This is beta site. https://beta.mathcity.org/ 2026-06-05T14:25:28+00:00 https://beta.mathcity.org/ https://beta.mathcity.org/_media/logo.png text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/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 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 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 2025-01-21T16:13:54+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 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-1-p1?rev=1737476036&do=diff Question 1, Exercise 1.1 Solutions of Question 1 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) ${{i}^{9}}+{{i}^{19}}$\begin{align}{{i}^{9}}+{{i}^{19}}&=i\cdot{{i}^{8}}+i\cdot{{i}^{18}}\\ &=i\cdot{{\left( {{i}^{2}} \right)}^{4}}+i\cdot{{\left( {{i}^{2}} \right)}^{9}}\\ &=i\cdot{{\left( -1 \right)}^{4}}+i\cdot{{\left( -1 \right)}^{9}}\\ &=i\cdo… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-1-p2?rev=1737476036&do=diff Question 2 & 3, Exercise 1.1 Solutions of Question 2 & 3 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2 ${{i}^{107}}+{{i}^{112}}+{{i}^{122}}+{{i}^{153}}=0$\begin{align}L.H.S.&={{i}^{107}}+{{i}^{112}}+{{i}^{122}}+{{i}^{153}}\\ &=i\cdot i^{106}+i^{112}+i^{122}+i\cdot i^{152}\\ &=i.{{\left( {{i}^{2}} \right)}^{53}}+{{\left( {{i}^{2}} \right)}^{56}}+… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-1-p6?rev=1737476036&do=diff Question 7, Exercise 1.1 Solutions of Question 7 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7(i) ${{z}_{1}}=1+2i$${{z}_{2}}=2+3i$$|{{z}_{1}}+{{z}_{2}}|$$z_1=1+2i$$z_2=2+3i$\begin{align} {{z}_{1}}+{{z}_{2}}&=1+2i+2+3i\\ &=1+2+2i+3i\\ &=3+5i \end{align}\begin{align} |z_1+z_2|&=\sqrt{3^2+5^2}\\ &=\sqrt{9+25}\\ &=\sqrt{34}\end{align}${{z}_{1}}=1+2… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-1-p7?rev=1737476036&do=diff Question 8, Exercise 1.1 Solutions of Question 8 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 8(i) $\dfrac{1-2i}{2+i}+\dfrac{4-i}{3+2i}$$a+ib.$\begin{align}&\dfrac{1-2i}{2+i}+\dfrac{4-i}{3+2i}\\ &=\dfrac{\left( 3+2i \right)\left( 1-2i \right)+\left( 2+i \right)\left( 4-i \right)}{\left( 2+i \right)\left( 3+2i \right)}\\ &=\dfrac{\left( 3+4+2i-6i … text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-1-p8?rev=1737476036&do=diff Question 9 & 10, Exercise 1.1 Solutions of Question 9 & 10 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9 $\dfrac{\left( 3-2i \right)\left( 2+3i \right)}{\left( 1+2i \right)\left( 2-i \right)}$\begin{align}z&=\dfrac{\left( 3-2i \right)\left( 2+3i \right)}{\left( 1+2i \right)\left( 2-i \right)}\\ &=\dfrac{6+6+9i-4i}{2+2+4i-i}\\ &=\dfrac{12+5i}{4+3… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-1-p9?rev=1737476036&do=diff Question 11, Exercise 1.1 Solutions of Question 11 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 11(i) ${{z}_{1}}=2-i$${{z}_{2}}=-2+i$${\rm Re}\left( \dfrac{{{z}_{1}}{{z}_{2}}}{\overline{{{z}_{1}}}} \right)$$z_1=2-i$$z_2=-2+i$$\overline{z_1}=2+i$\begin{align} z_1 z_2&=(2-i)(-2+i)\\ &=-4+1+2i+2i\\ &=-3+4i \end{align}\begin{align} \dfrac{z_1 z_2}{\… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-2-p1?rev=1737476036&do=diff Question 1, Exercise 1.2 Solutions of Question 1 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1 If ${{z}_{1}}=2+i$${{z}_{2}}=1-i$${{z}_{1}}=2+i$${{z}_{2}}=1-i$$$z_1+z_2=z_2+z_1.$$\begin{align}z_1+z_2&=(2+i)+(1-i)\\ &=3 \ldots (i) \end{align}\begin{align} z_2+z_1&=(1-i)+(2+i)\\ &=3 \ldots (ii)\end{align}$$z_1 z_2=z_2 z_1.$$\begin{align}z_1 z_2 … text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-2-p2?rev=1737476036&do=diff Question 2, Exercise 1.2 Solutions of Question 2 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2 $z_1=-1+i$, $z_2=3-2i$${{z}_{3}}=2-2i$${{z}_{1}}=-1+i$${{z}_{2}}=3-2i$${{z}_{3}}=2-2i$$$(z_1+z_2)+z_3=z_1+(z_2+z_3).$$\begin{align} {{z}_{1}}+{{z}_{2}}&=\left( -1+i \right)+\left( 3-2i \right)\\ &=2-i\end{align}\begin{align} \left( {{z}_{1}}+{{z}_{2}… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-2-p3?rev=1737476036&do=diff Question 3 & 4, Exercise 1.2 Solutions of Question 3 & 4 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 ${{z}_{1}}=\sqrt{3}+\sqrt{2}i$${{z}_{2}}=\sqrt{2}-\sqrt{3}i$${{z}_{3}}=2+3i$${{z}_{1}}=\sqrt{3}+\sqrt{2}i$${{z}_{2}}=\sqrt{2}-\sqrt{3}i$${{z}_{3}}=2+3i$\begin{align}{{z}_{1}}\left( {{z}_{2}}+{{z}_{3}} \right)&={{z}_{1}}{{z}_{2}}+{{z}_{1}}{{z}_{… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-2-p4?rev=1737476036&do=diff Question 5, Exercise 1.2 Solutions of Question 5 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) ${{z}_{1}}=2+4i$${{z}_{2}}=1-3i$$\overline{{{z}_{1}}+{{z}_{2}}}=\overline{{{z}_{1}}}+\overline{{{z}_{2}}}$${{z}_{1}}=2+4i$${{z}_{2}}=1-3i$$\overline{{{z}_{1}}}=2-4i$$\overline{{{z}_{2}}}=1+3i$\begin{align}z_1+z_2&=2+4i+1-3i\\ &=3+i \end{align}\begin… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-2-p5?rev=1737476036&do=diff Question 6, Exercise 1.2 Solutions of Question 6 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6(i) ${{z}_{1}}$${{z}_{2}}$$|{{z}_{1}}{{z}_{2}}|=|{{z}_{1}}||{{z}_{2}}|$${{z}_{1}}=a+bi$${{z}_{2}}=c+di$$|z_1=\sqrt{a^2+b^2}|$$|z_2=\sqrt{c^2+d^2}|$\begin{align} L.H.S.&=|{{z}_{1}}{{z}_{2}}|\\ &=|(a+bi)(c+di)|\\ &=|ac-bd+(ad+bc)i|\\ &=\sqrt{{{(ac-bd)}^{… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-2-p6?rev=1737476036&do=diff Question 7, Exercise 1.2 Solutions of Question 7 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7(i) $\dfrac{2+3i}{5-2i}$\begin{align}&\dfrac{2+3i}{5-2i} \\ =&\dfrac{2+3i}{5-2i}\times \dfrac{5+2i}{5+2i} \quad \text{by rationalizing} \\ =&\dfrac{10-6+15i+4i}{25+4}\\ =&\dfrac{4+19i}{29}\\ =&\dfrac{4}{29}+\dfrac{19}{29}i \end{align}$=\dfrac{4}{29}$$=\… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-2-p7?rev=1737476036&do=diff Question 8, Exercise 1.2 Solutions of Question 8 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 8(i) $z+\overline{z}=2\operatorname{Re}\left( z \right)$$z=a+ib$$\overline{z}=a-ib$\begin{align}z+\overline{z}&=\left( a+ib \right)+\left( a-ib \right)\\ &=a+ib+a-ib\\ &=2a\\ z+\overline{z}&=2\operatorname{Re}\left( z \right)\end{align}$z-\overline{z}=2i… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-2-p8?rev=1737476036&do=diff Question 9, Exercise 1.2 Solutions of Question 9 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9(i) $z=3+2i,$$-|z|\leq \operatorname{Re}\left( z \right)\leq |z|$$z=3+2i$$|z|=\sqrt{9+4}=\sqrt{13}$${\rm Re}z=3=\sqrt{9}$\begin{align} &-\sqrt{13} \leq \sqrt{9} \leq \sqrt{13}\\ \implies &-|z|\leq \operatorname{Re}\left( z \right)\leq |z|\end{align}$z=3… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-3-p1?rev=1737476036&do=diff Question 1, Exercise 1.3 Solutions of Question 1 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) \begin{align}&z-4w=3i\\ &2z+3w=11-5i\end{align}\begin{align}z-4w&=3i …(i)\\ 2z+3w&=11-5i …(ii)\end{align}$2$\begin{align}2z-8w&=6i …(iii)\end{align}\[\begin{array}{cccc} 2z&-8w&=6i \\ \mathop+\limits_{-}2z&\mathop+\limits_{-}3w&=\mathop-\limit… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-3-p2?rev=1737476036&do=diff Question 2, Exercise 1.3 Solutions of Question 2 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2(i) $P(z)$$$P\left( z \right)={{z}^{3}}+6z+20$$$$p\left( z \right)={{z}^{3}}+6z+20$$$(z-a)$$P(z)$$P(a)=0$$z=-2$\begin{align} P(-2)&=(-2)^3+6(-2)+20\\ &=-8-12+20\\ &=0\end{align}$z+2$${{z}^{3}}+6z+20$$$\begin{array}{c|cccc} -2 & 1 & 0 & 6 & 20 \\ & \d… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-3-p3?rev=1737476036&do=diff Question 3 & 4, Exercise 1.3 Solutions of Question 3 & 4 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 ${{z}_{1}}=-1+i$${{z}_{2}}=-1-i$${{z}^{2}}+2z+2=0$$$z^2+2z_1+2=0\quad \ldots (i)$$$z_1=-1+i$\begin{align}L.H.S &= (-1+i)^2+2(-1+i)+2\\ &=1-2i-1-2+2i+2\\ &=0=R.H.S\end{align}$z_1=-1+i$$z_2=-1-i$\begin{align} L.H.S&=(-1-i)^2+2(-1-i)+2\\ &=1+2i-1-… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-3-p4?rev=1737476036&do=diff Question 5, Exercise 1.3 Solutions of Question 5 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) ${{z}^{2}}+z+3=0$${{z}^{2}}+z+3=0$$a=1,\,\,\,b=1$$c=3$\begin{align}z&=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\\ z&=\dfrac{-\left( 1 \right)\pm \sqrt{{{\left( 1 \right)}^{2}}-4\left( 1 \right)\left( 3 \right)}}{2\left( 1 \right)}\\ z&=\dfrac{-1\pm \s… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-3-p5?rev=1737476036&do=diff Question 6, Exercise 1.3 Solutions of Question 6 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6(i) ${{z}^{4}}+{{z}^{2}}+1=0$\begin{align}{{z}^{4}}+{{z}^{2}}+1&=0\\ {{z}^{4}}+2\left( \dfrac{1}{2} \right){{z}^{2}}+\dfrac{1}{4}-\dfrac{1}{4}+1&=0\\ {{\left( {{z}^{2}}+\dfrac{1}{2} \right)}^{2}}+\dfrac{4-1}{4}&=0\\ {{\left( {{z}^{2}}+\dfrac{1}{2} \righ… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/review-ex-1-p1?rev=1737476036&do=diff Question 1, Review Exercise 1 Solutions of Question 1 of Review Exercise 1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1 ${{\left( \dfrac{2i}{1+i} \right)}^{2}}$$i$$2i$$1-i$$i+1$$\dfrac{5+2i}{4-3i}$$-\dfrac{7}{25}+\dfrac{26}{25}i$$\dfrac{5}{4}-\dfrac{2}{3}i$$\dfrac{14}{25}+\dfrac{23}{25}i$$\dfrac{26}{7}+\dfrac{23}{7}i$${{i}^{57}}+\frac{1}{{{i}^{25}}}$$0$$2i$$-2… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/review-ex-1-p2?rev=1737476036&do=diff Question 2 & 3, Review Exercise 1 Solutions of Question 2 & 3 of Review Exercise 1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${{i}^{n}}+{{i}^{n+1}}+{{i}^{n+2}}+{{i}^{n+3}}=0$$\forall n\in N$\begin{align}{{i}^{n}}+{{i}^{n+1}}+{{i}^{n+2}}+{{i}^{n+3}}&=0\\ L.H.S.&={{i}^{n}}+{{i}^{n}}\cdot i+{{i}^{n}}\cdot {{i}^{2}}+{{i}^{n}}\cdot {{i}^{3}}\\ &={{i}^{n}}\left( 1+i+{{i}^{2}}… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/review-ex-1-p3?rev=1737476036&do=diff Question 4 & 5, Review Exercise 1 Solutions of Question 4 & 5 of Review Exercise 1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${{z}_{1}}=2-i$${{z}_{2}}=1+i,$$|\dfrac{{{z}_{1}}+{{z}_{2}}+1}{{{z}_{1}}-{{z}_{2}}+1}|$\begin{align}{{z}_{1}}&=2-i,\\ {{z}_{2}}&=1+i,\\ \dfrac{{{z}_{1}}+{{z}_{2}}+1}{{{z}_{1}}-{{z}_{2}}+1}&=\dfrac{\left( 2-i \right)+\left( 1+i \right)+1}{\left( 2-… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/review-ex-1-p4?rev=1737476036&do=diff Question 6, 7 & 8, Review Exercise 1 Solutions of Question 6, 7 & 8 of Review Exercise 1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{1}{3+4i}$\begin{align}\dfrac{1}{3+4i}&=\dfrac{1}{3+4i}\times \dfrac{3-4i}{3-4i}\\ &=\dfrac{3-4i}{9+16}\\ &=\dfrac{3-4i}{25}\\ &=\dfrac{3}{25}-\dfrac{4i}{25}\end{align}$\dfrac{3i+2}{3-2i}$\begin{align}\dfrac{3i+2}{3-2i}\\ \dfrac{3i+2}… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-1-p2?rev=1737476036&do=diff Question 2, Exercise 10.1 Solutions of Question 2 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin \dfrac{\pi }{12}$$\dfrac{\pi }{12}$$\dfrac{\pi }{3}-\dfrac{\pi }{4}$\begin{align}\sin (\alpha -\beta )=\sin \alpha \cos \beta -\cos \alpha \sin.\end{align}\begin{align} \Rightarrow \quad \sin \left( \frac{\pi }{3}-\f… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-1-p8?rev=1737476036&do=diff Question 8, Exercise 10.1 Solutions of Question 8 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\tan \left( \dfrac{\pi }{4}+\theta \right)=\dfrac{\cos \theta +\sin \theta }{\cos \theta -\sin \theta }$\begin{align}L.H.S.&=\tan \left( \dfrac{\pi }{4}+\theta \right)\\ &=\dfrac{\sin \left( \dfrac{\pi }{4}+\theta \ri… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-3-p3?rev=1737476036&do=diff Question 3, Exercise 10.3 Solutions of Question 3 of Exercise 10.3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$\dfrac{\cos {{75}^{\circ }}+\cos {{15}^{\circ }}}{\sin {{75}^{\circ }}-\sin {{15}^{\circ }}}=\sqrt{3}.$$$$\cos \alpha +\cos \beta =2\cos \left( \dfrac{\alpha +\beta }{2} \right)\cos \left( \dfrac{\alpha -\beta }{2} \righ… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-3-p4?rev=1737476037&do=diff Question 5, Exercise 10.3 Solutions of Question 5 of Exercise 10.3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$\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)$\begin{align}L.H.S.&=\cos {{20… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/re-ex10-p1?rev=1737476037&do=diff Question 1, Review Exercise 10 Solutions of Question 1 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos {{50}^{\circ }}5{0}'\cos {{9}^{\circ }}1{0}'-\sin {{50}^{\circ }}5{0}'\sin {{9}^{\circ }}1{0}'=$$0$$\dfrac{1}{2}$$1$$\dfrac{\sqrt{3}}{2}$$\tan {{15}^{\circ }}=2-\sqrt{3}$${{\cot }^{2}}{{75}^{\circ }}$$7+\sq… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/re-ex10-p2?rev=1737476037&do=diff Question 2 and 3, Review Exercise 10 Solutions of Question 2 and 3 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{2\sin \theta \sin 2\theta }{\cos \theta +\cos 3\theta }=\tan 2\theta \tan \theta $\begin{align}L.H.S.&=\dfrac{2\sin \theta \sin 2\theta }{\cos \theta +\cos 3\theta }\\ &=\dfrac{2\sin \theta \s… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/re-ex10-p3?rev=1737476037&do=diff Question 4 & 5, Review Exercise 10 Solutions of Question 4 & 5 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${{\sin }^{2}}\dfrac{\theta }{2}=\dfrac{\sin \theta \tan \dfrac{\theta }{2}}{2}$\begin{align}R.H.S.&=\dfrac{\sin \theta \tan \dfrac{\theta }{2}}{2}\\ &=\dfrac{\sin \theta \sin \dfrac{\theta }{2}}{2\cos \d… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/re-ex10-p4?rev=1737476037&do=diff Question 6 & 7, Review Exercise 10 Solutions of Question 6 & 7 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos 4\theta =1-8{{\sin }^{2}}\theta {{\cos }^{2}}\theta $\begin{align}L.H.S&=\cos 4\theta \\ &=\cos 2\left( 2\theta \right)\\ &=1-2si{{n}^{2}}2\theta \\ &=1-2{{\left( 2sin\theta \cos \theta \right)}^{… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-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/math-11-kpk/sol/unit01/ex1-1-p1?rev=1737476037&do=diff Question 1, Exercise 1.1 Solutions of Question 1 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) ${{i}^{9}}+{{i}^{19}}$\begin{align}{{i}^{9}}+{{i}^{19}}&=i\cdot{{i}^{8}}+i\cdot{{i}^{18}}\\ &=i\cdot{{\left( {{i}^{2}} \right)}^{4}}+i\cdot{{\left( {{i}^{2}} \right)}^{9}}\\ &=i\cdot{{\left( -1 \right)}^{4}}+i\cdot{{\left( -1 \right)}^{9}}\\ &=i\cdo… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-1-p2?rev=1737476037&do=diff Question 2 & 3, Exercise 1.1 Solutions of Question 2 & 3 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2 ${{i}^{107}}+{{i}^{112}}+{{i}^{122}}+{{i}^{153}}=0$\begin{align}L.H.S.&={{i}^{107}}+{{i}^{112}}+{{i}^{122}}+{{i}^{153}}\\ &=i\cdot i^{106}+i^{112}+i^{122}+i\cdot i^{152}\\ &=i.{{\left( {{i}^{2}} \right)}^{53}}+{{\left( {{i}^{2}} \right)}^{56}}+… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-1-p6?rev=1737476037&do=diff Question 7, Exercise 1.1 Solutions of Question 7 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7(i) ${{z}_{1}}=1+2i$${{z}_{2}}=2+3i$$|{{z}_{1}}+{{z}_{2}}|$$z_1=1+2i$$z_2=2+3i$\begin{align} {{z}_{1}}+{{z}_{2}}&=1+2i+2+3i\\ &=1+2+2i+3i\\ &=3+5i \end{align}\begin{align} |z_1+z_2|&=\sqrt{3^2+5^2}\\ &=\sqrt{9+25}\\ &=\sqrt{34}\end{align}${{z}_{1}}=1+2… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-1-p7?rev=1737476037&do=diff Question 8, Exercise 1.1 Solutions of Question 8 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 8(i) $\dfrac{1-2i}{2+i}+\dfrac{4-i}{3+2i}$$a+ib.$\begin{align}&\dfrac{1-2i}{2+i}+\dfrac{4-i}{3+2i}\\ &=\dfrac{\left( 3+2i \right)\left( 1-2i \right)+\left( 2+i \right)\left( 4-i \right)}{\left( 2+i \right)\left( 3+2i \right)}\\ &=\dfrac{\left( 3+4+2i-6i … text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-1-p8?rev=1737476037&do=diff Question 9 & 10, Exercise 1.1 Solutions of Question 9 & 10 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9 $\dfrac{\left( 3-2i \right)\left( 2+3i \right)}{\left( 1+2i \right)\left( 2-i \right)}$\begin{align}z&=\dfrac{\left( 3-2i \right)\left( 2+3i \right)}{\left( 1+2i \right)\left( 2-i \right)}\\ &=\dfrac{6+6+9i-4i}{2+2+4i-i}\\ &=\dfrac{12+5i}{4+3… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-1-p9?rev=1737476037&do=diff Question 11, Exercise 1.1 Solutions of Question 11 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 11(i) ${{z}_{1}}=2-i$${{z}_{2}}=-2+i$${\rm Re}\left( \dfrac{{{z}_{1}}{{z}_{2}}}{\overline{{{z}_{1}}}} \right)$$z_1=2-i$$z_2=-2+i$$\overline{z_1}=2+i$\begin{align} z_1 z_2&=(2-i)(-2+i)\\ &=-4+1+2i+2i\\ &=-3+4i \end{align}\begin{align} \dfrac{z_1 z_2}{\… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-2-p1?rev=1737476037&do=diff Question 1, Exercise 1.2 Solutions of Question 1 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1 If ${{z}_{1}}=2+i$${{z}_{2}}=1-i$${{z}_{1}}=2+i$${{z}_{2}}=1-i$$$z_1+z_2=z_2+z_1.$$\begin{align}z_1+z_2&=(2+i)+(1-i)\\ &=3 \ldots (i) \end{align}\begin{align} z_2+z_1&=(1-i)+(2+i)\\ &=3 \ldots (ii)\end{align}$$z_1 z_2=z_2 z_1.$$\begin{align}z_1 z_2 … text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-2-p2?rev=1737476037&do=diff Question 2, Exercise 1.2 Solutions of Question 2 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2 $z_1=-1+i$, $z_2=3-2i$${{z}_{3}}=2-2i$${{z}_{1}}=-1+i$${{z}_{2}}=3-2i$${{z}_{3}}=2-2i$$$(z_1+z_2)+z_3=z_1+(z_2+z_3).$$\begin{align} {{z}_{1}}+{{z}_{2}}&=\left( -1+i \right)+\left( 3-2i \right)\\ &=2-i\end{align}\begin{align} \left( {{z}_{1}}+{{z}_{2}… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-2-p3?rev=1737476037&do=diff Question 3 & 4, Exercise 1.2 Solutions of Question 3 & 4 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 ${{z}_{1}}=\sqrt{3}+\sqrt{2}i$${{z}_{2}}=\sqrt{2}-\sqrt{3}i$${{z}_{3}}=2+3i$${{z}_{1}}=\sqrt{3}+\sqrt{2}i$${{z}_{2}}=\sqrt{2}-\sqrt{3}i$${{z}_{3}}=2+3i$\begin{align}{{z}_{1}}\left( {{z}_{2}}+{{z}_{3}} \right)&={{z}_{1}}{{z}_{2}}+{{z}_{1}}{{z}_{… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-2-p4?rev=1737476037&do=diff Question 5, Exercise 1.2 Solutions of Question 5 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) ${{z}_{1}}=2+4i$${{z}_{2}}=1-3i$$\overline{{{z}_{1}}+{{z}_{2}}}=\overline{{{z}_{1}}}+\overline{{{z}_{2}}}$${{z}_{1}}=2+4i$${{z}_{2}}=1-3i$$\overline{{{z}_{1}}}=2-4i$$\overline{{{z}_{2}}}=1+3i$\begin{align}z_1+z_2&=2+4i+1-3i\\ &=3+i \end{align}\begin… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-2-p5?rev=1737476037&do=diff Question 6, Exercise 1.2 Solutions of Question 6 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6(i) ${{z}_{1}}$${{z}_{2}}$$|{{z}_{1}}{{z}_{2}}|=|{{z}_{1}}||{{z}_{2}}|$${{z}_{1}}=a+bi$${{z}_{2}}=c+di$$|z_1=\sqrt{a^2+b^2}|$$|z_2=\sqrt{c^2+d^2}|$\begin{align} L.H.S.&=|{{z}_{1}}{{z}_{2}}|\\ &=|(a+bi)(c+di)|\\ &=|ac-bd+(ad+bc)i|\\ &=\sqrt{{{(ac-bd)}^{… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-2-p6?rev=1737476037&do=diff Question 7, Exercise 1.2 Solutions of Question 7 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7(i) $\dfrac{2+3i}{5-2i}$\begin{align}&\dfrac{2+3i}{5-2i} \\ =&\dfrac{2+3i}{5-2i}\times \dfrac{5+2i}{5+2i} \quad \text{by rationalizing} \\ =&\dfrac{10-6+15i+4i}{25+4}\\ =&\dfrac{4+19i}{29}\\ =&\dfrac{4}{29}+\dfrac{19}{29}i \end{align}$=\dfrac{4}{29}$$=\… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-2-p7?rev=1737476037&do=diff Question 8, Exercise 1.2 Solutions of Question 8 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 8(i) $z+\overline{z}=2\operatorname{Re}\left( z \right)$$z=a+ib$$\overline{z}=a-ib$\begin{align}z+\overline{z}&=\left( a+ib \right)+\left( a-ib \right)\\ &=a+ib+a-ib\\ &=2a\\ z+\overline{z}&=2\operatorname{Re}\left( z \right)\end{align}$z-\overline{z}=2i… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-2-p8?rev=1737476037&do=diff Question 9, Exercise 1.2 Solutions of Question 9 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9(i) $z=3+2i,$$-|z|\leq \operatorname{Re}\left( z \right)\leq |z|$$z=3+2i$$|z|=\sqrt{9+4}=\sqrt{13}$${\rm Re}z=3=\sqrt{9}$\begin{align} &-\sqrt{13} \leq \sqrt{9} \leq \sqrt{13}\\ \implies &-|z|\leq \operatorname{Re}\left( z \right)\leq |z|\end{align}$z=3… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-3-p1?rev=1737476037&do=diff Question 1, Exercise 1.3 Solutions of Question 1 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) \begin{align}&z-4w=3i\\ &2z+3w=11-5i\end{align}\begin{align}z-4w&=3i …(i)\\ 2z+3w&=11-5i …(ii)\end{align}$2$\begin{align}2z-8w&=6i …(iii)\end{align}\[\begin{array}{cccc} 2z&-8w&=6i \\ \mathop+\limits_{-}2z&\mathop+\limits_{-}3w&=\mathop-\limit… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-3-p2?rev=1737476037&do=diff Question 2, Exercise 1.3 Solutions of Question 2 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2(i) $P(z)$$$P\left( z \right)={{z}^{3}}+6z+20$$$$p\left( z \right)={{z}^{3}}+6z+20$$$(z-a)$$P(z)$$P(a)=0$$z=-2$\begin{align} P(-2)&=(-2)^3+6(-2)+20\\ &=-8-12+20\\ &=0\end{align}$z+2$${{z}^{3}}+6z+20$$$\begin{array}{c|cccc} -2 & 1 & 0 & 6 & 20 \\ & \d… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-3-p3?rev=1737476037&do=diff Question 3 & 4, Exercise 1.3 Solutions of Question 3 & 4 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 ${{z}_{1}}=-1+i$${{z}_{2}}=-1-i$${{z}^{2}}+2z+2=0$$$z^2+2z_1+2=0\quad \ldots (i)$$$z_1=-1+i$\begin{align}L.H.S &= (-1+i)^2+2(-1+i)+2\\ &=1-2i-1-2+2i+2\\ &=0=R.H.S\end{align}$z_1=-1+i$$z_2=-1-i$\begin{align} L.H.S&=(-1-i)^2+2(-1-i)+2\\ &=1+2i-1-… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-3-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/unit01/ex1-3-p5?rev=1737476037&do=diff Question 6, Exercise 1.3 Solutions of Question 6 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6(i) ${{z}^{4}}+{{z}^{2}}+1=0$$$z^4+z^2+1=0$$$$z^4+2z^2+1-z^2=0$$$$( z^2+1 )^2-z^2=0$$$$( z^2+1+z)( z^2+1-z )=0$$$$( z^2+z+1 )( z^2-z+1 )=0$$$$(z^2+z+1 )=0$$$$z=\dfrac{-1\pm \sqrt{1-4}}{2}$$$$z=\dfrac{-1\pm \sqrt{3}i}{2}$$$$(z^2-z+1 )=0$$$$z=\dfrac{1\pm … text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/review-ex-1-p1?rev=1737476037&do=diff Question 1, Review Exercise 1 Solutions of Question 1 of Review Exercise 1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1 ${{\left( \dfrac{2i}{1+i} \right)}^{2}}$$i$$2i$$1-i$$i+1$$\dfrac{5+2i}{4-3i}$$-\dfrac{7}{25}+\dfrac{26}{25}i$$\dfrac{5}{4}-\dfrac{2}{3}i$$\dfrac{14}{25}+\dfrac{23}{25}i$$\dfrac{26}{7}+\dfrac{23}{7}i$${{i}^{57}}+\frac{1}{{{i}^{25}}}$$0$$2i$$-2… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/review-ex-1-p2?rev=1737476037&do=diff Question 2 & 3, Review Exercise 1 Solutions of Question 2 & 3 of Review Exercise 1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${{i}^{n}}+{{i}^{n+1}}+{{i}^{n+2}}+{{i}^{n+3}}=0$$\forall n\in N$\begin{align}{{i}^{n}}+{{i}^{n+1}}+{{i}^{n+2}}+{{i}^{n+3}}&=0\\ L.H.S.&={{i}^{n}}+{{i}^{n}}\cdot i+{{i}^{n}}\cdot {{i}^{2}}+{{i}^{n}}\cdot {{i}^{3}}\\ &={{i}^{n}}\left( 1+i+{{i}^{2}}… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/review-ex-1-p3?rev=1737476037&do=diff Question 4 & 5, Review Exercise 1 Solutions of Question 4 & 5 of Review Exercise 1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${{z}_{1}}=2-i$${{z}_{2}}=1+i,$$\left|\dfrac{{{z}_{1}}+{{z}_{2}}+1}{{{z}_{1}}-{{z}_{2}}+1}\right|$$z_1=2-i$$z_2=1+i$\begin{align} \dfrac{{{z}_{1}}+{{z}_{2}}+1}{{{z}_{1}}-{{z}_{2}}+1}&=\dfrac{\left( 2-i \right)+\left( 1+i \right)+1}{\left( 2-i \rig… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/review-ex-1-p4?rev=1737476037&do=diff Question 6, 7 & 8, Review Exercise 1 Solutions of Question 6, 7 & 8 of Review Exercise 1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{1}{3+4i}$$$z=\dfrac{1}{3+4i}.$$\begin{align}z&=\dfrac{1}{3+4i}\times \dfrac{3-4i}{3-4i}\\ &=\dfrac{3-4i}{9+16}\\ &=\dfrac{3-4i}{25}\\ &=\dfrac{3}{25}-\dfrac{4}{25}i\end{align}$$\bar{z}=\dfrac{3}{25}+\dfrac{4}{25}i.$$$\dfrac{3i+2}{3-2… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p1?rev=1737476037&do=diff Question 1, Exercise 2.1 Solutions of Question 1 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) $$\left[ \begin{matrix} 1 & 2 & 4 \\ \end{matrix} \right] \left[ \begin{matrix} 1 & 0 & 2 \\ 2 & 0 & 1 \\ 0 & 1 & 2 \\ \end{matrix} \right] \left[ \begin{matrix} 2 \\ 4 \\ 6 \\ \end{matrix} \right]$$\begin{align}&\left[ \begin{matri… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p2?rev=1737476037&do=diff Question 2, Exercise 2.1 Solutions of Question 2 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2 $A=\begin{bmatrix}2 & -5 & 1\\ 3 & 0 & -4\end{bmatrix}$$B=\begin{bmatrix}1 & -2 & -3 \\ 0 & -1 & 5\end{bmatrix}$$C=\begin{bmatrix}0 & 1 & -2\\0 & -1 & -1\end{bmatrix}$$2A+3B-4C.