476036&do=diff Question 2 & 3, Review Exercise 1 Solutions of Question 2 & 3 of Review Exercise 1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${{i}^{n}}+{{i}^{n+1}}+{{i}^{n+2}}+{{i}^{n+3}}=0$$\forall n\in N$\begin{align}{{i}^{n}}+{{i}^{n+1}}+{{i}^{n+2}}+{{i}^{n+3}}&=0\\ L.H.S.&={{i}^{n}}+{{i}^{n}}\cdot i+{{i}^{n}}\cdot {{i}^{2}}+{{i}^{n}}\cdot {{i}^{3}}\\ &={{i}^{n}}\left( 1+i+{{i}^{2}}… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/review-ex-1-p3?rev=1737476036&do=diff Question 4 & 5, Review Exercise 1 Solutions of Question 4 & 5 of Review Exercise 1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${{z}_{1}}=2-i$${{z}_{2}}=1+i,$$|\dfrac{{{z}_{1}}+{{z}_{2}}+1}{{{z}_{1}}-{{z}_{2}}+1}|$\begin{align}{{z}_{1}}&=2-i,\\ {{z}_{2}}&=1+i,\\ \dfrac{{{z}_{1}}+{{z}_{2}}+1}{{{z}_{1}}-{{z}_{2}}+1}&=\dfrac{\left( 2-i \right)+\left( 1+i \right)+1}{\left( 2-… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-1-p2?rev=1737476036&do=diff Question 2, Exercise 10.1 Solutions of Question 2 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin \dfrac{\pi }{12}$$\dfrac{\pi }{12}$$\dfrac{\pi }{3}-\dfrac{\pi }{4}$\begin{align}\sin (\alpha -\beta )=\sin \alpha \cos \beta -\cos \alpha \sin.\end{align}\begin{align} \Rightarrow \quad \sin \left( \frac{\pi }{3}-\f… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-1-p3?rev=1737476036&do=diff Question 3, Exercise 10.1 Solutions of Question 3 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin u=\dfrac{3}{5}$$\sin v=\dfrac{4}{5}$$u$$v$$0$$\dfrac{\pi }{2}$$\cos \left( u+v \right)$$\sin u=\dfrac{3}{5},$$0\le u\le \dfrac{\pi }{2}.$$\sin v=\dfrac{4}{5},$$0\le v\le \dfrac{\pi }{2}.$$\cos u=\pm \sqrt{1-{{\sin }^… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-1-p4?rev=1737476036&do=diff Question, Exercise 10.1 Solutions of Question 4 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin \alpha =-\dfrac{4}{5}$$\cos \beta =-\dfrac{12}{13}$$\alpha $$\beta $$\sin \left( \alpha -\beta \right)$$\sin \alpha=-\dfrac{4}{5}$$\alpha$$\sin \beta=-\dfrac{12}{13}$$\beta$$$\cos \alpha=\pm \sqrt{1-\sin^2\alpha}.$$$\… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-1-p5?rev=1737476036&do=diff Question 5, Exercise 10.1 Solutions of Question 5 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\tan \alpha =\dfrac{3}{4}$$\sec \beta =\dfrac{13}{5}$$\alpha$$\beta$$\sin \left( \alpha +\beta \right)$$\tan\alpha =\dfrac{3}{4}$$\tan\alpha$$\alpha$\begin{align}{{\sec}^{2}}\alpha &=1+{{\tan}^{2}}\alpha\\ \Rightarrow \q… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-1-p6?rev=1737476036&do=diff Question 6, Exercise 10.1 Solutions of Question 1 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos \alpha =2{{\cos }^{2}}\dfrac{\alpha }{2}-1=1-2{{\sin }^{2}}\dfrac{\alpha }{2}$\begin{align}L.H.S&=\cos \alpha \\ \cos \alpha &=\cos 2\dfrac{\alpha }{2}\\ &={{\cos }^{2}}\dfrac{\alpha }{2}-{{\sin }^{2}}\dfrac{\alpha }… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-1-p7?rev=1737476036&do=diff Question 7, Exercise 10.1 Solutions of Question 1 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cot \left( \alpha +\beta \right)=\dfrac{\cot \alpha \cot \beta -1}{\cot \alpha +\cot \beta }$\begin{align}L.H.S.&=\cot (\alpha +\beta )\\ &=\dfrac{1}{\tan (\alpha +\beta )}\\ &=\dfrac{1}{\,\dfrac{\tan \alpha +\tan \beta… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-1-p8?rev=1737476036&do=diff Question 8, Exercise 10.1 Solutions of Question 8 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\tan \left( \dfrac{\pi }{4}+\theta \right)=\dfrac{\cos \theta +\sin \theta }{\cos \theta -\sin \theta }$\begin{align}L.H.S.&=\tan \left( \dfrac{\pi }{4}+\theta \right)\\ &=\dfrac{\sin \left( \dfrac{\pi }{4}+\theta \ri… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-1-p9?rev=1737476036&do=diff Question 9 and 10, Exercise 10.1 Solutions of Question 9 and 10 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{\sin \theta }{\sec 4\theta }+\dfrac{\cos \theta }{\cos ec4\theta }=\sin 5\theta $\begin{align}L.H.S.&=\dfrac{\sin \theta }{\sec 4\theta }+\dfrac{\cos \theta }{\cos ec4\theta }\\ &=\dfrac{\sin \theta }… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-1-p10?rev=1737476036&do=diff Question11 and 12, Exercise 10.1 Solutions of Question 11 and 12 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\alpha$$\beta$$\gamma$$ABC$$\cot \dfrac{\alpha }{2}+\cot \dfrac{\beta }{2}+\cot \dfrac{\gamma }{2}=\cot \dfrac{\alpha }{2}\cot \dfrac{\beta }{2}\cot \dfrac{\gamma }{2}$$\alpha$$\beta$$\gamma$\begin{align}&\… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-2-p2?rev=1737476036&do=diff Question 2, Exercise 10.2 Solutions of Question 2 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin \theta =\dfrac{5}{13}$$\theta $$\sin 2\theta $$\sin \theta =\dfrac{5}{13}$$$\cos \theta =\pm \sqrt{1-{{\sin }^{2}}\theta }.$$$\theta$$\cos$\begin{align}\cos\theta &=-\sqrt{1-{{\sin }^{2}}\theta }\\ &=-\sqrt{1-\left(\… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-2-p3?rev=1737476036&do=diff Question 3, Exercise 10.2 Solutions of Question 3 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin \theta =\dfrac{4}{5}$$\theta$$\sin2\theta$$\sin \theta =\dfrac{4}{5}$$\theta$$\cos \theta =-\dfrac{3}{5}$\begin{align}\sin 2\theta &=2\sin \theta \cos \theta \\ &=2\left( \dfrac{4}{5} \right)\left( -\dfrac{3}{5} \rig… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-2-p4?rev=1737476036&do=diff Question 4 and 5, Exercise 10.2 Solutions of Question 4 and 5 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos \theta =-\dfrac{3}{7}$$\theta $$\sin \dfrac{\theta }{2}$$\cos \theta =-\dfrac{3}{7}$$\theta$\begin{align}&\pi < \theta < \dfrac{3\pi}{2} \\ \implies &\frac{\pi}{2} < \frac{\theta}{2} < \dfrac{3\pi}{4}\end… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-2-p5?rev=1737476036&do=diff Question 6, Exercise 10.2 Solutions of Question 6 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos {{15}^{\circ }}$${{15}^{\circ }}=\dfrac{{{30}^{\circ }}}{2}$$\dfrac{\theta }{2}=\dfrac{{{30}^{\circ }}}{2}$$\cos {{15}^{\circ }}$\begin{align}\cos {{15}^{\circ }}&=\cos \dfrac{{{30}^{\circ }}}{2}=\sqrt{\dfrac{1+\cos … text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-3-p2?rev=1737476036&do=diff Question 2, Exercise 10.3 Solutions of Question 2 of Exercise 10.3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$\sin {{37}^{\circ }}+\sin {{43}^{\circ }}.$$$$\sin \alpha +\sin \beta =2\sin \left( \dfrac{\alpha +\beta }{2} \right)\cos \left( \dfrac{\alpha -\beta }{2} \right).$$$\alpha ={{37}^{\circ }}$$\beta ={{43}^{\circ }}$\begin… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-3-p3?rev=1737476036&do=diff Question 3, Exercise 10.3 Solutions of Question 3 of Exercise 10.3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$\dfrac{\cos {{75}^{\circ }}+\cos {{15}^{\circ }}}{\sin {{75}^{\circ }}-\sin {{15}^{\circ }}}=\sqrt{3}.$$$$\cos \alpha +\cos \beta =2\cos \left( \dfrac{\alpha +\beta }{2} \right)\cos \left( \dfrac{\alpha -\beta }{2} \righ… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/re-ex10-p2?rev=1737476037&do=diff Question 2 and 3, Review Exercise 10 Solutions of Question 2 and 3 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{2\sin \theta \sin 2\theta }{\cos \theta +\cos 3\theta }=\tan 2\theta \tan \theta $\begin{align}L.H.S.&=\dfrac{2\sin \theta \sin 2\theta }{\cos \theta +\cos 3\theta }\\ &=\dfrac{2\sin \theta \s… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/re-ex10-p3?rev=1737476037&do=diff Question 4 & 5, Review Exercise 10 Solutions of Question 4 & 5 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${{\sin }^{2}}\dfrac{\theta }{2}=\dfrac{\sin \theta \tan \dfrac{\theta }{2}}{2}$\begin{align}R.H.S.&=\dfrac{\sin \theta \tan \dfrac{\theta }{2}}{2}\\ &=\dfrac{\sin \theta \sin \dfrac{\theta }{2}}{2\cos \d… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/re-ex10-p4?rev=1737476037&do=diff Question 6 & 7, Review Exercise 10 Solutions of Question 6 & 7 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos 4\theta =1-8{{\sin }^{2}}\theta {{\cos }^{2}}\theta $\begin{align}L.H.S&=\cos 4\theta \\ &=\cos 2\left( 2\theta \right)\\ &=1-2si{{n}^{2}}2\theta \\ &=1-2{{\left( 2sin\theta \cos \theta \right)}^{… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-ptb/sol/ch01/ex1-2?rev=1737476037&do=diff Exercise 1.2 (Solutions) <lead>Notes (Solutions) of Exercise 1.2: Textbook of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board (PTB) Lahore.</lead> The main topics of this exercise are complex numbers, real part and imaginary part of complex numbers, properties of the fundamental operation on complex numbers, complex number as ordered pair of real numbers and special subset of complex numbers. These notes are based on the new Student Learning Outcomes… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-1-p2?rev=1737476037&do=diff Question 2 & 3, Exercise 1.1 Solutions of Question 2 & 3 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2 ${{i}^{107}}+{{i}^{112}}+{{i}^{122}}+{{i}^{153}}=0$\begin{align}L.H.S.&={{i}^{107}}+{{i}^{112}}+{{i}^{122}}+{{i}^{153}}\\ &=i\cdot i^{106}+i^{112}+i^{122}+i\cdot i^{152}\\ &=i.{{\left( {{i}^{2}} \right)}^{53}}+{{\left( {{i}^{2}} \right)}^{56}}+… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-1-p3?rev=1737476037&do=diff Question 4, Exercise 1.1 Solutions of Question 4 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4(i) $\left( a,0 \right)\left( 2,-b \right)$\begin{align}&\left( a,0 \right)-\left( 2,-b \right)\\ &=\left( a+0i \right)-\left( 2-bi \right)\\ &=\left( a-2 \right)+\left( 0+b \right)i\\ &=\left( a-2 \right)+bi\end{align}$\left( -3,\dfrac{1}{2} \right)\le… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-1-p4?rev=1737476037&do=diff Question 5, Exercise 1.1 Solutions of Question 5 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) $8i+11,-7+5i$\begin{align}&(8i+11)\times (-7+5i)\\ &=\left( 11+8i \right)\times \left( -7+5i \right)\\ &=\left( -77+40{{i}^{2}} \right)+\left( 55-56 \right)i\\ &=\left( -77+40\left( -1 \right) \right)+\left( 55-56 \right)i\\ &=\left( -77-40 \right)+… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-1-p5?rev=1737476037&do=diff Question 6, Exercise 1.1 Solutions of Question 6 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6(i) $\dfrac{4+i}{3+5i}$$a+ib$\begin{align}\dfrac{4+i}{3+5i}&=\dfrac{4+i}{3+5i}\times \dfrac{3-5i}{3-5i}\\ &=\dfrac{\left( 12+5 \right)+\left( 3-20 \right)i}{9-25{{i}^{2}}}\\ &=\dfrac{17-17i}{9+25}\\ &=\dfrac{17}{34}-\dfrac{17}{34}i\\ &=\dfrac{1}{2}-\dfr… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-1-p6?rev=1737476037&do=diff Question 7, Exercise 1.1 Solutions of Question 7 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7(i) ${{z}_{1}}=1+2i$${{z}_{2}}=2+3i$$|{{z}_{1}}+{{z}_{2}}|$$z_1=1+2i$$z_2=2+3i$\begin{align} {{z}_{1}}+{{z}_{2}}&=1+2i+2+3i\\ &=1+2+2i+3i\\ &=3+5i \end{align}\begin{align} |z_1+z_2|&=\sqrt{3^2+5^2}\\ &=\sqrt{9+25}\\ &=\sqrt{34}\end{align}${{z}_{1}}=1+2… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-1-p7?rev=1737476037&do=diff Question 8, Exercise 1.1 Solutions of Question 8 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 8(i) $\dfrac{1-2i}{2+i}+\dfrac{4-i}{3+2i}$$a+ib.$\begin{align}&\dfrac{1-2i}{2+i}+\dfrac{4-i}{3+2i}\\ &=\dfrac{\left( 3+2i \right)\left( 1-2i \right)+\left( 2+i \right)\left( 4-i \right)}{\left( 2+i \right)\left( 3+2i \right)}\\ &=\dfrac{\left( 3+4+2i-6i … text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-1-p8?rev=1737476037&do=diff Question 9 & 10, Exercise 1.1 Solutions of Question 9 & 10 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9 $\dfrac{\left( 3-2i \right)\left( 2+3i \right)}{\left( 1+2i \right)\left( 2-i \right)}$\begin{align}z&=\dfrac{\left( 3-2i \right)\left( 2+3i \right)}{\left( 1+2i \right)\left( 2-i \right)}\\ &=\dfrac{6+6+9i-4i}{2+2+4i-i}\\ &=\dfrac{12+5i}{4+3… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-2-p2?rev=1737476037&do=diff Question 2, Exercise 1.2 Solutions of Question 2 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2 $z_1=-1+i$, $z_2=3-2i$${{z}_{3}}=2-2i$${{z}_{1}}=-1+i$${{z}_{2}}=3-2i$${{z}_{3}}=2-2i$$$(z_1+z_2)+z_3=z_1+(z_2+z_3).$$\begin{align} {{z}_{1}}+{{z}_{2}}&=\left( -1+i \right)+\left( 3-2i \right)\\ &=2-i\end{align}\begin{align} \left( {{z}_{1}}+{{z}_{2}… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-2-p3?rev=1737476037&do=diff Question 3 & 4, Exercise 1.2 Solutions of Question 3 & 4 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 ${{z}_{1}}=\sqrt{3}+\sqrt{2}i$${{z}_{2}}=\sqrt{2}-\sqrt{3}i$${{z}_{3}}=2+3i$${{z}_{1}}=\sqrt{3}+\sqrt{2}i$${{z}_{2}}=\sqrt{2}-\sqrt{3}i$${{z}_{3}}=2+3i$\begin{align}{{z}_{1}}\left( {{z}_{2}}+{{z}_{3}} \right)&={{z}_{1}}{{z}_{2}}+{{z}_{1}}{{z}_{… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-3-p2?rev=1737476037&do=diff Question 2, Exercise 1.3 Solutions of Question 2 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2(i) $P(z)$$$P\left( z \right)={{z}^{3}}+6z+20$$$$p\left( z \right)={{z}^{3}}+6z+20$$$(z-a)$$P(z)$$P(a)=0$$z=-2$\begin{align} P(-2)&=(-2)^3+6(-2)+20\\ &=-8-12+20\\ &=0\end{align}$z+2$${{z}^{3}}+6z+20$$$\begin{array}{c|cccc} -2 & 1 & 0 & 6 & 20 \\ & \d… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-3-p3?rev=1737476037&do=diff Question 3 & 4, Exercise 1.3 Solutions of Question 3 & 4 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 ${{z}_{1}}=-1+i$${{z}_{2}}=-1-i$${{z}^{2}}+2z+2=0$$$z^2+2z_1+2=0\quad \ldots (i)$$$z_1=-1+i$\begin{align}L.H.S &= (-1+i)^2+2(-1+i)+2\\ &=1-2i-1-2+2i+2\\ &=0=R.H.S\end{align}$z_1=-1+i$$z_2=-1-i$\begin{align} L.H.S&=(-1-i)^2+2(-1-i)+2\\ &=1+2i-1-… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-3-p5?rev=1737476037&do=diff Question 6, Exercise 1.3 Solutions of Question 6 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6(i) ${{z}^{4}}+{{z}^{2}}+1=0$$$z^4+z^2+1=0$$$$z^4+2z^2+1-z^2=0$$$$( z^2+1 )^2-z^2=0$$$$( z^2+1+z)( z^2+1-z )=0$$$$( z^2+z+1 )( z^2-z+1 )=0$$$$(z^2+z+1 )=0$$$$z=\dfrac{-1\pm \sqrt{1-4}}{2}$$$$z=\dfrac{-1\pm \sqrt{3}i}{2}$$$$(z^2-z+1 )=0$$$$z=\dfrac{1\pm … text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/review-ex-1-p2?rev=1737476037&do=diff Question 2 & 3, Review Exercise 1 Solutions of Question 2 & 3 of Review Exercise 1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${{i}^{n}}+{{i}^{n+1}}+{{i}^{n+2}}+{{i}^{n+3}}=0$$\forall n\in N$\begin{align}{{i}^{n}}+{{i}^{n+1}}+{{i}^{n+2}}+{{i}^{n+3}}&=0\\ L.H.S.&={{i}^{n}}+{{i}^{n}}\cdot i+{{i}^{n}}\cdot {{i}^{2}}+{{i}^{n}}\cdot {{i}^{3}}\\ &={{i}^{n}}\left( 1+i+{{i}^{2}}… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/review-ex-1-p3?rev=1737476037&do=diff Question 4 & 5, Review Exercise 1 Solutions of Question 4 & 5 of Review Exercise 1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${{z}_{1}}=2-i$${{z}_{2}}=1+i,$$\left|\dfrac{{{z}_{1}}+{{z}_{2}}+1}{{{z}_{1}}-{{z}_{2}}+1}\right|$$z_1=2-i$$z_2=1+i$\begin{align} \dfrac{{{z}_{1}}+{{z}_{2}}+1}{{{z}_{1}}-{{z}_{2}}+1}&=\dfrac{\left( 2-i \right)+\left( 1+i \right)+1}{\left( 2-i \rig… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p2?rev=1737476037&do=diff Question 2, Exercise 2.1 Solutions of Question 2 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2 $A=\begin{bmatrix}2 & -5 & 1\\ 3 & 0 & -4\end{bmatrix}$$B=\begin{bmatrix}1 & -2 & -3 \\ 0 & -1 & 5\end{bmatrix}$$C=\begin{bmatrix}0 & 1 & -2\\0 & -1 & -1\end{bmatrix}$$2A+3B-4C.$$A=\begin{bmatrix}2 & -5 & 1\\ 3 & 0 & -4\end{bmatrix}$$B=\begin… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p3?rev=1737476037&do=diff Question 3, Exercise 2.1 Solutions of Question 3 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3(i) $A=\begin{bmatrix}x & y & z\end{bmatrix}$$B=\begin{bmatrix}a & h & g\\h & b & f\\g & f & c\end{bmatrix}$$C=\begin{bmatrix}x\\y\\z\end{bmatrix}$$\left( AB \right)C=A\left( BC \right)$$A=\begin{bmatrix}x & y & z\end{bmatrix}$$B=\begin{bmatri… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p4?rev=1737476037&do=diff Question 4, Exercise 2.1 Solutions of Question 4 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4 $A= \begin{bmatrix}1 & 4 & 4 \\ 4 & 1 & 4 \\ 4 & 4 & 1 \end{bmatrix}$$\dfrac{1}{3}A^2-2A-9I=0$$A=\begin{bmatrix} 1 & 4 & 4 \\ 4 & 1 & 4 \\ 4 & 4 & 1 \end{bmatrix}$\begin{align}\frac{1}{3}A^2&=\frac{1}{3}\left[ \begin{matrix} 1 & 4 & 4 … text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p6?rev=1737476037&do=diff Question 7, Exercise 2.1 Solutions of Question 7 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7 $ A=\begin{bmatrix}1 & 0 & -1 & 2 \\3 & 1 & 2 & \quad 5 \\0 & -2 & 1 & 6\end{bmatrix}$$ B=\begin{bmatrix} 2 & -1 & 3 & 1 \\1 & 3 & -1 & 4 \\3 & 1 & 2 & -1 \end{bmatrix}$$( A+B )^t=A^t+B^t$$A=\left[ \begin{matrix}1 & 0 & -1 & 2 \\3 & 1 &… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p7?rev=1737476037&do=diff Question 8, Exercise 2.1 Solutions of Question 8 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 8(i) $A=\begin{bmatrix}1 & 2 & 0 \\3 & -1 & 4 \end{bmatrix}$$( A^t )^t=A$$$A=\left[ \begin{matrix} 1 & 2 & 0 \\ 3 & -1 & 4 \\ \end{matrix} \right]$$$$A^t=\left[ \begin{matrix} 1 & 3 \\ 2 & -1 \\ 0 & 4 \\ \end{matrix} \rig… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p8?rev=1737476037&do=diff Question 9, Exercise 2.1 Solutions of Question 9 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9(i) $A=\begin{bmatrix}2 & -1 & 3 \\1 & \quad 0 & 1 \end{bmatrix},$$B=\begin{bmatrix}1 & 2 \\2 & 2 \\ 3 & 0 \end{bmatrix}$$( AB )^t=B^tA^t$$$A=\left[ \begin{matrix} 2 & -1 & 3 \\ 1 & \quad 0 & 1 \\ \end{matrix} \right],$$$$B=\left[… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p9?rev=1737476037&do=diff Question 10, Exercise 2.1 Solutions of Question 10 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 10 $A=\begin{bmatrix}1 & -3 & 4 \\-3 & 2 & -5 \\4 & -5 & 0 \end{bmatrix}$$B=\begin{bmatrix}5 & 6 & 7 \\6 & -8 & 3 \\7 & 3 & 1 \end{bmatrix}$$A$$B$$A+B$$$A=\left[ \begin{matrix} 1 & -3 & 4 \\ -3 & 2 & -5 \\ 4 & -5 & 0 \\ \end{ma… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p10?rev=1737476037&do=diff Question 11, Exercise 2.1 Solutions of Question 11 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 11 $A=\begin{bmatrix}0 & 1 & -2 \\-1 & 0 & 3 \\2 & -3 & 0 \end{bmatrix}$$B=\begin{bmatrix}0 & -6 & 11 \\6 & 0 & -7 \\-11 & 7 & 0 \end{bmatrix}$$A+B$$$A=\left[ \begin{matrix} 0 & 1 & -2 \\ -1 & 0 & 3 \\ 2 & -3 & 0 \\ \end{matri… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p11?rev=1737476037&do=diff Question 12, Exercise 2.1 Solutions of Question 12 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 12(i) $A=\begin{bmatrix}3 & 2 & 1 \\4 & 5 & 6 \\-2 & 3 & 4\end{bmatrix}$$A+A^t$$$A=\left[ \begin{matrix} 3 & 2 & 1 \\ 4 & 5 & 6 \\ -2 & 3 & 4 \\ \end{matrix} \right]$$$$A^t=\left[ \begin{matrix} 3 & 4 & -2 \\ 2 & 5 & 3 \… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p2?rev=1737476037&do=diff Question 2, Exercise 2.2 Solutions of Question 2 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2(i) $$\left| \begin{matrix} 1 & 2 & 0 \\ 3 & 1 & 0 \\ -1 & 2 & 0 \\ \end{matrix} \right|=0$$$\left| \begin{matrix}1 & 2 & 3 \\-8 & 4 & -12 \\2 & -1 & 3 \end{matrix} \right|=0$$$\left| \begin{matrix} 1 & 2 & 3 \\ -8 & 4 & -… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p3?rev=1737476037&do=diff Question 3, Exercise 2.2 Solutions of Question 3 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 $A$$3,$$|A^t|=|A|$$$A=\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{bmatrix}$$\begin{align}|A|&=a_{11} \left( a_{22} a_{33}-a_{23} a_{32} \right)-a_{12}\left( a_{21}a_{33}… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p4?rev=1737476037&do=diff Question 4, Exercise 2.2 Solutions of Question 4 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4(i) $\left| \begin{matrix}0 & 1 & 3 \\-1 & 2 & 1 \\2 & 1 & 1 \end{matrix} \right|.$\begin{align}&\left| \begin{matrix} 0 & 1 & 3 \\ -1 & 2 & 1 \\ 2 & 1 & 1 \\ \end{matrix} \right| \\ =&0\left( 2-1 \right)-1\left( -1-2 \right)+3\l… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p6?rev=1737476037&do=diff Question 6, Exercise 2.2 Solutions of Question 6 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Questiopn 6(i) $\left| \begin{matrix}a-b & b-c & c-a \\b-c & c-a & a-b \\c-a & a-b & b-c \end{matrix} \right|=0$\begin{align} L.H.S&=\left| \begin{matrix} a-b & b-c & c-a \\ b-c & c-a & a-b \\ c-a & a-b & b-c \\ \end{matrix} \right| \\ &=\left| \… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p7?rev=1737476037&do=diff Question 7, Exercise 2.2 Solutions of Question 7 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7(i) $\left| \begin{matrix}3860 & 3861 \\3862 & 3863 \end{matrix} \right|$$$\left| \begin{matrix} 3860 & 3861 \\ 3862 & 3863 \\ \end{matrix} \right|=14911180-14911182$$$$=-2$$$\left| \begin{matrix}81 & 82 & 83 \\84 & 85 & 86 \\87 & 8… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p8?rev=1737476037&do=diff Question 8,9 & 10, Exercise 2.2 Solutions of Questions 8,9 & 10 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\left| \begin{matrix}1+x & y & z \\x & 1+y & z \\x & y & 1+z \end{matrix} \right|=1+x+y+z$$$L.H.S.=\left| \begin{matrix} 1+x & y & z \\ x & 1+y & z \\ x & y & 1+z \\ \end{matrix} \right|$$$$=\left| \begin{matrix} 1 & 0 & -… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p9?rev=1737476037&do=diff Question 11, Exercise 2.2 Solutions of Question 11 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 11(i) $\left[ \begin{matrix}7 & 1 & 3 \\6 & 2 & -2 \\5 & 1 & 1\end{matrix} \right]$$$A=\left[ \begin{matrix} 7 & 1 & 3 \\ 6 & 2 & -2 \\ 5 & 1 & 1 \\ \end{matrix} \right]$$$$|A|=7(2+2)-1(6+10)+3(6-10)$$$$=28-16-12$$$$|A|=0$$$A$$\… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p10?rev=1737476037&do=diff Question 12, Exercise 2.2 Solutions of Question 12 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 12 $\lambda $$A$$A=\begin{bmatrix}-\lambda & 1 & 0 \\1 & -\lambda & 1 \\0 & 1 & -\lambda \end{bmatrix}$$$A=\left[ \begin{matrix} -\lambda & 1 & 0 \\ 1 & -\lambda & 1 \\ 0 & 1 & -\lambda \\ \end{matrix} \right]$$$$|A|=-\lamb… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p11?rev=1737476037&do=diff Question 13, Exercise 2.2 Solutions of Question 13 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 13(i) $x,$$\left| \begin{matrix}x & 2 & 3 \\0 & -1 & 1 \\0 & 4 & 5 \end{matrix} \right|=9$$$\left| \begin{matrix} x & 2 & 3 \\ 0 & -1 & 1 \\ 0 & 4 & 5 \\ \end{matrix} \right|=9$$$$x(-5-4)-2(0)+3(0)=9$$$$-9x=9$$$$x=-1$$$x,$$\left… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p12?rev=1737476037&do=diff Question 14 & 15, Exercise 2.2 Solutions of Questions 14 & 15 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$A=\begin{bmatrix}0 & 2 & 2 \\-1 & 3 & 2 \\1 & 0 & 5\end{bmatrix}$$A^{-1}$$$A=\left[ \begin{matrix} 0 & 2 & 2 \\ -1 & 3 & 2 \\ 1 & 0 & 5 \\ \end{matrix} \right]$$$A^{-1}$$$A^{-1}=\dfrac{Adj\,\,A}{|A|}$$$$Adj\,\,A={{\left[ \begin… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p13?rev=1737476037&do=diff Question 16 & 17, Exercise 2.2 Solutions of Questions 16 & 17 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$A=\begin{bmatrix}3 & -1 \\4 & 2\end{bmatrix}$$|A^{-1}|=\dfrac{1}{|A|}$$$A=\left[ \begin{matrix} 3 & -1 \\ 4 & 2 \\ \end{matrix} \right]$$$$|A|=6+4$$$$\Rightarrow |A|=10\ldots (1)$$$$A^{-1}=\dfrac{1}{|A|}AdjA$$$$AdjA=\left[ \begin{ma… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p14?rev=1737476037&do=diff Question 18, Exercise 2.2 Solutions of Question 18 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 18(i) $A$$B$$( A^{-1})^{-1}=A$$A$$2\times 2$$$A=\left[ \begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{matrix} \right]$$$$|A|=a_{11}a_{22}-a_{12}a_{21}$$$$AdjA=\left[ \begin{matrix} a_{22} & -a_{12} \\ -a_{21} & a_{11… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-2-p2?rev=1737476037&do=diff Question 2, Exercise 3.2 Solutions of Question 2 of Exercise 3.2 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2(i) Find unit vector having the same direction as the vector $3\hat{i}.$$$\overset{\scriptscriptstyle\rightharpoonup}{a}=3\hat{i}$$$$|\overset{\scriptscriptstyle\rightharpoonup}{a}|=\sqrt{{{(3)}^{2}}}=3$$$$\hat{a}=\dfrac{{\overset{\scriptscriptstyle\rightharpo… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-2-p3?rev=1737476037&do=diff Question 3 & 4, Exercise 3.2 Solutions of Question 3 & 4 of Exercise 3.2 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 If $\vec{r}=\hat{i}-9\hat{j}$$\vec{a}=\hat{i}+2\hat{j}$$\vec{b}=5\hat{i}-\hat{j}$$p$$q$$\vec{r}=p\vec{a}+q\vec{b}$$$\vec{r}=p\vec{a}+q\vec{b}.$$$\vec{r},\vec{a}$$\vec{b}$$$\hat{i}-9\hat{j}=p(\hat{i}+2\hat{j})+q(5\hat{i}-\hat{j})$$$$\implies \hat{i}-9\… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-2-p5?rev=1737476037&do=diff Question 7, Exercise 3.2 Solutions of Question 7 of Exercise 3.2 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7(i) Find the components and the magnitude of $\overrightarrow{PQ}$$P(-1,2)$$Q(2,-1)$\begin{align}\overrightarrow{PQ}&=\overrightarrow{OQ}-\overrightarrow{OP}\\ &=(2\hat{i}-\hat{j})-(-\hat{i}+2\hat{j})\\ &=3\hat{i}-3\hat{j}\end{align}\begin{align}|\overrighta… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-2-p6?rev=1737476037&do=diff Question 7, Exercise 3.2 Solutions of Question 7 of Exercise 3.2 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7(i) Find the components and the magnitude of $\overrightarrow{PQ}$$P(-1,2)$$Q(2,-1)$\begin{align}\overrightarrow{PQ}&=\overrightarrow{OQ}-\overrightarrow{OP}\\ &=(2\hat{i}-\hat{j})-(-\hat{i}+2\hat{j})\\ &=3\hat{i}-3\hat{j}\end{align}\begin{align}|\overrighta… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-2-p8?rev=1737476037&do=diff Question 11, Exercise 3.2 Solutions of Question 11 of Exercise 3.2 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 11(i) Find the position vectors of the point of division of the line segments joining point $C$$5\hat{j}$$D$$4\hat{i}+\hat{j}$$2:5$$C$$\overrightarrow{OC}=5\hat{j}$$D$$\overrightarrow{OD}=4\hat{i}+\hat{j}$$H$$\overline{CD}$$2:5$$H$\begin{align}\overrightarrow… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-2-p9?rev=1737476037&do=diff Question 12, 13 & 14, Exercise 3.2 Solutions of Question 12, 13 & 14 of Exercise 3.2 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 12 $\alpha ,$$|\alpha \hat{i}+(\alpha +1)\hat{j}+2\hat{k}|=3$\begin{align}|\alpha \hat{i}+(\alpha +1)\hat{j}+2\hat{k}|&=3.