This is beta site. https://beta.mathcity.org/ 2026-06-05T20:14:44+00:00 https://beta.mathcity.org/ https://beta.mathcity.org/_media/logo.png 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$$$\sin\theta = \pm \sq… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-3-p4?rev=1737476039&do=diff Question 4, Exercise 1.3 Solutions of Question 4 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 4(i) $(1-i) z+(1+i) \omega=3 ; 2 z-(2+5 i) \omega=2+3 i$\begin{align} &(1-i) z+(1+i) \omega=3 \quad \cdots(1)\\ &2 z-(2+5 i) \omega=2+3i \quad\cdots(2) \end{align}$2$\begin{align} &(2-2i)z+(2+2i) \omega=6 \quad \cdots (3) \end{align}$(1-i)$\b… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p9?rev=1737476039&do=diff Question 9, Exercise 1.2 Solutions of Question 9 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 9(i) $(2+4 i)^{-1}$$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 2025-01-21T16:14:00+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 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes?rev=1737476042&do=diff Notes of Mathematics [Notes of Mathematics] Mathematics is a language of science and is a basic need for physical or natural sciences as well as social sciences. On this page, notes on different subjects related to mathematics are listed. These notes or resources might be helpful for ADS or BS or MSc or MPhil Mathematics. These notes are send by different students or teachers. We are very thankful to them for sending us these notes. These notes are provided as it is as open educational resource… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/matric/10th_science?rev=1737476040&do=diff Mathematics 10 (Science Group) [Matric Science 10th Book Cover] The notes/solutions, definitions, MCQs and important question for Mathematics 10 (Science Group), published by Ilmi Kitab Khana, Urdu Bazar, Lahore, Pakistan are available on this page. Whenever we found the notes we will update this page and will upload notes here. If you wish to contribute and send us the notes please contact us via our $(b^2-4ac)$$ax^2+bx+c$$\mathbb{N}$$\mathbb{W}$$\mathbb{Z}$$E$$O$$P$$\mathbb{Q}$$\cup$$\cap$$\s… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-3-p5?rev=1737476037&do=diff Question 6, Exercise 1.3 Solutions of Question 6 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6(i) ${{z}^{4}}+{{z}^{2}}+1=0$$$z^4+z^2+1=0$$$$z^4+2z^2+1-z^2=0$$$$( z^2+1 )^2-z^2=0$$$$( z^2+1+z)( z^2+1-z )=0$$$$( z^2+z+1 )( z^2-z+1 )=0$$$$(z^2+z+1 )=0$$$$z=\dfrac{-1\pm \sqrt{1-4}}{2}$$$$z=\dfrac{-1\pm \sqrt{3}i}{2}$$$$(z^2-z+1 )=0$$$$z=\dfrac{1\pm … text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/dyk/1?rev=1737476035&do=diff Are the functions are same? [Are the functions are same?] Consider two functions $f(x)=x+3$ and $\displaystyle g(x)=\frac{x^2-9}{x-3}$. Is $f=g$? Answer <WRAP center round red 70%> A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. </WRAP>$f:A \to B$$f = A$$B$$A$$A$$B$$A=\mathbb{N}$$B=\mathbb{R}$$f(x)=x+1$$1 \mapsto 2$$\quad 2 \mapsto 3$$\quad 3 \mapsto 4$$f = \mathbb{N}$$A=\mathbb{R}$$B=\mathb… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-3-p2?rev=1737476038&do=diff Question 2 Exercise 4.3 Solutions of Question 2 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2(i) $a_1, a_n, n, d$$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$$… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p10?rev=1737476039&do=diff Question 10, Exercise 1.2 Solutions of Question 10 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 10(i) $z_{1}=-3+2 i$$$\left|z_{1}\right|=\left|-z_{1}\right|=\left|\overline{z_{!}}\right|=\left|-\overline{z_{!}}\right|.$$\begin{align} |z_1| &= \sqrt{(-3)^2 + (2)^2} \\ &= \sqrt{9 + 4} = \sqrt{13} \,\, -- (1) \end{align}\begin{align} -z_… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-5-p1?rev=1737476038&do=diff Question 1 and 2 Exercise 6.5 Solutions of Question 1 and 2 of Exercise 6.5 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$A$$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)$$$P(A \cap… text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/privacy_cookies_policy?rev=1737476042&do=diff Privacy & Cookies Policy The privacy of our visitors to mathcity.org is important to us. At mathcity.org, we recognize that privacy of your personal information is important. Here is information on what types of personal information we receive and collect when you use and visit mathcity.org, and how we safeguard your information. We never sell your personal information to third parties. text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/mathcraft/sample-01-latex?rev=1737476040&do=diff MathCraft: PDF to LaTeX file: Sample-01 If the PDF file provided by you as follows: Then the output LaTeX file is as follows: \documentclass[10pt]{amsart} %\usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} %\usepackage[version=4]{mhchem} \usepackage{stmaryrd} \usepackage{bbold} \usepackage{hyperref} \usepackage{enumerate} \hypersetup{colorlinks=true, linkcolor=blue, filecolor=magenta, urlcolor=cyan,} \urlstyle{same} \title{… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p3?rev=1737476037&do=diff Question 3, Exercise 2.1 Solutions of Question 3 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3(i) $A=\begin{bmatrix}x & y & z\end{bmatrix}$$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 2025-01-21T16:13:58+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$$$A=1$\begin{align}1+B&=0\\ B&=-1\end{align}\beg… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p10?rev=1737476039&do=diff Question 12 Exercise 7.3 Solutions of Question 12 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$2 y=\frac{1}{2^2}+\frac{1.3}{2 !} \cdot \frac{1}{2^4}+\frac{1.3 \cdot 5}{3 !} \cdot \frac{1}{2^6}+\ldots$$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 2025-01-21T16:14:00+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 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/dyk/3?rev=1737476035&do=diff Its about square root [DYK] The reason is not difficult if one knows about the definition of square root of real numbers. Definition: Let $x$ be a non-negative number. Then a non-negative number $r$ is called square root of $x$ iff $r^2=x$. Square root of $x$ is denoted by $\sqrt{x}$$2^2=4$$3^2=9$$x$$r$$x$$r^2=x$$2^2=4$$(-2)^2=4$$\sqrt{4}=\sqrt{2^2}=\sqrt{(-2)^2}=2$$\sqrt{4}=\sqrt{2^2}=\sqrt{(-2)^2}=\pm 2$$\sqrt{4}$$\sqrt{x}$$x\geq0$$\sqrt{x}=x^{\frac{1}{2}}$ text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-2-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}$$$\hat{i}-9\hat{j}=p(\hat{i}+2\hat{j})+q(5\hat{i}-\hat{j})$$$$\implies \hat{i}-9\… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-4-p8?rev=1737476038&do=diff Question 9 Exercise 3.4 Solutions of Question 9 of Exercise 3.4 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9(i) Find the area of parallelogram whose diagonals are $\vec{a}=4 \hat{i}+\hat{j}-2 \hat{k}\quad$$\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/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$$\begin{align}& \sum_{j=1}^n T_j=\sum_{j=1}^n(2 j-1)^2 \\ & =\sum_{j=1}^n(4 j^2-4 j+1)\\ & =4 \sum_{j=1}^n j^2-4 \sum_{j=1}^n j+\sum_{j=1}^n 1 \\ & =4 \dfr… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-4-p7?rev=1737476038&do=diff Question 7 Exercise 6.4 Solutions of Question 7 of Exercise 6.4 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.\begin{align}S&=\{(i, j) ; i, j=1,2,3,4,5,6\}\\ &=\left[\begin{array}{llllll} (1,1) & (1,2) & (1,3) & (1,4) & (1,5) & (1,6) \\ (2,1) & (2,2) & (2,3) & (2,4) & (2,5) & (2,6) \\ (3,1) & (3,2) & (3,3) & (3,4) & (3,5) & (3,6) \\ (4,1) & (4,2) & (… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p8?rev=1737476039&do=diff Question 10 Exercise 7.3 Solutions of Question 10 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$1-\frac{1}{2^2}+\frac{1.3}{2 !} \cdot \frac{1}{2^4}+\ldots$$(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}$$\frac{n(n-1)}{2 !} x^2=\frac{1.3}{2 !} \cdot … text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/dyk/4?rev=1737476035&do=diff Is it difficult to answer? 1−−−0.999… [Is 1=0.999...?] This is a simple and common question asked to students, who have passed higher secondary school certificate. Most of the student unable to give the correct answer. Well, we cannot predict the exact reason but one simple reason seems to be a lack of logical reasoning. For example, if we wish to round off $0.999...$$x=0.999...$$10x=9.99...$$10x=9+0.999...$$10x=9+x$$9x=9$$x=1$ text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/matric/9th_science?rev=1737476041&do=diff Mathematics 9 (Science Group) [Mathematics 9 (Science Group)] Mathematics 9 is written by Dr. Karamat H. Dar and Prof. Irfan-ul-Haq and this book is published by Carvan Book House, Lahore, Pakistan. This book consist of 302 pages and there are 17 units. Notes of Unit 1 and 3 are provided by $ka + kb + kc$$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$$(x + a) (x + b) (x + c) … text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/quote-of-the-day/mar?rev=1737476042&do=diff Quotes for the March “”“”“” “” “” --- “”“”“” --- “”“”“” --- “”“”“” --- “”“”“” --- “”“”“” --- “”“”“” --- “”“”“” --- “” text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-3-p8?rev=1737476037&do=diff Question 12 & 13, Exercise 3.3 Solutions of Question 12 & 13 of Exercise 3.3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 12 $\overrightarrow{B A} \cdot \overrightarrow{A C}=0$$|\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 2025-01-21T16:13:58+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}$$a_1=0.8, \… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p10?rev=1737476038&do=diff Question 10 Exercise 7.2 Solutions of Question 10 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$n=2 ;$$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$$(1 \div 1… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p8?rev=1737476039&do=diff Question 8, Exercise 1.2 Solutions of Question 8 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 8(i) $|2 z-i|=4$$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/ex1-4-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$$\left(x_{1}^{2}+y_{1}^{2}\right)\left(x_{2}^{2}+y_{2}^{2}\right)\left(x_{3}^{2}+y_{3}^{2}\right) \ldots\left(x_{n}^{2}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p2?rev=1737476039&do=diff Question 2, Exercise 2.6 Solutions of Question 2 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\lambda$$\lambda$$2 x_{1}-\lambda x_{2}+x_{3}=0$$2 x_{1}+3 x_{2}-x_{3}=0$$3 x_{1}-2 x_{2}+4 x_{3}=0$\begin{align*} &2 x_{1}-\lambda x_{2}+x_{3}=0 \cdots(i)\\ &2 x_{1}+3 x_{2}-x_{3}=0\cdots(ii)\\ &3 x_{1}-2 x_{2}+4 x_{3}=0\cdots(iii)\\ \end{ali… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p10?rev=1737476039&do=diff Question 20 and 21, Exercise 4.4 Solutions of Question 20 and 21 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$3 , \_\_\_ , \_\_\_ , \_\_\_ , 48$$$a_1=3$$a_5=48$$r$$$ a_n=ar^{n-1}. $$\begin{align*} &a_5=a_1 r^4 \\ \implies & 48=3r^4 \\ \implies & r^4 = 16 \\ \implies & r^4 = 2^4 \\ \implies & r = 2. \end{align*}\begin{align*} & a_2=a_1 r= (3… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p4?rev=1737476040&do=diff Question 6 Exercise 8.2 Solutions of Question 6 of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin 15^{\circ} \cos 15^{\circ}$$$\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 2025-01-21T16:14:02+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 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/dyk/2?rev=1737476035&do=diff What is the value of pi? <div><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Pi-unrolled-720.gif/220px-Pi-unrolled-720.gif" title="Did you know?" class="mediaright" alt="What is pi??" /></div> What is the value of $\pi$? * (A) 3.1415 * (B) $\frac{22}{7}$ * (C) $\frac{333}{160}$ * (D) None of these ✔ Answer <WRAP center round red 70%> The number π is a mathematical constant, the ratio of a circle's circumference to its diameter. It is an irrational number, meanin… 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 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 2025-01-21T16:13:56+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/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}$\begin{align}\sin (\alpha -\beta )=\sin \alpha \cos \beta -\cos \alpha \sin.\end{align}\begin{align} \Rightarrow \quad \sin \left( \frac{\pi }{3}-\f… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-2-p5?rev=1737476036&do=diff Question 6, Exercise 10.2 Solutions of Question 6 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\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 2025-01-21T16:13:57+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/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/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/ex3-4-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/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$$a_n=a_1 r^{n-1}$\begin{align}a_{10}&=a_1 r^9=l \\ a_{13}&=a_1 r^{12}=m\\ \text{and} \quad a_{16}&=a_1 e^{\mathbf{A 5}}=n\end{align}\begin{align}a_{10} \cdot a_{16}&=\ln =(a_1 r^9)(a_1 r^{15})\\ … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-5-p1?rev=1737476038&do=diff Question 1 Exercise 4.5 Solutions of Question 1 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) $3+6+12+\ldots+3.2^9$$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-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$$a_n=a_1 r^{n-1}$\begin{align}64&=(-2)^{n-1}\\ \Rightarrow(-2)^{n-1}&=(-2)^6 \\ \Rightarrow n-1&=6 \\ \Rightarrow n&=7\\ S_7&=\dfrac{a_1[r^{\prime \prime}-1]}{r-1}\\ \text{then}\\ S_7… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-5-p5?rev=1737476038&do=diff Question 5 & 6 Exercise 4.5 Solutions of Question 5 & 6 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5 $r$$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}$$S_S$\begin{align}\dfrac{a_1(r^{10}-1)}{r-1}&=244 \dfrac{a_1(r^5-1)}{r-1} \\ \Rightarrow r^{10}-… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-2-p1?rev=1737476038&do=diff Question 1 Exercise 5.2 Solutions of Question 1 of Exercise 5.2 of Unit 05: Mascellaneous series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) $n$$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/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}$\begin{align}S&=(HHII,HHT.HTH.HTT.THII.THT.TTH,TT7),\\ \text{then} n(S)&=2^3=8\end{align}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p2?rev=1737476039&do=diff Question 2, Exercise 10.1 Solutions of Question 2 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin \dfrac{\pi }{12}$$\dfrac{\pi }{12}$$\dfrac{\pi }{3}-\dfrac{\pi }{4}$\begin{align}\sin (\alpha -\beta )=\sin \alpha \cos \beta -\cos \alpha \sin.\end{align}\begin{align} \Rightarrow \quad \sin \left( \frac{\pi }{3}-\f… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-2-p5?rev=1737476039&do=diff Question 6, Exercise 10.2 Solutions of Question 6 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\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 }}$$\beta ={{43}^{\circ }}$\begin… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p2?rev=1737476039&do=diff Question 2, Exercise 1.2 Solutions of Question 2 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 2 $$ \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/ex2-3-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/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$$d$\begin{align} &S_n=\frac{n}{2}[a_1+a_n]\\ \implies & 876=\frac{n}{2}[7+139]\\ \implies & 1752=146n\\ \implies & n=\frac{1752}{146}=12. \end{align}\begin{align… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p11?rev=1737476039&do=diff Question 22 and 23, Exercise 4.4 Solutions of Question 22 and 23 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$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 2025-01-21T16:14:00+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/ex4-8-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 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part2-ptb/definitions?rev=1737476037&do=diff Definitions: FSc Part 2 (Mathematics): PTB On this page, all the definitions of “Calculus and Analytic Geometry, MATHEMATICS 12” (Mathematics FSc Part 2 or HSSC-II), Punjab Textbook Board (PTB) Lahore, Pakistan are given. We are very thankful to $A=x^2$$f:X\to Y$$X$$f:X\to Y$$y$$Y$$y=ax+b$$x$$y$$f(x)=2x-6$$p(x) = {a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + {a_{n - 2}}{x^{n - 2}} + ... + {a_1}x + {a_0}$${a_0},\,{a_1},\,{a_2},...,{a_n}$$f(x)=ax+b$$X$$I:X\to X$$X$$Y$$C:X \rightarrow Y$$C(x)=a$$x \in X$$… 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 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/matric/9th_science/ex-6-1?rev=1737476041&do=diff Exercise 6.1 On the following page we have given the solution of Exercise 6.1 of Mathematics 9 (Science) published by Caravan Book House, Lahore. <WRAP center round info 60%> We have created this page and it will be updated to add new solutions occasionally. Please stay in touch with this page. </WRAP>$39x^7y^3z$$91x^5y^6 z^7$$102xy^2z$$85x^2yz$$187xyz^2$$39x^7y^3z=13\times 3\times x^7 y^3 z$$91x^5y^6 z^7=13\times 7\times x^5 y^6 z^7$$13 x^5y^3z$$102xy^2z=2\times 3\times 17 xy^2z$$85x^2yz=3\tim… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-2-p1?rev=1737476036&do=diff Question 1, Exercise 1.2 Solutions of Question 1 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1 If ${{z}_{1}}=2+i$${{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-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-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 2025-01-21T16:13:57+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.$$\begin{align}z_1 z_2 … text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-2-p2?rev=1737476037&do=diff Question 2, Exercise 1.2 Solutions of Question 2 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2 $z_1=-1+i$, $z_2=3-2i$${{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).$$\begin{align} {{z}_{1}}+{{z}_{2}}&=\left( -1+i \right)+\left( 3-2i \right)\\ &=2-i\end{align}\begin{align} \left( {{z}_{1}}+{{z}_{2}… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-2-p7?rev=1737476037&do=diff Question 8, Exercise 1.2 Solutions of Question 8 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 8(i) $z+\overline{z}=2\operatorname{Re}\left( z \right)$$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/ex3-2-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 ,$$|\alpha \hat{i}+(\alpha +1)\hat{j}+2\hat{k}|=3$\begin{align}|\alpha \hat{i}+(\alpha +1)\hat{j}+2\hat{k}|&=3.\end{align}\begin{align}\sqrt{(\alpha )^2+(\alpha +1)^2+(2)^2}&=3.\end{align}\begin{align}&{\alpha ^2+(\alpha +1)^2}+4=9… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p13?rev=1737476038&do=diff Question 17 Exercise 4.2 Solutions of Question 17 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 17 $n$$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/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 2025-01-21T16:13:58+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)}$$\begin{align} \dfrac{1}{(3 n-2)(3 n+1)(3 n+4)}&=\dfrac{A}{3 n-2}+\dfrac{B}{3 n+1}+\dfrac{C}{3 n+4}\end{align}$(3 n-2)(3 n+… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-3-p1?rev=1737476038&do=diff Question 1 Exercise 6.3 Solutions of Question 1 of Exercise 6.3 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$^n C_2=36$$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 2025-01-21T16:13:58+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/ex6-4-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$$$P(B)=\d… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p5?rev=1737476038&do=diff Question 5 Exercise 7.2 Solutions of Question 5 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$(\dfrac{a}{x}+b x)^8$$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$$r=4$\begin{align}T_5&=\dfrac{8 !}{(8-4) ! 4 !}(\dfrac{a}{x})^{8-4}(b … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p9?rev=1737476039&do=diff Question 11 Exercise 7.3 Solutions of Question 11 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$y=\frac{1}{2^2}+\frac{1.3}{2 !} \cdot \frac{1}{2^4}+\frac{1 \cdot 3 \cdot 5}{3 !