This is beta site. https://beta.mathcity.org/ 2026-06-05T17:05:43+00:00 https://beta.mathcity.org/ https://beta.mathcity.org/_media/logo.png text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-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/ex8-2-p2?rev=1737476040&do=diff Question 4 Exercise 8.2 Solutions of Question 4 of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin 2 \theta$$\cos 2 \theta$$\tan 2 \theta$$\sin \frac{\theta}{2}$$\cos \frac{\theta}{2}$$\tan \frac{\theta}{2}$$\cos \theta=\frac{3}{5}$$0<\theta<\frac{\pi}{2}$$\cos\theta=\dfrac{3}{5}$$0<\theta<\dfrac{\pi}{2}$$\theta$$$\sin\theta = \pm \sq… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p9?rev=1737476039&do=diff Question 9, Exercise 1.2 Solutions of Question 9 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 9(i) $(2+4 i)^{-1}$$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 2025-01-21T16:13:57+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/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 $$2\sin \alpha \sin \beta =\cos \left( \alpha -\beta \right)-\cos \left( \alpha +\beta \r… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/re-ex10-p5?rev=1737476039&do=diff Question 8 & 9, Review Exercise 10 Solutions of Question 8 & 9 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin \left( \dfrac{\pi }{4}-\theta \right)\sin \left( \dfrac{\pi }{4}+\theta \right)=\dfrac{1}{2}\cos 2\theta $$2\sin \alpha \sin \beta =\cos \left( \alpha -\beta \right)-\cos \left( \alpha +\beta \r… text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/quote-of-the-day/mar?rev=1737476042&do=diff Quotes for the March “”“”“” “” “” --- “”“”“” --- “”“”“” --- “”“”“” --- “”“”“” --- “”“”“” --- “”“”“” --- “”“”“” --- “” text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/quote-of-the-day/may?rev=1737476042&do=diff Quotes for the May “”“”“” مورس کلائن (1908-1992) ---Morris Kline (1908-1992) --- “”“”“” ڈی آرسی تھامسن (1860-1948) ---D'Arcy Thompson (1860-1948) --- “”“”“” Vito Volterra (1860-1940) ---Vito Volterra (1860-1940) text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-ptb/sol/ch01/ex1-2?rev=1737476037&do=diff Exercise 1.2 (Solutions) <lead>Notes (Solutions) of Exercise 1.2: Textbook of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board (PTB) Lahore.</lead> The main topics of this exercise are complex numbers, real part and imaginary part of complex numbers, properties of the fundamental operation on complex numbers, complex number as ordered pair of real numbers and special subset of complex numbers. These notes are based on the new Student Learning Outcomes… text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/quote-of-the-day/apr?rev=1737476042&do=diff Quotes for the April “”“”“” لیو لینڈاؤ (1908-1968) ---Lev Landau (1908-1968) --- “”“”“” Hannes Alfvén (1908-1995) ---Hannes Alfvén (1908-1995) --- “”“”“” جوزف برٹرینڈ (1822-1900) ---Joseph Bertrand (1822-1900) text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/quote-of-the-day/feb?rev=1737476042&do=diff Quotes for the February “”“”“” --- “”“”“” --- “”“”“” --- “”“”“” --- “”“”“” --- “”“”“” --- “”“”“” --- “”“”“” --- “”“”“ text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/matric/9th_science/unit_02/exercise_2.4?rev=1737476041&do=diff Exercise 2.4 (Solutions) Question 1 Use law of exponent to simplify. * (i) $\frac{(243)^{\frac{-2}{3}}(32)^{\frac{-1}{5}}}{\sqrt(196)^{-1}}$ * (ii) $\left(2x^5y^{-4}\right)\left(-8x^{-3}y^2\right)$ * (iii) $\left(\frac{x^{-2}y^{-1}z^{-4}}{x^4y^{-3}z^0}\right)^{-3}$ * (iv) $\frac{\left(81\right)^n.3^5-\left(3\right)^{4n-1}\left(243\right)}{\left(9^2n\right)\left(3^3\right)}$ Solution (i) $$\begin{array}{cl} \begin{array}{cl} \frac{(243)^{\frac{-2}{3}}(32)^{\frac{-1… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p3?rev=1737476037&do=diff Question 3, Exercise 2.1 Solutions of Question 3 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3(i) $A=\begin{bmatrix}x & y & z\end{bmatrix}$$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 2025-01-21T16:13:56+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}$\begin{align}\sin (\alpha -\beta )=\sin \alpha \cos \beta -\cos \alpha \sin.\end{align}\begin{align} \Rightarrow \quad \sin \left( \frac{\pi }{3}-\f… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p2?rev=1737476039&do=diff Question 2, Exercise 10.1 Solutions of Question 2 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\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 2025-01-21T16:13:57+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 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-3-p4?rev=1737476039&do=diff Question 5, Exercise 10.3 Solutions of Question 5 of Exercise 10.3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$\cos {{20}^{\circ }}\cos {{40}^{\circ }}\cos {{60}^{\circ }}\cos {{80}^{\circ }}=\dfrac{1}{16}.