\end{align}\begin{align}\sqrt{(\alpha )^2+(\alpha +1)^2+(2)^2}&=3.\end{align}\begin{align}&{\alpha ^2+(\alpha +1)^2}+4=9… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-3-p4?rev=1737476037&do=diff Question 6 Exercise 3.3 Solutions of Question 6 of Exercise 3.3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6(i) Let $\vec{a}=\hat{i}+3 \hat{j}-4 \hat{k}$ and $\vec{b}=2 \hat{i}-3 \hat{j}-5 \hat{k}$$m$$\vec{a}+m \vec{b}$$\vec{a}$\begin{align} \vec{a}+m \vec{b}& =\hat{i}+3 \hat{j}-4 \hat{k}+m(2 \hat{i}-3 \hat{j}+5 \hat{k}) \\ & =(1+2 m) \hat{i}+(3-3 m) \hat{j}+(5 m-4) … text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-4-p4?rev=1737476037&do=diff Question 4 Exercise 3.4 Solutions of Question 4 of Exercise 3.4 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4(i) If $\vec{a}=3 \hat{i}-6 \hat{j}+5 \hat{k},\quad\vec{b}=2\hat{i}-\hat{j}+4 \hat{k} \quad$ and $\quad \vec{c}=\hat{i}+\hat{j} \quad \hat{k},\quad$$\vec{a} \times \vec{b}$\begin{align}\vec{a} \times \vec{b}&=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k}… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-5-p5?rev=1737476038&do=diff Question 6 Exercise 3.5 Solutions of Question 6 of Exercise 3.5 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6 Do the points $(4. 2.1)$$(5,1,6)$$(2.2,-5)$$(3.5 .0)$$A(4,-2,1), B(5,1,6)$$C(2,2,-5)$$D(3,5.0)$$A, \overrightarrow{O A}=4 \hat{i}-2 \hat{j}+\hat{k}$$B, \overrightarrow{O B}=5 \hat{i}+\hat{j}+6 \hat{k}$$C, \overrightarrow{O C}=2 \hat{i}+2 \hat{i}-5 \hat{k}$$D, … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-5-p6?rev=1737476038&do=diff Question 7 Exercise 3.5 Solutions of Question 7 of Exercise 3.5 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7(i) For what value of $c$$\vec{u}=\hat{i}+2 \hat{j}+3 \hat{k}$$\vec{v}=2 \hat{i}-3 \hat{j}+4 \hat{k} \cdot \vec{w}=3 \hat{i}+\hat{j}+c \hat{k}$\begin{align}\vec{u} \cdot \vec{v} \times \vec{w}&=0\\ \vec{u} \cdot \vec{v} \times \vec{w}&=0\\ \Rightarrow\left|\beg… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-5-p8?rev=1737476038&do=diff Question 9 Exercise 3.5 Solutions of Question 9 of Exercise 3.5 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9 (i) Write the value of $(\hat{i} \times \hat{j}). \hat{k}+\hat{i}. \hat{j}$\begin{align} (\hat{i} \times \hat{j}) \cdot \hat{k}&=\left|\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right|&=1 ....(1)\\ \text { and } \hat{i} \cdot \hat{j}&=0… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-1-p2?rev=1737476038&do=diff Question 3 and 4 Exercise 4.1 Solutions of Question 3 and 4 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{1}{2}, \dfrac{2}{3} \dfrac{3}{4}, \dfrac{4}{5}, \ldots$$$\dfrac{1}{1+1}, \dfrac{2}{2+1}, \dfrac{3}{3+1}, \dfrac{4}{4+1},...$$$\dfrac{n}{n+1}$$2,-4,6,-8,10, \ldots$\begin{align} &(-1)^2 \cdot 2 \cdot 1, (-1)^3 \cdot 2 \cdot 2, (-1)^4 \cdot 2 \… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-1-p3?rev=1737476038&do=diff Question 5 Exercise 4.1 Solutions of Question 5 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) $\sum_{j=1}^6(2 j-3)$\begin{align}\sum_{j=1}^6(2 j-3)&=(2.1-3)+(2.2-3)+(2.3-3)+(2.4-3)\\&+(2.5-3)+(2.6-3) \\ \implies \sum_{j=1}^6(2 j-3)&=-1+1+3+5+7+9 .\end{align}$\sum_{k=1}^5(-1)^k 2^{k-1}$\begin{align}\sum_{k=1}^5(-1)^k 2^{k-1}& =(-1)^1 2^{1-… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p2?rev=1737476038&do=diff Question 3 and 4 Exercise 4.2 Solutions of Question 3 and 4 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$6,9,12, \ldots, 78$$a_1=6$$d=9-6=3$$a_n=78$$$a_n=a_1+(n-1) d$$\begin{align}&78=6+(n-1) 3 \\ \implies &3(n-1)=78-6 \\ \implies &n-1=\dfrac{72}{3} \\ \implies &n=24+1=25.\end{align}$25$$n$$a_n=2n+7$$$a_n=2 n+7. --- (1)$$\begin{align}a_{n+1}=2(n+1)+7=2… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p3?rev=1737476038&do=diff Question 5 and 6 Exercise 4.2 Solutions of Question 5 and 6 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$\log a, \log (a b), \log \left(a b^2\right), \log \left(a b^3\right), \ldots$$$n$$\log$$a$$b$$b$$$a_n=\log (a b^{n-1}).$$\begin{align}a_n&=\log(a b^{n-1}). \end{align}\begin{align} d&=a_{n+1}-a_n \\ &=\log (a b^n)-\log (a b^{n-1}) \\ &=\log \left(\… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p4?rev=1737476038&do=diff Question 7 Exercise 4.2 Solutions of Question 7 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7 $a_6+a_4=6$$a_6-a_4=\dfrac{2}{3}$$a_1$$d$\begin{align} &a_6+a_4=6 \\ \implies & a_1+5d+a_1+3d=6\\ \implies & 2a_1+8d=6\\ \implies & a_1+4d=3 --- (1) \end{align}\begin{align} &a_6-a_4=\dfrac{2}{3} \\ \implies & a_1+5d-a_1-3d=\dfrac{2}{3}\\ \implies &… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p6?rev=1737476038&do=diff Question 9 Exercise 4.2 Solutions of Question 9 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9 $24m$$21m$$18m$$$a_1=24,$$$$a_2=21,$$$$a_3=18.$$$$d=21-24=18-21=-3,$$\begin{align} a_8&=a_1+7d\\ &=24+7(-3)=3. \end{align}$3m$ text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p7?rev=1737476038&do=diff Question 10 Exercise 4.2 Solutions of Question 10 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 10 $500$$a_1$$$a_1=20135.$$$d=-500$$a_{11}$\begin{align} a_{11}&=a_1+10d \\ &=20135+10(-500)\\ &=15135. \end{align}$1070$$15135$ text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p8?rev=1737476038&do=diff Question 11 Exercise 4.2 Solutions of Question 11 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 11 $a_1$$$a_1=1000.$$$= d=100$$a_n=5400$$n$\begin{align} &a_n=a_1+(n-1)d \\ \implies &5400=1000+(n-1)100\\ \implies &5400=900+100n \\ \implies &100n=5400-900\\ \implies &100n=4500\\ \implies &n=45.\end{align} text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p9?rev=1737476038&do=diff Question 12 & 13 Exercise 4.2 Solutions of Question 12 & 13 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$a_1$$$a_1=3500.$$$=d=750$$a_{21}$\begin{align} a_{21}&=a_1+20d\\ &=3500+20(750) \\ &=18500. \end{align}$12$$18$$a=12, b=18$$A$\begin{align}A&=\dfrac{a+b}{2}\\&=\dfrac{12+18}{2}\\&=\dfrac{30}{2}=15.\end{align}$\dfrac{1}{3}$$\dfrac{1}{4}$$a=\dfrac{1}{… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p11?rev=1737476038&do=diff Question 15 Exercise 4.2 Solutions of Question 15 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 15 $n, \dfrac{a^{n+1}+b^{n+1}}{a^n+b^n}$$a$$b$$a$$b$$A$$a$$b$$$ A=\dfrac{a+b}{2}. --- (1) $$$$ A=\dfrac{a^{n+1}+b^{n+1}}{a^n+b^n}. --- (2) $$\begin{align}&\dfrac{a+b}{2}=\dfrac{a^{n+1}+b^{n+1}}{a^n+b^n}, --- (3) \\ \implies &(a^n+b^n)(a+b)=2(a^{n+1… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-4-p8?rev=1737476038&do=diff Question 11 Exercise 4.4 Solutions of Question 11 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 11 $\mathrm{n}$$a$$b$$nth$$G_1, G_2, G_9, \ldots, G_n$$n$$a$$b$$a, G_1, G_2, G_3, \ldots, G_n, b$$n+2$$a_{n+2}=b$$a_n=a_1 r^{n-1}$$n$$n+2$\begin{align}a_{n+2}&=a_1 r^{n i 1}=a r^{n+1}=b \\ \because a_1&=a \\ \Rightarrow \quad r^{n+1}&=\dfrac{b}{a} .… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-1-p1?rev=1737476038&do=diff Question 1 Exercise 5.1 Solutions of Question 1 of Exercise 5.1 of Unit 05: Mascellaneous series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) $1^2+3^2+5^2+7^2+\ldots$$n$$1+3+5+\ldots$$n^{\text {th }}$$2 n-1$$n^{t h}$$$T_j=(2 j-1)^2$$\begin{align}& \sum_{j=1}^n T_j=\sum_{j=1}^n(2 j-1)^2 \\ & =\sum_{j=1}^n(4 j^2-4 j+1)\\ & =4 \sum_{j=1}^n j^2-4 \sum_{j=1}^n j+\sum_{j=1}^n 1 \\ & =4 \dfr… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-1-p3?rev=1737476038&do=diff Question 4 & 5 Exercise 5.1 Solutions of Question 4 & 5 of Exercise 5.1 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4 $2+(2+5)+(2+5+8)+\ldots$$n$\begin{align}& T_j=\dfrac{j}{2}[2(2)+3(j-1)]\\ &=\dfrac{j(3 j+1)}{2} \\ & =\dfrac{1}{2}(3 j^2+j)\end{align}\begin{align}& \sum_{j=1}^n T_i=\dfrac{1}{2}[3 \sum_{j=1}^n j^2+\sum_{j=1}^n j] \\ & =\dfrac{1}{2}[3 \dfra… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-1-p4?rev=1737476038&do=diff Question 6 Exercise 5.1 Solutions of Question 6 of Exercise 5.1 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6 $1.2 \cdot 3+2 \cdot 3.4+3.4 .5+\ldots$$n$$1+2+3+\ldots, \quad 2+3+4+5+\ldots$$3+4+5+6+7+\ldots$$n^{t h}$$j, j+1$$j+2$$n^{t h}$\begin{align} & T_j=j(j+1)(j+2)-j(j^2+3 j+2) \\ & =j^3+3 j^2+2 j\end{align}\begin{align} & \sum_{j=1}^n T_j=\sum_{j=1}^n … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-1-p5?rev=1737476038&do=diff Question 7 & 8 Exercise 5.1 Solutions of Question 7 & 8 of Exercise 5.1 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7 $n$$1.5 .9+2.6 .10+3.7 .11+\ldots$$T_j=j(j+4)(j+8)$\begin{align} & =j(j^2+12 j+32) \\ & =j^3+12 j^2+32 j\end{align}\begin{align} & \sum_{j=1}^n T_j=\sum_{j=1}^n j^3+12 \sum_{j=1}^n j^2+32 \sum_{j=1}^n j \\ & =(\dfrac{n(n+1)}{2})^2+12 \dfrac… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-3-p1?rev=1737476038&do=diff Question 1 Exercise 5.3 Solutions of Question 1 of Exercise 5.3 of Unit 05: Mascellaneous series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1 $n$$n$$4+13+28+49+76+\ldots$\begin{align} & a_2-a_1=13-4=9 \\ & a_3-a_2=28-13=15 \\ & a_4-a_3=49-28=21 \\ & \cdots \quad \cdots \quad \cdots \\ & \cdots \quad \cdots \quad \cdots \\ & a_n-a_{n-1}=(n-1)th \quad\text{term of sequence}\quad 9,15,21,..… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/re-ex5-p3?rev=1737476038&do=diff Question 4 Review Exercise Solutions of Question 4 of Review Exercise of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4 $\dfrac{1}{1.4 .7}+\dfrac{1}{4.7 .10}+\dfrac{1}{7.10 .13}+\ldots$$1,4,7, \ldots$$$a_n=\dfrac{1}{(3 n-2)(3 n+1)(3 n+4)}$$\begin{align} \dfrac{1}{(3 n-2)(3 n+1)(3 n+4)}&=\dfrac{A}{3 n-2}+\dfrac{B}{3 n+1}+\dfrac{C}{3 n+4}\end{align}$(3 n-2)(3 n+… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/re-ex5-p8?rev=1737476038&do=diff Question 10 Review Exercise Solutions of Question 10 of Review Exercise of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 10 $n^{\text {th }}$$n$$1+(1+\dfrac{1}{2})+(1+\dfrac{1}{2}+\dfrac{1}{4})+(1+\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8})+\ldots$\begin{align} a_n&=1+\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+\cdots+\dfrac{1}{2^{n-1}} \\ a_n&=\dfrac{1[1-(\dfrac{1}{2})… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-1-p4?rev=1737476038&do=diff Question 4 Exercise 6.1 Solutions of Question 4 of Exercise 6.1 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p2?rev=1737476038&do=diff Question 3 and 4 Exercise 6.2 Solutions of Question 3 and 4 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$^n P_r=n(^{n-1} P_{r-1})$$$^n P_r=n({ }^{n-1} P_{r-1})$$\begin{align}n(^{n-1} P_{r-1})&=n \dfrac{(n-1) !}{((n-1)-(r-1)) !} \\ & =\dfrac{n(n-1) !}{(n-r) !}\\ &=\dfrac{n !}{(n-r) !}\\ &=^n P_r\end{align}$^n P_r=^{n-1} P_r+r(^{n-1} … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p4?rev=1737476038&do=diff Question 7 and 8 Exercise 6.2 Solutions of Question 7 and 8 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$1,2,3,4$$E_1$$m_1=5$$E_2$$\cdot m_2=5$$E_3$$m_3=5$$$m_1 \cdot m_2 \cdot m_3=5.5 \cdot 5=125$$$1,2,3,4$$E_1$$m_1=5$$E_2$$m_2=4$$E_3$$m_3=3$$$m_1 \cdot m_2 \cdot m_3=5 \cdot 4 \cdot 3=60$$$8$$5$$=4$$=4$$=5$$=3$$4 ! \cdot 5 ! \cdot … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p5?rev=1737476038&do=diff Question 9 Exercise 6.2 Solutions of Question 9 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$=^6 P_1=6$$s=^6 P_2=30$$=^6 P_3=120$$=^6 P_4=360$$=^6 P_5=720$$=^6 P_6=720$$6+30+120+360+720+720=1956$ text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p7?rev=1737476038&do=diff Question 11 Exercise 6.2 Solutions of Question 11 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$10$$1000$$2.3,4,0,8,9$$10$$1000$$10$$100$$E_1$$m_1=5$$E_2$$m_2=5$$10$$100$$$m_1 \cdot m_2=5.5=25$$$100$$1000$$0$$E_1$$m_1=5$$E_2$$\boldsymbol{m}_2=5$$E_3$$m_3=4$$100$$1000$$$m_1 \cdot m_2 \cdot m_3=5.5 \cdot 4=100$$$10$$1000$$$100 + 25=125… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p8?rev=1737476038&do=diff Question 12 Exercise 6.2 Solutions of Question 12 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$8.$$n=8$$\mathrm{O}$$m_1=3$\begin{align} \left(\begin{array}{c} n \\ m 1 \end{array}\right)&=\left(\begin{array}{l} 8 \\ 3 \end{array}\right) \\ & =\dfrac{8 !}{3 !}\\ &=\dfrac{8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 !}{3 !}\\ &=6,720 \e… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-3-p3?rev=1737476038&do=diff Question 3 Exercise 6.3 Solutions of Question 3 of Exercise 6.3 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$n$$^{2 n} C_3:^n C_2=36: 3$\begin{align} & { }^{2 n} C_3:{ }^n C_2=36: 3 . \\ & \Rightarrow \dfrac{(2 n) !}{(2 n-3) ! 3 !} \times \dfrac{(n-2) ! 2 !}{n !}=12 \\ & \Rightarrow \dfrac{2 n(2 n-1)(2 n-2)(2 n-3) !}{(2 n-3) ! 3 !}\times\dfrac{(n-2… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-3-p4?rev=1737476038&do=diff Question 4 Exercise 6.3 Solutions of Question 4 of Exercise 6.3 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${ }^{n-1} C_r+{ }^{n-1} C_{r-1}={ }^n C_r$$${ }^n{ }^1 C_r+{ }^n{ }^1 C_{r-1}={ }^n C_s$$\begin{align} { }^{n-1} C_r+{ }^{n-1} C_{r-1}&=\dfrac{(n-1) !}{(n-r-1) ! r !}+\dfrac{(n-1) !}{(n-1-(r-1)) !(r-1) !} \\ & =\dfrac{(n-1) !}{(n-r-1) ! r(r-… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-3-p5?rev=1737476038&do=diff Question 5 and 6 Exercise 6.3 Solutions of Question 5 and 6 of Exercise 6.3 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$12$$n=12$${ }^{12} C_2=66$$12$$n=12$${ }^{12} C_3=220$$${ }^6 C_2=\dfrac{6 !}{(6-2) ! 2 !}=15 $$$6$$\quad 15-6=9$ text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-3-p6?rev=1737476038&do=diff Question 7 and 8 Exercise 6.3 Solutions of Question 7 and 8 of Exercise 6.3 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$20$\begin{align}{ }^{20} C_2&=\dfrac{20 !}{(20-2)2!}!\\ &=\dfrac{20!}{18!\cdot 2!}\\ &=190\end{align}$7$$10$$3$$7$$10$$${ }^{10} C_7=\dfrac{10 !}{(10-7) ! 7 !}=120$$$7$$4.$$4$$${ }^7 C_4=\dfrac{7 !}{(7-4) ! 4 !}=35.$$$35$$10.$ text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-4-p2?rev=1737476038&do=diff Question 2 Exercise 6.4 Solutions of Question 2 of Exercise 6.4 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$4$$5$$6$$3$$4+5+6=15$$${ }^{15} C_3=\dfrac{15 !}{(15-3) ! 3 !}=455 $$$${ }^6 C_4=\dfrac{6 !}{(6-4) ! 4 !}=15$$$$=\dfrac{15}{455}=\dfrac{3}{91}$$$4$$5$$6$$3$$4+5+6=15$$${ }^{15} C_3=\dfrac{15 !}{(15-3) ! 3 !}=455 $$$${ }^4 C_3=\dfrac{4 !}{(4-… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-4-p5?rev=1737476038&do=diff Question 5 Exercise 6.4 Solutions of Question 5 of Exercise 6.4 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$6$$4$$3$$2$$=6+4=10$$5$$10$\begin{align}{ }^{10)} C_5 &=\dfrac{10 !}{(10-5) ! 5 !}\\ &=252\\ n(S)&=252\end{align}$3$$2$$3$$2$\begin{align}{ }^6 \mathrm{C}_3\cdot{ }^{4} \mathrm{C}_2&=\dfrac{6 !}{(6-3) ! 3 !} \cdot \dfrac{4 !}{(4-2) ! 2 !}\\… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-4-p6?rev=1737476038&do=diff Question 6 Exercise 6.4 Solutions of Question 6 of Exercise 6.4 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$52$$$=52$$$=4$$$=\dfrac{4}{52}=\dfrac{1}{13}$$$52$$=52$$13$$13$$$\dfrac{13}{52}+ \dfrac{13}{52}=\dfrac{1}{4}+\dfrac{1}{4}=\dfrac{2}{4}=\dfrac{1}{2}$$$52$$=52$$13.$$$=\dfrac{13}{52}=\dfrac{1}{4}$$$52$$=52$$12.$$$=\dfrac{12}{52}=\dfrac{3}{13}$… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-5-p4?rev=1737476038&do=diff Question 7 Exercise 6.5 Solutions of Question 7 of Exercise 6.5 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$52$$52$$26$$26$$13$$13$$13$$13$$13$$10,9,8,7,6,5,4,3$$2.$$$=\dfrac{13}{52}=\dfrac{1}{4}$$$$=\dfrac{4}{52}=\dfrac{1}{13}$$\begin{align} P(A \cup B)&=P(A)+P(B) \\ & =\dfrac{1}{4}+\dfrac{1}{13}=\dfrac{17}{52} \end{align}$$=1-\dfrac{17}{52}=\dfr… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-5-p5?rev=1737476038&do=diff Question 8 Exercise 6.5 Solutions of Question 8 of Exercise 6.5 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$7$$11.$\begin{align}s&=(i i, j): i, j-1,2,3,4,5,6\}\\ &=\left[\begin{array}{llllll} (1.1) & (1.2) & (1.3) & (1.4) & (1.5) & (1.6) \\ (2.1) & (2.2) & (2.3) & (2.4) & (2.5) & (2.6) \\ (3.1) & (3.2) & (3.3) & (3.4) & (3.5) & (3.6) \\ (4.1) & (4… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p3?rev=1737476038&do=diff Question 3 & 4 Review Exercise 6 Solutions of Question 3 & 4 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${ }^{56} P_{r+6}:{ }^{54} P_{r+3}=30800: 1$$r$\begin{align} { }^{56} P_{r+6}:{ }^{54} P_r+3&=30800: 1 \\ \Rightarrow \dfrac{\dfrac{56 !}{[56-(r+6)] !}}{\dfrac{54 !}{[54-(r+3)] !}}&=\dfrac{30800}{1} \\ \Rightarrow \dfrac{56… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p4?rev=1737476038&do=diff Question 5 & 6 Review Exercise 6 Solutions of Question 5 & 6 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$n=6$$$$(n-1) !=(6-1) !=5 !=120$$$120-24=96$$n=6$$(n-1) !=(6-1) !=5 !=120$$$(n-1) !=(5-1) !=4 !=24$$$$(n-1) !=(6-1) !=5 !=120$$$$4 ! \cdot 2 !=48$$$(5-1) !$ text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p5?rev=1737476038&do=diff Question 7 & 8 Review Exercise 6 Solutions of Question 7 & 8 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$P(A)=0.8, P(B)=0.5$$P(B / A)=0.4$$P(A \cap B)$\begin{align} P(B \mid A)&=\dfrac{P(A \cap B)}{P(A)} \\ \Rightarrow P(A \cap B)&=P(B \mid A) \cdot P(A)\\ &=0.4 \times 0.8=0.32\end{align}$P(A)=0.8, P(B)=0.5$$P(B / A)=0.4$$P(A … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p6?rev=1737476038&do=diff Question 9 & 10 Review Exercise 6 Solutions of Question 9 & 10 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$2,3,0,3,4,2,3$$1$$=100,0000$$$=\dfrac{7 !}{3 ! \cdot 2 !}=420 $$$1$$0$$7$$0$$$=\dfrac{6 !}{2 ! 3 !}=60 $$$1$$420-50=360$$n$$n$$(n-1)$$(n - 1)$$(n-1)$$(n-2) !$$2$$2 !$$n$$$(n-2) ! \cdot 2 !=2(n-2) ! $$ text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p2?rev=1737476038&do=diff Question 2 Exercise 7.1 Solutions of Question 2 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$1+5+9+\ldots+(4 n-3)=n(2 n-1)$$n=1$$$1=1(2.1-1)=1$$$n=1$$n=k$\begin{align}1+5+9+\ldots+(4 k-3)\\ & =k(2 k-1)....(i) \\ \end{align}$n=k+1$$k+1$$$a_{k-1}=4(k+1)-3=4 k+1 $$$(k+1)^{t h}$\begin{align}1+5+9+\ldots+(4 k-3)+(4 k+1)& =k(2 k-1)+4 k+1 … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p3?rev=1737476038&do=diff Question 3 Exercise 7.1 Solutions of Question 3 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$3+6+9+\ldots+3 n=\dfrac{3 n(n+1)}{2}$$n=1$$3=\dfrac{3.1(1+1)}{2}=3$$n=1$$n=k$$$3+6+9+\ldots+3 k=\dfrac{3 k(k+1)}{2}....(i)$$$n=k+1$$(k+1)$$a_{k+1}=3(k+1)$$(k+1)^{t h}$\begin{align}3+6+9+\ldots+3 k+3(k+1) & =\dfrac{3 k(k+1)}{2}+3(k+1) \\ & =3… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p4?rev=1737476038&do=diff Question 4 Exercise 7.1 Solutions of Question 4 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$3+7+11+\cdots+(4 n-1)=n(2 n+1)$$n=1$$$3=1(2+1)=3 $$$n=1$$n=k$\begin{align}3+7+11+\cdots+(4 k-1) & =k(2 k+1)....(i) \end{align}$n=k+1$$(k+1)$$a_{k+1}=4(k+1)-1$$(k+1)^{t h}$\begin{align} 3+7+11+\cdots+(4 k-1)+[4(k+1)-1] & =k(2 k+1)+4(k+1)-1 \… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p5?rev=1737476038&do=diff Question 5 Exercise 7.1 Solutions of Question 5 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$1^3+2^3+3^3+\ldots+n^3=\left[\dfrac{n(n+1)}{2}\right]^2$$n=1$$1^3=1=\left[\dfrac{1(1+1)}{2}\right]^2=1$$n=1$$n=k_1$\begin{align}1^3+2^3+3^3+\ldots+k^3& =[\dfrac{k(k+1)}{2}]^2....(i)\end{aligned} 3. Now $$ the $$ term of the given series on l… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p8?rev=1737476038&do=diff Question 8 Exercise 7.1 Solutions of Question 8 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$1+2+2^2+2^3+\ldots+2^n 1=2^n-1$$n=1$$1=2^1-1=1$$n=1$$n-k>1$\begin{align}1+2+2^2+2^3+\ldots+2^{k-1} \\ & =2^k-1 ....(i)\end{align}$n-k-1$$(k+1)^{t h}$$a_{k+1}=2^k$$a_{k+1}$\begin{align}1+2+2^2+2^3+\ldots+2^{k-1}-2^k & =2^k-12^k \\ & =2^k+2^k-… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p9?rev=1737476038&do=diff Question 9 Exercise 7.1 Solutions of Question 9 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{1}{3}+\dfrac{1}{9}+\dfrac{1}{27}+\ldots+\dfrac{1}{3^n}=\dfrac{1}{2}[1-\dfrac{1}{3^n}]$$n=1$$$\dfrac{1}{3}-\dfrac{1}{2}[1-\dfrac{1}{3}]-\dfrac{1}{2} \dfrac{2}{3}=\dfrac{1}{3} $$$n=1$$n=k$$$\dfrac{1}{3}+\dfrac{1}{9}+\dfrac{1}{27}+\ldots… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p10?rev=1737476038&do=diff Question 10 Exercise 7.1 Solutions of Question 10 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\left(\begin{array}{1}5 \\5 \end{array}\right)+\left(\begin{array}{l}6 \\ 5\end{array}\right)+\left(\begin{array}{l}7 \\ 5\end{array}\right)+\ldots+\left(\begin{array}{c}n+4 \\ 5\end{array}\right)=\left(\begin{array}{c}n+5 \\ 6\end{array}\… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p11?rev=1737476038&do=diff Question 11 Exercise 7.1 Solutions of Question 11 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.\begin{align} & \left(\begin{array}{l} 2 \\ 2 \end{array}\right)+\left(\begin{array}{l} 3 \\ 2 \end{array}\right)+\left(\begin{array}{l} 4 \\ 2 \end{array}\right)+\ldots+\left(\begin{array}{l} n \\ 2 \end{array}\right)=\left(\begin{array}{c… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p12?rev=1737476038&do=diff Question 12 Exercise 7.1 Solutions of Question 12 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{5^{2 n}-1}{24}$$n=1$$$\dfrac{5^{2 n}-1}{24}=\dfrac{5^{2.1}-1}{24}=\dfrac{24}{24}=1 \in \mathbb{Z}$$$n=1$$n=k>1$$$\dfrac{5^{2 k}-1}{24} \in \mathbb{Z}$$$n=k+1$\begin{align}\dfrac{5^{2(k+1)}-1}{24}&=\dfrac{5^{2 k+2}-1}{24} \\ & =\dfra… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p3?rev=1737476038&do=diff Question 3 Exercise 7.2 Solutions of Question 3 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$x$$(\dfrac{4 x^2}{3}-\dfrac{3}{2 x})$$n=9, \quad a=\dfrac{4 x^2}{3}$$b=-\dfrac{3}{2 x}$$T_{r+1}$$x$$T_{r+1}$\begin{align}T_{r+1}&=\dfrac{9 !}{(9-r) ! r !}(\dfrac{4 x^2}{3})^{9-r}(-\dfrac{3}{2 x})^r \\ & =\dfrac{9 !}{(9-r) ! r !} \cdot \dfrac… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p6?rev=1737476038&do=diff Question 6 Exercise 7.2 Solutions of Question 6 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$(2 \sqrt{x}-\dfrac{3}{x \sqrt{x}})^{23}$$a=2 \sqrt{x}$$b=-\dfrac{3}{x \sqrt{x}}$$n=23$$x$\begin{align} T_{r+1}&=\dfrac{23 !}{(23-r) ! r !}(2 \sqrt{x})^{23-r}(-\dfrac{3}{x \sqrt{x}})^r \\ & =\dfrac{23 !}{(23-r) ! r !} \cdot 2^{23-r} \cdot(-3)… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p10?rev=1737476038&do=diff Question 10 Exercise 7.2 Solutions of Question 10 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$n=2 ;$$s=2^{n-1}$$$ \left.(1+x)^n=\left(\begin{array}{l} n \\ \vdots \end{array}\right)+\left(\begin{array}{l} m \\ 1 \end{array}\right) x+\left(\begin{array}{l} n \\ 2 \end{array}\right) x^2-\ldots+i_n^*\right) x^n \cdot $$$x=1$$(1 \div 1… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p4?rev=1737476039&do=diff Question 4 Exercise 7.3 Solutions of Question 4 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$x$$x^2$$x$$$ \sqrt{\frac{1-3 x}{1+4 x}}=1-\frac{7 x}{2} $$$$ \sqrt{\frac{1-3 x}{1-4 x}}=(1-3 x)^{\frac{1}{2}}(1+4 x)^{-\frac{1}{2}} $$$x^2$$x$$$ \begin{aligned} & =\left(1-\frac{3 x}{2}\right) \times\left(1-\frac{4 x}{2}\right) \\ & =\left(1… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p7?rev=1737476039&do=diff Question 9 Exercise 7.3 Solutions of Question 9 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$x^{\prime \prime}$$\left(\frac{1+x}{1-x}\right)^2$$$ \begin{aligned} & \left(\frac{1+x}{1-x}\right)^2=(1+x)^2(1-x)^{-2} \\ & =\left(x^2+2 x+1\right)(1-x)^2 \end{aligned} $$$$ \begin{aligned} & =\left(x^2+2 x+1\right)[1+2 x+ \\ & \frac{-2(-2-… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p12?rev=1737476039&do=diff Question 14 Exercise 7.3 Solutions of Question 14 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$x$$p x^p-q x^q=(p-q) x^{p+q}$$x$$x=1+h$$h \longrightarrow 0$$$ p x^p-q x^q=p(1+h)^p-q(1+h)^q $$$$ \begin{aligned} & p x^p-q x^q \\ & =p(1+p h+\text { higher powers h) } \\ & -q(1+q h+\text { higher powcrs } h) \\ & \Rightarrow p x^p-q x^q=… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/re-ex7-p2?rev=1737476039&do=diff Question 2 Review Exercise 7 Solutions of Question 2 of Review Exercise 7 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\left(2 x^3+3 y\right)^8$$a=2 x^3$$b=3 y$$n=8$$n=8$$\frac{8+2}{2}=5$$$ \begin{aligned} & T_5=\frac{8 !}{(8-4) ! 4 !}\left(2 x^3\right)^{8-4}(3 y)^4 \\ & T_5=70.2^4 \cdot 3^4 \cdot x^{12} \cdot y^4 \\ & =90720 x^{12} y^4 \end{aligne… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/re-ex7-p5?rev=1737476039&do=diff Question 7 & 8 Review Exercise 7 Solutions of Question 7 & 8 of Review Exercise 7 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$7^n-3^n$$n=1$$7^k-3^k=7-4=4$$n=1$$n=k>1$$7^n-3^n=4 Q$$Q$$n=k+1$$$ \begin{aligned} & 7^{k+1}-3^{k+1}=7.7^k-3.3^k \\ & =(4+3) \cdot 7^k-3.3^k \\ & =4.7^k+3.7^k-3.3^k \end{aligned} $$$$ \begin{aligned} & =4.7^k+3\left[7^k-3^k\… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/re-ex7-p6?rev=1737476039&do=diff Question 9 and 10 Review Exercise 7 Solutions of Question 9 and 10 of Review Exercise 7 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p2?rev=1737476039&do=diff Question 2, Exercise 10.1 Solutions of Question 2 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin \dfrac{\pi }{12}$$\dfrac{\pi }{12}$$\dfrac{\pi }{3}-\dfrac{\pi }{4}$\begin{align}\sin (\alpha -\beta )=\sin \alpha \cos \beta -\cos \alpha \sin.