$$A=\begin{bmatrix}2 & -5 & 1\\ 3 & 0 & -4\end{bmatrix}$$B=\begin… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p3?rev=1737476037&do=diff Question 3, Exercise 2.1 Solutions of Question 3 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3(i) $A=\begin{bmatrix}x & y & z\end{bmatrix}$$B=\begin{bmatrix}a & h & g\\h & b & f\\g & f & c\end{bmatrix}$$C=\begin{bmatrix}x\\y\\z\end{bmatrix}$$\left( AB \right)C=A\left( BC \right)$$A=\begin{bmatrix}x & y & z\end{bmatrix}$$B=\begin{bmatri… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p4?rev=1737476037&do=diff Question 4, Exercise 2.1 Solutions of Question 4 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4 $A= \begin{bmatrix}1 & 4 & 4 \\ 4 & 1 & 4 \\ 4 & 4 & 1 \end{bmatrix}$$\dfrac{1}{3}A^2-2A-9I=0$$A=\begin{bmatrix} 1 & 4 & 4 \\ 4 & 1 & 4 \\ 4 & 4 & 1 \end{bmatrix}$\begin{align}\frac{1}{3}A^2&=\frac{1}{3}\left[ \begin{matrix} 1 & 4 & 4 … text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p5?rev=1737476037&do=diff Question 5 & 6, Exercise 2.1 Solutions of Question 5 & 6 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$A= \begin{bmatrix} 0 & 2b & -2 \\ 3 & 1 & 3 \\ 3a & 3 & -1 \end{bmatrix}$$a$$b$$A=\begin{bmatrix} 0 & 2b & -2 \\ 3 & 1 & 3 \\ 3a & 3 & -1 \end{bmatrix}$$$A^t=\left[ \begin{matrix} 0 & 3 & 3a \\ 2b & 1 & 3 \\ -2 & 3 & -1 \\ \end{ma… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p6?rev=1737476037&do=diff Question 7, Exercise 2.1 Solutions of Question 7 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7 $ A=\begin{bmatrix}1 & 0 & -1 & 2 \\3 & 1 & 2 & \quad 5 \\0 & -2 & 1 & 6\end{bmatrix}$$ B=\begin{bmatrix} 2 & -1 & 3 & 1 \\1 & 3 & -1 & 4 \\3 & 1 & 2 & -1 \end{bmatrix}$$( A+B )^t=A^t+B^t$$A=\left[ \begin{matrix}1 & 0 & -1 & 2 \\3 & 1 &… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p7?rev=1737476037&do=diff Question 8, Exercise 2.1 Solutions of Question 8 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 8(i) $A=\begin{bmatrix}1 & 2 & 0 \\3 & -1 & 4 \end{bmatrix}$$( A^t )^t=A$$$A=\left[ \begin{matrix} 1 & 2 & 0 \\ 3 & -1 & 4 \\ \end{matrix} \right]$$$$A^t=\left[ \begin{matrix} 1 & 3 \\ 2 & -1 \\ 0 & 4 \\ \end{matrix} \rig… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p8?rev=1737476037&do=diff Question 9, Exercise 2.1 Solutions of Question 9 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9(i) $A=\begin{bmatrix}2 & -1 & 3 \\1 & \quad 0 & 1 \end{bmatrix},$$B=\begin{bmatrix}1 & 2 \\2 & 2 \\ 3 & 0 \end{bmatrix}$$( AB )^t=B^tA^t$$$A=\left[ \begin{matrix} 2 & -1 & 3 \\ 1 & \quad 0 & 1 \\ \end{matrix} \right],$$$$B=\left[… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p9?rev=1737476037&do=diff Question 10, Exercise 2.1 Solutions of Question 10 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 10 $A=\begin{bmatrix}1 & -3 & 4 \\-3 & 2 & -5 \\4 & -5 & 0 \end{bmatrix}$$B=\begin{bmatrix}5 & 6 & 7 \\6 & -8 & 3 \\7 & 3 & 1 \end{bmatrix}$$A$$B$$A+B$$$A=\left[ \begin{matrix} 1 & -3 & 4 \\ -3 & 2 & -5 \\ 4 & -5 & 0 \\ \end{ma… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p10?rev=1737476037&do=diff Question 11, Exercise 2.1 Solutions of Question 11 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 11 $A=\begin{bmatrix}0 & 1 & -2 \\-1 & 0 & 3 \\2 & -3 & 0 \end{bmatrix}$$B=\begin{bmatrix}0 & -6 & 11 \\6 & 0 & -7 \\-11 & 7 & 0 \end{bmatrix}$$A+B$$$A=\left[ \begin{matrix} 0 & 1 & -2 \\ -1 & 0 & 3 \\ 2 & -3 & 0 \\ \end{matri… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p11?rev=1737476037&do=diff Question 12, Exercise 2.1 Solutions of Question 12 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 12(i) $A=\begin{bmatrix}3 & 2 & 1 \\4 & 5 & 6 \\-2 & 3 & 4\end{bmatrix}$$A+A^t$$$A=\left[ \begin{matrix} 3 & 2 & 1 \\ 4 & 5 & 6 \\ -2 & 3 & 4 \\ \end{matrix} \right]$$$$A^t=\left[ \begin{matrix} 3 & 4 & -2 \\ 2 & 5 & 3 \… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p12?rev=1737476037&do=diff Question 13, Exercise 2.1 Solutions of Question 13 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 13(i) $A$$3$$A+A^t$$$A=\left[ \begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{matrix} \right]$$$$A^t=\left[ \begin{matrix} a_{11} & a_{21} & a_{31} \\ a_{12} & a_{22} … text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p1?rev=1737476037&do=diff Question 1, Exercise 2.2 Solutions of Question 1 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1 $A=\begin{bmatrix}1 & 3 & 1 \\-1 & 2 & 0 \\2 & 0 & -2 \end{bmatrix}$$A_{11},A_{21},A_{23},A_{31},A_{32},A_{33}.$$|A|.$$$A=\left[ \begin{matrix} 1 & 3 & 1 \\ -1 & 2 & 0 \\ 2 & 0 & -2 \\ \end{matrix} \right]$$$${{A}_{11}}={{\left(… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p2?rev=1737476037&do=diff Question 2, Exercise 2.2 Solutions of Question 2 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2(i) $$\left| \begin{matrix} 1 & 2 & 0 \\ 3 & 1 & 0 \\ -1 & 2 & 0 \\ \end{matrix} \right|=0$$$\left| \begin{matrix}1 & 2 & 3 \\-8 & 4 & -12 \\2 & -1 & 3 \end{matrix} \right|=0$$$\left| \begin{matrix} 1 & 2 & 3 \\ -8 & 4 & -… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p3?rev=1737476037&do=diff Question 3, Exercise 2.2 Solutions of Question 3 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 $A$$3,$$|A^t|=|A|$$$A=\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{bmatrix}$$\begin{align}|A|&=a_{11} \left( a_{22} a_{33}-a_{23} a_{32} \right)-a_{12}\left( a_{21}a_{33}… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p4?rev=1737476037&do=diff Question 4, Exercise 2.2 Solutions of Question 4 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4(i) $\left| \begin{matrix}0 & 1 & 3 \\-1 & 2 & 1 \\2 & 1 & 1 \end{matrix} \right|.$\begin{align}&\left| \begin{matrix} 0 & 1 & 3 \\ -1 & 2 & 1 \\ 2 & 1 & 1 \\ \end{matrix} \right| \\ =&0\left( 2-1 \right)-1\left( -1-2 \right)+3\l… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p5?rev=1737476037&do=diff Question 5, Exercise 2.2 Solutions of Question 5 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) $\begin{vmatrix}a & b & c\\l & m & n\\x & y & z \end{vmatrix}=\begin{vmatrix}a & l & x\\b & m & y\\c & n & z \end{vmatrix}$\begin{align}L.H.S.&=\begin{vmatrix} a & b & c \\ l & m & n \\ x & y & z \end{vmatrix}\\ &=\begin{vmatrix} a & b &… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p6?rev=1737476037&do=diff Question 6, Exercise 2.2 Solutions of Question 6 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Questiopn 6(i) $\left| \begin{matrix}a-b & b-c & c-a \\b-c & c-a & a-b \\c-a & a-b & b-c \end{matrix} \right|=0$\begin{align} L.H.S&=\left| \begin{matrix} a-b & b-c & c-a \\ b-c & c-a & a-b \\ c-a & a-b & b-c \\ \end{matrix} \right| \\ &=\left| \… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p7?rev=1737476037&do=diff Question 7, Exercise 2.2 Solutions of Question 7 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7(i) $\left| \begin{matrix}3860 & 3861 \\3862 & 3863 \end{matrix} \right|$$$\left| \begin{matrix} 3860 & 3861 \\ 3862 & 3863 \\ \end{matrix} \right|=14911180-14911182$$$$=-2$$$\left| \begin{matrix}81 & 82 & 83 \\84 & 85 & 86 \\87 & 8… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p8?rev=1737476037&do=diff Question 8,9 & 10, Exercise 2.2 Solutions of Questions 8,9 & 10 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\left| \begin{matrix}1+x & y & z \\x & 1+y & z \\x & y & 1+z \end{matrix} \right|=1+x+y+z$$$L.H.S.=\left| \begin{matrix} 1+x & y & z \\ x & 1+y & z \\ x & y & 1+z \\ \end{matrix} \right|$$$$=\left| \begin{matrix} 1 & 0 & -… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p9?rev=1737476037&do=diff Question 11, Exercise 2.2 Solutions of Question 11 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 11(i) $\left[ \begin{matrix}7 & 1 & 3 \\6 & 2 & -2 \\5 & 1 & 1\end{matrix} \right]$$$A=\left[ \begin{matrix} 7 & 1 & 3 \\ 6 & 2 & -2 \\ 5 & 1 & 1 \\ \end{matrix} \right]$$$$|A|=7(2+2)-1(6+10)+3(6-10)$$$$=28-16-12$$$$|A|=0$$$A$$\… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p10?rev=1737476037&do=diff Question 12, Exercise 2.2 Solutions of Question 12 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 12 $\lambda $$A$$A=\begin{bmatrix}-\lambda & 1 & 0 \\1 & -\lambda & 1 \\0 & 1 & -\lambda \end{bmatrix}$$$A=\left[ \begin{matrix} -\lambda & 1 & 0 \\ 1 & -\lambda & 1 \\ 0 & 1 & -\lambda \\ \end{matrix} \right]$$$$|A|=-\lamb… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p11?rev=1737476037&do=diff Question 13, Exercise 2.2 Solutions of Question 13 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 13(i) $x,$$\left| \begin{matrix}x & 2 & 3 \\0 & -1 & 1 \\0 & 4 & 5 \end{matrix} \right|=9$$$\left| \begin{matrix} x & 2 & 3 \\ 0 & -1 & 1 \\ 0 & 4 & 5 \\ \end{matrix} \right|=9$$$$x(-5-4)-2(0)+3(0)=9$$$$-9x=9$$$$x=-1$$$x,$$\left… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p12?rev=1737476037&do=diff Question 14 & 15, Exercise 2.2 Solutions of Questions 14 & 15 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$A=\begin{bmatrix}0 & 2 & 2 \\-1 & 3 & 2 \\1 & 0 & 5\end{bmatrix}$$A^{-1}$$$A=\left[ \begin{matrix} 0 & 2 & 2 \\ -1 & 3 & 2 \\ 1 & 0 & 5 \\ \end{matrix} \right]$$$A^{-1}$$$A^{-1}=\dfrac{Adj\,\,A}{|A|}$$$$Adj\,\,A={{\left[ \begin… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p13?rev=1737476037&do=diff Question 16 & 17, Exercise 2.2 Solutions of Questions 16 & 17 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$A=\begin{bmatrix}3 & -1 \\4 & 2\end{bmatrix}$$|A^{-1}|=\dfrac{1}{|A|}$$$A=\left[ \begin{matrix} 3 & -1 \\ 4 & 2 \\ \end{matrix} \right]$$$$|A|=6+4$$$$\Rightarrow |A|=10\ldots (1)$$$$A^{-1}=\dfrac{1}{|A|}AdjA$$$$AdjA=\left[ \begin{ma… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p14?rev=1737476037&do=diff Question 18, Exercise 2.2 Solutions of Question 18 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 18(i) $A$$B$$( A^{-1})^{-1}=A$$A$$2\times 2$$$A=\left[ \begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{matrix} \right]$$$$|A|=a_{11}a_{22}-a_{12}a_{21}$$$$AdjA=\left[ \begin{matrix} a_{22} & -a_{12} \\ -a_{21} & a_{11… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p15?rev=1737476037&do=diff Question 19, Exercise 2.2 Solutions of Question 19 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 19 $A=\begin{bmatrix}2 & 3 \\-1 & 1\end{bmatrix}$$( A^{-1})^t=( A^t)^{-1}$$$A=\left[ \begin{matrix} 2 & 3 \\ -1 & 1 \\ \end{matrix} \right]$$$$A^t=\left[ \begin{matrix} 2 & -1 \\ 3 & 1 \\ \end{matrix} \right]$$$$|A^t|=5$$$$Ad… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-3-p1?rev=1737476037&do=diff Question 1, Exercise 2.3 Solutions of Question 1 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) $\begin{bmatrix}1 & 3 & -1 \\2 & 1 & 4 \\3 & 4 & -5\end{bmatrix}$\begin{align}&\begin{bmatrix} 1 & 3 & -1 \\ 2 & 1 & 4 \\ 3 & 4 & -5 \end{bmatrix}\\ \underset{\sim}{R}&\begin{bmatrix} 1 & 3 & -1 \\ 0 & -5 & 6 \\ 0 & -5 & -2 \end{bmat… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-3-p2?rev=1737476037&do=diff Question 2, Exercise 2.3 Solutions of Question 2 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2(i) $$\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/unit02/ex2-3-p3?rev=1737476037&do=diff Question 3, Exercise 2.3 Solutions of Question 3 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3(i) $$\left[ \begin{matrix} 1 & 0 & -2 \\ 2 & 2 & 1 \\ -1 & 2 & 3 \\ \end{matrix} \right]$$\begin{align}&\begin{bmatrix} 1 & 0 & -2 \\ 2 & 2 & 1 \\ -1 & 2 & 3 \end{bmatrix}\\ \underset{\sim}{R}& \begin{bmatrix} 1 & 0 & -2 \\ 0 &… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-3-p4?rev=1737476037&do=diff Question 4, Exercise 2.3 Solutions of Question 4 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4 $\begin{bmatrix}2 & 3 & 4 & 5 \\3 & 4 & 5 & 6 \\4 & 5 & 6 & 7 \\9 & 10 & 11 & 12\end{bmatrix}$\begin{align}&\begin{bmatrix} 2 & 3 & 4 & 5 \\ 3 & 4 & 5 & 6 \\ 4 & 5 & 6 & 7 \\ 9 & 10 & 11 & 12 \end{bmatrix}\\ \underset{\sim}{R}&\begin{bm… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-2-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/ex3-2-p2?rev=1737476037&do=diff Question 2, Exercise 3.2 Solutions of Question 2 of Exercise 3.2 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2(i) Find unit vector having the same direction as the vector $3\hat{i}.$$$\overset{\scriptscriptstyle\rightharpoonup}{a}=3\hat{i}$$$$|\overset{\scriptscriptstyle\rightharpoonup}{a}|=\sqrt{{{(3)}^{2}}}=3$$$$\hat{a}=\dfrac{{\overset{\scriptscriptstyle\rightharpo… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-2-p3?rev=1737476037&do=diff Question 3 & 4, Exercise 3.2 Solutions of Question 3 & 4 of Exercise 3.2 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 If $\vec{r}=\hat{i}-9\hat{j}$$\vec{a}=\hat{i}+2\hat{j}$$\vec{b}=5\hat{i}-\hat{j}$$p$$q$$\vec{r}=p\vec{a}+q\vec{b}$$$\vec{r}=p\vec{a}+q\vec{b}.$$$\vec{r},\vec{a}$$\vec{b}$$$\hat{i}-9\hat{j}=p(\hat{i}+2\hat{j})+q(5\hat{i}-\hat{j})$$$$\implies \hat{i}-9\… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-2-p4?rev=1737476037&do=diff Question 5 & 6, Exercise 3.2 Solutions of Question 5 & 6 of Exercise 3.2 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5 Find the length of the vector $\overrightarrow{AB}$$\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/ex3-2-p5?rev=1737476037&do=diff Question 7, Exercise 3.2 Solutions of Question 7 of Exercise 3.2 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7(i) Find the components and the magnitude of $\overrightarrow{PQ}$$P(-1,2)$$Q(2,-1)$\begin{align}\overrightarrow{PQ}&=\overrightarrow{OQ}-\overrightarrow{OP}\\ &=(2\hat{i}-\hat{j})-(-\hat{i}+2\hat{j})\\ &=3\hat{i}-3\hat{j}\end{align}\begin{align}|\overrighta… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-2-p6?rev=1737476037&do=diff Question 7, Exercise 3.2 Solutions of Question 7 of Exercise 3.2 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7(i) Find the components and the magnitude of $\overrightarrow{PQ}$$P(-1,2)$$Q(2,-1)$\begin{align}\overrightarrow{PQ}&=\overrightarrow{OQ}-\overrightarrow{OP}\\ &=(2\hat{i}-\hat{j})-(-\hat{i}+2\hat{j})\\ &=3\hat{i}-3\hat{j}\end{align}\begin{align}|\overrighta… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-2-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 $$ unit which is parallel to the vector $\begin{align}2\overrightarrow{a}-\overrightarrow{b}+3\overrightarrow{c}&=2(\hat{i}+\hat{j}+\hat{k})-(4\hat{i}-2\hat{j}+3\h… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-2-p8?rev=1737476037&do=diff Question 11, Exercise 3.2 Solutions of Question 11 of Exercise 3.2 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 11(i) Find the position vectors of the point of division of the line segments joining point $C$$5\hat{j}$$D$$4\hat{i}+\hat{j}$$2:5$$C$$\overrightarrow{OC}=5\hat{j}$$D$$\overrightarrow{OD}=4\hat{i}+\hat{j}$$H$$\overline{CD}$$2:5$$H$\begin{align}\overrightarrow… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-2-p9?rev=1737476037&do=diff Question 12, 13 & 14, Exercise 3.2 Solutions of Question 12, 13 & 14 of Exercise 3.2 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 12 $\alpha ,$$|\alpha \hat{i}+(\alpha +1)\hat{j}+2\hat{k}|=3$\begin{align}|\alpha \hat{i}+(\alpha +1)\hat{j}+2\hat{k}|&=3.\end{align}\begin{align}\sqrt{(\alpha )^2+(\alpha +1)^2+(2)^2}&=3.\end{align}\begin{align}&{\alpha ^2+(\alpha +1)^2}+4=9… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-3-p1?rev=1737476037&do=diff Question 1, Exercise 3.3 Solutions of Question 1 of Exercise 3.3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) If $\vec{a}=3 \hat{i}+4 \hat{j}-\hat{k}$, $\vec{b}=\hat{i}-\hat{j}+3 \hat{k}$ and $\vec{c}=2\hat{i}+\hat{j}-5 \hat{k}$$\vec{a}\cdot \vec{b}$\begin{align}\vec{a} \cdot \vec{b}&=(3 \hat{i}+4 \hat{j}-\hat{k}) \cdot(\hat{i}-\hat{j}+3 \hat{k})\\ \Rightarrow &=(… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-3-p2?rev=1737476037&do=diff Question 2 and 3 Exercise 3.3 Solutions of Question 2 and 3 of Exercise 3.3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2 $$\vec{a}=2 \hat{i} + 2 \hat{j}-5 \hat{k}, \quad \vec{b}=2 \hat{i}+\hat{j}-7 \hat{k}$$\begin{align}\vec{a}+\vec{b}&=(2 \hat{i}+2 \hat{j}-5 \hat{k})+(2 \hat{i}+\hat{j}-7 \hat{k}) \\ \Rightarrow &=4 \hat{i}+3 \hat{j}-12 \hat{k}\\ \Rightarrow|\vec{a}+\… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-3-p3?rev=1737476037&do=diff Question 4 and 5 Exercise 3.3 Solutions of Question 4 and 5 of Exercise 3.3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4 $\hat{i}+7 \hat{j} + 3 \hat{k}$$\hat{i}-\hat{j}+2 \hat{k}$$2 \hat{i}-$$\hat{j}+3 \hat{k}$$\vec{a}=\hat{i}+7 \hat{j}+3 \hat{k}$$\vec{b}=\hat{i}-\hat{j}+2 \hat{k}$$\vec{c} = 2 \hat{i}-\hat{j}-3 \hat{k}$\begin{align}\vec{a} \cdot \vec{b}&=(\hat{i}+7 \h… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-3-p4?rev=1737476037&do=diff Question 6 Exercise 3.3 Solutions of Question 6 of Exercise 3.3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6(i) Let $\vec{a}=\hat{i}+3 \hat{j}-4 \hat{k}$ and $\vec{b}=2 \hat{i}-3 \hat{j}-5 \hat{k}$$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-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-p6?rev=1737476037&do=diff Question 9 & 10, Exercise 3.3 Solutions of Question 9 & 10 of Exercise 3.3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9 $\vec{k}-2 \hat{i}+3 \hat{j}+\hat{k}$$\vec{S}=2 \hat{i}+\hat{j}-\hat{k}$\begin{align}W &=\vec{F} \cdot s \\ \Rightarrow W &=(2 \hat{i}+3 \hat{j}+\hat{k}) \cdot(2 \hat{i}+\hat{j}-\hat{k}) \\ \Rightarrow W &=2(2) \div 3(1)+1(-1) \\ \Rightarrow W &=4+3 … text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-3-p7?rev=1737476037&do=diff Question 11, Exercise 3.3 Solutions of Question 11 of Exercise 3.3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 11 (i) Show that the vectors $3 \hat{i}-2 \hat{j}+$$\hat{k} . \quad \hat{i}-3 \hat{j}-5 \hat{k}$$2 \hat{i}+\hat{j}-4 \hat{k}$$\vec{a}=3 \hat{i}-2 \hat{j}+\hat{k}$$\vec{b}=\hat{i}-3 \hat{j}+5 \hat{k}$$\vec{c}=2 \hat{i}+\hat{j}-4 \hat{k}$\begin{align}|\vec{a}|&… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-3-p8?rev=1737476037&do=diff Question 12 & 13, Exercise 3.3 Solutions of Question 12 & 13 of Exercise 3.3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 12 $\overrightarrow{B A} \cdot \overrightarrow{A C}=0$$|\vec{a}|=\vec{b}|=| \vec{c} \mid=$$\vec{b}=-\vec{c}$$\triangle A B O$\begin{align}\overrightarrow{O B}+\overrightarrow{A B}&=\overrightarrow{O A}\\ \Rightarrow \overrightarrow{B A}&=\overrightar… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-4-p1?rev=1737476037&do=diff Question 1 Exercise 3.4 Solutions of Question 1 of Exercise 3.4 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) Find the cross product $\hat{j} \times(2 \hat{j}+3 \hat{k})$\begin{align}\vec{a}=\hat{j}&=0 \hat{i}+\hat{j}+0 \hat{k}\\ \vec{b}&=0 \hat{i}+2 \hat{j}-3 \hat{k}\\ \vec{a} \times \vec{b}&=\hat{j} \times(2 \hat{j}+3 \hat{k})\\ &=\left|\begin{array}{lll}\hat{i}… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-4-p2?rev=1737476037&do=diff Question 2 Exercise 3.4 Solutions of Question 2 of Exercise 3.4 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2(i) Show in two different ways that the vectors $\vec{a}$$\vec{b}$$\vec{a}=-\hat{i}+2 \hat{j}-3 \hat{k}, \quad \vec{b}=2 \hat{i}-4 \hat{j}+$$6 \hat{k}$\begin{align}\vec{a} \times \vec{b}&=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ -1 & 2 & -3 \\ 2 … text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-4-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}$$\vec{b}$\begin{align}\hat{n}&=\dfrac{\vec{a} \times \vec{b}}{\mid \vec{a} \times \vec{b}} \\ \text { … text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-4-p4?rev=1737476037&do=diff Question 4 Exercise 3.4 Solutions of Question 4 of Exercise 3.4 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4(i) If $\vec{a}=3 \hat{i}-6 \hat{j}+5 \hat{k},\quad\vec{b}=2\hat{i}-\hat{j}+4 \hat{k} \quad$ and $\quad \vec{c}=\hat{i}+\hat{j} \quad \hat{k},\quad$$\vec{a} \times \vec{b}$\begin{align}\vec{a} \times \vec{b}&=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k}… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-4-p5?rev=1737476037&do=diff Question 5 Exercise 3.4 Solutions of Question 5 of Exercise 3.4 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) Use the vector product to compute the area of the triangle with the given vertices $P(-2,-3), \quad Q(3,2)\quad$$\quad R(-1,-8)$$P Q$$\bar{P} R$\begin{align}\text{Area of triangle}&=\dfrac{1}{2}|\overrightarrow{P Q} \times \overrightarrow{P R}| \\ \text { S… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-4-p6?rev=1737476037&do=diff Question 6 Exercise 3.4 Solutions of Question 6 of Exercise 3.4 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6(i) A force $\vec{F}=3 \hat{i}-2 \hat{j}+5 \hat{k}$$(1,-2,2)$$\vec{r}$$P(1,-2.2)$$O(0,0,0)$\begin{align}\vec{r}&=\overrightarrow{O P}\\ &=(1,-2,2)-(0,0,0) \\ \Rightarrow \vec{r}&=(1,-2,2).\\ \text { Hence } \vec{M}-\vec{r} \times \vec{F}&=\left|\begin{array}{cc… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-4-p7?rev=1737476037&do=diff Question 7 & 8 Exercise 3.4 Solutions of Question 7 & 8 of Exercise 3.4 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7 If $\vec{A}+\vec{B}+\vec{C}=\vec{O}$$$\vec{A} \times \vec{B}=\vec{B} \times \vec{C}=\vec{C} \times \vec{A}.$$$$\vec{A}+\vec{B}+\vec{C}=\vec{O} \text {. }$$$\vec{A}$$$\vec{A} \times(\vec{A}+\vec{B}+\vec{C})=0$$\begin{align}\Rightarrow \vec{A} \times \ve… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-4-p8?rev=1737476038&do=diff Question 9 Exercise 3.4 Solutions of Question 9 of Exercise 3.4 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9(i) Find the area of parallelogram whose diagonals are $\vec{a}=4 \hat{i}+\hat{j}-2 \hat{k}\quad$$\quad\vec{b}=-2 \hat{i}+3 \hat{j}+4 \hat{k}$$\vec{c}$$\vec{d}$$E$$E$\begin{align}\overrightarrow{A E}&=\overrightarrow{E C}\\ &=\dfrac{1}{2} \vec{a}\\ &=2 \hat{i}+… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-5-p1?rev=1737476038&do=diff Question 1 & 2 Exercise 3.5 Solutions of Question 1 & 2 of Exercise 3.5 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1 Find $\vec{a} \cdot \vec{b} \times \vec{c}$$\vec{a}=2 \hat{i}+\hat{j}+3 \hat{k}$$\vec{b}=-\hat{i}+2 \hat{j}+\hat{k} \quad \text { and }\quad \vec{c}=3 \hat{i}+\hat{j}+2 \hat{k} \text {. }$\begin{align}V&=\vec{a} \cdot \vec{b} \times \vec{c}\\ &=\left|\… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-5-p2?rev=1737476038&do=diff Question 3 & 4 Exercise 3.5 Solutions of Question 3 & 4 of Exercise 3.5 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 For the vectors $\vec{a}=3 \hat{i}+2 \hat{k}$$\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}$$\vec{a} \cdot \vec{b}… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-5-p3?rev=1737476038&do=diff Question 5(i) & 5(ii) Exercise 3.5 Solutions of Question 5(i) & 5(ii) of Exercise 3.5 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) $\vec{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}\quad$$\quad\vec{b}=b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k}\quad$$\vec{a} \times \vec{b}\quad$$\vec{a} \times \vec{b}$$\vec{a}$$\vec{b}$$\vec{a} \times \vec{b}$$\vec{a}$$\vec{b}$$\vec{a} \times \v… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-5-p4?rev=1737476038&do=diff Question 5(iii) & 5(iv) Exercise 3.5 Solutions of Question 5(iii) & 5(iv) of Exercise 3.5 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\vec{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}\quad$$\quad\vec{b}=b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k}\quad$$(\vec{a}. \vec{b})^2,\quad|a|^2,\quad|b|^2$\begin{align}\vec{a} \cdot \vec{b}&=(a_1 \hat{i}+a_2 \hat{j} + a_3 \hat{k}) \cdot(b_1 \hat{i}+b_2 \… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-5-p5?rev=1737476038&do=diff Question 6 Exercise 3.5 Solutions of Question 6 of Exercise 3.5 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6 Do the points $(4. 2.1)$$(5,1,6)$$(2.2,-5)$$(3.5 .0)$$A(4,-2,1), B(5,1,6)$$C(2,2,-5)$$D(3,5.0)$$A, \overrightarrow{O A}=4 \hat{i}-2 \hat{j}+\hat{k}$$B, \overrightarrow{O B}=5 \hat{i}+\hat{j}+6 \hat{k}$$C, \overrightarrow{O C}=2 \hat{i}+2 \hat{i}-5 \hat{k}$$D, … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-5-p6?rev=1737476038&do=diff Question 7 Exercise 3.5 Solutions of Question 7 of Exercise 3.5 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7(i) For what value of $c$$\vec{u}=\hat{i}+2 \hat{j}+3 \hat{k}$$\vec{v}=2 \hat{i}-3 \hat{j}+4 \hat{k} \cdot \vec{w}=3 \hat{i}+\hat{j}+c \hat{k}$\begin{align}\vec{u} \cdot \vec{v} \times \vec{w}&=0\\ \vec{u} \cdot \vec{v} \times \vec{w}&=0\\ \Rightarrow\left|\beg… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-5-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/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-p1?rev=1737476038&do=diff Question 1 Review Exercise 3 Solutions of Question 1 of Review Exercise 3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1 $\hat{i} \cdot(\hat{j} \times \hat{k})+\hat{j} \cdot(\hat{i} \times \hat{k})+\hat{k} \cdot(\hat{i} \times \hat{j})$$0$$1$$1$$3$$3 \hat{i}+5 \hat{j}+2 \hat{k}$$2 \hat{i}-3 \hat{j}-5 \hat{k}$$5 \hat{i}+2 \hat{j}-3 \hat{k}$$\hat{i}-2 \hat{i}+\hat{j}+3 \… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/review-ex3-p2?rev=1737476038&do=diff Question 2 & 3 Review Exercise 3 Solutions of Question 2 & 3 of Review Exercise 3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2 $\lambda$$\mu$$$(\hat{i}+3 \hat{j}+9 \hat{k}) \times(3 \hat{i}-\lambda \hat{j}+\mu \hat{k})=\overrightarrow{0} \text {. }$$\begin{align}(\hat{i}+3 \hat{j}+9 \hat{k}) \times(3 \hat{i}-\lambda \hat{j}+\mu \hat{k})&=\vec{O} \\ \Rightarrow\left|\b… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/review-ex3-p3?rev=1737476038&do=diff Question 4 & 5 Review Exercise 3 Solutions of Question 4 & 5 of Review Exercise 3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4 $\vec{r}=x \hat{i}+y \hat{j}+z \hat{k}$$(\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-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}$$\vec{c}=\lambda \hat{j}+3 \hat{k}$\begin{align}\vec{a} \cdot \vec{b} \times \vec{c}&=0 \\ \Rightarrow\left|\begin{array}{ccc} 1 & 3 & 1 \\ 2 & -1 & -1 \\ 0 & \lamb… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/review-ex3-p5?rev=1737476038&do=diff Question 8 & 9 Review Exercise 3 Solutions of Question 8 & 9 of Review Exercise 3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 8 $(0,0,2),(-1,3,2),(1,0,4)$$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)$$$ \text{Area of triangle} =\dfr… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/review-ex3-p6?rev=1737476038&do=diff Question 10 Review Exercise 3 Solutions of Question 10 of Review Exercise 3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 10(i) $A B C$$|\vec{a}|^2=|\vec{b}|^2+|\vec{c}|^2 -2|\vec{b}|| \vec{c}| \cos A$$A B C$$\vec{a}, \vec{b}$$\vec{c}$\begin{align} \vec{b}&=\vec{a}+\vec{c} \\ \Rightarrow \vec{a}&=\vec{b}-\vec{c} \\ \Rightarrow \vec{a} \cdot \vec{a}&=(\vec{b}-\vec{c}) \cd… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-1-p1?rev=1737476038&do=diff Question 1 and 2 Exercise 4.1 Solutions of Question 1 and 2 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$2,4,6,8, \ldots ,50$$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$$$a_2=\dfrac{2(2… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-1-p2?rev=1737476038&do=diff Question 3 and 4 Exercise 4.1 Solutions of Question 3 and 4 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{1}{2}, \dfrac{2}{3} \dfrac{3}{4}, \dfrac{4}{5}, \ldots$$$\dfrac{1}{1+1}, \dfrac{2}{2+1}, \dfrac{3}{3+1}, \dfrac{4}{4+1},...