$$$2\cos \alpha \cos \beta =\cos \left( \alpha +\beta \right)+\cos \left( \alpha -\beta \right)$\begin{align}L.H.S.&=\cos {… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-5-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 2025-01-21T16:13:57+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}$$2\cos \alpha \cos \beta =\cos \left( \alpha +\beta \right)+\cos \left( \alpha -\beta \right)$\begin{align}L.H.S.&=\cos {{20… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-3-p5?rev=1737476039&do=diff Question 5, Exercise 10.3 Solutions of Question 5 of Exercise 10.3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos {{20}^{\circ }}\cos {{40}^{\circ }}\cos {{60}^{\circ }}\cos {{80}^{\circ }}=\dfrac{1}{16}$$2\cos \alpha \cos \beta =\cos \left( \alpha +\beta \right)+\cos \left( \alpha -\beta \right)$\begin{align}L.H.S.&=\cos {{20… text/html 2025-01-21T16:13:57+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 2025-01-21T16:13:59+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}$$z=x+iy$\begin{align*} &\arg (z-1)=-\dfrac{\pi}{4} \\ \implies & \arg(x+iy-1) = -\dfrac{\pi}{4} \\ \implies & \arg(x-1+iy) = -\dfrac{\pi}{4} \\ \implies & \tan^{-1}\left(\dfrac{y}{x-1}\right) = -\dfrac{\pi}{4} … text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-3-p5?rev=1737476036&do=diff Question 6, Exercise 1.3 Solutions of Question 6 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6(i) ${{z}^{4}}+{{z}^{2}}+1=0$\begin{align}{{z}^{4}}+{{z}^{2}}+1&=0\\ {{z}^{4}}+2\left( \dfrac{1}{2} \right){{z}^{2}}+\dfrac{1}{4}-\dfrac{1}{4}+1&=0\\ {{\left( {{z}^{2}}+\dfrac{1}{2} \right)}^{2}}+\dfrac{4-1}{4}&=0\\ {{\left( {{z}^{2}}+\dfrac{1}{2} \righ… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-5-p3?rev=1737476039&do=diff Question 3, Exercise 2.5 Solutions of Question 3 of Exercise 2.5 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left[\begin{array}{ccc}0 & -1 & -1 \\ -1 & 3 & 0 \\ 1 & -1 & 4\end{array}\right]$$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 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-ptb/sol/ch02/ex2-8?rev=1737476037&do=diff Exercise 2.8 (Solutions) <lead>Notes (Solutions) of Exercise 2.8: Textbook of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board (PTB) Lahore.</lead> The main topic of this exercise are binary operation, semi-group, monoid, groups and abelian groups. These notes are based on the new Student Learning Outcomes (SLOs). Version: 4.1, Available at MathCity.org $\oplus$$G=\{0,1\}$\[ \begin{array}{|c|c|c|} \hline \oplus & 0 & 1 \\ \hline 0 & 1 & 1 \\ \hl… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p10?rev=1737476038&do=diff Question 10 Exercise 7.2 Solutions of Question 10 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$n=2 ;$$s=2^{n-1}$$$ \left.(1+x)^n=\left(\begin{array}{l} n \\ \vdots \end{array}\right)+\left(\begin{array}{l} m \\ 1 \end{array}\right) x+\left(\begin{array}{l} n \\ 2 \end{array}\right) x^2-\ldots+i_n^*\right) x^n \cdot $$$x=1$$(1 \div 1… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/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/ex2-2-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/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 2025-01-21T16:13:56+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$${{z}_{3}}=2+3i$\begin{align}{{z}_{1}}\left( {{z}_{2}}+{{z}_{3}} \right)&={{z}_{1}}{{z}_{2}}+{{z}_{1}}{{z}_{… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-2-p3?rev=1737476037&do=diff Question 3 & 4, Exercise 1.2 Solutions of Question 3 & 4 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 ${{z}_{1}}=\sqrt{3}+\sqrt{2}i$${{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$\begin{align}{{z}_{1}}\left( {{z}_{2}}+{{z}_{3}} \right)&={{z}_{1}}{{z}_{2}}+{{z}_{1}}{{z}_{… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p5?rev=1737476039&do=diff Question 5 and 6 Exercise 7.3 Solutions of Question 5 and 6 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$x$$x^2$$x$$$ \frac{(8+3 x)^{\frac{2}{3}}}{(2+3 x) \sqrt{4-5 x}}=1-\frac{5 x}{8} $$$$ \frac{\sqrt[4]{3}-3 x j^{\frac{2}{3}}}{2 \cdot 3 x+4-5 x} $$$$ \begin{aligned} & =\frac{8^{\frac{2}{3}}\left(1+\frac{3 