\end{align}\begin{align} \Rightarrow \quad \sin \left( \frac{\pi }{3}-\f… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p3?rev=1737476039&do=diff Question 3, Exercise 10.1 Solutions of Question 3 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin u=\dfrac{3}{5}$$\sin v=\dfrac{4}{5}$$u$$v$$0$$\dfrac{\pi }{2}$$\cos \left( u+v \right)$$\sin u=\dfrac{3}{5},$$0\le u\le \dfrac{\pi }{2}.$$\sin v=\dfrac{4}{5},$$0\le v\le \dfrac{\pi }{2}.$$\cos u=\pm \sqrt{1-{{\sin }^… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p4?rev=1737476039&do=diff Question, Exercise 10.1 Solutions of Question 4 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin \alpha =-\dfrac{4}{5}$$\cos \beta =-\dfrac{12}{13}$$\alpha $$\beta $$\sin \left( \alpha -\beta \right)$$\sin \alpha=-\dfrac{4}{5}$$\alpha$$\sin \beta=-\dfrac{12}{13}$$\beta$$$\cos \alpha=\pm \sqrt{1-\sin^2\alpha}.$$$\… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p5?rev=1737476039&do=diff Question 5, Exercise 10.1 Solutions of Question 5 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\tan \alpha =\dfrac{3}{4}$$\sec \beta =\dfrac{13}{5}$$\alpha$$\beta$$\sin \left( \alpha +\beta \right)$$\tan\alpha =\dfrac{3}{4}$$\tan\alpha$$\alpha$\begin{align}{{\sec}^{2}}\alpha &=1+{{\tan}^{2}}\alpha\\ \Rightarrow \q… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p6?rev=1737476039&do=diff Question 6, Exercise 10.1 Solutions of Question 1 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos \alpha =2{{\cos }^{2}}\dfrac{\alpha }{2}-1=1-2{{\sin }^{2}}\dfrac{\alpha }{2}$\begin{align}\cos \alpha &=\cos 2\dfrac{\alpha }{2}\\ &={{\cos }^{2}}\dfrac{\alpha }{2}-{{\sin }^{2}}\dfrac{\alpha }{2}\\ &={{\cos }^{2}}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p7?rev=1737476039&do=diff Question 7, Exercise 10.1 Solutions of Question 1 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cot \left( \alpha +\beta \right)=\dfrac{\cot \alpha \cot \beta -1}{\cot \alpha +\cot \beta }$\begin{align}L.H.S.&=\cot (\alpha +\beta )\\ &=\dfrac{1}{\tan (\alpha +\beta )}\\ &=\dfrac{1}{\,\dfrac{\tan \alpha +\tan \beta… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p8?rev=1737476039&do=diff Question 8, Exercise 10.1 Solutions of Question 8 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\tan \left( \dfrac{\pi }{4}+\theta \right)=\dfrac{\cos \theta +\sin \theta }{\cos \theta -\sin \theta }$\begin{align}L.H.S.&=\tan \left( \dfrac{\pi }{4}+\theta \right)\\ &=\dfrac{\sin \left( \dfrac{\pi }{4}+\theta \ri… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p9?rev=1737476039&do=diff Question 9 and 10, Exercise 10.1 Solutions of Question 9 and 10 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{\sin \theta }{\sec 4\theta }+\dfrac{\cos \theta }{\cos ec4\theta }=\sin 5\theta $\begin{align}L.H.S.&=\dfrac{\sin \theta }{\sec 4\theta }+\dfrac{\cos \theta }{\cos ec4\theta }\\ &=\dfrac{\sin \theta }… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p10?rev=1737476039&do=diff Question11 and 12, Exercise 10.1 Solutions of Question 11 and 12 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\alpha$$\beta$$\gamma$$ABC$$\cot \dfrac{\alpha }{2}+\cot \dfrac{\beta }{2}+\cot \dfrac{\gamma }{2}=\cot \dfrac{\alpha }{2}\cot \dfrac{\beta }{2}\cot \dfrac{\gamma }{2}$$\alpha$$\beta$$\gamma$\begin{align}&\… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-2-p2?rev=1737476039&do=diff Question 2, Exercise 10.2 Solutions of Question 2 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin \theta =\dfrac{5}{13}$$\theta $$\sin 2\theta $$\sin \theta =\dfrac{5}{13}$$$\cos \theta =\pm \sqrt{1-{{\sin }^{2}}\theta }.$$$\theta$$\cos$\begin{align}\cos\theta &=-\sqrt{1-{{\sin }^{2}}\theta }\\ &=-\sqrt{1-\left(\… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-2-p3?rev=1737476039&do=diff Question 3, Exercise 10.2 Solutions of Question 3 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin \theta =\dfrac{4}{5}$$\theta$$\sin2\theta$$\sin \theta =\dfrac{4}{5}$$\theta$$\cos \theta =-\dfrac{3}{5}$\begin{align}\sin 2\theta &=2\sin \theta \cos \theta \\ &=2\left( \dfrac{4}{5} \right)\left( -\dfrac{3}{5} \rig… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-2-p4?rev=1737476039&do=diff Question 4 and 5, Exercise 10.2 Solutions of Question 4 and 5 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos \theta =-\dfrac{3}{7}$$\theta $$\sin \dfrac{\theta }{2}$$\cos \theta =-\dfrac{3}{7}$$\theta$\begin{align}&\pi < \theta < \dfrac{3\pi}{2} \\ \implies &\frac{\pi}{2} < \frac{\theta}{2} < \dfrac{3\pi}{4}\end… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-2-p5?rev=1737476039&do=diff Question 6, Exercise 10.2 Solutions of Question 6 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos {{15}^{\circ }}$${{15}^{\circ }}=\dfrac{{{30}^{\circ }}}{2}$$\dfrac{\theta }{2}=\dfrac{{{30}^{\circ }}}{2}$$\cos {{15}^{\circ }}$\begin{align}\cos {{15}^{\circ }}&=\cos \dfrac{{{30}^{\circ }}}{2}=\sqrt{\dfrac{1+\cos … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-3-p2?rev=1737476039&do=diff Question 2, Exercise 10.3 Solutions of Question 2 of Exercise 10.3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$\sin {{37}^{\circ }}+\sin {{43}^{\circ }}.$$$$\sin \alpha +\sin \beta =2\sin \left( \dfrac{\alpha +\beta }{2} \right)\cos \left( \dfrac{\alpha -\beta }{2} \right).$$$\alpha ={{37}^{\circ }}$$\beta ={{43}^{\circ }}$\begin… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-3-p3?rev=1737476039&do=diff Question 3, Exercise 10.3 Solutions of Question 3 of Exercise 10.3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$\dfrac{\cos {{75}^{\circ }}+\cos {{15}^{\circ }}}{\sin {{75}^{\circ }}-\sin {{15}^{\circ }}}=\sqrt{3}.$$$$\cos \alpha +\cos \beta =2\cos \left( \dfrac{\alpha +\beta }{2} \right)\cos \left( \dfrac{\alpha -\beta }{2} \righ… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/re-ex10-p2?rev=1737476039&do=diff Question 2 and 3, Review Exercise 10 Solutions of Question 2 and 3 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{2\sin \theta \sin 2\theta }{\cos \theta +\cos 3\theta }=\tan 2\theta \tan \theta $\begin{align}L.H.S.&=\dfrac{2\sin \theta \sin 2\theta }{\cos \theta +\cos 3\theta }\\ &=\dfrac{2\sin \theta \s… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/re-ex10-p3?rev=1737476039&do=diff Question 4 & 5, Review Exercise 10 Solutions of Question 4 & 5 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${{\sin }^{2}}\dfrac{\theta }{2}=\dfrac{\sin \theta \tan \dfrac{\theta }{2}}{2}$\begin{align}R.H.S.&=\dfrac{\sin \theta \tan \dfrac{\theta }{2}}{2}\\ &=\dfrac{\sin \theta \sin \dfrac{\theta }{2}}{2\cos \d… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/re-ex10-p4?rev=1737476039&do=diff Question 6 & 7, Review Exercise 10 Solutions of Question 6 & 7 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos 4\theta =1-8{{\sin }^{2}}\theta {{\cos }^{2}}\theta $\begin{align}L.H.S&=\cos 4\theta \\ &=\cos 2\left( 2\theta \right)\\ &=1-2\sin^2 2\theta \\ &=1-2{{\left( 2\sin\theta \cos \theta \right)}^{2}}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-1-p2?rev=1737476039&do=diff Question 2, Exercise 1.1 Solutions of Question 2 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 2(i) $x+iy$$(3+2i)+(2+4i)$\begin{align}&(3+i2)+(2+i4)\\ =&(3+2)+(i2+i4)\\ =&5+i6\end{align}$x+iy$$(4+3i)-(2+5i)$\begin{align}&(4+3i)-(2+5i)\\ =&(4-2)+(3i-5i)\\ =&2-2i\end{align}$x+iy$$(4+7i)+(4-7i)$\begin{align} &(4+7i)+(4-7i)\\ =&(4+4)+(7i-7i… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-1-p3?rev=1737476039&do=diff Question 3, Exercise 1.1 Solutions of Question 3 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 3(i) $\dfrac{(2+i)(3-2i)}{1+i}$\begin{align}&\dfrac{(2+i)(3-2i)}{1+i}\\ =&\dfrac{6-2i^2+3i-4i}{1+i}\\ =&\dfrac{8-i}{1+i}\\ =&\dfrac{8-i}{1+i}\times \dfrac{1-i}{1-i}\\ =&\dfrac{8+i^2-8i-i}{1^2-i^2}\\ =&\dfrac{7-9i}{2}\\ =&\dfrac{7}{2}-\dfrac{9}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-1-p4?rev=1737476039&do=diff Question 4, Exercise 1.1 Solutions of Question 4 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 4(i) $x$$y$$(2+3i)x+(1+3i)y+2=0$\begin{align}&(2+3i)x+(1+3i)y+2=0\\ \implies &(2x+y+2)+(3x+3y)i=0.\end{align}\begin{align} 2x+y+2&=0 \quad \cdots(1)\\ 3x+3y&=0\quad \cdots (2) \end{align}\begin{align} &3x=-3y \\ x=-y \quad ... (3) \end{align}$… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-1-p6?rev=1737476039&do=diff Question 6, Exercise 1.1 Solutions of Question 6 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 6(i) $4-3 i$$z=4-3 i$$\bar{z}=4+3i$$3 i+8$$2+\sqrt{\dfrac{-1}{5}}$\begin{align}z=&2+\sqrt{\dfrac{-1}{5}}\\ =&2+\sqrt{\dfrac{1}{5}}i,\end{align}$$\bar{z}=2-\sqrt{\dfrac{1}{5}}i$$$\dfrac{5 }{2}i-\dfrac{7}{8}$$z=\dfrac{5 }{2}i-\dfrac{7}{8},$$\bar… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p4?rev=1737476039&do=diff Question 4, Exercise 1.2 Solutions of Question 4 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 4 $z_{1}=2-3 i$$\left|z_{1} z_{2}\right|=16$$\left|z_{2}\right|$$$z_{1}=2-3i$$\begin{align}|z_1|&=\sqrt{(2)^2+(-3)^2}\\ &=\sqrt{13}\end{align}\begin{align}&|z_{1} z_{2}|=16\\ \Rightarrow \quad &|z_{1}|| z_{2}|=16\\ \Rightarrow \quad & \sqrt{13… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p7?rev=1737476039&do=diff Question 7, Exercise 1.2 Solutions of Question 7 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 7 $\sqrt{2}|z| \geq|\operatorname{Re}(z)|+|\operatorname{Im}(z)| \quad$$\left(|x|-|y|)^{2} \geq 0\right)$\begin{align} &\left(|x|-|y|)^{2} \geq 0\right) \\ \implies & |x|^2+|y|^2-2|x||y| \geq 0 \\ \implies & |x|^2+|y|^2 \geq 2|x||y| \\ \implie… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p8?rev=1737476039&do=diff Question 8, Exercise 1.2 Solutions of Question 8 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 8(i) $|2 z-i|=4$$x$$y$$z=x+i y$$$|2z-i|=4.$$$z=x+i y$\begin{align} & |2(x+iy)-i|=4 \\ \implies & |2x+i(2y-1)|=4 \\ \implies & \sqrt{(2x)^2+(2y-1)^2}=4 \end{align}\begin{align} & (2x)^2+(2y-1)^2 = 16\\ \implies & 4x^2+4y^2-4y+1-16=0 \\ \implies… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-3-p2?rev=1737476039&do=diff Question 2, Exercise 1.3 Solutions of Question 2 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 2(i) $z^{2}-6 z+2=0$\begin{align} & z^2 - 6z + 2 = 0 \\ \implies & z^2 - 2(3)(z)+9-9+2=0 \\ \implies & (z - 3)^2+7= 0 \\ \implies & (z - 3)^2 = 7. \end{align}\begin{align} &z - 3 = \pm \sqrt{7} \\ \implies &z = 3 \pm \sqrt{7}\end{align}$\{3 … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p2?rev=1737476039&do=diff Question 2, Exercise 1.4 Solutions of Question 2 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 2(i) $\left(\cos \dfrac{\pi}{6}+i \sin \dfrac{\pi}{6}\right)\left(\cos \dfrac{\pi}{3}+i \sin \dfrac{\pi}{3}\right)$$z_1=\cos \dfrac{\pi}{6}+i \sin \dfrac{\pi}{6}=e^{i\frac{\pi}{6}}$$z_2=\cos \dfrac{\pi}{3}+i \sin \dfrac{\pi}{3}=e^{i\frac{\pi}{… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p4?rev=1737476039&do=diff Question 4, Exercise 1.4 Solutions of Question 4 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 4 $\dfrac{1+z}{1-z}=\cos 2 \theta+i \sin 2 \theta$$z=i \tan \theta$\begin{align}&\dfrac{1+z}{1-z}=\cos 2 \theta+i \sin 2 \theta\\ \implies &\dfrac{1+z}{1-z}=e^{i2\theta}\\ \implies &(1+z)=(1-z)e^{i2\theta}\\ \implies &z+z e^{i2\theta}=e^{i2\th… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p5?rev=1737476039&do=diff Question 5, Exercise 1.4 Solutions of Question 5 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 5 $\cos \alpha+\cos \beta+\cos \gamma=\sin \alpha+\sin \beta+\sin \gamma=0$$\cos 3 \alpha+\cos 3 \beta+\cos 3 \gamma=3 \cos (\alpha+\beta+\gamma)$$\sin 3 \alpha+\sin 3 \beta+\sin 3 \gamma=3 \sin (\alpha+\beta+\gamma)$\begin{align} \cos \alpha … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p6?rev=1737476039&do=diff Question 6(i-ix), Exercise 1.4 Solutions of Question 6(i-ix) of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sqrt{2}\left(\cos 315^{\circ}+i \sin 315^{\circ}\right)$\begin{align} &\sqrt{2}\left(\cos 315^{\circ}+i \sin 315^{\circ}\right) \\ =& \sqrt{2} \left(\dfrac{1}{\sqrt{2}}-\dfrac{i}{\sqrt{2}} \right) \\ =& 1-i. \end{align}$5\left(\cos 210^{\ci… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p7?rev=1737476039&do=diff Question 6(x-xvii), Exercise 1.4 Solutions of Question 6(x-xvii) of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $7 \sqrt{2}\left(\cos \dfrac{5 \pi}{4}+i \sin \dfrac{5 \pi}{4}\right)$$10 \sqrt{2}\left(\cos \dfrac{7 \pi}{4}+i \sin \dfrac{7 \pi}{4}\right)$$2\left(\cos\dfrac{5\pi}{2}+i \sin \dfrac{5\pi}{2}\right)$$\dfrac{1}{\sqrt{2}}\left(\cos \dfrac{\… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p9?rev=1737476039&do=diff Question 8, Exercise 1.4 Solutions of Question 8 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 8(i) $0.004 \mathrm{~mm}$$\dfrac{\pi}{4}$$$x_{\max}=0.004, \quad \theta=\dfrac{\pi}{4}.$$\begin{align} x&=x_{\max} e^{i\theta} \\ &=0.004 e^{i\dfrac{\pi}{4}} \\ &=\frac{4}{1000} \left(\cos\left(\dfrac{\pi}{4}\right) +i \sin\left(\dfrac{\pi}{4}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p10?rev=1737476039&do=diff Question 9, Exercise 1.4 Solutions of Question 9 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 9(i) $x=2+3 i$$x_{\max }=1+4 i$$\mathrm{t}=0$$$x=2+3i$$$$x_{\max}=1+4 i$$$$\implies x=x_{\max} e^{i\theta}$$$$2+3i=(1+4 i) e^{i\theta}$$\begin{align} \implies e^{i\theta}&=\dfrac{2+3i}{1+4i} \\ &=\dfrac{(2+3i)(1-4i)}{(1+4i)(1-4i)} \\ &=\dfrac{… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/re-ex-p2?rev=1737476039&do=diff Question 2, Review Exercise Solutions of Question 2 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $i^{2}+i^{4}+i^{6}+\cdots+i^{100}$\begin{align*} & i^{2}+i^{4}+i^{6}+\ldots+i^{100} \\ =& i^2 + (i^2)^2 + (i^2)^3 + (i^2)^4 + \ldots +(i^2)^{49} +(i^2)^{50} \\ =& -1 + (-1)^2 + (-1)^3 + (-1)^4 + \ldots + (-1)^{49}+(-1)^{50} \\ =& -1+1-1+1- \ldots -… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/re-ex-p3?rev=1737476039&do=diff Question 3, Review Exercise Solutions of Question 3 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $3 x^{2}+108$\begin{align*} & 3 x^{2}+108\\ =&3 (x^{2}+36)\\ =&3 (x^{2}-(6i)^2)\\ =&3 (x+6i)(x-6i) \end{align*}$4 x^{2}+40$\begin{align*} &4 x^{2}+40\\ =&4 (x^{2}+10)\\ =&4 (x^{2}+(\sqrt{10}i)^2)\\ =&4 (x+\sqrt{10}i)(x-\sqrt{10}i) \end{align*} text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/re-ex-p4?rev=1737476039&do=diff Question 4, Review Exercise Solutions of Question 4 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $z=x+i y$$\left|\dfrac{z+2 i}{z-2 i}\right|=1$$z = x + iy$\begin{align*} & \left|\dfrac{z + 2i}{z - 2i}\right| = 1\\ \implies & |z + 2i| = |z - 2i|\\ \implies & |x + i(y + 2)| = |x + i(y - 2)|\\ \implies & \sqrt{x^2 + (y + 2)^2} = \sqrt{x^2 + (y -… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/re-ex-p5?rev=1737476039&do=diff Question 5, Review Exercise Solutions of Question 5 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $z$$(z-3 i)(2+5 i)=3-4 i$$z$$(z-3 i)(2+5 i)=3-4 i$\begin{align*} &(z-3 i)(2+5 i)=3-4 i \\ \implies & z-3 i=\dfrac{3-4 i}{2+5 i} \\ \implies & z-3 i=\dfrac{(3-4 i)(2-5i)}{(2+5 i)(2-5i)}\\ \implies & z-3 i=\dfrac{6-20-15i-8i}{4+25}\\ \implies & z-3 i… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/re-ex-p6?rev=1737476039&do=diff Question 6, Review Exercise Solutions of Question 6 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left[\dfrac{1}{i^{10}}+(2-i)^{2}+\sqrt{-25}\right]^{3}$\begin{align*} &\left[\dfrac{1}{i^{10}} + (2 - i)^2 + \sqrt{-25}\right]^3\\ =&\left[\dfrac{1}{(i^2)^5} + ( 4 - 4i + i^2) + 5i \right]^3\\ =&\left[\dfrac{1}{(-1)^5} + ( 4 - 4i -1) + 5i \right]… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/re-ex-p7?rev=1737476039&do=diff Question 7, Review Exercise Solutions of Question 7 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 z^{2}-11 z+16=0$\begin{align*} &2 z^{2}-11 z+16=0\\ \implies&z^2 - \dfrac{11}{2}z + 8 = 0\\ \implies& z^2 - \dfrac{11}{2}z = -8\\ \implies& z^2 - 2z\dfrac{11}{4}z + \dfrac{121}{16} = -8 + \dfrac{121}{16}\\ \implies&\left(z-\dfrac{11}{4}\right)^2… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-1-p2?rev=1737476039&do=diff Question 2, Exercise 2.1 Solutions of Question 2 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\quad A=\left[\begin{array}{lll}3 & 6 & 2 \\ 2 & 1 & 9\end{array}\right]$$B=\left[\begin{array}{ll}\frac{1}{3} & 1 \\ 2 & 6\end{array}\right]$$C=\left[\begin{array}{l}3 \\ 2 \\ 8\end{array}\right]$$D=\left[\begin{array}{lll}1 & 6 & 9 \\ 2 & 0 … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-1-p3?rev=1737476039&do=diff Question 3, Exercise 2.1 Solutions of Question 3 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$A=\begin{bmatrix} 3 & 0 & 0 \\ 0 & 1 & 0 \\ 2 & 6 & 0 \end{bmatrix}$$$$B=\begin{bmatrix} -6 & 0 & 0 \\ 0 & -6 & 0 \\ 0 & 0 & -6 \end{bmatrix}$$$$C=\begin{bmatrix} 1 & 0 \\ 2 & 0 \end{bmatrix}$$$$D=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 &… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p2?rev=1737476039&do=diff Question 3, Exercise 2.2 Solutions of Question 3 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{ccc}3 & -1 & 2 \\ 0 & 6 & 1 \\ -1 & 0 & -3\end{array}\right]$$B=\left[\begin{array}{ccc}2 & 1 & 7 \\ 0 & 2 & -1 \\ -3 & 4 & 2\end{array}\right]$$C$$A+B+C=0$$$A+B+C=0,$$$$C=-A-B.$$\begin{align*} C&=-\begin{bmatrix}3 & -1 &… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p3?rev=1737476039&do=diff Question 3, Exercise 2.2 Solutions of Question 3 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{ccc}3 & -1 & 2 \\ 0 & 6 & 1 \\ -1 & 0 & -3\end{array}\right]$$B=\left[\begin{array}{ccc}2 & 1 & 7 \\ 0 & 2 & -1 \\ -3 & 4 & 2\end{array}\right]$$C$$A+B+C=0$$$A+B+C=0,$$$$C=-A-B.$$\begin{align*} C&=-\begin{bmatrix}3 & -1 &… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p4?rev=1737476039&do=diff Question 4, Exercise 2.2 Solutions of Question 4 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A$\begin{align}\left[\begin{array}{cc} 2 & 1 \\ 3 & 2 \end{array}\right]A\left[\begin{array}{cc} 1 & 3 \\ 2 & 4 \end{array}\right]&=\left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right]\end{align}$ B = \left[\begin{array}{cc} 2 & 1 \\ 3… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p5?rev=1737476039&do=diff Question 5, Exercise 2.2 Solutions of Question 5 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $X=\left[\begin{array}{lll}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{array}\right]$$X^{2}-4 X-5 I=0$\begin{align}L.H.S. & =X^{2}-4 X-5 I \\ &=\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p6?rev=1737476039&do=diff Question 6, Exercise 2.2 Solutions of Question 6 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{cc}2 & 1 \\ 3 & -3\end{array}\right]$$\alpha$$\beta$$A^{2}+\alpha I=\beta A$\begin{align} & A^{2}+\alpha I=\beta A\\ \implies &\begin{bmatrix} 2 & 1 \\ 3 & -3 \end{bmatrix} \begin{bmatrix} 2 & 1 \\ 3 & -3 \end{bmatrix}+\a… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p7?rev=1737476039&do=diff Question 7, Exercise 2.2 Solutions of Question 7 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{ll}x & 0 \\ y & 1\end{array}\right]$$n, A^{n}=\left[\begin{array}{cc}x^{n} & 0 \\ \dfrac{y\left(x^{n}-1\right)}{x-1} & 1\end{array}\right]$$$A = \begin{bmatrix} x & 0 \\ y & 1 \end{bmatrix}.$$$n = 1$\begin{align}A^1 =\beg… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p8?rev=1737476039&do=diff Question 8, Exercise 2.2 Solutions of Question 8 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A$$B$$2 \times 3$$3 \times 2$$(A B)^{t}=B^{t} A^{t}$\( A \)\( B \)\( 2 \times 3 \)\( 3 \times 2 \)\begin{align*} A &= \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix}\\ B &= \begin{bmatrix} b_{11} & b_{12}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p10?rev=1737476039&do=diff Question 10, Exercise 2.2 Solutions of Question 10 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A$$B$$A B=B$$B A=A$$A^{2}+B^{2}$$$AB = B$$$$BA = A$$\begin{align*} A^2 &= AA\\ & = A(BA)\\ &=(AB)A\\ &=BA\\ &=A \end{align*}\begin{align*} B^2&= BB \\ &=B(AB)\\ & = (BA)B\\ &=AB\\ &=B\end{align*}$$A^2 + B^2 = A + B$$$AB = B$$BA = A$$$A^2 + B… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p11?rev=1737476039&do=diff Question 11, Exercise 2.2 Solutions of Question 11 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[a_{i j}\right]$$3 \times 3$$a_{i j}=i^{2}-j^{2}$$A$$A=\left[a_{i j}\right]$$a_{ij}=a+{ji}$$a_{ij}=-a_{ji}$$a_{i j}=i^{2}-j^{2}$\begin{align} a_{ji} & = j^2 -i^2 \\ &= - (i^2 -j^2) \\ &= - a_{ij} \end{align}$a_{ij}=-a_{ji}$ text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p12?rev=1737476039&do=diff Question 12, Exercise 2.2 Solutions of Question 12 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A$$\left(A^{n}\right)^{t}=\left(A^{t}\right)^{n}$ text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-3-p2?rev=1737476039&do=diff Question 2, Exercise 2.3 Solutions of Question 2 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left[\begin{array}{lll}3 & 2 & 3 \\ 4 & 5 & 1 \\ 2 & 1 & 0\end{array}\right]$\(R_1\)\(a_{11} = 3\)\(a_{12} = 2\)\(a_{13} = 3\)\begin{align*} A &= \left[\begin{array}{ccc} 3 & 2 & 3 \\ 4 & 5 & 1 \\ 2 & 1 & 0 \end{array}\right]\\ & A_{11} = (-1… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-3-p3?rev=1737476039&do=diff Question 3, Exercise 2.3 Solutions of Question 3 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left[\begin{array}{ccc}3 & 1 & 2 \\ 2 & 3 & 1 \\ -4 & 1 & -3\end{array}\right]$\begin{align*} A &= \left[\begin{array}{ccc} 3 & 1 & 2 \\ 2 & 3 & 1 \\ -4 & 1 & -3\end{array}\right]\end{align*}\(3 \times 3\)\begin{align*} |A| &= 3(3 \cdot (-3) … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-3-p4?rev=1737476039&do=diff Question 4, Exercise 2.3 Solutions of Question 4 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\lambda$$\left[\begin{array}{lll}\lambda & 1 & 3 \\ 2 & 1 & 8 \\ 0 & 3 & 1\end{array}\right]$\begin{align*} A &= \left[\begin{array}{ccc} \lambda & 1 & 3 \\ 2 & 1 & 8 \\ 0 & 3 & 1 \end{array}\right]\\ |A| &= \lambda \cdot (-23) - 1 \cdot 2 + 3… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-3-p5?rev=1737476039&do=diff Question 5, Exercise 2.3 Solutions of Question 5 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left[\begin{array}{ccc}1 & -1 & 1 \\ 2 & 1 & -1 \\ 1 & -2 & -1\end{array}\right]$\begin{align*} A &= \left[\begin{array}{ccc} 1 & -1 & 1 \\ 2 & 1 & -1 \\ 1 & -2 & -1 \end{array}\right]\\ |A|&= 1 [-1 - 2] + 1 [-2 + 1] + 1 [-4 - 1] \\ &= 1 \cd… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p5?rev=1737476039&do=diff Question 5, Exercise 2.6 Solutions of Question 5 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $x_{1}+x_{2}+2 x_{3}=8$$-x_{1}-2 x_{2}+3 x_{3}=1$$3 x_{1}-7 x_{2}+4 x_{3}=10$$A X=B$\begin{align*} &A = \begin{bmatrix} 1 & 1 & 2 \\ -1 & -2 & 3 \\ 3 & -7 & 4 \end{bmatrix}, \quad X = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}, \quad B = \… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/re-ex-p2?rev=1737476039&do=diff Question 2 and 3, Review Exercise Solutions of Question 2 and 3 of Review Exercise of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{ccc}1 & 2 & 0 \\ -3 & 4 & 9 \\ 2 & 1 & 6\end{array}\right]$$A_{13}, A_{23}$$A_{33}$$|A|$\begin{align*} A&=\left[\begin{array}{ccc}1 & 2 & 0 \\ -3 & 4 & 9 \\ 2 & 1 & 6\end{array}\right]\\ A_{13} &= (-1)^{… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p2?rev=1737476039&do=diff Question 3 and 4, Exercise 4.1 Solutions of Question 3 and 4 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$a_{10}$$a_{n}=\frac{n}{n+1}$$$a_n = \frac{n}{n+1}.$$\begin{align*} a_1 &= \frac{1}{1+1} = \frac{1}{2}\\ a_2 &= \frac{2}{2+1} = \frac{2}{3}\\ a_3 &= \frac{3}{3+1} = \frac{3}{4}\\ a_4 &= \frac{4}{4+1} = \frac{4}{5}\\ \end{align*}\begin… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p3?rev=1737476039&do=diff Question 5 and 6, Exercise 4.1 Solutions of Question 5 and 6 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$a_{10}$$a_{15}$$a_{n}=n^{2}-2 n$$$a_n = n^2 - 2n.$$\begin{align*} a_1 &= (1)^2 - 2(1) = 1 - 2 = -1\\ a_2 &= (2)^2 - 2(2) = 4 - 4 = 0\\ a_3 &= (3)^2 - 2(3) = 9 - 6 = 3\\ a_4 &= (4)^2 - 2(4) = 16 - 8 = 8\\ \end{align*}\begin{align*} a_{… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p4?rev=1737476039&do=diff Question 7 and 8, Exercise 4.1 Solutions of Question 7 and 8 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$a_{10}$$a_{15}$$a_{n}=\left(\frac{-1}{2}\right)^{n-1}$$$a_n = \left( \frac{-1}{2} \right)^{n-1}.$$\begin{align*}a_1 &= \left( \frac{-1}{2} \right)^{1-1} = \left( \frac{-1}{2} \right)^0 = 1 \\ a_2 &= \left( \frac{-1}{2} \right)^{2-1} =… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p5?rev=1737476039&do=diff Question 9 and 10, Exercise 4.1 Solutions of Question 9 and 10 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$a_{10}$$a_{15}$$a_{n}=(-1)^{n}(n+3)$$n$$a_{10}$$a_{15}$$$a_{n}=(-1)^{n+1}(3 n-5).$$$$a_n = (-1)^{n+1}(3n - 5).$$\begin{align*} a_1 &= (-1)^{1+1}(3(1) - 5) = (1)(3 - 5) = -2 \\ a_2 &= (-1)^{2+1}(3(2) - 5) = (-1)(6 - 5) = -1 \\ a_3 &=… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p6?rev=1737476039&do=diff Question 11 and 12, Exercise 4.1 Solutions of Question 11 and 12 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{n}=4 n-3; a_8$$$a_n = 4n - 3.$$\begin{align*} a_8 &= 4(8) - 3 \\ &= 32 - 3 \\ &= 29 \end{align*}$a_8 = 29$$a_{n}=5 n+11 ; a_{9}$ text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p7?rev=1737476039&do=diff Question 13 and 14, Exercise 4.1 Solutions of Question 13 and 14 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{n}=(3 n+4)(2 n-5) ; a_{7}$$a_{n}=(-1)^{n-1}(3.4 n-17.3) ; a_{12}$$$a_n = (-1)^{n-1}(3.4n - 17.3).