} \cdot \frac{1}{2^6}+\ldots$$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/ex1-4-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/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 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/about_us?rev=1737476042&do=diff About Us The only purpose of this website is to help students to learn mathematics. This site contains material for the students of F.Sc., B.Sc., M.Sc., M.Phil. and Ph.D, in the subject of mathematics. The website division has been made according to the different classes for simple navigation. For example, student or teacher searching for notes of F.Sc may see text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/error?rev=1737476042&do=diff Report Error <div> <img src="../images/thankyou_math.gif" alt=" Thank you" title=" Thank you" class="mediacenter" /><br> </div> Thank you very much for coming on this page as we understand that you come to this page for helping us and community of mathematics. We really appreciate your effort to give us your quality time. In case of “Broken Link”, please provide us the link by copying the link from address bar and paste it in comments box. text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/errors?rev=1737476042&do=diff Report Error <div> <img src="../images/thankyou_math.gif" alt=" Thank you" title=" Thank you" class="mediacenter" /><br> </div> Thank you very much for coming on this page as we understand that you come to this page for helping us and community of mathematics who are using this website. We really appreciate your effort to give us your quality time. <form> Action mail admin@mathcity.org Thanks 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 2025-01-21T16:13:55+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-part1-ptb/definitions-aurang-zaib?rev=1737476037&do=diff Definitions: FSc Part 1 (Mathematics): PTB by Aurang Zaib Definitions from Textbook of Algebra and Trigonometry Class XI, published by Punjab Textbook Board (PTB) Lahore, Pakistan. We are very thankful to Aurang Zaib for his valuable contribution. Chapter 01: Number System \( \dfrac{p}{q} \)\( p, q \in \mathbb{Z} \)\( q \neq 0 \)\( \dfrac{3}{4} \)\( \dfrac{7}{2} \)\( \sqrt{2} \)\( \pi \)\( \mathbb{R} \)\( 0.25 \)\( 3.75 \)\( 0.3333... \)\( 1.234234... \)\( \pi \)\( 3.1415... \)\( \sqrt{2} \)\(… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/msc/notes/targets?rev=1737476041&do=diff Targets Here we have listed the notes for MSc or BS Mathematics, which will be published on MathCity.org. We are working hard to find these notes. Whenever we found these notes we will put them on our website. Here are our targets. * Fluid Mechanics 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$$c=3$\begin{align}z&=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\\ z&=\dfrac{-\left( 1 \right)\pm \sqrt{{{\left( 1 \right)}^{2}}-4\left( 1 \right)\left( 3 \right)}}{2\left( 1 \right)}\\ z&=\dfrac{-1\pm \s… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-1-p3?rev=1737476036&do=diff Question 3, Exercise 10.1 Solutions of Question 3 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin u=\dfrac{3}{5}$$\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-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)$… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-2-p2?rev=1737476036&do=diff Question 2, Exercise 10.2 Solutions of Question 2 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\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$$\cos \theta =-\dfrac{3}{5}$\begin{align}\sin 2\theta &=2\sin \theta \cos \theta \\ &=2\left( \dfrac{4}{5} \right)\left( -\dfrac{3}{5} \rig… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-3-p1?rev=1737476036&do=diff Question 1, Exercise 10.3 Solutions of Question 1 of Exercise 10.3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. There are four parts in Question 1.$2\sin 6x\sin x$$$-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-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/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}.$$$\overset{\scriptscriptstyle\rightharpoonup}{a}=3\hat{i}$$$$|\overset{\scriptscriptstyle\rightharpoonup}{a}|=\sqrt{{{(3)}^{2}}}=3$$$$\hat{a}=\dfrac{{\overset{\scriptscriptstyle\rightharpo… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-5-p4?rev=1737476038&do=diff Question 5(iii) & 5(iv) Exercise 3.5 Solutions of Question 5(iii) & 5(iv) of Exercise 3.5 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\vec{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}\quad$$\quad\vec{b}=b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k}\quad$$(\vec{a}. \vec{b})^2,\quad|a|^2,\quad|b|^2$\begin{align}\vec{a} \cdot \vec{b}&=(a_1 \hat{i}+a_2 \hat{j} + a_3 \hat{k}) \cdot(b_1 \hat{i}+b_2 \… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p3?rev=1737476038&do=diff Question 5 and 6 Exercise 4.2 Solutions of Question 5 and 6 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$\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}).$$\begin{align}a_n&=\log(a b^{n-1}). \end{align}\begin{align} d&=a_{n+1}-a_n \\ &=\log (a b^n)-\log (a b^{n-1}) \\ &=\log \left(\… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p11?rev=1737476038&do=diff Question 15 Exercise 4.2 Solutions of Question 15 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 15 $n, \dfrac{a^{n+1}+b^{n+1}}{a^n+b^n}$$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/ex4-3-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-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}$$a_1, a_1 r, a_1 r^2, a_1 r^3, a_1 r^4, \ld… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-5-p3?rev=1737476038&do=diff Question 3 Exercise 4.5 Solutions of Question 3 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 $a_2=2$$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-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/ex5-4-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/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$$n$\begin{align}S_n&=5+12 x+19 x^2+26 x^3+\cdots+(7 n-2) x^{n-1}...(i)\\ x S_n&=5 x+12 x^2+19 x^3+\cdots+(7 n-9) x^{n-1}+(7 n-1) x^n....(ii)\end{align}\begin{align}(1-x) S_n&=5+(12-5) x+(19-12) x^2+\cdots\\ &+[7 n-2-(… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-1-p2?rev=1737476038&do=diff Question 3 & 4 Exercise 6.1 Solutions of Question 3 & 4 of Exercise 6.1 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{1}{6 !}+\dfrac{2}{7 !}+\dfrac{3}{8 !}=\dfrac{75}{8 !}$\begin{align}\dfrac{1}{6 !}+\dfrac{2}{7 !}+\dfrac{3}{8 !}&=\dfrac{1}{6 !}+\dfrac{2}{7.6 !}+\dfrac{3}{8.7 .6 !} \\ & =\dfrac{56+16+3}{8 !}\\ &=\dfrac{75}{8 !}\end{align}$\df… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p9?rev=1737476038&do=diff Question 13 Exercise 6.2 Solutions of Question 13 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\mathrm{E}$$n=10$$m_1=4$$E, m_2=2$$L$$m_3=2$$C$\begin{align}\text{total number of permutations are} &=\left(\begin{array}{c} n \\ m_1, m_2, m_3 \end{array}\right)\\&=\left(\begin{array}{c} 10 \\ 4,2,2 \end{array}\right) \\ & =\dfrac{10 !}… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p6?rev=1737476038&do=diff Question 9 & 10 Review Exercise 6 Solutions of Question 9 & 10 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$2,3,0,3,4,2,3$$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 2025-01-21T16:13:59+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-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} $$$$ \begin{aligned} & =4.7^k+3\left[7^k-3^k\… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p3?rev=1737476039&do=diff Question 3, Exercise 10.1 Solutions of Question 3 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin u=\dfrac{3}{5}$$\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-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)$… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-2-p2?rev=1737476039&do=diff Question 2, Exercise 10.2 Solutions of Question 2 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\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$$\cos \theta =-\dfrac{3}{5}$\begin{align}\sin 2\theta &=2\sin \theta \cos \theta \\ &=2\left( \dfrac{4}{5} \right)\left( -\dfrac{3}{5} \rig… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-3-p1?rev=1737476039&do=diff Question 1, Exercise 10.3 Solutions of Question 1 of Exercise 10.3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. There are four parts in Question 1.$2\sin 6x\sin x$$$-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-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-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}$$z=x+iy$\begin{align*} &\arg (z-1)=-\dfrac{\pi}{4} \\ \implies & \arg(x+iy-1) = -\dfrac{\pi}{4} \\ \implies & \arg(x-1+iy) = -\dfrac{\pi}{4} \\ \implies & \tan^{-1}\left(\dfrac{y}{x-1}\right) = -\dfrac{\pi}{4} … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p1?rev=1737476039&do=diff Question 1, Exercise 2.2 Solutions of Question 1 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[a_{i j}\right]$$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/ex2-6-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 2025-01-21T16:13:59+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.$$\begin{align*} … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p2?rev=1737476039&do=diff Question 2, Exercise 4.2 Solutions of Question 2 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $5,9,13, \ldots$$$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$$… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p6?rev=1737476039&do=diff Question 9 and 10, Exercise 4.2 Solutions of Question 9 and 10 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\dfrac{1}{a}, b, \dfrac{1}{c}$$\dfrac{a-c}{2 a c}$$\dfrac{1}{a}, b, \dfrac{1}{c}$\begin{align*} d&=b-\frac{1}{a}\cdots (i)\\ \end{align*}\begin{align*} d&=\frac{1}{c}-b \cdots (ii) \end{align*}\begin{align*} b-\frac{1}{a}&=\frac{1}{c}-… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p11?rev=1737476039&do=diff Question 23 and 24, Exercise 4.3 Solutions of Question 23 and 24 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$ 14+16+18+...+a_{25}.$$$a_1=14$$d=16-14=2$$n=25$$a_25$$S_25$\begin{align} a_n&=a_1+(n-1)d\\ \implies a_{25}&= 14+(25-1)(2)\\ &=62. \end{align}\begin{align} S_n&=\frac{n}{2}[a_1+a_n]\\ \implies S_{25}& =\frac{25}{2}[14+62]\\ & =25 \t… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p12?rev=1737476039&do=diff Question 25 and 26, Exercise 4.3 Solutions of Question 25 and 26 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$ 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-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. $$\begin{align*} S_6&=\frac{(1)\left(1-2^7 \right)}{1-2} \\ &=\frac{1-128}{-2}\\ &=127 \end{align… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-8-p1?rev=1737476040&do=diff Question 1 and 2, Exercise 4.8 Solutions of Question 1 and 2 of Exercise 4.8 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $3+7+13+21+\ldots$$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 2025-01-21T16:14:00+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 2025-01-21T16:14:00+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 2025-01-21T16:14:00+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 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/matric/9th_science/unit_02/exercise_2.1?rev=1737476041&do=diff Exercise 2.1 (Solutions) Question 1 Identify which of the following are rational and irrational numbers: (i) $\sqrt{3}$ (ii) $\frac{1}{6}$ (iii) $\pi$ (iv) $\frac{15}{2}$ (v) $7.25$ (vi)$\sqrt{29}$ Solution * Rational: $\frac{1}{6}$, $\frac{15}{2}$, $7.25$ * Irrational: $\sqrt{3}$, $\pi$, $\sqrt{29}$ Question 2 Convert the following fraction into decimal fraction.$\frac{17}{25}$$\frac{19}{4}$$\frac{57}{8}$$\frac{205}{18}$$\frac{5}{8}$$\frac{25}{38}$$\frac{2}{3}$$\pi$$\frac{1}{9}$$\fr… text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/bsc?rev=1737476042&do=diff ADS/BSc One this page we have listed notes/resources widely used in BSc or in Associated Degree of Science (ADS). This includes notes of some famous books and other resources. Notes of Famous Books Notes of Calculus with Analytic Geometry Notes of Calculus with Analytic Geometry Notes of Mathematical Method Notes of Mathematical Method Introduction to Mechanics Introduction to Mechanics Other notes Notes of Mechanics text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/bsc/notes_of_mathematical_method?rev=1737476035&do=diff Notes of Mathematical Method [BSc Mathematical Method] Notes of the Mathematical Method written by by S.M. Yusuf, A. Majeed and M. Amin and published by Ilmi Kitab Khana, Lahore. This is an old and good book of mathematical method. The notes given here are provided by awesome peoples, who dare to help others. Some of the notes are send by the authors of these notes and other are send by people who didn't write but share these notes as Open Educational Resources (OER). We are thankful to text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-ptb/definitions?rev=1737476037&do=diff Definitions: FSc Part 1 (Mathematics): PTB On this page, all the definitions of “Textbook of Algebra and Trigonometry Class XI, published by Punjab Textbook Board (PTB) Lahore, Pakistan are given. We are very thankful to Muhammad Waqas Sulaiman for his valuable contribution.$\frac{p}{q}$$p,q \in \mathbb{Z}$$q\neq 0$$\frac{p}{q}$$p,q \in \mathbb{Z}$$q\neq 0$$\mathbb{R}$$0.3333....,21.134134$$\pi = 3.1415...$$\divideontimes$$z=x+iy$$x,y \in \mathbb{R}, i = \sqrt{-1}$$x$$y$$z$$2, 3+\sqrt{3}i, \fra… text/html 2025-01-21T16:13:54+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/atiq/math-608/what_is_mathematics?rev=1737476034&do=diff What is Mathematics? Different people would gave different answers of the above title. A student in elementary school would probably say it was about adding, subtracting, multiplying and dividing. Oh yes--- about functions and decimals too. A student in high school would probably say that it is about learning rules and formulas to solve equations. Oh yes 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 2025-01-21T16:13:56+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-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$$\overline{z_1}=2+i$\begin{align} z_1 z_2&=(2-i)(-2+i)\\ &=-4+1+2i+2i\\ &=-3+4i \end{align}\begin{align} \dfrac{z_1 z_2}{\… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-2-p4?rev=1737476036&do=diff Question 5, Exercise 1.2 Solutions of Question 5 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) ${{z}_{1}}=2+4i$${{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}|$$|z_2=\sqrt{c^2+d^2}|$\begin{align} L.H.S.&=|{{z}_{1}}{{z}_{2}}|\\ &=|(a+bi)(c+di)|\\ &=|ac-bd+(ad+bc)i|\\ &=\sqrt{{{(ac-bd)}^{… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-3-p5?rev=1737476036&do=diff Question 6, Exercise 1.3 Solutions of Question 6 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6(i) ${{z}^{4}}+{{z}^{2}}+1=0$\begin{align}{{z}^{4}}+{{z}^{2}}+1&=0\\ {{z}^{4}}+2\left( \dfrac{1}{2} \right){{z}^{2}}+\dfrac{1}{4}-\dfrac{1}{4}+1&=0\\ {{\left( {{z}^{2}}+\dfrac{1}{2} \right)}^{2}}+\dfrac{4-1}{4}&=0\\ {{\left( {{z}^{2}}+\dfrac{1}{2} \righ… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-1-p4?rev=1737476036&do=diff Question, Exercise 10.1 Solutions of Question 4 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin \alpha =-\dfrac{4}{5}$$\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}.$$$\… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-3-p4?rev=1737476037&do=diff Question 5, Exercise 10.3 Solutions of Question 5 of Exercise 10.3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$\cos {{20}^{\circ }}\cos {{40}^{\circ }}\cos {{60}^{\circ }}\cos {{80}^{\circ }}=\dfrac{1}{16}.$$$2\cos \alpha \cos \beta =\cos \left( \alpha +\beta \right)+\cos \left( \alpha -\beta \right)$\begin{align}L.H.S.&=\cos {… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-3-p5?rev=1737476037&do=diff Question 5, Exercise 10.3 Solutions of Question 5 of Exercise 10.3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos {{20}^{\circ }}\cos {{40}^{\circ }}\cos {{60}^{\circ }}\cos {{80}^{\circ }}=\dfrac{1}{16}$$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/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-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$$\overline{z_1}=2+i$\begin{align} z_1 z_2&=(2-i)(-2+i)\\ &=-4+1+2i+2i\\ &=-3+4i \end{align}\begin{align} \dfrac{z_1 z_2}{\… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-2-p4?rev=1737476037&do=diff Question 5, Exercise 1.2 Solutions of Question 5 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) ${{z}_{1}}=2+4i$${{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}|$$|z_2=\sqrt{c^2+d^2}|$\begin{align} L.H.S.&=|{{z}_{1}}{{z}_{2}}|\\ &=|(a+bi)(c+di)|\\ &=|ac-bd+(ad+bc)i|\\ &=\sqrt{{{(ac-bd)}^{… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p6?rev=1737476037&do=diff Question 6, Exercise 2.2 Solutions of Question 6 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Questiopn 6(i) $\left| \begin{matrix}a-b & b-c & c-a \\b-c & c-a & a-b \\c-a & a-b & b-c \end{matrix} \right|=0$\begin{align} L.H.S&=\left| \begin{matrix} a-b & b-c & c-a \\ b-c & c-a & a-b \\ c-a & a-b & b-c \\ \end{matrix} \right| \\ &=\left| \… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-2-p4?rev=1737476037&do=diff Question 5 & 6, Exercise 3.2 Solutions of Question 5 & 6 of Exercise 3.2 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5 Find the length of the vector $\overrightarrow{AB}$$\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-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 2025-01-21T16:13:57+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}$$\vec{c}=2 \hat{i}+\hat{j}-4 \hat{k}$\begin{align}|\vec{a}|&… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-4-p3?rev=1737476037&do=diff Question 3 Exercise 3.4 Solutions of Question 3 of Exercise 3.4 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3(i) Find a unit vector that is orthogonal to the given vector $\vec{a}=\hat{i}- 2 \hat{j}+3 \hat{k}, \quad \vec{b}=2 \hat{i}+\hat{j}-\hat{k}$$\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/ex3-5-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/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}$$\vec{c}=\lambda \hat{j}+3 \hat{k}$\begin{align}\vec{a} \cdot \vec{b} \times \vec{c}&=0 \\ \Rightarrow\left|\begin{array}{ccc} 1 & 3 & 1 \\ 2 & -1 & -1 \\ 0 & \lamb… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/review-ex3-p6?rev=1737476038&do=diff Question 10 Review Exercise 3 Solutions of Question 10 of Review Exercise 3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 10(i) $A B C$$|\vec{a}|^2=|\vec{b}|^2+|\vec{c}|^2 -2|\vec{b}|| \vec{c}| \cos A$$A B C$$\vec{a}, \vec{b}$$\vec{c}$\begin{align} \vec{b}&=\vec{a}+\vec{c} \\ \Rightarrow \vec{a}&=\vec{b}-\vec{c} \\ \Rightarrow \vec{a} \cdot \vec{a}&=(\vec{b}-\vec{c}) \cd… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-1-p2?rev=1737476038&do=diff Question 3 and 4 Exercise 4.1 Solutions of Question 3 and 4 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{1}{2}, \dfrac{2}{3} \dfrac{3}{4}, \dfrac{4}{5}, \ldots$$$\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-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.$$$r=0$\begin{align}&P_{0+1… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p12?rev=1737476038&do=diff Question 16 Exercise 4.2 Solutions of Question 16 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 16 $5$$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/ex4-3-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$$120$\begin{align}(a-3 d)^2+(a-d)^2+(a+d)^2+(a+2 d)^2&=120 \\ \Rightarrow a^2-6 a d+… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-3-p7?rev=1737476038&do=diff Question 11 & 12 Exercise 4.3 Solutions of Question 11 & 12 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$16 \mathrm{ft}$$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/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 {.}$$\begin… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-4-p7?rev=1737476038&do=diff Question 10 Exercise 4.4 Solutions of Question 10 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 10 $48$$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-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/ex5-1-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/ex5-2-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/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/ex6-2-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})$$\begin{align}n(^{n-1} P_{r-1})&=n \dfrac{(n-1) !}{((n-1)-(r-1)) !} \\ & =\dfrac{n(n-1) !}{(n-r) !}\\ &=\dfrac{n !}{(n-r) !}\\ &=^n P_r\end{align}$^n P_r=^{n-1} P_r+r(^{n-1} … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p3?rev=1737476038&do=diff Question 5 and 6 Exercise 6.2 Solutions of Question 5 and 6 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$7.