x}{8}\right)^{\frac{2}{3}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p3?rev=1737476039&do=diff Question 3, Exercise 1.4 Solutions of Question 3 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 3(i) $\left(x_{1}+i y_{1}\right)\left(x_{2}+i y_{2}\right)\left(x_{3}+i y_{3}\right) \ldots\left(x_{n}+i y_{n}\right)=a+i b$$\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 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-1-p5?rev=1737476036&do=diff Question 5, Exercise 10.1 Solutions of Question 5 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\tan \alpha =\dfrac{3}{4}$$\sec \beta =\dfrac{13}{5}$$\alpha$$\beta$$\sin \left( \alpha +\beta \right)$$\tan\alpha =\dfrac{3}{4}$$\tan\alpha$$\alpha$\begin{align}{{\sec}^{2}}\alpha &=1+{{\tan}^{2}}\alpha\\ \Rightarrow \q… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-2-p6?rev=1737476036&do=diff Question 7, Exercise 10.2 Solutions of Question 7 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${{\cos }^{4}}\theta -{{\sin }^{4}}\theta =\dfrac{1}{\sec 2\theta }$\begin{align}L.H.S&={{\cos }^{4}}\theta -{{\sin }^{4}}\theta \\ &=\left( {{\cos }^{2}}\theta -{{\sin }^{2}}\theta \right)\left( {{\cos }^{2}}\theta +{{\… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p5?rev=1737476039&do=diff Question 5, Exercise 10.1 Solutions of Question 5 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\tan \alpha =\dfrac{3}{4}$$\sec \beta =\dfrac{13}{5}$$\alpha$$\beta$$\sin \left( \alpha +\beta \right)$$\tan\alpha =\dfrac{3}{4}$$\tan\alpha$$\alpha$\begin{align}{{\sec}^{2}}\alpha &=1+{{\tan}^{2}}\alpha\\ \Rightarrow \q… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-2-p6?rev=1737476039&do=diff Question 7, Exercise 10.2 Solutions of Question 7 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${{\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 2025-01-21T16:13:56+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/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$$\forall n\in N$\begin{align}{{i}^{n}}+{{i}^{n+1}}+{{i}^{n+2}}+{{i}^{n+3}}&=0\\ L.H.S.&={{i}^{n}}+{{i}^{n}}\cdot i+{{i}^{n}}\cdot {{i}^{2}}+{{i}^{n}}\cdot {{i}^{3}}\\ &={{i}^{n}}\left( 1+i+{{i}^{2}}… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-1-p2?rev=1737476037&do=diff Question 2 & 3, Exercise 1.1 Solutions of Question 2 & 3 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2 ${{i}^{107}}+{{i}^{112}}+{{i}^{122}}+{{i}^{153}}=0$\begin{align}L.H.S.&={{i}^{107}}+{{i}^{112}}+{{i}^{122}}+{{i}^{153}}\\ &=i\cdot i^{106}+i^{112}+i^{122}+i\cdot i^{152}\\ &=i.{{\left( {{i}^{2}} \right)}^{53}}+{{\left( {{i}^{2}} \right)}^{56}}+… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-3-p7?rev=1737476039&do=diff Question 7, Exercise 2.3 Solutions of Question 7 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $(A B)^{-1}=B^{-1} A^{-1}$$A=\left[\begin{array}{ll}2 & 1 \\ 8 & 6\end{array}\right]$$B=\left[\begin{array}{ll}3 & 2 \\ 0 & 2\end{array}\right]$\begin{align*} A &= \left[\begin{array}{ll}2 & 1 \\ 8 & 6\end{array}\right] \\ |A|& = 12 - 8 = 4\\ … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-5-p2?rev=1737476039&do=diff Question 2, Exercise 2.5 Solutions of Question 2 of Exercise 2.5 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left[\begin{array}{ccc}5 & 9 & 3 \\ 3 & -5 & 6 \\ 2 & 10 & 6\end{array}\right]$\begin{align*}&\quad\left[ \begin{array}{ccc} 5 & 9 & 3 \\ 3 & -5 & 6 \\ 2 & 10 & 6 \end{array} \right]\\ \sim & \text{R}\left[ \begin{array}{ccc} 1 & \frac{9}{… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-8-p4?rev=1737476040&do=diff Question 7 and 8, Exercise 4.8 Solutions of Question 7 and 8 of Exercise 4.8 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$$\frac{1}{1 \times 4}+\frac{1}{4 \times 7}+\frac{1}{7 \times 10}+\ldots$$$$\frac{1}{1 \times 4}+\frac{1}{4 \times 7}+\frac{1}{7 \times 10}+\dots$$$T_k$\begin{align*} T_k &=\frac{1}{(3k-2)(3k+1)}. \end{align*}\begin{align*} \frac{1}{(3… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-1-p8?rev=1737476036&do=diff Question 8, Exercise 10.1 Solutions of Question 8 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\tan \left( \dfrac{\pi }{4}+\theta \right)=\dfrac{\cos \theta +\sin \theta }{\cos \theta -\sin \theta }$\begin{align}L.H.S.&=\tan \left( \dfrac{\pi }{4}+\theta \right)\\ &=\dfrac{\sin \left( \dfrac{\pi }{4}+\theta \ri… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-1-p9?rev=1737476036&do=diff Question 9 and 10, Exercise 10.1 Solutions of Question 9 and 10 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{\sin \theta }{\sec 4\theta }+\dfrac{\cos \theta }{\cos ec4\theta }=\sin 5\theta $\begin{align}L.H.S.