$$\begin{align*} a_{12} &= (-1)^{12-1}(3.4 \cdot 12 - 17.3) \\ &= (-1)^{11}(40.8 - 17.3) \\ &= (-1)^{11}(23.5) \\ &= -23.5 \end{align… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p8?rev=1737476039&do=diff Question 15 and 16, Exercise 4.1 Solutions of Question 15 and 16 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{n}=4 n^{2}(11 n+31) ; a_{22}$$$a_n = 4n^2(11n + 31).$$\begin{align*} a_{22} &= 4 \cdot 22^2 \cdot (11 \cdot 22 + 31) \\ &= 4 \cdot 484 \cdot (242 + 31) \\ &= 4 \cdot 484 \cdot 273 \\ &= 4 \cdot 132132 \\ &= 528528 \end{align*}$a_{… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p9?rev=1737476039&do=diff Question 17 and 18, Exercise 4.1 Solutions of Question 17 and 18 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{n}=\log 10^{n} ; a_{43}$$$a_n = \log 10^n.$$\begin{align*} a_{43} &= \log 10^{43} \\ &= 43 \cdot \log 10 \\ &= 43 \cdot 1 \\ &= 43 \end{align*}$a_{43}= 43$$a_{n}=\ln e^{n} ; a_{67}$$$a_n = \ln e^n.$$\begin{align*} a_{67} &= \ln e^… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p10?rev=1737476039&do=diff Question 19 and 20, Exercise 4.1 Solutions of Question 19 and 20 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{n}$$1,3,5,7,9, \ldots$$$1, 3, 5, 7, 9, \ldots$$$a_1=1$$d=3-1=2$$$a_n = a_1 + (n - 1) d$$\begin{align*} \implies a_n &= 1 + (n - 1) \cdot 2\\ &= 1 + 2n - 2\\ &= 2n - 1 \end{align*}$a_n = 2n - 1$$a_{n}$$3,9,27,81,243, \ldots$\begin… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p2?rev=1737476039&do=diff Question 2, Exercise 4.2 Solutions of Question 2 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $5,9,13, \ldots$$$5, 9, 13, \ldots $$$a_1=5$$d=9-5=4$$$a_n=a_1+(n-1)d.$$\begin{align*} a_4 &=5+(4-1)(4)=5+12=17\\ a_5 &=5+(5-1)(4)=5+16=21\\ a_6 &=5+(6-1)(4)=5+20=25 \end{align*}$17$$21$$25$$11,14,17, \ldots$$$11, 14, 17, \ldots$$$a_1=11$$d=14-11=3$$… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p3?rev=1737476039&do=diff Question 3 and 4, Exercise 4.2 Solutions of Question 3 and 4 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $0.07,0.12,0.7, \ldots$$$0.07,0.12,0.7, \ldots$$$a_1 = 0.07$$d=0.05$$a_{11}=?$\begin{align*} a_n&=a_1+(n-1)d \\ \implies a_{11}&= 0.07+(11-1)(0.05)\\ &=0.07+(10)(0.05)\\ &=0.57 \end{align*}$a_{11}=0.57.$$a_3 = 14$$a_9 = -1$$$a_n = a_1 + (… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p4?rev=1737476039&do=diff Question 5 and 6, Exercise 4.2 Solutions of Question 5 and 6 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{17}=-40$$a_{28}=-73$$a_{1}$$d$$$a_n=a_1+(n-1)d$$\begin{align*} & a_{17} = -40 \\ \implies &a_1 + 16d = -40 \quad \cdots (1) \end{align*}\begin{align*} &a_{28}=-73\\ \implies &a_1 + 27d = -73 \quad \cdots (2) \end{align*}\begin{align*}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p5?rev=1737476039&do=diff Question 7 and 8, Exercise 4.2 Solutions of Question 7 and 8 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $-6,-2,2, \ldots$$70$$-6,-2,2, \ldots$$a_1=-6$$d=-2+6=4$$a_n=70$$n=?$$$a_n=a_1+(n-1)d.$$\begin{align*} &70=-6+(n-1)4\\ \implies &70=-6+4n-4\\ \implies &70=4n-10\\ \implies &4n=80\\ \implies & n=20 \end{align*}$a_{20}=70$$\dfrac{5}{2}, \df… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p6?rev=1737476039&do=diff Question 9 and 10, Exercise 4.2 Solutions of Question 9 and 10 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\dfrac{1}{a}, b, \dfrac{1}{c}$$\dfrac{a-c}{2 a c}$$\dfrac{1}{a}, b, \dfrac{1}{c}$\begin{align*} d&=b-\frac{1}{a}\cdots (i)\\ \end{align*}\begin{align*} d&=\frac{1}{c}-b \cdots (ii) \end{align*}\begin{align*} b-\frac{1}{a}&=\frac{1}{c}-… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p2?rev=1737476039&do=diff Question 3 and 4, Exercise 4.3 Solutions of Question 3 and 4 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=5$$a_{n}=100$$n=200$$a_{1}=5$$a_{n}=100$$n=200$$a_{1}=5$$a_{n}=100$$n=200$$S_n$\begin{align} S_n&=\frac{n}{2}[a_1+a_n] \\ \implies S_{200}&=\frac{200}{2}[5+100]\\ &=10500. \end{align}$S_{200}=10500$$a_{1}=4$$n=15$$d=3$$a_{1}=4$$n=1… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p3?rev=1737476039&do=diff Question 5 and 6, Exercise 4.3 Solutions of Question 5 and 6 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=50$$n=20$$d=-4$$a_{1}=50$$n=20$$d=-4$$a_{1}=50$$n=20$$d=-4$$S_n$\begin{align} S_n&=\frac{n}{2}[2a_1+(n-1)d] \\ \implies S_{20}&=\frac{20}{2}[2(50)+(20-1)(-4)]\\ &=10\times [100-76]\\ &=240. \end{align}$S_{20}=240$$-3+(-7)+(-11)+\cd… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p6?rev=1737476039&do=diff Question 11 and 12, Exercise 4.3 Solutions of Question 11 and 12 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $S_{\boldsymbol{n}}$$a_{1}=3$$a_{n}=-38$$n=8$$a_{1}=3$$a_{n}=-38$$n=8$\begin{align} S_n&=\frac{n}{2}[a_1+a_n] \\ \implies S_{8}&=\frac{8}{2}[3-38]\\ &=4\times[-35] \\ &=-140. \end{align}$S_{8}=-140$$S_n$$a_{1}=85$$n=21$$a_{n}=25$$a_{1… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p7?rev=1737476039&do=diff Question 13 and 14, Exercise 4.3 Solutions of Question 13 and 14 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $S_s$$a_{1}=34$$n=9$$a_{n}=2$$a_{1}=34$$n=9$$a_{n}=2$\begin{align} S_n&=\frac{n}{2}[a_1+a_n] \\ \implies S_{9}&=\frac{9}{2}[34+2]\\ &=162. \end{align}$S_{9}=162$$S_n$$a_{1}=5$$d=\frac{1}{2}$$n=13$$a_{1}=5$$d=\frac{1}{2}$$n=13$\begin{a… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p8?rev=1737476039&do=diff Question 15 and 16, Exercise 4.3 Solutions of Question 15 and 16 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $S_n$$a_{1}=91$$d=-4$$a_{n}=15$$a_{1}=91$$d=-4$$a_{n}=15$$n=?$\begin{align} & a_n=a_1+(n-1)d \\ \implies & 15=91+(n-1)(-4) \\ \implies & 15=91-4n+4 \\ \implies & 4n=95-15 \\ \implies & 4n = 80\\ \implies & n = 20. \end{align}\begin{… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p9?rev=1737476039&do=diff Question 17, 18 and 19, Exercise 4.3 Solutions of Question 17, 18 and 19 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $6+12+18+\ldots+96$$$6+12+18+\ldots+96.$$$a_{1}=6$$d=12-6=6$$a_{n}=96$$n=?$\begin{align} & a_n=a_1+(n-1)d \\ \implies & 96=6+(n-1)(6) \\ \implies & 96=6+6n-6 \\ \implies & 6n=96 \\ \implies & n = 24. \end{align}\begin{align}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p10?rev=1737476039&do=diff Question 20, 21 and 22, Exercise 4.3 Solutions of Question 20, 21 and 22 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=7$$a_{n}=139$$S_{n}=876$$a_{1}=7$$a_{n}=139$$S_{n}=876$$n$$d$\begin{align} &S_n=\frac{n}{2}[a_1+a_n]\\ \implies & 876=\frac{n}{2}[7+139]\\ \implies & 1752=146n\\ \implies & n=\frac{1752}{146}=12. \end{align}\begin{align… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p11?rev=1737476039&do=diff Question 23 and 24, Exercise 4.3 Solutions of Question 23 and 24 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$ 14+16+18+...+a_{25}.$$$a_1=14$$d=16-14=2$$n=25$$a_25$$S_25$\begin{align} a_n&=a_1+(n-1)d\\ \implies a_{25}&= 14+(25-1)(2)\\ &=62. \end{align}\begin{align} S_n&=\frac{n}{2}[a_1+a_n]\\ \implies S_{25}& =\frac{25}{2}[14+62]\\ & =25 \t… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p2?rev=1737476039&do=diff Question 3 and 4, Exercise 4.4 Solutions of Question 3 and 4 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\frac{3}{2}, \frac{9}{4}, \frac{27}{8}, \frac{81}{16}, \ldots$\(\frac{3}{2}, \frac{9}{4}, \frac{27}{8}, \frac{81}{16}, \ldots\)\begin{align*} r_1&=\frac{9/4}{3/2} = \frac{9}{4} \times \frac{2}{3} = \frac{3}{2} \\ r_2&=\frac{27/8}{9/4} = … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p3?rev=1737476039&do=diff Question 5, 6 and 7, Exercise 4.4 Solutions of Question 5, 6 and 7 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=3, r=-2$$a_{1}=3$$r=-2$$$a_{n}=a_{1} r^{n-1}.$$\begin{align*} & a_{2}=a_{1} r=(3)(-2)= -6 \\ & a_{3}=a_{1} r^{2}=(3)(-2)^{2}=3 (4)= 12 \\ & a_{4}=a_{1} r^{3}=(3)(-2)^{3}=3 (-8) = -24 \end{align*}$a_1=3$$a_2=-6$$a_3=12$$a_4=-… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p4?rev=1737476039&do=diff Question 8 and 9, Exercise 4.4 Solutions of Question 8 and 9 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$90,30,10 \ldots$$$a_1=90$$r=\dfrac{30}{90}=\dfrac{1}{3}$$$a_{n}=a_{1} r^{n-1}.$$\begin{align*} & a_{4}=a_{1} r^3=(90)\left(\dfrac{1}{3} \right)^3=90 \times\dfrac{1}{27}=\dfrac{10}{3}\\ & a_{5}=a_{1} r^3=(90)\left(\dfrac{1}{4} \right)^4=… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p5?rev=1737476039&do=diff Question 10 and 11, Exercise 4.4 Solutions of Question 10 and 11 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$20,30,45 \ldots$$\(a_1=20\)\(r=\frac{30}{20}=\frac{3}{2}\)$$a_{n}=a_{1} r^{n-1}.$$\begin{align*} & a_{4}=a_{1} r^3=(20)\left(\frac{3}{2}\right)^3=20 \times \frac{27}{8} = \frac{540}{8} = 67.5 \\ & a_{5}=a_{1} r^4=(20)\left(\frac{3}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p6?rev=1737476039&do=diff Question 12 and 13, Exercise 4.4 Solutions of Question 12 and 13 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$\frac{1}{27}, \frac{1}{9}, \frac{1}{3}, \ldots$$\(a_1=\frac{1}{27}\)\(r=\frac{\frac{1}{9}}{\frac{1}{27}}=3\)$a_{n}=a_{1} r^{n-1}.$\begin{align*} & a_{4}=a_{1} r^3=\left(\frac{1}{27}\right)(3)^3=\frac{1}{27} \times 27 = 1 \\ & a_{5}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p7?rev=1737476039&do=diff Question 14 and 15, Exercise 4.4 Solutions of Question 14 and 15 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=4, n=3, r=5$$a_{1}=4, n=3, r=5$$$a_{n}=a_{1} r^{n-1}.$$\begin{align*} a_3&= 4\times 5^2 \\ &=4\times 25 = 100. \end{align*}$a_3=100$$a_{1}=2, n=5, r=2$$a_{1}=2$$n=5$$r=2$$a_{n}=a_{1} r^{n-1}.$\begin{align*} a_5 &= 2 \times 2^{… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p8?rev=1737476039&do=diff Question 16 and 17, Exercise 4.4 Solutions of Question 16 and 17 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=7, n=4, r=2$$a_{1}=7$$n=4$$r=2$$a_{n}=a_{1} r^{n-1}.$\begin{align*} a_4 &= 7 \times 2^{4-1} \\ &= 7 \times 2^3 \\ &= 7 \times 8 = 56. \end{align*}$a_4=56$$a_{1}=243, n=5, r=-\frac{1}{3}$$a_{1}=243$$n=5$$r=-\frac{1}{3}$$a_{n}=… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p9?rev=1737476039&do=diff Question 18 and 19, Exercise 4.4 Solutions of Question 18 and 19 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=32, n=6, r=-\frac{1}{2}$$a_{1}=32$$n=6$$r=-\frac{1}{2}$$a_{n}=a_{1} r^{n-1}.$\begin{align*} a_6 &= 32 \times \left(-\frac{1}{2}\right)^{6-1} \\ &= 32 \times \left(-\frac{1}{2}\right)^{5} \\ &= 32 \times \left(-\frac{1}{32}\ri… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p10?rev=1737476039&do=diff Question 20 and 21, Exercise 4.4 Solutions of Question 20 and 21 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$3 , \_\_\_ , \_\_\_ , \_\_\_ , 48$$$a_1=3$$a_5=48$$r$$$ a_n=ar^{n-1}. $$\begin{align*} &a_5=a_1 r^4 \\ \implies & 48=3r^4 \\ \implies & r^4 = 16 \\ \implies & r^4 = 2^4 \\ \implies & r = 2. \end{align*}\begin{align*} & a_2=a_1 r= (3… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p11?rev=1737476039&do=diff Question 22 and 23, Exercise 4.4 Solutions of Question 22 and 23 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$8 , \_\_\_, \_\_\_, \_\_\_, \_\_\_, \dfrac{1}{4}$$$a_1=8$$a_6=\frac{1}{4}$$r$$n$$a_n = a_1 r^{n-1}.$\begin{align*} a_6 &= a_1 r^5 \\ \implies \frac{1}{4} &= 8 \cdot r^5 \\ \implies r^5 &= \frac{1}{4 \cdot 8} \\ \implies r^5 &= \frac… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p12?rev=1737476039&do=diff Question 24 and 25, Exercise 4.4 Solutions of Question 24 and 25 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$5 , \_\_\_, \_\_\_, \_\_\_, 80$$$a_1=5$$a_5=80$$r$$n$$$a_n = a_1 r^{n-1}.$$\begin{align*} a_5 &= a_1 r^4 \\ \implies 80 &= 5 \cdot r^4 \\ \implies r^4 &= \frac{80}{5} \\ \implies r^4 &= 16 \\ \implies r &= 2. \end{align*}\begin{alig… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p14?rev=1737476039&do=diff Question 28 and 29, Exercise 4.4 Solutions of Question 28 and 29 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $=a_1= 1$$= a_2 = 2$$= a_3 = 2(2)=4$$= a_7$$$ 1+2+4+...+a_7 $$$a_1=1$$r=2$$n=7$$$ S_n=\frac{a_1\left(1-r^n \right)}{1-r}, \quad r\neq 1. $$\begin{align*} S_6&=\frac{(1)\left(1-2^7 \right)}{1-2} \\ &=\frac{1-128}{-2}\\ &=127 \end{align… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p2?rev=1737476040&do=diff Question 3 and 4, Exercise 4.5 Solutions of Question 3 and 4 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=5$$r=3$$n=12$$a_{1}=5$$r=3$$n=12$$n$\[ S_n = \frac{a_1 \left(1 - r^n\right)}{1 - r}, \quad r\neq 1. \]\begin{align*} S_{12} &= \frac{5\left(1 - 3^{12}\right)}{1 - 3} \\ &= \frac{5\left(1 - 531441\right)}{-2} \\ &= \frac{5(-531440)}… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p3?rev=1737476040&do=diff Question 5 and 6, Exercise 4.5 Solutions of Question 5 and 6 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=7, r=2, n=14$$a_1 = 7$$r = 2$$n = 14$$n$$$S_n = \frac{a_1 \left(1 - r^n\right)}{1 - r}, \quad r \neq 1.$$\begin{align*} S_{14} &= \frac{7 \left(1 - 2^{14}\right)}{1 - 2} \\ &= \frac{7 \left(1 - 16384\right)}{-1} \\ &= \frac{7 \time… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p4?rev=1737476040&do=diff Question 7 and 8, Exercise 4.5 Solutions of Question 7 and 8 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=16, r=-\frac{1}{2}, n=10$$a_1 = 16$$r = -\frac{1}{2}$$n = 10$$n$$$S_n = \frac{a_1 \left(1 - r^n\right)}{1 - r}, \quad r \neq 1.$$\begin{align*} S_{10} &= \frac{16 \left(1 - \left(-\frac{1}{2}\right)^{10}\right)}{1 - \left(-\frac{1}… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p5?rev=1737476040&do=diff Question 9 and 10, Exercise 4.5 Solutions of Question 9 and 10 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=343, a_{4}=-1, r=-\frac{1}{7}$$a_{1}=343$$a_{4}=-1$$r=-\frac{1}{7}$$S_n$$$ S_n =\frac{a_1-a_n r}{1-r}, \quad r\neq 1.$$\begin{align*} S_4 & =\frac{343-(-1)\left(-\frac{1}{7}\right)}{1+\frac{1}{7}} \\ &=\frac{\frac{2400}{7}}{\frac… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p6?rev=1737476040&do=diff Question 11, 12 and 13, Exercise 4.5 Solutions of Question 11, 12 and 13 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}$$S_{n}=244, r=-3, n=5$$S_{n}=244$$r=-3$$n=5$$$ S_n =\frac{a_1(1-r^n)}{1-r}, \quad r\neq 1.$$\begin{align*} & 244=\frac{a_1(1-(-3)^5)}{1-(-3)} \\ \implies & 244=\frac{a_1(1+243)}{4} \\ \implies & 976=244a_1\\ \implies & … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p7?rev=1737476040&do=diff Question 14, Exercise 4.5 Solutions of Question 14 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $0.444...$$$0.444... = 0.4+0.04+0.004+...$$$a_1=0.4$$r=\frac{0.04}{0.4}=0.1$$|r|=0.1 < 1$\begin{align*} S-\infty & = \frac{a_1}{1-r} \\ & = \frac{0.4}{1.0.1} = \frac{0.4}{0.9} \\ & = \frac{4}{9}. \end{align*}$S_{\infty} =\dfrac{4}{9}$$9.99999 ...$$… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p8?rev=1737476040&do=diff Question 15, Exercise 4.5 Solutions of Question 15 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $30 ft$$\frac{2}{5}$$= 30 ft$$= 30 \times \frac{2}{5} = 12 ft$$= 12 \times \frac{2}{5} = \frac{24}{5} ft$$= \frac{24}{5} \times \frac{2}{5} = \frac{48}{25} ft$$D$$$D=30+2\left(12+\frac{24}{5}+\frac{24}{5}+... \right)$$$$ 12+\frac{24}{5}+\frac{24}{5… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-6-p7?rev=1737476040&do=diff Question 12, Exercise 4.6 Solutions of Question 12 of Exercise 4.6 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\dfrac{1}{3}$$\dfrac{1}{11}$$H_1, H_2, H_3, H_4$$H.Ms$$\dfrac{1}{3}$$\dfrac{1}{11}$$$\dfrac{1}{3},H_1, H_2, H_3, H_4, \dfrac{1}{11} \text{ are in H.P.}$$$$\quad 3,\dfrac{1}{H_1},\dfrac{1}{H_2}, \dfrac{1}{H_3}, \dfrac{1}{H_4},11 \text{ are in A.P.}… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p2?rev=1737476040&do=diff Question 3 and 4, Exercise 4.7 Solutions of Question 3 and 4 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sum_{k=0}^{5} 2^{k}$\begin{align*} \sum_{k=0}^{5} 2^{k} &= 2^0 + 2^1 + 2^2 + 2^3 + 2^4 + 2^5 \\ &= 1 + 2 + 4 + 8 + 16 + 32 \\ &= 63 \end{align*}$\sum_{k=0}^{9} \pi k$\begin{align*} \sum_{k=0}^{9} \pi k &= \pi(0) + \pi(1) + \pi(2) + \pi(… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p3?rev=1737476040&do=diff Question 5 and 6, Exercise 4.7 Solutions of Question 5 and 6 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sum_{k=1}^{8} \frac{k}{k+1}$\begin{align*} \sum_{k=1}^{8} \frac{k}{k+1} &= \frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} + \frac{5}{6}\\ &+ \frac{6}{7} + \frac{7}{8} + \frac{8}{9} \\ &= 0.5 + 0.6667 + 0.75 + 0.8 + 0.8333\\ &+ 0.… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p4?rev=1737476040&do=diff Question 7 and 8, Exercise 4.7 Solutions of Question 7 and 8 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sum_{k=0}^{5}\left(k^{2}-2 k+3\right)$\begin{align*} \sum_{k=0}^{5} (k^{2} - 2k + 3) &= (0^{2} - 2(0) + 3) + (1^{2} - 2(1) + 3) + (2^{2} - 2 (2) + 3) \\ &+ (3^{2} - 2 (3) + 3) + (4^{2} - 2 (4) + 3) + (5^{2} - 2 (5) + 3) \\ &= (0 - 0 + 3… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p5?rev=1737476040&do=diff Question 9 and 10, Exercise 4.7 Solutions of Question 9 and 10 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\frac{4}{5}+\frac{5}{6}+\dots$$$ \frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} + \frac{5}{6} +... = \sum_{k=1}^{\infty}\frac{k}{k+1} $$$3+6+9+12+15$$$3+6+9+12+15=\sum_{k=1}^{5}3k$$ text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p6?rev=1737476040&do=diff Question 11, 12 and 13, Exercise 4.7 Solutions of Question 11, 12 and 13 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $-2+4-8+16-32+64$$$ -2 + 4 - 8 + 16 - 32 + 64 = \sum_{k=1}^{6} (-1)^k 2^k $$$\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\frac{1}{4 \cdot 5}+$$$ \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} +… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p7?rev=1737476040&do=diff Question 14, 15 and 16, Exercise 4.7 Solutions of Question 14, 15 and 16 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$n$$n+1$$T_n$$n$$$ T_{n} = n+1. $$\begin{align*}\sum_{n=1}^{\infty} T_{n} &= \sum_{n=1}^{\infty} (n+1)\\ & = \sum_{n=1}^{\infty} n + \sum_{n=1}^{\infty} 1 \\ & = \frac{n(n+1)}{2} + n \\ & = \frac{n(n+1)}{2} + \frac{2n}{2} \… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p8?rev=1737476040&do=diff Question 17 and 18, Exercise 4.7 Solutions of Question 17 and 18 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$2^{2}+5^{2}+8^{2}+\ldots$$2+5+8+\ldots$$a_k=2+(k-1)(3)=2+3k-3=3k-1$$T_k$$k$\begin{align*}T_k&=(3k-1)^2 \\ &=9k^2-6k+1. \end{align*}\begin{align*}\sum_{k=1}^{n} T_{k} &= \sum_{k=1}^{n} (9k^{2} - 6k + 1)\\ & = 9\sum_{k=1}^{n} k^{2} … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p9?rev=1737476040&do=diff Question 19 and 20, Exercise 4.7 Solutions of Question 19 and 20 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$1^{3}+3^{3}+5^{3}+$$1+3+5+\ldots$$a_k=1+(k-1)(2)=1+2k-2=2k-1$$T_k$$k$\begin{align*}T_k&=(2k-1)^3 \\ &=(2k)^3+3(2k)^2(-1)+3(2k)(-1)^2+(-1)^3 \\ &=8k^3-12k^2+6k+1 \end{align*}\begin{align*}\sum_{k=1}^{n} T_{k} &= \sum_{k=1}^{n} (8k^… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p10?rev=1737476040&do=diff Question 19 and 20, Exercise 4.7 Solutions of Question 19 and 20 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$1^{3}+3^{3}+5^{3}+$$1+3+5+\ldots$$a_k=1+(k-1)(2)=1+2k-2=2k-1$$T_k$$k$\begin{align*}T_k&=(2k-1) \\ &=9k^2-6k+1. \end{align*}\begin{align*}\sum_{k=1}^{n} T_{k} &= \sum_{k=1}^{n} (2k - 1)\\ & = 2 \sum_{k=1}^{n} k - \sum_{k=1}^{n} 1 \… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p11?rev=1737476040&do=diff Question 21 and 22, Exercise 4.7 Solutions of Question 21 and 22 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$1 \times 4+2 \times 7+3 \times 10+\cdots$$4+7+10+\ldots$$a_k=4+(k-1)(3)=4+3k-3=3k+1$$1+2+3+...$$k$$k(3k+1)$$T_k$$k$\begin{align*}T_k&=k(3k+1) \\ &=3k^2+k. \end{align*}\begin{align*}\sum_{k=1}^{n} T_{k} &= \sum_{k=1}^{n} (3k^2 +k)\… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p12?rev=1737476040&do=diff Question 23 and 24, Exercise 4.7 Solutions of Question 23 and 24 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$$1+2 \times 2+3 \times 2^{2}+4 \times 2^{3}+\ldots.$$$$1+2 \times 2+3 \times 2^{2}+4 \times 2^{3}+\ldots$$$$ 1\times 1+2 \times 2+3 \times 2^{2}+4 \times 2^{3}+\ldots $$$1,2,3,4,\ldots$$a=1$$d=1$$1, 2, 2^2, 2^3, \ldots$$r=\frac{2}… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p13?rev=1737476040&do=diff Question 25 and 26, Exercise 4.7 Solutions of Question 25 and 26 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$1+\frac{4}{7}+\frac{7}{7^{2}}+\frac{10}{7^{3}}+\ldots$\[ 1 + \frac{4}{7} + \frac{7}{7^2} + \frac{10}{7^3} + \ldots \]\(1, 4, 7, 10, \ldots\)\(a = 1\)\(d = 3\)\(1, \frac{1}{7}, \frac{1}{7^2}, \frac{1}{7^3}, \ldots\)\(1\)\(r = \frac… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p14?rev=1737476040&do=diff Question 27 and 28, Exercise 4.7 Solutions of Question 27 and 28 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$5+\frac{7}{3}+\frac{9}{9}+\frac{11}{27}+\ldots$$$$5+\frac{7}{3}+\frac{9}{9}+\frac{11}{27}+\ldots$$$$ 5\times 1+7\times\frac{1}{3}+9\times\frac{1}{9}+11\times\frac{1}{27}+\ldots $$$5,7,9,11,4,\ldots$$a=5$$d=7-5=2$$1, \dfrac{1}{3}, \d… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-8-p4?rev=1737476040&do=diff Question 7 and 8, Exercise 4.8 Solutions of Question 7 and 8 of Exercise 4.8 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$$\frac{1}{1 \times 4}+\frac{1}{4 \times 7}+\frac{1}{7 \times 10}+\ldots$$$$\frac{1}{1 \times 4}+\frac{1}{4 \times 7}+\frac{1}{7 \times 10}+\dots$$$T_k$\begin{align*} T_k &=\frac{1}{(3k-2)(3k+1)}. \end{align*}\begin{align*} \frac{1}{(3… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-8-p6?rev=1737476040&do=diff Question 11 and 12, Exercise 4.8 Solutions of Question 11 and 12 of Exercise 4.8 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sum_{k=1}^{n} \frac{1}{k(k+2)}$$T_k$$k$\begin{align*} T_k &= \frac{1}{k(k+2)}. \end{align*}\begin{align*} \frac{1}{k(k+2)} = \frac{A}{k} + \frac{B}{k+2} \ldots (1) \end{align*}$k(k+2)$\begin{align*} 1 = A(k+2) + Bk \ldots (2) \end{… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-1-p1?rev=1737476040&do=diff Question 1, Exercise 5.1 Solutions of Question 1 of Exercise 5.1 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 1(i) $2 x^{3}+3 x^{2}-4 x+1$$x+2$$p(x)=2 x^{3}+3 x^{2}-4 x+1$$x-c=x+2 \implies c=-2$\begin{align*} \text{Remainder} & = p(c) = p(-2) \\ & = 2(-2)^{3}+3 (-2)^{2}-4 (-2)+1 \\ & = -16+12+8+1 \\ &= 5. \end{align*}$x^{4}+2 x^{3}-x^{2}+2 x+3$$x-2$\( p(x… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-1-p2?rev=1737476040&do=diff Question 2 and 3, Exercise 5.1 Solutions of Question 2 and 3 of Exercise 5.1 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $x-3$$x^{3}-2 x^{2}-5 x+6$$p(x)=x^{3}-2 x^{2}-5 x+6$$x-c=x-3$$\implies c=3$$x-3$$p(x)$$p(3)=0$\begin{align*} p(3)&=3^3-2(3)^2-5(3)+6 \\ & = 27-18-15+6 \\ & = 0. \end{align*}$x-3$$p(x)$$x-3$$x^{3}-2 x^{2}-5 x+1$$p(x)=x^{3}-2 x^{2}-5 x+1$$x-c=x-3$$… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-1-p3?rev=1737476040&do=diff Question 4 and 5, Exercise 5.1 Solutions of Question 4 and 5 of Exercise 5.1 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $4 y^{3}-4 y^{2}+10+2 y$$4 y^{2}-8 y+10$$q$$x^{3}+q x^{2}-7 x+6$$(x+1)$$p(x)=x^{3}+q x^{2}-7 x+6$$x-c=x+1$$\implies c=-1$$x+1$$p(x)$$p(-1)=0$\begin{align*} &(-1)^3+q(-1)^2-7(-1)+6=0 \\ -&1+q+7+6=0\\ &q+12=0\\ &q=-12 \end{align*} text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-1-p5?rev=1737476040&do=diff Question 8 and 9, Exercise 5.1 Solutions of Question 8 and 9 of Exercise 5.1 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 x^{3}+3 x^{2}-11 x-6$$p(x)=2x^3+3x^2-11x-6$\begin{align} p(2) &= 2(2)^3+3(2)^2-11(2)-6 \\ &=16+12-22-6 = 0 \end{align}$p(x)$\begin{align} \begin{array}{r|rrrr} 2 & 2 & 3 & -11 & -6 \\ & \downarrow & 4 & 14 & 6 \\ \hline & 2 & 7 & 3 & 0 \\ \… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-2-p2?rev=1737476040&do=diff Question 3 and 4, Exercise 5.2 Solutions of Question 3 and 4 of Exercise 5.2 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 x^{3}+5 x^{2}-9 x-18$\( f(x) = 2x^{3} + 5x^{2} - 9x - 18 \)\begin{align*} f(-2) &= 2(-2)^{3} + 5(-2)^{2} - 9(-2) - 18 \\ &= 2(-8) + 5(4) + 18 - 18 \\ &= -16 + 20 + 18 - 18 = 0. \end{align*}\( x + 2 \)\( f(x) \)\[ \begin{array}{r|rrrr} -2 & 2 &… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-2-p3?rev=1737476040&do=diff ; Question 5 and 6, Exercise 5.2 Solutions of Question 5 and 6 of Exercise 5.2 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $t^{3}+t^{2}+3 t-5$\( f(t) = t^{3} + t^{2} + 3t - 5 \)\begin{align*} f(1) &= (1)^{3} + (1)^{2} + 3(1) - 5 \\ &= 1 + 1 + 3 - 5 \\ &= 0. \end{align*}\( t - 1 \)\( f(t) \)\begin{align} \begin{array}{r|rrrr} 1 & 1 & 1 & 3 & -5 \\ & & 1 & 2 & 5 \… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-3-p2?rev=1737476040&do=diff Question 2, Exercise 5.3 Solutions of Question 2 of Exercise 5.3 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 2 $t(x)=x^{3}-12 x^{2}+48 x+74$$x$$$t(x)=x^{3}-12 x^{2}+48 x+74.$$$t=12$\begin{align*} t(12)&=(12)^3-12(12)^2+48(12)+74 \\ &=650. \end{align*} text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-3-p3?rev=1737476040&do=diff Question 3, Exercise 5.3 Solutions of Question 3 of Exercise 5.3 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 3 $x$$2x$$2x+2$\begin{align*} & x(2x)(2x+2) = 144 \\ \implies & 4x^2(x+1)=144 \\ \implies & x^2(x+1)=36 \\ \implies & x^3+x^2-36=0 \end{align*}$$p(x)=x^3+x^2-36.$$\begin{align*} p(3)&=3^3+3^2-36 \\ &=27+9-36 = 0 \end{align*}$x=3$$p(x)$$2(3)$$2(3)+… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-3-p4?rev=1737476040&do=diff Question 4, Exercise 5.3 Solutions of Question 4 of Exercise 5.3 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 4 $x$$2x+3$$x-2$\begin{align*} & x(2x+3)(x-2) = 2475 \\ \implies & x(2x^2+3x-4x-6)=2475 \\ \implies & x(2x^2-x-6)-2475=0 \\ \implies & 2x^3-x^2-6x-2475=0 \end{align*}$$p(x)=2x^3-x^2-6x-2475.