$$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/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/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}$$x^{23}$\begin{align}T_{r-1}&=\dfrac{20 !}{(20-r) ! r !}(x^2)^{20 r}(-x)^r \\ & =\dfrac{20 !}{(20-r) ! r !}(-1)^r \cdot x^{40-2 r+r} \\ & =\dfrac{20 !}{(20-r) ! r !}(-1)^r x^{40-r}\end{… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p7?rev=1737476038&do=diff Question 7 Exercise 7.2 Solutions of Question 7 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$(2+\sqrt{3})^5+(2-\sqrt{3})^5$\begin{align}(2+\sqrt{3})^5+(2 \cdot \sqrt{3})^5& =[(2)^5+{ }^5 C_1 \cdot 2^4 \cdot \sqrt{3}+{ }^5 C_2 \cdot 2^3 \cdot(\sqrt{3})^2 \\ & +^5 C_3 \cdot 2^2 \cdot(\sqrt{3})^4+{ }^5 C_4 \cdot 2 \cdot(\sqrt{3})^4 \\ … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p8?rev=1737476038&do=diff Question 8 Exercise 7.2 Solutions of Question 8 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$(3-2 x)^{10}$$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/ex7-3-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$$$ \frac{(8+3 x)^{\frac{2}{3}}}{(2+3 x) \sqrt{4-5 x}}=1-\frac{5 x}{8} $$$$ \frac{\sqrt[4]{3}-3 x j^{\frac{2}{3}}}{2 \cdot 3 x+4-5 x} $$$$ \begin{aligned} & =\frac{8^{\frac{2}{3}}\left(1+\frac{3 x}{8}\right)^{\frac{2}{3}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/re-ex7-p4?rev=1737476039&do=diff Question 5 & 6 Review Exercise 7 Solutions of Question 5 & 6 of Review Exercise 7 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\left(\frac{2}{x^2}+\frac{x^2}{2}\right)^{10}$$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/ex10-1-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/ex10-3-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)$\begin{align}L.H.S.&=\cos {… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-3-p5?rev=1737476039&do=diff Question 5, Exercise 10.3 Solutions of Question 5 of Exercise 10.3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos {{20}^{\circ }}\cos {{40}^{\circ }}\cos {{60}^{\circ }}\cos {{80}^{\circ }}=\dfrac{1}{16}$$2\cos \alpha \cos \beta =\cos \left( \alpha +\beta \right)+\cos \left( \alpha -\beta \right)$\begin{align}L.H.S.&=\cos {{20… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p4?rev=1737476039&do=diff Question 4, Exercise 2.2 Solutions of Question 4 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A$\begin{align}\left[\begin{array}{cc} 2 & 1 \\ 3 & 2 \end{array}\right]A\left[\begin{array}{cc} 1 & 3 \\ 2 & 4 \end{array}\right]&=\left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right]\end{align}$ B = \left[\begin{array}{cc} 2 & 1 \\ 3… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p7?rev=1737476039&do=diff Question 7, Exercise 2.2 Solutions of Question 7 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{ll}x & 0 \\ y & 1\end{array}\right]$$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/ex2-5-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]$$A A^{-1}=A^{-1} A=I$\begin{align*} A&=\left[ \begin{array}{ccc} 0 & -1 & -1 \\ -1 & 3 & 0 \\ 1 & -1 & 4 \end{array} \right]\\ |A|&=0+1(-4)-1(1-3)\\ &=-4+3\… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p3?rev=1737476039&do=diff Question 3, Exercise 2.6 Solutions of Question 3 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 x+3 y+4 z=2$$2 x+y+z=5$$3 x-2 y+z=-3$\begin{align*} \begin{aligned} 2x + 3y + 4z &= 2 \\ 2x + y + z &= 5 \\ 3x - 2y + z &= -3 \end{aligned}\end{align*}\begin{align*} A_{b} &=\quad \left[\begin{array}{cccc} 2 & 3 & 4 & 2 \\ 2 & 1 & 1 & 5 \\ 3… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p9?rev=1737476039&do=diff Question 17, 18 and 19, Exercise 4.3 Solutions of Question 17, 18 and 19 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $6+12+18+\ldots+96$$$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 2025-01-21T16:14:00+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/ex4-7-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/ex4-8-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$$k$\begin{align*} T_k &= \frac{1}{(2k+3)(2k+9)}. \end{align*}\begin{align*} \frac{1}{(2k+3)(2k+9)} = \frac{A}{2k+3} + \frac{B}{2k+9} \ldots … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p2?rev=1737476040&do=diff Question 2, Exercise 8.1 Solutions of Question 2 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\cos 15^{\circ}$$\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-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$$\cos$\begin{align*… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/re-ex-p2?rev=1737476040&do=diff Question 2, Review Exercise Solutions of Question 2 of Review Exercise of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin \theta=\dfrac{3}{5}, \sin \phi=\dfrac{5}{13}$$\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 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk?rev=1737476042&do=diff FSc Part 1 (Mathematics): KPK [A Textbook of Mathematics for Class XI] <lead>A Textbook of Mathematics for Class XI is published by Khybar Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. The book has total of twelve (12) chapters. </lead> <callout type=“info” icon=“true text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-ptb?rev=1737476042&do=diff FSc/ICS Part 1 (Mathematics): PTB [Textbook of Algebra and Trigonometry Class XI] <lead>Textbook of Algebra and Trigonometry Class XI is published by Punjab Textbook Board (PTB) Lahore, Pakistan. The book has total of 14 chapters.</lead> One this page, we have posted Notes (Solutions), MCQs, short question, sample papers and old papers related to this subject. This book has wide scope and it is part of syllabus of Mathematics in FSc/ICS from all board (Board of Intermediate and Secondary Educa… text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part2-ptb?rev=1737476042&do=diff FSc/ICS Part 2 (Mathematics): PTB [Calculus and Analytic Geometry, MATHEMATICS 12] <lead>Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc/ICS Part 2 or HSSC-II), Punjab Textbook Board (PTB) Lahore, Pakistan. There are total seven (7) units in this book.</lead> One this page we have posted Notes (Solutions), MCQs, short question, sample papers and old papers related to this subject. This book has wide scope and it is part of syllabus of Mathematics in FSc/ICS from all board (Boar… text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk?rev=1737476042&do=diff FSc/ICS Part 1 (Mathematics): KPK [A Textbook of Mathematics for Class XI] <lead>A Textbook of Mathematics for Class XI is published by Khybar Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. The book has total of twelve (12) chapters. </lead> <callout type=“info” icon= text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/matric?rev=1737476042&do=diff Matric <callout type=“success” icon=“fa fa-info”> This is a new section. We need to do lot of work to complete this section. We will add notes of mathematics for 9th and 10th classes (general and science). </callout> Mathematics (Science Group) text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/participate?rev=1737476042&do=diff Participate <callout type=“tip” icon=“true”> If you have written notes, then why not share on MathCity.org with others. This is your help to mathematics students and teachers (یہ صدقہ جاریہ ہے). You may send us hard copies, we will return your notes after scanning and publish it on MathCity.org. Please contact text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/report_abuse?rev=1737476042&do=diff Report Abuse All the resources published/posted on MathCity.org are sent by different people to help the other people for the promotion of mathematics. These are open educational resources (OER). OER are teaching, learning, and research resources that reside in the public domain or have been released under an intellectual property license that permits their free use or re-purposing by others. Open educational resources include full courses, course materials, modules, textbooks, streaming vide… text/html 2025-01-21T16:13:54+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/atiq/fa21-mth322?rev=1737476034&do=diff MTH322: Real Analysis II (Fall 2021) This course is offered to MSc, Semester II at Department of Mathematics, COMSATS University Islamabad, Attock campus. This course need rigorous knowledge of continuity, differentiation, integration, sequences and series of numbers, that is many notion included in $\int_{1}^{\infty }{{{x}^{-p}} dx}$$p$$f\in \mathcal{R}[a,b]$$b\ge a$$f(x)\ge 0$$x\ge a$$\int_{a}^{\infty }{f(x) dx}$$M>0$$\int\limits_{a}^{b}{f(x)\,dx} \le M$$b\ge a$$f\in \mathcal{R}[a,b]$$b\ge a$… text/html 2025-01-21T16:13:54+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/atiq/sp23-mth322?rev=1737476034&do=diff MTH322: Real Analysis II (Spring 2023) [MTH322: Real Analysis II (Spring 2023)] This course is offered to BS, Semester VI at Department of Mathematics, COMSATS University Islamabad, Attock campus. This course need rigorous knowledge of continuity, differentiation, integration, sequences and series of numbers, that is many notions included in $f\in \mathcal{R}[a,b]$$b\ge a$$f(x)\ge 0$$x\ge a$$\int_{\,a}^{\,\infty }{f(x)\,dx}$$M>0$$\int\limits_{a}^{b}{f(x)\,dx}\leq M$$b\ge a$$f(x)$$g(x)$$x>a$$\li… text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/bsc/number-theory-by-prof-asghar-ali?rev=1737476035&do=diff Number Theory by Prof. Asghar Ali [Number Theory by M Asghar Ali] We are very thankful to Prof. Asghar Ali for send these notes. These notes are very helpful to prepare BSc or ADS mathematics portion of Number Theory. Number theory is a subject in which students learn different concepts created on the set of integers. For example, the concept of divisibilty exists in the set of integer. Let a and b be any two integers suhc that $a\neq 0$ text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part2-ptb/definitions-muzzammil-subhan?rev=1737476037&do=diff Definitions: Mathematics 12: PTB by Muzzammil Subhan Definitions from Calculus and Analytic Geometry, MATHEMATICS 12, published by Punjab Textbook Board (PTB) Lahore, Pakistan. We are very thankful to Muzzammil Subhan for his valuable contribution. Download or view PDF for all definitions. Samples is given below$P(x)=a_0 x^0+a_1 x^1+a_2 x^2+\ldots . .+a_{n-1} x^{n-1}+a_n x^n$$n \in W$$a_0, a_1, a_2, \ldots, a_n \in R$$f(x)=a x+b$$a, b \in R$$a \neq 0$$f(x)=x$$f(x)=c$$c \in R$$\frac{P(x)}{Q(x)}$… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/analysis?rev=1737476041&do=diff Notes of Analysis advanced-analysis-handwritten-notes Advanced Analysis: Handwritten Notes Advanced-Analysis-iqra-liaqat Advance Analysis by Ms. Iqra Liaqat Complex-Analysis-Dr-Amir-Mahmood Complex Analysis (Easy Notes of Complex Analysis) Complex-Analysis-M-Usman-Hamid Complex Analysis by M Usman Hamid complex-analysis-iqra-liaqat Complex Analysis (Notes) by Ms. Iqra Liaqat Complex-Analysis-Quick-Review Complex Analysis (Quick Review) Akhtar Abbas fundamental-of-complex-… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/mathematical-method-muzammil-tanveer?rev=1737476041&do=diff Mathematical Method by Sir Muhammad Awais Aun [Mathematical Method by Muzammil Tanveer] Mathematical methods are the approaches employed by mathematicians to address issues in mathematics and science. Algebra, functions, relations and associated graphs, calculus, and statistics are examples of mathematical techniques. Through their usage in resolving practical issues, they are applied to modelling. text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/measure-thoery-muhsa?rev=1737476041&do=diff Measure Theory by M Usman Hamid & Saima Akram [Measure Theory by M Usman Hamid & Saima Akram] The study of measures on sets is the focus of the mathematical field known as measure theory. A measure is a function that gives specific subsets of a given set a non-negative real integer, indicating their size inferentially. The concept of measure is a formalisation and generalisation of common concepts like mass and event probability as well as geometrical measurements (length, area, and volume). text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/multivariable-calculus-sheikh-muhammad-saleem-shahzad?rev=1737476041&do=diff Multivariable Calculus by Sheikh Muhammad Saleem Shahzad [Multivariable Calculus by Sheikh Muhammad Saleem Shahzad] * Have you ever wondered how we can understand the speed of a moving object at any instant of time? * Did you know that Calculus can help us predict future trends by analyzing patterns in data? text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/numerical-analysis-by-m-usman-hamid?rev=1737476041&do=diff Numerical Analysis by M Usman Hamid These notes are initially provided by Mr. Anwar Khan. Later the updated version is send by Muhammad Tahir. We are really very thankful to Mr. Anwar Khan and Muhammad Tahir for providing these notes and appreciates their effort to publish these notes on MathCity.org text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/vector-and-tensor-analysis-by-dr-nawazish?rev=1737476042&do=diff Vector & Tensor Analysis by Dr Nawazish Ali (Solutions) [Vector & Tensor Analysis by Dr Nawazish Ali (Solutions)] We are very thankful to Prof. Fazal Abbas Sajid for sharing these solutions. Vector & Tensor Analysis for Scientists and Engineers, by Prof. Dr. Nawazish Ali Shah is a famous book taught in different universities of the Pakistan. On this page, we have added the solutions of the exercises of the book. Solutions of Chapter 5 is written by text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/people/muzzammil-subhan?rev=1737476042&do=diff Muzzammil Subhan [Muzzammil Subhan] We'd like to express our deepest gratitude to Mr. Muzzammil Subhan from Narowal city! A true pioneer contributor, he's not only a fantastic teacher but also a brilliant mathematician. His contributions have been invaluable to MathCity.org, and we're so lucky to have him as part of our community. 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 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/ppsc/ppsc-maths-2015?rev=1737476042&do=diff PPSC Paper 2015 (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 Kaushef Salamat. We are very thankful to her for providing this paper.\(\displaystyle \int_{-4}^{0}\frac{tdt}{\sqrt{16-t62}}\)$0$$-4$$4$\(A\cos wt+B\sin wt\)$\… text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/ppsc/ppsc-mock-interview-mathematics?rev=1737476042&do=diff PPSC Mock Interview Lecturer Mathematics [PPSC Mock Interview Lecturer Mathematics] This handout is shared by Mr. Rashad Wattu and written by Sawaira Sikandar. We are really very thankful to him for providing this handout and appreciates his efforts to publish it on MathCity.org. This handout contains the questions collected from the different interviews of the Lecturer in Mathematics conducted by Public Service Commission (PSC). This is help full to prepare interview of all types of jobs whic… 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 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/bsc/notes_of_mechanics/introduction_to_mechanics?rev=1737476035&do=diff Introduction to Mechanics [Introduction to Mechanics by Q.K Ghori] Notes of Introduction to Mechanics by Q. K. Ghori published by West Pak Publishing Company (Pvt) Ltd. There are thirteen chapters in this book. We don't have all the notes of this book but the notes which we have are listed below. Please choose your require chapter to see the notes. text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/matric/9th_science/ex-4-1?rev=1737476041&do=diff Exercise 4.1 On the following page we have given the solution of Exercise 4.1 of Mathematics 9 (Science) published by Caravan Book House, Lahore. <WRAP center round info 60%> We have created this page and it will be updated to add new solutions occasionally. Please stay in touch with this page. </WRAP>$3x^2+\frac{1}{x}-5$$3x^3-4x^2-x\sqrt{x}+3$$x^2-3x+\sqrt{2}$$\frac{3x}{2x-1}+8$$3x^2+\frac{1}{x}-5$$No (Reason:\frac{1}{x})$$3x^3-4x^2-x\sqrt{x}+3$$No (Reasons \sqrt{x})$$x^2-3x+\sqrt{2}$$Yes$$\f… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/matric/9th_science/ex-6-2?rev=1737476041&do=diff Exercise 6.2 On the following page we have given the solution of Exercise 6.2 of Mathematics 9 (Science) published by Caravan Book House, Lahore. <WRAP center round info 60%> We have created this page and it will be updated to add new solutions occasionally. Please stay in touch with this page. </WRAP>$\frac{x^2-x-6}{x^2-9}+\frac{x^2+2x-24}{x^2-x-12}$\begin{align} \frac{x^2-x-6}{x^2-9}&+\frac{x^2+2x-24}{x^2-x-12}\\ &=\frac{x^2-3x+2x-6}{(x)^2-(3)^2}+\frac{x^2+6x-4x-24}{x^2-4x+3x-12}\\&= \frac{x(… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/matric/9th_science/ex-6-3?rev=1737476041&do=diff Exercise 6.3 On the following page we have given the solution of Exercise 6.3 of Mathematics 9 (Science) published by Caravan Book House, Lahore. <WRAP center round info 60%> We have created this page and it will be updated to add new solutions occasionally. Please stay in touch with this page. </WRAP>$4x^2-12xy +9y^2$$x^2-1+\frac{1}{4x^2}, (x\neq 0)$$\frac{1}{16}x^2-\frac{1}{12}xy+ \frac{1}{36}y^2$$4(a+b)^2-12(a^2-b^2)+9(a-b)^2$$\frac{4x^6-12x^3y^3+9y^6}{9x^4-24x^2y^2+16y^4},(x \neq 0)$$\left(… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/matric/9th_science/review_exercise?rev=1737476041&do=diff Review exercise On the following page we have given the solution of Review exercise of Mathematics 9 (Science) published by Caravan Book House, Lahore. <WRAP center round info 60%> We have created this page and it will be updated to add new solutions occasionally. Please stay in touch with this page. </WRAP>$x-2$$x^2+x-6$$x^2+x-6$$x+3$$x-2$$x+2$$c$$a^3+b^3$$a^2-ab+b^2$$a+b$$a^2-ab+b^2$$(a-b)^2$$a^2+b^2$$c$$x^2-5x+6$$x^2-x-6$$x-3$$x+2$$x^2-4$$x-2$$a$$a^2-b^2$$a^3-b^3$$a-b$$a+b$$a^2+ab+b^2$$a^2-a… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/msc/mcqs_short_questions/real_analysis?rev=1737476041&do=diff Real Analysis: Short Questions and MCQs <callout type=“info” icon=“true”> We are going to add short questions and MCQs for Real Analysis. The subject is similar to calculus but little bit more abstract. This is a compulsory subject in MSc and BS Mathematics in most of the universities of Pakistan. The author of this page is Dr. $\left\{\frac{1}{n+1} \right\}$$\left\{\frac{n+2}{n+1} \right\}$$\{x_n\}$$\{y_n\}$$\lim_{n\to\infty z_n}$$z_n=x_n-2y_n$$\{x_n\}$$\{y_n\}$$\lim_{n\to\infty z_n}$$x_n=2y_n… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/msc/notes/mathematical-method-by-khalid-latif-mir?rev=1737476041&do=diff Mathematical Method by Khalid Latif Mir (Solutions) [Mathematical Method by Khalid Latif Mir (Solutions)] We are very thankful to Prof. Fazal Abbas Sajid for sharing these solutions. Problems & Methods Mathematical Method by Khalid Latif Mir is a famous book taught in different universities of the Pakistan at BS and Master level. On this page, we have added the solutions of the exercises of the book. The solutions of Chapter 06: The Laplace Transform and its Applications is written by Marrium … text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/msc/notes/theoretical-mechanics-by-khalid-latif-mir?rev=1737476041&do=diff Theoretical Mechanics by Khalid Latif Mir (Solutions) [Theoretical Mechanics by Khalid Latif Mir (Solutions)] We are very thankful to Prof. Fazal Abbas Sajid for sharing these solutions. An Intermediate Course in Theoretical Mechanics By Khalid Latif Mir is a famous book taught in different universities of the Pakistan. On this page, we have added the solutions of the exercises of the book. 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$$$\begin{array}{c|cccc} -2 & 1 & 0 & 6 & 20 \\ & \d… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-2-p1?rev=1737476036&do=diff Question 1, Exercise 10.2 Solutions of Question 1 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. There are four parts in Question 1.