$$$\dfrac{n}{n+1}$$2,-4,6,-8,10, \ldots$\begin{align} &(-1)^2 \cdot 2 \cdot 1, (-1)^3 \cdot 2 \cdot 2, (-1)^4 \cdot 2 \… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-1-p3?rev=1737476038&do=diff Question 5 Exercise 4.1 Solutions of Question 5 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) $\sum_{j=1}^6(2 j-3)$\begin{align}\sum_{j=1}^6(2 j-3)&=(2.1-3)+(2.2-3)+(2.3-3)+(2.4-3)\\&+(2.5-3)+(2.6-3) \\ \implies \sum_{j=1}^6(2 j-3)&=-1+1+3+5+7+9 .\end{align}$\sum_{k=1}^5(-1)^k 2^{k-1}$\begin{align}\sum_{k=1}^5(-1)^k 2^{k-1}& =(-1)^1 2^{1-… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-1-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.$$$r=0$\begin{align}&P_{0+1… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p1?rev=1737476038&do=diff Question 1 and 2 Exercise 4.2 Solutions of Question 1 and 2 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$15$$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-p3?rev=1737476038&do=diff Question 5 and 6 Exercise 4.2 Solutions of Question 5 and 6 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$\log a, \log (a b), \log \left(a b^2\right), \log \left(a b^3\right), \ldots$$$n$$\log$$a$$b$$b$$$a_n=\log (a b^{n-1}).$$\begin{align}a_n&=\log(a b^{n-1}). \end{align}\begin{align} d&=a_{n+1}-a_n \\ &=\log (a b^n)-\log (a b^{n-1}) \\ &=\log \left(\… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p4?rev=1737476038&do=diff Question 7 Exercise 4.2 Solutions of Question 7 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7 $a_6+a_4=6$$a_6-a_4=\dfrac{2}{3}$$a_1$$d$\begin{align} &a_6+a_4=6 \\ \implies & a_1+5d+a_1+3d=6\\ \implies & 2a_1+8d=6\\ \implies & a_1+4d=3 --- (1) \end{align}\begin{align} &a_6-a_4=\dfrac{2}{3} \\ \implies & a_1+5d-a_1-3d=\dfrac{2}{3}\\ \implies &… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p5?rev=1737476038&do=diff Question 8 Exercise 4.2 Solutions of Question 8 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 8 $\dfrac{b+c-a}{a}, \dfrac{c+a-b}{b}, \dfrac{a+b-c}{c}$$\dfrac{1}{a}, \dfrac{1}{b}, \dfrac{1}{c}$$\dfrac{b+c-a}{a}, \dfrac{c+a-b}{b}, \dfrac{a+b-c}{c}$\begin{align}\therefore \dfrac{c+a-b}{b}-\dfrac{b+c-a}{a}&=\dfrac{a+b-c}{c}-\dfrac{c+a-b}{b} \\ \te… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p6?rev=1737476038&do=diff Question 9 Exercise 4.2 Solutions of Question 9 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9 $24m$$21m$$18m$$$a_1=24,$$$$a_2=21,$$$$a_3=18.$$$$d=21-24=18-21=-3,$$\begin{align} a_8&=a_1+7d\\ &=24+7(-3)=3. \end{align}$3m$ text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p7?rev=1737476038&do=diff Question 10 Exercise 4.2 Solutions of Question 10 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 10 $500$$a_1$$$a_1=20135.$$$d=-500$$a_{11}$\begin{align} a_{11}&=a_1+10d \\ &=20135+10(-500)\\ &=15135. \end{align}$1070$$15135$ text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p8?rev=1737476038&do=diff Question 11 Exercise 4.2 Solutions of Question 11 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 11 $a_1$$$a_1=1000.$$$= d=100$$a_n=5400$$n$\begin{align} &a_n=a_1+(n-1)d \\ \implies &5400=1000+(n-1)100\\ \implies &5400=900+100n \\ \implies &100n=5400-900\\ \implies &100n=4500\\ \implies &n=45.\end{align} text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p9?rev=1737476038&do=diff Question 12 & 13 Exercise 4.2 Solutions of Question 12 & 13 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$a_1$$$a_1=3500.$$$=d=750$$a_{21}$\begin{align} a_{21}&=a_1+20d\\ &=3500+20(750) \\ &=18500. \end{align}$12$$18$$a=12, b=18$$A$\begin{align}A&=\dfrac{a+b}{2}\\&=\dfrac{12+18}{2}\\&=\dfrac{30}{2}=15.\end{align}$\dfrac{1}{3}$$\dfrac{1}{4}$$a=\dfrac{1}{… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p10?rev=1737476038&do=diff Question 14 Exercise 4.2 Solutions of Question 14 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 14(i) $A_1, A_2, A_3$$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/ex4-2-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/ex4-2-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.$$\begin{align}&a_7=a+6d\\ \implies &8=5+6d\\ \implies &6d=8-5\\ \implies &d=\dfrac{3}{6}=\dfrac{1}{2}. \end{align}\begin{align} A_1&=a+d=5+\dfra… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p13?rev=1737476038&do=diff Question 17 Exercise 4.2 Solutions of Question 17 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 17 $n$$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/ex4-3-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$$n$\begin{align} S_n&=\dfrac{n}{2}[a_1+a_n], \\ \implies S_{20}&=\dfrac{20}{2}[9-29] … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-3-p2?rev=1737476038&do=diff Question 2 Exercise 4.3 Solutions of Question 2 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2(i) $a_1, a_n, n, d$$S_n$$a_1=2, n=17, d=3$$a_1=2, n=17, d=3$$a_{17}$$S_{17}$$$a_{n}=a_1+(n-1)d.$$$$a_{17}=2+(17-1)(3)=50.$$$$S_n=\dfrac{n}{2}[a_1+a_n]$$\begin{align}S_{17}&=\dfrac{17}{2}(a_1+a_17) \\ &=\dfrac{17}{2}(2+50)=442.\end{align}$a_{17}=50$$… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-3-p3?rev=1737476038&do=diff Question 3 & 4 Exercise 4.3 Solutions of Question 3 & 4 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 $5$$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-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$$120$\begin{align}(a-3 d)^2+(a-d)^2+(a+d)^2+(a+2 d)^2&=120 \\ \Rightarrow a^2-6 a d+… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-3-p5?rev=1737476038&do=diff Question 7 & 8 Exercise 4.3 Solutions of Question 7 & 8 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7 $1+3-5+7+9-11+13+15-$$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/ex4-3-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/ex4-3-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-p1?rev=1737476038&do=diff Question 1 Exercise 4.4 Solutions of Question 1 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question(i) $a_1=5, \quad r=3$$a_1, a_1 r, a_1 r^2, a_1 r^3, a_1 r^4, \ldots$$a_1=5 ; r=3$\begin{align}&5,5.3,5.3^2, 5.3^3, 5.3^4, \ldots\\ \Rightarrow &5,15,45,135,405, \ldots\end{align}$a_1=8, \quad r=-\dfrac{1}{2}$$a_1, a_1 r, a_1 r^2, a_1 r^3, a_1 r^4, \ld… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-4-p2?rev=1737476038&do=diff Question 2 & 3 Exercise 4.4 Solutions of Question 2 & 3 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2 $27$$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}$$a_1(9)=27 \quad \te… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-4-p3?rev=1737476038&do=diff Question 4 & 5 Exercise 4.4 Solutions of Question 4 & 5 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4 $\dfrac{1}{64}$$r=\dfrac{1}{2}$$a_1=16$$a_n=\dfrac{1}{64}$$r=\dfrac{1}{2}$$n$$$a_n=a_1 r^{n-1} \quad \text{then}$$\begin{align}\dfrac{1}{64}&=16(\dfrac{1}{2})^{n-1} \\ \Rightarrow(\dfrac{1}{2})^{n-1}&=\dfrac{1}{64 \times 16}=\dfrac{1}{1024} … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-4-p4?rev=1737476038&do=diff Question 6 & 7 Exercise 4.4 Solutions of Question 6 & 7 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6 $a_{10}=l, a_{13}=m$$a_{16}=n;\quad$$\ln =m^2$$a_n=a_1 r^{n-1}$\begin{align}a_{10}&=a_1 r^9=l \\ a_{13}&=a_1 r^{12}=m\\ \text{and} \quad a_{16}&=a_1 e^{\mathbf{A 5}}=n\end{align}\begin{align}a_{10} \cdot a_{16}&=\ln =(a_1 r^9)(a_1 r^{15})\\ … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-4-p5?rev=1737476038&do=diff Question 8 Exercise 4.4 Solutions of Question 8 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 8(i) $3.14$$2.71$$a=3.14$$b=2.71$$$G= \pm \sqrt{(3.14)(2.71)}= \pm 2.94$$$$G=2.94 \quad \text{or} \quad -2.94$$$-6$$-216$$a=-6$$b=-216$\begin{align}G&= \pm \sqrt{(-6)(-216)}= \pm \sqrt{1296} \\ \Rightarrow G&= \pm 36\end{align}$$G=36 \quad \text{or} \… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-4-p6?rev=1737476038&do=diff Question 9 Exercise 4.4 Solutions of Question 9 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9(i) $3 \dfrac{5}{9}=\dfrac{32}{9}\quad$$\quad40 \dfrac{1}{2}=\dfrac{81}{2}$$G_1, G_2, G_3, G_4$$G_5$$\dfrac{32}{9}$$\dfrac{81}{2}$$\dfrac{32}{9}, G_1, G_2, G_3, G_4, G_5, \dfrac{81}{2}$$a_7=\dfrac{81}{2}$$a_1=\dfrac{32}{9}$\begin{align}a_1 r^6&=\dfra… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-4-p7?rev=1737476038&do=diff Question 10 Exercise 4.4 Solutions of Question 10 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 10 $48$$18$$a$$b$$1$$48$$$\quad a-b=48....(i)$$$a$$b$$$G=\sqrt{a b}$$$a$$b$$$A=\dfrac{a+b}{2}$$$2$$A \cdot M=G \cdot M+18$$A \cdot M-G \cdot M=18$$$\Rightarrow \dfrac{a+b}{2}-\sqrt{a b}=18$$$$(a+b)-2 \sqrt{a b}=36 \text {. }$$$a=b+48$\begin{align}(b… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-4-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$$n+2$\begin{align}a_{n+2}&=a_1 r^{n i 1}=a r^{n+1}=b \\ \because a_1&=a \\ \Rightarrow \quad r^{n+1}&=\dfrac{b}{a} .… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-4-p9?rev=1737476038&do=diff Question 12 Exercise 4.4 Solutions of Question 12 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 12 $n, . \dfrac{a^{n+1}+b^{n+1}}{a^n+b^n}$$a$$b$$a$$b$$\dfrac{a^{n+1}+b^{n-1}}{a^n+b^n}$$a$$b$\begin{align}\dfrac{a^{n+1}+b^{n+1}}{a^n+b^n}&=\sqrt{a b}\quad \because G \cdot M=\sqrt{a b} \\ \Rightarrow \dfrac{a^{n+1}+b^{n+1}}{a^n+b^n}&=a^{\dfrac{1}{… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/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-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-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}$$a_1=0.8, \… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-5-p5?rev=1737476038&do=diff Question 5 & 6 Exercise 4.5 Solutions of Question 5 & 6 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5 $r$$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}$$S_S$\begin{align}\dfrac{a_1(r^{10}-1)}{r-1}&=244 \dfrac{a_1(r^5-1)}{r-1} \\ \Rightarrow r^{10}-… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-5-p6?rev=1737476038&do=diff Question 7 & 8 Exercise 4.5 Solutions of Question 7 & 8 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7 $\operatorname{sum} S_n$$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/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}$\begin{align}a^{q-r}&=(a_1 r^{p-1})^{q-r} . \\ b^{r-p}&=(a_1 r^{q-1})^{r-p}, \text { and } \\ c^{p-q}&=(a_1 r^… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-5-p9?rev=1737476038&do=diff Question 13 & 14 Exercise 4.5 Solutions of Question 13 & 14 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$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/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$$a_1=1$\begi… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-1-p1?rev=1737476038&do=diff Question 1 Exercise 5.1 Solutions of Question 1 of Exercise 5.1 of Unit 05: Mascellaneous series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) $1^2+3^2+5^2+7^2+\ldots$$n$$1+3+5+\ldots$$n^{\text {th }}$$2 n-1$$n^{t h}$$$T_j=(2 j-1)^2$$\begin{align}& \sum_{j=1}^n T_j=\sum_{j=1}^n(2 j-1)^2 \\ & =\sum_{j=1}^n(4 j^2-4 j+1)\\ & =4 \sum_{j=1}^n j^2-4 \sum_{j=1}^n j+\sum_{j=1}^n 1 \\ & =4 \dfr… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-1-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$$$ \begin{aligned} & \sum_{j=1}^{99} \tau_j=\sum_{j=1}^{99} j^2+\sum_{j=1}^{99} j \\ & =\frac{99… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-1-p3?rev=1737476038&do=diff Question 4 & 5 Exercise 5.1 Solutions of Question 4 & 5 of Exercise 5.1 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4 $2+(2+5)+(2+5+8)+\ldots$$n$\begin{align}& T_j=\dfrac{j}{2}[2(2)+3(j-1)]\\ &=\dfrac{j(3 j+1)}{2} \\ & =\dfrac{1}{2}(3 j^2+j)\end{align}\begin{align}& \sum_{j=1}^n T_i=\dfrac{1}{2}[3 \sum_{j=1}^n j^2+\sum_{j=1}^n j] \\ & =\dfrac{1}{2}[3 \dfra… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-1-p4?rev=1737476038&do=diff Question 6 Exercise 5.1 Solutions of Question 6 of Exercise 5.1 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6 $1.2 \cdot 3+2 \cdot 3.4+3.4 .5+\ldots$$n$$1+2+3+\ldots, \quad 2+3+4+5+\ldots$$3+4+5+6+7+\ldots$$n^{t h}$$j, j+1$$j+2$$n^{t h}$\begin{align} & T_j=j(j+1)(j+2)-j(j^2+3 j+2) \\ & =j^3+3 j^2+2 j\end{align}\begin{align} & \sum_{j=1}^n T_j=\sum_{j=1}^n … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-1-p5?rev=1737476038&do=diff Question 7 & 8 Exercise 5.1 Solutions of Question 7 & 8 of Exercise 5.1 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7 $n$$1.5 .9+2.6 .10+3.7 .11+\ldots$$T_j=j(j+4)(j+8)$\begin{align} & =j(j^2+12 j+32) \\ & =j^3+12 j^2+32 j\end{align}\begin{align} & \sum_{j=1}^n T_j=\sum_{j=1}^n j^3+12 \sum_{j=1}^n j^2+32 \sum_{j=1}^n j \\ & =(\dfrac{n(n+1)}{2})^2+12 \dfrac… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-1-p6?rev=1737476038&do=diff Question 9 Exercise 5.1 Solutions of Question 9 of Exercise 5.1 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9(i) $n$$n$$n$\begin{align} & T_n=n^2(2 n+3)=2 n^3+3 n^2 \\ & \Rightarrow T_j=2 j^3+3 j^2\end{align}\begin{align} & \sum_{j=1}^n T_j=2 \sum_{j=1}^n j^3+3 \sum_{j=1}^n j^2 \\ & =2(\dfrac{n(n+1)}{2})^2+3 \dfrac{n(n+1)(2 n+1)}{6} \\ & =\dfrac{n(n+1)}{2}… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-2-p1?rev=1737476038&do=diff Question 1 Exercise 5.2 Solutions of Question 1 of Exercise 5.2 of Unit 05: Mascellaneous series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) $n$$1.2+2.2^2+3.2^3+4.2^4+\ldots$\begin{align} & S_n=1.2+2.2^2+3 \cdot 2^3+4 \cdot 2^4+\ldots +n \cdot 2^n....(i) \\ & 2 S_n=1.2^2+2.2^3+3.2^4+4.2^5+\ldots +n \cdot 2^n....(ii)\end{align}\begin{align} (1-2) S_n&=1 \cdot 2+(2-1) 2^2+(3-2) 2^2+(4-… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-2-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/ex5-2-p3?rev=1737476038&do=diff Question 4 & 5 Exercise 5.2 Solutions of Question 4 & 5 of Exercise 5.2 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4 $5+\dfrac{7}{3}+\dfrac{9}{3^2}+\dfrac{11}{3^3}+\ldots$\begin{align} & S_{\infty}=5+\dfrac{7}{3}+\dfrac{9}{3^2}+\dfrac{11}{3^3}+\ldots.(i) \\ & \dfrac{1}{3} S_{\infty}=\dfrac{5}{3}+\dfrac{7}{3^2}+\dfrac{9}{3^2}+\dfrac{11}{3^3}+\ldots.(ii) \e… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/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-p3?rev=1737476038&do=diff Question 3 Exercise 5.3 Solutions of Question 3 of Exercise 5.3 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 $n$$n$$4+10+18+28+40+\ldots$\begin{align} & a_2-a_1=10-4=6 \\ & a_3-a_2=18-10=8 \\ & a_4-a_3=28-18=10 \\ & \text {... ... ... } \\ & \text {... ... ... } \\ & a_n-a_{n \quad 1}=(\mathrm{n}-1) \text { term of the sequence } \end{align}$6,10,8, \ldot… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-3-p4?rev=1737476038&do=diff Question 4 Exercise 5.3 Solutions of Question 4 of Exercise 5.3 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4 $n$$n$$3+5+11+29+83+245+\ldots$\begin{align} & a_2-a_1=5-3=2 \\ & a_3-a_2=11-5=6 \\ & a_4-a_3=29-11=18 \\ & \text {... ... ... } \\ & \text {... ... ... } \\ & a_n-a_{n-1}=(\mathrm{n}-1) \text { term ofthe sequence }\end{align}$6,10,18, \ldots$\beg… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-3-p5?rev=1737476038&do=diff Question 5 Exercise 5.3 Solutions of Question 5 of Exercise 5.3 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5 $n$$n$$3+9+21+45+93+189+\ldots$\begin{align} & a_2-a_1=9-3=6 \\ & a_3-a_2=21-9=12 \\ & a_4-a_3=45-21=24\\ & \text {... ... ... } \\ & \text {... ... ... } \\ &a_n-a_{n-1}=(\mathrm{n}-1)\quad \text{ term of the sequence}\quad 6,12,24, \ldots\end{ali… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/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-p1?rev=1737476038&do=diff Question 1 Exercise 5.3 Solutions of Question 1 of Exercise 5.4 of Unit 05: Mascellaneous series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) $\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+\ldots$$n$$$T_n=\dfrac{1}{n(n+1)}$$$T_n$$$\dfrac{1}{n(n+1)}=\dfrac{A}{n}+\dfrac{B}{(n+1)}$$$n(n+1)$$$1=A(n+1)+B n=(A+B) n+A$$$n$$$A+B=0 \text{and} A=1$$$A=1$\begin{align}1+B&=0\\ B&=-1\end{align}\beg… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-4-p2?rev=1737476038&do=diff Question 2 & 3 Exercise 5.4 Solutions of Question 2 & 3 of Exercise 5.4 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2 $\sum_{k=1}^n \dfrac{1}{9 k^2+3 k-2}$\begin{align}\text { Let } S_n&=\sum_{k=1}^n \dfrac{1}{9 k^2+3 k-2} \\ S_n&=\sum_{k=1}^n \dfrac{1}{9 k^2+6 k-3 k-2} \\ & =\sum_{k=1}^n \dfrac{1}{3 k(3 k+2)-1(3 k+2)} \\ S_n&=\sum_{k=1}^n \dfrac{1}{(3 k-1… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-4-p3?rev=1737476038&do=diff Question 4 Exercise 5.4 Solutions of Question 4 of Exercise 5.4 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4 $\sum_{k=1}^n \dfrac{1}{k^2+7 k+12}$\begin{align}S_n &=\sum_{k=1}^n \dfrac{1}{k^2+7 k+12} \\ & =\sum_{k=1}^n \dfrac{1}{(k+3)(k+4)}\end{align}$n^{\text {th }}$$$u_n=\dfrac{1}{(n+3)(n+4)}$$$$\dfrac{1}{(n+3)(n+4)}=\dfrac{A}{n+3}+\dfrac{B}{n+4}$$$A$$B$… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/re-ex5-p1?rev=1737476038&do=diff Question 1 Review Exercise 5 Solutions of Question 1 of Review Exercise 5 of Unit 05: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1 $t_n=6 n+5$$t_{n+1}=$$6 n-1$$6 n+11$$6 n+6$$6 n-5$$1+\dfrac{2}{3}+\dfrac{6}{3^2}+\dfrac{10}{3^3}+\dfrac{14}{3^4}+\ldots$$6$$2$$3$$4$$1+2.2+3.2^2+\cdots+100.2^{\prime \prime}$$99.2^{100}$$100.2^{100}$$99.2^{100}+1$$1000.2^{100}$$n^{t h}$$1.2+2.3+3.4+\… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/re-ex5-p2?rev=1737476038&do=diff Question 2 & 3 Review Exercise Solutions of Question 2 & 3 of Review Exercise of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$n$$1.2+2.3+3.4+\ldots$$n^{\text {th }}$$$a_n=n(n+1)=n^2+n$$\begin{align} \sum_{r=1}^n a_r&=\sum_{r=1}^n r^2+\sum_{r=1}^n r \\ & =\dfrac{n(n+1)(2 n+1)}{6}+\dfrac{n(n+1)}{2} \\ & =\dfrac{n(n+1)}{2}[\dfrac{2 n+1}{3}+1] \\ & =\dfrac{n(n+1)}{2} \cdot … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/re-ex5-p3?rev=1737476038&do=diff Question 4 Review Exercise Solutions of Question 4 of Review Exercise of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4 $\dfrac{1}{1.4 .7}+\dfrac{1}{4.7 .10}+\dfrac{1}{7.10 .13}+\ldots$$1,4,7, \ldots$$$a_n=\dfrac{1}{(3 n-2)(3 n+1)(3 n+4)}$$\begin{align} \dfrac{1}{(3 n-2)(3 n+1)(3 n+4)}&=\dfrac{A}{3 n-2}+\dfrac{B}{3 n+1}+\dfrac{C}{3 n+4}\end{align}$(3 n-2)(3 n+… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/re-ex5-p4?rev=1737476038&do=diff Question 5 & 6 Review Exercise Solutions of Question 5 & 6 of Review Exercise of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$5+12 x+19 x^2+26 x^3+\ldots$$n$\begin{align}S_n&=5+12 x+19 x^2+26 x^3+\cdots+(7 n-2) x^{n-1}...(i)\\ x S_n&=5 x+12 x^2+19 x^3+\cdots+(7 n-9) x^{n-1}+(7 n-1) x^n....(ii)\end{align}\begin{align}(1-x) S_n&=5+(12-5) x+(19-12) x^2+\cdots\\ &+[7 n-2-(… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/re-ex5-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)$$2^2, 3^2, 4^2, \ldots,(n+1)^2$\begin{align} & a_n=(2 n-1)(n+1)^2 \\ & a_n=(2 n-1)(n^2+2 n+1) \\ & a_n=2 n^3+3 n^2-1\end{align}\begin{align} \sum_{r=1}^n a_r&=2 \sum_{r=1}^n r^3+\sum_{r=1… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/re-ex5-p6?rev=1737476038&do=diff Question 8 Review Exercise Solutions of Question 8 of Review Exercise of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 8(i) $n$$n^{t h}$$n^3+3^n.$$n^h$$$a_n=n^3+3^n$$\begin{align}\sum_{r=1}^n a_r&=\sum_{r=1}^n r^3+\sum_{r=1}^n 3^r \\ & =[\dfrac{n(n+1)}{2}]^2+\dfrac{3(3^n-1)}{3-1} \\ & =\dfrac{n^2(n+1)^2}{4}+\dfrac{3}{2}(3^n-1) \end{align}$n$$$S_n=\dfrac{n^2(n+1… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/re-ex5-p7?rev=1737476038&do=diff Question 9 Review Exercise Solutions of Question 9 of Review Exercise of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9(i) $n$$3+7+13+21+31+\ldots$\begin{align} & a_2-a_1=7-3=4 \\ & a_3-a_2=13-7=6 \\ & a_4-a_3=21-13=8 \\ & \ldots \quad \ldots \quad \ldots \\ & \ldots \quad \cdots \quad \ldots \\ & a_n-a_{n-1}=(n-1) \text { term of the series } \\ & 4,6,8, \ldo… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/re-ex5-p8?rev=1737476038&do=diff Question 10 Review Exercise Solutions of Question 10 of Review Exercise of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 10 $n^{\text {th }}$$n$$1+(1+\dfrac{1}{2})+(1+\dfrac{1}{2}+\dfrac{1}{4})+(1+\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8})+\ldots$\begin{align} a_n&=1+\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+\cdots+\dfrac{1}{2^{n-1}} \\ a_n&=\dfrac{1[1-(\dfrac{1}{2})… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-1-p1?rev=1737476038&do=diff Question 1 and 2 Exercise 6.1 Solutions of Question 1 and 2 of Exercise 6.1 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{10 !}{3 ! .3 ! \cdot 4 !}$\begin{align}\dfrac{10 !}{3 ! \cdot 3 ! \cdot 4 !}&=\dfrac{10.9 .8 \cdot 7 \cdot 6 \cdot 5.4 !}{3 ! \cdot 3 ! \cdot 4 !}\\ &=\dfrac{10.9 .8 .7 .5}{3.2 .1}\\ &=4200 \end{align}$\dfrac{3 !+4 !}{5 !-… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-1-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}$\df… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-1-p3?rev=1737476038&do=diff Question 5 Exercise 6.1 Solutions of Question 5 of Exercise 6.1 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{(2 n) !}{n !}=2^n(1.3 .5 \ldots(2 n-1))$\begin{align}\dfrac{(2 n) !}{n !}&=\dfrac{1}{n !}[(2 n)(2 n-1)(2 n-2) \\ &=(2 n-3)(2 n-4)(2 n-5) \ldots(2 n-(2 n-4))\\ &(2 n-(2 n-3))(2 n-(2 n-2))(2 n-(2 n-1))]\end{align}$2 n$\begin{align}\dfra… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-1-p4?rev=1737476038&do=diff Question 4 Exercise 6.1 Solutions of Question 4 of Exercise 6.1 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-1-p5?rev=1737476038&do=diff Question 5 Exercise 6.1 Solutions of Question 5 of Exercise 6.1 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-2-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/ex6-2-p4?rev=1737476038&do=diff Question 7 and 8 Exercise 6.2 Solutions of Question 7 and 8 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$1,2,3,4$$E_1$$m_1=5$$E_2$$\cdot m_2=5$$E_3$$m_3=5$$$m_1 \cdot m_2 \cdot m_3=5.5 \cdot 5=125$$$1,2,3,4$$E_1$$m_1=5$$E_2$$m_2=4$$E_3$$m_3=3$$$m_1 \cdot m_2 \cdot m_3=5 \cdot 4 \cdot 3=60$$$8$$5$$=4$$=4$$=5$$=3$$4 ! \cdot 5 ! \cdot … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p5?rev=1737476038&do=diff Question 9 Exercise 6.2 Solutions of Question 9 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$=^6 P_1=6$$s=^6 P_2=30$$=^6 P_3=120$$=^6 P_4=360$$=^6 P_5=720$$=^6 P_6=720$$6+30+120+360+720+720=1956$ text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p6?rev=1737476038&do=diff Question 10 Exercise 6.2 Solutions of Question 10 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$n=8$$r=5$\begin{align}^8 P_5&=\dfrac{8 !}{(8-5) !}\\ &=\dfrac{8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 !}{3 !}\\ &=6720\end{align}\begin{align}^2 P_2 \times^7 P_4&=2 \times \dfrac{7 !}{(7-4) !}\\ &=2 \times\dfrac{7.6 .5 .4 .3 !}{3 !}\\ &=… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p7?rev=1737476038&do=diff Question 11 Exercise 6.2 Solutions of Question 11 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$10$$1000$$2.3,4,0,8,9$$10$$1000$$10$$100$$E_1$$m_1=5$$E_2$$m_2=5$$10$$100$$$m_1 \cdot m_2=5.5=25$$$100$$1000$$0$$E_1$$m_1=5$$E_2$$\boldsymbol{m}_2=5$$E_3$$m_3=4$$100$$1000$$$m_1 \cdot m_2 \cdot m_3=5.5 \cdot 4=100$$$10$$1000$$$100 + 25=125… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p8?rev=1737476038&do=diff Question 12 Exercise 6.2 Solutions of Question 12 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$8.$$n=8$$\mathrm{O}$$m_1=3$\begin{align} \left(\begin{array}{c} n \\ m 1 \end{array}\right)&=\left(\begin{array}{l} 8 \\ 3 \end{array}\right) \\ & =\dfrac{8 !}{3 !}\\ &=\dfrac{8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 !}{3 !}\\ &=6,720 \e… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p9?rev=1737476038&do=diff Question 13 Exercise 6.2 Solutions of Question 13 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\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/ex6-2-p10?rev=1737476038&do=diff Question 14 and 15 Exercise 6.2 Solutions of Question 14 and 15 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$\dfrac{(n-1) !}{2}=\dfrac{(5-1) !}{2}=\dfrac{24}{2}=12 $$$7$$7$$6 !$$6$$5!$$2 !=2$$7$$$2 \times 5 !=240$$ text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-3-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$$n$\begin{align}&^n C_2=36\\ & \Rightarrow \dfrac{n !}{(n-2) ! 2 !}=36 \\ & \Rightarrow \dfrac{n(n-1)(n-2) !}{(n-2) ! \cdot 2}=36 \\ & \Rightarrow n(n-1)=72 \\ & \Rightarrow n^2-n-72=0 \\ & \Rightarrow n^2-9 n+8 n-72=0\\ & \Rightar… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-3-p2?rev=1737476038&do=diff Question 2 Exercise 6.3 Solutions of Question 2 of Exercise 6.3 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$n$$r$${ }^n P_r=840$${ }^n C_r=35$\begin{align} &^n P_r=\dfrac{n !