$$\begin{align*} p(11)&=2(11)^3-11^2-6(11)-2475 \\ &=2662… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-3-p5?rev=1737476040&do=diff Question 5, Exercise 5.3 Solutions of Question 5 of Exercise 5.3 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 5 $6 x^{2}+38 x+56$$2 x+8$$ACED$$ABFG$$ACED$$6 x^{2}+38 x+56$$2 x+8$\begin{align*} & 6 x^{2}+38 x+56 \\ = & 2(3x^2+19x+28) \\ = & 2(3x^2+12x+7x+28) \\ = & 2(3x(x+4)+7(x+4)) \\ =& 2(x+4)(3x+7) \\ =& (2x+8)(3x+7) \end{align*}\begin{align*} & Length … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-4-p2?rev=1737476040&do=diff Question 2, Review Exercise Solutions of Question 2 of Review Exercise of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Go to text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/re-ex-p2?rev=1737476040&do=diff Question 2 & 3, Review Exercise Solutions of Question 2 & 3 of Review Exercise of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left(64 y^{3}-8\right) \div(4 y-2) \quad$\begin{align*} \frac{(64 y^{3}-8)}{(4 y-2)}&= \frac{(4y - 2)(16y^{2} + 8y + 4)}{4y - 2}\\ & = 16y^{2} + 8y + 4 .\end{align*}$\left(125 y^{3}-8\right) \div(5 y-2)$\begin{align*} \frac{(125 y^{3}-8)}{(5 … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/re-ex-p3?rev=1737476040&do=diff Question 4 & 5, Review Exercise Solutions of Question 4 & 5 of Review Exercise of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $3 y-2$$6 y^{3}-y^{2}-5 y+2$\begin{align*}3y-2&=0\\ 3y&=2\\ y&=\frac{2}{3}\end{align*}\begin{align*} f(y) &= 6y^{3} - y^{2} - 5y + 2\\ f\left(\frac{2}{3}\right) &= 6\left(\frac{2}{3}\right)^{3} - \left(\frac{2}{3}\right)^{2} - 5\left(\frac{2}{3… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/re-ex-p4?rev=1737476040&do=diff Question 6 & 7, Review Exercise Solutions of Question 6 & 7 of Review Exercise of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $k$$\left(x^{2}+8 x+k\right)$$(x-4)$\( p(x) = x^{2} + 8x + k \)\( p(x) \)\( (x - 4) \)\( p(4) \)\( p(4) = 0 \)\begin{align*} p(4) &= (4)^2 + 8(4) + k \\ &= 16 + 32 + k \\ &= 48 + k. \end{align*}\[ 48 + k = 0. \]\[ k = -48. \]$3 x^{2}-x+32-\frac… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/re-ex-p6?rev=1737476040&do=diff Question 6, Review Exercise Solutions of Question 6 of Review Exercise of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 6 $k$$\left(x^{2}+8 x+k\right)$$(x-4)$ text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/re-ex-p7?rev=1737476040&do=diff Question 7, Review Exercise Solutions of Question 7 of Review Exercise of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 7 $3 x^{2}-x+32-\frac{121}{x+4}$ text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p1?rev=1737476040&do=diff Question 1, Exercise 8.1 Solutions of Question 1 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\cos (\alpha \pm \beta), \sin (\alpha \pm \beta)$$\tan (\alpha \pm \beta)$$\alpha=180^{\circ}, \beta=60^{\circ}$$\alpha=180^{\circ}$$\beta=60^{\circ}$\begin{align*} \cos (\alpha + \beta) & = \cos \alpha \cos \beta - \sin \alpha \sin \beta \… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p2?rev=1737476040&do=diff Question 2, Exercise 8.1 Solutions of Question 2 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\cos 15^{\circ}$$\cos \left(45^{\circ}-30^{\circ}\right)$\begin{align*} \cos 15^{\circ} & = \cos \left(45^{\circ}-30^{\circ}\right)\\ &= \cos 45 \cos 30 + \sin 45 \sin 30 \\ &= \dfrac{1}{\sqrt{2}}\cdot \dfrac{\sqrt{3}}{2} + \dfrac{1}{\sqrt{2… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p3?rev=1737476040&do=diff Question 3, Exercise 8.1 Solutions of Question 3 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\cos 120^{\circ}$$\cos \left(180^{\circ}-60^{\circ}\right)$$\cos \left(90^{\circ}+30^{\circ}\right)$\begin{align*} \cos 120^{\circ} & = \cos \left(180^{\circ}-60^{\circ}\right) \\ &= - \cos 60 ^{\circ}\\ &= -\dfrac{1}{2}. \end{align*}\begin{… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p4?rev=1737476040&do=diff Question 4, Exercise 8.1 Solutions of Question 4 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\cos 6 \theta \cos 3 \theta-\sin 6 \theta \sin 3 \theta$\begin{align*} & \cos 6 \theta \cos 3 \theta-\sin 6 \theta \sin 3 \theta \\ & = \cos (6\theta +3\theta) \\ & = \cos 9\theta . \end{align*}$\cos 7 \theta \cos 2 \theta+\sin 7 \theta \sin… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p6?rev=1737476040&do=diff Question 7, Exercise 8.1 Solutions of Question 7 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\alpha$$\beta$$\sin \alpha=\dfrac{12}{13}$$\tan \beta=\dfrac{4}{3}$$\sin(\alpha+\beta)$$\cos(\alpha+\beta)$$\tan(\alpha+\beta)$$\sin \alpha=\dfrac{12}{13}$$\alpha$$\tan \beta=\dfrac{4}{3}$$\beta$$$\cos \alpha=\pm \sqrt{1-\sin^2\alpha}.$$\(\a… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p7?rev=1737476040&do=diff Question 8, Exercise 8.1 Solutions of Question 8 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin \alpha=\dfrac{3}{5}$$0<\alpha<\dfrac{\pi}{2}$$\cos \beta=\dfrac{12}{13}$$\dfrac{3 \pi}{2}<\beta<2 \pi$$\csc (\alpha+\beta)$$\sec (\alpha+\beta)$$\cot (\alpha+\beta)$$\sin \alpha=\dfrac{3}{5}$$0<\alpha<\dfrac{\pi}{2}$$\alpha$\begin{align… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p8?rev=1737476040&do=diff Question 9, Exercise 8.1 Solutions of Question 9 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\alpha$$\beta$$\sin \alpha=\dfrac{1}{\sqrt{2}}$$\cos \beta=-\dfrac{3}{5}$$\sin (\alpha \pm \beta)$$\sin \alpha=\dfrac{1}{\sqrt{2}}$$\alpha$$\cos \beta=-\dfrac{3}{5}$$\beta$$$\cos \alpha=\pm \sqrt{1-\sin^2\alpha}.$$$\alpha$$\cos$\begin{align*… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p11?rev=1737476040&do=diff Question 12, Exercise 8.1 Solutions of Question 12 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\alpha+\beta+\gamma=180^{\circ}$$\tan \alpha+\tan \beta+\tan \gamma=\tan \alpha \tan \beta \tan \gamma$$$\alpha+\beta+\gamma=180^{\circ}$$\begin{align*} & \alpha+\beta=180^{\circ}-\gamma \\ \implies & \tan(\alpha+\beta) = \tan(180^{\circ}-… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p12?rev=1737476040&do=diff Question 13, Exercise 8.1 Solutions of Question 13 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $r \sin (\theta+\phi)$$12 \sin \theta-5 \cos \theta$$12=r\cos \varphi $$-5=r\sin \varphi$\begin{align*} & (12)^2+(-5)^2=r^2 \cos^2\varphi+r^2 \sin^2 \varphi \\ \implies & 144+25={{r}^{2}}\left( {{\cos }^{2}}\varphi +{{\sin }^{2}}\varphi \r… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p2?rev=1737476040&do=diff Question 4 Exercise 8.2 Solutions of Question 4 of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin 2 \theta$$\cos 2 \theta$$\tan 2 \theta$$\sin \frac{\theta}{2}$$\cos \frac{\theta}{2}$$\tan \frac{\theta}{2}$$\cos \theta=\frac{3}{5}$$0<\theta<\frac{\pi}{2}$$\cos\theta=\dfrac{3}{5}$$0<\theta<\dfrac{\pi}{2}$$\theta$$$\sin\theta = \pm \sq… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p3?rev=1737476040&do=diff Question 5 Exercise 8.2 Solutions of Question 5 of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin \theta$$\cos \theta$$\tan \theta$$\sin 2 \theta=\frac{24}{25}, 2 \theta$$\sin 2\theta=\dfrac{24}{25}$$2\theta$$$\cos 2\theta = \pm \sqrt{1-\sin^2 2\theta}$$$2\theta$$\cos 2\theta$\begin{align*}\cos 2\theta & = - \sqrt{1-\sin^2 2\theta}\\… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p4?rev=1737476040&do=diff Question 6 Exercise 8.2 Solutions of Question 6 of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin 15^{\circ} \cos 15^{\circ}$$$\sin 2 \theta = 2\sin\theta \cos\theta$$$$\sin\theta \cos\theta = \frac{1}{2}\sin 2\theta$$$\theta = 15^{\circ}$\begin{align*} \sin 15^{\circ} \cos 15^{\circ} & = \frac{1}{2}\sin 2(15^{\circ}) \\ & \frac{1}{2… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p5?rev=1737476040&do=diff Question 7 Exercise 8.2 Solutions of Question 7 of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$\sin ^{2} \alpha \cos ^{2} \alpha$$\begin{align*} \sin ^{2} \alpha \cos ^{2} \alpha &= \left(\frac{1-\cos 2\alpha}{2} \right)\left(\frac{1+\cos 2\alpha}{2} \right)\\ &= \frac{1}{4}(1-\cos^2 2\alpha) \\ &=\frac{1}{4}\left(1-\frac{1+\cos 4\alp… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p7?rev=1737476040&do=diff Question 8(iv, v & vi) Exercise 8.2 Solutions of Question 8(iv, v & vi) of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\csc 2 \alpha=\dfrac{\tan \alpha+\cot \alpha}{2}$\begin{align*} RHS & = \dfrac{\tan \alpha+\cot \alpha}{2} \\ & = \dfrac{1}{2}\left(\frac{\sin\alpha}{\cos\alpha}+\frac{\cos\alpha}{\sin\alpha} \right)\\ \end{align*}$8 \… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p8?rev=1737476040&do=diff Question 8(vii, viii & ix) Exercise 8.2 Solutions of Question 8(vii, viii & ix) of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin 2 \theta=2 \cot \theta \sin ^{2} \theta$\begin{align*} RHS &= 2 \cot \theta \sin ^{2} \theta\\ &= 2 \frac{\cos \theta }{\sin \theta} \sin ^{2} \theta\\ &= 2 \cos \theta \sin\theta\\ &= \sin2 \theta\\ &=LH… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p9?rev=1737476040&do=diff Question 8(x, xi & xii) Exercise 8.2 Solutions of Question 8(x, xi & xii) of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sec 2 x=\dfrac{\cos x}{\cos x+\sin x}+\dfrac{\sin x}{\cos x-\sin x}$\begin{align*} RHS &= \dfrac{\cos x}{\cos x+\sin x}+\dfrac{\sin x}{\cos x-\sin x}\\ &=\dfrac{\cos x(\cos x-\sin x)+\sin x(\cos x+\sin x)}{(\cos x+\… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p10?rev=1737476040&do=diff Question 8(xiii, xiv & xv) Exercise 8.2 Solutions of Question 8(xiii, xiv & xv) of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\csc 2 \alpha-\cot 2 \alpha=\tan \alpha$\begin{align*} LHS &= \csc 2 \alpha-\cot 2 \alpha\\ &=\frac{1}{\sin 2 \alpha}- \frac{\cos2 \alpha}{\sin 2\alpha }\\ &=\frac{1-\cos2 \alpha}{\sin2 \alpha}\\ &= \frac{2\si… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-3-p2?rev=1737476040&do=diff Question 1(v, vi, vii & viii) Exercise 8.3 Solutions of Question 1(v, vi, vii & viii) of Exercise 8.3 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $ \sin(-u) \sin 5u$\begin{align*} &\sin(-u) \sin 5u \\ =& -\sin u \sin 5u \\ =& -\frac{1}{2}[\cos(u - 5u) - \cos(u + 5u)] \\ = &-\frac{1}{2}[\cos(-4u) - \cos(6u)] \\ =& \frac{1}{2}[\cos(6u) - \cos(4u) ] \e… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-3-p3?rev=1737476040&do=diff Question 1(ix, x & xi) Exercise 8.3 Solutions of Question 1(ix, x & xi) of Exercise 8.3 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 \sin 75{\circ} \sin 15{\circ}$\begin{align*} &\quad2 \sin 75^{\circ} \sin 15^{\circ} \\ &= \cos(75^{\circ} - 15^{\circ}) - \cos(75^{\circ} + 15^{\circ}) \\ &= \cos 60^{\circ} - \cos 90^{\circ} \\ \end{align*}$4 \sin … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-3-p4?rev=1737476040&do=diff Question 2(i, ii, iii, iv and v) Exercise 8.3 Solutions of Question 2(i, ii, iii, iv and v) of Exercise 8.3 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin 70^{\circ} + \sin 30^{\circ}$\begin{align*} & \quad \sin 70^{\circ} + \sin 30^{\circ} \\ & = 2 \sin \left(\frac{70+30}{2} \right) \cos \left(\frac{70-30}{2} \right) \\ & = 2 \sin \left(\frac{1… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-3-p6?rev=1737476040&do=diff Question 3(vi, vii, viii, ix & x) Exercise 8.3 Solutions of Question 3(vi, vii, viii, ix & x) of Exercise 8.3 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2\tan y \cos 3y= \sec y(\sin 4y-\sin 2y)$\begin{align*} LHS & = 2\tan y \cos 3y \\ & = 2 \cdot \frac{\sin y}{\cos y} \cos 3y \\ & = \sec y (2 \cos 3y \sin y) \\ & = \sec y \left(\sin (3y+y)-\sin (… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-3-p7?rev=1737476040&do=diff Question 3(xi, xii & xiii) Exercise 8.3 Solutions of Question 3(xi, xii & xiii) of Exercise 8.3 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2\cos2u \cos u-\sin 2u \sin u=2\cos^3 u$\begin{align*} LHS & = 2\cos 2u \cos u - \sin 2u \sin u \\ & = 2\left(\cos^2 u - \sin^2 u\right) \cos u - 2\sin u \cos u \sin u \\ & = 2\cos^3 u - 2\sin^2 u \cos u \\ & =… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/re-ex-p2?rev=1737476040&do=diff Question 2, Review Exercise Solutions of Question 2 of Review Exercise of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin \theta=\dfrac{3}{5}, \sin \phi=\dfrac{5}{13}$$\theta$$\phi$$\sin (\theta-\phi)$$\sin \theta=\dfrac{3}{5}$$\sin \phi=\dfrac{5}{13}$$\theta$$\phi$\begin{align*} \cos^2 \theta &= 1-\sin^2\theta\\ &= 1-\left(\frac{3}{5}\right)^2\\ & =… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/re-ex-p3?rev=1737476040&do=diff Question 3, Review Exercise Solutions of Question 3 of Review Exercise of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\frac{1}{\sqrt{2}}(\sin \beta+\cos \beta)$\begin{align*} &\frac{1}{\sqrt{2}}(\sin \beta+\cos \beta)\\ =& \sin \frac{\pi}{4}\sin \beta+\cos \frac{\pi}{4}\cos \beta\\ =& \cos(\beta -\frac{\pi}{4}) \end{align*}$\frac{1}{\sqrt{2}} \sin 75^… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/re-ex-p4?rev=1737476040&do=diff Question 4, Review Exercise Solutions of Question 4 of Review Exercise of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\dfrac{1+\tan 15^{\circ}}{1-\tan 15^{\circ}}$\begin{align*} &\frac{1+\tan 15^{\circ}}{1-\tan 15^{\circ}}\\ =&\frac{1+\tan 15^{\circ}}{1-1 \cdot \tan 15^{\circ}}\\ =&\frac{\tan 45^{\circ} + \tan 15^{\circ}}{1 - \tan 45^{\circ} \tan 15^{… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/re-ex-p5?rev=1737476040&do=diff Question 5 and 6, Review Exercise Solutions of Question 5 and 6 of Review Exercise of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\tan \theta$$\tan \left(\theta-45^{\circ}\right)=\frac{1}{3}$\begin{align*} & \frac{\tan \theta - \tan 45^{\circ}}{1 + \tan \theta \cdot \tan 45^{\circ}} =\frac{1}{3}\\ \implies & \frac{\tan \theta - 1}{1 + \tan \theta}= \f… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/re-ex-p6?rev=1737476040&do=diff Question 7, Review Exercise Solutions of Question 7 of Review Exercise of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\dfrac{4 \sin ^{2} \theta \cos \theta}{\cos 3 \theta+\cos \theta}=\tan 2 \theta \tan \theta$\begin{align*} LHS&=\frac{4 \sin^2 \theta \cos \theta}{\cos 3 \theta + \cos \theta}\\ &=\frac{4 \sin \theta\sin \theta \cos \theta}{4\cos^ 3 \t… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/re-ex-p7?rev=1737476040&do=diff Question 8, Review Exercise Solutions of Question 8 of Review Exercise of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$\sqrt{\frac{\cos \left(90^{\circ}+x\right) \sec (-x) \tan \left(180^{\circ}-x\right)}{\sec \left(360^{\circ}-x\right) \sin \left(180^{\circ}+x\right) \cot \left(90^{\circ}-x\right)}}=i .$$\begin{align*} LHS&= \sqrt{\frac{\cos \left(90… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/re-ex-p8?rev=1737476040&do=diff Question 9, Review Exercise Solutions of Question 9 of Review Exercise of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$\sqrt{\frac{\left(1-\tan ^{2} x \cos (-x) \cos \left(360^{\circ}-x\right)\right) \tan 45^{\circ}}{\left\{\sin 90^{\circ}-\sin \left(180^{\circ}+x\right)\right\}\left\{\sin 90^{\circ}-\cos \left(90^{\circ}-x\right)\right\}}}$$\begin{al… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p2?rev=1737476040&do=diff Question 2, Exercise 9.1 Solutions of Question 2 of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y=\dfrac{1}{4+3 \operatorname{Sin} \theta}$\begin{align*} -1 \leq \operatorname{Sin} \theta \leq 1 \end{align*}$3$\begin{align*} -3 \leq 3 \operatorname{Sin} \theta \leq 3 \end{align*}$4$\begin{align*} & 1 \leq 4+3 \operatorname{Sin} \theta \l… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p4?rev=1737476040&do=diff Question 4(i-iv), Exercise 9.1 Solutions of Question 4(i-iv) of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y=\sin x+x \cdot \cos x$$f(x)=\sin x+x \cdot \cos x$\begin{align*} f(-x) = \sin (-x) + (-x)\cdot \cos (-x) \end{align*}$\sin(-x)=-\sin x$$\cos (-x) = \cos x$\begin{align*} f(x) & = -\sin x - x \cdot \cos x \\ & = -(\sin x + x \cdot … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p5?rev=1737476040&do=diff Question 4(v-viii), Exercise 9.1 Solutions of Question 4(v-viii) of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y=\dfrac{\sin ^{2} x}{x+\tan x}$\[y = \frac{\sin^2 x}{x + \tan x}\]\begin{align*} y(-x) &= \frac{\big(-\sin x\big)^2}{-x - \tan x} \\ &= \frac{\sin^2 x}{-x - \tan x}\\ & = \frac{\sin^2 x}{-(x + \tan x)}\\ & = -\frac{\sin^2 x}{x +… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p6?rev=1737476040&do=diff Question 5(i-v), Exercise 9.1 Solutions of Question 5(i-v) of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y=2 \operatorname{Sin} x$$y=2 \operatorname{Cos} 3 x$$y=2 \operatorname{Tan} 2 x$$\mathrm{y}=\operatorname{Cos} \frac{\mathrm{x}}{2}$ text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p7?rev=1737476040&do=diff Question 5(vi-x), Exercise 9.1 Solutions of Question 5(vi-x) of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y=2 \operatorname{Sin} 3 x$$y=3 \operatorname{Cos} x$$y=\operatorname{Cos}^{2} x$$y=\operatorname{Sin}^{2} x$$y=\operatorname{Tan}^{2} x$$y=\operatorname{Sin} \frac{x}{2}$ text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p8?rev=1737476040&do=diff Question 6, Exercise 9.1 Solutions of Question 6 of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y=6 \sec(2 x-3)$$\sec$$2\pi$\begin{align*} 6 \sec(2 x-3) & = 6 \sec(2 x-3+2\pi) \\ & = 6 \sec(2(x+\pi)-3) \end{align*}$6 \sec(2 x-3)$$\pi$$y=\cos (5 x+4)$$\cos$$2\pi$\begin{align*} \cos (5 x+4) & = 6 \cos(5x+4+2\pi) \\ & = \cos\left(5\left(x+\fr… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p9?rev=1737476040&do=diff Question 7 & 8, Exercise 9.1 Solutions of Question 7 & 8 of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y=\operatorname{Sin} x$$y=\operatorname{Sin} 2 x$$[0,2 \pi]$$y=\operatorname{Cos} x$$y=\operatorname{Cos} 2 x$$[0,2 \pi]$ text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p10?rev=1737476040&do=diff Question 9, Exercise 9.1 Solutions of Question 9 of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin x=\cos x$$\cos x=x$$\sin x=x$$\tan x=x$ text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/re-ex-1-p2?rev=1737476040&do=diff Question 2 and 3,Review Exercise Solutions of Question 2 and 3 of Review Exercise of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/re-ex-1-p3?rev=1737476040&do=diff Question 4, Review Exercise Solutions of Question 4 of Review Exercise of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/re-ex-p3?rev=1737476040&do=diff Question 4, Review Exercise Solutions of Question 4 of Review Exercise of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/re-ex-p4?rev=1737476040&do=diff Question 5 and 6, Review Exercise Solutions of Question 5 and 6 of Review Exercise of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/re-ex-p5?rev=1737476040&do=diff Question 7, Review Exercise Solutions of Question 7 of Review Exercise of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/re-ex-p6?rev=1737476040&do=diff Question 8, Review Exercise Solutions of Question 8 of Review Exercise of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/re-ex-p7?rev=1737476040&do=diff Question 9, Review Exercise Solutions of Question 9 of Review Exercise of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/re-ex-p8?rev=1737476040&do=diff Question 10(i-v), Review Exercise Solutions of Question 10(i-v) of Review Exercise of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/re-ex-p9?rev=1737476040&do=diff Question 10(vi-x), Review Exercise Solutions of Question 10(vi-x) of Review Exercise of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/ppsc/ppsc-maths-2011?rev=1737476042&do=diff PPSC Paper 2011 (Lecturer in Mathematics) [PPSC Paper 2011 (Lecturer in Mathematics)] On this page, we have given question from old (past) paper of Lecturer in Mathematics conducted in year 2011. This is a MCQs paper and answers are given at the end of the paper. At the end of the PDF is also given to download. This paper is provided by Ms. $R$$x\in R$$x^2=x$$x^2=-x$$x^2=0$$x^2=1$$6$$8$$10$$4$$G$$H$$H$$G$$2$$4$$nZ$$Z$$n$$G$$24$$a$$a^{10}$$2$$12$$10$$V$$n$$V$$n+1$$n$$n-1$$v_1,v_2,v_3,....,v_r$$… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-2-p7?rev=1737476036&do=diff Question 8 and 9, Exercise 10.2 Solutions of Question 8 and 9 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${{\cos }^{4}}\theta $\begin{align}{{\cos}^{4}}\theta &={{\left( {{\cos }^{2}}\theta \right)}^{2}}\\ &={{\left( \dfrac{1+\cos 2\theta }{2} \right)}^{2}}\\ &=\dfrac{1+2\cos 2\theta +{{\cos }^{2}}2\theta }{4}\\… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-1-p1?rev=1737476038&do=diff Question 1 and 2 Exercise 4.1 Solutions of Question 1 and 2 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$2,4,6,8, \ldots ,50$$50 $$1,0,1,0,1, \ldots$$0$$1$$...,-4,0,4,8, \ldots, 60$$1,-\dfrac{1}{3}, \dfrac{1}{9},-\dfrac{1}{27}, \ldots,-\dfrac{1}{2187}$$a_n=\dfrac{n(n+1)}{2}$$$a_n=\dfrac{n(n+1)}{2}$$$n=1,$$$a_1=\dfrac{1(1+1)}{2}=1$$$n=2$$$a_2=\dfrac{2(2… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p13?rev=1737476038&do=diff Question 17 Exercise 4.2 Solutions of Question 17 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 17 $n$$7: 13$$n$$A_1, A_2, A_3, \ldots, A_n$$n$$5, A_1, A_2, A_3, \ldots, A_n, 32$$$a_1=5 \text{ and } a_{n+2}=32.$$$a_n=a_1+(n-1) d$$n$$n+2$\begin{align}a_{n+2}&=a_1+(n+2-1) d \\ & =a_1+(n+1) d \\ \implies 32&=5+(n+1)d \\ \implies (n+1)d&=32-5\\… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-4-p3?rev=1737476038&do=diff Question 4 Exercise 5.4 Solutions of Question 4 of Exercise 5.4 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4 $\sum_{k=1}^n \dfrac{1}{k^2+7 k+12}$\begin{align}S_n &=\sum_{k=1}^n \dfrac{1}{k^2+7 k+12} \\ & =\sum_{k=1}^n \dfrac{1}{(k+3)(k+4)}\end{align}$n^{\text {th }}$$$u_n=\dfrac{1}{(n+3)(n+4)}$$$$\dfrac{1}{(n+3)(n+4)}=\dfrac{A}{n+3}+\dfrac{B}{n+4}$$$A$$B$… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p1?rev=1737476038&do=diff Question 1 Exercise 7.2 Solutions of Question 1 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$(x^2-\dfrac{1}{y})^4$\begin{align}(x^2-\dfrac{1}{y})^4&=(x^2)^4+{ }^4 C_1(x^2)^3(-\dfrac{1}{y})+ \\ & { }^4 C_2(x^2)^2(-\dfrac{1}{y})^2+{ }^4 C_3(x^2)(-\dfrac{1}{y})^3 + { }^4 C_4(-\dfrac{1}{y})^4 \\ & =x^8- \dfrac{4x^6}{y}+\dfrac{6x^4}{y^2}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p1?rev=1737476039&do=diff Question 1 Exercise 7.3 Solutions of Question 1 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\frac{1}{2}$$$ \begin{aligned} & (1-x)^{\frac{1}{2}}=1+\frac{1}{2} x+ \\ & \frac{\frac{1}{2}\left(-\frac{1}{2}-1\right)}{2 !}(-x)^2 \end{aligned} $$$$ \begin{aligned} & +\frac{-\frac{1}{2}\left(-\frac{1}{2}-1\right)\left(-\frac{1}{2}-2\right… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-2-p7?rev=1737476039&do=diff Question 8 and 9, Exercise 10.2 Solutions of Question 8 and 9 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${{\cos }^{4}}\theta $\begin{align}{{\cos}^{4}}\theta &={{\left( {{\cos }^{2}}\theta \right)}^{2}}\\ &={{\left( \dfrac{1+\cos 2\theta }{2} \right)}^{2}}\\ &=\dfrac{1+2\cos 2\theta +{{\cos }^{2}}2\theta }{4}\\… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-3-p1?rev=1737476039&do=diff Question 1, Exercise 2.3 Solutions of Question 1 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left[\begin{array}{ccc}2 & 3 & 1 \\ 1 & -1 & 2 \\ 4 & 1 & 2\end{array}\right]$\begin{align*} A &= \left[\begin{array}{ccc}2 & 3 & 1 \\ 1 & -1 & 2 \\ 4 & 1 & 2\end{array}\right]\\ |A|&=2(-2-2)-3(2-8)+1(1+4)\\ \implies |A|&=-8+18+5\\ \implies |… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p1?rev=1737476039&do=diff Question 1 and 2, Exercise 4.5 Solutions of Question 1 and 2 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $16+16+16+\ldots$$a_1=16$$r=\dfrac{16}{16}=1$$r\neq 1$\begin{align*} &16+16+16+\ldots \text{ to 11 terms}\\ =&11(16) \\ =& 176 \end{align*}$75+15+3+...$$75+15+3+...$$a_1= 75$$r = \frac{15}{75} = \frac{1}{5}$$n = 10$$n$$$ S_n = \frac{a_1 \… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p1?rev=1737476040&do=diff Question 1, 2 and 3 Exercise 8.2 Solutions of Question 1, 2 and 3 of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $P(-3,4)$$\theta$$\theta$$\cos 2 \theta$$\sin 2 \theta$$2 \theta$$x=-3$$y=4$\begin{align*} r&= \sqrt{(-3)^2+4^2} \\ &=\sqrt{25} = 5. \end{align*}$$\sin\theta = \frac{4}{5} \text{ and } \cos\theta = -\frac{3}{5}.$$\begin{align… text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc?rev=1737476042&do=diff FSc Notes (Solutions), MCQs/Objective type questions, model papers and old/previous papers (of FBISE and BISE) given here, are useful for FSc Part 1 and Part 2 (HSSC). Text Book of Algebra and Trigonometry Class XI and Calculus and Analytic Geometry, MATHEMATICS 12, Punjab Text Book Board Lahore text/html 2025-01-21T18:32:03+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/home?rev=1737484323&do=diff Home Welcome to MathCity.org. Please browse the website by using navigation bar or search the website. “”“”“” More Quotes--- “”“”“” “” Updates * | Math 11 (NBF) | Notes of unit 08 for FSc/ICS part 1 mathematics by NBF has been added text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/software?rev=1737476042&do=diff Software On this page, we have listed document viewers, graphing tools, calculators and other useful software for Mathematics. As most of the visitors of our websites are using android mobiles, therefore first of all we are giving some apps for Android.