&=\dfrac{\sin \theta }{\sec 4\theta }+\dfrac{\cos \theta }{\cos ec4\theta }\\ &=\dfrac{\sin \theta }… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p1?rev=1737476039&do=diff Question 1 Exercise 7.3 Solutions of Question 1 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\frac{1}{2}$$$ \begin{aligned} & (1-x)^{\frac{1}{2}}=1+\frac{1}{2} x+ \\ & \frac{\frac{1}{2}\left(-\frac{1}{2}-1\right)}{2 !}(-x)^2 \end{aligned} $$$$ \begin{aligned} & +\frac{-\frac{1}{2}\left(-\frac{1}{2}-1\right)\left(-\frac{1}{2}-2\right… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p8?rev=1737476039&do=diff Question 8, Exercise 10.1 Solutions of Question 8 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\tan \left( \dfrac{\pi }{4}+\theta \right)=\dfrac{\cos \theta +\sin \theta }{\cos \theta -\sin \theta }$\begin{align}L.H.S.&=\tan \left( \dfrac{\pi }{4}+\theta \right)\\ &=\dfrac{\sin \left( \dfrac{\pi }{4}+\theta \ri… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p9?rev=1737476039&do=diff Question 9 and 10, Exercise 10.1 Solutions of Question 9 and 10 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{\sin \theta }{\sec 4\theta }+\dfrac{\cos \theta }{\cos ec4\theta }=\sin 5\theta $\begin{align}L.H.S.&=\dfrac{\sin \theta }{\sec 4\theta }+\dfrac{\cos \theta }{\cos ec4\theta }\\ &=\dfrac{\sin \theta }… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p8?rev=1737476037&do=diff Question 9, Exercise 2.1 Solutions of Question 9 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9(i) $A=\begin{bmatrix}2 & -1 & 3 \\1 & \quad 0 & 1 \end{bmatrix},$$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],$$$$B=\left[… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p10?rev=1737476038&do=diff Question 10 Exercise 7.1 Solutions of Question 10 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\left(\begin{array}{1}5 \\5 \end{array}\right)+\left(\begin{array}{l}6 \\ 5\end{array}\right)+\left(\begin{array}{l}7 \\ 5\end{array}\right)+\ldots+\left(\begin{array}{c}n+4 \\ 5\end{array}\right)=\left(\begin{array}{c}n+5 \\ 6\end{array}\… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p8?rev=1737476040&do=diff Question 9, Exercise 8.1 Solutions of Question 9 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\alpha$$\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$\begin{align*… text/html 2025-01-21T16:13:54+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/atiq/sp23-mth321?rev=1737476034&do=diff MTH321: Real Analysis I (Spring 2023) ~~DISCUSSION~~ [Photo-illustration of Zeno's Paradox] At the end of this course the students will be able to understand the basic set theoretic statements and emphasize the proofs’ development of various statements by induction. Define the limit of, a function at a value, a sequence and the Cauchy criterion. Prove various theorems about limits of sequences and functions and emphasize the proofs’ development. Define continuity of a function and uniform con… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/definitions?rev=1737476039&do=diff Definitions: Mathematics 11 NBF Model Textbook of Mathematics for Class XI is published by National Book Foundation (NBF), Islamabad, Pakistan. NBF can be considered as Federal Textbook Board Islamabad. The book has total of nine (9) chapters. Definition of the book provide the quick overview of the book.$x+iy$$x,y\in\mathbb{R}$$i^2=1$$\mathbb{C}$$x+i y$$x$$y$$x$$y$$Re(z)=x$$Im(z)=y$$z=x+i y$$\bar{z}$$\bar{z}=x-i y$$z=x+i y$$|z|$$|z|=\sqrt{x^{2}+y^{2}}$$z=x+iy$$Re(z)=x$$Im(z)=y$$Re(z^{-1})= \d… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-ptb/important-questions/ch03-matrices-and-determinants?rev=1737476037&do=diff Ch 03: Matrices and Determinants <list-group> * Fin $x$ and $y$ if $ \left[ {\begin{array}{c} x+3&1\\ -3& 3y-4 \end{array}} \right]= \left[ {\begin{array}{c} 2&1\\ -3&2 \end{array}} \right]$ --- BISE Gujrawala(2015) * Solve for matrix $A$ if $\left[ {\begin{array}{c}4&3\\ 2&2 \end{array}} \right]A-\left[ {\begin{array}{c} 2&3\\ -1&-2 \end{array}} \right]= \left[ {\begin{array}{c} -1&-4\\ 3&6 \end{array}} \right]$ --- BISE Gujrawala(2015) * Prove without expansion $ \left[ {\begin{… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-1-p8?rev=1737476036&do=diff Question 9 & 10, Exercise 1.1 Solutions of Question 9 & 10 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9 $\dfrac{\left( 3-2i \right)\left( 2+3i \right)}{\left( 1+2i \right)\left( 2-i \right)}$\begin{align}z&=\dfrac{\left( 3-2i \right)\left( 2+3i \right)}{\left( 1+2i \right)\left( 2-i \right)}\\ &=\dfrac{6+6+9i-4i}{2+2+4i-i}\\ &=\dfrac{12+5i}{4+3… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-2-p6?rev=1737476036&do=diff Question 7, Exercise 1.2 Solutions of Question 7 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7(i) $\dfrac{2+3i}{5-2i}$\begin{align}&\dfrac{2+3i}{5-2i} \\ =&\dfrac{2+3i}{5-2i}\times \dfrac{5+2i}{5+2i} \quad \text{by rationalizing} \\ =&\dfrac{10-6+15i+4i}{25+4}\\ =&\dfrac{4+19i}{29}\\ =&\dfrac{4}{29}+\dfrac{19}{29}i \end{align}$=\dfrac{4}{29}$$=\… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-1-p8?rev=1737476037&do=diff Question 9 & 10, Exercise 1.1 Solutions of Question 9 & 10 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9 $\dfrac{\left( 3-2i \right)\left( 2+3i \right)}{\left( 1+2i \right)\left( 2-i \right)}$\begin{align}z&=\dfrac{\left( 3-2i \right)\left( 2+3i \right)}{\left( 1+2i \right)\left( 2-i \right)}\\ &=\dfrac{6+6+9i-4i}{2+2+4i-i}\\ &=\dfrac{12+5i}{4+3… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-2-p6?rev=1737476037&do=diff Question 7, Exercise 1.2 Solutions of Question 7 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7(i) $\dfrac{2+3i}{5-2i}$\begin{align}&\dfrac{2+3i}{5-2i} \\ =&\dfrac{2+3i}{5-2i}\times \dfrac{5+2i}{5+2i} \quad \text{by rationalizing} \\ =&\dfrac{10-6+15i+4i}{25+4}\\ =&\dfrac{4+19i}{29}\\ =&\dfrac{4}{29}+\dfrac{19}{29}i \end{align}$=\dfrac{4}{29}$$=\… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p2?rev=1737476039&do=diff Question 2 Exercise 7.3 Solutions of Question 2 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sqrt{26}$$$ \begin{aligned} & \sqrt{26}=\sqrt{25+1} \\ & =\sqrt{25} \sqrt{1+\frac{1}{25}}=5\left[1+\frac{1}{25}\right]^{\frac{1}{2}} \end{aligned} $$$$ \begin{aligned} & \sqrt{26}=5\left[1+\frac{1}{25}\right]^{\frac{1}{2}} \\ & =5\left[1+\f… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p11?rev=1737476039&do=diff Question 13 Exercise 7.3 Solutions of Question 13 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$x$$x^3$$x$$n^{\text {th }}$$1+x$$\frac{2 n+(n+1) x}{2 n+(n-1) x}$$$ \begin{aligned} & (1+x)^{\frac{1}{n}}=\frac{2 n+(n+1) x}{2 n+(n-1) x} \\ & \frac{2 n+(n+1) x}{2 n+(n-1) x} \\ & =1+\frac{1}{n} x+\frac{\frac{1}{n}\left(\frac{1}{n}-1\right… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p2?rev=1737476039&do=diff Question 2, Exercise 1.4 Solutions of Question 2 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 2(i) $\left(\cos \dfrac{\pi}{6}+i \sin \dfrac{\pi}{6}\right)\left(\cos \dfrac{\pi}{3}+i \sin \dfrac{\pi}{3}\right)$$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 2025-01-21T16:14:00+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 2025-01-21T16:13:56+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 2025-01-21T16:13:57+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/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 2025-01-21T16:13:59+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/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 2025-01-21T16:14:00+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$$\cos$\begin{alig… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-ptb/trigonometric-formulas?rev=1737476037&do=diff Trigonometric Formulas These are the common formulas used in Chapter 9 to 14 of Textbook of Algebra and Trigonometry Class XI, Punjab Textbook Board Lahore. This handout is very helpful to remember the formulas. All these formulas are given for real valued and defined trigonometric functions. A PDF file can be downloaded for high quality printing and a word file is also given if you wish to modify the contents or credit as you need. <panel> ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$$1+{{\tan… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/matric/9th_science/ex-6-3?rev=1737476041&do=diff Exercise 6.3 On the following page we have given the solution of Exercise 6.3 of Mathematics 9 (Science) published by Caravan Book House, Lahore. <WRAP center round info 60%> We have created this page and it will be updated to add new solutions occasionally. Please stay in touch with this page. </WRAP>$4x^2-12xy +9y^2$$x^2-1+\frac{1}{4x^2}, (x\neq 0)$$\frac{1}{16}x^2-\frac{1}{12}xy+ \frac{1}{36}y^2$$4(a+b)^2-12(a^2-b^2)+9(a-b)^2$$\frac{4x^6-12x^3y^3+9y^6}{9x^4-24x^2y^2+16y^4},(x \neq 0)$$\left(… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p13?rev=1737476037&do=diff Question 16 & 17, Exercise 