$\sin 2\theta ,\,\,\cos 2\theta$$\tan 2\theta$$\tan \theta =-\dfrac{1}{5}$$\theta$$\sin \theta =\dfrac{1}{\sqrt{26}}$$\cos \theta =\dfrac{-5}{\sqrt{26}}$\begin{align}\sin 2\theta &=2\sin… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-2-p4?rev=1737476036&do=diff Question 4 and 5, Exercise 10.2 Solutions of Question 4 and 5 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\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 2025-01-21T16:13:57+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$$$\begin{array}{c|cccc} -2 & 1 & 0 & 6 & 20 \\ & \d… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p4?rev=1737476037&do=diff Question 4, Exercise 2.1 Solutions of Question 4 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4 $A= \begin{bmatrix}1 & 4 & 4 \\ 4 & 1 & 4 \\ 4 & 4 & 1 \end{bmatrix}$$\dfrac{1}{3}A^2-2A-9I=0$$A=\begin{bmatrix} 1 & 4 & 4 \\ 4 & 1 & 4 \\ 4 & 4 & 1 \end{bmatrix}$\begin{align}\frac{1}{3}A^2&=\frac{1}{3}\left[ \begin{matrix} 1 & 4 & 4 … text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p11?rev=1737476037&do=diff Question 12, Exercise 2.1 Solutions of Question 12 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 12(i) $A=\begin{bmatrix}3 & 2 & 1 \\4 & 5 & 6 \\-2 & 3 & 4\end{bmatrix}$$A+A^t$$$A=\left[ \begin{matrix} 3 & 2 & 1 \\ 4 & 5 & 6 \\ -2 & 3 & 4 \\ \end{matrix} \right]$$$$A^t=\left[ \begin{matrix} 3 & 4 & -2 \\ 2 & 5 & 3 \… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p12?rev=1737476037&do=diff Question 13, Exercise 2.1 Solutions of Question 13 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 13(i) $A$$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]$$$$A^t=\left[ \begin{matrix} a_{11} & a_{21} & a_{31} \\ a_{12} & a_{22} … text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p8?rev=1737476037&do=diff Question 8,9 & 10, Exercise 2.2 Solutions of Questions 8,9 & 10 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\left| \begin{matrix}1+x & y & z \\x & 1+y & z \\x & y & 1+z \end{matrix} \right|=1+x+y+z$$$L.H.S.=\left| \begin{matrix} 1+x & y & z \\ x & 1+y & z \\ x & y & 1+z \\ \end{matrix} \right|$$$$=\left| \begin{matrix} 1 & 0 & -… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p12?rev=1737476037&do=diff Question 14 & 15, Exercise 2.2 Solutions of Questions 14 & 15 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$A=\begin{bmatrix}0 & 2 & 2 \\-1 & 3 & 2 \\1 & 0 & 5\end{bmatrix}$$A^{-1}$$$A=\left[ \begin{matrix} 0 & 2 & 2 \\ -1 & 3 & 2 \\ 1 & 0 & 5 \\ \end{matrix} \right]$$$A^{-1}$$$A^{-1}=\dfrac{Adj\,\,A}{|A|}$$$$Adj\,\,A={{\left[ \begin… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-2-p5?rev=1737476037&do=diff Question 7, Exercise 3.2 Solutions of Question 7 of Exercise 3.2 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7(i) Find the components and the magnitude of $\overrightarrow{PQ}$$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)$$Q(2,-1)$\begin{align}\overrightarrow{PQ}&=\overrightarrow{OQ}-\overrightarrow{OP}\\ &=(2\hat{i}-\hat{j})-(-\hat{i}+2\hat{j})\\ &=3\hat{i}-3\hat{j}\end{align}\begin{align}|\overrighta… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-3-p3?rev=1737476037&do=diff Question 4 and 5 Exercise 3.3 Solutions of Question 4 and 5 of Exercise 3.3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4 $\hat{i}+7 \hat{j} + 3 \hat{k}$$\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}$$\vec{c} = 2 \hat{i}-\hat{j}-3 \hat{k}$\begin{align}\vec{a} \cdot \vec{b}&=(\hat{i}+7 \h… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-5-p3?rev=1737476038&do=diff Question 5(i) & 5(ii) Exercise 3.5 Solutions of Question 5(i) & 5(ii) of Exercise 3.5 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) $\vec{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}\quad$$\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-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-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/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 $$\begin{align}\text { Now } \vec{r} \times \hat{i}&=\left|\begin{array}{ccc} \hat{… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/review-ex3-p5?rev=1737476038&do=diff Question 8 & 9 Review Exercise 3 Solutions of Question 8 & 9 of Review Exercise 3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 8 $(0,0,2),(-1,3,2),(1,0,4)$$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)$$$ \text{Area of triangle} =\dfr… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-1-p3?rev=1737476038&do=diff Question 5 Exercise 4.1 Solutions of Question 5 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) $\sum_{j=1}^6(2 j-3)$\begin{align}\sum_{j=1}^6(2 j-3)&=(2.1-3)+(2.2-3)+(2.3-3)+(2.4-3)\\&+(2.5-3)+(2.6-3) \\ \implies \sum_{j=1}^6(2 j-3)&=-1+1+3+5+7+9 .\end{align}$\sum_{k=1}^5(-1)^k 2^{k-1}$\begin{align}\sum_{k=1}^5(-1)^k 2^{k-1}& =(-1)^1 2^{1-… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p1?rev=1737476038&do=diff Question 1 and 2 Exercise 4.2 Solutions of Question 1 and 2 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$15$$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.$$\begin{align} &a_{21}=8+(21-1) d \\ \implies &108=8+20 d\\ \implies &20 d=108-8=100 \\ \imp… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p2?rev=1737476038&do=diff Question 3 and 4 Exercise 4.2 Solutions of Question 3 and 4 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$6,9,12, \ldots, 78$$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-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$$d$\begin{align} &a_6+a_4=6 \\ \implies & a_1+5d+a_1+3d=6\\ \implies & 2a_1+8d=6\\ \implies & a_1+4d=3 --- (1) \end{align}\begin{align} &a_6-a_4=\dfrac{2}{3} \\ \implies & a_1+5d-a_1-3d=\dfrac{2}{3}\\ \implies &… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p7?rev=1737476038&do=diff Question 10 Exercise 4.2 Solutions of Question 10 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 10 $500$$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/ex4-3-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$$n$\begin{align} S_n&=\dfrac{n}{2}[a_1+a_n], \\ \implies S_{20}&=\dfrac{20}{2}[9-29] … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-3-p3?rev=1737476038&do=diff Question 3 & 4 Exercise 4.3 Solutions of Question 3 & 4 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 $5$$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-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/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}$$\begin{align}\dfrac{1}{64}&=16(\dfrac{1}{2})^{n-1} \\ \Rightarrow(\dfrac{1}{2})^{n-1}&=\dfrac{1}{64 \times 16}=\dfrac{1}{1024} … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-5-p10?rev=1737476038&do=diff Question 15 & 16 Exercise 4.5 Solutions of Question 15 & 16 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$2$$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$$a_1=1$\begi… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-1-p2?rev=1737476038&do=diff Question 2 & 3 Exercise 5.1 Solutions of Question 2 & 3 of Exercise 5.1 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Q2 Find the sum $1.2+2.3+3.4+\ldots+99.100$$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/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$$n^{t h}$\begin{align} & T_j=j(j+1)(j+2)-j(j^2+3 j+2) \\ & =j^3+3 j^2+2 j\end{align}\begin{align} & \sum_{j=1}^n T_j=\sum_{j=1}^n … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-1-p6?rev=1737476038&do=diff Question 9 Exercise 5.1 Solutions of Question 9 of Exercise 5.1 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9(i) $n$$n$$n$\begin{align} & T_n=n^2(2 n+3)=2 n^3+3 n^2 \\ & \Rightarrow T_j=2 j^3+3 j^2\end{align}\begin{align} & \sum_{j=1}^n T_j=2 \sum_{j=1}^n j^3+3 \sum_{j=1}^n j^2 \\ & =2(\dfrac{n(n+1)}{2})^2+3 \dfrac{n(n+1)(2 n+1)}{6} \\ & =\dfrac{n(n+1)}{2}… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-3-p3?rev=1737476038&do=diff Question 3 Exercise 5.3 Solutions of Question 3 of Exercise 5.3 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 $n$$n$$4+10+18+28+40+\ldots$\begin{align} & a_2-a_1=10-4=6 \\ & a_3-a_2=18-10=8 \\ & a_4-a_3=28-18=10 \\ & \text {... ... ... } \\ & \text {... ... ... } \\ & a_n-a_{n \quad 1}=(\mathrm{n}-1) \text { term of the sequence } \end{align}$6,10,8, \ldot… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/re-ex5-p7?rev=1737476038&do=diff Question 9 Review Exercise Solutions of Question 9 of Review Exercise of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9(i) $n$$3+7+13+21+31+\ldots$\begin{align} & a_2-a_1=7-3=4 \\ & a_3-a_2=13-7=6 \\ & a_4-a_3=21-13=8 \\ & \ldots \quad \ldots \quad \ldots \\ & \ldots \quad \cdots \quad \ldots \\ & a_n-a_{n-1}=(n-1) \text { term of the series } \\ & 4,6,8, \ldo… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-1-p3?rev=1737476038&do=diff Question 5 Exercise 6.1 Solutions of Question 5 of Exercise 6.1 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\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/ex6-2-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/ex6-3-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$${ }^n C_r=35$\begin{align} &^n P_r=\dfrac{n !}{(n-r) !}=840 ....(i)\\ &^n C_r=\dfrac{n !}{(n-r) ! r !}=35....(ii)\end{align}\begin{align}\dfrac{n !}{(n-r) !} \cdot \dfrac{(n-r) ! r !}{n !}&=\dfrac{840}{35}\\ r!&=24\\ \te… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-5-p3?rev=1737476038&do=diff Question 5 and 6 Exercise 6.5 Solutions of Question 5 and 6 of Exercise 6.5 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{8}{9}$$$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$$4$\begin{align}S… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-5-p4?rev=1737476038&do=diff Question 7 Exercise 6.5 Solutions of Question 7 of Exercise 6.5 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$52$$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}$$=1-\dfrac{17}{52}=\dfr… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p4?rev=1737476038&do=diff Question 5 & 6 Review Exercise 6 Solutions of Question 5 & 6 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$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/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-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) !]$$a_{k-1}$\begin{ali… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p8?rev=1737476038&do=diff Question 8 Exercise 7.1 Solutions of Question 8 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$1+2+2^2+2^3+\ldots+2^n 1=2^n-1$$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-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$\begin{align} & 2^{k+1}=2^k \cdot 2>k \cdot 2 \quad \text { by (i) } \\ & \Rightarrow 2^{k+1}>2 k=k+k \\ &\Rightarrow 2^{k+1}>k+1 \text {. as } k>1… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p3?rev=1737476038&do=diff Question 3 Exercise 7.2 Solutions of Question 3 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$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$\begin{align} T_{r+1}&=\dfrac{23 !}{(23-r) ! r !}(2 \sqrt{x})^{23-r}(-\dfrac{3}{x \sqrt{x}})^r \\ & =\dfrac{23 !}{(23-r) ! r !} \cdot 2^{23-r} \cdot(-3)… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p9?rev=1737476039&do=diff Question 9 Exercise 7.2 Solutions of Question 9 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$(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-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}$$$ \begin{aligned} & \sqrt{26}=\sqrt{25+1} \\ & =\sqrt{25} \sqrt{1+\frac{1}{25}}=5\left[1+\frac{1}{25}\right]^{\frac{1}{2}} \end{aligned} $$$$ \begin{aligned} & \sqrt{26}=5\left[1+\frac{1}{25}\right]^{\frac{1}{2}} \\ & =5\left[1+\f… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p11?rev=1737476039&do=diff Question 13 Exercise 7.3 Solutions of Question 13 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$x$$x^3$$x$$n^{\text {th }}$$1+x$$\frac{2 n+(n+1) x}{2 n+(n-1) x}$$$ \begin{aligned} & (1+x)^{\frac{1}{n}}=\frac{2 n+(n+1) x}{2 n+(n-1) x} \\ & \frac{2 n+(n+1) x}{2 n+(n-1) x} \\ & =1+\frac{1}{n} x+\frac{\frac{1}{n}\left(\frac{1}{n}-1\right… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p6?rev=1737476039&do=diff Question 6, Exercise 10.1 Solutions of Question 1 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos \alpha =2{{\cos }^{2}}\dfrac{\alpha }{2}-1=1-2{{\sin }^{2}}\dfrac{\alpha }{2}$\begin{align}\cos \alpha &=\cos 2\dfrac{\alpha }{2}\\ &={{\cos }^{2}}\dfrac{\alpha }{2}-{{\sin }^{2}}\dfrac{\alpha }{2}\\ &={{\cos }^{2}}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-2-p1?rev=1737476039&do=diff Question 1, Exercise 10.2 Solutions of Question 1 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. There are four parts in Question 1.$\sin 2\theta ,\,\,\cos 2\theta$$\tan 2\theta$$\tan \theta =-\dfrac{1}{5}$$\theta$$\sin \theta =\dfrac{1}{\sqrt{26}}$$\cos \theta =\dfrac{-5}{\sqrt{26}}$\begin{align}\sin 2\theta &=2\sin… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-2-p4?rev=1737476039&do=diff Question 4 and 5, Exercise 10.2 Solutions of Question 4 and 5 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\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-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-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$$\bar{z}=x-iy$\begin{align}&4z-3\bar{z}=\dfrac{1-18i}{2-i}\\ \implies &4(x+iy)-3(x-iy)=\dfrac{1-18i}{2-i}\times \dfrac{2+i}{2+i}\\ \implies &4x+4iy-3x+3iy=\dfrac{(1-18i)(2+i)}{2^2-i^2} \end{align}\b… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/re-ex-p4?rev=1737476039&do=diff Question 4, Review Exercise Solutions of Question 4 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $z=x+i y$$\left|\dfrac{z+2 i}{z-2 i}\right|=1$$z = x + iy$\begin{align*} & \left|\dfrac{z + 2i}{z - 2i}\right| = 1\\ \implies & |z + 2i| = |z - 2i|\\ \implies & |x + i(y + 2)| = |x + i(y - 2)|\\ \implies & \sqrt{x^2 + (y + 2)^2} = \sqrt{x^2 + (y -… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/re-ex-p8?rev=1737476039&do=diff Question 8, Review Exercise Solutions of Question 8 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sqrt{2}+i \sqrt{2}$$\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/ex2-2-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$$A^{2}+\alpha I=\beta A$\begin{align} & A^{2}+\alpha I=\beta A\\ \implies &\begin{bmatrix} 2 & 1 \\ 3 & -3 \end{bmatrix} \begin{bmatrix} 2 & 1 \\ 3 & -3 \end{bmatrix}+\a… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p13?rev=1737476039&do=diff Question 13, Exercise 2.2 Solutions of Question 13 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $X$$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-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} $$ A $\begin{align*} … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p7?rev=1737476039&do=diff Question 7 and 8, Exercise 2.6 Solutions of Question 7 and 8 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{ccc}3 & 2 & 1 \\ 4 & -1 & 2 \\ 7 & 3 & -3\end{array}\right]$$A^{-1}$$3 x+4 y+7 z=14 ; 2 x-y+3 z=4 ; \quad x+2 y-3 z=0$\begin{align*} A &= \begin{bmatrix} 3 & 2 & 1 \\ 4 & -1 & 2 \\ 7 & 3 & -3 \end{bmatrix}\\ |… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p7?rev=1737476039&do=diff Question 11 and 12, Exercise 4.2 Solutions of Question 11 and 12 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $1000$$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/ex4-3-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$$n=?$\begin{align} & a_n=a_1+(n-1)d \\ \implies & 15=91+(n-1)(-4) \\ \implies & 15=91-4n+4 \\ \implies & 4n=95-15 \\ \implies & 4n = 80\\ \implies & n = 20. \end{align}\begin{… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p13?rev=1737476039&do=diff Question 26 and 27, Exercise 4.4 Solutions of Question 26 and 27 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $16\,\, ft$$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}.$$\begin{align*} a_{6}&=a_{1} r^5 \\ &=(4)\left(\dfrac{1}{4} \right)^5 \\ & = \dfrac{1}{256} \end{align*}… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p3?rev=1737476040&do=diff Question 5 and 6, Exercise 4.5 Solutions of Question 5 and 6 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=7, r=2, n=14$$a_1 = 7$$r = 2$$n = 14$$n$$$S_n = \frac{a_1 \left(1 - r^n\right)}{1 - r}, \quad r \neq 1.$$\begin{align*} S_{14} &= \frac{7 \left(1 - 2^{14}\right)}{1 - 2} \\ &= \frac{7 \left(1 - 16384\right)}{-1} \\ &= \frac{7 \time… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p7?rev=1737476040&do=diff Question 14, 15 and 16, Exercise 4.7 Solutions of Question 14, 15 and 16 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$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-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-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-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/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/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$$\cos$\begin{alig… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p13?rev=1737476040&do=diff Question 14, Exercise 8.1 Solutions of Question 14 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\theta$$\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$$\theta$\begin{align*} … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p4?rev=1737476040&do=diff Question 4(i-iv), Exercise 9.1 Solutions of Question 4(i-iv) of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y=\sin x+x \cdot \cos x$$f(x)=\sin x+x \cdot \cos x$\begin{align*} f(-x) = \sin (-x) + (-x)\cdot \cos (-x) \end{align*}$\sin(-x)=-\sin x$$\cos (-x) = \cos x$\begin{align*} f(x) & = -\sin x - x \cdot \cos x \\ & = -(\sin x + x \cdot … text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/matric/9th_science/unit11/11-1?rev=1737476041&do=diff Exercise 11.1 (Solutions) On this page solutions of Exercise of Unit 11: Parallelograms and Triangles of Mathematics 9 written by Dr. Karamat H. Dar and Prof. Irfan-ul-Haq has been given. There are two questions in this exercise and solution of both the questions are given below. $ABCD$$m\angle B=130^\circ$$m\angle B=m\angle D$$m\angle B=m\angle D=130^\circ$\begin{align} & m\angle A +\,\,m\angle B=180^\circ \\ & m\angle A+\,{{130}^{\circ }}={{180}^{\circ }}\\ & m\angle A={{180}^{\circ }}-{{130… text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/conferences?rev=1737476042&do=diff Mathematics Conferences <callout type=“warning” title=“Page Moved to Mathematical Events” icon=“true”> This page will no longer be updated. To spread the scope of this page, it has been merged with Mathematics Events. </callout> Conferences can be ideal places to meet with professionals and present our work to live audience to get engage with them. It might be best place to find collaborators from with in country and outside of the country. On this page we have posted mathematics conferences oc… text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/cui?rev=1737476042&do=diff Mathematics CUI: LaTeX Resources [Department of Mathematics, COMSATS University Islamabad, Attock Campus] This page contains LaTeX template of CIIT Mathematics, MSc Project and MS Thesis templates. Templates Download a zip file given below and extract it by right clicking on the file. BS Project Template: (Version 1.5, Uploaded: Sep 29, 2022)$\$$I$$\mathbb{R}$$f:I\to \mathbb{R}$$(\$$$\sin^2 \theta + \cos^2 \theta =1$$\begin{equation} \sin^2 \theta + \cos^2 \theta =1 \end{equation} text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/people?rev=1737476042&do=diff People (Contributor) We're so grateful to all the amazing contributors on MathCity.org! This page lists all the students, teachers, and anyone who has shared their knowledge with us. Thanks to all of you, MathCity is a fantastic place to learn! * text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/privacy_cookies_banner?rev=1737476042&do=diff This website uses cookies to improve your experience. We'll assume you're ok with this, but you can opt-out if you wish. text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/todo?rev=1737476042&do=diff Things to do Here is the list of things which we need to do. * Chapter 1 of FSc Part 2 need your attention. * <http://www.mathcity.org/fsc-part1-ptb/mcqs> * Objective FSc Part 2 need to be added. * Old paper of FBISE need to be added. * Old paper page need to be redefine. * Old papers of FSc Part 1 need to be uploaded. text/html 2025-01-21T16:13:54+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/atiq/fa20-mth211?rev=1737476034&do=diff MTH211: Discrete Mathematics (Fall 2020) Course Objectives: Discrete Mathematics is branch of Mathematics which deals with discrete structures like logic. sequences, graphs, relations in contrast to Calculus. where we enjoy the continuity of functions and the set of real numbers. This course is introduction to discrete structures which are not the part of main stream courses. Discrete Mathematics has applications in Computer Science. Economics and Decision Making etc. This course will help t… text/html 2025-01-21T16:13:54+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/atiq/fa20-mth424?rev=1737476034&do=diff MTH424: Convex Analysis (Fall 2020) [Convex Analysis] Objectives: At the end of this course the students will be able to understand the concept of Convex Analysis, convex sets, convex functions, Differential of the convex function. Developing ability to study the Hadamard-Hermite inequalities and their applications. Prepare students to be self independent and enhance their mathematical ability by giving them home work and projects.