}{(n-r) !}=840 ....(i)\\ &^n C_r=\dfrac{n !}{(n-r) ! r !}=35....(ii)\end{align}\begin{align}\dfrac{n !}{(n-r) !} \cdot \dfrac{(n-r) ! r !}{n !}&=\dfrac{840}{35}\\ r!&=24\\ \te… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-3-p3?rev=1737476038&do=diff Question 3 Exercise 6.3 Solutions of Question 3 of Exercise 6.3 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$n$$^{2 n} C_3:^n C_2=36: 3$\begin{align} & { }^{2 n} C_3:{ }^n C_2=36: 3 . \\ & \Rightarrow \dfrac{(2 n) !}{(2 n-3) ! 3 !} \times \dfrac{(n-2) ! 2 !}{n !}=12 \\ & \Rightarrow \dfrac{2 n(2 n-1)(2 n-2)(2 n-3) !}{(2 n-3) ! 3 !}\times\dfrac{(n-2… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-3-p4?rev=1737476038&do=diff Question 4 Exercise 6.3 Solutions of Question 4 of Exercise 6.3 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${ }^{n-1} C_r+{ }^{n-1} C_{r-1}={ }^n C_r$$${ }^n{ }^1 C_r+{ }^n{ }^1 C_{r-1}={ }^n C_s$$\begin{align} { }^{n-1} C_r+{ }^{n-1} C_{r-1}&=\dfrac{(n-1) !}{(n-r-1) ! r !}+\dfrac{(n-1) !}{(n-1-(r-1)) !(r-1) !} \\ & =\dfrac{(n-1) !}{(n-r-1) ! r(r-… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-3-p5?rev=1737476038&do=diff Question 5 and 6 Exercise 6.3 Solutions of Question 5 and 6 of Exercise 6.3 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$12$$n=12$${ }^{12} C_2=66$$12$$n=12$${ }^{12} C_3=220$$${ }^6 C_2=\dfrac{6 !}{(6-2) ! 2 !}=15 $$$6$$\quad 15-6=9$ text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-3-p6?rev=1737476038&do=diff Question 7 and 8 Exercise 6.3 Solutions of Question 7 and 8 of Exercise 6.3 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$20$\begin{align}{ }^{20} C_2&=\dfrac{20 !}{(20-2)2!}!\\ &=\dfrac{20!}{18!\cdot 2!}\\ &=190\end{align}$7$$10$$3$$7$$10$$${ }^{10} C_7=\dfrac{10 !}{(10-7) ! 7 !}=120$$$7$$4.$$4$$${ }^7 C_4=\dfrac{7 !}{(7-4) ! 4 !}=35.$$$35$$10.$ text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-3-p7?rev=1737476038&do=diff Question 9 Exercise 6.3 Solutions of Question 9 of Exercise 6.3 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$8$$6$$7$$7$$6.$$=7+6=13$${ }^7 C_4$${ }^6 C_4$\begin{align}{ }^7 C_4 \cdot{ }^6 C_4&=\dfrac{7 !}{(7-4) ! 4 !} \cdot \dfrac{6 !}{(6-4)}\\\ &= 525\end{align}$8$$6$$7$$7$$6$$=7+6=13$$3,4,5,6$$6$\begin{align}{ }^7 C_2 \cdot{ }^6 C_6&=\dfrac{7 !}… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-3-p8?rev=1737476038&do=diff Question 9 Exercise 6.3 Solutions of Question 9 of Exercise 6.3 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-4-p1?rev=1737476038&do=diff Question 1 Exercise 6.4 Solutions of Question 1 of Exercise 6.4 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$S=\{1,2,3,4,5,6\}$$5$$5$\begin{align}A&=\{5\}\\ P(A)&=\dfrac{n(A)}{n(S)}\\ &=\dfrac{1}{6} \end{align}$S=\{1,2,3,4,5,6\}$$1$$1$\begin{align}B&=\{\}\\ &=\phi \text{then}\\ P(B)&=\dfrac{n(B)}{n(S)}\\ &=\dfrac{0}{6}\\ &=0\end{align}$S=\{1,2,3,4,… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-4-p2?rev=1737476038&do=diff Question 2 Exercise 6.4 Solutions of Question 2 of Exercise 6.4 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$4$$5$$6$$3$$4+5+6=15$$${ }^{15} C_3=\dfrac{15 !}{(15-3) ! 3 !}=455 $$$${ }^6 C_4=\dfrac{6 !}{(6-4) ! 4 !}=15$$$$=\dfrac{15}{455}=\dfrac{3}{91}$$$4$$5$$6$$3$$4+5+6=15$$${ }^{15} C_3=\dfrac{15 !}{(15-3) ! 3 !}=455 $$$${ }^4 C_3=\dfrac{4 !}{(4-… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-4-p3?rev=1737476038&do=diff Question 3 Exercise 6.4 Solutions of Question 3 of Exercise 6.4 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$8$$8$$2^8$$$n(S)=256$$$$\dfrac{1}{256}$$$8$$$A=\{8\}$$$${ }^8 C_8=\dfrac{8 !}{(8-8) ! 8 !}=1$$$8$$$P(A)=\dfrac{1}{256}$$$7$$8$$2^8$$$n(S)=256$$$$\dfrac{1}{256}$$$7$$$B=\{7\}$$$7$$8$$$n(B)={ }^8 C_7=\dfrac{8 !}{(8-7) ! 7 !}=8$$$7$$8$$$P(B)=\d… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-4-p4?rev=1737476038&do=diff Question 4 Exercise 6.4 Solutions of Question 4 of Exercise 6.4 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.\begin{align}S&=(HHII,HHT.HTH.HTT.THII.THT.TTH,TT7),\\ \text{then} n(S)&=2^3=8\end{align}$$A=\{H H H\}$$$$n(A)=1$$$P(A)=\dfrac{n(A)}{n(S)}=\dfrac{1}{8}$\begin{align}S&=(HHII,HHT.HTH.HTT.THII.THT.TTH,TT7),\\ \text{then} n(S)&=2^3=8\end{align}… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-4-p5?rev=1737476038&do=diff Question 5 Exercise 6.4 Solutions of Question 5 of Exercise 6.4 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$6$$4$$3$$2$$=6+4=10$$5$$10$\begin{align}{ }^{10)} C_5 &=\dfrac{10 !}{(10-5) ! 5 !}\\ &=252\\ n(S)&=252\end{align}$3$$2$$3$$2$\begin{align}{ }^6 \mathrm{C}_3\cdot{ }^{4} \mathrm{C}_2&=\dfrac{6 !}{(6-3) ! 3 !} \cdot \dfrac{4 !}{(4-2) ! 2 !}\\… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-4-p6?rev=1737476038&do=diff Question 6 Exercise 6.4 Solutions of Question 6 of Exercise 6.4 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$52$$$=52$$$=4$$$=\dfrac{4}{52}=\dfrac{1}{13}$$$52$$=52$$13$$13$$$\dfrac{13}{52}+ \dfrac{13}{52}=\dfrac{1}{4}+\dfrac{1}{4}=\dfrac{2}{4}=\dfrac{1}{2}$$$52$$=52$$13.$$$=\dfrac{13}{52}=\dfrac{1}{4}$$$52$$=52$$12.$$$=\dfrac{12}{52}=\dfrac{3}{13}$… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-4-p7?rev=1737476038&do=diff Question 7 Exercise 6.4 Solutions of Question 7 of Exercise 6.4 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.\begin{align}S&=\{(i, j) ; i, j=1,2,3,4,5,6\}\\ &=\left[\begin{array}{llllll} (1,1) & (1,2) & (1,3) & (1,4) & (1,5) & (1,6) \\ (2,1) & (2,2) & (2,3) & (2,4) & (2,5) & (2,6) \\ (3,1) & (3,2) & (3,3) & (3,4) & (3,5) & (3,6) \\ (4,1) & (4,2) & (… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-5-p1?rev=1737476038&do=diff Question 1 and 2 Exercise 6.5 Solutions of Question 1 and 2 of Exercise 6.5 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$A$$B$$P(A)=\dfrac{2}{5}, P(B)=\dfrac{2}{5}$$P(A \cup B)=\dfrac{1}{2}$$P(A \cap B)$\begin{align} P(A \cup B)&=P(A)+P(B)-P(A \cap B) \\ \Rightarrow P(A \cap B)&=P(A)+P(B)-P(A \cup B) \end{align}$P(A), P(B)$$P(A \cup B)$$$P(A \cap… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-5-p2?rev=1737476038&do=diff Question 3 and 4 Exercise 6.5 Solutions of Question 3 and 4 of Exercise 6.5 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$P(A)=0.5$$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.$\begin{align}S&=\{1,2,3, \ldots, 50\} \tex… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-5-p3?rev=1737476038&do=diff Question 5 and 6 Exercise 6.5 Solutions of Question 5 and 6 of Exercise 6.5 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\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}$$=1-\dfrac{17}{52}=\dfr… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-5-p5?rev=1737476038&do=diff Question 8 Exercise 6.5 Solutions of Question 8 of Exercise 6.5 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$7$$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$$r$\begin{align} { }^{56} P_{r+6}:{ }^{54} P_r+3&=30800: 1 \\ \Rightarrow \dfrac{\dfrac{56 !}{[56-(r+6)] !}}{\dfrac{54 !}{[54-(r+3)] !}}&=\dfrac{30800}{1} \\ \Rightarrow \dfrac{56… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p4?rev=1737476038&do=diff Question 5 & 6 Review Exercise 6 Solutions of Question 5 & 6 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$n=6$$$$(n-1) !=(6-1) !=5 !=120$$$120-24=96$$n=6$$(n-1) !=(6-1) !=5 !=120$$$(n-1) !=(5-1) !=4 !=24$$$$(n-1) !=(6-1) !=5 !=120$$$$4 ! \cdot 2 !=48$$$(5-1) !$ text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p5?rev=1737476038&do=diff Question 7 & 8 Review Exercise 6 Solutions of Question 7 & 8 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$P(A)=0.8, P(B)=0.5$$P(B / A)=0.4$$P(A \cap B)$\begin{align} P(B \mid A)&=\dfrac{P(A \cap B)}{P(A)} \\ \Rightarrow P(A \cap B)&=P(B \mid A) \cdot P(A)\\ &=0.4 \times 0.8=0.32\end{align}$P(A)=0.8, P(B)=0.5$$P(B / A)=0.4$$P(A … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p6?rev=1737476038&do=diff Question 9 & 10 Review Exercise 6 Solutions of Question 9 & 10 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$2,3,0,3,4,2,3$$1$$=100,0000$$$=\dfrac{7 !}{3 ! \cdot 2 !}=420 $$$1$$0$$7$$0$$$=\dfrac{6 !}{2 ! 3 !}=60 $$$1$$420-50=360$$n$$n$$(n-1)$$(n - 1)$$(n-1)$$(n-2) !$$2$$2 !$$n$$$(n-2) ! \cdot 2 !=2(n-2) ! $$ text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p7?rev=1737476038&do=diff Question 11 Review Exercise 6 Solutions of Question 11 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$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-p1?rev=1737476038&do=diff Question 1 Exercise 7.1 Solutions of Question 1 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$2+4+6+\cdots+2 n=n(n+1)$$n=1$$$2=1(1+1)=2 $$$n=1$$n=k$$$2+4+6+\cdots+2 k=k(k+1)....(i)$$$n=k+1$$(k+1)^{t h}$$$a_{k+1}=\mathbf{2}(k+1)=2 k+2 $$$k+1$\begin{align}2+4+6+\cdots+2 k+2(k+1)& =k(k+1)+2(k+1) \\ & =(k+1)[k+2] \\ & =(k+1)(k+1+1)\end{a… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p2?rev=1737476038&do=diff Question 2 Exercise 7.1 Solutions of Question 2 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$1+5+9+\ldots+(4 n-3)=n(2 n-1)$$n=1$$$1=1(2.1-1)=1$$$n=1$$n=k$\begin{align}1+5+9+\ldots+(4 k-3)\\ & =k(2 k-1)....(i) \\ \end{align}$n=k+1$$k+1$$$a_{k-1}=4(k+1)-3=4 k+1 $$$(k+1)^{t h}$\begin{align}1+5+9+\ldots+(4 k-3)+(4 k+1)& =k(2 k-1)+4 k+1 … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p3?rev=1737476038&do=diff Question 3 Exercise 7.1 Solutions of Question 3 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$3+6+9+\ldots+3 n=\dfrac{3 n(n+1)}{2}$$n=1$$3=\dfrac{3.1(1+1)}{2}=3$$n=1$$n=k$$$3+6+9+\ldots+3 k=\dfrac{3 k(k+1)}{2}....(i)$$$n=k+1$$(k+1)$$a_{k+1}=3(k+1)$$(k+1)^{t h}$\begin{align}3+6+9+\ldots+3 k+3(k+1) & =\dfrac{3 k(k+1)}{2}+3(k+1) \\ & =3… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p4?rev=1737476038&do=diff Question 4 Exercise 7.1 Solutions of Question 4 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$3+7+11+\cdots+(4 n-1)=n(2 n+1)$$n=1$$$3=1(2+1)=3 $$$n=1$$n=k$\begin{align}3+7+11+\cdots+(4 k-1) & =k(2 k+1)....(i) \end{align}$n=k+1$$(k+1)$$a_{k+1}=4(k+1)-1$$(k+1)^{t h}$\begin{align} 3+7+11+\cdots+(4 k-1)+[4(k+1)-1] & =k(2 k+1)+4(k+1)-1 \… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p5?rev=1737476038&do=diff Question 5 Exercise 7.1 Solutions of Question 5 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$1^3+2^3+3^3+\ldots+n^3=\left[\dfrac{n(n+1)}{2}\right]^2$$n=1$$1^3=1=\left[\dfrac{1(1+1)}{2}\right]^2=1$$n=1$$n=k_1$\begin{align}1^3+2^3+3^3+\ldots+k^3& =[\dfrac{k(k+1)}{2}]^2....(i)\end{aligned} 3. Now $$ the $$ term of the given series on l… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p6?rev=1737476038&do=diff Question 6 Exercise 7.1 Solutions of Question 6 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$1(1 !)+2(2 !)+3(3 !)+\ldots+n(n !)= -(n+1) !-1$$n=1$$$1(1 !)=1=(1+1) !-1=2 !-1=1 $$$n=1$$n=k$\begin{align}1(1 !)+2(2 !)+3(3 !)+\ldots+k(k !)& =(k+1) !-1 \ldots . .(i)\end{align}$n=k+1$$(k+1)^{t h}$$a_{k+1}=(k+1)[(k+1) !]$$a_{k-1}$\begin{ali… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p7?rev=1737476038&do=diff Question 7 Exercise 7.1 Solutions of Question 7 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$1.2+2.3+3.4+\ldots+n(n+1)=\dfrac{n(n+1)(n+2)}{3}$$n=1$$$1.2=2=\dfrac{1(1+1)(1+2)}{3}=2 $$$n=1$$n=k$\begin{align}1.2+2.3+3.4+\ldots+k(k+1)& =\dfrac{k(k+1)(k+2)}{3}....(i)\end{align}$n=k+1$$(k-1)^{t h}$$a_{k+1}=(k+1)(k+ 2)$$(k+1)^{\text {th }}… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p8?rev=1737476038&do=diff Question 8 Exercise 7.1 Solutions of Question 8 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$1+2+2^2+2^3+\ldots+2^n 1=2^n-1$$n=1$$1=2^1-1=1$$n=1$$n-k>1$\begin{align}1+2+2^2+2^3+\ldots+2^{k-1} \\ & =2^k-1 ....(i)\end{align}$n-k-1$$(k+1)^{t h}$$a_{k+1}=2^k$$a_{k+1}$\begin{align}1+2+2^2+2^3+\ldots+2^{k-1}-2^k & =2^k-12^k \\ & =2^k+2^k-… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p9?rev=1737476038&do=diff Question 9 Exercise 7.1 Solutions of Question 9 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{1}{3}+\dfrac{1}{9}+\dfrac{1}{27}+\ldots+\dfrac{1}{3^n}=\dfrac{1}{2}[1-\dfrac{1}{3^n}]$$n=1$$$\dfrac{1}{3}-\dfrac{1}{2}[1-\dfrac{1}{3}]-\dfrac{1}{2} \dfrac{2}{3}=\dfrac{1}{3} $$$n=1$$n=k$$$\dfrac{1}{3}+\dfrac{1}{9}+\dfrac{1}{27}+\ldots… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p10?rev=1737476038&do=diff Question 10 Exercise 7.1 Solutions of Question 10 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\left(\begin{array}{1}5 \\5 \end{array}\right)+\left(\begin{array}{l}6 \\ 5\end{array}\right)+\left(\begin{array}{l}7 \\ 5\end{array}\right)+\ldots+\left(\begin{array}{c}n+4 \\ 5\end{array}\right)=\left(\begin{array}{c}n+5 \\ 6\end{array}\… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p11?rev=1737476038&do=diff Question 11 Exercise 7.1 Solutions of Question 11 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.\begin{align} & \left(\begin{array}{l} 2 \\ 2 \end{array}\right)+\left(\begin{array}{l} 3 \\ 2 \end{array}\right)+\left(\begin{array}{l} 4 \\ 2 \end{array}\right)+\ldots+\left(\begin{array}{l} n \\ 2 \end{array}\right)=\left(\begin{array}{c… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p12?rev=1737476038&do=diff Question 12 Exercise 7.1 Solutions of Question 12 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{5^{2 n}-1}{24}$$n=1$$$\dfrac{5^{2 n}-1}{24}=\dfrac{5^{2.1}-1}{24}=\dfrac{24}{24}=1 \in \mathbb{Z}$$$n=1$$n=k>1$$$\dfrac{5^{2 k}-1}{24} \in \mathbb{Z}$$$n=k+1$\begin{align}\dfrac{5^{2(k+1)}-1}{24}&=\dfrac{5^{2 k+2}-1}{24} \\ & =\dfra… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-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/ex7-1-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$$m=k+1$\begin{align}a^{2(k+1)}-b^{2(k-1)} & =a^{2 k+2}-b^{2 k+2} \\ & =… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p1?rev=1737476038&do=diff Question 1 Exercise 7.2 Solutions of Question 1 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$(x^2-\dfrac{1}{y})^4$\begin{align}(x^2-\dfrac{1}{y})^4&=(x^2)^4+{ }^4 C_1(x^2)^3(-\dfrac{1}{y})+ \\ & { }^4 C_2(x^2)^2(-\dfrac{1}{y})^2+{ }^4 C_3(x^2)(-\dfrac{1}{y})^3 + { }^4 C_4(-\dfrac{1}{y})^4 \\ & =x^8- \dfrac{4x^6}{y}+\dfrac{6x^4}{y^2}… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p2?rev=1737476038&do=diff Question 2 Exercise 7.2 Solutions of Question 2 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$4^{th}$$(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 }}$$r=3$\begin{align} & T_{3+1}=\dfrac{7 !}{(7-3) ! 3 !} 2^{7-3} a^3 \\ & \Rightarrow T_4=\dfrac{7 !}{4 ! 3 !} \cdot 2^4 a^3 \\ & \Rightarrow… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p3?rev=1737476038&do=diff Question 3 Exercise 7.2 Solutions of Question 3 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$x$$(\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$$T_{r+1}$\begin{align}T_{r+1}&=\dfrac{9 !}{(9-r) ! r !}(\dfrac{4 x^2}{3})^{9-r}(-\dfrac{3}{2 x})^r \\ & =\dfrac{9 !}{(9-r) ! r !} \cdot \dfrac… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p4?rev=1737476038&do=diff Question 4 Exercise 7.2 Solutions of Question 4 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$x^{23}$$(x^2-x)^{20}$$n=20, \quad a=x^2$$b=-x$$T_{r, 1}$$x^{23}$\begin{align}T_{r-1}&=\dfrac{20 !}{(20-r) ! r !}(x^2)^{20 r}(-x)^r \\ & =\dfrac{20 !}{(20-r) ! r !}(-1)^r \cdot x^{40-2 r+r} \\ & =\dfrac{20 !}{(20-r) ! r !}(-1)^r x^{40-r}\end{… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p5?rev=1737476038&do=diff Question 5 Exercise 7.2 Solutions of Question 5 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$(\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$\begin{align}T_5&=\dfrac{8 !}{(8-4) ! 4 !}(\dfrac{a}{x})^{8-4}(b … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p6?rev=1737476038&do=diff Question 6 Exercise 7.2 Solutions of Question 6 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$(2 \sqrt{x}-\dfrac{3}{x \sqrt{x}})^{23}$$a=2 \sqrt{x}$$b=-\dfrac{3}{x \sqrt{x}}$$n=23$$x$\begin{align} T_{r+1}&=\dfrac{23 !}{(23-r) ! r !}(2 \sqrt{x})^{23-r}(-\dfrac{3}{x \sqrt{x}})^r \\ & =\dfrac{23 !}{(23-r) ! r !} \cdot 2^{23-r} \cdot(-3)… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p7?rev=1737476038&do=diff Question 7 Exercise 7.2 Solutions of Question 7 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$(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}$$x=… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p9?rev=1737476039&do=diff Question 9 Exercise 7.2 Solutions of Question 9 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$(x-y)="$$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|}$$\left(\frac{1}{12}\right)^2 \cdot… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p10?rev=1737476038&do=diff Question 10 Exercise 7.2 Solutions of Question 10 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$n=2 ;$$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$$(1 \div 1… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p11?rev=1737476038&do=diff Question 11 Exercise 7.2 Solutions of Question 11 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$(1+x)^n$$\left(\begin{array}{l}n \\ r\end{array}\right)=\mathrm{C}_r$$\mathrm{C}_1+2 \mathrm{C}_2 x+3 \mathrm{C}_3 x^2+\ldots \ldots . .+\mathrm{nC}_{\mathrm{n}} x^{\mathrm{n}-1}=\mathrm{n}(1+x)^{\mathrm{n}-1}$ text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p1?rev=1737476039&do=diff Question 1 Exercise 7.3 Solutions of Question 1 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\frac{1}{2}$$$ \begin{aligned} & (1-x)^{\frac{1}{2}}=1+\frac{1}{2} x+ \\ & \frac{\frac{1}{2}\left(-\frac{1}{2}-1\right)}{2 !}(-x)^2 \end{aligned} $$$$ \begin{aligned} & +\frac{-\frac{1}{2}\left(-\frac{1}{2}-1\right)\left(-\frac{1}{2}-2\right… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p2?rev=1737476039&do=diff Question 2 Exercise 7.3 Solutions of Question 2 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sqrt{26}$$$ \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} $$$$ \begin{aligned} & \sqrt{26}=5\left[1+\frac{1}{25}\right]^{\frac{1}{2}} \\ & =5\left[1+\f… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p3?rev=1737476039&do=diff Question 3 Exercise 7.3 Solutions of Question 3 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sqrt{\frac{1-x}{1+x}}$$x^3$$\sqrt{\frac{1-x}{1+x}}$$$ =(1-x)^{\frac{1}{2}}(1+x)^{-\frac{1}{2}} \text {. } $$$$ \begin{aligned} & (1-x)^{\frac{1}{2}}(1+x)^{\frac{1}{2}} \\ & =\left[1-\frac{x}{2}+\frac{\frac{1}{2}\left(\frac{1}{2}-1\right)}{2… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p4?rev=1737476039&do=diff Question 4 Exercise 7.3 Solutions of Question 4 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$x$$x^2$$x$$$ \sqrt{\frac{1-3 x}{1+4 x}}=1-\frac{7 x}{2} $$$$ \sqrt{\frac{1-3 x}{1-4 x}}=(1-3 x)^{\frac{1}{2}}(1+4 x)^{-\frac{1}{2}} $$$x^2$$x$$$ \begin{aligned} & =\left(1-\frac{3 x}{2}\right) \times\left(1-\frac{4 x}{2}\right) \\ & =\left(1… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p5?rev=1737476039&do=diff Question 5 and 6 Exercise 7.3 Solutions of Question 5 and 6 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$x$$x^2$$x$$$ \frac{(8+3 x)^{\frac{2}{3}}}{(2+3 x) \sqrt{4-5 x}}=1-\frac{5 x}{8} $$$$ \frac{\sqrt[4]{3}-3 x j^{\frac{2}{3}}}{2 \cdot 3 x+4-5 x} $$$$ \begin{aligned} & =\frac{8^{\frac{2}{3}}\left(1+\frac{3 x}{8}\right)^{\frac{2}{3}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p6?rev=1737476039&do=diff Question 7 and 8 Exercise 7.3 Solutions of Question 7 and 8 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$x^4$$(1-x)^{\frac{1}{4}}+(1-x)^{\frac{1}{4}}=a-b x^2$$a$$b$$$ \begin{aligned} & (1+x)^{\frac{1}{4}}+(1-x)^{\frac{1}{4}} \\ & =\left[1+\frac{x}{4}+\frac{\frac{1}{4}\left(\frac{1}{4}-1\right)}{2 !} x^2+\right. \\ & \left.\frac{\fra… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p7?rev=1737476039&do=diff Question 9 Exercise 7.3 Solutions of Question 9 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$x^{\prime \prime}$$\left(\frac{1+x}{1-x}\right)^2$$$ \begin{aligned} & \left(\frac{1+x}{1-x}\right)^2=(1+x)^2(1-x)^{-2} \\ & =\left(x^2+2 x+1\right)(1-x)^2 \end{aligned} $$$$ \begin{aligned} & =\left(x^2+2 x+1\right)[1+2 x+ \\ & \frac{-2(-2-… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p8?rev=1737476039&do=diff Question 10 Exercise 7.3 Solutions of Question 10 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$1-\frac{1}{2^2}+\frac{1.3}{2 !} \cdot \frac{1}{2^4}+\ldots$$(1+x)^n$$$ \begin{aligned} & 1+n x+\frac{n(n-1)}{2 !} x^2 \\ & +\frac{n(n-1(n-2))}{3 !} x^3+\ldots \end{aligned} $$$n x=-\frac{1}{4}$$\frac{n(n-1)}{2 !} x^2=\frac{1.3}{2 !} \cdot … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p9?rev=1737476039&do=diff Question 11 Exercise 7.3 Solutions of Question 11 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$y=\frac{1}{2^2}+\frac{1.3}{2 !} \cdot \frac{1}{2^4}+\frac{1 \cdot 3 \cdot 5}{3 !} \cdot \frac{1}{2^6}+\ldots$$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-kpk/sol/unit07/ex7-3-p10?rev=1737476039&do=diff Question 12 Exercise 7.3 Solutions of Question 12 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$2 y=\frac{1}{2^2}+\frac{1.3}{2 !} \cdot \frac{1}{2^4}+\frac{1.3 \cdot 5}{3 !} \cdot \frac{1}{2^6}+\ldots$$4 y^2+4 y-1=0$$$ 2 y=\frac{1}{2^2}+\frac{1.3}{2 !} \cdot \frac{1}{2^4}-\frac{1.3 \cdot 5}{3 !} \cdot \frac{1}{2^6}+\ldots $$$S=2 y+1=… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p11?rev=1737476039&do=diff Question 13 Exercise 7.3 Solutions of Question 13 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$x$$x^3$$x$$n^{\text {th }}$$1+x$$\frac{2 n+(n+1) x}{2 n+(n-1) x}$$$ \begin{aligned} & (1+x)^{\frac{1}{n}}=\frac{2 n+(n+1) x}{2 n+(n-1) x} \\ & \frac{2 n+(n+1) x}{2 n+(n-1) x} \\ & =1+\frac{1}{n} x+\frac{\frac{1}{n}\left(\frac{1}{n}-1\right… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p12?rev=1737476039&do=diff Question 14 Exercise 7.3 Solutions of Question 14 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$x$$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)$$\dfrac{n+1}{n… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/re-ex7-p2?rev=1737476039&do=diff Question 2 Review Exercise 7 Solutions of Question 2 of Review Exercise 7 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\left(2 x^3+3 y\right)^8$$a=2 x^3$$b=3 y$$n=8$$n=8$$\frac{8+2}{2}=5$$$ \begin{aligned} & T_5=\frac{8 !}{(8-4) ! 4 !}\left(2 x^3\right)^{8-4}(3 y)^4 \\ & T_5=70.2^4 \cdot 3^4 \cdot x^{12} \cdot y^4 \\ & =90720 x^{12} y^4 \end{aligne… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/re-ex7-p3?rev=1737476039&do=diff Question 3 & 4 Review Exercise 7 Solutions of Question 3 & 4 of Review Exercise 7 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$(2 x-4 y)^7$$n=7, a=2 x$$b=-4 y$$$ \begin{aligned} & T_{3+1}=\frac{7 !}{(7-3) ! 3 !}(2 x)^{7 \cdot 3}(-4 y)^3 \\ & =\frac{7 !}{(7-3) ! 3 !} \cdot\left(2^4\right) \cdot(-4)^3 \cdot x^4 y^3 \\ & \Rightarrow T_4=-35840 x^4 y^3… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/re-ex7-p4?rev=1737476039&do=diff Question 5 & 6 Review Exercise 7 Solutions of Question 5 & 6 of Review Exercise 7 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\left(\frac{2}{x^2}+\frac{x^2}{2}\right)^{10}$$n=10, a^{\prime}=\frac{2}{x^2}$$b=\frac{x^2}{2}$$T_{r+1}$$x$$$ \begin{aligned} & T_{r+1}=\frac{10 !}{(10-r) ! r !}\left(\frac{2}{x^2}\right)^{10 r}\left(\frac{x^2}{2}\right)^r … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/re-ex7-p5?rev=1737476039&do=diff Question 7 & 8 Review Exercise 7 Solutions of Question 7 & 8 of Review Exercise 7 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$7^n-3^n$$n=1$$7^k-3^k=7-4=4$$n=1$$n=k>1$$7^n-3^n=4 Q$$Q$$n=k+1$$$ \begin{aligned} & 7^{k+1}-3^{k+1}=7.7^k-3.3^k \\ & =(4+3) \cdot 7^k-3.3^k \\ & =4.7^k+3.7^k-3.3^k \end{aligned} $$$$ \begin{aligned} & =4.7^k+3\left[7^k-3^k\… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/re-ex7-p6?rev=1737476039&do=diff Question 9 and 10 Review Exercise 7 Solutions of Question 9 and 10 of Review Exercise 7 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/re-ex7-p7?rev=1737476039&do=diff Question 11 Review Exercise 7 Solutions of Question 11 of Review Exercise 7 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p2?rev=1737476039&do=diff Question 2, Exercise 10.1 Solutions of Question 2 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin \dfrac{\pi }{12}$$\dfrac{\pi }{12}$$\dfrac{\pi }{3}-\dfrac{\pi }{4}$\begin{align}\sin (\alpha -\beta )=\sin \alpha \cos \beta -\cos \alpha \sin.\end{align}\begin{align} \Rightarrow \quad \sin \left( \frac{\pi }{3}-\f… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p8?rev=1737476039&do=diff Question 8, Exercise 10.1 Solutions of Question 8 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\tan \left( \dfrac{\pi }{4}+\theta \right)=\dfrac{\cos \theta +\sin \theta }{\cos \theta -\sin \theta }$\begin{align}L.H.S.&=\tan \left( \dfrac{\pi }{4}+\theta \right)\\ &=\dfrac{\sin \left( \dfrac{\pi }{4}+\theta \ri… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p10?rev=1737476039&do=diff Question11 and 12, Exercise 10.1 Solutions of Question 11 and 12 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\alpha$$\beta$$\gamma$$ABC$$\cot \dfrac{\alpha }{2}+\cot \dfrac{\beta }{2}+\cot \dfrac{\gamma }{2}=\cot \dfrac{\alpha }{2}\cot \dfrac{\beta }{2}\cot \dfrac{\gamma }{2}$$\alpha$$\beta$$\gamma$\begin{align}&\… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p11?rev=1737476039&do=diff Question 13, Exercise 10.1 Solutions of Question 13 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$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)$… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-2-p1?rev=1737476039&do=diff Question 1, Exercise 10.2 Solutions of Question 1 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. There are four parts in Question 1.