$\LaTeX$$\LaTeX$$\LaTeX$$\LaTeX$ text/html 2025-01-21T16:13:54+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/atiq/sp23-mth321?rev=1737476034&do=diff MTH321: Real Analysis I (Spring 2023) ~~DISCUSSION~~ [Photo-illustration of Zeno's Paradox] At the end of this course the students will be able to understand the basic set theoretic statements and emphasize the proofs’ development of various statements by induction. Define the limit of, a function at a value, a sequence and the Cauchy criterion. Prove various theorems about limits of sequences and functions and emphasize the proofs’ development. Define continuity of a function and uniform con… text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/bsc/notes_of_calculus_with_analytic_geometry?rev=1737476035&do=diff Notes of Calculus with Analytic Geometry [Calculus with Analytic Geometry by Dr. S. M. Yusuf and Prof. Muhammad Amin] Calculus with Analytic Geometry by Dr. S. M. Yusuf and Prof. Muhammad Amin, published by Ilmi Kitab Khana, Lahore-Pakistan is one of the books studied widely in Bachelor and undergraduate classes inclduing different engineering programs. There are total of ten chapters. Solutions of the books are given in the chapters listed below. The only aim to publish the soltuions is to pro… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part2-ptb/derivatives-integration-formulas-rules-muzzammil-subhan?rev=1737476037&do=diff math_12 formula_pages muzzammil_subhan Derivatives, Integration Formulas & Rules This page contains all the important derivative and integration formulas & rules used in chapter 2 and 3 of FSc Part 2. This page is send by Muzzammil Subhan. [Derivative and Integration Formulas and Rules] [Download PDF] text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_trigonometric_review?rev=1737476036&do=diff Trigonometric Review Here the review of the formulas are given, which are used in Chapter 9 to 14 of Text Book of Algebra and Trigonometry Class XI, Punjab Text Book Board Lahore. This handout is very helpful to remember the formulas. All these formulas are given for real valued and defined trigonometric functions. A PDF file can be downloaded for high quality printing. We are very thankful to <div> <img src="../files/fsc/fsc_part1/fsc-trignometric-review.gif" title="FSc Trigonometric Review" a… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/kpk_fsc_part_1?rev=1737476036&do=diff FSc Part 1 (KPK Boards) AVAILABLE HERE [FSc Part 2 KPTP] Notes of FSc Part 1 of “A Textbook of Mathematics For Class XI” published by Khyber Pakhtunkhwa Textbook Board, Peshawar. We are posting the notes chapter-wise. These notes are shared as open educational resources. This page will be continuously updated.$P(z)$$(\sum)$$\sum n$$\sum n^2$$\sum n^3$$n$$n$$$\frac{a}{a(a+d)}+\frac{a}{(a+d)(a+2d)}+...$$$^nP_r$$^nC_r=\left(\begin{smallmatrix}n\\ r\end{smallmatrix} \right)=\frac{n!}{r!(n-r)!}$$P(… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/matric/9th_science?rev=1737476041&do=diff Mathematics 9 (Science Group) [Mathematics 9 (Science Group)] Mathematics 9 is written by Dr. Karamat H. Dar and Prof. Irfan-ul-Haq and this book is published by Carvan Book House, Lahore, Pakistan. This book consist of 302 pages and there are 17 units. Notes of Unit 1 and 3 are provided by $ka + kb + kc$$ac + ad + bc + bd$$a^2 + 2ab + b^2$$a^2 – b^2$$a^2 + 2ab + b^2 – c^2$$a^4 + a^2b^2 + b^4$$a^4 + 4b^4$$x^2 + px + q$$ax^2 + bx + c$$(ax^2 + bx + c) (ax2 + bx + d) + k$$(x + a) (x + b) (x + c) … text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-ptb/mcq-bank/ch01?rev=1737476037&do=diff MCQs: Ch 01 Number Systems High quality MCQs of Chapter 01 Number System of Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. The answers are given at the end of the page. MCQs * If $*$$A$$a, b \in A$$a+b \in A$$a-b \in A$$a \times b \in A$$a * b \in A$$z=(1,3)$$z^{-1}= $$(\displaystyle{\frac{1}{10}},\displaystyle{\frac{3}{10}})$$(-\displaystyle{\frac{1}{10}},\displaystyle{\frac{3}{10}})$$(\displaystyle{\frac{1}{10}},-\display… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-ptb/mcq-bank/ch02?rev=1737476037&do=diff MCQs: Ch 02 Sets, Functions and Groups High quality MCQs of Chapter 02 Sets, Functions and Groups of Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. The answers are given at the end of the page.$\forall$$\wedge$$<$$\in$$A$$B$$A\cap B=\phi$$A=B$$B\subseteq A$$A \subseteq B$$A$$B$$A-B \neq \phi$$A=B$$A \subseteq B$$B\subseteq A$$A$$B$$A\cap B=A$$B \subseteq A$$A\cap B=\phi$$A\subseteq B$$B\subseteq A$$A=\phi$$A \cup B=A$$A \cap B=… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-ptb/mcq-bank/ch04?rev=1737476037&do=diff MCQs: Ch 04 Quadratic Equations High quality MCQs of Chapter 01 Number System of Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. The answers are given at the end of the page. MCQs $ax^2+bx+c=0$$ax^2+bx+c=0$$b \neq 0$$c \neq 0$$a \neq 0$$x$$ax^2+bx+c$$ax^2+bx+c=0$$\{a,b\}$$ax^2+bx+c=0$$a\neq 0$$x= \frac{b \pm \sqrt{b^2-4ac}}{a}$$x= \frac{-b \pm \sqrt{b^2+4ac}}{2a}$$x= \frac{-b \pm \sqrt{4ac-b^2}}{2a}$$x= \frac{-b \pm \sqrt{b^2-… text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_old_papers/fbise_annual_2009?rev=1737476035&do=diff FBISE Annual 2009 * FSc part 1 (HSSC-I) mathematics paper conducted by Federal Board of Intermediate and Secondary Education (FBISE), Islamabad has been analyse on this web page with the help of chart. Three type of chart are given in which one includes bar chart between chapters and marks, 2nd one include relation between algebraic and trigonometric portion and 3rd one contains pie chart which show the portion of questions from exercises to non-exercise question from book a <div> <center> </… text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_old_papers/fbise_annual_2011?rev=1737476035&do=diff FBISE Annual 2011 * FSc part 1 (HSSC-I) mathematics paper conducted by Federal Board of Intermediate and Secondary Education (FBISE), Islamabad has been analyse on this web page with the help of chart. Three type of chart are given in which one includes bar chart between chapters and marks, 2nd one include relation between algebraic and trigonometric portion and 3rd one contains pie chart which show the portion of questions from exercises to non-exercise question from book a <div> <center> </… text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_old_papers/fbise_annual_2012?rev=1737476035&do=diff FBISE Annual 2012 * FSc part 1 (HSSC-I) mathematics paper conducted by Federal Board of Intermediate and Secondary Education (FBISE), Islamabad has been analyse on this web page with the help of chart. Three type of chart are given in which one includes bar chart between chapters and marks, 2nd one include relation between algebraic and trigonometric portion and 3rd one contains pie chart which show the portion of questions from exercises to non-exercise question from book a <div> <center> </… text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch02?rev=1737476035&do=diff Chapter 02: Sets, Functions and Groups Notes (Solutions) of Chapter 02: Sets, Functions and Groups, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. [Chapter 02: Sets, Functions and Groups] Contents & summary * Introduction$p\leftrightarrow q$ text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch03?rev=1737476035&do=diff Chapter 03: Matrices and Determinants [Chapter 03: Matrices and Determinants] Notes (Solutions) of Chapter 03: Matrices and Determinants, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary * Introduction$2\times2$$2\times2$$2\times2$$n\geq 3$$n\geq 3$ text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch05?rev=1737476035&do=diff Chapter 05: Partial Fractions [Chapter 05: Partial Fractions] Notes (Solutions) of Chapter 05: Partial Fractions, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary * Introduction * Rational Fraction$\frac {P(x)}{Q(x)}$ text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch06?rev=1737476035&do=diff Chapter 06: Sequences and Series [Chapter 06: Sequences and Series] Notes (Solutions) of Chapter 06: Sequences and Series, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary * Introduction * Types of Sequences$l,m,n$$p$$q$$r$$$l(q-r)+m(r-p)+n(p-q)=0$$$a_1$$d$$$\begin{align}l=a_1+(p-1)d,\\ m=a_1+(q-1)d,\\ n=a_1+(r-1)d.\end{align}$$ Now $$\begin{align}L.H.S &= l(q-r)+m(r-p)+n(p-q)\\ &= lq-lr+mr-mp+np-nq\\ &=… text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch07?rev=1737476035&do=diff Chapter 07: Permutation, Combination and Probability [Chapter 07: Permutation , Combination and Probability] Notes (Solutions) of Chapter 07: Permutation , Combination and Probability, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch08?rev=1737476036&do=diff Chapter 08: Mathematical Induction and Binomial Theorem [Chapter 08 Mathematical Induction and Binomial Theorem] Notes (Solutions) of Chapter 08: Mathematical Induction and Binomial Theorem, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore.$(a+x)^n$$(a+x)^n$ text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch09?rev=1737476036&do=diff Chapter 09: Fundamentals of Trigonometry [Chapter 09: Fundamentals of Trigonometry] Notes (Solutions) of Chapter 09: Fundamentals of Trigonometry, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. This chapter has four exercise and solutions of those exercises are given below which can be downloaded in PDF format or can be viewed online.$D^\circ M'S''$$45^\circ , 30^\circ , 60^\circ$$0^\circ , 90^\circ , 180^\circ , 270^\circ , 36… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch10?rev=1737476036&do=diff Chapter 10: Trigonometric Identities [Chapter 10: Trigonometric Identities] Notes (Solutions) of Chapter 10: Trigonometric Identities, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. There are four exercise in this chapter. text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch11?rev=1737476036&do=diff Chapter 11: Trigonometric Functions and their Graphs [Chapter 11: Trigonometric Functions and their Graphs] Notes (Solutions) of Chapter 11: Trigonometric Functions and their Graphs, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary $ y = \sin x$$-2\pi \hbox{ to } 2\pi$$ y = \cos x$$-2\pi \hbox{ to } 2\pi$$ y = \tan x$$-\pi \hbox{ to } \pi$$ y = \cot x$$-2\pi \hbox{ to } \pi$$ y = \sec x$$-2\pi \hbox{ to } 2\pi… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch12?rev=1737476036&do=diff Chapter 12: Application of Trigonometry [Chapter 12: Application of Trigonometry] Notes (Solutions) of Chapter 12: Application of Trigonometry, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary * Introduction text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch13?rev=1737476036&do=diff Chapter 13: Inverse Trigonometric Functions [Chapter 13: Inverse Trigonometric Functions] Notes (Solutions) of Chapter 13: Inverse Trigonometric Functions, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary * ${\sin ^{ - 1}}A + {\sin ^{ - 1}}B = {\sin ^{ - 1}}\left( {A\sqrt {1 - {B^2}} + B\sqrt {1 - {A^2}} } \right)$${\sin ^{ - 1}}A - {\sin ^{ - 1}}B = {\sin ^{ - 1}}\left( {A\sqrt {1 - {B^2}} - B\sqrt {1 - {… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch14?rev=1737476036&do=diff Chapter 14: Solutions of Trigonometric Equation [Chapter 14: Solutions of Trigonometric Equation] Notes (Solutions) of Chapter 14: Solutions of Trigonometric Equation, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary ${\sin ^{ - 1}}A + {\sin ^{ - 1}}B = {\sin ^{ - 1}}\left( {A\sqrt {1 - {B^2}} + B\sqrt {1 - {A^2}} } \right)$${\sin ^{ - 1}}A - {\sin ^{ - 1}}B = {\sin ^{ - 1}}\left( {A\sqrt {1 - {B^2}} - B\sqr… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/kpk-fsc-part1-km/view?rev=1737476036&do=diff FSc-I Mathematics KPK: View Online AVAILABLE HERE On this page one can view the PDF of solutions of the A Textbook of Mathematics For Class XI” published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. These notes are written by khalid. Link to PDF is given at the end of preview. text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-1-p1?rev=1737476036&do=diff Question 1, Exercise 1.1 Solutions of Question 1 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) ${{i}^{9}}+{{i}^{19}}$\begin{align}{{i}^{9}}+{{i}^{19}}&=i\cdot{{i}^{8}}+i\cdot{{i}^{18}}\\ &=i\cdot{{\left( {{i}^{2}} \right)}^{4}}+i\cdot{{\left( {{i}^{2}} \right)}^{9}}\\ &=i\cdot{{\left( -1 \right)}^{4}}+i\cdot{{\left( -1 \right)}^{9}}\\ &=i\cdo… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-1-p9?rev=1737476036&do=diff Question 11, Exercise 1.1 Solutions of Question 11 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 11(i) ${{z}_{1}}=2-i$${{z}_{2}}=-2+i$${\rm Re}\left( \dfrac{{{z}_{1}}{{z}_{2}}}{\overline{{{z}_{1}}}} \right)$$z_1=2-i$$z_2=-2+i$$\overline{z_1}=2+i$\begin{align} z_1 z_2&=(2-i)(-2+i)\\ &=-4+1+2i+2i\\ &=-3+4i \end{align}\begin{align} \dfrac{z_1 z_2}{\… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-2-p1?rev=1737476036&do=diff Question 1, Exercise 1.2 Solutions of Question 1 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1 If ${{z}_{1}}=2+i$${{z}_{2}}=1-i$${{z}_{1}}=2+i$${{z}_{2}}=1-i$$$z_1+z_2=z_2+z_1.$$\begin{align}z_1+z_2&=(2+i)+(1-i)\\ &=3 \ldots (i) \end{align}\begin{align} z_2+z_1&=(1-i)+(2+i)\\ &=3 \ldots (ii)\end{align}$$z_1 z_2=z_2 z_1.$$\begin{align}z_1 z_2 … text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-2-p8?rev=1737476036&do=diff Question 9, Exercise 1.2 Solutions of Question 9 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9(i) $z=3+2i,$$-|z|\leq \operatorname{Re}\left( z \right)\leq |z|$$z=3+2i$$|z|=\sqrt{9+4}=\sqrt{13}$${\rm Re}z=3=\sqrt{9}$\begin{align} &-\sqrt{13} \leq \sqrt{9} \leq \sqrt{13}\\ \implies &-|z|\leq \operatorname{Re}\left( z \right)\leq |z|\end{align}$z=3… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-3-p1?rev=1737476036&do=diff Question 1, Exercise 1.3 Solutions of Question 1 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) \begin{align}&z-4w=3i\\ &2z+3w=11-5i\end{align}\begin{align}z-4w&=3i …(i)\\ 2z+3w&=11-5i …(ii)\end{align}$2$\begin{align}2z-8w&=6i …(iii)\end{align}\[\begin{array}{cccc} 2z&-8w&=6i \\ \mathop+\limits_{-}2z&\mathop+\limits_{-}3w&=\mathop-\limit… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-3-p5?rev=1737476036&do=diff Question 6, Exercise 1.3 Solutions of Question 6 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6(i) ${{z}^{4}}+{{z}^{2}}+1=0$\begin{align}{{z}^{4}}+{{z}^{2}}+1&=0\\ {{z}^{4}}+2\left( \dfrac{1}{2} \right){{z}^{2}}+\dfrac{1}{4}-\dfrac{1}{4}+1&=0\\ {{\left( {{z}^{2}}+\dfrac{1}{2} \right)}^{2}}+\dfrac{4-1}{4}&=0\\ {{\left( {{z}^{2}}+\dfrac{1}{2} \righ… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/review-ex-1-p1?rev=1737476036&do=diff Question 1, Review Exercise 1 Solutions of Question 1 of Review Exercise 1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1 ${{\left( \dfrac{2i}{1+i} \right)}^{2}}$$i$$2i$$1-i$$i+1$$\dfrac{5+2i}{4-3i}$$-\dfrac{7}{25}+\dfrac{26}{25}i$$\dfrac{5}{4}-\dfrac{2}{3}i$$\dfrac{14}{25}+\dfrac{23}{25}i$$\dfrac{26}{7}+\dfrac{23}{7}i$${{i}^{57}}+\frac{1}{{{i}^{25}}}$$0$$2i$$-2… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/review-ex-1-p4?rev=1737476036&do=diff Question 6, 7 & 8, Review Exercise 1 Solutions of Question 6, 7 & 8 of Review Exercise 1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{1}{3+4i}$\begin{align}\dfrac{1}{3+4i}&=\dfrac{1}{3+4i}\times \dfrac{3-4i}{3-4i}\\ &=\dfrac{3-4i}{9+16}\\ &=\dfrac{3-4i}{25}\\ &=\dfrac{3}{25}-\dfrac{4i}{25}\end{align}$\dfrac{3i+2}{3-2i}$\begin{align}\dfrac{3i+2}{3-2i}\\ \dfrac{3i+2}… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-1-p1?rev=1737476036&do=diff Question 1, Exercise 10.1 Solutions of Question 1 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. There are four parts in Question 1.$\sin {{37}^{\circ }}\cos {{22}^{\circ }}+\cos {{37}^{\circ }}\sin {{22}^{\circ }}$\begin{align} \sin (\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta, \end{align}\begin{a… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-2-p1?rev=1737476036&do=diff Question 1, Exercise 10.2 Solutions of Question 1 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. There are four parts in Question 1.$\sin 2\theta ,\,\,\cos 2\theta$$\tan 2\theta$$\tan \theta =-\dfrac{1}{5}$$\theta$$\sin \theta =\dfrac{1}{\sqrt{26}}$$\cos \theta =\dfrac{-5}{\sqrt{26}}$\begin{align}\sin 2\theta &=2\sin… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-3-p1?rev=1737476036&do=diff Question 1, Exercise 10.3 Solutions of Question 1 of Exercise 10.3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. There are four parts in Question 1.$2\sin 6x\sin x$$$-2\sin \alpha \sin \beta =\cos (\alpha +\beta )-\cos (\alpha -\beta ).$$$\alpha =6x$$\beta =x$\begin{align}-\,2\sin 6x\sin x&=\cos (6x+x)-\cos (6x-x)\\ &=\cos 7x-\cos x… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-3-p5?rev=1737476037&do=diff Question 5, Exercise 10.3 Solutions of Question 5 of Exercise 10.3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos {{20}^{\circ }}\cos {{40}^{\circ }}\cos {{60}^{\circ }}\cos {{80}^{\circ }}=\dfrac{1}{16}$$2\cos \alpha \cos \beta =\cos \left( \alpha +\beta \right)+\cos \left( \alpha -\beta \right)$\begin{align}L.H.S.&=\cos {{20… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/re-ex10-p1?rev=1737476037&do=diff Question 1, Review Exercise 10 Solutions of Question 1 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos {{50}^{\circ }}5{0}'\cos {{9}^{\circ }}1{0}'-\sin {{50}^{\circ }}5{0}'\sin {{9}^{\circ }}1{0}'=$$0$$\dfrac{1}{2}$$1$$\dfrac{\sqrt{3}}{2}$$\tan {{15}^{\circ }}=2-\sqrt{3}$${{\cot }^{2}}{{75}^{\circ }}$$7+\sq… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/re-ex10-p5?rev=1737476037&do=diff Question 8 & 9, Review Exercise 10 Solutions of Question 8 & 9 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin \left( \dfrac{\pi }{4}-\theta \right)\sin \left( \dfrac{\pi }{4}+\theta \right)=\dfrac{1}{2}\cos 2\theta $$2\sin \alpha \sin \beta =\cos \left( \alpha -\beta \right)-\cos \left( \alpha +\beta \r… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch07/viewer?rev=1737476036&do=diff View Online (Solutions of Chapter 07) Notes (Solutions) of Chapter 07: Permutation , Combination and Probability, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Here is the list of all exercises of Chapter 07 text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch10/viewer?rev=1737476036&do=diff View Online (Solutions of Chapter 10) Notes (Solutions) of Chapter 10: Trigonometric Identities, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. There are total of four exercise in this chapter. text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch11/viewer?rev=1737476036&do=diff View Online (Solutions of Chapter 11) Notes (Solutions) of Chapter 11: Trigonometric Functions and their Graphs, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. There are two exercise in this chapter. text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch12/viewer?rev=1737476036&do=diff View Online (Solutions of Chapter 12) Notes (Solutions) of Chapter 12: Application of Trigonometry, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. There are eight exercises in this chapter. text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch13/viewer?rev=1737476036&do=diff View Online (Solutions of Chapter 13) Notes (Solutions) of Chapter 13: Inverse Trigonometric Functions, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. In this chpater, there are two exercise. text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch14/viewer?rev=1737476036&do=diff View Online (Solutions of Chapter 14) Notes (Solutions) of Chapter 14: Solutions of Trigonometric Equation of Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. In this chapter there is only one exercise. text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-1-p1?rev=1737476037&do=diff Question 1, Exercise 1.1 Solutions of Question 1 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) ${{i}^{9}}+{{i}^{19}}$\begin{align}{{i}^{9}}+{{i}^{19}}&=i\cdot{{i}^{8}}+i\cdot{{i}^{18}}\\ &=i\cdot{{\left( {{i}^{2}} \right)}^{4}}+i\cdot{{\left( {{i}^{2}} \right)}^{9}}\\ &=i\cdot{{\left( -1 \right)}^{4}}+i\cdot{{\left( -1 \right)}^{9}}\\ &=i\cdo… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-1-p9?rev=1737476037&do=diff Question 11, Exercise 1.1 Solutions of Question 11 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 11(i) ${{z}_{1}}=2-i$${{z}_{2}}=-2+i$${\rm Re}\left( \dfrac{{{z}_{1}}{{z}_{2}}}{\overline{{{z}_{1}}}} \right)$$z_1=2-i$$z_2=-2+i$$\overline{z_1}=2+i$\begin{align} z_1 z_2&=(2-i)(-2+i)\\ &=-4+1+2i+2i\\ &=-3+4i \end{align}\begin{align} \dfrac{z_1 z_2}{\… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-2-p1?rev=1737476037&do=diff Question 1, Exercise 1.2 Solutions of Question 1 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1 If ${{z}_{1}}=2+i$${{z}_{2}}=1-i$${{z}_{1}}=2+i$${{z}_{2}}=1-i$$$z_1+z_2=z_2+z_1.$$\begin{align}z_1+z_2&=(2+i)+(1-i)\\ &=3 \ldots (i) \end{align}\begin{align} z_2+z_1&=(1-i)+(2+i)\\ &=3 \ldots (ii)\end{align}$$z_1 z_2=z_2 z_1.$$\begin{align}z_1 z_2 … text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-2-p8?rev=1737476037&do=diff Question 9, Exercise 1.2 Solutions of Question 9 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9(i) $z=3+2i,$$-|z|\leq \operatorname{Re}\left( z \right)\leq |z|$$z=3+2i$$|z|=\sqrt{9+4}=\sqrt{13}$${\rm Re}z=3=\sqrt{9}$\begin{align} &-\sqrt{13} \leq \sqrt{9} \leq \sqrt{13}\\ \implies &-|z|\leq \operatorname{Re}\left( z \right)\leq |z|\end{align}$z=3… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-3-p1?rev=1737476037&do=diff Question 1, Exercise 1.3 Solutions of Question 1 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) \begin{align}&z-4w=3i\\ &2z+3w=11-5i\end{align}\begin{align}z-4w&=3i …(i)\\ 2z+3w&=11-5i …(ii)\end{align}$2$\begin{align}2z-8w&=6i …(iii)\end{align}\[\begin{array}{cccc} 2z&-8w&=6i \\ \mathop+\limits_{-}2z&\mathop+\limits_{-}3w&=\mathop-\limit… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/review-ex-1-p1?rev=1737476037&do=diff Question 1, Review Exercise 1 Solutions of Question 1 of Review Exercise 1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1 ${{\left( \dfrac{2i}{1+i} \right)}^{2}}$$i$$2i$$1-i$$i+1$$\dfrac{5+2i}{4-3i}$$-\dfrac{7}{25}+\dfrac{26}{25}i$$\dfrac{5}{4}-\dfrac{2}{3}i$$\dfrac{14}{25}+\dfrac{23}{25}i$$\dfrac{26}{7}+\dfrac{23}{7}i$${{i}^{57}}+\frac{1}{{{i}^{25}}}$$0$$2i$$-2… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/review-ex-1-p4?rev=1737476037&do=diff Question 6, 7 & 8, Review Exercise 1 Solutions of Question 6, 7 & 8 of Review Exercise 1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{1}{3+4i}$$$z=\dfrac{1}{3+4i}.$$\begin{align}z&=\dfrac{1}{3+4i}\times \dfrac{3-4i}{3-4i}\\ &=\dfrac{3-4i}{9+16}\\ &=\dfrac{3-4i}{25}\\ &=\dfrac{3}{25}-\dfrac{4}{25}i\end{align}$$\bar{z}=\dfrac{3}{25}+\dfrac{4}{25}i.$$$\dfrac{3i+2}{3-2… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p1?rev=1737476037&do=diff Question 1, Exercise 2.1 Solutions of Question 1 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) $$\left[ \begin{matrix} 1 & 2 & 4 \\ \end{matrix} \right] \left[ \begin{matrix} 1 & 0 & 2 \\ 2 & 0 & 1 \\ 0 & 1 & 2 \\ \end{matrix} \right] \left[ \begin{matrix} 2 \\ 4 \\ 6 \\ \end{matrix} \right]$$\begin{align}&\left[ \begin{matri… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p12?rev=1737476037&do=diff Question 13, Exercise 2.1 Solutions of Question 13 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 13(i) $A$$3$$A+A^t$$$A=\left[ \begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{matrix} \right]$$$$A^t=\left[ \begin{matrix} a_{11} & a_{21} & a_{31} \\ a_{12} & a_{22} … text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p1?rev=1737476037&do=diff Question 1, Exercise 2.2 Solutions of Question 1 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1 $A=\begin{bmatrix}1 & 3 & 1 \\-1 & 2 & 0 \\2 & 0 & -2 \end{bmatrix}$$A_{11},A_{21},A_{23},A_{31},A_{32},A_{33}.$$|A|.$$$A=\left[ \begin{matrix} 1 & 3 & 1 \\ -1 & 2 & 0 \\ 2 & 0 & -2 \\ \end{matrix} \right]$$$${{A}_{11}}={{\left(… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p15?rev=1737476037&do=diff Question 19, Exercise 2.2 Solutions of Question 19 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 19 $A=\begin{bmatrix}2 & 3 \\-1 & 1\end{bmatrix}$$( A^{-1})^t=( A^t)^{-1}$$$A=\left[ \begin{matrix} 2 & 3 \\ -1 & 1 \\ \end{matrix} \right]$$$$A^t=\left[ \begin{matrix} 2 & -1 \\ 3 & 1 \\ \end{matrix} \right]$$$$|A^t|=5$$$$Ad… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-2-p1?rev=1737476037&do=diff Question 1, Exercise 3.2 Solutions of Question 1 of Exercise 3.2 of Unit 03: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question.1(i) $\vec{a}=3\hat{i}-5\hat{j}$$\vec{b}=-2\hat{i}+3\hat{j}$$\vec{a}+2\vec{b}$\begin{align}\vec{a}+2\vec{b}&=3\hat{i}-5\hat{j}+2(-2\hat{i}+3\hat{j})\\ &=3\hat{i}-5\hat{j}-4\hat{i}+6\hat{j}\\ &=-\hat{i}+\hat{j}\end{align}$\vec{a}=3\hat{i}-5\hat{… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-3-p1?rev=1737476037&do=diff Question 1, Exercise 3.3 Solutions of Question 1 of Exercise 3.3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) If $\vec{a}=3 \hat{i}+4 \hat{j}-\hat{k}$, $\vec{b}=\hat{i}-\hat{j}+3 \hat{k}$ and $\vec{c}=2\hat{i}+\hat{j}-5 \hat{k}$$\vec{a}\cdot \vec{b}$\begin{align}\vec{a} \cdot \vec{b}&=(3 \hat{i}+4 \hat{j}-\hat{k}) \cdot(\hat{i}-\hat{j}+3 \hat{k})\\ \Rightarrow &=(… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-4-p1?rev=1737476037&do=diff Question 1 Exercise 3.4 Solutions of Question 1 of Exercise 3.4 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) Find the cross product $\hat{j} \times(2 \hat{j}+3 \hat{k})$\begin{align}\vec{a}=\hat{j}&=0 \hat{i}+\hat{j}+0 \hat{k}\\ \vec{b}&=0 \hat{i}+2 \hat{j}-3 \hat{k}\\ \vec{a} \times \vec{b}&=\hat{j} \times(2 \hat{j}+3 \hat{k})\\ &=\left|\begin{array}{lll}\hat{i}… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/review-ex3-p1?rev=1737476038&do=diff Question 1 Review Exercise 3 Solutions of Question 1 of Review Exercise 3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1 $\hat{i} \cdot(\hat{j} \times \hat{k})+\hat{j} \cdot(\hat{i} \times \hat{k})+\hat{k} \cdot(\hat{i} \times \hat{j})$$0$$1$$1$$3$$3 \hat{i}+5 \hat{j}+2 \hat{k}$$2 \hat{i}-3 \hat{j}-5 \hat{k}$$5 \hat{i}+2 \hat{j}-3 \hat{k}$$\hat{i}-2 \hat{i}+\hat{j}+3 \… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/review-ex3-p6?rev=1737476038&do=diff Question 10 Review Exercise 3 Solutions of Question 10 of Review Exercise 3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 10(i) $A B C$$|\vec{a}|^2=|\vec{b}|^2+|\vec{c}|^2 -2|\vec{b}|| \vec{c}| \cos A$$A B C$$\vec{a}, \vec{b}$$\vec{c}$\begin{align} \vec{b}&=\vec{a}+\vec{c} \\ \Rightarrow \vec{a}&=\vec{b}-\vec{c} \\ \Rightarrow \vec{a} \cdot \vec{a}&=(\vec{b}-\vec{c}) \cd… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p1?rev=1737476038&do=diff Question 1 and 2 Exercise 4.2 Solutions of Question 1 and 2 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$15$$2,5,8, \ldots$$a_1=2$$d=5-2=3$$n=15$$$a_n=a_1+(n-1) d$$\begin{align}a_{15}&=2+(15-1) 3 \\ &=2+42=44 \end{align}$44$$a_1=8$$a_{21}=108$$$a_n=a_1+(n-1) d.