$f(x)=x$$\mathbb{R}$$f(x)=x^2$$\mathbb{R}$$f:[a,b]\to \mathbb{… text/html 2025-01-21T16:13:54+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/atiq/sp20-mth211?rev=1737476034&do=diff MTH211: Discrete Mathematics (Spring 2020) Course Objectives: Discrete Mathematics is branch of Mathematics which deals with discrete structures like logic. sequences, graphs, relations in contrast to Calculus. where we enjoy the continuity of functions and the set of real numbers. This course is introduction to discrete structures which are not the part of main stream courses. Discrete Mathematics has applications in Computer Science. Economics and Decision Making etc. This course will help… text/html 2025-01-21T16:13:54+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/atiq/sp21-mth211?rev=1737476034&do=diff MTH211: Discrete Mathematics (Fall 2020) Course Objectives: Discrete Mathematics is branch of Mathematics which deals with discrete structures like logic. sequences, graphs, relations in contrast to Calculus. where we enjoy the continuity of functions and the set of real numbers. This course is introduction to discrete structures which are not the part of main stream courses. Discrete Mathematics has applications in Computer Science. Economics and Decision Making etc. This course will help t… text/html 2025-01-21T16:13:54+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/atiq/sp22-mth211?rev=1737476034&do=diff MTH211: Discrete Mathematics (Spring 2022) Course Objectives: Discrete Mathematics is branch of Mathematics which deals with discrete structures like logic. sequences, graphs, relations in contrast to Calculus. where we enjoy the continuity of functions and the set of real numbers. This course is introduction to discrete structures which are not the part of main stream courses. Discrete Mathematics has applications in Computer Science. Economics and Decision Making etc. This course will help… text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/bsc/notes_of_metric_spaces?rev=1737476035&do=diff Notes of Metric Spaces These notes are related to Section IV of B Course of Mathematics, paper B. We are very thankful to Mr. Tahir Aziz for sending these notes. Name Notes of Metric Space Author Prof. Shahzad Ahmad Khan Send by Tahir Aziz text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/bsc/notes-fouries-series?rev=1737476035&do=diff Notes of Fourier Series These notes are provided by Mr. Muhammad Ashfaq. We are really very thankful to him for providing these notes and appreciates his effort to publish these notes on MathCity.org Name Notes of Fluid Mechanics Author Qayyum Ullah Khan text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/bsc/notes-of-metric-spaces-umer-asghar?rev=1737476035&do=diff Notes of Metric Spaces by Umer Asghar These notes are related to Section IV of B Course of Mathematics, paper B. We are very thankful to Mr. Umer Asghar for sending these notes. Name Notes of Metric Space Author Mr. Umer Asghar Pages 19 pages text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/bsc/number-theory-by-prof-m-tanveer?rev=1737476035&do=diff Number Theory by Prof. M. Tanveer [Number Theory by Prof. M. Tanveer] These notes are very helpful to prepare one of the sections of mathematics for BSc. Also these notes can be used for other classes. Author: Prof. M. Tanveer Type: Handwritten Format: PDF (972 kB) Pages: $a,b, \in \mathbb{Z}$$a \neq 0$$a$$b$$c\in \mathbb{Z}$$b=ac$$a,b, \in \mathbb{Z}$$c \in \mathbb{Z}$$a$$b$$c | a$$c | b$ text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/bsc/series_of_important_functions?rev=1737476035&do=diff Some important series of functions On this page we are going to post some series of functions, which are used in mathematics at undergraduate level. * $\sinh x = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} +... \qquad -\infty <x< \infty$ * $\cosh x = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!} +... \qquad -\infty <x< \infty$ * $\tanh x = x - \frac{x^3}{3} + 2\frac{x^5}{15} - 17\frac{x^7}{315} +... \qquad \left |x\right |< \frac{\pi}{2}$ * $\coth x = \frac{1}{x} … text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/bsc/vector_analysis_by_hameed_ullah?rev=1737476035&do=diff Vector Analysis by Hameed Ullah: Notes [right triangle in semi circle] Note of vector analysis by Hammed Ullah. These notes are send by Umer Asghar, we are very thankful to him for providing these notes. These notes are for helpful for undergraduate level (BSc or BS). <div> <center> </div> Name Notes of vector analysis <div> </center> </div> text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/conferences/18th-pmc-islamabad?rev=1737476035&do=diff 18th International Pure Mathematics Conference 2017, Islamabad (4-6 August 2017) [18th PMC Margalla Islamabad, 2017] It will provide a stimulating opportunity to meet experts from various countries in a variety of branches of mathematics. The entire conference will be organized under one roof at a hotel, in the modern, peaceful, and beautiful federal capital of Pakistan located at the footsteps of the scenic Margalla Hills. We will be able to offer you subsidized accommodation, free meals, fre… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/definitions?rev=1737476036&do=diff Definitions: FSc Part1 KPK A Textbook of Mathematics for Class XI is published by Khybar Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. The book has total of twelve (12) chapters. Definition of the book provide the quick overview of the book.$360^\circ$$\theta$$90^{\circ} \pm \theta, 180^{\circ} \pm \theta, 270^{\circ} \pm \theta, 360^{\circ} \pm \theta$$16^\circ 13' 9''$$sin(\alpha+2\pi)=sin\alpha$$sin x=\frac{2}{7}$$cos x-tan x=0$ text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-ptb/bise-papers?rev=1737476037&do=diff Question Paper/Model Paper/Paper Pattern HSSC-I: BISE On this page, we have discussed the paper pattern of the HSSC-I or FSc Part 1 paper pattern of the mathematics subject. All the boards follow the same pattern. Old (past) question papers and model papers of mathematics for HSSC-I (FSc Part 1) conducted by Board of Intermediate and Secondary Education (BISE) in Punjab. There are lot of boards (e.g. Multan Board, Faisalabad Board, Sargodha Board, Gujranwala Board, DG Khan Board, Rawalpindi Boa… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-ptb/definitions-and-review?rev=1737476037&do=diff Definitions and Review Textbook of Algebra and Trigonometry Class XI is published by Punjab Textbook Board (PTB) Lahore, Pakistan. The book has total of 14 chapters. On this page, we have posted material related to definitions, reviews and quick reviews. text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-ptb/definitions-muzzammil-subhan?rev=1737476037&do=diff Definitions: Mathematics 11: PTB by Muzzammil Subhan Definitions from Textbook of Algebra and Trigonometry Class XI, published by Punjab Textbook Board (PTB) Lahore, Pakistan. We are very thankful to Muzzammil Subhan for his valuable contribution. Download or view PDF for all definitions. Samples is given below$y=2^x$$y=e^x$$f(x)=\log_a x$$f(x)=\log_e x$$y$$x$$y$$y=x^2+3x$$x^2+xy+y^2=4$$f(-x)=f(x)$$f(-x)=-f(x)$ text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-ptb/sol?rev=1737476037&do=diff FSc Part 1 Mathematics Notes/Solutions [FSc Part1 PTB Book Cover] <lead>Notes (Solutions) of Textbook of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board (PTB) Lahore.</lead> There are fourteen chapters in this book and we have work hard to make easy and suitable solution for students and teachers so that it help them learn things quickly and easily. Please click on a desire chapter to view the solution of any particular exercise. This work is licensed… text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions?rev=1737476035&do=diff FSc Part 1 Mathematics Notes/Solutions [FSc Part1 PTB Book Cover] <lead>Notes (Solutions) of Textbook of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board (PTB) Lahore.</lead> There are fourteen chapters in this book and we have work hard to make easy and suitable solution for students and teachers so that it help them learn things quickly and easily. Please click on a desire chapter to view the solution of any particular exercise. This work is licensed… 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/fsc_part_2_model_papers?rev=1737476036&do=diff FSc Part 2 Model Papers Federal Board of Intermediate & Secondary Education, Islamabad <div> <center> </div> ARW 1st Model Paper 2009 View Online Download PDF (84KB) ARW Official Model Paper (with solution) Download PDF (233KB) <div> </center> </div> Board of Intermediate & Secondary Education. All the boards in Punjab expect Federal Board have the same paper pattern.<div> <center> </div><div> </center> </div> 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 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/kpk?rev=1737476036&do=diff FSc (Kyber Pakhtunkhwa (KPK) Boards) Notes (resources) Textbook of Mathematics for Class XI“ and “Textbook of Mathematics Grade 12” published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan are available on this page. At the moment we are going to publish notes of FSc Part 1 and Part 2. 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 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/kpk_fsc_part_2?rev=1737476036&do=diff FSc Part 2 (KPK Boards) [A Textbook of Mathematics For Class XII] Notes of FSc Part 2 of “A Textbook of Mathematics For Class XII” 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.$y=x^n$$y=(ax+b)^n$ text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/definitions?rev=1737476037&do=diff Definitions: FSc Part1 KPK A Textbook of Mathematics for Class XI is published by Khybar Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. The book has total of twelve (12) chapters. Definition of the book provide the quick overview of the book.$360^\circ$$\theta$$90^{\circ} \pm \theta, 180^{\circ} \pm \theta, 270^{\circ} \pm \theta, 360^{\circ} \pm \theta$$16^\circ 13' 9''$$sin(\alpha+2\pi)=sin\alpha$$sin x=\frac{2}{7}$$cos x-tan x=0$ text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/definitions?rev=1737476039&do=diff Definitions: Mathematics 11 NBF Model Textbook of Mathematics for Class XI is published by National Book Foundation (NBF), Islamabad, Pakistan. NBF can be considered as Federal Textbook Board Islamabad. The book has total of nine (9) chapters. Definition of the book provide the quick overview of the book.$x+iy$$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 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/mathcraft/sample-02-latex?rev=1737476040&do=diff MathCraft: PDF to LaTeX file: Sample-02 If the PDF file provided by you as follows: Then the output LaTeX file is as follows: \documentclass[4pt]{article} \usepackage[utf8]{inputenc} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage[version=4]{mhchem} \usepackage{stmaryrd} \usepackage{bbold} \usepackage[a4paper]{geometry} \linespread{1.3} % double spaces lines \textwidth 6.3truein % These 4 commands define more efficient margins \textheight 9.9truein \oddsidemargi… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/matric/9th_general?rev=1737476041&do=diff General Mathematics 9 [General Mathematics 9th Class] There are ten chapters in General Mathematics 9 for Punjab Textbook Board, Lahore. Solutions of all the chapters are given below. One can download the PDF file of the notes. Please remember that, to view these notes one must have PDF reader installed in their system. We will try our best to add online view of the notes very soon. text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/advanced-analysis-handwritten-notes?rev=1737476041&do=diff Advanced Analysis: Handwritten Notes [Advanced Analysis: Handwritten Notes] These notes are provided by Mr. Anwar Khan. We are really very thankful to Mr. Anwar Khan for providing these notes and appreciates his effort to publish these notes on MathCity.org It covers the complete syllabus of Advanced Analysis paper of MSc Mathematics. See the contents of the notes given below to see the topics covered by these notes. text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/advanced-analysis-iqra-liaqat?rev=1737476041&do=diff Advance Analysis by Ms. Iqra Liaqat [Advance Analysis by Ms. Iqra Liaqat] These notes are send by Ms. Iqra Liaqat. We are really very thankful to her for providing these notes and appreciates her effort to publish these notes on MathCity.org Name Advance Analysis Sender Ms. Iqra Liaqat Author $\sigma$ text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/affine-and-euclidean-geometry-shahzad-idress?rev=1737476041&do=diff Affine and Euclidean Geometry by Shahzad Idress [Affine and Euclidean Geometry by Shahzad Idress] These notes are sent by shahzad-idress. We acknowledged his efforts to published these notes on MathCity.org. These are short notes containing topics related to Affine and Euclidean Geometry. The main sections includes $R^n$ text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/algebra?rev=1737476041&do=diff Notes of Algebra All the notes of algebra are given here. [All the notes of algebra] <well> <caption> Algebra II by Syed Sheraz Asghar </caption> A notes of Algebra-II composed by Muzammil Tanveer. We are very thankful to him for sending these notes. </well><well> <caption> Elementary Linear Algebra by Muhammad Usman Hamid </caption> Linear Algebra is the study of vectors and linear transformations. Best notes to prepare the subject written by text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/algebra-ii-by-syed-sheraz-asghar?rev=1737476041&do=diff Algebra II by Syed Sheraz Asghar [Algebra II by Syed Sheraz Asghar] These notes are provided and composed by Mr. Muzammil Tanveer. We are really very thankful to him for providing these notes and appreciates his effort to publish these notes on MathCity.org * Name: Algebra II * Provider: Mr. Muzammil Tanveer text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/algebraic-number-theory-notes-anwar-khan?rev=1737476041&do=diff Algebraic Number Theory Notes by Anwar Khan [Algebraic Number Theory Notes by Anwar Khan] Algebraic number theory is a subfield of number theory that studies integers, rational numbers, and their generalisations using abstract algebra techniques. It covers Galois theory, ideals and units in rings of integers, unique factorization, and algebraic number fields and related rings of integers. It is a complex and in-depth subject with numerous linkages to other branches of mathematics.$\mathbb{R}$ text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/classical-mechanics-muhammad-usman-hamid?rev=1737476041&do=diff Classical Mechanics by Muhammad Usman Hamid [Classical Mechanics by Muhammad Usman Hamid] Objectives of the course: To provide solid understanding of classical mechanics and enable the students to use this understanding while studying courses on quantum mechanics, statistical mechanics, electromagnetism, fluid dynamics, space-flight dynamics, astrodynamics and continuum mechanics. text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/complex-analysis-dr-amir-mahmood?rev=1737476041&do=diff Complex Analysis (Easy Notes of Complex Analysis) [Complex Analysis (Easy Notes of Complex Analysis) ] The study of complex numbers, together with their manipulation, derivatives, and other characteristics, is known as complex integration. Contour integration is a technique for analysing specific integrals along pathways in the complex plane in the field of complex analysis mathematics. The complex analytic technique known as the calculus of residues and contour integration are closely linked. text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/complex-analysis-iqra-liaqat?rev=1737476041&do=diff Complex Analysis (Notes) by Ms. Iqra Liaqat [Notes of Complex Analysis by Ms. Iqra Liaqat] These notes are send by Ms. Iqra Liaqat. We are really very thankful to her for providing these notes and appreciates her effort to publish these notes on MathCity.org Complex analysis is the study of functions of complex numbers. It is useful in a variety of mathematical fields, such as algebraic geometry, number theory, analytic combinatorics, and applied mathematics, as well as in physics fields like… 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 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/complex-analysis-quick-review?rev=1737476041&do=diff Complex Analysis (Quick Review) [Complex Analysis: Quick Review] These notes are made and shared by Mr. Akhtar Abbas. We are really very thankful to him for providing these notes and appreciates his efforts to publish these notes on MathCity.org. Important definitions and important results are the part of these notes, these might be helpful to prepare interviews or any other written test after graduation like PPSC, FPSC or etc.$z_1, z_2 \in S$$S$$v(x,y)$$u(x,y)$$f(z)=u(x,y)+iv(x,y)$$f$$D$$C$$D$… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/computing-tools-for-mathematics-asif-arshad?rev=1737476041&do=diff Computing Tools for Mathematics by Asif Arshad [omputing Tools for Mathematics by Asif Arshad] Computing tools for mathematics are algorithms that use computers to solve mathematical issues. They are employed in a number of scientific, technical, industrial, and technological domains where computer is crucial and central. They may assist in creating precise and effective numerical techniques, for instance, to solve physical or biological models. text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/differential-geometry-kaushef_salamat?rev=1737476041&do=diff Differential Geometry (Notes) by Ms. Kaushef Salamat [Differential Geometry by Ms. Kaushef Salamat] Differential Geometry is an important field in the mathematical sciences. Usually, this is a compulsory or core subject for BS mathematics students. These notes are send by Ms. Kaushef Salamat. We are really very thankful to her for providing these notes and appreciates her effort to publish these notes on MathCity.org text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/differential-geometry-m-usman-hamid?rev=1737476041&do=diff Differential Geometry by M Usman Hamid [Differential Geometry by M Usman Hamid] These notes are initially provided by Mr. Anwar Khan. Later send by many persons. We are really very thankful to all for providing these notes and appreciates their effort to publish these notes on MathCity.org text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/differential-geometry-syed-hassan-waqas?rev=1737476041&do=diff Differential Geometry by Syed Hassan Waqas [Differential Geometry by Syed Hassan Waqas] In the field of mathematics known as differential geometry, smooth manifolds—also known as smooth shapes and spaces—and their geometry are studied. Investigating the geometric characteristics of curves and surfaces involves the use of the tools of differential and integral calculus, linear algebra, and multilinear algebra. Differential calculus and mathematical analysis are related to differential geometry.… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/fluid-dynamics-i-m-usman-hamid?rev=1737476041&do=diff Fluid Dynamics I by Muhammad Usman Hamid [Fluid Dynamics I by Muhammad Usman Hamid] These notes are send by Mr. Anwar Khan, PhD. We are really very thankful to him for providing these notes and appreciates his effort to publish these notes on MathCity.org. These notes are written by Muhammad Usman Hamid. * Name: Fluid Dynamics I text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/fluid-mechanics-ali-raza?rev=1737476041&do=diff Fluid Mechanics by Ali Raza [Fluid Mechanics by Ali Raza] Fluid mechanics is the branch of physics that studies how fluids (liquids, gases, and plasmas) behave and interact with forces and energy. Fluid mechanics has many applications in engineering, geophysics, biology, and other fields. text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/fluid-mechanics-i-muzammil-tanveer?rev=1737476041&do=diff Fluid Mechanics I by Dr Rao Muzamal Hussain [Fluid Mechanics I by Muzammil Tanveer] These notes are provided and composed by Mr. Muzammil Tanveer. We are really very thankful to him for providing these notes and appreciates his effort to publish these notes on MathCity.org. These notes are based on lectures delivered by Mr. Muzammil Hussain at GC University Faisalabad. text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/fluid-mechanics-ii-muzammil-tanveer?rev=1737476041&do=diff Fluid Mechanics II by Dr Rao Muzamal Hussain [Fluid Mechanics I by Muzammil Tanveer] These notes are provided and composed by Mr. Muzammil Tanveer. We are really very thankful to him for providing these notes and appreciates his effort to publish these notes on MathCity.org. These notes are based on lectures delivered by Mr. Muzammil Hussain at GC University Faisalabad. text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/functional-analysis-by-prof-mumtaz-ahmad?rev=1737476041&do=diff Functional Analysis by Prof Mumtaz Ahmad [Functional Analysis by Prof Mumtaz Ahmad] Functional analysis is a subfield of mathematics that deals with vector space theory and linear algebra. It entails researching the connections between roles, things, incidents, actions, and outcomes. The word $l^\infty$$l^\infty$ text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/functional-analysis-by-tahir-hussain-jaffery?rev=1737476041&do=diff Functional Analysis by Mr. Tahir Hussain Jaffery [Functional Analysis by Prof Mumtaz Ahmad] Functional analysis is a branch of mathematics concerned with vector space theory and linear algebra. It requires looking into the relationships between various roles, objects, incidents, actions, and results. The term $M$$M+N$ text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/functional-analysis-m-usman-hamid-and-zeeshan-ahmad?rev=1737476041&do=diff Functional Analysis by M Usman Hamid and Zeeshan Ahmad [Functional Analysis by M Usman Hamid and Zeeshan Ahmad] These notes are send by Muhammad Usman Hamid and written by Muhammad Usman Hamid and Zeeshan Ahmad. We are really very thankful to him for providing these notes and appreciate his effort to publish these notes on MathCity.org. Usman is dedicated and committed mathematician, who is working very hard for better understanding of mathematics to it readers. text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/fundamental-of-complex-analysis-prof-m-saleem?rev=1737476041&do=diff Fundamental of Complex Analysis (Solutions of Some Exercises) [Fundamental of Complex Analysis, Solutions of Some Exercises] Complex analysis is the study of functions that exist in the complex plane, that is, functions with complex arguments and complex outputs. With roots in the 18th century and the years just before, it is one of the classical branches of mathematics. In the 20th century, significant figures in mathematics who are connected to complex numbers include Euler, Gauss, Riemann, … text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/fuzzy-sets-theory-umar-saeed-bunarai?rev=1737476041&do=diff Fuzzy Sets Theory by Umar Saeed Bunarai [Fuzzy Sets Theory by Umar Saeed Bunarai] In the field of mathematics known as fuzzy set theory, sets having varying degrees of membership are studied. This implies that an element can be a member of a set only partially or not at all. In several disciplines, including logic, control, data analysis, and decision-making, fuzzy set theory is used to model uncertainty, vagueness, and imprecision. text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/general-topolgy-raheel-ahmad?rev=1737476041&do=diff General Topology by Raheel Ahmad A handwritten notes of Topology by Mr. Raheel Ahmad. We are very thankful to him for sending these notes. Name General Topology Author Mr. Raheel Ahmad Pages 87 pages Format Mobile Scanned PDF Size 8.0 MB What is in the notes? 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 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/groups-theory-m-iftikhar?rev=1737476041&do=diff Group Theory by Mr. Muhammad Iftikhar [Group Theory by Mr. Muhammad Iftikhar] These notes are send by Mr. Muhammad Iftikhar. We are really very thankful to him for providing these notes and appreciates his effort to publish these notes on MathCity.org. Name Lecture Notes on Group Theory Author Mr. Muhammad Iftikhar text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/handwritten-notes-real-analysis-asim-marwat?rev=1737476041&do=diff Handwritten Notes of Real Analysis by Asim Marwat [Handwritten Notes of Real Analysis by Asim Marwat] Real analysis is a branch of mathematics that analyses how real numbers, sequences and series, and real functions behave. It focuses on real numbers and frequently extends the real line by including positive and negative infinity. Real analysis investigates a number of the properties of real-valued sequences and functions, including convergence, limits, continuity, smoothness, differentiabilit… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/history-of-mathematics-m-usman-hamid?rev=1737476041&do=diff History of Mathematics by Muhammad Usman Hamid [History of Mathematics by Muhammad Usman Hamid] Mathematics is a unique aspect of human thought, and its history differs in essence from all other histories. These notes are written by Muhammad Usman Hamid. We are very thankful to him for sharing these notes on our website. text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/linear-algebra-important-definitions-and-results?rev=1737476041&do=diff Linear Algebra: Important Definitions and Results [Linear Algebra: 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 efforts 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.$V$$n$$n$$V$$W$$V$$A$$A$$A… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/mathematical-method-usman-hamid?rev=1737476041&do=diff Mathematical Method by Muhammad Usman Hamid [Mathematical Method by Muhammad Usman Hamid] These notes are send by Muhammad Usman Hamid. We acknowledged his efforts to published these notes on MathCity.org. The main objective of this course is to provide the students with a range of mathematical methods that are essential to the solution of advanced problems encountered in the fields of applied physics and engineering. In addition this course is intended to prepare the students with mathematic… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/mathematical-statistics-i-muzammil-tanveer?rev=1737476041&do=diff Mathematical Statistics I by Muzammil Tanveer [Mathematical Statistics I] These notes are provided and composed by Mr. Muzammil Tanveer. We are really very thankful to him for providing these notes and appreciates his effort to publish these notes on MathCity.org Name Mathematical Statistics text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/mathematical-statistics-ii-muzammil-tanveer?rev=1737476041&do=diff Mathematical Statistics II by Sir Haidar Ali [Mathematical Statistics II] A subfield of mathematics called mathematical statistics is concerned with using mathematical techniques to solve statistical problems. It involves using mathematical analysis and probability theory to the study of statistical issues like estimate, hypothesis testing, and confidence intervals. Financial, engineering, and scientific fields all benefit from the use of mathematical statistics, which is a significant area of… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/mathematical-statistics-iqra-liaqat?rev=1737476041&do=diff Mathematical Statistics by Ms. Iqra Liaqat [Mathematical Statistics by Ms. Iqra Liaqat] These notes are send by Ms. Iqra Liaqat. We are really very thankful to her for providing these notes and appreciates her effort to publish these notes on MathCity.org Name Mathematical Statistics Provided by text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/measure-theory-notes-asim-marwat?rev=1737476041&do=diff Measure Theory Handwritten Notes by Asim Marwat [Measure Theory Notes by Asim Marwat] These notes are made and shared by Mr. Asim Marwat. He has our sincere gratitude for supplying these notes, and we value his effort in having them published on MathCity.org. Measure Theory is an important subject in BS Mathematics. These notes contain topic from base level, like equivalence set, to advance level, like convergent in measure.$L_p$$L_p$$L^\infty$$L_p$ text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/mechanics-by-dr-amir-mahmood?rev=1737476041&do=diff Mechanics (Easy Notes of Mechanics) [Easy Notes of Mechanics] These notes are provided and prepared by Dr. Amir Mahmood. We are really very thankful to him for providing these notes and appreciates his effort to publish these notes on MathCity.org If there is error or mistake in the notes, then please directly contact to Dr. Amir Mahmood for correction. He can be accessed via his email address at text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/mechanics-by-sir-nouman-siddique?rev=1737476041&do=diff Mechanics by Sir Nouman Siddique These notes are provided and composed by Mr. Muzammil Tanveer. We are really very thankful to him for providing these notes and appreciates his effort to publish these notes on MathCity.org * Name: Mechanics * Provider: Mr. Muzammil Tanveer text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/mechanics-i-statics-dr-babar-ahmad?rev=1737476041&do=diff Mechanics I (Statics) by Dr Babar Ahmad [Mechanics I (Statics) by Dr Babar Ahmad] We are very thankful to Dr Babar Ahmad for sharing his book on MathCity.org. This book is very helpful for undergraduate students of Science and Engineering Programs. This book is shared by the permission of the author and he keeps the copyright of the book. text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/method-of-mathematical-physics-m-usman-hamid?rev=1737476041&do=diff Method of Mathematical Physics by Mr. Muhammad Usman Hamid [Method of Mathematical Physics by Mr. Muhammad Usman Hamid] Method of Mathematical Physics is an area of mathematics concerned with the application of mathematical methods to physics problems. Mathematical techniques for statistical mechanics, quantum mechanics, and classical mechanics are all included in this large field. text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/multiple-choice-questions-_bsc-bs-ppsc_-akhtar-abbas?rev=1737476041&do=diff Multiple Choice Questions (BSc/BS/PPSC) by Akhtar Abbas [Multiple Choice Questions (BSc/BS/PPSC)] These notes are made and shared by Mr. Akhtar Abbas. We are really very thankful to him for providing these notes and appreciates his efforts to publish these notes on MathCity.org. Multiple Choice Questions (MCQs) are given in these notes, which might be helpful in BSc, BS or Punjab Public Service Commission (PPSC) exams.$a$$b$$n$$na > b$$(p − 1)! \equiv −1(mod p)$$p$$p$$p$ text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/note-for-numerical-methods-m-usman-hamid?rev=1737476041&do=diff Notes for Numerical Methods by M Usman Hamid [Notes for Numerical Methods by M Usman Hamid] These notes are initially provided by Mr. Anwar Khan. Later the updated version is send by Muhammad Tahir. We are really very thankful to Mr. Anwar Khan and Muhammad Tahir for providing these notes and appreciates their effort to publish these notes on MathCity.org$\left(\frac{1}{3}\right)$$\left(\frac{3}{8}\right)$ text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/number-theory-notes-anwar-khan?rev=1737476041&do=diff Number Theory Notes by Anwar Khan [Number Theory Notes by Anwar Khan] Mathematicians who specialize in number theory examine the characteristics and connections between integers. “Higher arithmetic” and “the queen of mathematics” are some names for it. Because it examines the characteristics and connections between integers and arithmetic functions, number theory is interesting. It has numerous uses in coding theory, combinatorics, cryptography, and other branches of mathematics. Like the ones… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/number-theory-umer-shuaib?rev=1737476041&do=diff Number Theory by Dr Muhammad Umer Shuaib [Number Theory Notes] A subfield of mathematics called number theory studies the characteristics of positive integers. Higher arithmetic is another name for it. The study of the relationships between various types of numbers, including prime numbers, rational numbers, and algebraic integers, is done using number theory, one of the oldest fields of mathematics.$\phi$ text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/numerical-analysis-ii?rev=1737476041&do=diff Numerical Analysis II [Numerical Analysis by Muzammil Tanveer] These notes are provided and composed by Mr. Muzammil Tanveer. We are really very thankful to him for providing these notes and appreciates his effort to publish these notes on MathCity.org. Name Numerical Analysis II Compiled by Muzammil Tanveer text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/operation-research-haidar-ali?rev=1737476041&do=diff Operation Research by Sir Haidar Ali [Notes of Operation Research by Sir Haidar Ali] These notes are provided and composed by Mr. Muzammil Tanveer. We are really very thankful to him for providing these notes and appreciates his effort to publish these notes on MathCity.org. These are based on lectures delivered by Sir Haidar Ali at GC University Faisalabad. text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/partial-differential-equations-m-usman-hamid?rev=1737476041&do=diff Partial Differential Equations by M Usman Hamid The course provides a foundation to solve PDE’s with special emphasis on wave, heat and Laplace equations, formulation and some theory of these equations are also intended. We are really very thankful to Prof. Muhammad Usman Hamid for providing these notes and appreciates his effort to publish these notes on MathCity.org text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/partial-differential-equations-muzammil-tanveer?rev=1737476041&do=diff Partial Differential Equations by Muzammil Tanveer [Partial Differential Equations] These notes are provided and composed by Mr. Muzammil Tanveer. We are really very thankful to him for providing these notes and appreciates his effort to publish these notes on MathCity.org Name Partial Differential Equations or PDEs text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/real-analysis-hand-written-kaushef?rev=1737476041&do=diff Real Analysis Handwritten Notes by Kaushef Salamat [Real Analysis Handwritten Notes by Kaushef Salamat] We are very thankful to Ms. Kaushef Salamat for providing these notes. Real Analysis is a core subject in BS or MSc Mathematics. Without a fundamental grasp of Real Analysis, one cannot claim to be a mathematician. These notes are very comprehensive containing almost all the notions of Real Analysis. For providing these notes, Ms. $\infty$ text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/rings-and-modules-iqra-liaqat?rev=1737476041&do=diff Rings & Modules by Ms. Iqra Liaqat Ring is a mathematical structure with two operations. With one operation it is abelian group and with other operation it is semi-group with distributive law holds with first operation w.r.t the other operation. This concept is of pure in nature in mathematics. These advanced level notes are typically taken as an elective in a mathematics undergraduate degree. text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/special-functions-muzammil-tanveer?rev=1737476042&do=diff Special Functions by Dr. Muhey-U-Din These notes are provided and composed by Mr. Muzammil Tanveer. We are really very thankful to him for providing these notes and appreciates his effort to publish these notes on MathCity.org. Thease notes are based on the lectures by Dr. Muhey-U-Din. text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/special-theory-of-relativity-usman-hameed-and-zeeshan-ahmad?rev=1737476042&do=diff Special Theory of Relativity by M Usman Hamid and M Zeeshan Ahmad [Special Theory of Relativity by M Usman Hamid and M Zeeshan Ahmad] Special theory of relativity developed by Einstein in 1905, after the failure of Newton‘s laws, when he was dealing with the relative motion in space. The theory of relativity deals with relations which exist between physical quantities (such as mass of a particle, length of a rod, electric field at a point etc.) as they appear to different observers in relative … text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/theory-of-optimization-muzammil-tanveer?rev=1737476042&do=diff Theory of Optimization by Ma'am Iqra Razzaq [Special Theory of Optimization by Ma'am Iqra Razzaq] These notes are provided and composed by Mr. Muzammil Tanveer. We are really very thankful to him for providing these notes and appreciates his effort to publish these notes on MathCity.org. These notes are based on the lectures by Ma'am Iqra Razzaq. text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/topology-and-functional-analysis-solved-pu?rev=1737476042&do=diff Topology and Functional Analysis Solved Paper by Noman Khalid These solved papers are written and provided by Mr. noman-khalid. We are very thankful to him for providing these notes to publish on MathCity.org. Topology and Functional Analysis is considered as one the tough subject in BS/MSc Mathematics in different universities. Here solved paper for the said subject is given for the University of the Punjab (PU), Lahore. text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/vector-spaces-review-by-rashad-wattu?rev=1737476042&do=diff Vector Space (Review) by Rashad Wattu Mathematical vector space is a crucial and fundamental idea. It depends on the concept of field. These notes were primarily written to help maths students understand vector space, which is a concept that they all need to be familiar with. text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/papers/old_papers_of_m.phil._university_of_sargodha?rev=1737476042&do=diff Old Papers of M.Phil. University of Sargodha M.Phil. (Mathematics) and MS (Mathematics) are equivalent programs. University of Sargodha (SU) is offering M.Phil in Mathematics. The reason might be that this institution was offering M.Sc program in Mathematics while BS was not launched in this prestigious institute. For the benefits of students and teacher, we are here sharing old Admission Test, M. Phil. (Mathematics), University of Sargodha, Sargodha.