2.2 Solutions of Questions 16 & 17 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$A=\begin{bmatrix}3 & -1 \\4 & 2\end{bmatrix}$$|A^{-1}|=\dfrac{1}{|A|}$$$A=\left[ \begin{matrix} 3 & -1 \\ 4 & 2 \\ \end{matrix} \right]$$$$|A|=6+4$$$$\Rightarrow |A|=10\ldots (1)$$$$A^{-1}=\dfrac{1}{|A|}AdjA$$$$AdjA=\left[ \begin{ma… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p1?rev=1737476039&do=diff Question 1, Exercise 1.4 Solutions of Question 1 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 1(i) $2+i 2 \sqrt{3}$$z=x+iy=2 + i 2 \sqrt{3}$\begin{align} r & = \sqrt{x^2 + y^2} = \sqrt{2^2 + (2\sqrt{3})^2} \\ & = \sqrt{4 + 12} = \sqrt{16} = 4. \end{align}\begin{align} \alpha & = \tan^{-1}\left|\frac{y}{x}\right| = \tan^{-1}\left|\fra… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/matric/9th_science/review_exercise?rev=1737476041&do=diff Review exercise On the following page we have given the solution of Review exercise of Mathematics 9 (Science) published by Caravan Book House, Lahore. <WRAP center round info 60%> We have created this page and it will be updated to add new solutions occasionally. Please stay in touch with this page. </WRAP>$x-2$$x^2+x-6$$x^2+x-6$$x+3$$x-2$$x+2$$c$$a^3+b^3$$a^2-ab+b^2$$a+b$$a^2-ab+b^2$$(a-b)^2$$a^2+b^2$$c$$x^2-5x+6$$x^2-x-6$$x-3$$x+2$$x^2-4$$x-2$$a$$a^2-b^2$$a^3-b^3$$a-b$$a+b$$a^2+ab+b^2$$a^2-a… text/html 2025-01-21T16:13:57+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$$$\left| \begin{matrix} 1 & 2 & 3 \\ -8 & 4 & -… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p5?rev=1737476040&do=diff Question 7 Exercise 8.2 Solutions of Question 7 of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$\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 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-1-p3?rev=1737476036&do=diff Question 4, Exercise 1.1 Solutions of Question 4 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4(i) $\left( a,0 \right)\left( 2,-b \right)$\begin{align}&\left( a,0 \right)-\left( 2,-b \right)\\ &=\left( a+0i \right)-\left( 2-bi \right)\\ &=\left( a-2 \right)+\left( 0+b \right)i\\ &=\left( a-2 \right)+bi\end{align}$\left( -3,\dfrac{1}{2} \right)\le… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-1-p4?rev=1737476036&do=diff Question 5, Exercise 1.1 Solutions of Question 5 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) $8i+11,-7+5i$\begin{align}&(8i+11)\times (-7+5i)\\ &=\left( 11+8i \right)\times \left( -7+5i \right)\\ &=\left( -77+40{{i}^{2}} \right)+\left( 55-56 \right)i\\ &=\left( -77+40\left( -1 \right) \right)+\left( 55-56 \right)i\\ &=\left( -77-40 \right)+… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-2-p7?rev=1737476036&do=diff Question 8, Exercise 1.2 Solutions of Question 8 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 8(i) $z+\overline{z}=2\operatorname{Re}\left( z \right)$$z=a+ib$$\overline{z}=a-ib$\begin{align}z+\overline{z}&=\left( a+ib \right)+\left( a-ib \right)\\ &=a+ib+a-ib\\ &=2a\\ z+\overline{z}&=2\operatorname{Re}\left( z \right)\end{align}$z-\overline{z}=2i… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-1-p4?rev=1737476036&do=diff Question, Exercise 10.1 Solutions of Question 4 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin \alpha =-\dfrac{4}{5}$$\cos \beta =-\dfrac{12}{13}$$\alpha $$\beta $$\sin \left( \alpha -\beta \right)$$\sin \alpha=-\dfrac{4}{5}$$\alpha$$\sin \beta=-\dfrac{12}{13}$$\beta$$$\cos \alpha=\pm \sqrt{1-\sin^2\alpha}.$$$\… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-1-p3?rev=1737476037&do=diff Question 4, Exercise 1.1 Solutions of Question 4 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4(i) $\left( a,0 \right)\left( 2,-b \right)$\begin{align}&\left( a,0 \right)-\left( 2,-b \right)\\ &=\left( a+0i \right)-\left( 2-bi \right)\\ &=\left( a-2 \right)+\left( 0+b \right)i\\ &=\left( a-2 \right)+bi\end{align}$\left( -3,\dfrac{1}{2} \right)\le… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-1-p4?rev=1737476037&do=diff Question 5, Exercise 1.1 Solutions of Question 5 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) $8i+11,-7+5i$\begin{align}&(8i+11)\times (-7+5i)\\ &=\left( 11+8i \right)\times \left( -7+5i \right)\\ &=\left( -77+40{{i}^{2}} \right)+\left( 55-56 \right)i\\ &=\left( -77+40\left( -1 \right) \right)+\left( 55-56 \right)i\\ &=\left( -77-40 \right)+… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-2-p7?rev=1737476037&do=diff Question 8, Exercise 1.2 Solutions of Question 8 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 8(i) $z+\overline{z}=2\operatorname{Re}\left( z \right)$$z=a+ib$$\overline{z}=a-ib$\begin{align}z+\overline{z}&=\left( a+ib \right)+\left( a-ib \right)\\ &=a+ib+a-ib\\ &=2a\\ z+\overline{z}&=2\operatorname{Re}\left( z \right)\end{align}$z-\overline{z}=2i… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p4?rev=1737476039&do=diff Question, Exercise 10.1 Solutions of Question 4 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin \alpha =-\dfrac{4}{5}$$\cos \beta =-\dfrac{12}{13}$$\alpha $$\beta $$\sin \left( \alpha -\beta \right)$$\sin \alpha=-\dfrac{4}{5}$$\alpha$$\sin \beta=-\dfrac{12}{13}$$\beta$$$\cos \alpha=\pm \sqrt{1-\sin^2\alpha}.