$$\begin{align} &a_{21}=8+(21-1) d \\ \implies &108=8+20 d\\ \implies &20 d=108-8=100 \\ \imp… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-3-p1?rev=1737476038&do=diff Question 1 Exercise 4.3 Solutions of Question 1 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) $9,7,5,3, \ldots$$a_1$$d$\begin{align}&a_1=9 \\ &d=7-9=-2 \\ &n=20. \end{align}\begin{align}&a_n=a_1+(n-1)d \\ \implies &a_20=9+(20-1)(-2)=-29. \end{align}$S_n$$n$\begin{align} S_n&=\dfrac{n}{2}[a_1+a_n], \\ \implies S_{20}&=\dfrac{20}{2}[9-29] … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-4-p1?rev=1737476038&do=diff Question 1 Exercise 4.4 Solutions of Question 1 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question(i) $a_1=5, \quad r=3$$a_1, a_1 r, a_1 r^2, a_1 r^3, a_1 r^4, \ldots$$a_1=5 ; r=3$\begin{align}&5,5.3,5.3^2, 5.3^3, 5.3^4, \ldots\\ \Rightarrow &5,15,45,135,405, \ldots\end{align}$a_1=8, \quad r=-\dfrac{1}{2}$$a_1, a_1 r, a_1 r^2, a_1 r^3, a_1 r^4, \ld… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-4-p9?rev=1737476038&do=diff Question 12 Exercise 4.4 Solutions of Question 12 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 12 $n, . \dfrac{a^{n+1}+b^{n+1}}{a^n+b^n}$$a$$b$$a$$b$$\dfrac{a^{n+1}+b^{n-1}}{a^n+b^n}$$a$$b$\begin{align}\dfrac{a^{n+1}+b^{n+1}}{a^n+b^n}&=\sqrt{a b}\quad \because G \cdot M=\sqrt{a b} \\ \Rightarrow \dfrac{a^{n+1}+b^{n+1}}{a^n+b^n}&=a^{\dfrac{1}{… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-1-p6?rev=1737476038&do=diff Question 9 Exercise 5.1 Solutions of Question 9 of Exercise 5.1 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9(i) $n$$n$$n$\begin{align} & T_n=n^2(2 n+3)=2 n^3+3 n^2 \\ & \Rightarrow T_j=2 j^3+3 j^2\end{align}\begin{align} & \sum_{j=1}^n T_j=2 \sum_{j=1}^n j^3+3 \sum_{j=1}^n j^2 \\ & =2(\dfrac{n(n+1)}{2})^2+3 \dfrac{n(n+1)(2 n+1)}{6} \\ & =\dfrac{n(n+1)}{2}… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-2-p1?rev=1737476038&do=diff Question 1 Exercise 5.2 Solutions of Question 1 of Exercise 5.2 of Unit 05: Mascellaneous series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) $n$$1.2+2.2^2+3.2^3+4.2^4+\ldots$\begin{align} & S_n=1.2+2.2^2+3 \cdot 2^3+4 \cdot 2^4+\ldots +n \cdot 2^n....(i) \\ & 2 S_n=1.2^2+2.2^3+3.2^4+4.2^5+\ldots +n \cdot 2^n....(ii)\end{align}\begin{align} (1-2) S_n&=1 \cdot 2+(2-1) 2^2+(3-2) 2^2+(4-… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-2-p3?rev=1737476038&do=diff Question 4 & 5 Exercise 5.2 Solutions of Question 4 & 5 of Exercise 5.2 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4 $5+\dfrac{7}{3}+\dfrac{9}{3^2}+\dfrac{11}{3^3}+\ldots$\begin{align} & S_{\infty}=5+\dfrac{7}{3}+\dfrac{9}{3^2}+\dfrac{11}{3^3}+\ldots.(i) \\ & \dfrac{1}{3} S_{\infty}=\dfrac{5}{3}+\dfrac{7}{3^2}+\dfrac{9}{3^2}+\dfrac{11}{3^3}+\ldots.(ii) \e… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/re-ex5-p1?rev=1737476038&do=diff Question 1 Review Exercise 5 Solutions of Question 1 of Review Exercise 5 of Unit 05: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1 $t_n=6 n+5$$t_{n+1}=$$6 n-1$$6 n+11$$6 n+6$$6 n-5$$1+\dfrac{2}{3}+\dfrac{6}{3^2}+\dfrac{10}{3^3}+\dfrac{14}{3^4}+\ldots$$6$$2$$3$$4$$1+2.2+3.2^2+\cdots+100.2^{\prime \prime}$$99.2^{100}$$100.2^{100}$$99.2^{100}+1$$1000.2^{100}$$n^{t h}$$1.2+2.3+3.4+\… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-1-p1?rev=1737476038&do=diff Question 1 and 2 Exercise 6.1 Solutions of Question 1 and 2 of Exercise 6.1 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{10 !}{3 ! .3 ! \cdot 4 !}$\begin{align}\dfrac{10 !}{3 ! \cdot 3 ! \cdot 4 !}&=\dfrac{10.9 .8 \cdot 7 \cdot 6 \cdot 5.4 !}{3 ! \cdot 3 ! \cdot 4 !}\\ &=\dfrac{10.9 .8 .7 .5}{3.2 .1}\\ &=4200 \end{align}$\dfrac{3 !+4 !}{5 !-… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-1-p3?rev=1737476038&do=diff Question 5 Exercise 6.1 Solutions of Question 5 of Exercise 6.1 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{(2 n) !}{n !}=2^n(1.3 .5 \ldots(2 n-1))$\begin{align}\dfrac{(2 n) !}{n !}&=\dfrac{1}{n !}[(2 n)(2 n-1)(2 n-2) \\ &=(2 n-3)(2 n-4)(2 n-5) \ldots(2 n-(2 n-4))\\ &(2 n-(2 n-3))(2 n-(2 n-2))(2 n-(2 n-1))]\end{align}$2 n$\begin{align}\dfra… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-1-p5?rev=1737476038&do=diff Question 5 Exercise 6.1 Solutions of Question 5 of Exercise 6.1 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p10?rev=1737476038&do=diff Question 14 and 15 Exercise 6.2 Solutions of Question 14 and 15 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$\dfrac{(n-1) !}{2}=\dfrac{(5-1) !}{2}=\dfrac{24}{2}=12 $$$7$$7$$6 !$$6$$5!$$2 !=2$$7$$$2 \times 5 !=240$$ text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-3-p7?rev=1737476038&do=diff Question 9 Exercise 6.3 Solutions of Question 9 of Exercise 6.3 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$8$$6$$7$$7$$6.$$=7+6=13$${ }^7 C_4$${ }^6 C_4$\begin{align}{ }^7 C_4 \cdot{ }^6 C_4&=\dfrac{7 !}{(7-4) ! 4 !} \cdot \dfrac{6 !}{(6-4)}\\\ &= 525\end{align}$8$$6$$7$$7$$6$$=7+6=13$$3,4,5,6$$6$\begin{align}{ }^7 C_2 \cdot{ }^6 C_6&=\dfrac{7 !}… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-3-p8?rev=1737476038&do=diff Question 9 Exercise 6.3 Solutions of Question 9 of Exercise 6.3 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-4-p7?rev=1737476038&do=diff Question 7 Exercise 6.4 Solutions of Question 7 of Exercise 6.4 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.\begin{align}S&=\{(i, j) ; i, j=1,2,3,4,5,6\}\\ &=\left[\begin{array}{llllll} (1,1) & (1,2) & (1,3) & (1,4) & (1,5) & (1,6) \\ (2,1) & (2,2) & (2,3) & (2,4) & (2,5) & (2,6) \\ (3,1) & (3,2) & (3,3) & (3,4) & (3,5) & (3,6) \\ (4,1) & (4,2) & (… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-5-p1?rev=1737476038&do=diff Question 1 and 2 Exercise 6.5 Solutions of Question 1 and 2 of Exercise 6.5 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$A$$B$$P(A)=\dfrac{2}{5}, P(B)=\dfrac{2}{5}$$P(A \cup B)=\dfrac{1}{2}$$P(A \cap B)$\begin{align} P(A \cup B)&=P(A)+P(B)-P(A \cap B) \\ \Rightarrow P(A \cap B)&=P(A)+P(B)-P(A \cup B) \end{align}$P(A), P(B)$$P(A \cup B)$$$P(A \cap… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-5-p7?rev=1737476038&do=diff Question 10 Exercise 6.5 Solutions of Question 10 of Exercise 6.5 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$20$$10$$5$$3$$2$$=20$$=10$$=5$$=3$$=15$$=5$$=10$$=3$$=22$$E$$a A$$B$$2$\begin{align}n(S)&={ }^{30} C_2\\ &=435\\ P(A)&=\dfrac{^{20} C_2}{^{30} C_2}\\ &=\dfrac{190}{435}=\dfrac{38}{87}\\ P(B)&=\dfrac{^{22} C_2}{^{30} C_2}\\ &=\dfrac{231}{43… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p1?rev=1737476038&do=diff Question 1 Review Exercise 6 Solutions of Question 1 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$O, P, Q, R, S, T, U$$2520$$9040$$5140$$4880$$\{1,2,3,4,5,6,7\}$$14$$42$$28$$21$$\{1,2,3,4,6,7,8\}$$3$$7$$120$$180$$144$$96$$\dfrac{(n+2) !(n-2) !}{(n+1) !(n-1) !}$$(n-3)$$(\dot{n}-1)$$\dfrac{n+1}{n+2}$$\dfrac{n+2}{n-1}$$5$$768$$724… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p1?rev=1737476038&do=diff Question 1 Exercise 7.1 Solutions of Question 1 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$2+4+6+\cdots+2 n=n(n+1)$$n=1$$$2=1(1+1)=2 $$$n=1$$n=k$$$2+4+6+\cdots+2 k=k(k+1)....(i)$$$n=k+1$$(k+1)^{t h}$$$a_{k+1}=\mathbf{2}(k+1)=2 k+2 $$$k+1$\begin{align}2+4+6+\cdots+2 k+2(k+1)& =k(k+1)+2(k+1) \\ & =(k+1)[k+2] \\ & =(k+1)(k+1+1)\end{a… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p15?rev=1737476038&do=diff Question 15 Exercise 7.1 Solutions of Question 15 of Exercise 7.1 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$a+b$$a^n-b^n$$n$$n$$n=2 n, \quad m \in \mathbb{Z}^{+}$$m=1$$$a^{2 n}-b^{2 m}=a^2-b^2=(a+b)(a-b)$$$\Rightarrow(a+b)$$a^2-b^2$$m=1$$n=2$$m=k$$$a^{2 k}-b^{2 k}=Q(a+b)$$$Q$$m=k+1$\begin{align}a^{2(k+1)}-b^{2(k-1)} & =a^{2 k+2}-b^{2 k+2} \\ & =… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p11?rev=1737476038&do=diff Question 11 Exercise 7.2 Solutions of Question 11 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$(1+x)^n$$\left(\begin{array}{l}n \\ r\end{array}\right)=\mathrm{C}_r$$\mathrm{C}_1+2 \mathrm{C}_2 x+3 \mathrm{C}_3 x^2+\ldots \ldots . .+\mathrm{nC}_{\mathrm{n}} x^{\mathrm{n}-1}=\mathrm{n}(1+x)^{\mathrm{n}-1}$ text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/re-ex7-p1?rev=1737476039&do=diff Question 1 Review Exercise 7 Solutions of Question 1 of Review Exercise 7 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Chose the correct option. <panel>$O, P, Q, R, S, T, U$$2520$$9040$$5140$$4880$$\{1,2,3,4,5,6,7\}$$14$$42$$28$$21$$\{1,2,3,4,6,7,8\}$$3$$7$$120$$180$$144$$96$$\dfrac{(n+2) !(n-2) !}{(n+1) !(n-1) !}$$(n-3)$$(\dot{n}-1)$$\dfrac{n+1}{n… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/re-ex7-p7?rev=1737476039&do=diff Question 11 Review Exercise 7 Solutions of Question 11 of Review Exercise 7 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p1?rev=1737476039&do=diff Question 1, Exercise 10.1 Solutions of Question 1 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. There are four parts in Question 1.$\sin {{37}^{\circ }}\cos {{22}^{\circ }}+\cos {{37}^{\circ }}\sin {{22}^{\circ }}$\begin{align} \sin (\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta, \end{align}\begin{a… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-2-p1?rev=1737476039&do=diff Question 1, Exercise 10.2 Solutions of Question 1 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. There are four parts in Question 1.$\sin 2\theta ,\,\,\cos 2\theta$$\tan 2\theta$$\tan \theta =-\dfrac{1}{5}$$\theta$$\sin \theta =\dfrac{1}{\sqrt{26}}$$\cos \theta =\dfrac{-5}{\sqrt{26}}$\begin{align}\sin 2\theta &=2\sin… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-3-p1?rev=1737476039&do=diff Question 1, Exercise 10.3 Solutions of Question 1 of Exercise 10.3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. There are four parts in Question 1.$2\sin 6x\sin x$$$-2\sin \alpha \sin \beta =\cos (\alpha +\beta )-\cos (\alpha -\beta ).$$$\alpha =6x$$\beta =x$\begin{align}-\,2\sin 6x\sin x&=\cos (6x+x)-\cos (6x-x)\\ &=\cos 7x-\cos x… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-3-p5?rev=1737476039&do=diff Question 5, Exercise 10.3 Solutions of Question 5 of Exercise 10.3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos {{20}^{\circ }}\cos {{40}^{\circ }}\cos {{60}^{\circ }}\cos {{80}^{\circ }}=\dfrac{1}{16}$$2\cos \alpha \cos \beta =\cos \left( \alpha +\beta \right)+\cos \left( \alpha -\beta \right)$\begin{align}L.H.S.&=\cos {{20… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/re-ex10-p1?rev=1737476039&do=diff Question 1, Review Exercise 10 Solutions of Question 1 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos {{50}^{\circ }}5{0}'\cos {{9}^{\circ }}1{0}'-\sin {{50}^{\circ }}5{0}'\sin {{9}^{\circ }}1{0}'=$$0$$\dfrac{1}{2}$$1$$\dfrac{\sqrt{3}}{2}$$\tan {{15}^{\circ }}=2-\sqrt{3}$${{\cot }^{2}}{{75}^{\circ }}$$7+\sq… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/re-ex10-p5?rev=1737476039&do=diff Question 8 & 9, Review Exercise 10 Solutions of Question 8 & 9 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin \left( \dfrac{\pi }{4}-\theta \right)\sin \left( \dfrac{\pi }{4}+\theta \right)=\dfrac{1}{2}\cos 2\theta $$2\sin \alpha \sin \beta =\cos \left( \alpha -\beta \right)-\cos \left( \alpha +\beta \r… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-1-p1?rev=1737476039&do=diff Question 1, Exercise 1.1 Solutions of Question 1 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 1(i) ${{i}^{31}}$\begin{align}{{i}^{31}}&=i\cdot{{i}^{30}}\\ &=i\cdot{{\left( {{i}^{2}} \right)}^{15}}\\ &=i\cdot{{\left( -1 \right)}^{15}} \quad \because i^2=-1\\ &=i\cdot(-1)\\ &=-i.\end{align}${{\left( -i \right)}^{6}}$\begin{align} {{\left… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-1-p7?rev=1737476039&do=diff Question 7, Exercise 1.1 Solutions of Question 7 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 7(i) $11+12 i$$$z=11+12i$$\begin{align}|z|&= \sqrt{(11)^2+(12)^2}\\ &=\sqrt{265}\end{align}$|11+12 i|=\sqrt{265}$$(2+3 i)-(2+6 i)$$z=(2+3i)−(2+6i)$\begin{align}z&=2+3i−2−6i\\ &=-3i \end{align}\begin{align} |z| &= \sqrt{0^2+(-3)^2} \\ &= \sqrt{… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p1?rev=1737476039&do=diff Question 1, Exercise 1.2 Solutions of Question 1 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 1(i) $\operatorname{Re}(i z)=-\operatorname{Im}(z)$$$z=x+iy$$\begin{align} iz&=i(x+iy)\\ &=ix-y\end{align}\begin{align}Re(iz)&=-y\\ \implies Re(iz)&=-Im(z)\end{align}$\operatorname{Im}(i z)=\operatorname{Re}(z)$$$z=x+iy$$\begin{align}iz&=i(x+i… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p10?rev=1737476039&do=diff Question 10, Exercise 1.2 Solutions of Question 10 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 10(i) $z_{1}=-3+2 i$$$\left|z_{1}\right|=\left|-z_{1}\right|=\left|\overline{z_{!}}\right|=\left|-\overline{z_{!}}\right|.$$\begin{align} |z_1| &= \sqrt{(-3)^2 + (2)^2} \\ &= \sqrt{9 + 4} = \sqrt{13} \,\, -- (1) \end{align}\begin{align} -z_… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-3-p1?rev=1737476039&do=diff Question 1, Exercise 1.3 Solutions of Question 1 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 1(i) $z^{2}+169$\begin{align} & z^{2} + 169 \\ = & z^{2} - (13i)^2 \\ = &(z + 13i)(z - 13i). \end{align}$2 z^{2}+18$\begin{align} & 2z^2 + 18 \\ = &2(z^2 - (3i)^2)\\ = &2(z + 3i)(z - 3i) \end{align}$3 z^{2}+363$\begin{align} & 3z^2 + 363 \\ … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-3-p4?rev=1737476039&do=diff Question 4, Exercise 1.3 Solutions of Question 4 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 4(i) $(1-i) z+(1+i) \omega=3 ; 2 z-(2+5 i) \omega=2+3 i$\begin{align} &(1-i) z+(1+i) \omega=3 \quad \cdots(1)\\ &2 z-(2+5 i) \omega=2+3i \quad\cdots(2) \end{align}$2$\begin{align} &(2-2i)z+(2+2i) \omega=6 \quad \cdots (3) \end{align}$(1-i)$\b… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p1?rev=1737476039&do=diff Question 1, Exercise 1.4 Solutions of Question 1 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 1(i) $2+i 2 \sqrt{3}$$z=x+iy=2 + i 2 \sqrt{3}$\begin{align} r & = \sqrt{x^2 + y^2} = \sqrt{2^2 + (2\sqrt{3})^2} \\ & = \sqrt{4 + 12} = \sqrt{16} = 4. \end{align}\begin{align} \alpha & = \tan^{-1}\left|\frac{y}{x}\right| = \tan^{-1}\left|\fra… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p11?rev=1737476039&do=diff Question 10, Exercise 1.4 Solutions of Question 10 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 10(i) $Z$$E=(-50+100 i)$$I=(-6-2 i)$$E=(-50+100 i)$$I=(-6-2 i)$$$ E = I \times Z $$$$(-50+100 i)= (-6-2 i) \times Z $$\begin{align} \implies Z & = \dfrac{-50+100 i}{-6-2 i} \\ & = \dfrac{(-50+100 i)(-6+2i)}{(-6-2 i)(-6+2i)}\\ & = \dfrac{300-… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/re-ex-p1?rev=1737476039&do=diff Question 1, Review Exercise Solutions of Question 1 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\operatorname{part}(\mathrm{s})$$z$$z$$(0,0)$$(1,0)$$(0,1)$$(1,1)$$z$$|z|$$1 / z$$-z$$\bar{z}$$x$$y$$x y$$z_{1}=3+2 i$$z_{2}=5+6 i$$z_{1}>z_{2}$$z_{1}<z_{2}$$\overline{z_{1}}=\overline{z_{2}}$$\overline{z_{1}}=-\overline{z_{2}}$$\mathrm{z}=3+4 i$$… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/re-ex-p8?rev=1737476039&do=diff Question 8, Review Exercise Solutions of Question 8 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sqrt{2}+i \sqrt{2}$$\theta=45^{\circ}$$$x= \sqrt{2} + i \sqrt{2}, \quad \theta=\dfrac{\pi}{4}.$$$x_{\max}$\begin{align} &x=x_{\max} e^{i\theta} \\ \implies & \sqrt{2} + i \sqrt{2}=x_{\max} e^{i\dfrac{\pi}{4}} \\ \implies & x_{\max} \left(\cos\dfr… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-1-p4?rev=1737476039&do=diff Question 4, Exercise 2.1 Solutions of Question 4 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$ A=\left[\begin{array}{ccc} 2 & 0 \\ \sqrt{5} & 6 \\ 1 & 9 \end{array}\right]$$$$ A^t=\begin{bmatrix} 2 & \sqrt{5} & 1 \\ 0 & 6 & 9 \end{bmatrix}$$$$B=\left[\begin{array}{cccc} 1 & 6 & 2 & 0 \end{array}\right] $$$$B^t=\left[\begin{array}{c} 1… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p1?rev=1737476039&do=diff Question 1, Exercise 2.2 Solutions of Question 1 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[a_{i j}\right]$$2 \times 2$$a_{i j}=\dfrac{i+3 j}{2}$\( a_{ij} = \dfrac{i + 3j}{2} \)\( i = 1, j = 1 \)\[ a_{11} = \dfrac{1 + 3 \cdot 1}{2} = \dfrac{1 + 3}{2} = \dfrac{4}{2} = 2 \]\( i = 1, j = 2 \)\[ a_{12} = \dfrac{1 + 3 \cdot 2}{2} … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p13?rev=1737476039&do=diff Question 13, Exercise 2.2 Solutions of Question 13 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $X$$Y$$2 X-Y=\left[\begin{array}{ccc}1 & 6 & -3 \\ 2 & 1 & 7\end{array}\right]$$X+3 Y=\left[\begin{array}{ccc}4 & 3 & 2 \\ 1 & -3 & 0\end{array}\right]$\begin{align*} 2X - Y = \begin{pmatrix} 1 & 6 & -3 \\ 2 & 1 & 7 \end{pmatrix} \cdots (i)\\… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-3-p7?rev=1737476039&do=diff Question 7, Exercise 2.3 Solutions of Question 7 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $(A B)^{-1}=B^{-1} A^{-1}$$A=\left[\begin{array}{ll}2 & 1 \\ 8 & 6\end{array}\right]$$B=\left[\begin{array}{ll}3 & 2 \\ 0 & 2\end{array}\right]$\begin{align*} A &= \left[\begin{array}{ll}2 & 1 \\ 8 & 6\end{array}\right] \\ |A|& = 12 - 8 = 4\\ … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p8?rev=1737476039&do=diff Question 9 and 10, Exercise 2.6 Solutions of Question 9 and 10 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 x-y+3 z=\alpha ; 3 x+y-5 z=\beta ;-5 x-5 y+21 z=\gamma$$\gamma \neq 2 \alpha-3 \beta$$2$$2$$3$$3$ text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/re-ex-p1?rev=1737476039&do=diff Question 1, Review Exercise Solutions of Question 1 of Review Exercise of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A$$m \times n$$B$$n \times p$$A B$$n \times p$$m \times p$$p \times m$$n \times n$$A$$1 \times n$$A^{t} A$$1 \times n$$n \times 1$$1 \times 1$$n \times n$$a_{i j}$$A$$a_{i j}=(-1)^{i+j} A_{i j}$$a_{i j}=(-1)^{i+j} M_{i j}$$\frac{A_{i j}}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/re-ex-p3?rev=1737476039&do=diff Question 4 and 5, Review Exercise Solutions of Question 4 and 5 of Review Exercise of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left|\begin{array}{ccc}a+1 & l & l \\ l & a+1 & l \\ l & l & a+1\end{array}\right|=(a+1+2 l)(a+1-l)^{2}$\begin{align*} L.H.S &= \left|\begin{array}{ccc}a+1 & l & l \\ l & a+1 & l \\ l & l & a+1\end{array}\right|\\ &=\left|\b… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p1?rev=1737476039&do=diff Question 1 and 2, Exercise 4.1 Solutions of Question 1 and 2 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$a_{10}$$a_{15}$$$a_{n}=3 n+1$$$$a_{n}=3 n+1$$\begin{align*} a_1 &= 3(1) + 1 = 3 + 1 = 4\\ a_2 &= 3(2) + 1 = 6 + 1 = 7\\ a_3 &= 3(3) + 1 = 9 + 1 = 10\\ a_4 &= 3(4) + 1 = 12 + 1 = 13\\ \end{align*}\begin{align*} a_{10} &= 3(10) + 1 = 30… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p11?rev=1737476039&do=diff Question 21 and 22, Exercise 4.1 Solutions of Question 21 and 22 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{n}$$\sqrt{2}, \sqrt{4}, \sqrt{6}, \sqrt{8}, \sqrt{10}, \ldots$$$\sqrt{2}, \sqrt{4}, \sqrt{6}, \sqrt{8}, \sqrt{10}, \ldots$$\begin{align*} &a_1=\sqrt{2 \cdot 1}, \\ &a_2=\sqrt{4}=\sqrt{2 \cdot 2} \\ &a_3=\sqrt{6}=\sqrt{2 \cdot 3}\\… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p1?rev=1737476039&do=diff Question 1, Exercise 4.2 Solutions of Question 1 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=4, d=3$$a_1= 4$$d=3$$$a_n = a_1 + (n - 1)d.$$\begin{align*} a_2&=4+(2-1)3=4+3=7\\ a_3 &= 4+ (3-1) 3 = 4 + 6 = 10\\ a_4&=4+(4-1)3=4+9=13 \end{align*}$a_1=4$$a_2=7$$a_3=10$$a_4=13$$a_1=7$$d=5$$a_1= 7$$d=5$$$a_n = a_1 + (n - 1)d.$$\begin{align*} … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p10?rev=1737476039&do=diff Question 16 and 17, Exercise 4.2 Solutions of Question 16 and 17 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $5$$17$$A_1$$A_2$$5$$17$$5$$A_1$$A_2$$17$$a_1=5$$a_4=17$$$a_n=a_1+(n-1)d.$$\begin{align*} &a_4 = a_1 + 3d \\ \implies & 17=5+3d\\ \implies & 3d=12\\ \implies & \boxed{d=4}.\end{align*}\begin{align*} A_1 &= a_2= a_1+d \\ &=5+4=9 \end{a… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p1?rev=1737476039&do=diff Question 1 and 2, Exercise 4.3 Solutions of Question 1 and 2 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $4+7+10+13+16+19+22+25$$4+7+10+13+16+19+22+25$$a_1=4$$d=7-4=3$$n=8$\begin{align} S_n&=\frac{n}{2}[2a_1+(n-1)d]\\ \implies S_8&=\frac{8}{2}[2(4)+(8-1)(3)]\\ &=4[8+7\times 3] = 116 \end{align}$a_{1}=2$$a_{n}=200$$n=100$$a_{1}=2$$a_{n}=200$$… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p12?rev=1737476039&do=diff Question 25 and 26, Exercise 4.3 Solutions of Question 25 and 26 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$ 6000+70,000+...+a_{20}.$$$a_1=6,000$$d=70,000-6,000=64,000$$n=20$$S_n$\begin{align} S_n&=\frac{n}{2}[2a_1+(n-1)d]\\ \implies S_{20}& =\frac{20}{2}[2(6,000)+(20-1)(64,000)]\\ & =10 \times [12,000+1,216,000]\\ & =12,280,000. \end{ali… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p1?rev=1737476039&do=diff Question 1 and 2, Exercise 4.4 Solutions of Question 1 and 2 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $5,20,100,500, \ldots$$5, 20, 100, 500, \ldots $\begin{align*} \frac{20}{5} = 4\neq \frac{100}{20} = 5.\end{align*}$5, 20, 100, 500, \ldots $\begin{align*} r_1& =\frac{20}{5} = 4\\ r_2&=\frac{100}{20} = 5\\ r_3&=\frac{500}{100} = 5. \end{… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p15?rev=1737476039&do=diff Question 30, Exercise 4.4 Solutions of Question 30 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $=a_1= 1$$=a_2= 3$$=a_3=3\times 3 = 9$$=a_4=3\times 9 = 27$$=a_5=3\times 27 = 81$$81$$a_1=1$$r=3$$a_5=?$$$a_n=a_1 r^{n-1}.$$\begin{align*} a_5&=a_1 r^4 \\ &=(1)(3)^4 = 81 \end{align*}$$S_n=a_1+a_2+a_3+a_4+a_5.$$ text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p9?rev=1737476040&do=diff Question 16, Exercise 4.5 Solutions of Question 16 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $80 ft$$90\%$$a_1$$a_1 r$$a_1 r^2$$=a_1= 80 ft$$r=90% = \frac{90}{100} =0.9$$A$\begin{align} A &= a_1+a_1r+a_1r^2+... \\ & = \frac{a_1}{1-r} \\ & = \frac{80}{1-0.9}\\ &= 800 \end{align}$800 ft$ text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p1?rev=1737476040&do=diff Question 1 and 2, Exercise 4.7 Solutions of Question 1 and 2 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sum_{k=1}^{5} \frac{1}{2 k}$\begin{align*} \sum_{k=1}^{5} \frac{1}{2k} &= \frac{1}{2(1)} + \frac{1}{2(2)} + \frac{1}{2(3)} + \frac{1}{2(4)} + \frac{1}{2(5)}\\ &= \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \frac{1}{10}\\ &= … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p15?rev=1737476040&do=diff Question 29 and 30, Exercise 4.7 Solutions of Question 29 and 30 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$1+4 x+7 x^{2}+10 x^{3}+\ldots$$\[ 1 + 4x + 7x^2 + 10x^3 + \ldots \]\[ 1 \times 1 + 4 \times x + 7 \times x^2 + 10 \times x^3 + \ldots \]\(1, 4, 7, 10, \ldots\)\(a = 1\)\(d = 4 - 1 = 3\)\(1, x, x^2, x^3, \ldots\)\(1\)\(r = x\)\[ S_{\… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-8-p7?rev=1737476040&do=diff Question 13, 14 and 15, Exercise 4.8 Solutions of Question 13, 14 and 15 of Exercise 4.8 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\frac{1}{5 \cdot 11}+\frac{1}{7 \cdot 13}+\frac{1}{9 \cdot 15}+\ldots \ldots$$n$$T_k$$k$\begin{align*} T_k &= \frac{1}{(2k+3)(2k+9)}. \end{align*}\begin{align*} \frac{1}{(2k+3)(2k+9)} = \frac{A}{2k+3} + \frac{B}{2k+9} \ldots … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-1-p6?rev=1737476040&do=diff Question 10, Exercise 5.1 Solutions of Question 10 of Exercise 5.1 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 10 $\left(x^{3}+11 x^{2}+34 x+24\right)$$(x+1)$$p(x)=x^{3}+11 x^{2}+34 x+24$\begin{align} \begin{array}{r|rrrr} -1 & 1 & 11 & 34 & 24 \\ & \downarrow & -1 & -10 & -24 \\ \hline & 1 & 10 & 24 & 0 \\ \end{array}\end{align}$$ p(x) = (x+1)(x^2+10… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-2-p1?rev=1737476040&do=diff Question 1 and 2, Exercise 5.2 Solutions of Question 1 and 2 of Exercise 5.2 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y^{3}-7 y-6$$f(y)=y^{3}-7 y-6$\begin{align*} f(-1)&=(-1)^{3}-7 (-1)-6 \\ &= -1+7-6 =0. \end{align*}$y+1$$f(y)$\begin{align} \begin{array}{r|rrrr} -1 & 1 & 0 & -7 & -6 \\ & \downarrow & -1 & 1 & 6 \\ \hline & 1 & -1 & -6 & 0 \\ \end{array}\end… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-2-p4?rev=1737476040&do=diff Question 7 and 8, Exercise 5.2 Solutions of Question 7 and 8 of Exercise 5.2 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 x^{3}-15 x^{2}+27 x-10$$\dfrac{1}{2}$\( f(x) \)\( x - \frac{1}{2} \)\begin{align} \begin{array}{r|rrrr} \frac{1}{2} & 2 & -15 & 27 & -10 \\ & & 1 & -7 & 10 \\ \hline & 2 & -14 & 20 & 0 \\ \end{array} \end{align}\begin{align*} f(x) &= \left… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-3-p1?rev=1737476040&do=diff Question 1, Exercise 5.3 Solutions of Question 1 of Exercise 5.3 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 1 $x$$x+3$$x+3+7=x+10$$120 cm^3$\begin{align*} & x(x+3)(x+10)=120 \\ \implies & x(x^2+3x+10x+30)-120=0\\ \implies & x^3+13x^2+30x-120=0. \end{align*}$$p(x)=x^3+13x^2+30x-120$$\begin{align*} p(2)&=2^3+13(2)^2+30(2)-120 \\ &=8+52+60-120 =0 \end{ali… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-3-p6?rev=1737476040&do=diff Question 6, Exercise 5.3 Solutions of Question 6 of Exercise 5.3 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 6 $y^3-2y^2-y+2$$y-2$$=p(y)=y^3-2y^2-y+2$$y-2$$y-2$$p(y)$$2$$p(y)$\[ \begin{array}{r|rrrr} 2 & 1 & -2 & -1 & 2 \\ & \downarrow & 2 & 0 & -2 \\ \hline & 1 & 0 & -1 & 0 \\ \end{array} \]\begin{align*} p(y) & = (y-2)(y^2-1) \\ & = (y-2)(y+1)(y-1)… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/re-ex-p1?rev=1737476040&do=diff Question 1, Review Exercise Solutions of Question 1 of Review Exercise of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 1 $-2-x+x^{2}$$(x-2)(x-1)$$(x+1)(x+2)$$(x+2)(x-1)$$(x+1)(x-2)$$9 y^{2}+9 y-10$$3 y-2$$ 0$$1$$2$$3$$\frac{x^{2}-x-9}{x-3}=x+2+\frac{?