<img src=../images/math_uos.jpg class="medi… text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/papers/old_ppsc_written_tests_for_lecturer_mathematics?rev=1737476042&do=diff Old PPSC Written tests for Lecturer (Mathematics) * Previous Written tests (papers) of Punjab Public Service Commission, Lahore for Lecturer in Mathematics. <callout type=“info” icon=“true”> * From now the PPSC will conduct a MCQs test for this post. <div> <center> </div><div> </center> </div> text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/people/akhtar?rev=1737476042&do=diff Akhtar Abbas <image shape=“rounded”></image> Mr. Akhtar Abbas is Lecture in Mathematics at University of Jhang, Jhang, Punjab, Pakistan. He is a dedicated and hardworking teacher. We are very thankful to him for his great contribution to our website. text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/people/amir?rev=1737476042&do=diff Amir Shehzad We are very thankful to Mr. Amir Shehzad for his contribution to the website. <image shape=“rounded”>[Mr. Amir Shehzad]</image> * Email: <amirshehzad569@gmail.com> * Cell: +92-343-4443214 * YouTube Chanel: <https://www.youtube.com/channel/UCAci3yf20CcDotwdTJOD8WQ> Contribution: 9th (Science) (PTB) * Unit 04 (10th Science PTB) | VIEW text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/people/anwar-khan?rev=1737476042&do=diff Anwar Khan, PhD [Dr. Anwar Khan] There are several advantages to being a strong mathematics student. It not only offers a solid basis for logical reasoning and problem-solving abilities, but it also widens the range of career options. Persistence and discipline are skills that are beneficial in all facets of life, and they are qualities that are required of maths students. A sense of satisfaction and self-confidence that come from excelling in mathematics can also have a good impact on one's a… text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/people/asim-marwat?rev=1737476042&do=diff Asim Marwat <image shape=“rounded”></image> We are very thankful to Mr. Asim Marwat for providing notes of functional analysis. His notes are attached below, which might be helpful in BS Mathematics. Mr. Asim has done MS (Mathematics) from University of Peshawar. He is an excellent student and mathematician whose mentors include Dr. Muhammad Adil, an important researcher in the field of analysis. text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/people/babar-ahmad?rev=1737476042&do=diff Dr Babar Ahmad <image shape=“rounded”>[Mr. Raheel Ahmad]</image> We are very thankful to him for his great contribution to our website. Dr. Ahmad is an Assistant Professor at COMSTAS University Islamabad. He is talented teacher and researcher, who wrote many book on Mechanics. text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/people/bahadar-ali-khan?rev=1737476042&do=diff Bahadar Ali Khan [Bahadar Ali Khan] We are very thankful to Mr. Bahadar Ali Khan for contributing to this website. * Email: <bahadarkhan639@yahoo.com> * Cell: +92-305-8069878 * Civil Engineer from University of Engineering & Technology Lahore (Narowal Campus) Contribution: bahadar_ali_khan-contributor text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/people/farooq?rev=1737476042&do=diff Prof. Muhammad Farooq We are very thankful to Prof. M. Farooq for contributing to this website. * Punjab College Sarogha Contribution: prof_m_farooq contributor text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/people/fazal-abbas?rev=1737476042&do=diff Prof. Fazal Abbas Sajid <image shape=“rounded”>[Prof Fazal Abbas Sajid]</image> We are very thankful to him for his contribution to the website. Prof. Fazal is a talented teacher contributing to the knowledge of mathematics by making videos and delivering online lectures by Zoom. text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/people/fiaz?rev=1737476042&do=diff Muhammad Fiaz Hussain We are very thankful to Muhammad Fiaz Hussain for contributing to our website. <image shape=“rounded”></image> * M.Sc, M.Phil. in Applied & Analytic Mathematics (COMSATS Institute of Information Technology Islamabad) * text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/people/idrees?rev=1737476042&do=diff Muhammad Idrees <callout type=“info” icon=“true”> Muhammad Idrees; M.Sc, MSC, M.Phil Department of Mathematics Govt: Boys Degree College, Nushki-Balochistan. Emails: <idrees.math@hotmail.com>, <idrees@idrees.pk> </callout> We are very thankful to Muhammad Idrees for contributing to the resources of MathCity.org text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/people/iftikhar?rev=1737476042&do=diff Muhammad Iftikhar We are very thankful to Mr. Muhammad Iftikhar for his contribution to the website. * Email: <sidram0520@gmail.com> * Department of Mathematics, PMAS Arid Agriculture University, Rawalpindi, Pakistan Contribution: muhammad_iftikhar contributor text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/people/ihsan-danish?rev=1737476042&do=diff Ihsan Danish <image shape=“rounded”></image> We are very thankful to Mr. Ihsan Danish for providing notes of complex analysis. His notes are attached below, which might be helpful in BS Mathematics. Mr. Ihsan Danish has done is BS Mathematics from Abdul Wali Khan University, Mardan. text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/people/iqra-liaqat?rev=1737476042&do=diff Iqra Liaqat <image shape=“rounded”>[ Publication Certificate]</image> We are very thankful to Ms. Iqra Liaqat for her contribution to the website. She works as a freelancer to help students in solving their assignments and past question papers of Mathematics and Statistics for all levels up to MSc. text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/people/kaushef?rev=1737476042&do=diff Kaushef Salamat [Kaushef Salamat] [Solved Paper by Kaushef Salamat] We are very thankful to Ms. Kaushef Salamat for providing us notes. The author has done MPhil in Mathematics from Lahore Garrison University with Gold Medal. She is a lecturer in Queen Mary College Lahore. She also works as an online tutor for O Level and A Level Students. text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/people/maryam-jabeen?rev=1737476042&do=diff Maryam Jabeen We are very thankful to Ms. Maryam Jabeen for her contribution to the website. * Email: <maryamjabeen312@gmail.com> Contribution: contributor maryam_jabeen text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/people/moin?rev=1737476042&do=diff Engr. Moin Latif We are very thankful to Engr. Moin Latif for his contribution to the website. <image shape=“rounded”>[Moin Latif]</image> * Email: <moinlatif@gmail.com> * Cell: +92-342-7673137 * Qualification: B.E (UET Lahore, Pakistan) * Currently working as Assistant Manager at Millat Tractors Limited. text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/people/muzammil?rev=1737476042&do=diff Muzammil Tanveer We are very thankful to Mr. Muzammil Tanveer for his contribution to the website. <callout type=“info” icon=“true”> Mr. Muzammil Tanveer is also an expert to compose mathematics notes, books or papers. You may contact him via his email or cell number given below for professional paid composing. </callout> text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/people/nawab-ali?rev=1737476042&do=diff Nawab Ali <image shape=“rounded”></image> We are very thankful to Mr. Nawab Ali for providing notes of functional analysis. His notes are attached below, which might be helpful in BS Mathematics. * Name: Nawab Ali * Father Name: Muhammad Ayar text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/people/raheel?rev=1737476042&do=diff Raheel Ahmad <image shape=“rounded”>[Mr. Raheel Ahmad]</image> We are very thankful to Mr. Raheel Ahmad for his contribution to the website. * Email: <ruh242@gmail.com> * Facebook: www.facebook.com/ruh141 * Currently working as Lecturer in the Department of Mathematics at GCUF Chiniot Campus. Contribution: text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/people/rashad?rev=1737476042&do=diff Rashad Wattu <image shape=“rounded”></image> We are very thankful to him for providing notes to share with others. * Email: <rashadwattu@gmail.com> * Cell: +923037728892 * YouTube Channel: www.youtube.com/c/mathlogic.pk * Facebook page: www.facebook.com/rashad.wattu Contribution: rashad_wattu contributor text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/people/saleem-shahzad?rev=1737476042&do=diff Sheikh Muhammad Saleem Shahzad <image shape=“rounded”></image> “Mr. Saleem, an accomplished mathematician with an M.Phil in Processor Optimization through Memoryless Computations from the University of Sargodha, currently serves as a dedicated Mathematics Lecturer at Government Graduate College Jauharabad. His teaching experience includes a role as a Visiting Lecturer at the University of Education, Lahore, where he instructed courses such as Calculus, Set Theory, Real Analysis, Group Theory, a… text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/people/salman-sherazi?rev=1737476042&do=diff Muhammad Salman Sherazi <image shape=“rounded”>[Muhammad Salman Sherazin]</image> We are very thankful to Mr. Muhammad Salman Sherazi for contributing to this website. * Email: <salmansherazi3@gmail.com> * Cell: +92-333-7727666 or +92-306-7856232 * Teaching Mathematics at Mailsi , (Vehari) text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/people/tanveer?rev=1737476042&do=diff Prof. Muhammad Tanveer We are very thankful to Prof. M. 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Currently, he is doing teaching in this field. text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/people/umer?rev=1737476042&do=diff Umer Asghar We are very thankful to Mr. Umer Asghar for his contribution to the website. <image shape=“rounded”></image> * Email: <umermth2016@gmail.com> * Skype ID: sp15mmth06678 * Cell: +92-307-4896454 Contribution: * Notes of Metric Spaces by Umer Asghar NEW * Notes of Number Theory by Umer Asghar * Exercise 9.2 (BSc Mathematical Method) text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/people/usman?rev=1737476042&do=diff Muhammad Usman Hamid We are very thankful to him for his contribution to the website. Mr. Usman is a very talented and dedicated teacher contributing to the knowledge of mathematics by many ways. His composed notes, videos are worth mentioning in this regards. Mr. Usman is making video on different topics of mathematics which are available on his YouTube channel. 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 2024-08-05T06:27:23+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/wiki/welcome?rev=1722839243&do=diff Welcome to your new DokuWiki Congratulations, your wiki is now up and running. Here are a few more tips to get you started. Enjoy your work with DokuWiki, -- the developers Create your first pages Your wiki needs to have a start page. As long as it doesn't exist, this link will be red: 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/ch03_general_theorem_intermediate_forms?rev=1737476035&do=diff Chapter 03: General Theorem, Intermediate Forms [BSc Calculus 3rd Chapter] What is in the this chapter? * Rolle's theorem * Geometrical interpretation of Rolle's theorem * The mean value theorems * Another form of mean value theorem * Increasing and decreasing functions$\frac{0}{0}$$\frac{\infty}{\infty}$$0\times \infty$$\infty \times 0$$\infty-\infty$$0^\infty, 1^\infty, \infty^0$ 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/ch04_techniques_of_integration?rev=1737476035&do=diff Chapter 04: Techniques of Integration These notes are written by Mr. Aqeel Nawaz. We are very thankful to him for providing these notes. * Anti-derivative * Table of integrals * Integration by substitution * Integration by parts * Column (or tabular) integration 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/ch04_techniques_of_integration_farooq?rev=1737476035&do=diff Chapter 04: Techniques of Integration <div><img src="https://dl.dropbox.com/u/64787761/integration.jpg" title="Integral of the one variable function" class="mediaright" alt="Integral of the one variable function" /></div> These notes are written by Prof. Muhammad Farooq. We are very thankful to him for providing these notes. * Anti-derivative * Table of integrals * Integration by substitution * Integration by parts * Column (or tabular) integration 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/ch05_the_definite_integral?rev=1737476035&do=diff Chapter 05: The Definite Integral These notes are written by Mr. Abrar Mustafa. We are very thankful to him for providing these notes. * The definite integral as limit of a sum * Evaluation of limit of a sum * Evaluation of definite integral 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/ch08_analytic_geometry_of_three_dimensions?rev=1737476035&do=diff Chapter 08: Analytic Geometry of Three Dimensions Notes of the book Calculus with Analytic Geometry written by Dr. S. M. Yusuf and Prof. Muhammad Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN. Contents & Summary * Distance between two points$\mathbb{R}^3$ text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/bsc/notes_of_mathematical_method/ch01_complex_numbers?rev=1737476035&do=diff Chapter 01: Complex Numbers [Chapter 01 Complex Numbers Methods] Notes of the book Mathematical Method written by S.M. Yusuf, A. Majeed and M. Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN. A complex number is an element $(x,y)$ of the set $$ \mathbb{R}^2=\{(x,y): x,y \in \mathbb{R}\} $$ obeying the following rules of addition and multiplication.$z_1=(x_1,y_1)$$z_2=(x_2,y_2)$$z_1+z_2= (x_1+x_2, y_1+y_2)$$z_1 z_2 = (x_1 x_2 - y_1 y_2, x_1 y_2+y_1 x_2)$$\mathbb{R}^2$$\mathbb{C}$ text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/bsc/notes_of_mathematical_method/ch06_vector_spaces?rev=1737476035&do=diff Chapter 06: Vector Spaces Notes of Chapter 06 Vector Spaces of the book Mathematical Method written by S.M. Yusuf, A. Majeed and M. Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN. Contents and summary * Subspaces * Linear combinations and spanning sets text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/bsc/notes_of_mathematical_method/ch11_the_laplace_transform?rev=1737476035&do=diff Chapter 11: The Laplace Transform Notes of the book Mathematical Method written by S.M. Yusuf, A. Majeed and M. Amin. This book is published by Ilmi Kitab Khana, Lahore - PAKISTAN. Solutions of Chapter 11: The Laplace Transform are given here in pdf form. $f$$[0,\infty)$$f$$\mathcal{L}(f)$$F$$ provided the above improper integral converges. We have $ text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/bsc/notes_of_mechanics/tariq_mahmood_qadri?rev=1737476035&do=diff Notes of Mechanics by Tariq Mahmood Qadri [Notes of Mechanics by Tariq Mahmood Qadri] We are very thankful to Tariq Mahmood Qadri for providing these notes. These notes are helpful at BSc or BS level of Mathematics. Vector and Mechanics is essential part of the B Course of Mathematics in BSc or in Associate Degree of Science (ADS). text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01?rev=1737476036&do=diff Unit 1: Complex Numbers (Solutions) This is a first unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the students will be able to$z$$z=a+ib$$(a,b)$$a$$b$$i=\sqrt{-1}$$a$$z$$b$$z$$\bar{z} = a —ib$$z=a+ib$$|z| = \sqrt{a^2+b^2}$$z=a+ib$$'+'$$'\times'$$z$$|z|=|-z|=|\bar{z}=|-\bar{z}|$$pz^2+ qz+ r = 0$$p,q,r$$z$ text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10?rev=1737476036&do=diff Unit 10: Trigonometric Identities of Sum and Difference of Angles (Solutions) This is a tenth unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions.$a\sin\theta + b\cos \theta$$r\sin(\theta +\psi )$$a = r\cos\psi$$b=r\sin\psi$ text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-ptb/mcq-bank/ch02?rev=1737476037&do=diff MCQs: Ch 02 Sets, Functions and Groups High quality MCQs of Chapter 02 Sets, Functions and Groups of Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. The answers are given at the end of the page.$\forall$$\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 2025-01-21T16:13:55+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/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/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_mcqs/short_questions_by_mr._akhtar_abbas?rev=1737476036&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.$\sqrt{x^2-4}$$f(x)=\frac{2x}{x-2}$$x=2$$x^{100}$ text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_2_solutions/ch02?rev=1737476036&do=diff Unit 02: Differentiation [Unit 02: Differentiation] Notes (Solutions) of Unit 02: Differentiation, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. You can view online or download PDF. To view PDF, you must have PDF Reader installed on your system and it can be downloaded from Software section.$f'(x)$$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 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01?rev=1737476037&do=diff Unit 01: Complex Numbers (Solutions) This is a first unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the students will be able to$z$$z=a+ib$$(a,b)$$a$$b$$i=\sqrt{-1}$$a$$z$$b$$z$$\bar{z} = a —ib$$z=a+ib$$|z| = \sqrt{a^2+b^2}$$z=a+ib$$'+'$$'\times'$$z$$|z|=|-z|=|\bar{z}=|-\bar{z}|$$pz^2+ qz+ r = 0$$p,q,r$$z$ text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02?rev=1737476037&do=diff Unit 02: Matrices and Determinants (Solutions) This is a second unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the students will be able to$z$$z=a+ib$$(a,b)$$a$$b$$i=\sqrt{-1}$$a$$z$$b$$z$$\bar{z} = a —ib$$z=a+ib$$|z| = \sqrt{a^2+b^2}$$z=a+ib$$'+'$$'\times'$$z$$|z|=|-z|=|\bar{z}=|-\bar{z}|$$pz^2+ qz+ r = 0$$p,q,r$$z$ text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03?rev=1737476037&do=diff Unit 03: Vectors (Solutions) This is a third unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the students will be able to$i$$j$$i$$j$$k$$O$$-A$$A$$i.i=j.j=k.k=1$$i.j=j.k=k.i=0$$i\times i =j\times j =k\times k=0$$i\times j = k$$j\times k =k\times j = i$$A \times B$$A$$B$$i.j\times k =j.k\times i=k.i\times j=1$$i.k\times j = J.i\times k=k.j\times i=-1$ text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04?rev=1737476038&do=diff Unit 04: Sequence and Series (Solutions) This is a forth unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the students will be able to$n$$n$$n$$n$$n$ text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05?rev=1737476038&do=diff Unit 05: Miscellaneous Series (Solutions) This is a fifth unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the students will be able to$n$$n$$n$$n$$n$$\dfrac{1}{a(a+d)}+\dfrac{1}{(a+d)(a+2 d)}+ \cdots $ text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06?rev=1737476038&do=diff Unit 06: Permutation, Combination and Probability (Solutions) This is a sixth unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the students will be able to$n$$n!$$n$$r$$^nP_r$$n$$r$$n$$r$ text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07?rev=1737476038&do=diff Unit 07: Mathmatical Induction and Binomial Theorem (Solutions) This is a seventh unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the students will be able to$(x+y)^n$$n$$(x+y)^n$$(x+ y)^n.$$(1 +x)^n$$n$$n.$$(l +x)^n$$x$$|x| < 1$$n$ text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10?rev=1737476039&do=diff Unit 10: Trigonometric Identities of Sum and Difference of Angles (Solutions) This is a tenth unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions.$a\sin\theta + b\cos \theta$$r\sin(\theta +\psi )$$a = r\cos\psi$$b=r\sin\psi$ text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01?rev=1737476039&do=diff Unit 01: Complex Numbers (Solutions) [Unit 01: Complex Numbers (Solutions)] This is a first unit of the book Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. On this page we have provided the solutions of the questions.$z$$z^2+a^2$$z^3-3z^2+z=5$$pz^2+qz+r=0$$p,q,r$$z$ text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02?rev=1737476039&do=diff Unit 02: Matrices and Determinants (Solutions) This is a second unit of the book Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. On this page we have provided the solutions of the questions. text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04?rev=1737476039&do=diff Unit 04: Sequences and Seeries This is a forth unit of the book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. On this page we have provided the solutions of the questions.$n$$n$$n$$n$$n$$n$$n$ text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05?rev=1737476040&do=diff Unit 05: Polynomials [Unit 05: Polynomials] This is a fifth unit of the book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. On this page we have provided the solutions of the questions. text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08?rev=1737476040&do=diff Unit 08: Fundamental of Trigonometry [Unit 08: Fundamental of Trigonometry] This is a eight unit of the book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. On this page we have provided the solutions of the questions.$\cos(\alpha -\beta)=\cos \alpha \cos\beta+\sin\alpha \sin\beta$$\cos(\alpha +\beta)=\cos \alpha \cos\beta-\sin\alpha \sin\beta$$\sin(\alpha \pm \beta)=\sin \alpha \cos\beta \pm \sin\alpha \co… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09?rev=1737476040&do=diff Unit 09: Trigonometric Functions This is a ninth unit of the book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. On this page we have provided the solutions of the questions.$a+b \sin \theta$$a+b \cos \theta$$a+b \sin(c \theta+d)$$a+b \cos(c \theta+d)$$a, b, c$$d$$\sin \theta$$\cos \theta$$\tan \theta$ text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/msc/mcqs_short_questions/toplogy?rev=1737476041&do=diff Topology: Short Questions and MCQs <callout type=“info” icon=“true”> We are going to add short questions and MCQs for Topology. This is a compulsory subject in MSc and BS Mathematics in most of the universities of Pakistan. The author of this page is Dr. $\mathbb{R}$$X=\{a\}$$X$$X$$X$$\tau$$\mathbb{N}$$\tau$$(\mathbb{Z}, \tau)$$\mathbb{N}$$\tau$$A=\{\pm 100,\pm 101, \pm 102, ... \}$$\tau$$E=\{0,\pm 2,\pm 4,...