$\sin 2\theta ,\,\,\cos 2\theta$$\tan 2\theta$$\tan \theta =-\dfrac{1}{5}$$\theta$$\sin \theta =\dfrac{1}{\sqrt{26}}$$\cos \theta =\dfrac{-5}{\sqrt{26}}$\begin{align}\sin 2\theta &=2\sin… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-2-p2?rev=1737476039&do=diff Question 2, Exercise 10.2 Solutions of Question 2 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin \theta =\dfrac{5}{13}$$\theta $$\sin 2\theta $$\sin \theta =\dfrac{5}{13}$$$\cos \theta =\pm \sqrt{1-{{\sin }^{2}}\theta }.$$$\theta$$\cos$\begin{align}\cos\theta &=-\sqrt{1-{{\sin }^{2}}\theta }\\ &=-\sqrt{1-\left(\… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-2-p3?rev=1737476039&do=diff Question 3, Exercise 10.2 Solutions of Question 3 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin \theta =\dfrac{4}{5}$$\theta$$\sin2\theta$$\sin \theta =\dfrac{4}{5}$$\theta$$\cos \theta =-\dfrac{3}{5}$\begin{align}\sin 2\theta &=2\sin \theta \cos \theta \\ &=2\left( \dfrac{4}{5} \right)\left( -\dfrac{3}{5} \rig… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-2-p4?rev=1737476039&do=diff Question 4 and 5, Exercise 10.2 Solutions of Question 4 and 5 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos \theta =-\dfrac{3}{7}$$\theta $$\sin \dfrac{\theta }{2}$$\cos \theta =-\dfrac{3}{7}$$\theta$\begin{align}&\pi < \theta < \dfrac{3\pi}{2} \\ \implies &\frac{\pi}{2} < \frac{\theta}{2} < \dfrac{3\pi}{4}\end… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-2-p5?rev=1737476039&do=diff Question 6, Exercise 10.2 Solutions of Question 6 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos {{15}^{\circ }}$${{15}^{\circ }}=\dfrac{{{30}^{\circ }}}{2}$$\dfrac{\theta }{2}=\dfrac{{{30}^{\circ }}}{2}$$\cos {{15}^{\circ }}$\begin{align}\cos {{15}^{\circ }}&=\cos \dfrac{{{30}^{\circ }}}{2}=\sqrt{\dfrac{1+\cos … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-2-p6?rev=1737476039&do=diff Question 7, Exercise 10.2 Solutions of Question 7 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${{\cos }^{4}}\theta -{{\sin }^{4}}\theta =\dfrac{1}{\sec 2\theta }$\begin{align}L.H.S&={{\cos }^{4}}\theta -{{\sin }^{4}}\theta \\ &=\left( {{\cos }^{2}}\theta -{{\sin }^{2}}\theta \right)\left( {{\cos }^{2}}\theta +{{\… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-2-p7?rev=1737476039&do=diff Question 8 and 9, Exercise 10.2 Solutions of Question 8 and 9 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${{\cos }^{4}}\theta $\begin{align}{{\cos}^{4}}\theta &={{\left( {{\cos }^{2}}\theta \right)}^{2}}\\ &={{\left( \dfrac{1+\cos 2\theta }{2} \right)}^{2}}\\ &=\dfrac{1+2\cos 2\theta +{{\cos }^{2}}2\theta }{4}\\… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-3-p1?rev=1737476039&do=diff Question 1, Exercise 10.3 Solutions of Question 1 of Exercise 10.3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. There are four parts in Question 1.$2\sin 6x\sin x$$$-2\sin \alpha \sin \beta =\cos (\alpha +\beta )-\cos (\alpha -\beta ).$$$\alpha =6x$$\beta =x$\begin{align}-\,2\sin 6x\sin x&=\cos (6x+x)-\cos (6x-x)\\ &=\cos 7x-\cos x… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-3-p2?rev=1737476039&do=diff Question 2, Exercise 10.3 Solutions of Question 2 of Exercise 10.3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$\sin {{37}^{\circ }}+\sin {{43}^{\circ }}.$$$$\sin \alpha +\sin \beta =2\sin \left( \dfrac{\alpha +\beta }{2} \right)\cos \left( \dfrac{\alpha -\beta }{2} \right).$$$\alpha ={{37}^{\circ }}$$\beta ={{43}^{\circ }}$\begin… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-3-p3?rev=1737476039&do=diff Question 3, Exercise 10.3 Solutions of Question 3 of Exercise 10.3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$\dfrac{\cos {{75}^{\circ }}+\cos {{15}^{\circ }}}{\sin {{75}^{\circ }}-\sin {{15}^{\circ }}}=\sqrt{3}.$$$$\cos \alpha +\cos \beta =2\cos \left( \dfrac{\alpha +\beta }{2} \right)\cos \left( \dfrac{\alpha -\beta }{2} \righ… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-3-p4?rev=1737476039&do=diff Question 5, Exercise 10.3 Solutions of Question 5 of Exercise 10.3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$\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)$\begin{align}L.H.S.&=\cos {{20… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/re-ex10-p1?rev=1737476039&do=diff Question 1, Review Exercise 10 Solutions of Question 1 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos {{50}^{\circ }}5{0}'\cos {{9}^{\circ }}1{0}'-\sin {{50}^{\circ }}5{0}'\sin {{9}^{\circ }}1{0}'=$$0$$\dfrac{1}{2}$$1$$\dfrac{\sqrt{3}}{2}$$\tan {{15}^{\circ }}=2-\sqrt{3}$${{\cot }^{2}}{{75}^{\circ }}$$7+\sq… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/re-ex10-p2?rev=1737476039&do=diff Question 2 and 3, Review Exercise 10 Solutions of Question 2 and 3 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{2\sin \theta \sin 2\theta }{\cos \theta +\cos 3\theta }=\tan 2\theta \tan \theta $\begin{align}L.H.S.&=\dfrac{2\sin \theta \sin 2\theta }{\cos \theta +\cos 3\theta }\\ &=\dfrac{2\sin \theta \s… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/re-ex10-p3?rev=1737476039&do=diff Question 4 & 5, Review Exercise 10 Solutions of Question 4 & 5 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${{\sin }^{2}}\dfrac{\theta }{2}=\dfrac{\sin \theta \tan \dfrac{\theta }{2}}{2}$\begin{align}R.H.S.&=\dfrac{\sin \theta \tan \dfrac{\theta }{2}}{2}\\ &=\dfrac{\sin \theta \sin \dfrac{\theta }{2}}{2\cos \d… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/re-ex10-p4?rev=1737476039&do=diff Question 6 & 7, Review Exercise 10 Solutions of Question 6 & 7 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos 4\theta =1-8{{\sin }^{2}}\theta {{\cos }^{2}}\theta $\begin{align}L.H.S&=\cos 4\theta \\ &=\cos 2\left( 2\theta \right)\\ &=1-2\sin^2 2\theta \\ &=1-2{{\left( 2\sin\theta \cos \theta \right)}^{2}}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/re-ex10-p5?rev=1737476039&do=diff Question 8 & 9, Review Exercise 10 Solutions of Question 8 & 9 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin \left( \dfrac{\pi }{4}-\theta \right)\sin \left( \dfrac{\pi }{4}+\theta \right)=\dfrac{1}{2}\cos 2\theta $$2\sin \alpha \sin \beta =\cos \left( \alpha -\beta \right)-\cos \left( \alpha +\beta \r… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-1-p1?rev=1737476039&do=diff Question 1, Exercise 1.1 Solutions of Question 1 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 1(i) ${{i}^{31}}$\begin{align}{{i}^{31}}&=i\cdot{{i}^{30}}\\ &=i\cdot{{\left( {{i}^{2}} \right)}^{15}}\\ &=i\cdot{{\left( -1 \right)}^{15}} \quad \because i^2=-1\\ &=i\cdot(-1)\\ &=-i.\end{align}${{\left( -i \right)}^{6}}$\begin{align} {{\left… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-1-p2?rev=1737476039&do=diff Question 2, Exercise 1.1 Solutions of Question 2 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 2(i) $x+iy$$(3+2i)+(2+4i)$\begin{align}&(3+i2)+(2+i4)\\ =&(3+2)+(i2+i4)\\ =&5+i6\end{align}$x+iy$$(4+3i)-(2+5i)$\begin{align}&(4+3i)-(2+5i)\\ =&(4-2)+(3i-5i)\\ =&2-2i\end{align}$x+iy$$(4+7i)+(4-7i)$\begin{align} &(4+7i)+(4-7i)\\ =&(4+4)+(7i-7i… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-1-p3?rev=1737476039&do=diff Question 3, Exercise 1.1 Solutions of Question 3 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 3(i) $\dfrac{(2+i)(3-2i)}{1+i}$\begin{align}&\dfrac{(2+i)(3-2i)}{1+i}\\ =&\dfrac{6-2i^2+3i-4i}{1+i}\\ =&\dfrac{8-i}{1+i}\\ =&\dfrac{8-i}{1+i}\times \dfrac{1-i}{1-i}\\ =&\dfrac{8+i^2-8i-i}{1^2-i^2}\\ =&\dfrac{7-9i}{2}\\ =&\dfrac{7}{2}-\dfrac{9}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-1-p4?rev=1737476039&do=diff Question 4, Exercise 1.1 Solutions of Question 4 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 4(i) $x$$y$$(2+3i)x+(1+3i)y+2=0$\begin{align}&(2+3i)x+(1+3i)y+2=0\\ \implies &(2x+y+2)+(3x+3y)i=0.\end{align}\begin{align} 2x+y+2&=0 \quad \cdots(1)\\ 3x+3y&=0\quad \cdots (2) \end{align}\begin{align} &3x=-3y \\ x=-y \quad ... (3) \end{align}$… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-1-p5?rev=1737476039&do=diff Question 5, Exercise 1.1 Solutions of Question 5 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 5 $z$$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/ex1-1-p6?rev=1737476039&do=diff Question 6, Exercise 1.1 Solutions of Question 6 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 6(i) $4-3 i$$z=4-3 i$$\bar{z}=4+3i$$3 i+8$$2+\sqrt{\dfrac{-1}{5}}$\begin{align}z=&2+\sqrt{\dfrac{-1}{5}}\\ =&2+\sqrt{\dfrac{1}{5}}i,\end{align}$$\bar{z}=2-\sqrt{\dfrac{1}{5}}i$$$\dfrac{5 }{2}i-\dfrac{7}{8}$$z=\dfrac{5 }{2}i-\dfrac{7}{8},$$\bar… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-1-p7?rev=1737476039&do=diff Question 7, Exercise 1.1 Solutions of Question 7 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 7(i) $11+12 i$$$z=11+12i$$\begin{align}|z|&= \sqrt{(11)^2+(12)^2}\\ &=\sqrt{265}\end{align}$|11+12 i|=\sqrt{265}$$(2+3 i)-(2+6 i)$$z=(2+3i)−(2+6i)$\begin{align}z&=2+3i−2−6i\\ &=-3i \end{align}\begin{align} |z| &= \sqrt{0^2+(-3)^2} \\ &= \sqrt{… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p1?rev=1737476039&do=diff Question 1, Exercise 1.2 Solutions of Question 1 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 1(i) $\operatorname{Re}(i z)=-\operatorname{Im}(z)$$$z=x+iy$$\begin{align} iz&=i(x+iy)\\ &=ix-y\end{align}\begin{align}Re(iz)&=-y\\ \implies Re(iz)&=-Im(z)\end{align}$\operatorname{Im}(i z)=\operatorname{Re}(z)$$$z=x+iy$$\begin{align}iz&=i(x+i… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-2-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/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/ex1-2-p4?rev=1737476039&do=diff Question 4, Exercise 1.2 Solutions of Question 4 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 4 $z_{1}=2-3 i$$\left|z_{1} z_{2}\right|=16$$\left|z_{2}\right|$$$z_{1}=2-3i$$\begin{align}|z_1|&=\sqrt{(2)^2+(-3)^2}\\ &=\sqrt{13}\end{align}\begin{align}&|z_{1} z_{2}|=16\\ \Rightarrow \quad &|z_{1}|| z_{2}|=16\\ \Rightarrow \quad & \sqrt{13… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p5?rev=1737476039&do=diff Question 5, Exercise 1.2 Solutions of Question 5 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 5 $z_1$$z_2$$|z_1+z_2|^2-|z_1-z_2|^2=4Re(z_1)Re(z_2)$\begin{align}z_1&=x_1+iy_1 \text{ and } z_2&=x_2+iy_2\end{align}\begin{align}z_1+z_2&=x_1+iy_1+x_2+iy_2\\ &=x_1+x_2+i(y_1+y_2)\\ |z_1+z_2|^2&=(x_1+x_2)^2+(y_1+y_2)^2\\ &=x^2_1+x^2_2+2x_1x_… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p6?rev=1737476039&do=diff Question 6, Exercise 1.2 Solutions of Question 6 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 6 $\lambda$$\left|\dfrac{z_{1}}{z_{2}}+\lambda\right|=\sqrt{\lambda+2}$$z_{1}=3+i$$z_{2}=1+i$\begin{align} &z_{1}=3+i\text{ and } z_{2}=1+i.\end{align}\begin{align} \dfrac{z_1}{z_2} &= \dfrac{3+i}{1+i}\\ &=\dfrac{(3+i)(1-i)}{(1+i)(1-i)} \\ &=\… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p7?rev=1737476039&do=diff Question 7, Exercise 1.2 Solutions of Question 7 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 7 $\sqrt{2}|z| \geq|\operatorname{Re}(z)|+|\operatorname{Im}(z)| \quad$$\left(|x|-|y|)^{2} \geq 0\right)$\begin{align} &\left(|x|-|y|)^{2} \geq 0\right) \\ \implies & |x|^2+|y|^2-2|x||y| \geq 0 \\ \implies & |x|^2+|y|^2 \geq 2|x||y| \\ \implie… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p8?rev=1737476039&do=diff Question 8, Exercise 1.2 Solutions of Question 8 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 8(i) $|2 z-i|=4$$x$$y$$z=x+i y$$$|2z-i|=4.$$$z=x+i y$\begin{align} & |2(x+iy)-i|=4 \\ \implies & |2x+i(2y-1)|=4 \\ \implies & \sqrt{(2x)^2+(2y-1)^2}=4 \end{align}\begin{align} & (2x)^2+(2y-1)^2 = 16\\ \implies & 4x^2+4y^2-4y+1-16=0 \\ \implies… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p9?rev=1737476039&do=diff Question 9, Exercise 1.2 Solutions of Question 9 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 9(i) $(2+4 i)^{-1}$$z=2+4i$\begin{align} Re(2+4i)^{-1} & = Re(z^{-1}) = \dfrac{Re(z)}{|z|^2} \\ & =\dfrac{2}{2^2+4^2} = \dfrac{2}{20}\\ &= \dfrac{1}{10}. \end{align}\begin{align} Im(2+4i)^{-1} & = Im(z^{-1}) = -\dfrac{Im(z)}{|z|^2} \\ & =-\df… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p10?rev=1737476039&do=diff Question 10, Exercise 1.2 Solutions of Question 10 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 10(i) $z_{1}=-3+2 i$$$\left|z_{1}\right|=\left|-z_{1}\right|=\left|\overline{z_{!}}\right|=\left|-\overline{z_{!}}\right|.$$\begin{align} |z_1| &= \sqrt{(-3)^2 + (2)^2} \\ &= \sqrt{9 + 4} = \sqrt{13} \,\, -- (1) \end{align}\begin{align} -z_… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-3-p1?rev=1737476039&do=diff Question 1, Exercise 1.3 Solutions of Question 1 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 1(i) $z^{2}+169$\begin{align} & z^{2} + 169 \\ = & z^{2} - (13i)^2 \\ = &(z + 13i)(z - 13i). \end{align}$2 z^{2}+18$\begin{align} & 2z^2 + 18 \\ = &2(z^2 - (3i)^2)\\ = &2(z + 3i)(z - 3i) \end{align}$3 z^{2}+363$\begin{align} & 3z^2 + 363 \\ … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-3-p2?rev=1737476039&do=diff Question 2, Exercise 1.3 Solutions of Question 2 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 2(i) $z^{2}-6 z+2=0$\begin{align} & z^2 - 6z + 2 = 0 \\ \implies & z^2 - 2(3)(z)+9-9+2=0 \\ \implies & (z - 3)^2+7= 0 \\ \implies & (z - 3)^2 = 7. \end{align}\begin{align} &z - 3 = \pm \sqrt{7} \\ \implies &z = 3 \pm \sqrt{7}\end{align}$\{3 … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-3-p3?rev=1737476039&do=diff Question 3, Exercise 1.3 Solutions of Question 3 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 3(i) $\dfrac{1}{3} z^{2}+2 z-16=0$\begin{align}&\dfrac{1}{3}z^{2}+2 z-16=0\\ \implies &z^{2} + 6z - 48 = 0 \end{align}$$ z = \dfrac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a},$$$$a = 1,\quad b = 6,\quad \text{and}\quad c = -48.$$\begin{align} z& = \d… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-3-p4?rev=1737476039&do=diff Question 4, Exercise 1.3 Solutions of Question 4 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 4(i) $(1-i) z+(1+i) \omega=3 ; 2 z-(2+5 i) \omega=2+3 i$\begin{align} &(1-i) z+(1+i) \omega=3 \quad \cdots(1)\\ &2 z-(2+5 i) \omega=2+3i \quad\cdots(2) \end{align}$2$\begin{align} &(2-2i)z+(2+2i) \omega=6 \quad \cdots (3) \end{align}$(1-i)$\b… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p1?rev=1737476039&do=diff Question 1, Exercise 1.4 Solutions of Question 1 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 1(i) $2+i 2 \sqrt{3}$$z=x+iy=2 + i 2 \sqrt{3}$\begin{align} r & = \sqrt{x^2 + y^2} = \sqrt{2^2 + (2\sqrt{3})^2} \\ & = \sqrt{4 + 12} = \sqrt{16} = 4. \end{align}\begin{align} \alpha & = \tan^{-1}\left|\frac{y}{x}\right| = \tan^{-1}\left|\fra… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p2?rev=1737476039&do=diff Question 2, Exercise 1.4 Solutions of Question 2 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 2(i) $\left(\cos \dfrac{\pi}{6}+i \sin \dfrac{\pi}{6}\right)\left(\cos \dfrac{\pi}{3}+i \sin \dfrac{\pi}{3}\right)$$z_1=\cos \dfrac{\pi}{6}+i \sin \dfrac{\pi}{6}=e^{i\frac{\pi}{6}}$$z_2=\cos \dfrac{\pi}{3}+i \sin \dfrac{\pi}{3}=e^{i\frac{\pi}{… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p3?rev=1737476039&do=diff Question 3, Exercise 1.4 Solutions of Question 3 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 3(i) $\left(x_{1}+i y_{1}\right)\left(x_{2}+i y_{2}\right)\left(x_{3}+i y_{3}\right) \ldots\left(x_{n}+i y_{n}\right)=a+i b$$\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/unit01/ex1-4-p4?rev=1737476039&do=diff Question 4, Exercise 1.4 Solutions of Question 4 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 4 $\dfrac{1+z}{1-z}=\cos 2 \theta+i \sin 2 \theta$$z=i \tan \theta$\begin{align}&\dfrac{1+z}{1-z}=\cos 2 \theta+i \sin 2 \theta\\ \implies &\dfrac{1+z}{1-z}=e^{i2\theta}\\ \implies &(1+z)=(1-z)e^{i2\theta}\\ \implies &z+z e^{i2\theta}=e^{i2\th… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p5?rev=1737476039&do=diff Question 5, Exercise 1.4 Solutions of Question 5 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 5 $\cos \alpha+\cos \beta+\cos \gamma=\sin \alpha+\sin \beta+\sin \gamma=0$$\cos 3 \alpha+\cos 3 \beta+\cos 3 \gamma=3 \cos (\alpha+\beta+\gamma)$$\sin 3 \alpha+\sin 3 \beta+\sin 3 \gamma=3 \sin (\alpha+\beta+\gamma)$\begin{align} \cos \alpha … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p6?rev=1737476039&do=diff Question 6(i-ix), Exercise 1.4 Solutions of Question 6(i-ix) of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sqrt{2}\left(\cos 315^{\circ}+i \sin 315^{\circ}\right)$\begin{align} &\sqrt{2}\left(\cos 315^{\circ}+i \sin 315^{\circ}\right) \\ =& \sqrt{2} \left(\dfrac{1}{\sqrt{2}}-\dfrac{i}{\sqrt{2}} \right) \\ =& 1-i. \end{align}$5\left(\cos 210^{\ci… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p7?rev=1737476039&do=diff Question 6(x-xvii), Exercise 1.4 Solutions of Question 6(x-xvii) of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $7 \sqrt{2}\left(\cos \dfrac{5 \pi}{4}+i \sin \dfrac{5 \pi}{4}\right)$$10 \sqrt{2}\left(\cos \dfrac{7 \pi}{4}+i \sin \dfrac{7 \pi}{4}\right)$$2\left(\cos\dfrac{5\pi}{2}+i \sin \dfrac{5\pi}{2}\right)$$\dfrac{1}{\sqrt{2}}\left(\cos \dfrac{\… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-4-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}$$z=x+iy$\begin{align*} &\arg (z-1)=-\dfrac{\pi}{4} \\ \implies & \arg(x+iy-1) = -\dfrac{\pi}{4} \\ \implies & \arg(x-1+iy) = -\dfrac{\pi}{4} \\ \implies & \tan^{-1}\left(\dfrac{y}{x-1}\right) = -\dfrac{\pi}{4} … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p9?rev=1737476039&do=diff Question 8, Exercise 1.4 Solutions of Question 8 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 8(i) $0.004 \mathrm{~mm}$$\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/unit01/re-ex-p1?rev=1737476039&do=diff Question 1, Review Exercise Solutions of Question 1 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\operatorname{part}(\mathrm{s})$$z$$z$$(0,0)$$(1,0)$$(0,1)$$(1,1)$$z$$|z|$$1 / z$$-z$$\bar{z}$$x$$y$$x y$$z_{1}=3+2 i$$z_{2}=5+6 i$$z_{1}>z_{2}$$z_{1}<z_{2}$$\overline{z_{1}}=\overline{z_{2}}$$\overline{z_{1}}=-\overline{z_{2}}$$\mathrm{z}=3+4 i$$… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/re-ex-p2?rev=1737476039&do=diff Question 2, Review Exercise Solutions of Question 2 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $i^{2}+i^{4}+i^{6}+\cdots+i^{100}$\begin{align*} & i^{2}+i^{4}+i^{6}+\ldots+i^{100} \\ =& i^2 + (i^2)^2 + (i^2)^3 + (i^2)^4 + \ldots +(i^2)^{49} +(i^2)^{50} \\ =& -1 + (-1)^2 + (-1)^3 + (-1)^4 + \ldots + (-1)^{49}+(-1)^{50} \\ =& -1+1-1+1- \ldots -… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/re-ex-p3?rev=1737476039&do=diff Question 3, Review Exercise Solutions of Question 3 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $3 x^{2}+108$\begin{align*} & 3 x^{2}+108\\ =&3 (x^{2}+36)\\ =&3 (x^{2}-(6i)^2)\\ =&3 (x+6i)(x-6i) \end{align*}$4 x^{2}+40$\begin{align*} &4 x^{2}+40\\ =&4 (x^{2}+10)\\ =&4 (x^{2}+(\sqrt{10}i)^2)\\ =&4 (x+\sqrt{10}i)(x-\sqrt{10}i) \end{align*} text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/re-ex-p4?rev=1737476039&do=diff Question 4, Review Exercise Solutions of Question 4 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $z=x+i y$$\left|\dfrac{z+2 i}{z-2 i}\right|=1$$z = x + iy$\begin{align*} & \left|\dfrac{z + 2i}{z - 2i}\right| = 1\\ \implies & |z + 2i| = |z - 2i|\\ \implies & |x + i(y + 2)| = |x + i(y - 2)|\\ \implies & \sqrt{x^2 + (y + 2)^2} = \sqrt{x^2 + (y -… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/re-ex-p5?rev=1737476039&do=diff Question 5, Review Exercise Solutions of Question 5 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $z$$(z-3 i)(2+5 i)=3-4 i$$z$$(z-3 i)(2+5 i)=3-4 i$\begin{align*} &(z-3 i)(2+5 i)=3-4 i \\ \implies & z-3 i=\dfrac{3-4 i}{2+5 i} \\ \implies & z-3 i=\dfrac{(3-4 i)(2-5i)}{(2+5 i)(2-5i)}\\ \implies & z-3 i=\dfrac{6-20-15i-8i}{4+25}\\ \implies & z-3 i… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/re-ex-p6?rev=1737476039&do=diff Question 6, Review Exercise Solutions of Question 6 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left[\dfrac{1}{i^{10}}+(2-i)^{2}+\sqrt{-25}\right]^{3}$\begin{align*} &\left[\dfrac{1}{i^{10}} + (2 - i)^2 + \sqrt{-25}\right]^3\\ =&\left[\dfrac{1}{(i^2)^5} + ( 4 - 4i + i^2) + 5i \right]^3\\ =&\left[\dfrac{1}{(-1)^5} + ( 4 - 4i -1) + 5i \right]… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/re-ex-p7?rev=1737476039&do=diff Question 7, Review Exercise Solutions of Question 7 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 z^{2}-11 z+16=0$\begin{align*} &2 z^{2}-11 z+16=0\\ \implies&z^2 - \dfrac{11}{2}z + 8 = 0\\ \implies& z^2 - \dfrac{11}{2}z = -8\\ \implies& z^2 - 2z\dfrac{11}{4}z + \dfrac{121}{16} = -8 + \dfrac{121}{16}\\ \implies&\left(z-\dfrac{11}{4}\right)^2… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/re-ex-p8?rev=1737476039&do=diff Question 8, Review Exercise Solutions of Question 8 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sqrt{2}+i \sqrt{2}$$\theta=45^{\circ}$$$x= \sqrt{2} + i \sqrt{2}, \quad \theta=\dfrac{\pi}{4}.$$$x_{\max}$\begin{align} &x=x_{\max} e^{i\theta} \\ \implies & \sqrt{2} + i \sqrt{2}=x_{\max} e^{i\dfrac{\pi}{4}} \\ \implies & x_{\max} \left(\cos\dfr… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-1-p1?rev=1737476039&do=diff Question 1, Exercise 2.1 Solutions of Question 1 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{lll}1 & 3 & 0 \\ 2 & 0 & 1\end{array}\right]$\begin{align}\text{Order of A}&= 2\times 3\end{align}$B=\left[\begin{array}{ll}1 & 2 \\ 2 & 3 \\ 3 & 4\end{array}\right]$\begin{align}\text{Order of B}&= 3\times 2\end{align}$C… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-1-p2?rev=1737476039&do=diff Question 2, Exercise 2.1 Solutions of Question 2 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\quad A=\left[\begin{array}{lll}3 & 6 & 2 \\ 2 & 1 & 9\end{array}\right]$$B=\left[\begin{array}{ll}\frac{1}{3} & 1 \\ 2 & 6\end{array}\right]$$C=\left[\begin{array}{l}3 \\ 2 \\ 8\end{array}\right]$$D=\left[\begin{array}{lll}1 & 6 & 9 \\ 2 & 0 … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-1-p3?rev=1737476039&do=diff Question 3, Exercise 2.1 Solutions of Question 3 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$A=\begin{bmatrix} 3 & 0 & 0 \\ 0 & 1 & 0 \\ 2 & 6 & 0 \end{bmatrix}$$$$B=\begin{bmatrix} -6 & 0 & 0 \\ 0 & -6 & 0 \\ 0 & 0 & -6 \end{bmatrix}$$$$C=\begin{bmatrix} 1 & 0 \\ 2 & 0 \end{bmatrix}$$$$D=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 &… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-1-p4?rev=1737476039&do=diff Question 4, Exercise 2.1 Solutions of Question 4 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$ A=\left[\begin{array}{ccc} 2 & 0 \\ \sqrt{5} & 6 \\ 1 & 9 \end{array}\right]$$$$ A^t=\begin{bmatrix} 2 & \sqrt{5} & 1 \\ 0 & 6 & 9 \end{bmatrix}$$$$B=\left[\begin{array}{cccc} 1 & 6 & 2 & 0 \end{array}\right] $$$$B^t=\left[\begin{array}{c} 1… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p1?rev=1737476039&do=diff Question 1, Exercise 2.2 Solutions of Question 1 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[a_{i j}\right]$$2 \times 2$$a_{i j}=\dfrac{i+3 j}{2}$\( a_{ij} = \dfrac{i + 3j}{2} \)\( i = 1, j = 1 \)\[ a_{11} = \dfrac{1 + 3 \cdot 1}{2} = \dfrac{1 + 3}{2} = \dfrac{4}{2} = 2 \]\( i = 1, j = 2 \)\[ a_{12} = \dfrac{1 + 3 \cdot 2}{2} … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p2?rev=1737476039&do=diff Question 3, Exercise 2.2 Solutions of Question 3 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{ccc}3 & -1 & 2 \\ 0 & 6 & 1 \\ -1 & 0 & -3\end{array}\right]$$B=\left[\begin{array}{ccc}2 & 1 & 7 \\ 0 & 2 & -1 \\ -3 & 4 & 2\end{array}\right]$$C$$A+B+C=0$$$A+B+C=0,$$$$C=-A-B.$$\begin{align*} C&=-\begin{bmatrix}3 & -1 &… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p3?rev=1737476039&do=diff Question 3, Exercise 2.2 Solutions of Question 3 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{ccc}3 & -1 & 2 \\ 0 & 6 & 1 \\ -1 & 0 & -3\end{array}\right]$$B=\left[\begin{array}{ccc}2 & 1 & 7 \\ 0 & 2 & -1 \\ -3 & 4 & 2\end{array}\right]$$C$$A+B+C=0$$$A+B+C=0,$$$$C=-A-B.$$\begin{align*} C&=-\begin{bmatrix}3 & -1 &… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p4?rev=1737476039&do=diff Question 4, Exercise 2.2 Solutions of Question 4 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A$\begin{align}\left[\begin{array}{cc} 2 & 1 \\ 3 & 2 \end{array}\right]A\left[\begin{array}{cc} 1 & 3 \\ 2 & 4 \end{array}\right]&=\left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right]\end{align}$ B = \left[\begin{array}{cc} 2 & 1 \\ 3… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p5?rev=1737476039&do=diff Question 5, Exercise 2.2 Solutions of Question 5 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $X=\left[\begin{array}{lll}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{array}\right]$$X^{2}-4 X-5 I=0$\begin{align}L.H.S. & =X^{2}-4 X-5 I \\ &=\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p6?rev=1737476039&do=diff Question 6, Exercise 2.2 Solutions of Question 6 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{cc}2 & 1 \\ 3 & -3\end{array}\right]$$\alpha$$\beta$$A^{2}+\alpha I=\beta A$\begin{align} & A^{2}+\alpha I=\beta A\\ \implies &\begin{bmatrix} 2 & 1 \\ 3 & -3 \end{bmatrix} \begin{bmatrix} 2 & 1 \\ 3 & -3 \end{bmatrix}+\a… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p7?rev=1737476039&do=diff Question 7, Exercise 2.2 Solutions of Question 7 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{ll}x & 0 \\ y & 1\end{array}\right]$$n, A^{n}=\left[\begin{array}{cc}x^{n} & 0 \\ \dfrac{y\left(x^{n}-1\right)}{x-1} & 1\end{array}\right]$$$A = \begin{bmatrix} x & 0 \\ y & 1 \end{bmatrix}.