$$$\… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p3?rev=1737476040&do=diff Question 3, Exercise 8.1 Solutions of Question 3 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\cos 120^{\circ}$$\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/ex8-3-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/mathcraft/sample-01-latex?rev=1737476040&do=diff MathCraft: PDF to LaTeX file: Sample-01 If the PDF file provided by you as follows: Then the output LaTeX file is as follows: \documentclass[10pt]{amsart} %\usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} %\usepackage[version=4]{mhchem} \usepackage{stmaryrd} \usepackage{bbold} \usepackage{hyperref} \usepackage{enumerate} \hypersetup{colorlinks=true, linkcolor=blue, filecolor=magenta, urlcolor=cyan,} \urlstyle{same} \title{… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/mathcraft/sample-02-latex?rev=1737476040&do=diff MathCraft: PDF to LaTeX file: Sample-02 If the PDF file provided by you as follows: Then the output LaTeX file is as follows: \documentclass[4pt]{article} \usepackage[utf8]{inputenc} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage[version=4]{mhchem} \usepackage{stmaryrd} \usepackage{bbold} \usepackage[a4paper]{geometry} \linespread{1.3} % double spaces lines \textwidth 6.3truein % These 4 commands define more efficient margins \textheight 9.9truein \oddsidemargi… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/matric/9th_science/ex-6-2?rev=1737476041&do=diff Exercise 6.2 On the following page we have given the solution of Exercise 6.2 of Mathematics 9 (Science) published by Caravan Book House, Lahore. <WRAP center round info 60%> We have created this page and it will be updated to add new solutions occasionally. Please stay in touch with this page. </WRAP>$\frac{x^2-x-6}{x^2-9}+\frac{x^2+2x-24}{x^2-x-12}$\begin{align} \frac{x^2-x-6}{x^2-9}&+\frac{x^2+2x-24}{x^2-x-12}\\ &=\frac{x^2-3x+2x-6}{(x)^2-(3)^2}+\frac{x^2+6x-4x-24}{x^2-4x+3x-12}\\&= \frac{x(… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-3-p4?rev=1737476036&do=diff Question 5, Exercise 1.3 Solutions of Question 5 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) ${{z}^{2}}+z+3=0$${{z}^{2}}+z+3=0$$a=1,\,\,\,b=1$$c=3$\begin{align}z&=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\\ z&=\dfrac{-\left( 1 \right)\pm \sqrt{{{\left( 1 \right)}^{2}}-4\left( 1 \right)\left( 3 \right)}}{2\left( 1 \right)}\\ z&=\dfrac{-1\pm \s… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-3-p2?rev=1737476036&do=diff Question 2, Exercise 10.3 Solutions of Question 2 of Exercise 10.3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$\sin {{37}^{\circ }}+\sin {{43}^{\circ }}.$$$$\sin \alpha +\sin \beta =2\sin \left( \dfrac{\alpha +\beta }{2} \right)\cos \left( \dfrac{\alpha -\beta }{2} \right).$$$\alpha ={{37}^{\circ }}$$\beta ={{43}^{\circ }}$\begin… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-3-p5?rev=1737476037&do=diff Question 6, Exercise 1.3 Solutions of Question 6 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6(i) ${{z}^{4}}+{{z}^{2}}+1=0$$$z^4+z^2+1=0$$$$z^4+2z^2+1-z^2=0$$$$( z^2+1 )^2-z^2=0$$$$( z^2+1+z)( z^2+1-z )=0$$$$( z^2+z+1 )( z^2-z+1 )=0$$$$(z^2+z+1 )=0$$$$z=\dfrac{-1\pm \sqrt{1-4}}{2}$$$$z=\dfrac{-1\pm \sqrt{3}i}{2}$$$$(z^2-z+1 )=0$$$$z=\dfrac{1\pm … text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/review-ex-1-p2?rev=1737476037&do=diff Question 2 & 3, Review Exercise 1 Solutions of Question 2 & 3 of Review Exercise 1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${{i}^{n}}+{{i}^{n+1}}+{{i}^{n+2}}+{{i}^{n+3}}=0$$\forall n\in N$\begin{align}{{i}^{n}}+{{i}^{n+1}}+{{i}^{n+2}}+{{i}^{n+3}}&=0\\ L.H.S.