}{x-3}$$-27$$-3$$\frac{3}{x-3}+x+2$$ 3$$3 x^{3}-2 x^{2}+5$$x+1$$x+1$$x^{3}+5 x^{2}-4 x+k$$k$$-4$$-20$$20$$0$$… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/re-ex-p5?rev=1737476040&do=diff Question 8, Review Exercise Solutions of Question 8 of Review Exercise of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 8 $y^{3}+6 y^{2}-y-30$$(y-2)$$(y+3)$$p(y)=y^{3}+6 y^{2}-y-30$$(y-2)$$(y+3)$$p(y)$$2$$-3$$p(y)$\[ \begin{array}{r|rrrr} 2 & 1 & 6 & -1 & -30 \\ & \downarrow & 2 & 16 & 30 \\ \hline -3 & 1 & 8 & 15 & 0 \\ & \downarrow & -3 & -15 & … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/re-ex-p8?rev=1737476040&do=diff Question 8, Review Exercise Solutions of Question 8 of Review Exercise of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 8 $y^{3}+6 y^{2}-y-30$$(y-2)$$(y+3)$ text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p13?rev=1737476040&do=diff Question 14, Exercise 8.1 Solutions of Question 14 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\theta$$\sin \theta$$\cos \theta$$\alpha$\begin{align*} &\tan\alpha = \frac{\overline{BC}}{\overline{AB}} \\ \implies &\tan\alpha = \frac{3}{3} = 1 \\ \implies &\alpha = \tan^{-1}(1) = 45^\circ \end{align*}$45^\circ$$\theta$\begin{align*} … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p12?rev=1737476040&do=diff Question 8(xix, xx, xxi & xxii) Exercise 8.2 Solutions of Question 8(xix, xx, xxi & xxii) of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$\frac{\sin 2 \alpha}{\sin \alpha}-\frac{\cos 2 \alpha}{\cos \alpha}=\sec \alpha$$\begin{align*} LHS &= \dfrac{\sin 2 \alpha}{\sin \alpha}-\frac{\cos 2 \alpha}{\cos \alpha}\\ &= \dfrac{\sin 2 \alpha … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-3-p1?rev=1737476040&do=diff Question 1(i, ii, iii & iv) Exercise 8.3 Solutions of Question 1(i, ii, iii & iv) of Exercise 8.3 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$4 \sin 16x \cos 10x $$\begin{align*} &4 \sin 16x \cos 10x \\ & = 2 (2\sin 16x \cos 10x) \\ &= 2[\sin(16x+10x)+\sin(16x-10x)]\\ &= 2[\sin (26x)+\sin(6x)] \end{align*}$10 \cos 10y \cos 6y$\begin{align*} &10 \… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-3-p8?rev=1737476040&do=diff Question 4 Exercise 8.3 Solutions of Question 4 of Exercise 8.3 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\cos 80^{\circ} \cos 60^{\circ} \cos 40^{\circ} \cos 20^{\circ}=\dfrac{1}{16}$\begin{align*} LHS &= \cos 80^\circ \cos 60^\circ \cos 40^\circ \cos 20^\circ \\ &= \cos 80^\circ \left(\frac{1}{2}\right) \cos 40^\circ \cos 20^\circ \\ &= \frac{1… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/re-ex-p1?rev=1737476040&do=diff Question 1, Review Exercise Solutions of Question 1 of Review Exercise of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin \left(45^{\circ}-30^{\circ}\right)=\ldots$$\frac{\sqrt{6}-\sqrt{2}}{4}$$\frac{\sqrt{6}+\sqrt{2}}{4}$$\frac{\sqrt{6}-\sqrt{2}}{2}$$\frac{\sqrt{3}-\sqrt{2}}{2}$$\tan \left(\frac{\pi}{6}+\frac{\pi}{4}\right)=\ldots$$\frac{\sqrt{3}-1}… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/re-ex-p9?rev=1737476040&do=diff Question 10, Review Exercise Solutions of Question 10 of Review Exercise of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin (16 x)=16 \sin (x) \cos (x) \cos (2 x) \cos (4 x) \cos (8 x)$\begin{align*} RHS&=16 \sin (x) \cos (x) \cos (2 x) \cos (4 x) \cos (8 x) \\ &= 8(2 \sin (x) \cos (x) )\cos (2 x) \cos (4 x) \cos (8 x) \\ &= 8 \sin2 (x) \cos (2 x) \… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p11?rev=1737476040&do=diff Question 10, Exercise 9.1 Solutions of Question 10 of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $ V(t)=a \operatorname{Sin}(k(t-d))+c$$56 \mathrm{~Hz} A C$$k$ text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/re-ex-p1?rev=1737476040&do=diff Question 1,Review Exercise Solutions of Question 1 of Review Exercise of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\cos \theta=\frac{\sqrt{3}}{2}$$\sin \theta=$$\frac{1}{2}$$-\frac{1}{2}$$\sqrt{3}$$-\frac{2}{\sqrt{3}}$$\tan (-15 \pi)=$$ 0$$-1$$1$$2 \sin \theta+\frac{1}{2}cosec \theta \theta $$\theta=45^{\circ}$$\frac{1}{\sqrt{2}}$$\frac{1}{3}$$\frac{3}{… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/re-ex-p10?rev=1737476040&do=diff Question 10(xi-xv), Review Exercise Solutions of Question 10(xi-xv) of Review Exercise of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/mathcraft?rev=1737476042&do=diff <jumbotron> MathCraft Introducing “MathCraft”: Your Solution for Document Transformation! [MathCraft] We are thrilled to unveil our latest service, MathCraft, tailored exclusively for the mathematics community. With MathCraft, you can easily get code from PDFs and pictures into LaTeX or Word files without spending too much money. Whether you're a student, researcher, or teacher, MathCraft can help you create your math documents in the format you want. text/html 2025-01-21T16:13:54+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/atiq/fa23-mth480?rev=1737476034&do=diff MTH480: Introductory Quantum Mechanics Objective The physical principles and mathematical formalism of quantum theory, with emphasis on applications to atomic, molecular, and many-body physics; scattering phenomena; and electromagnetism (photon physics). $x(t)={{t}^{3}}+2\sin t$$t=\dfrac{\pi }{6}$$v(t)={{t}^{2}}+t{{e}^{t}}$ text/html 2025-01-21T16:13:54+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/atiq/math-301?rev=1737476034&do=diff MATH-301: Complex Analysis <div> <img src="http://www.mathematicianspictures.com/Images/Eul_300w_post_us1.jpg" alt="image" title="image" class="mediaright" /><br> <center> </div> Objectives of the course This is an introductory course in complex analysis, giving the basics of the theory along with applications, with an emphasis on applications of complex analysis and especially conformal mappings. Students should have a background in real analysis (as in the course Real Analysis I), includin… text/html 2025-01-21T16:13:54+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/atiq/math-505?rev=1737476034&do=diff MATH-505: Complex Analysis Provisional Results <WRAP third column> MMAF13E101 = 65 MMAF13E102 = 65 MMAF13E103 = 58 MMAF13E104 = 58 MMAF13E105 = 78 MMAF13E106 = 62 MMAF13E107 = 50 MMAF13E108 = 75 MMAF13E109 = 61 MMAF13E110 = 50 $\cot 2z$<div> <center> </div><div> </center> </div> text/html 2025-01-21T16:13:54+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/atiq/sp24-mth480?rev=1737476034&do=diff MTH480: Introductory Quantum Mechanics Objective The physical principles and mathematical formalism of quantum theory, with emphasis on applications to atomic, molecular, and many-body physics; scattering phenomena; and electromagnetism (photon physics). text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_2_mcqs?rev=1737476036&do=diff MCQs/Objective: HSSC-II On this page, MCQ/Objective for FSc-II (HSSC-II) or FSc Part 2 are given. <div> <img src=http://mathcity.org/images/mcq2.jpg class="mediacenter" /> </div> <WRAP center round box 80%> * Short Questions by Mr. Akhtar Abbas NEW * Short Questions without answers by Mr. Akhtar Abbas for FSc Part 2. </WRAP> <WRAP center round box 80%> text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_2_solutions?rev=1737476036&do=diff FSc/ICS Part 2 Solutions [Calculus and Analytic Geometry, MATHEMATICS 12] <lead>Notes (Solutions) of Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc/ICS Part 2 or HSSC-II), Punjab Text Book Board Lahore.</lead> There are seven units in this book and we have work hard to make easy and suitable solutions for students and teachers so that it help them learn things quickly and easily. Please click on a desire unit to view the solution of any particular exercise. This work is licens… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/complex-analysis-m-usman-hamid?rev=1737476041&do=diff Complex Analysis by M Usman Hamid [Complex Analysis by M Usman Hamid] We are really very thankful to Muhammad Usman Hamid for providing these notes and appreciates his effort to publish these notes on MathCity.org It covers the one part of the syllabus of Complex Analysis paper of MSc Mathematics. See the contents of the notes given below to see the topics covered by these notes.$$ x^2+4=0, x^2+x+1=0 \text{ and } x^2-2x+3=0 $$$i=\sqrt{-1}$$i^2=-1$ text/html 2024-08-05T06:27:23+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/wiki/dokuwiki?rev=1722839243&do=diff DokuWiki wiki:dokuwiki DokuWiki is a simple to use and highly versatile Open Source wiki software that doesn't require a database. It is loved by users for its clean and readable Formatting Syntax. The ease of maintenance, backup and integration makes it an administrator's favorite. Built in text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_mcqs/fill_in_the_blanks_by_rauf_nabeel?rev=1737476035&do=diff Fill in the blanks by Rauf & Nabeel * Fill in the blanks, provided by Adil Rauf & Muhammad Nabil (F.Sc. Part I, Session: 2003-05, FAZMIC Sargodha) * Text Book of Algebra and Trigonometry Class XI (Punjab Textbook Board, Lahore) <div> <center> </div> Chapter 02 Download PDF (93KB) <div> </center> </div> text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_mcqs/mcqs_by_nauman_idrees?rev=1737476035&do=diff MCQs by Nauman Idrees <div> <img src=http://mathcity.org/images/mcqs.jpg class="mediacenter" /> </div> * Text Book of Algebra and Trigonometry, Class XI (Punjab Textbook Board, Lahore). Chapter 01 View Online | Download PDF (52KB) | Ch 01: Key Chapter 02 View Online | Download PDF (68KB) | Ch 02: Key Chapter 03 View Online | Download PDF (41KB) | Ch 03: Key Chapter 04 View Online | Download PDF (43KB) | Ch 04: Key Chapter 05 View Online | Download PDF (39KB) | text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_mcqs/mcqs-short_questions_by_mr._parvez_khan?rev=1737476035&do=diff MCQs-Short Questions by Mr Parvez Khan * MCQs and Short Question written by Mr. Parvez Khan, composed by Mr. Momin Ali from Text Book of Algebra and Trigonometry Class XI (Punjab Textbook Board, Lahore) * <wrap hi>Key to the MCQs is given at page 57.</wrap><div> <div align="center"> <iframe src="http://docs.google.com/viewer?url=http%3A%2F%2Fwww.mathcity.org%2Ffiles%2Ffsc%2Ffsc_part1%2FMCQs-Short_Questions_Math_FSc_Part1.pdf&embedded=true" style='width: 100%; height: 550px; border: none;'><… text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_mcqs/short_questions_by_mr._akhtar_abbas?rev=1737476035&do=diff Short Questions by Mr. Akhtar Abbas * We are very thankful to Mr. Akhtar Abbas for sharing these short questions. * These short questions are selected from previous 5 years papers of different boards. Solve these at your own to perform well in annual exams. text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch01?rev=1737476035&do=diff Chapter 01: Number System [Chapter 01: Number System] Notes (Solutions) of Chapter 01: Number System, Textbook of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary * Rational numbers and irrational numbers$\mathbb{C}$$(x+iy)^n$$\left(\frac{x_1+iy_1}{x_2+iy_2}\right)^n, x_2+iy_2\neq 0$$\sqrt{-1}=i$$\sqrt{-1}$$i$$-i$$i$$-i$$-1$$i^2=-1$$\sqrt{-1}=i$$\sqrt{-1}$ text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch04?rev=1737476035&do=diff Chapter 04: Quadratic Equations [Chapter 04: Quadratic Equations] Notes (Solutions) of Chapter 04: Quadratic Equations, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board, Lahore. Contents & summary * Introduction * Solutions of Quadratic Equations text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_2_solutions/ch01?rev=1737476036&do=diff Unit 01: Functions and Limits [Unit 01: Functions and Limits] Notes (Solutions) of Unit 01: Functions and Limits, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. There are five exercises in this chapter. You can view online or download PDF. To view PDF, you must have PDF Reader installed on your system and it can be downloaded from $\lim_{x\to a}\frac{x^n-a^n}{x-a} = na^{n-1}$$\lim_{x\to0}\frac{\sqrt{x+a} - \sqrt{a}}{x} = \frac{… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_2_solutions/ch02?rev=1737476036&do=diff Unit 02: Differentiation [Unit 02: Differentiation] Notes (Solutions) of Unit 02: Differentiation, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. You can view online or download PDF. To view PDF, you must have PDF Reader installed on your system and it can be downloaded from Software section.$f'(x)$$x^n$$n \in \mathbb{Z}$$\frac{x+1}{x-1}$$x$$$ \begin{aligned} \frac{d}{dx}\left(\frac{x+1}{x-1}\right) &= \frac{(x-1)\frac{d}{dx}(x… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_2_solutions/ch03?rev=1737476036&do=diff Unit 03: Integration [Unit 03: Integration] Notes (Solutions) of Unit 03: Integration, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. You can view online or download PDF. To view PDF, you must have PDF Reader installed on your system and it can be downloaded from Software section.$dy$$\delta{y}$$[f(x)]^n f'(x)$$[f(x)]^{-1}f'(x)$ text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_2_solutions/ch04?rev=1737476036&do=diff Unit 04: Introduction to Analytic Geometry [Unit 01: Functions and Limits] Notes (Solutions) of Unit 04: Introduction to Analytic Geometry, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. You can view online or download PDF. To view PDF, you must have PDF Reader installed on your system and it can be downloaded from Software section.$ax^2+ 2hxy+by^2=0$ text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_2_solutions/ch05?rev=1737476036&do=diff Unit 05: Linear Inequalities and Linear Programming [Unit 05: Linear Inequalities and Linear Programming] Notes (Solutions) of Unit 05: Linear Inequalities and Linear Programming, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. You can view online or download PDF. To view PDF, you must have PDF Reader installed on your system and it can be downloaded from Software section. text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_2_solutions/ch06?rev=1737476036&do=diff Unit 06: Conic Section [Unit 06: Conic Section] Notes (Solutions) of Unit 06: Conic Section, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. You can view online or download PDF. To view PDF, you must have PDF Reader installed on your system and it can be downloaded from Software section. text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_2_solutions/ch07?rev=1737476036&do=diff Unit 07: Vectors [Unit 07: Vectors] Notes (Solutions) of Unit 07: Vectors, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. You can view online or download PDF. To view PDF, you must have PDF Reader installed on your system and it can be downloaded from Software section.$u\cdot v$$u\times v$$u\cdot(v\times w)$ text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/msc/syllabus/pu?rev=1737476041&do=diff Syllabus for PU <img src=http://www.mathcity.org/images/logopu.gif alt="University of the Punjab Logo" class=mediaright /> Syllabus and scheme of studies for Regular/Private students doing MSc Mathematics from University of the Punjab, Lahore. 2 years M.Sc Mathematics programme consists of two parts namely Part-I and Part II. The regulation, Syllabi and Courses of Reading for the M.Sc. (Mathematics) Part-I and Part-II (Regular Scheme) are given below. text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/bsc/paper_pattern/sargodha_university/applied_mathematics?rev=1737476035&do=diff Applied Mathematics (Paper A & B) This paper consista of two papers of 100 marks each. One paper is called “Paper A” and other is called “Paper B”. Paper A * NOTE: attempt two questions from each section. SECTION-I (4/12: 17,17,17,17) $(\lambda ,\mu )$ text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/bsc/paper_pattern/sargodha_university/b-course_of_mathematics?rev=1737476035&do=diff B-Course of Mathematics (Paper A & B) <WRAP center round info 60%> This subject is consists of two papers of 100 marks each. One is called “Paper A” and other is called “Paper B”. This page is updated on February 15, 2015. This syllabus is for 1st Annual 2015 and onward organized by University of Sargodha, Sargodha. </WRAP>$(\lambda ,\mu )$ text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_mcqs/short_questions_by_mr._akhtar_abbas/view?rev=1737476035&do=diff View * We are very thankful to Mr. Akhtar Abbas for sharing these short questions. These short questions are selected from previous five years papers of different boards. Solve these at your own to perform well in annual examination. Recommended book for these short questions is “Text Book of Algebra and Trigonometry Class XI (Punjab Textbook Board, Lahore)”. But any student Mathematics can get benefit from it. text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch01/view?rev=1737476035&do=diff Ch 01: Number System: Mathematics FSc Part 1 Notes (Solutions) of Chapter 01: Number System, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board (PTB), Lahore. There are three exercises in this chapter. Please see the main page of this chapter for MCQs and important question at text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch02/view?rev=1737476035&do=diff Ch 02: Sets, Functions and Groups: Mathematics FSc Part 1 Notes (Solutions) of Chapter 02: Sets, Functions and Groups, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board (PTB), Lahore. There are eight exercises in this chapter. Please see the main page of this chapter for MCQs and important question at text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch04/view?rev=1737476035&do=diff View Online (Notes of Chapter 04) Notes (Solutions) of Chapter 04: Quadratic Equations, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board, Lahore. These notes are provided by M. Shahid Nadeem, Lecturer in Mathematics, Punjab College Wah Cantt. One can also download PDF of the notes from this page. text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch05/view?rev=1737476035&do=diff Ch 05: Partial Fractions: Mathematics FSc Part 1 Notes (Solutions) of Chapter 05: Partial Fractions, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board (PTB), Lahore. There are four exercises in this chapter. Please see the main page of this chapter for MCQs and important question at text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch06/view?rev=1737476035&do=diff Ch 06: Sequences and Series: Mathematics FSc Part 1 Notes (Solutions) of Chapter 06: Sequences and Series, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board (PTB), Lahore. There are eleven exercises in this chapter. Please see the main page of this chapter for MCQs and important question at text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch07/view?rev=1737476036&do=diff Ch 07: Permutation, Combination and Probability: Mathematics FSc Part 1 Notes (Solutions) of Chapter 07: Permutation, Combination and Probability, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board (PTB), Lahore. There are eight exercises in this chapter. Please see the main page of this chapter for MCQs and important question at text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch08/view?rev=1737476036&do=diff Ch 08: Mathematical Induction and Binomial Theorem: Mathematics FSc Part 1 Notes (Solutions) of Chapter 08: Mathematical Induction and Binomial Theorem, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board (PTB), Lahore. There are three exercises in this chapter. Please see the main page of this chapter for MCQs and important question at text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch09/view?rev=1737476036&do=diff Ch 09: Fundamentals of Trigonometry: Mathematics FSc Part 1 Notes (Solutions) of Chapter 09: Fundamentals of Trigonometry, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board (PTB), Lahore. There are four exercises in this chapter. Please see the main page of this chapter for MCQs and important question at text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch10/view?rev=1737476036&do=diff Ch 10: Trigonometric Identities: Mathematics FSc Part 1 Notes (Solutions) of Chapter 10: Trigonometric Identities, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board (PTB), Lahore. There are four exercises in this chapter. Please see the main page of this chapter for MCQs and important question at text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch12/view?rev=1737476036&do=diff Ch 12: Application of Trigonometry: Mathematics FSc Part 1 Notes (Solutions) of Chapter 12: Application of Trigonometry, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board (PTB), Lahore. There are four exercises in this chapter. Please see the main page of this chapter for MCQs and important question at text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch13/view?rev=1737476036&do=diff Ch 13: Inverse Trigonometric Functions: Mathematics FSc Part 1 Notes (Solutions) of Chapter 10: Inverse Trigonometric Functions, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board (PTB), Lahore. There are two exercises in this chapter. Please see the main page of this chapter for MCQs and important question at text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch14/view?rev=1737476036&do=diff Ch 14: Solutions of Trigonometric Equation Notes (Solutions) of Chapter 14: Solutions of Trigonometric Equation, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board (PTB), Lahore. There are four exercises in this chapter. Please see the main page of this chapter for MCQs and important question at text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_2_solutions/ch02/view?rev=1737476036&do=diff Unit 02: Differentiation: Mathematics FSc part 2 Notes (Solutions) of Unit 02: Differentiation, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. There are ten exercises in this chapter. Please see the main page of this chapter for MCQs and important question text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_2_solutions/ch03/view?rev=1737476036&do=diff Unit 03: Differentiation: Mathematics FSc part 2 Notes (Solutions) of Unit 03: Integration, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. There are eight exercises in this chapter. Please see the main page of this chapter for MCQs and important question text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_2_solutions/ch03/viewer?rev=1737476036&do=diff View Online (Solutions of Unit 03) Notes (Solutions) of Unit 03: Integration, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. In this chapter, integration is defined and basic techniques of integration are given. text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_2_solutions/ch04/view?rev=1737476036&do=diff Unit 04: Introduction to Analytic Geometry: Mathematics FSc part 2 Notes (Solutions) of Unit 04: Introduction to Analytic Geometry, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. There are five exercises in this chapter. Please see the main page of this chapter for MCQs and important question text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_2_solutions/ch04/viewer?rev=1737476036&do=diff View Online (Solutions of Unit 04) Notes (Solutions) of Unit 04: Introduction to Analytic Geometry, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. This chapter have only five exercises but it covers lot of topics of analytic geometry in the plane. text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_2_solutions/ch05/view?rev=1737476036&do=diff Unit 05: Linear Inequalities and Linear Programming: Mathematics FSc part 2 Notes (Solutions) of Unit 05: Linear Inequalities and Linear Programming, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. There are three exercises in this chapter. Please see the main page of this chapter for MCQs and important question text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_2_solutions/ch06/view?rev=1737476036&do=diff Unit 06: Conic Section: Mathematics FSc part 2 Notes (Solutions) of Unit 06: Conic Section, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. There are nine exercises in this chapter. Please see the main page of this chapter for MCQs and important question text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_2_solutions/ch06/viewer?rev=1737476036&do=diff View Online (Solutions of Unit 06) Notes (Solutions) of Unit 06: Conic Section, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. From this page, you can also download PDF of the notes. text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_2_solutions/ch07/view?rev=1737476036&do=diff Unit 07: Vectors: Mathematics FSc part 2 Notes (Solutions) of Unit 07: Vectors, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. There are three exercises in this chapter. Please see the main page of this chapter for MCQs and important question