$$$n = 1$\begin{align}A^1 =\beg… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p8?rev=1737476039&do=diff Question 8, Exercise 2.2 Solutions of Question 8 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A$$B$$2 \times 3$$3 \times 2$$(A B)^{t}=B^{t} A^{t}$\( A \)\( B \)\( 2 \times 3 \)\( 3 \times 2 \)\begin{align*} A &= \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix}\\ B &= \begin{bmatrix} b_{11} & b_{12}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p9?rev=1737476039&do=diff Question 9, Exercise 2.2 Solutions of Question 9 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A$$B$$3 \times 3$$(A+B)^{t}=A^{t}+B^{t}$\begin{align*} A &= \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} \\ B &= \begin{pmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p10?rev=1737476039&do=diff Question 10, Exercise 2.2 Solutions of Question 10 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A$$B$$A B=B$$B A=A$$A^{2}+B^{2}$$$AB = B$$$$BA = A$$\begin{align*} A^2 &= AA\\ & = A(BA)\\ &=(AB)A\\ &=BA\\ &=A \end{align*}\begin{align*} B^2&= BB \\ &=B(AB)\\ & = (BA)B\\ &=AB\\ &=B\end{align*}$$A^2 + B^2 = A + B$$$AB = B$$BA = A$$$A^2 + B… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p11?rev=1737476039&do=diff Question 11, Exercise 2.2 Solutions of Question 11 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[a_{i j}\right]$$3 \times 3$$a_{i j}=i^{2}-j^{2}$$A$$A=\left[a_{i j}\right]$$a_{ij}=a+{ji}$$a_{ij}=-a_{ji}$$a_{i j}=i^{2}-j^{2}$\begin{align} a_{ji} & = j^2 -i^2 \\ &= - (i^2 -j^2) \\ &= - a_{ij} \end{align}$a_{ij}=-a_{ji}$ text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p12?rev=1737476039&do=diff Question 12, Exercise 2.2 Solutions of Question 12 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A$$\left(A^{n}\right)^{t}=\left(A^{t}\right)^{n}$ text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-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-p1?rev=1737476039&do=diff Question 1, Exercise 2.3 Solutions of Question 1 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left[\begin{array}{ccc}2 & 3 & 1 \\ 1 & -1 & 2 \\ 4 & 1 & 2\end{array}\right]$\begin{align*} A &= \left[\begin{array}{ccc}2 & 3 & 1 \\ 1 & -1 & 2 \\ 4 & 1 & 2\end{array}\right]\\ |A|&=2(-2-2)-3(2-8)+1(1+4)\\ \implies |A|&=-8+18+5\\ \implies |… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-3-p2?rev=1737476039&do=diff Question 2, Exercise 2.3 Solutions of Question 2 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left[\begin{array}{lll}3 & 2 & 3 \\ 4 & 5 & 1 \\ 2 & 1 & 0\end{array}\right]$\(R_1\)\(a_{11} = 3\)\(a_{12} = 2\)\(a_{13} = 3\)\begin{align*} A &= \left[\begin{array}{ccc} 3 & 2 & 3 \\ 4 & 5 & 1 \\ 2 & 1 & 0 \end{array}\right]\\ & A_{11} = (-1… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-3-p3?rev=1737476039&do=diff Question 3, Exercise 2.3 Solutions of Question 3 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left[\begin{array}{ccc}3 & 1 & 2 \\ 2 & 3 & 1 \\ -4 & 1 & -3\end{array}\right]$\begin{align*} A &= \left[\begin{array}{ccc} 3 & 1 & 2 \\ 2 & 3 & 1 \\ -4 & 1 & -3\end{array}\right]\end{align*}\(3 \times 3\)\begin{align*} |A| &= 3(3 \cdot (-3) … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-3-p4?rev=1737476039&do=diff Question 4, Exercise 2.3 Solutions of Question 4 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\lambda$$\left[\begin{array}{lll}\lambda & 1 & 3 \\ 2 & 1 & 8 \\ 0 & 3 & 1\end{array}\right]$\begin{align*} A &= \left[\begin{array}{ccc} \lambda & 1 & 3 \\ 2 & 1 & 8 \\ 0 & 3 & 1 \end{array}\right]\\ |A| &= \lambda \cdot (-23) - 1 \cdot 2 + 3… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-3-p5?rev=1737476039&do=diff Question 5, Exercise 2.3 Solutions of Question 5 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left[\begin{array}{ccc}1 & -1 & 1 \\ 2 & 1 & -1 \\ 1 & -2 & -1\end{array}\right]$\begin{align*} A &= \left[\begin{array}{ccc} 1 & -1 & 1 \\ 2 & 1 & -1 \\ 1 & -2 & -1 \end{array}\right]\\ |A|&= 1 [-1 - 2] + 1 [-2 + 1] + 1 [-4 - 1] \\ &= 1 \cd… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-3-p6?rev=1737476039&do=diff Question 6, Exercise 2.3 Solutions of Question 6 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{ccc}2 & 1 & -3 \\ 0 & 1 & 0 \\ 2 & 1 & 6\end{array}\right]$$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/ex2-3-p7?rev=1737476039&do=diff Question 7, Exercise 2.3 Solutions of Question 7 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $(A B)^{-1}=B^{-1} A^{-1}$$A=\left[\begin{array}{ll}2 & 1 \\ 8 & 6\end{array}\right]$$B=\left[\begin{array}{ll}3 & 2 \\ 0 & 2\end{array}\right]$\begin{align*} A &= \left[\begin{array}{ll}2 & 1 \\ 8 & 6\end{array}\right] \\ |A|& = 12 - 8 = 4\\ … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-5-p1?rev=1737476039&do=diff Question 1, Exercise 2.5 Solutions of Question 1 of Exercise 2.5 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left[\begin{array}{ccc}1 & 3 & 5 \\ -6 & 8 & 3 \\ -4 & 6 & 5\end{array}\right]$\begin{align*} & \quad \left[\begin{array}{ccc}1 & 3 & 5 \\ -6 & 8 & 3 \\ -4 & 6 & 5\end{array}\right]\\ \sim & \text{R} \left[\begin{array}{ccc} 1 & 3 & 5 \\ 0 & … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-5-p2?rev=1737476039&do=diff Question 2, Exercise 2.5 Solutions of Question 2 of Exercise 2.5 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left[\begin{array}{ccc}5 & 9 & 3 \\ 3 & -5 & 6 \\ 2 & 10 & 6\end{array}\right]$\begin{align*}&\quad\left[ \begin{array}{ccc} 5 & 9 & 3 \\ 3 & -5 & 6 \\ 2 & 10 & 6 \end{array} \right]\\ \sim & \text{R}\left[ \begin{array}{ccc} 1 & \frac{9}{… text/html 2025-01-21T16:13:59+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]$$A A^{-1}=A^{-1} A=I$\begin{align*} A&=\left[ \begin{array}{ccc} 0 & -1 & -1 \\ -1 & 3 & 0 \\ 1 & -1 & 4 \end{array} \right]\\ |A|&=0+1(-4)-1(1-3)\\ &=-4+3\… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p1?rev=1737476039&do=diff Question 1, Exercise 2.6 Solutions of Question 1 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $ 2 x_{1}-3 x_{2}+4 x_{3}=0$$x_{1}-2 x_{2}+3 x_{3}=0$$4 x_{1}+x_{2}-6 x_{3}=0$\begin{align*} &2 x_{1}-3 x_{2}+4 x_{3}=0\cdots (i)\\ &x_{1}-2 x_{2}+3 x_{3}=0\cdots (ii)\\ &4 x_{1}+x_{2}-6 x_{3}=0\cdots (iii)\\ \end{align*}\begin{align*} A &= \le… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p2?rev=1737476039&do=diff Question 2, Exercise 2.6 Solutions of Question 2 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\lambda$$\lambda$$2 x_{1}-\lambda x_{2}+x_{3}=0$$2 x_{1}+3 x_{2}-x_{3}=0$$3 x_{1}-2 x_{2}+4 x_{3}=0$\begin{align*} &2 x_{1}-\lambda x_{2}+x_{3}=0 \cdots(i)\\ &2 x_{1}+3 x_{2}-x_{3}=0\cdots(ii)\\ &3 x_{1}-2 x_{2}+4 x_{3}=0\cdots(iii)\\ \end{ali… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p3?rev=1737476039&do=diff Question 3, Exercise 2.6 Solutions of Question 3 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 x+3 y+4 z=2$$2 x+y+z=5$$3 x-2 y+z=-3$\begin{align*} \begin{aligned} 2x + 3y + 4z &= 2 \\ 2x + y + z &= 5 \\ 3x - 2y + z &= -3 \end{aligned}\end{align*}\begin{align*} A_{b} &=\quad \left[\begin{array}{cccc} 2 & 3 & 4 & 2 \\ 2 & 1 & 1 & 5 \\ 3… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p4?rev=1737476039&do=diff Question 4, Exercise 2.6 Solutions of Question 4 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 x_{1}-x_{2}-x_{3}=2$$3 x_{1}-4 x_{2}+3 x_{3}=7$$4 x_{1}+2 x_{2}-5 x_{3}=10$\begin{align*} 2x_1 - x_2 - x_3 &= 2, \\ 3x_1 - 4x_2 + 3x_3 &= 7, \\ 4x_1 + 2x_2 - 5x_3 &= 10, \end{align*}\begin{align*} A_b &= \begin{bmatrix} 2 & -1 & -1 & : & 2 … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p5?rev=1737476039&do=diff Question 5, Exercise 2.6 Solutions of Question 5 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $x_{1}+x_{2}+2 x_{3}=8$$-x_{1}-2 x_{2}+3 x_{3}=1$$3 x_{1}-7 x_{2}+4 x_{3}=10$$A X=B$\begin{align*} &A = \begin{bmatrix} 1 & 1 & 2 \\ -1 & -2 & 3 \\ 3 & -7 & 4 \end{bmatrix}, \quad X = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}, \quad B = \… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/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/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/unit02/ex2-6-p8?rev=1737476039&do=diff Question 9 and 10, Exercise 2.6 Solutions of Question 9 and 10 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 x-y+3 z=\alpha ; 3 x+y-5 z=\beta ;-5 x-5 y+21 z=\gamma$$\gamma \neq 2 \alpha-3 \beta$$2$$2$$3$$3$ text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/re-ex-p1?rev=1737476039&do=diff Question 1, Review Exercise Solutions of Question 1 of Review Exercise of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A$$m \times n$$B$$n \times p$$A B$$n \times p$$m \times p$$p \times m$$n \times n$$A$$1 \times n$$A^{t} A$$1 \times n$$n \times 1$$1 \times 1$$n \times n$$a_{i j}$$A$$a_{i j}=(-1)^{i+j} A_{i j}$$a_{i j}=(-1)^{i+j} M_{i j}$$\frac{A_{i j}}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/re-ex-p2?rev=1737476039&do=diff Question 2 and 3, Review Exercise Solutions of Question 2 and 3 of Review Exercise of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{ccc}1 & 2 & 0 \\ -3 & 4 & 9 \\ 2 & 1 & 6\end{array}\right]$$A_{13}, A_{23}$$A_{33}$$|A|$\begin{align*} A&=\left[\begin{array}{ccc}1 & 2 & 0 \\ -3 & 4 & 9 \\ 2 & 1 & 6\end{array}\right]\\ A_{13} &= (-1)^{… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/re-ex-p3?rev=1737476039&do=diff Question 4 and 5, Review Exercise Solutions of Question 4 and 5 of Review Exercise of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left|\begin{array}{ccc}a+1 & l & l \\ l & a+1 & l \\ l & l & a+1\end{array}\right|=(a+1+2 l)(a+1-l)^{2}$\begin{align*} L.H.S &= \left|\begin{array}{ccc}a+1 & l & l \\ l & a+1 & l \\ l & l & a+1\end{array}\right|\\ &=\left|\b… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p1?rev=1737476039&do=diff Question 1 and 2, Exercise 4.1 Solutions of Question 1 and 2 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$a_{10}$$a_{15}$$$a_{n}=3 n+1$$$$a_{n}=3 n+1$$\begin{align*} a_1 &= 3(1) + 1 = 3 + 1 = 4\\ a_2 &= 3(2) + 1 = 6 + 1 = 7\\ a_3 &= 3(3) + 1 = 9 + 1 = 10\\ a_4 &= 3(4) + 1 = 12 + 1 = 13\\ \end{align*}\begin{align*} a_{10} &= 3(10) + 1 = 30… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p2?rev=1737476039&do=diff Question 3 and 4, Exercise 4.1 Solutions of Question 3 and 4 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$a_{10}$$a_{n}=\frac{n}{n+1}$$$a_n = \frac{n}{n+1}.$$\begin{align*} a_1 &= \frac{1}{1+1} = \frac{1}{2}\\ a_2 &= \frac{2}{2+1} = \frac{2}{3}\\ a_3 &= \frac{3}{3+1} = \frac{3}{4}\\ a_4 &= \frac{4}{4+1} = \frac{4}{5}\\ \end{align*}\begin… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p3?rev=1737476039&do=diff Question 5 and 6, Exercise 4.1 Solutions of Question 5 and 6 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$a_{10}$$a_{15}$$a_{n}=n^{2}-2 n$$$a_n = n^2 - 2n.$$\begin{align*} a_1 &= (1)^2 - 2(1) = 1 - 2 = -1\\ a_2 &= (2)^2 - 2(2) = 4 - 4 = 0\\ a_3 &= (3)^2 - 2(3) = 9 - 6 = 3\\ a_4 &= (4)^2 - 2(4) = 16 - 8 = 8\\ \end{align*}\begin{align*} a_{… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p4?rev=1737476039&do=diff Question 7 and 8, Exercise 4.1 Solutions of Question 7 and 8 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$a_{10}$$a_{15}$$a_{n}=\left(\frac{-1}{2}\right)^{n-1}$$$a_n = \left( \frac{-1}{2} \right)^{n-1}.$$\begin{align*}a_1 &= \left( \frac{-1}{2} \right)^{1-1} = \left( \frac{-1}{2} \right)^0 = 1 \\ a_2 &= \left( \frac{-1}{2} \right)^{2-1} =… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p5?rev=1737476039&do=diff Question 9 and 10, Exercise 4.1 Solutions of Question 9 and 10 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$a_{10}$$a_{15}$$a_{n}=(-1)^{n}(n+3)$$n$$a_{10}$$a_{15}$$$a_{n}=(-1)^{n+1}(3 n-5).$$$$a_n = (-1)^{n+1}(3n - 5).$$\begin{align*} a_1 &= (-1)^{1+1}(3(1) - 5) = (1)(3 - 5) = -2 \\ a_2 &= (-1)^{2+1}(3(2) - 5) = (-1)(6 - 5) = -1 \\ a_3 &=… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p6?rev=1737476039&do=diff Question 11 and 12, Exercise 4.1 Solutions of Question 11 and 12 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{n}=4 n-3; a_8$$$a_n = 4n - 3.$$\begin{align*} a_8 &= 4(8) - 3 \\ &= 32 - 3 \\ &= 29 \end{align*}$a_8 = 29$$a_{n}=5 n+11 ; a_{9}$ text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p7?rev=1737476039&do=diff Question 13 and 14, Exercise 4.1 Solutions of Question 13 and 14 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{n}=(3 n+4)(2 n-5) ; a_{7}$$a_{n}=(-1)^{n-1}(3.4 n-17.3) ; a_{12}$$$a_n = (-1)^{n-1}(3.4n - 17.3).$$\begin{align*} a_{12} &= (-1)^{12-1}(3.4 \cdot 12 - 17.3) \\ &= (-1)^{11}(40.8 - 17.3) \\ &= (-1)^{11}(23.5) \\ &= -23.5 \end{align… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p8?rev=1737476039&do=diff Question 15 and 16, Exercise 4.1 Solutions of Question 15 and 16 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{n}=4 n^{2}(11 n+31) ; a_{22}$$$a_n = 4n^2(11n + 31).$$\begin{align*} a_{22} &= 4 \cdot 22^2 \cdot (11 \cdot 22 + 31) \\ &= 4 \cdot 484 \cdot (242 + 31) \\ &= 4 \cdot 484 \cdot 273 \\ &= 4 \cdot 132132 \\ &= 528528 \end{align*}$a_{… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p9?rev=1737476039&do=diff Question 17 and 18, Exercise 4.1 Solutions of Question 17 and 18 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{n}=\log 10^{n} ; a_{43}$$$a_n = \log 10^n.$$\begin{align*} a_{43} &= \log 10^{43} \\ &= 43 \cdot \log 10 \\ &= 43 \cdot 1 \\ &= 43 \end{align*}$a_{43}= 43$$a_{n}=\ln e^{n} ; a_{67}$$$a_n = \ln e^n.$$\begin{align*} a_{67} &= \ln e^… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p10?rev=1737476039&do=diff Question 19 and 20, Exercise 4.1 Solutions of Question 19 and 20 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{n}$$1,3,5,7,9, \ldots$$$1, 3, 5, 7, 9, \ldots$$$a_1=1$$d=3-1=2$$$a_n = a_1 + (n - 1) d$$\begin{align*} \implies a_n &= 1 + (n - 1) \cdot 2\\ &= 1 + 2n - 2\\ &= 2n - 1 \end{align*}$a_n = 2n - 1$$a_{n}$$3,9,27,81,243, \ldots$\begin… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p11?rev=1737476039&do=diff Question 21 and 22, Exercise 4.1 Solutions of Question 21 and 22 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{n}$$\sqrt{2}, \sqrt{4}, \sqrt{6}, \sqrt{8}, \sqrt{10}, \ldots$$$\sqrt{2}, \sqrt{4}, \sqrt{6}, \sqrt{8}, \sqrt{10}, \ldots$$\begin{align*} &a_1=\sqrt{2 \cdot 1}, \\ &a_2=\sqrt{4}=\sqrt{2 \cdot 2} \\ &a_3=\sqrt{6}=\sqrt{2 \cdot 3}\\… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p1?rev=1737476039&do=diff Question 1, Exercise 4.2 Solutions of Question 1 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=4, d=3$$a_1= 4$$d=3$$$a_n = a_1 + (n - 1)d.$$\begin{align*} a_2&=4+(2-1)3=4+3=7\\ a_3 &= 4+ (3-1) 3 = 4 + 6 = 10\\ a_4&=4+(4-1)3=4+9=13 \end{align*}$a_1=4$$a_2=7$$a_3=10$$a_4=13$$a_1=7$$d=5$$a_1= 7$$d=5$$$a_n = a_1 + (n - 1)d.$$\begin{align*} … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p2?rev=1737476039&do=diff Question 2, Exercise 4.2 Solutions of Question 2 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $5,9,13, \ldots$$$5, 9, 13, \ldots $$$a_1=5$$d=9-5=4$$$a_n=a_1+(n-1)d.$$\begin{align*} a_4 &=5+(4-1)(4)=5+12=17\\ a_5 &=5+(5-1)(4)=5+16=21\\ a_6 &=5+(6-1)(4)=5+20=25 \end{align*}$17$$21$$25$$11,14,17, \ldots$$$11, 14, 17, \ldots$$$a_1=11$$d=14-11=3$$… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p3?rev=1737476039&do=diff Question 3 and 4, Exercise 4.2 Solutions of Question 3 and 4 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $0.07,0.12,0.7, \ldots$$$0.07,0.12,0.7, \ldots$$$a_1 = 0.07$$d=0.05$$a_{11}=?$\begin{align*} a_n&=a_1+(n-1)d \\ \implies a_{11}&= 0.07+(11-1)(0.05)\\ &=0.07+(10)(0.05)\\ &=0.57 \end{align*}$a_{11}=0.57.$$a_3 = 14$$a_9 = -1$$$a_n = a_1 + (… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p4?rev=1737476039&do=diff Question 5 and 6, Exercise 4.2 Solutions of Question 5 and 6 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{17}=-40$$a_{28}=-73$$a_{1}$$d$$$a_n=a_1+(n-1)d$$\begin{align*} & a_{17} = -40 \\ \implies &a_1 + 16d = -40 \quad \cdots (1) \end{align*}\begin{align*} &a_{28}=-73\\ \implies &a_1 + 27d = -73 \quad \cdots (2) \end{align*}\begin{align*}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p5?rev=1737476039&do=diff Question 7 and 8, Exercise 4.2 Solutions of Question 7 and 8 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $-6,-2,2, \ldots$$70$$-6,-2,2, \ldots$$a_1=-6$$d=-2+6=4$$a_n=70$$n=?$$$a_n=a_1+(n-1)d.$$\begin{align*} &70=-6+(n-1)4\\ \implies &70=-6+4n-4\\ \implies &70=4n-10\\ \implies &4n=80\\ \implies & n=20 \end{align*}$a_{20}=70$$\dfrac{5}{2}, \df… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p6?rev=1737476039&do=diff Question 9 and 10, Exercise 4.2 Solutions of Question 9 and 10 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\dfrac{1}{a}, b, \dfrac{1}{c}$$\dfrac{a-c}{2 a c}$$\dfrac{1}{a}, b, \dfrac{1}{c}$\begin{align*} d&=b-\frac{1}{a}\cdots (i)\\ \end{align*}\begin{align*} d&=\frac{1}{c}-b \cdots (ii) \end{align*}\begin{align*} b-\frac{1}{a}&=\frac{1}{c}-… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p7?rev=1737476039&do=diff Question 11 and 12, Exercise 4.2 Solutions of Question 11 and 12 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $1000$$3000$$2$$5000$$3$$20$$$1000, 3000, 5000, \dots, \text{ upto 20 terms}.$$$a_1 = 1000$$d=3000-1000=2000$$S_20=?$$$S_n =\frac{n}{2}[2a_1+(n-1)d],$$\begin{align*} S_{20} &= \frac{20}{2}[2(1000)+(20-1)2000]\\ &= 10 [2000+(19)2000] \… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p8?rev=1737476039&do=diff Question 13, Exercise 4.2 Solutions of Question 13 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $7$$17$$a=7$$b=17$\begin{align*} \text{A.M.} &= \frac{a + b}{2}\\ &= \frac{7 + 17}{2} \\ &= \frac{24}{2} = 12. \end{align*}$12$$3+3 \sqrt{2}$$7-3 \sqrt{2}$$a=3+3\sqrt{2}$$b=7-3\sqrt{2}$\begin{align*} \text{A.M.} &= \frac{a + b}{2}\\ &= \frac{(3 + 3… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p9?rev=1737476039&do=diff Question 14 and 15, Exercise 4.2 Solutions of Question 14 and 15 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $b$$10$$b$$20$$a= b$$b=20$\begin{align*} &\text{A.M.} = \frac{a + b}{2} \\ \implies & 10 = \frac{b + 20}{2} \\ \implies & 20 = b + 20 \\ \implies & b = 20 - 20 \\ \implies & b = 0 \end{align*}$b = 0$$b$$25$$b$$20$$b$$10$$b$$-10$$x$$y$… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p10?rev=1737476039&do=diff Question 16 and 17, Exercise 4.2 Solutions of Question 16 and 17 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $5$$17$$A_1$$A_2$$5$$17$$5$$A_1$$A_2$$17$$a_1=5$$a_4=17$$$a_n=a_1+(n-1)d.$$\begin{align*} &a_4 = a_1 + 3d \\ \implies & 17=5+3d\\ \implies & 3d=12\\ \implies & \boxed{d=4}.\end{align*}\begin{align*} A_1 &= a_2= a_1+d \\ &=5+4=9 \end{a… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p1?rev=1737476039&do=diff Question 1 and 2, Exercise 4.3 Solutions of Question 1 and 2 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $4+7+10+13+16+19+22+25$$4+7+10+13+16+19+22+25$$a_1=4$$d=7-4=3$$n=8$\begin{align} S_n&=\frac{n}{2}[2a_1+(n-1)d]\\ \implies S_8&=\frac{8}{2}[2(4)+(8-1)(3)]\\ &=4[8+7\times 3] = 116 \end{align}$a_{1}=2$$a_{n}=200$$n=100$$a_{1}=2$$a_{n}=200$$… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p2?rev=1737476039&do=diff Question 3 and 4, Exercise 4.3 Solutions of Question 3 and 4 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=5$$a_{n}=100$$n=200$$a_{1}=5$$a_{n}=100$$n=200$$a_{1}=5$$a_{n}=100$$n=200$$S_n$\begin{align} S_n&=\frac{n}{2}[a_1+a_n] \\ \implies S_{200}&=\frac{200}{2}[5+100]\\ &=10500. \end{align}$S_{200}=10500$$a_{1}=4$$n=15$$d=3$$a_{1}=4$$n=1… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p3?rev=1737476039&do=diff Question 5 and 6, Exercise 4.3 Solutions of Question 5 and 6 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=50$$n=20$$d=-4$$a_{1}=50$$n=20$$d=-4$$a_{1}=50$$n=20$$d=-4$$S_n$\begin{align} S_n&=\frac{n}{2}[2a_1+(n-1)d] \\ \implies S_{20}&=\frac{20}{2}[2(50)+(20-1)(-4)]\\ &=10\times [100-76]\\ &=240. \end{align}$S_{20}=240$$-3+(-7)+(-11)+\cd… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p4?rev=1737476039&do=diff Question 7 and 8, Exercise 4.3 Solutions of Question 7 and 8 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $9+11+13+15+\cdots$$n=12$$a_1=9$$d=11-9=2$$n=12$$S_n$\begin{align} S_n&=\frac{n}{2}[2a_1+(n-1)d] \\ \implies S_{12}&=\frac{12}{2}[2(9)+(12-1)(2)]\\ &=6\times [18+22]\\ &=240. \end{align}$S_{12}=240$$2$$100$$2$$100$$$2+4+6+...+100 (50 \tex… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p5?rev=1737476039&do=diff Question 9 and 10, Exercise 4.3 Solutions of Question 9 and 10 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $1$$99$$1$$99$$$1+3+5+...+99 (50 \text{ terms}).$$$a_{1}=1$$n=50$$d=3-1=2$$S_n$\begin{align} S_n&=\frac{n}{2}[2a_1+(n-1)d] \\ \implies S_{50}&=\frac{50}{2}[2(1)+(50-1)(2)]\\ &=25\times [2+98]\\ &=2500. \end{align}$1$$99$$2500$$14$$523$$… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p6?rev=1737476039&do=diff Question 11 and 12, Exercise 4.3 Solutions of Question 11 and 12 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $S_{\boldsymbol{n}}$$a_{1}=3$$a_{n}=-38$$n=8$$a_{1}=3$$a_{n}=-38$$n=8$\begin{align} S_n&=\frac{n}{2}[a_1+a_n] \\ \implies S_{8}&=\frac{8}{2}[3-38]\\ &=4\times[-35] \\ &=-140. \end{align}$S_{8}=-140$$S_n$$a_{1}=85$$n=21$$a_{n}=25$$a_{1… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p7?rev=1737476039&do=diff Question 13 and 14, Exercise 4.3 Solutions of Question 13 and 14 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $S_s$$a_{1}=34$$n=9$$a_{n}=2$$a_{1}=34$$n=9$$a_{n}=2$\begin{align} S_n&=\frac{n}{2}[a_1+a_n] \\ \implies S_{9}&=\frac{9}{2}[34+2]\\ &=162. \end{align}$S_{9}=162$$S_n$$a_{1}=5$$d=\frac{1}{2}$$n=13$$a_{1}=5$$d=\frac{1}{2}$$n=13$\begin{a… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p8?rev=1737476039&do=diff Question 15 and 16, Exercise 4.3 Solutions of Question 15 and 16 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $S_n$$a_{1}=91$$d=-4$$a_{n}=15$$a_{1}=91$$d=-4$$a_{n}=15$$n=?$\begin{align} & a_n=a_1+(n-1)d \\ \implies & 15=91+(n-1)(-4) \\ \implies & 15=91-4n+4 \\ \implies & 4n=95-15 \\ \implies & 4n = 80\\ \implies & n = 20. \end{align}\begin{… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p9?rev=1737476039&do=diff Question 17, 18 and 19, Exercise 4.3 Solutions of Question 17, 18 and 19 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $6+12+18+\ldots+96$$$6+12+18+\ldots+96.$$$a_{1}=6$$d=12-6=6$$a_{n}=96$$n=?$\begin{align} & a_n=a_1+(n-1)d \\ \implies & 96=6+(n-1)(6) \\ \implies & 96=6+6n-6 \\ \implies & 6n=96 \\ \implies & n = 24. \end{align}\begin{align}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p10?rev=1737476039&do=diff Question 20, 21 and 22, Exercise 4.3 Solutions of Question 20, 21 and 22 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=7$$a_{n}=139$$S_{n}=876$$a_{1}=7$$a_{n}=139$$S_{n}=876$$n$$d$\begin{align} &S_n=\frac{n}{2}[a_1+a_n]\\ \implies & 876=\frac{n}{2}[7+139]\\ \implies & 1752=146n\\ \implies & n=\frac{1752}{146}=12. \end{align}\begin{align… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p11?rev=1737476039&do=diff Question 23 and 24, Exercise 4.3 Solutions of Question 23 and 24 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$ 14+16+18+...+a_{25}.$$$a_1=14$$d=16-14=2$$n=25$$a_25$$S_25$\begin{align} a_n&=a_1+(n-1)d\\ \implies a_{25}&= 14+(25-1)(2)\\ &=62. \end{align}\begin{align} S_n&=\frac{n}{2}[a_1+a_n]\\ \implies S_{25}& =\frac{25}{2}[14+62]\\ & =25 \t… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p12?rev=1737476039&do=diff Question 25 and 26, Exercise 4.3 Solutions of Question 25 and 26 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$ 6000+70,000+...+a_{20}.$$$a_1=6,000$$d=70,000-6,000=64,000$$n=20$$S_n$\begin{align} S_n&=\frac{n}{2}[2a_1+(n-1)d]\\ \implies S_{20}& =\frac{20}{2}[2(6,000)+(20-1)(64,000)]\\ & =10 \times [12,000+1,216,000]\\ & =12,280,000. \end{ali… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p1?rev=1737476039&do=diff Question 1 and 2, Exercise 4.4 Solutions of Question 1 and 2 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $5,20,100,500, \ldots$$5, 20, 100, 500, \ldots $\begin{align*} \frac{20}{5} = 4\neq \frac{100}{20} = 5.\end{align*}$5, 20, 100, 500, \ldots $\begin{align*} r_1& =\frac{20}{5} = 4\\ r_2&=\frac{100}{20} = 5\\ r_3&=\frac{500}{100} = 5. \end{… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p2?rev=1737476039&do=diff Question 3 and 4, Exercise 4.4 Solutions of Question 3 and 4 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\frac{3}{2}, \frac{9}{4}, \frac{27}{8}, \frac{81}{16}, \ldots$\(\frac{3}{2}, \frac{9}{4}, \frac{27}{8}, \frac{81}{16}, \ldots\)\begin{align*} r_1&=\frac{9/4}{3/2} = \frac{9}{4} \times \frac{2}{3} = \frac{3}{2} \\ r_2&=\frac{27/8}{9/4} = … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p3?rev=1737476039&do=diff Question 5, 6 and 7, Exercise 4.4 Solutions of Question 5, 6 and 7 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=3, r=-2$$a_{1}=3$$r=-2$$$a_{n}=a_{1} r^{n-1}.$$\begin{align*} & a_{2}=a_{1} r=(3)(-2)= -6 \\ & a_{3}=a_{1} r^{2}=(3)(-2)^{2}=3 (4)= 12 \\ & a_{4}=a_{1} r^{3}=(3)(-2)^{3}=3 (-8) = -24 \end{align*}$a_1=3$$a_2=-6$$a_3=12$$a_4=-… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p4?rev=1737476039&do=diff Question 8 and 9, Exercise 4.4 Solutions of Question 8 and 9 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$90,30,10 \ldots$$$a_1=90$$r=\dfrac{30}{90}=\dfrac{1}{3}$$$a_{n}=a_{1} r^{n-1}.