\}$$\tau$$\tau$$B=\{1,2,3,...,99\}$$\tau$$C=\{10^{10}+n : n \in \mathbb{Z} \}$$\tau$$S… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/msc/notes/number-theory-iqra-liaqat?rev=1737476041&do=diff Number Theory by Ms. Iqra Liaqat [Number Theory by Ms. Iqra Liaqat] Notes of number theory provided Ms. Iqra Liaqat is a very good addition in the MSc notes section. We are actually quite grateful to her for giving these notes and likes her encouragement to distribute these notes on MathCity.org Name text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/papers/old_admission_test_of_assms_for_ph.d._mathematics/how_to_prepare_admission_test_a_short_guide?rev=1737476042&do=diff How to prepare admission test (A short guide) <callout type=“warning” icon=“true”> MathCity.org does not represent any official or government/semi-government/private educational institute or board or university. The resources given on the site holds no official position in government/semi-government/private educational institute or board or university. While using a resources given on this site you agreed to the term that we (MathCity.org or person related to MathCity.org) do not take any respo… text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/papers/old_papers_for_bsc_mathematics/university_of_gujrat?rev=1737476042&do=diff University of Gujrat, Gujrat, BSc Old Papers (Mathematics Only) Old/previous papers of Mathematics A-Course and B-Course (Double Math) of BSc for University of Gujrat (UoG), Gujrat. Each subject have four papers. We are very thankful to Mr. Ali Sahi for his contribution. text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/papers/old_papers_for_msc_mathematics/sargodha_university?rev=1737476042&do=diff University of Sargodha, Sargodha (Old Papers) <img src=http://www.mathcity.org/images/UoS_Gate.jpg class=mediacenter /> <callout type=“tip” icon=“true”> * To open or print a DjVu file, you must have some DjVu file viewer, e.g. WinDjVu. It can be downloaded from here * From 1st Annual 2013, the paper pattern has been changed. Check the complete syllabus <div> <center> </div><div> </center> </div><div> <center> </div><div> </center> </div><div> <center> </div><div> </center> </div><div> … text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/papers/old_papers_for_msc_mathematics/university_of_gujrat?rev=1737476042&do=diff University of Gujrat (Old Papers) [University of Gujrat] Old/previous papers of MSc (Mathematics) from University of Gujrat is given here. We don't know about year of these papers and also it is not mentioned on papers. But these are useful to use them as sample. <div> <center> </div> <div> </center> </div> 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/ch04_techniques_of_integration_farooq/viewer?rev=1737476035&do=diff Chapter 04: Viewer These notes are written by Prof. Muhammad Farooq. We are very thankful to him for providing these notes. List of all exercises of chapter 04 * Solution of Exercise 4.1 * Solution of Exercise 4.2 * Solution of Exercise 4.3 * Solution of Exercise 4.4 * Solution of Exercise 4.5 * Solution of Exercise 4.6 text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/bsc/notes_of_mechanics/introduction_to_mechanics/ch02-composition-of-forces?rev=1737476035&do=diff Chapter 02: Composition of Forces Notes of Chapter 02: Composition of Forces: Introduction to Mechanics by Q. K. Ghori published by West Pak Publishing Company (Pvt) Ltd. Notes of this chapter is send by Mr. Tahir Aziz. We are very thankful for his such effort.$(\lambda,\mu)$ text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/bsc/notes_of_mechanics/introduction_to_mechanics/ch06-virtual-work?rev=1737476035&do=diff Chapter 06: Virtual Work Notes of Chapter 06: Virtual Work: Introduction to Mechanics by Q. K. Ghori published by West Pak Publishing Company (Pvt) Ltd. Notes of this chapter is provide by *Prof. M. Tanveer*. We are very thankful to him for such a kind effort. text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/bsc/notes_of_mechanics/introduction_to_mechanics/ch07-kinematics?rev=1737476035&do=diff Chapter 07: Kinematics Notes of Chapter 07: Kinematics: Introduction to Mechanics by Q. K. Ghori published by West Pak Publishing Company (Pvt) Ltd. Notes of this chapter is send by Mr. Tahir Aziz and Mr. Umair Sabi Ullah. We are very thankful to them for such a kind effort. text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/bsc/paper_pattern/punjab_university/b.sc._paper_pattern_for_a_and_b_course_of_mathematics?rev=1737476035&do=diff Syllabus & Paper Pattern for A and B Course of Mathematics <WRAP center round info 60%> This is a new page created to discuss the syllabus or course outline of PU splitted into two part. It will take some time to complete this page. Please stay in touch with this page to be updated. </WRAP> 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$$z_2=-1-i$\begin{align} L.H.S&=(-1-i)^2+2(-1-i)+2\\ &=1+2i-1-… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/review-ex-1-p4?rev=1737476036&do=diff Question 6, 7 & 8, Review Exercise 1 Solutions of Question 6, 7 & 8 of Review Exercise 1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{1}{3+4i}$\begin{align}\dfrac{1}{3+4i}&=\dfrac{1}{3+4i}\times \dfrac{3-4i}{3-4i}\\ &=\dfrac{3-4i}{9+16}\\ &=\dfrac{3-4i}{25}\\ &=\dfrac{3}{25}-\dfrac{4i}{25}\end{align}$\dfrac{3i+2}{3-2i}$\begin{align}\dfrac{3i+2}{3-2i}\\ \dfrac{3i+2}… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-3-p3?rev=1737476036&do=diff Question 3, Exercise 10.3 Solutions of Question 3 of Exercise 10.3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$\dfrac{\cos {{75}^{\circ }}+\cos {{15}^{\circ }}}{\sin {{75}^{\circ }}-\sin {{15}^{\circ }}}=\sqrt{3}.$$$$\cos \alpha +\cos \beta =2\cos \left( \dfrac{\alpha +\beta }{2} \right)\cos \left( \dfrac{\alpha -\beta }{2} \righ… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/re-ex10-p5?rev=1737476037&do=diff Question 8 & 9, Review Exercise 10 Solutions of Question 8 & 9 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin \left( \dfrac{\pi }{4}-\theta \right)\sin \left( \dfrac{\pi }{4}+\theta \right)=\dfrac{1}{2}\cos 2\theta $$2\sin \alpha \sin \beta =\cos \left( \alpha -\beta \right)-\cos \left( \alpha +\beta \r… text/html 2025-01-21T16:13:57+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 2025-01-21T16:13:55+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 2025-01-21T16:13:57+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$$z_2=-1-i$\begin{align} L.H.S&=(-1-i)^2+2(-1-i)+2\\ &=1+2i-1-… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/review-ex-1-p3?rev=1737476037&do=diff Question 4 & 5, Review Exercise 1 Solutions of Question 4 & 5 of Review Exercise 1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${{z}_{1}}=2-i$${{z}_{2}}=1+i,$$\left|\dfrac{{{z}_{1}}+{{z}_{2}}+1}{{{z}_{1}}-{{z}_{2}}+1}\right|$$z_1=2-i$$z_2=1+i$\begin{align} \dfrac{{{z}_{1}}+{{z}_{2}}+1}{{{z}_{1}}-{{z}_{2}}+1}&=\dfrac{\left( 2-i \right)+\left( 1+i \right)+1}{\left( 2-i \rig… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/review-ex-1-p4?rev=1737476037&do=diff Question 6, 7 & 8, Review Exercise 1 Solutions of Question 6, 7 & 8 of Review Exercise 1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{1}{3+4i}$$$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.$$$\dfrac{3i+2}{3-2… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p6?rev=1737476037&do=diff Question 7, Exercise 2.1 Solutions of Question 7 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7 $ A=\begin{bmatrix}1 & 0 & -1 & 2 \\3 & 1 & 2 & \quad 5 \\0 & -2 & 1 & 6\end{bmatrix}$$ 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-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/ex2-2-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}$$\begin{align}|A|&=a_{11} \left( a_{22} a_{33}-a_{23} a_{32} \right)-a_{12}\left( a_{21}a_{33}… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-3-p2?rev=1737476037&do=diff Question 2, Exercise 2.3 Solutions of Question 2 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2(i) $$\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 2025-01-21T16:13:57+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/ex3-3-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 2025-01-21T16:13:57+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}$$\vec{a}$\begin{align} \vec{a}+m \vec{b}& =\hat{i}+3 \hat{j}-4 \hat{k}+m(2 \hat{i}-3 \hat{j}+5 \hat{k}) \\ & =(1+2 m) \hat{i}+(3-3 m) \hat{j}+(5 m-4) … text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-3-p6?rev=1737476037&do=diff Question 9 & 10, Exercise 3.3 Solutions of Question 9 & 10 of Exercise 3.3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9 $\vec{k}-2 \hat{i}+3 \hat{j}+\hat{k}$$\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/ex3-4-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}+$$6 \hat{k}$\begin{align}\vec{a} \times \vec{b}&=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ -1 & 2 & -3 \\ 2 … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-5-p1?rev=1737476038&do=diff Question 1 & 2 Exercise 3.5 Solutions of Question 1 & 2 of Exercise 3.5 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1 Find $\vec{a} \cdot \vec{b} \times \vec{c}$$\vec{a}=2 \hat{i}+\hat{j}+3 \hat{k}$$\vec{b}=-\hat{i}+2 \hat{j}+\hat{k} \quad \text { and }\quad \vec{c}=3 \hat{i}+\hat{j}+2 \hat{k} \text {. }$\begin{align}V&=\vec{a} \cdot \vec{b} \times \vec{c}\\ &=\left|\… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-5-p7?rev=1737476038&do=diff Question 8 Exercise 3.5 Solutions of Question 8 of Exercise 3.5 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 8(i) Find the volume of tetrahedron with the Vectors as coterminous edges \begin{align}\vec{a}&=\hat{i}+2 \hat{j}+3 \hat{k},\\ \vec{b}&=4 \hat{i}+5 \hat{j}+6 \hat{k}, \\ \vec{c}&=7 \hat{j}+8 \hat{k}\end{align}\begin{align}V&=\dfrac{1}{6}[\vec{u} \cdot \vec{v} \… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/review-ex3-p2?rev=1737476038&do=diff Question 2 & 3 Review Exercise 3 Solutions of Question 2 & 3 of Review Exercise 3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2 $\lambda$$\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/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-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.$$\begin{align}& a_5=11\\ \Rightarrow &a_1+4 d=41 \\ \Rightarrow &6+4 d=41 \\ \Rightarrow &d=\dfrac{41-6}{4}\\ &=\dfrac{35}{4}.\end{align}\begin{align} A_1&=a+d=6+\dfrac{35}{4} \… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-4-p8?rev=1737476038&do=diff Question 11 Exercise 4.4 Solutions of Question 11 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 11 $\mathrm{n}$$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/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 2025-01-21T16:13:58+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/ex5-2-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-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/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/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/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/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$$B$… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/re-ex5-p5?rev=1737476038&do=diff Question 7 Review Exercise Solutions of Question 7 of Review Exercise of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7(i) $1.2^2+3.3^2+5.4^2+\ldots$$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/ex6-2-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/ex6-3-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/ex6-5-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$$30.$\begin{align}S&=\{1,2,3, \ldots, 50\} \tex… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-5-p5?rev=1737476038&do=diff Question 8 Exercise 6.5 Solutions of Question 8 of Exercise 6.5 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$7$$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/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/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$$724… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p2?rev=1737476038&do=diff Question 2 Review Exercise 6 Solutions of Question 2 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${ }^{2 n} C_r={ }^{2 n} C_{r+2}$$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)$\begin{align} \Rightarrow \dfrac{1}{(2 n-r) ! r… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p3?rev=1737476038&do=diff Question 3 & 4 Review Exercise 6 Solutions of Question 3 & 4 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${ }^{56} P_{r+6}:{ }^{54} P_{r+3}=30800: 1$$r$\begin{align} { }^{56} P_{r+6}:{ }^{54} P_r+3&=30800: 1 \\ \Rightarrow \dfrac{\dfrac{56 !}{[56-(r+6)] !}}{\dfrac{54 !}{[54-(r+3)] !}}&=\dfrac{30800}{1} \\ \Rightarrow \dfrac{56… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p7?rev=1737476038&do=diff Question 11 Review Exercise 6 Solutions of Question 11 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$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/ex7-1-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-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)$$(k+1)^{\text {th }}… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p11?rev=1737476038&do=diff Question 11 Exercise 7.1 Solutions of Question 11 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.\begin{align} & \left(\begin{array}{l} 2 \\ 2 \end{array}\right)+\left(\begin{array}{l} 3 \\ 2 \end{array}\right)+\left(\begin{array}{l} 4 \\ 2 \end{array}\right)+\ldots+\left(\begin{array}{l} n \\ 2 \end{array}\right)=\left(\begin{array}{c… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p12?rev=1737476038&do=diff Question 12 Exercise 7.1 Solutions of Question 12 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{5^{2 n}-1}{24}$$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}$$$n=k+1$\begin{align}\dfrac{5^{2(k+1)}-1}{24}&=\dfrac{5^{2 k+2}-1}{24} \\ & =\dfra… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p14?rev=1737476038&do=diff Question 14 Exercise 7.1 Solutions of Question 14 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$5$$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$\begin{align} 3^{2(k+1)-1}+2^{2(k+1)-1} & =3^{2 k+2-1}+2^{2… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p15?rev=1737476038&do=diff Question 15 Exercise 7.1 Solutions of Question 15 of Exercise 7.1 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$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-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$\begin{align} & T_{3+1}=\dfrac{7 !}{(7-3) ! 3 !} 2^{7-3} a^3 \\ & \Rightarrow T_4=\dfrac{7 !}{4 ! 3 !} \cdot 2^4 a^3 \\ & \Rightarrow… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p12?rev=1737476039&do=diff Question 14 Exercise 7.3 Solutions of Question 14 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$x$$p x^p-q x^q=(p-q) x^{p+q}$$x$$x=1+h$$h \longrightarrow 0$$$ p x^p-q x^q=p(1+h)^p-q(1+h)^q $$$$ \begin{aligned} & p x^p-q x^q \\ & =p(1+p h+\text { higher powers h) } \\ & -q(1+q h+\text { higher powcrs } h) \\ & \Rightarrow p x^p-q x^q=… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/re-ex7-p1?rev=1737476039&do=diff Question 1 Review Exercise 7 Solutions of Question 1 of Review Exercise 7 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Chose the correct option. <panel>$O, P, Q, R, S, T, U$$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)$$\dfrac{n+1}{n… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-3-p3?rev=1737476039&do=diff Question 3, Exercise 10.3 Solutions of Question 3 of Exercise 10.3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$\dfrac{\cos {{75}^{\circ }}+\cos {{15}^{\circ }}}{\sin {{75}^{\circ }}-\sin {{15}^{\circ }}}=\sqrt{3}.$$$$\cos \alpha +\cos \beta =2\cos \left( \dfrac{\alpha +\beta }{2} \right)\cos \left( \dfrac{\alpha -\beta }{2} \righ… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/re-ex10-p5?rev=1737476039&do=diff Question 8 & 9, Review Exercise 10 Solutions of Question 8 & 9 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin \left( \dfrac{\pi }{4}-\theta \right)\sin \left( \dfrac{\pi }{4}+\theta \right)=\dfrac{1}{2}\cos 2\theta $$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/ex1-2-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$$z_{2}=1+i$\begin{align} &z_{1}=3+i\text{ and } z_{2}=1+i.\end{align}\begin{align} \dfrac{z_1}{z_2} &= \dfrac{3+i}{1+i}\\ &=\dfrac{(3+i)(1-i)}{(1+i)(1-i)} \\ &=\… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p1?rev=1737476039&do=diff Question 1, Exercise 1.4 Solutions of Question 1 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 1(i) $2+i 2 \sqrt{3}$$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-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-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-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/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/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/ex2-3-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$$\left[\begin{array}{lll}\lambda & 1 & 3 \\ 2 & 1 & 8 \\ 0 & 3 & 1\end{array}\right]$\begin{align*} A &= \left[\begin{array}{ccc} \lambda & 1 & 3 \\ 2 & 1 & 8 \\ 0 & 3 & 1 \end{array}\right]\\ |A| &= \lambda \cdot (-23) - 1 \cdot 2 + 3… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p5?rev=1737476039&do=diff Question 9 and 10, Exercise 4.1 Solutions of Question 9 and 10 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$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-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/ex4-1-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$$\begin{align*} &a_1=\sqrt{2 \cdot 1}, \\ &a_2=\sqrt{4}=\sqrt{2 \cdot 2} \\ &a_3=\sqrt{6}=\sqrt{2 \cdot 3}\\… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p3?rev=1737476039&do=diff Question 3 and 4, Exercise 4.2 Solutions of Question 3 and 4 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $0.07,0.12,0.7, \ldots$$$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/ex4-3-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$$… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p15?rev=1737476039&do=diff Question 30, Exercise 4.4 Solutions of Question 30 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $=a_1= 1$$=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.$$ text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p1?rev=1737476039&do=diff Question 1 and 2, Exercise 4.5 Solutions of Question 1 and 2 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $16+16+16+\ldots$$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/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$$$D=30+2\left(12+\frac{24}{5}+\frac{24}{5}+... \right)$$$$ 12+\frac{24}{5}+\frac{24}{5… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-6-p5?rev=1737476040&do=diff Question 9 & 10, Exercise 4.6 Solutions of Question 9 & 10 of Exercise 4.6 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\frac{1}{7}, \frac{1}{6},-1,-\frac{1}{3}, \ldots$$$\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 2025-01-21T16:14:00+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$$1, \dfrac{1}{3}, \d… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-1-p4?rev=1737476040&do=diff Question 6 and 7, Exercise 5.1 Solutions of Question 6 and 7 of Exercise 5.1 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $m$$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/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-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/ex5-3-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$$\begin{align*} p(2)&=2^3+13(2)^2+30(2)-120 \\ &=8+52+60-120 =0 \end{ali… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-3-p5?rev=1737476040&do=diff Question 5, Exercise 5.3 Solutions of Question 5 of Exercise 5.3 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 5 $6 x^{2}+38 x+56$$2 x+8$$ACED$$ABFG$$ACED$$6 x^{2}+38 x+56$$2 x+8$\begin{align*} & 6 x^{2}+38 x+56 \\ = & 2(3x^2+19x+28) \\ = & 2(3x^2+12x+7x+28) \\ = & 2(3x(x+4)+7(x+4)) \\ =& 2(x+4)(3x+7) \\ =& (2x+8)(3x+7) \end{align*}\begin{align*} & Length … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/re-ex-p3?rev=1737476040&do=diff Question 4 & 5, Review Exercise Solutions of Question 4 & 5 of Review Exercise of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $3 y-2$$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/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-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-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 2025-01-21T16:14:00+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 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/msc/syllabus/uos/preparation_guide?rev=1737476041&do=diff Preparation Guide This guide is made by Mr. Anwar Khan, PhD. We are very thankful to him for sharing. This guide is helpful to prepare papers for MSc Mathematics (annual system) from University of Sargodha. Part 1 1. REAL ANAYSIS * Real Analysis (Notes by Syed Gul Shah) * Chapter # 08 sequences and series of Mathematical Method by SM Yousaf (solutions are available $z= f(x,y)$