$$\begin{align*} & a_{4}=a_{1} r^3=(90)\left(\dfrac{1}{3} \right)^3=90 \times\dfrac{1}{27}=\dfrac{10}{3}\\ & a_{5}=a_{1} r^3=(90)\left(\dfrac{1}{4} \right)^4=… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p5?rev=1737476039&do=diff Question 10 and 11, Exercise 4.4 Solutions of Question 10 and 11 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$20,30,45 \ldots$$\(a_1=20\)\(r=\frac{30}{20}=\frac{3}{2}\)$$a_{n}=a_{1} r^{n-1}.$$\begin{align*} & a_{4}=a_{1} r^3=(20)\left(\frac{3}{2}\right)^3=20 \times \frac{27}{8} = \frac{540}{8} = 67.5 \\ & a_{5}=a_{1} r^4=(20)\left(\frac{3}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p6?rev=1737476039&do=diff Question 12 and 13, Exercise 4.4 Solutions of Question 12 and 13 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$\frac{1}{27}, \frac{1}{9}, \frac{1}{3}, \ldots$$\(a_1=\frac{1}{27}\)\(r=\frac{\frac{1}{9}}{\frac{1}{27}}=3\)$a_{n}=a_{1} r^{n-1}.$\begin{align*} & a_{4}=a_{1} r^3=\left(\frac{1}{27}\right)(3)^3=\frac{1}{27} \times 27 = 1 \\ & a_{5}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p7?rev=1737476039&do=diff Question 14 and 15, Exercise 4.4 Solutions of Question 14 and 15 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=4, n=3, r=5$$a_{1}=4, n=3, r=5$$$a_{n}=a_{1} r^{n-1}.$$\begin{align*} a_3&= 4\times 5^2 \\ &=4\times 25 = 100. \end{align*}$a_3=100$$a_{1}=2, n=5, r=2$$a_{1}=2$$n=5$$r=2$$a_{n}=a_{1} r^{n-1}.$\begin{align*} a_5 &= 2 \times 2^{… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p8?rev=1737476039&do=diff Question 16 and 17, Exercise 4.4 Solutions of Question 16 and 17 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=7, n=4, r=2$$a_{1}=7$$n=4$$r=2$$a_{n}=a_{1} r^{n-1}.$\begin{align*} a_4 &= 7 \times 2^{4-1} \\ &= 7 \times 2^3 \\ &= 7 \times 8 = 56. \end{align*}$a_4=56$$a_{1}=243, n=5, r=-\frac{1}{3}$$a_{1}=243$$n=5$$r=-\frac{1}{3}$$a_{n}=… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p9?rev=1737476039&do=diff Question 18 and 19, Exercise 4.4 Solutions of Question 18 and 19 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=32, n=6, r=-\frac{1}{2}$$a_{1}=32$$n=6$$r=-\frac{1}{2}$$a_{n}=a_{1} r^{n-1}.$\begin{align*} a_6 &= 32 \times \left(-\frac{1}{2}\right)^{6-1} \\ &= 32 \times \left(-\frac{1}{2}\right)^{5} \\ &= 32 \times \left(-\frac{1}{32}\ri… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p10?rev=1737476039&do=diff Question 20 and 21, Exercise 4.4 Solutions of Question 20 and 21 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$3 , \_\_\_ , \_\_\_ , \_\_\_ , 48$$$a_1=3$$a_5=48$$r$$$ a_n=ar^{n-1}. $$\begin{align*} &a_5=a_1 r^4 \\ \implies & 48=3r^4 \\ \implies & r^4 = 16 \\ \implies & r^4 = 2^4 \\ \implies & r = 2. \end{align*}\begin{align*} & a_2=a_1 r= (3… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p11?rev=1737476039&do=diff Question 22 and 23, Exercise 4.4 Solutions of Question 22 and 23 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$8 , \_\_\_, \_\_\_, \_\_\_, \_\_\_, \dfrac{1}{4}$$$a_1=8$$a_6=\frac{1}{4}$$r$$n$$a_n = a_1 r^{n-1}.$\begin{align*} a_6 &= a_1 r^5 \\ \implies \frac{1}{4} &= 8 \cdot r^5 \\ \implies r^5 &= \frac{1}{4 \cdot 8} \\ \implies r^5 &= \frac… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p12?rev=1737476039&do=diff Question 24 and 25, Exercise 4.4 Solutions of Question 24 and 25 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$5 , \_\_\_, \_\_\_, \_\_\_, 80$$$a_1=5$$a_5=80$$r$$n$$$a_n = a_1 r^{n-1}.$$\begin{align*} a_5 &= a_1 r^4 \\ \implies 80 &= 5 \cdot r^4 \\ \implies r^4 &= \frac{80}{5} \\ \implies r^4 &= 16 \\ \implies r &= 2. \end{align*}\begin{alig… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-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 2025-01-21T16:13:59+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. $$\begin{align*} S_6&=\frac{(1)\left(1-2^7 \right)}{1-2} \\ &=\frac{1-128}{-2}\\ &=127 \end{align… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p15?rev=1737476039&do=diff Question 30, Exercise 4.4 Solutions of Question 30 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $=a_1= 1$$=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-p2?rev=1737476040&do=diff Question 3 and 4, Exercise 4.5 Solutions of Question 3 and 4 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=5$$r=3$$n=12$$a_{1}=5$$r=3$$n=12$$n$\[ S_n = \frac{a_1 \left(1 - r^n\right)}{1 - r}, \quad r\neq 1. \]\begin{align*} S_{12} &= \frac{5\left(1 - 3^{12}\right)}{1 - 3} \\ &= \frac{5\left(1 - 531441\right)}{-2} \\ &= \frac{5(-531440)}… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p3?rev=1737476040&do=diff Question 5 and 6, Exercise 4.5 Solutions of Question 5 and 6 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=7, r=2, n=14$$a_1 = 7$$r = 2$$n = 14$$n$$$S_n = \frac{a_1 \left(1 - r^n\right)}{1 - r}, \quad r \neq 1.$$\begin{align*} S_{14} &= \frac{7 \left(1 - 2^{14}\right)}{1 - 2} \\ &= \frac{7 \left(1 - 16384\right)}{-1} \\ &= \frac{7 \time… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p4?rev=1737476040&do=diff Question 7 and 8, Exercise 4.5 Solutions of Question 7 and 8 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=16, r=-\frac{1}{2}, n=10$$a_1 = 16$$r = -\frac{1}{2}$$n = 10$$n$$$S_n = \frac{a_1 \left(1 - r^n\right)}{1 - r}, \quad r \neq 1.$$\begin{align*} S_{10} &= \frac{16 \left(1 - \left(-\frac{1}{2}\right)^{10}\right)}{1 - \left(-\frac{1}… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p5?rev=1737476040&do=diff Question 9 and 10, Exercise 4.5 Solutions of Question 9 and 10 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=343, a_{4}=-1, r=-\frac{1}{7}$$a_{1}=343$$a_{4}=-1$$r=-\frac{1}{7}$$S_n$$$ S_n =\frac{a_1-a_n r}{1-r}, \quad r\neq 1.$$\begin{align*} S_4 & =\frac{343-(-1)\left(-\frac{1}{7}\right)}{1+\frac{1}{7}} \\ &=\frac{\frac{2400}{7}}{\frac… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p6?rev=1737476040&do=diff Question 11, 12 and 13, Exercise 4.5 Solutions of Question 11, 12 and 13 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}$$S_{n}=244, r=-3, n=5$$S_{n}=244$$r=-3$$n=5$$$ S_n =\frac{a_1(1-r^n)}{1-r}, \quad r\neq 1.$$\begin{align*} & 244=\frac{a_1(1-(-3)^5)}{1-(-3)} \\ \implies & 244=\frac{a_1(1+243)}{4} \\ \implies & 976=244a_1\\ \implies & … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p7?rev=1737476040&do=diff Question 14, Exercise 4.5 Solutions of Question 14 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $0.444...$$$0.444... = 0.4+0.04+0.004+...$$$a_1=0.4$$r=\frac{0.04}{0.4}=0.1$$|r|=0.1 < 1$\begin{align*} S-\infty & = \frac{a_1}{1-r} \\ & = \frac{0.4}{1.0.1} = \frac{0.4}{0.9} \\ & = \frac{4}{9}. \end{align*}$S_{\infty} =\dfrac{4}{9}$$9.99999 ...$$… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p8?rev=1737476040&do=diff Question 15, Exercise 4.5 Solutions of Question 15 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $30 ft$$\frac{2}{5}$$= 30 ft$$= 30 \times \frac{2}{5} = 12 ft$$= 12 \times \frac{2}{5} = \frac{24}{5} ft$$= \frac{24}{5} \times \frac{2}{5} = \frac{48}{25} ft$$D$$$D=30+2\left(12+\frac{24}{5}+\frac{24}{5}+... \right)$$$$ 12+\frac{24}{5}+\frac{24}{5… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p9?rev=1737476040&do=diff Question 16, Exercise 4.5 Solutions of Question 16 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $80 ft$$90\%$$a_1$$a_1 r$$a_1 r^2$$=a_1= 80 ft$$r=90% = \frac{90}{100} =0.9$$A$\begin{align} A &= a_1+a_1r+a_1r^2+... \\ & = \frac{a_1}{1-r} \\ & = \frac{80}{1-0.9}\\ &= 800 \end{align}$800 ft$ text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-6-p1?rev=1737476040&do=diff Question 1 and 2, Exercise 4.6 Solutions of Question 1 and 2 of Exercise 4.6 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\frac{1}{9}, \frac{1}{12}, \frac{1}{15}, \cdots \quad 7$$$\frac{1}{9}, \frac{1}{12}, \frac{1}{15}, \cdots \text{ is in H.P.}$$$$9, 12, 15, ... \text{ is in A.P.}$$$a_1=9$$d=12-9=3$$a_7=?$$$ a_n=a_1+(n-1)d. $$\begin{align*} a_7&=9+(6)(3) … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-6-p2?rev=1737476040&do=diff Question 3 & 4, Exercise 4.6 Solutions of Question 3 & 4 of Exercise 4.6 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\frac{1}{18}, \frac{1}{13}, \frac{1}{8}, \ldots \quad 20$\begin{align*} &\frac{1}{18}, \frac{1}{13}, \frac{1}{8}, \ldots \quad \text{ is in H.P.} \\ &18, 13, 8, \ldots \quad \text{ is in A.P.} \end{align*}$a_1 = 18$$d = 13 - 18 = -5$$a_{20}.… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-6-p3?rev=1737476040&do=diff Question 5 & 6, Exercise 4.6 Solutions of Question 5 & 6 of Exercise 4.6 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\dfrac{1}{27}, \dfrac{1}{20}, \dfrac{1}{13}, \ldots \quad$\begin{align*} &\frac{1}{27}, \frac{1}{20}, \frac{1}{13}, \ldots \quad \text{ is in H.P.} \\ &27, 20, 13, \ldots \quad \text{ is in A.P.} \end{align*}$a_1 = 27$$d = 20 - 27 = -7$$a_n=… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-6-p4?rev=1737476040&do=diff Question 7 & 8, Exercise 4.6 Solutions of Question 7 & 8 of Exercise 4.6 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\frac{1}{4}, \frac{1}{7}, \frac{1}{10}, \frac{1}{13}, \ldots$$ \frac{1}{4}, \frac{1}{7}, \frac{1}{10}, \frac{1}{13}, \ldots $$ a_1 = \frac{1}{4} $$d = \frac{1}{7} - \frac{1}{4} = -\frac{3}{28},$$ n = 14$$$a_n = a_1 + (n-1)d.$$\begin{align*} … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/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/ex4-6-p6?rev=1737476040&do=diff Question 11, Exercise 4.6 Solutions of Question 11 of Exercise 4.6 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\dfrac{2}{3}$$\dfrac{4}{7}$$a=\dfrac{2}{3}$$b=\dfrac{4}{7}$\begin{align*} \text{H.M.}&=\frac{2ab}{a+b} \\ &=\frac{2\times\frac{2}{3}\times\frac{4}{7}}{\frac{2}{3}+\frac{4}{7}} \\ &=\frac{16/21}{26/21} \\ &=\frac{8}{13} \\ \end{align*}$\dfrac{8}{13… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-6-p7?rev=1737476040&do=diff Question 12, Exercise 4.6 Solutions of Question 12 of Exercise 4.6 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\dfrac{1}{3}$$\dfrac{1}{11}$$H_1, H_2, H_3, H_4$$H.Ms$$\dfrac{1}{3}$$\dfrac{1}{11}$$$\dfrac{1}{3},H_1, H_2, H_3, H_4, \dfrac{1}{11} \text{ are in H.P.}$$$$\quad 3,\dfrac{1}{H_1},\dfrac{1}{H_2}, \dfrac{1}{H_3}, \dfrac{1}{H_4},11 \text{ are in A.P.}… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p1?rev=1737476040&do=diff Question 1 and 2, Exercise 4.7 Solutions of Question 1 and 2 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sum_{k=1}^{5} \frac{1}{2 k}$\begin{align*} \sum_{k=1}^{5} \frac{1}{2k} &= \frac{1}{2(1)} + \frac{1}{2(2)} + \frac{1}{2(3)} + \frac{1}{2(4)} + \frac{1}{2(5)}\\ &= \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \frac{1}{10}\\ &= … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p2?rev=1737476040&do=diff Question 3 and 4, Exercise 4.7 Solutions of Question 3 and 4 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sum_{k=0}^{5} 2^{k}$\begin{align*} \sum_{k=0}^{5} 2^{k} &= 2^0 + 2^1 + 2^2 + 2^3 + 2^4 + 2^5 \\ &= 1 + 2 + 4 + 8 + 16 + 32 \\ &= 63 \end{align*}$\sum_{k=0}^{9} \pi k$\begin{align*} \sum_{k=0}^{9} \pi k &= \pi(0) + \pi(1) + \pi(2) + \pi(… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p3?rev=1737476040&do=diff Question 5 and 6, Exercise 4.7 Solutions of Question 5 and 6 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sum_{k=1}^{8} \frac{k}{k+1}$\begin{align*} \sum_{k=1}^{8} \frac{k}{k+1} &= \frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} + \frac{5}{6}\\ &+ \frac{6}{7} + \frac{7}{8} + \frac{8}{9} \\ &= 0.5 + 0.6667 + 0.75 + 0.8 + 0.8333\\ &+ 0.… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p4?rev=1737476040&do=diff Question 7 and 8, Exercise 4.7 Solutions of Question 7 and 8 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sum_{k=0}^{5}\left(k^{2}-2 k+3\right)$\begin{align*} \sum_{k=0}^{5} (k^{2} - 2k + 3) &= (0^{2} - 2(0) + 3) + (1^{2} - 2(1) + 3) + (2^{2} - 2 (2) + 3) \\ &+ (3^{2} - 2 (3) + 3) + (4^{2} - 2 (4) + 3) + (5^{2} - 2 (5) + 3) \\ &= (0 - 0 + 3… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p5?rev=1737476040&do=diff Question 9 and 10, Exercise 4.7 Solutions of Question 9 and 10 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\frac{4}{5}+\frac{5}{6}+\dots$$$ \frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} + \frac{5}{6} +... = \sum_{k=1}^{\infty}\frac{k}{k+1} $$$3+6+9+12+15$$$3+6+9+12+15=\sum_{k=1}^{5}3k$$ text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p6?rev=1737476040&do=diff Question 11, 12 and 13, Exercise 4.7 Solutions of Question 11, 12 and 13 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $-2+4-8+16-32+64$$$ -2 + 4 - 8 + 16 - 32 + 64 = \sum_{k=1}^{6} (-1)^k 2^k $$$\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\frac{1}{4 \cdot 5}+$$$ \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} +… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p7?rev=1737476040&do=diff Question 14, 15 and 16, Exercise 4.7 Solutions of Question 14, 15 and 16 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$n$$n+1$$T_n$$n$$$ T_{n} = n+1. $$\begin{align*}\sum_{n=1}^{\infty} T_{n} &= \sum_{n=1}^{\infty} (n+1)\\ & = \sum_{n=1}^{\infty} n + \sum_{n=1}^{\infty} 1 \\ & = \frac{n(n+1)}{2} + n \\ & = \frac{n(n+1)}{2} + \frac{2n}{2} \… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p8?rev=1737476040&do=diff Question 17 and 18, Exercise 4.7 Solutions of Question 17 and 18 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$2^{2}+5^{2}+8^{2}+\ldots$$2+5+8+\ldots$$a_k=2+(k-1)(3)=2+3k-3=3k-1$$T_k$$k$\begin{align*}T_k&=(3k-1)^2 \\ &=9k^2-6k+1. \end{align*}\begin{align*}\sum_{k=1}^{n} T_{k} &= \sum_{k=1}^{n} (9k^{2} - 6k + 1)\\ & = 9\sum_{k=1}^{n} k^{2} … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p9?rev=1737476040&do=diff Question 19 and 20, Exercise 4.7 Solutions of Question 19 and 20 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$1^{3}+3^{3}+5^{3}+$$1+3+5+\ldots$$a_k=1+(k-1)(2)=1+2k-2=2k-1$$T_k$$k$\begin{align*}T_k&=(2k-1)^3 \\ &=(2k)^3+3(2k)^2(-1)+3(2k)(-1)^2+(-1)^3 \\ &=8k^3-12k^2+6k+1 \end{align*}\begin{align*}\sum_{k=1}^{n} T_{k} &= \sum_{k=1}^{n} (8k^… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p10?rev=1737476040&do=diff Question 19 and 20, Exercise 4.7 Solutions of Question 19 and 20 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$1^{3}+3^{3}+5^{3}+$$1+3+5+\ldots$$a_k=1+(k-1)(2)=1+2k-2=2k-1$$T_k$$k$\begin{align*}T_k&=(2k-1) \\ &=9k^2-6k+1. \end{align*}\begin{align*}\sum_{k=1}^{n} T_{k} &= \sum_{k=1}^{n} (2k - 1)\\ & = 2 \sum_{k=1}^{n} k - \sum_{k=1}^{n} 1 \… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p11?rev=1737476040&do=diff Question 21 and 22, Exercise 4.7 Solutions of Question 21 and 22 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$1 \times 4+2 \times 7+3 \times 10+\cdots$$4+7+10+\ldots$$a_k=4+(k-1)(3)=4+3k-3=3k+1$$1+2+3+...$$k$$k(3k+1)$$T_k$$k$\begin{align*}T_k&=k(3k+1) \\ &=3k^2+k. \end{align*}\begin{align*}\sum_{k=1}^{n} T_{k} &= \sum_{k=1}^{n} (3k^2 +k)\… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p12?rev=1737476040&do=diff Question 23 and 24, Exercise 4.7 Solutions of Question 23 and 24 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$$1+2 \times 2+3 \times 2^{2}+4 \times 2^{3}+\ldots.$$$$1+2 \times 2+3 \times 2^{2}+4 \times 2^{3}+\ldots$$$$ 1\times 1+2 \times 2+3 \times 2^{2}+4 \times 2^{3}+\ldots $$$1,2,3,4,\ldots$$a=1$$d=1$$1, 2, 2^2, 2^3, \ldots$$r=\frac{2}… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p13?rev=1737476040&do=diff Question 25 and 26, Exercise 4.7 Solutions of Question 25 and 26 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$1+\frac{4}{7}+\frac{7}{7^{2}}+\frac{10}{7^{3}}+\ldots$\[ 1 + \frac{4}{7} + \frac{7}{7^2} + \frac{10}{7^3} + \ldots \]\(1, 4, 7, 10, \ldots\)\(a = 1\)\(d = 3\)\(1, \frac{1}{7}, \frac{1}{7^2}, \frac{1}{7^3}, \ldots\)\(1\)\(r = \frac… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p14?rev=1737476040&do=diff Question 27 and 28, Exercise 4.7 Solutions of Question 27 and 28 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$5+\frac{7}{3}+\frac{9}{9}+\frac{11}{27}+\ldots$$$$5+\frac{7}{3}+\frac{9}{9}+\frac{11}{27}+\ldots$$$$ 5\times 1+7\times\frac{1}{3}+9\times\frac{1}{9}+11\times\frac{1}{27}+\ldots $$$5,7,9,11,4,\ldots$$a=5$$d=7-5=2$$1, \dfrac{1}{3}, \d… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p15?rev=1737476040&do=diff Question 29 and 30, Exercise 4.7 Solutions of Question 29 and 30 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$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-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/unit04/ex4-8-p4?rev=1737476040&do=diff Question 7 and 8, Exercise 4.8 Solutions of Question 7 and 8 of Exercise 4.8 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$$\frac{1}{1 \times 4}+\frac{1}{4 \times 7}+\frac{1}{7 \times 10}+\ldots$$$$\frac{1}{1 \times 4}+\frac{1}{4 \times 7}+\frac{1}{7 \times 10}+\dots$$$T_k$\begin{align*} T_k &=\frac{1}{(3k-2)(3k+1)}. \end{align*}\begin{align*} \frac{1}{(3… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-8-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/unit04/ex4-8-p6?rev=1737476040&do=diff Question 11 and 12, Exercise 4.8 Solutions of Question 11 and 12 of Exercise 4.8 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sum_{k=1}^{n} \frac{1}{k(k+2)}$$T_k$$k$\begin{align*} T_k &= \frac{1}{k(k+2)}. \end{align*}\begin{align*} \frac{1}{k(k+2)} = \frac{A}{k} + \frac{B}{k+2} \ldots (1) \end{align*}$k(k+2)$\begin{align*} 1 = A(k+2) + Bk \ldots (2) \end{… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-8-p7?rev=1737476040&do=diff Question 13, 14 and 15, Exercise 4.8 Solutions of Question 13, 14 and 15 of Exercise 4.8 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\frac{1}{5 \cdot 11}+\frac{1}{7 \cdot 13}+\frac{1}{9 \cdot 15}+\ldots \ldots$$n$$T_k$$k$\begin{align*} T_k &= \frac{1}{(2k+3)(2k+9)}. \end{align*}\begin{align*} \frac{1}{(2k+3)(2k+9)} = \frac{A}{2k+3} + \frac{B}{2k+9} \ldots … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-1-p1?rev=1737476040&do=diff Question 1, Exercise 5.1 Solutions of Question 1 of Exercise 5.1 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 1(i) $2 x^{3}+3 x^{2}-4 x+1$$x+2$$p(x)=2 x^{3}+3 x^{2}-4 x+1$$x-c=x+2 \implies c=-2$\begin{align*} \text{Remainder} & = p(c) = p(-2) \\ & = 2(-2)^{3}+3 (-2)^{2}-4 (-2)+1 \\ & = -16+12+8+1 \\ &= 5. \end{align*}$x^{4}+2 x^{3}-x^{2}+2 x+3$$x-2$\( p(x… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-1-p2?rev=1737476040&do=diff Question 2 and 3, Exercise 5.1 Solutions of Question 2 and 3 of Exercise 5.1 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $x-3$$x^{3}-2 x^{2}-5 x+6$$p(x)=x^{3}-2 x^{2}-5 x+6$$x-c=x-3$$\implies c=3$$x-3$$p(x)$$p(3)=0$\begin{align*} p(3)&=3^3-2(3)^2-5(3)+6 \\ & = 27-18-15+6 \\ & = 0. \end{align*}$x-3$$p(x)$$x-3$$x^{3}-2 x^{2}-5 x+1$$p(x)=x^{3}-2 x^{2}-5 x+1$$x-c=x-3$$… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-1-p3?rev=1737476040&do=diff Question 4 and 5, Exercise 5.1 Solutions of Question 4 and 5 of Exercise 5.1 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $4 y^{3}-4 y^{2}+10+2 y$$4 y^{2}-8 y+10$$q$$x^{3}+q x^{2}-7 x+6$$(x+1)$$p(x)=x^{3}+q x^{2}-7 x+6$$x-c=x+1$$\implies c=-1$$x+1$$p(x)$$p(-1)=0$\begin{align*} &(-1)^3+q(-1)^2-7(-1)+6=0 \\ -&1+q+7+6=0\\ &q+12=0\\ &q=-12 \end{align*} text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-1-p4?rev=1737476040&do=diff Question 6 and 7, Exercise 5.1 Solutions of Question 6 and 7 of Exercise 5.1 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $m$$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-p5?rev=1737476040&do=diff Question 8 and 9, Exercise 5.1 Solutions of Question 8 and 9 of Exercise 5.1 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 x^{3}+3 x^{2}-11 x-6$$p(x)=2x^3+3x^2-11x-6$\begin{align} p(2) &= 2(2)^3+3(2)^2-11(2)-6 \\ &=16+12-22-6 = 0 \end{align}$p(x)$\begin{align} \begin{array}{r|rrrr} 2 & 2 & 3 & -11 & -6 \\ & \downarrow & 4 & 14 & 6 \\ \hline & 2 & 7 & 3 & 0 \\ \… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-1-p6?rev=1737476040&do=diff Question 10, Exercise 5.1 Solutions of Question 10 of Exercise 5.1 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 10 $\left(x^{3}+11 x^{2}+34 x+24\right)$$(x+1)$$p(x)=x^{3}+11 x^{2}+34 x+24$\begin{align} \begin{array}{r|rrrr} -1 & 1 & 11 & 34 & 24 \\ & \downarrow & -1 & -10 & -24 \\ \hline & 1 & 10 & 24 & 0 \\ \end{array}\end{align}$$ p(x) = (x+1)(x^2+10… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-2-p1?rev=1737476040&do=diff Question 1 and 2, Exercise 5.2 Solutions of Question 1 and 2 of Exercise 5.2 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y^{3}-7 y-6$$f(y)=y^{3}-7 y-6$\begin{align*} f(-1)&=(-1)^{3}-7 (-1)-6 \\ &= -1+7-6 =0. \end{align*}$y+1$$f(y)$\begin{align} \begin{array}{r|rrrr} -1 & 1 & 0 & -7 & -6 \\ & \downarrow & -1 & 1 & 6 \\ \hline & 1 & -1 & -6 & 0 \\ \end{array}\end… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-2-p2?rev=1737476040&do=diff Question 3 and 4, Exercise 5.2 Solutions of Question 3 and 4 of Exercise 5.2 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 x^{3}+5 x^{2}-9 x-18$\( f(x) = 2x^{3} + 5x^{2} - 9x - 18 \)\begin{align*} f(-2) &= 2(-2)^{3} + 5(-2)^{2} - 9(-2) - 18 \\ &= 2(-8) + 5(4) + 18 - 18 \\ &= -16 + 20 + 18 - 18 = 0. \end{align*}\( x + 2 \)\( f(x) \)\[ \begin{array}{r|rrrr} -2 & 2 &… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-2-p3?rev=1737476040&do=diff ; Question 5 and 6, Exercise 5.2 Solutions of Question 5 and 6 of Exercise 5.2 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $t^{3}+t^{2}+3 t-5$\( f(t) = t^{3} + t^{2} + 3t - 5 \)\begin{align*} f(1) &= (1)^{3} + (1)^{2} + 3(1) - 5 \\ &= 1 + 1 + 3 - 5 \\ &= 0. \end{align*}\( t - 1 \)\( f(t) \)\begin{align} \begin{array}{r|rrrr} 1 & 1 & 1 & 3 & -5 \\ & & 1 & 2 & 5 \… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-2-p4?rev=1737476040&do=diff Question 7 and 8, Exercise 5.2 Solutions of Question 7 and 8 of Exercise 5.2 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 x^{3}-15 x^{2}+27 x-10$$\dfrac{1}{2}$\( f(x) \)\( x - \frac{1}{2} \)\begin{align} \begin{array}{r|rrrr} \frac{1}{2} & 2 & -15 & 27 & -10 \\ & & 1 & -7 & 10 \\ \hline & 2 & -14 & 20 & 0 \\ \end{array} \end{align}\begin{align*} f(x) &= \left… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-3-p1?rev=1737476040&do=diff Question 1, Exercise 5.3 Solutions of Question 1 of Exercise 5.3 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 1 $x$$x+3$$x+3+7=x+10$$120 cm^3$\begin{align*} & x(x+3)(x+10)=120 \\ \implies & x(x^2+3x+10x+30)-120=0\\ \implies & x^3+13x^2+30x-120=0. \end{align*}$$p(x)=x^3+13x^2+30x-120$$\begin{align*} p(2)&=2^3+13(2)^2+30(2)-120 \\ &=8+52+60-120 =0 \end{ali… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-3-p2?rev=1737476040&do=diff Question 2, Exercise 5.3 Solutions of Question 2 of Exercise 5.3 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 2 $t(x)=x^{3}-12 x^{2}+48 x+74$$x$$$t(x)=x^{3}-12 x^{2}+48 x+74.$$$t=12$\begin{align*} t(12)&=(12)^3-12(12)^2+48(12)+74 \\ &=650. \end{align*} text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-3-p3?rev=1737476040&do=diff Question 3, Exercise 5.3 Solutions of Question 3 of Exercise 5.3 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 3 $x$$2x$$2x+2$\begin{align*} & x(2x)(2x+2) = 144 \\ \implies & 4x^2(x+1)=144 \\ \implies & x^2(x+1)=36 \\ \implies & x^3+x^2-36=0 \end{align*}$$p(x)=x^3+x^2-36.$$\begin{align*} p(3)&=3^3+3^2-36 \\ &=27+9-36 = 0 \end{align*}$x=3$$p(x)$$2(3)$$2(3)+… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-3-p4?rev=1737476040&do=diff Question 4, Exercise 5.3 Solutions of Question 4 of Exercise 5.3 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 4 $x$$2x+3$$x-2$\begin{align*} & x(2x+3)(x-2) = 2475 \\ \implies & x(2x^2+3x-4x-6)=2475 \\ \implies & x(2x^2-x-6)-2475=0 \\ \implies & 2x^3-x^2-6x-2475=0 \end{align*}$$p(x)=2x^3-x^2-6x-2475.$$\begin{align*} p(11)&=2(11)^3-11^2-6(11)-2475 \\ &=2662… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-3-p5?rev=1737476040&do=diff Question 5, Exercise 5.3 Solutions of Question 5 of Exercise 5.3 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 5 $6 x^{2}+38 x+56$$2 x+8$$ACED$$ABFG$$ACED$$6 x^{2}+38 x+56$$2 x+8$\begin{align*} & 6 x^{2}+38 x+56 \\ = & 2(3x^2+19x+28) \\ = & 2(3x^2+12x+7x+28) \\ = & 2(3x(x+4)+7(x+4)) \\ =& 2(x+4)(3x+7) \\ =& (2x+8)(3x+7) \end{align*}\begin{align*} & Length … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-3-p6?rev=1737476040&do=diff Question 6, Exercise 5.3 Solutions of Question 6 of Exercise 5.3 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 6 $y^3-2y^2-y+2$$y-2$$=p(y)=y^3-2y^2-y+2$$y-2$$y-2$$p(y)$$2$$p(y)$\[ \begin{array}{r|rrrr} 2 & 1 & -2 & -1 & 2 \\ & \downarrow & 2 & 0 & -2 \\ \hline & 1 & 0 & -1 & 0 \\ \end{array} \]\begin{align*} p(y) & = (y-2)(y^2-1) \\ & = (y-2)(y+1)(y-1)… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-4-p2?rev=1737476040&do=diff Question 2, Review Exercise Solutions of Question 2 of Review Exercise of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Go to text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/re-ex-p1?rev=1737476040&do=diff Question 1, Review Exercise Solutions of Question 1 of Review Exercise of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 1 $-2-x+x^{2}$$(x-2)(x-1)$$(x+1)(x+2)$$(x+2)(x-1)$$(x+1)(x-2)$$9 y^{2}+9 y-10$$3 y-2$$ 0$$1$$2$$3$$\frac{x^{2}-x-9}{x-3}=x+2+\frac{?}{x-3}$$-27$$-3$$\frac{3}{x-3}+x+2$$ 3$$3 x^{3}-2 x^{2}+5$$x+1$$x+1$$x^{3}+5 x^{2}-4 x+k$$k$$-4$$-20$$20$$0$$… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/re-ex-p2?rev=1737476040&do=diff Question 2 & 3, Review Exercise Solutions of Question 2 & 3 of Review Exercise of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left(64 y^{3}-8\right) \div(4 y-2) \quad$\begin{align*} \frac{(64 y^{3}-8)}{(4 y-2)}&= \frac{(4y - 2)(16y^{2} + 8y + 4)}{4y - 2}\\ & = 16y^{2} + 8y + 4 .\end{align*}$\left(125 y^{3}-8\right) \div(5 y-2)$\begin{align*} \frac{(125 y^{3}-8)}{(5 … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/re-ex-p3?rev=1737476040&do=diff Question 4 & 5, Review Exercise Solutions of Question 4 & 5 of Review Exercise of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $3 y-2$$6 y^{3}-y^{2}-5 y+2$\begin{align*}3y-2&=0\\ 3y&=2\\ y&=\frac{2}{3}\end{align*}\begin{align*} f(y) &= 6y^{3} - y^{2} - 5y + 2\\ f\left(\frac{2}{3}\right) &= 6\left(\frac{2}{3}\right)^{3} - \left(\frac{2}{3}\right)^{2} - 5\left(\frac{2}{3… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/re-ex-p4?rev=1737476040&do=diff Question 6 & 7, Review Exercise Solutions of Question 6 & 7 of Review Exercise of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $k$$\left(x^{2}+8 x+k\right)$$(x-4)$\( p(x) = x^{2} + 8x + k \)\( p(x) \)\( (x - 4) \)\( p(4) \)\( p(4) = 0 \)\begin{align*} p(4) &= (4)^2 + 8(4) + k \\ &= 16 + 32 + k \\ &= 48 + k. \end{align*}\[ 48 + k = 0. \]\[ k = -48. \]$3 x^{2}-x+32-\frac… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/re-ex-p5?rev=1737476040&do=diff Question 8, Review Exercise Solutions of Question 8 of Review Exercise of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 8 $y^{3}+6 y^{2}-y-30$$(y-2)$$(y+3)$$p(y)=y^{3}+6 y^{2}-y-30$$(y-2)$$(y+3)$$p(y)$$2$$-3$$p(y)$\[ \begin{array}{r|rrrr} 2 & 1 & 6 & -1 & -30 \\ & \downarrow & 2 & 16 & 30 \\ \hline -3 & 1 & 8 & 15 & 0 \\ & \downarrow & -3 & -15 & … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/re-ex-p6?rev=1737476040&do=diff Question 6, Review Exercise Solutions of Question 6 of Review Exercise of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 6 $k$$\left(x^{2}+8 x+k\right)$$(x-4)$ text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/re-ex-p7?rev=1737476040&do=diff Question 7, Review Exercise Solutions of Question 7 of Review Exercise of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 7 $3 x^{2}-x+32-\frac{121}{x+4}$ text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/re-ex-p8?rev=1737476040&do=diff Question 8, Review Exercise Solutions of Question 8 of Review Exercise of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 8 $y^{3}+6 y^{2}-y-30$$(y-2)$$(y+3)$ text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p1?rev=1737476040&do=diff Question 1, Exercise 8.1 Solutions of Question 1 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\cos (\alpha \pm \beta), \sin (\alpha \pm \beta)$$\tan (\alpha \pm \beta)$$\alpha=180^{\circ}, \beta=60^{\circ}$$\alpha=180^{\circ}$$\beta=60^{\circ}$\begin{align*} \cos (\alpha + \beta) & = \cos \alpha \cos \beta - \sin \alpha \sin \beta \… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p2?rev=1737476040&do=diff Question 2, Exercise 8.1 Solutions of Question 2 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\cos 15^{\circ}$$\cos \left(45^{\circ}-30^{\circ}\right)$\begin{align*} \cos 15^{\circ} & = \cos \left(45^{\circ}-30^{\circ}\right)\\ &= \cos 45 \cos 30 + \sin 45 \sin 30 \\ &= \dfrac{1}{\sqrt{2}}\cdot \dfrac{\sqrt{3}}{2} + \dfrac{1}{\sqrt{2… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p3?rev=1737476040&do=diff Question 3, Exercise 8.1 Solutions of Question 3 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\cos 120^{\circ}$$\cos \left(180^{\circ}-60^{\circ}\right)$$\cos \left(90^{\circ}+30^{\circ}\right)$\begin{align*} \cos 120^{\circ} & = \cos \left(180^{\circ}-60^{\circ}\right) \\ &= - \cos 60 ^{\circ}\\ &= -\dfrac{1}{2}. \end{align*}\begin{… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p4?rev=1737476040&do=diff Question 4, Exercise 8.1 Solutions of Question 4 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\cos 6 \theta \cos 3 \theta-\sin 6 \theta \sin 3 \theta$\begin{align*} & \cos 6 \theta \cos 3 \theta-\sin 6 \theta \sin 3 \theta \\ & = \cos (6\theta +3\theta) \\ & = \cos 9\theta . \end{align*}$\cos 7 \theta \cos 2 \theta+\sin 7 \theta \sin… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p5?rev=1737476040&do=diff Question 5 and 6, Exercise 8.1 Solutions of Question 5 and 6 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin \alpha=\dfrac{4}{5}, \tan \beta=-\dfrac{5}{12}$$\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-p6?rev=1737476040&do=diff Question 7, Exercise 8.1 Solutions of Question 7 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\alpha$$\beta$$\sin \alpha=\dfrac{12}{13}$$\tan \beta=\dfrac{4}{3}$$\sin(\alpha+\beta)$$\cos(\alpha+\beta)$$\tan(\alpha+\beta)$$\sin \alpha=\dfrac{12}{13}$$\alpha$$\tan \beta=\dfrac{4}{3}$$\beta$$$\cos \alpha=\pm \sqrt{1-\sin^2\alpha}.$$\(\a… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p7?rev=1737476040&do=diff Question 8, Exercise 8.1 Solutions of Question 8 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin \alpha=\dfrac{3}{5}$$0<\alpha<\dfrac{\pi}{2}$$\cos \beta=\dfrac{12}{13}$$\dfrac{3 \pi}{2}<\beta<2 \pi$$\csc (\alpha+\beta)$$\sec (\alpha+\beta)$$\cot (\alpha+\beta)$$\sin \alpha=\dfrac{3}{5}$$0<\alpha<\dfrac{\pi}{2}$$\alpha$\begin{align… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p8?rev=1737476040&do=diff Question 9, Exercise 8.1 Solutions of Question 9 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\alpha$$\beta$$\sin \alpha=\dfrac{1}{\sqrt{2}}$$\cos \beta=-\dfrac{3}{5}$$\sin (\alpha \pm \beta)$$\sin \alpha=\dfrac{1}{\sqrt{2}}$$\alpha$$\cos \beta=-\dfrac{3}{5}$$\beta$$$\cos \alpha=\pm \sqrt{1-\sin^2\alpha}.$$$\alpha$$\cos$\begin{align*… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-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/ex8-1-p10?rev=1737476040&do=diff Question 11, Exercise 8.1 Solutions of Question 11 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\dfrac{\sin \left(180^{\circ}+\lambda\right) \cos \left(270^{\circ}+\lambda\right)}{\sin \left(180^{\circ}-\lambda\right) \cos \left(270^{\circ}-\lambda\right)}=1$\begin{align*} L.H.S & = \dfrac{\sin \left(180^{\circ}+\lambda\right) \cos \… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p11?rev=1737476040&do=diff Question 12, Exercise 8.1 Solutions of Question 12 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\alpha+\beta+\gamma=180^{\circ}$$\tan \alpha+\tan \beta+\tan \gamma=\tan \alpha \tan \beta \tan \gamma$$$\alpha+\beta+\gamma=180^{\circ}$$\begin{align*} & \alpha+\beta=180^{\circ}-\gamma \\ \implies & \tan(\alpha+\beta) = \tan(180^{\circ}-… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p12?rev=1737476040&do=diff Question 13, Exercise 8.1 Solutions of Question 13 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $r \sin (\theta+\phi)$$12 \sin \theta-5 \cos \theta$$12=r\cos \varphi $$-5=r\sin \varphi$\begin{align*} & (12)^2+(-5)^2=r^2 \cos^2\varphi+r^2 \sin^2 \varphi \\ \implies & 144+25={{r}^{2}}\left( {{\cos }^{2}}\varphi +{{\sin }^{2}}\varphi \r… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p13?rev=1737476040&do=diff Question 14, Exercise 8.1 Solutions of Question 14 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\theta$$\sin \theta$$\cos \theta$$\alpha$\begin{align*} &\tan\alpha = \frac{\overline{BC}}{\overline{AB}} \\ \implies &\tan\alpha = \frac{3}{3} = 1 \\ \implies &\alpha = \tan^{-1}(1) = 45^\circ \end{align*}$45^\circ$$\theta$\begin{align*} … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-2-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}.$$\begin{align… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p2?rev=1737476040&do=diff Question 4 Exercise 8.2 Solutions of Question 4 of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin 2 \theta$$\cos 2 \theta$$\tan 2 \theta$$\sin \frac{\theta}{2}$$\cos \frac{\theta}{2}$$\tan \frac{\theta}{2}$$\cos \theta=\frac{3}{5}$$0<\theta<\frac{\pi}{2}$$\cos\theta=\dfrac{3}{5}$$0<\theta<\dfrac{\pi}{2}$$\theta$$$\sin\theta = \pm \sq… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p3?rev=1737476040&do=diff Question 5 Exercise 8.2 Solutions of Question 5 of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin \theta$$\cos \theta$$\tan \theta$$\sin 2 \theta=\frac{24}{25}, 2 \theta$$\sin 2\theta=\dfrac{24}{25}$$2\theta$$$\cos 2\theta = \pm \sqrt{1-\sin^2 2\theta}$$$2\theta$$\cos 2\theta$\begin{align*}\cos 2\theta & = - \sqrt{1-\sin^2 2\theta}\\… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p4?rev=1737476040&do=diff Question 6 Exercise 8.2 Solutions of Question 6 of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin 15^{\circ} \cos 15^{\circ}$$$\sin 2 \theta = 2\sin\theta \cos\theta$$$$\sin\theta \cos\theta = \frac{1}{2}\sin 2\theta$$$\theta = 15^{\circ}$\begin{align*} \sin 15^{\circ} \cos 15^{\circ} & = \frac{1}{2}\sin 2(15^{\circ}) \\ & \frac{1}{2… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p5?rev=1737476040&do=diff Question 7 Exercise 8.2 Solutions of Question 7 of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$\sin ^{2} \alpha \cos ^{2} \alpha$$\begin{align*} \sin ^{2} \alpha \cos ^{2} \alpha &= \left(\frac{1-\cos 2\alpha}{2} \right)\left(\frac{1+\cos 2\alpha}{2} \right)\\ &= \frac{1}{4}(1-\cos^2 2\alpha) \\ &=\frac{1}{4}\left(1-\frac{1+\cos 4\alp… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p6?rev=1737476040&do=diff Question 8(i, ii & iii) Exercise 8.2 Solutions of Question 8(i, ii & iii) of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $(\sin \theta+\cos \theta)^{2}=1+\sin 2 \theta$\begin{align*} LHS & = (\sin \theta+\cos \theta)^{2} \\ &=\sin^2\theta + \cos^2\theta +2\sin \theta \cos\theta\\ &= 1+2\sin \theta \cos\theta \quad (\because \sin^2\theta… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p7?rev=1737476040&do=diff Question 8(iv, v & vi) Exercise 8.2 Solutions of Question 8(iv, v & vi) of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\csc 2 \alpha=\dfrac{\tan \alpha+\cot \alpha}{2}$\begin{align*} RHS & = \dfrac{\tan \alpha+\cot \alpha}{2} \\ & = \dfrac{1}{2}\left(\frac{\sin\alpha}{\cos\alpha}+\frac{\cos\alpha}{\sin\alpha} \right)\\ \end{align*}$8 \… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p8?rev=1737476040&do=diff Question 8(vii, viii & ix) Exercise 8.2 Solutions of Question 8(vii, viii & ix) of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin 2 \theta=2 \cot \theta \sin ^{2} \theta$\begin{align*} RHS &= 2 \cot \theta \sin ^{2} \theta\\ &= 2 \frac{\cos \theta }{\sin \theta} \sin ^{2} \theta\\ &= 2 \cos \theta \sin\theta\\ &= \sin2 \theta\\ &=LH… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p9?rev=1737476040&do=diff Question 8(x, xi & xii) Exercise 8.2 Solutions of Question 8(x, xi & xii) of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sec 2 x=\dfrac{\cos x}{\cos x+\sin x}+\dfrac{\sin x}{\cos x-\sin x}$\begin{align*} RHS &= \dfrac{\cos x}{\cos x+\sin x}+\dfrac{\sin x}{\cos x-\sin x}\\ &=\dfrac{\cos x(\cos x-\sin x)+\sin x(\cos x+\sin x)}{(\cos x+\… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p10?rev=1737476040&do=diff Question 8(xiii, xiv & xv) Exercise 8.2 Solutions of Question 8(xiii, xiv & xv) of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\csc 2 \alpha-\cot 2 \alpha=\tan \alpha$\begin{align*} LHS &= \csc 2 \alpha-\cot 2 \alpha\\ &=\frac{1}{\sin 2 \alpha}- \frac{\cos2 \alpha}{\sin 2\alpha }\\ &=\frac{1-\cos2 \alpha}{\sin2 \alpha}\\ &= \frac{2\si… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p11?rev=1737476040&do=diff Question 8(xvi, xvii & xviii) Exercise 8.2 Solutions of Question 8(xvi, xvii & xviii) of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\dfrac{1-\cos ^{2} \beta}{2-2 \cos \beta}=\cos ^{2} \dfrac{\beta}{2}$\begin{align*} LHS &= \dfrac{1-\cos ^{2} \beta}{2-2 \cos \beta}\\ &= \dfrac{\sin ^{2} \beta}{2-2 \cos \beta}\\ &=\dfrac{4\sin ^{2} \fr… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p12?rev=1737476040&do=diff Question 8(xix, xx, xxi & xxii) Exercise 8.2 Solutions of Question 8(xix, xx, xxi & xxii) of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$\frac{\sin 2 \alpha}{\sin \alpha}-\frac{\cos 2 \alpha}{\cos \alpha}=\sec \alpha$$\begin{align*} LHS &= \dfrac{\sin 2 \alpha}{\sin \alpha}-\frac{\cos 2 \alpha}{\cos \alpha}\\ &= \dfrac{\sin 2 \alpha … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-3-p1?rev=1737476040&do=diff Question 1(i, ii, iii & iv) Exercise 8.3 Solutions of Question 1(i, ii, iii & iv) of Exercise 8.3 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$4 \sin 16x \cos 10x $$\begin{align*} &4 \sin 16x \cos 10x \\ & = 2 (2\sin 16x \cos 10x) \\ &= 2[\sin(16x+10x)+\sin(16x-10x)]\\ &= 2[\sin (26x)+\sin(6x)] \end{align*}$10 \cos 10y \cos 6y$\begin{align*} &10 \… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-3-p2?rev=1737476040&do=diff Question 1(v, vi, vii & viii) Exercise 8.3 Solutions of Question 1(v, vi, vii & viii) of Exercise 8.3 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $ \sin(-u) \sin 5u$\begin{align*} &\sin(-u) \sin 5u \\ =& -\sin u \sin 5u \\ =& -\frac{1}{2}[\cos(u - 5u) - \cos(u + 5u)] \\ = &-\frac{1}{2}[\cos(-4u) - \cos(6u)] \\ =& \frac{1}{2}[\cos(6u) - \cos(4u) ] \e… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-3-p3?rev=1737476040&do=diff Question 1(ix, x & xi) Exercise 8.3 Solutions of Question 1(ix, x & xi) of Exercise 8.3 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 \sin 75{\circ} \sin 15{\circ}$\begin{align*} &\quad2 \sin 75^{\circ} \sin 15^{\circ} \\ &= \cos(75^{\circ} - 15^{\circ}) - \cos(75^{\circ} + 15^{\circ}) \\ &= \cos 60^{\circ} - \cos 90^{\circ} \\ \end{align*}$4 \sin … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-3-p4?rev=1737476040&do=diff Question 2(i, ii, iii, iv and v) Exercise 8.3 Solutions of Question 2(i, ii, iii, iv and v) of Exercise 8.3 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin 70^{\circ} + \sin 30^{\circ}$\begin{align*} & \quad \sin 70^{\circ} + \sin 30^{\circ} \\ & = 2 \sin \left(\frac{70+30}{2} \right) \cos \left(\frac{70-30}{2} \right) \\ & = 2 \sin \left(\frac{1… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-3-p5?rev=1737476040&do=diff Question 3(i, ii, iii, iv & v) Exercise 8.3 Solutions of Question 3(i, ii, iii, iv & v) of Exercise 8.3 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\dfrac{\cos (\alpha + \beta)}{\cos(\alpha - \beta)}=\dfrac{1- \tan \alpha \tan \beta}{1+ \tan \alpha \tan \beta}$\begin{align*} RHS & = \dfrac{1- \tan \alpha \tan \beta}{1+ \tan \alpha \tan \beta} \\ & … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-3-p6?rev=1737476040&do=diff Question 3(vi, vii, viii, ix & x) Exercise 8.3 Solutions of Question 3(vi, vii, viii, ix & x) of Exercise 8.3 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2\tan y \cos 3y= \sec y(\sin 4y-\sin 2y)$\begin{align*} LHS & = 2\tan y \cos 3y \\ & = 2 \cdot \frac{\sin y}{\cos y} \cos 3y \\ & = \sec y (2 \cos 3y \sin y) \\ & = \sec y \left(\sin (3y+y)-\sin (… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-3-p7?rev=1737476040&do=diff Question 3(xi, xii & xiii) Exercise 8.3 Solutions of Question 3(xi, xii & xiii) of Exercise 8.3 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2\cos2u \cos u-\sin 2u \sin u=2\cos^3 u$\begin{align*} LHS & = 2\cos 2u \cos u - \sin 2u \sin u \\ & = 2\left(\cos^2 u - \sin^2 u\right) \cos u - 2\sin u \cos u \sin u \\ & = 2\cos^3 u - 2\sin^2 u \cos u \\ & =… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-3-p8?rev=1737476040&do=diff Question 4 Exercise 8.3 Solutions of Question 4 of Exercise 8.3 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\cos 80^{\circ} \cos 60^{\circ} \cos 40^{\circ} \cos 20^{\circ}=\dfrac{1}{16}$\begin{align*} LHS &= \cos 80^\circ \cos 60^\circ \cos 40^\circ \cos 20^\circ \\ &= \cos 80^\circ \left(\frac{1}{2}\right) \cos 40^\circ \cos 20^\circ \\ &= \frac{1… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/re-ex-p1?rev=1737476040&do=diff Question 1, Review Exercise Solutions of Question 1 of Review Exercise of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin \left(45^{\circ}-30^{\circ}\right)=\ldots$$\frac{\sqrt{6}-\sqrt{2}}{4}$$\frac{\sqrt{6}+\sqrt{2}}{4}$$\frac{\sqrt{6}-\sqrt{2}}{2}$$\frac{\sqrt{3}-\sqrt{2}}{2}$$\tan \left(\frac{\pi}{6}+\frac{\pi}{4}\right)=\ldots$$\frac{\sqrt{3}-1}… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/re-ex-p2?rev=1737476040&do=diff Question 2, Review Exercise Solutions of Question 2 of Review Exercise of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin \theta=\dfrac{3}{5}, \sin \phi=\dfrac{5}{13}$$\theta$$\phi$$\sin (\theta-\phi)$$\sin \theta=\dfrac{3}{5}$$\sin \phi=\dfrac{5}{13}$$\theta$$\phi$\begin{align*} \cos^2 \theta &= 1-\sin^2\theta\\ &= 1-\left(\frac{3}{5}\right)^2\\ & =… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/re-ex-p3?rev=1737476040&do=diff Question 3, Review Exercise Solutions of Question 3 of Review Exercise of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\frac{1}{\sqrt{2}}(\sin \beta+\cos \beta)$\begin{align*} &\frac{1}{\sqrt{2}}(\sin \beta+\cos \beta)\\ =& \sin \frac{\pi}{4}\sin \beta+\cos \frac{\pi}{4}\cos \beta\\ =& \cos(\beta -\frac{\pi}{4}) \end{align*}$\frac{1}{\sqrt{2}} \sin 75^… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/re-ex-p4?rev=1737476040&do=diff Question 4, Review Exercise Solutions of Question 4 of Review Exercise of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\dfrac{1+\tan 15^{\circ}}{1-\tan 15^{\circ}}$\begin{align*} &\frac{1+\tan 15^{\circ}}{1-\tan 15^{\circ}}\\ =&\frac{1+\tan 15^{\circ}}{1-1 \cdot \tan 15^{\circ}}\\ =&\frac{\tan 45^{\circ} + \tan 15^{\circ}}{1 - \tan 45^{\circ} \tan 15^{… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/re-ex-p5?rev=1737476040&do=diff Question 5 and 6, Review Exercise Solutions of Question 5 and 6 of Review Exercise of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\tan \theta$$\tan \left(\theta-45^{\circ}\right)=\frac{1}{3}$\begin{align*} & \frac{\tan \theta - \tan 45^{\circ}}{1 + \tan \theta \cdot \tan 45^{\circ}} =\frac{1}{3}\\ \implies & \frac{\tan \theta - 1}{1 + \tan \theta}= \f… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/re-ex-p6?rev=1737476040&do=diff Question 7, Review Exercise Solutions of Question 7 of Review Exercise of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\dfrac{4 \sin ^{2} \theta \cos \theta}{\cos 3 \theta+\cos \theta}=\tan 2 \theta \tan \theta$\begin{align*} LHS&=\frac{4 \sin^2 \theta \cos \theta}{\cos 3 \theta + \cos \theta}\\ &=\frac{4 \sin \theta\sin \theta \cos \theta}{4\cos^ 3 \t… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/re-ex-p7?rev=1737476040&do=diff Question 8, Review Exercise Solutions of Question 8 of Review Exercise of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$\sqrt{\frac{\cos \left(90^{\circ}+x\right) \sec (-x) \tan \left(180^{\circ}-x\right)}{\sec \left(360^{\circ}-x\right) \sin \left(180^{\circ}+x\right) \cot \left(90^{\circ}-x\right)}}=i .$$\begin{align*} LHS&= \sqrt{\frac{\cos \left(90… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/re-ex-p8?rev=1737476040&do=diff Question 9, Review Exercise Solutions of Question 9 of Review Exercise of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$\sqrt{\frac{\left(1-\tan ^{2} x \cos (-x) \cos \left(360^{\circ}-x\right)\right) \tan 45^{\circ}}{\left\{\sin 90^{\circ}-\sin \left(180^{\circ}+x\right)\right\}\left\{\sin 90^{\circ}-\cos \left(90^{\circ}-x\right)\right\}}}$$\begin{al… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/re-ex-p9?rev=1737476040&do=diff Question 10, Review Exercise Solutions of Question 10 of Review Exercise of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin (16 x)=16 \sin (x) \cos (x) \cos (2 x) \cos (4 x) \cos (8 x)$\begin{align*} RHS&=16 \sin (x) \cos (x) \cos (2 x) \cos (4 x) \cos (8 x) \\ &= 8(2 \sin (x) \cos (x) )\cos (2 x) \cos (4 x) \cos (8 x) \\ &= 8 \sin2 (x) \cos (2 x) \… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/ex9-1-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 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p3?rev=1737476040&do=diff Question 3, Exercise 9.1 Solutions of Question 3 of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y=7 \cos 4x$\begin{align*} & -1\leq \cos 4x \leq 1 \,\, \forall \,\, x\in \mathbb{R} \\ \implies & -7\leq 7 \cos 4x \leq 7 \\ \end{align*}$= ]-\infty, \infty[ = \mathbb{R}$$=[-7,7]$$y=\cos \frac{x}{3}$\begin{align*} & -1\leq \cos \frac{x}{3} \… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p4?rev=1737476040&do=diff Question 4(i-iv), Exercise 9.1 Solutions of Question 4(i-iv) of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y=\sin x+x \cdot \cos x$$f(x)=\sin x+x \cdot \cos x$\begin{align*} f(-x) = \sin (-x) + (-x)\cdot \cos (-x) \end{align*}$\sin(-x)=-\sin x$$\cos (-x) = \cos x$\begin{align*} f(x) & = -\sin x - x \cdot \cos x \\ & = -(\sin x + x \cdot … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p5?rev=1737476040&do=diff Question 4(v-viii), Exercise 9.1 Solutions of Question 4(v-viii) of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y=\dfrac{\sin ^{2} x}{x+\tan x}$\[y = \frac{\sin^2 x}{x + \tan x}\]\begin{align*} y(-x) &= \frac{\big(-\sin x\big)^2}{-x - \tan x} \\ &= \frac{\sin^2 x}{-x - \tan x}\\ & = \frac{\sin^2 x}{-(x + \tan x)}\\ & = -\frac{\sin^2 x}{x +… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p6?rev=1737476040&do=diff Question 5(i-v), Exercise 9.1 Solutions of Question 5(i-v) of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y=2 \operatorname{Sin} x$$y=2 \operatorname{Cos} 3 x$$y=2 \operatorname{Tan} 2 x$$\mathrm{y}=\operatorname{Cos} \frac{\mathrm{x}}{2}$ text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p7?rev=1737476040&do=diff Question 5(vi-x), Exercise 9.1 Solutions of Question 5(vi-x) of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y=2 \operatorname{Sin} 3 x$$y=3 \operatorname{Cos} x$$y=\operatorname{Cos}^{2} x$$y=\operatorname{Sin}^{2} x$$y=\operatorname{Tan}^{2} x$$y=\operatorname{Sin} \frac{x}{2}$ text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p8?rev=1737476040&do=diff Question 6, Exercise 9.1 Solutions of Question 6 of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y=6 \sec(2 x-3)$$\sec$$2\pi$\begin{align*} 6 \sec(2 x-3) & = 6 \sec(2 x-3+2\pi) \\ & = 6 \sec(2(x+\pi)-3) \end{align*}$6 \sec(2 x-3)$$\pi$$y=\cos (5 x+4)$$\cos$$2\pi$\begin{align*} \cos (5 x+4) & = 6 \cos(5x+4+2\pi) \\ & = \cos\left(5\left(x+\fr… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p9?rev=1737476040&do=diff Question 7 & 8, Exercise 9.1 Solutions of Question 7 & 8 of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y=\operatorname{Sin} x$$y=\operatorname{Sin} 2 x$$[0,2 \pi]$$y=\operatorname{Cos} x$$y=\operatorname{Cos} 2 x$$[0,2 \pi]$ text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p10?rev=1737476040&do=diff Question 9, Exercise 9.1 Solutions of Question 9 of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin x=\cos x$$\cos x=x$$\sin x=x$$\tan x=x$ text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p11?rev=1737476040&do=diff Question 10, Exercise 9.1 Solutions of Question 10 of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $ V(t)=a \operatorname{Sin}(k(t-d))+c$$56 \mathrm{~Hz} A C$$k$ text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/re-ex-1-p2?rev=1737476040&do=diff Question 2 and 3,Review Exercise Solutions of Question 2 and 3 of Review Exercise of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/re-ex-1-p3?rev=1737476040&do=diff Question 4, Review Exercise Solutions of Question 4 of Review Exercise of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/re-ex-p1?rev=1737476040&do=diff Question 1,Review Exercise Solutions of Question 1 of Review Exercise of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\cos \theta=\frac{\sqrt{3}}{2}$$\sin \theta=$$\frac{1}{2}$$-\frac{1}{2}$$\sqrt{3}$$-\frac{2}{\sqrt{3}}$$\tan (-15 \pi)=$$ 0$$-1$$1$$2 \sin \theta+\frac{1}{2}cosec \theta \theta $$\theta=45^{\circ}$$\frac{1}{\sqrt{2}}$$\frac{1}{3}$$\frac{3}{… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/re-ex-p2?rev=1737476040&do=diff Question 2 and 3, Review Exercise Solutions of Question 2 and 3 of Review Exercise of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\cos \theta -\sin \theta=\sqrt{2}\sin \theta,$$\cos \theta+ \sin \theta=\sqrt{2} \cos \theta$$$\cos \theta -\sin \theta=\sqrt{2}\sin \theta$$\begin{align*} & \cos \theta=\sqrt{2}\sin \theta + \sin \theta \\ \implies & \cos \the… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/re-ex-p3?rev=1737476040&do=diff Question 4, Review Exercise Solutions of Question 4 of Review Exercise of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/re-ex-p4?rev=1737476040&do=diff Question 5 and 6, Review Exercise Solutions of Question 5 and 6 of Review Exercise of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/re-ex-p5?rev=1737476040&do=diff Question 7, Review Exercise Solutions of Question 7 of Review Exercise of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/re-ex-p6?rev=1737476040&do=diff Question 8, Review Exercise Solutions of Question 8 of Review Exercise of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/re-ex-p7?rev=1737476040&do=diff Question 9, Review Exercise Solutions of Question 9 of Review Exercise of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/re-ex-p8?rev=1737476040&do=diff Question 10(i-v), Review Exercise Solutions of Question 10(i-v) of Review Exercise of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/re-ex-p9?rev=1737476040&do=diff Question 10(vi-x), Review Exercise Solutions of Question 10(vi-x) of Review Exercise of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/re-ex-p10?rev=1737476040&do=diff Question 10(xi-xv), Review Exercise Solutions of Question 10(xi-xv) of Review Exercise of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.