&={{i}^{n}}+{{i}^{n}}\cdot i+{{i}^{n}}\cdot {{i}^{2}}+{{i}^{n}}\cdot {{i}^{3}}\\ &={{i}^{n}}\left( 1+i+{{i}^{2}}… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/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 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p7?rev=1737476038&do=diff Question 7 Exercise 7.2 Solutions of Question 7 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$(2+\sqrt{3})^5+(2-\sqrt{3})^5$\begin{align}(2+\sqrt{3})^5+(2 \cdot \sqrt{3})^5& =[(2)^5+{ }^5 C_1 \cdot 2^4 \cdot \sqrt{3}+{ }^5 C_2 \cdot 2^3 \cdot(\sqrt{3})^2 \\ & +^5 C_3 \cdot 2^2 \cdot(\sqrt{3})^4+{ }^5 C_4 \cdot 2 \cdot(\sqrt{3})^4 \\ … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-3-p2?rev=1737476039&do=diff Question 2, Exercise 10.3 Solutions of Question 2 of Exercise 10.3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$\sin {{37}^{\circ }}+\sin {{43}^{\circ }}.$$$$\sin \alpha +\sin \beta =2\sin \left( \dfrac{\alpha +\beta }{2} \right)\cos \left( \dfrac{\alpha -\beta }{2} \right).$$$\alpha ={{37}^{\circ }}$$\beta ={{43}^{\circ }}$\begin… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p6?rev=1737476039&do=diff Question 6, Exercise 2.6 Solutions of Question 6 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $5 x+3 y+z=6$$2 x+y+3 z=19$$x+2 y+4 z=25$\begin{align*} A &= \begin{bmatrix} 5 & 3 & 1 \\ 2 & 1 & 3 \\ 1 & 2 & 4 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad B = \begin{bmatrix} 6 \\ 19 \\ 25 \end{bmatrix} \end{alig… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p14?rev=1737476037&do=diff Question 18, Exercise 2.2 Solutions of Question 18 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 18(i) $A$$B$$( A^{-1})^{-1}=A$$A$$2\times 2$$$A=\left[ \begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{matrix} \right]$$$$|A|=a_{11}a_{22}-a_{12}a_{21}$$$$AdjA=\left[ \begin{matrix} a_{22} & -a_{12} \\ -a_{21} & a_{11… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p9?rev=1737476038&do=diff Question 13 Exercise 6.2 Solutions of Question 13 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\mathrm{E}$$n=10$$m_1=4$$E, m_2=2$$L$$m_3=2$$C$\begin{align}\text{total number of permutations are} &=\left(\begin{array}{c} n \\ m_1, m_2, m_3 \end{array}\right)\\&=\left(\begin{array}{c} 10 \\ 4,2,2 \end{array}\right) \\ & =\dfrac{10 !}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-3-p1?rev=1737476039&do=diff Question 1, Exercise 2.3 Solutions of Question 1 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left[\begin{array}{ccc}2 & 3 & 1 \\ 1 & -1 & 2 \\ 4 & 1 & 2\end{array}\right]$\begin{align*} A &= \left[\begin{array}{ccc}2 & 3 & 1 \\ 1 & -1 & 2 \\ 4 & 1 & 2\end{array}\right]\\ |A|&=2(-2-2)-3(2-8)+1(1+4)\\ \implies |A|&=-8+18+5\\ \implies |… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-1-p7?rev=1737476036&do=diff Question 8, Exercise 1.1 Solutions of Question 8 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 8(i) $\dfrac{1-2i}{2+i}+\dfrac{4-i}{3+2i}$$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 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-ptb/sol/ch01/ex1-1?rev=1737476037&do=diff Exercise 1.1 (Solutions) <lead>Notes (Solutions) of Exercise 1.1: Textbook of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board (PTB) Lahore.</lead> The main topics of this exercise are properties of real numbers, binary operation, addition and multiplication law, properties of equality, properties of inequality (order properties), field, rule of fractions. These notes are based on the new Student Learning Outcomes (SLOs). Version: 4.0, Available at Ma… text/html 2025-01-21T16:13:57+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}$$a+ib.$\begin{align}&\dfrac{1-2i}{2+i}+\dfrac{4-i}{3+2i}\\ &=\dfrac{\left( 3+2i \right)\left( 1-2i \right)+\left( 2+i \right)\left( 4-i \right)}{\left( 2+i \right)\left( 3+2i \right)}\\ &=\dfrac{\left( 3+4+2i-6i … text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p5?rev=1737476037&do=diff Question 5 & 6, Exercise 2.1 Solutions of Question 5 & 6 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$A= \begin{bmatrix} 0 & 2b & -2 \\ 3 & 1 & 3 \\ 3a & 3 & -1 \end{bmatrix}$$a$$b$$A=\begin{bmatrix} 0 & 2b & -2 \\ 3 & 1 & 3 \\ 3a & 3 & -1 \end{bmatrix}$$$A^t=\left[ \begin{matrix} 0 & 3 & 3a \\ 2b & 1 & 3 \\ -2 & 3 & -1 \\ \end{ma… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p12?rev=1737476037&do=diff Question 13, Exercise 2.1 Solutions of Question 13 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 13(i) $A$$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]$$$$A^t=\left[ \begin{matrix} a_{11} & a_{21} & a_{31} \\ a_{12} & a_{22} … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p11?rev=1737476038&do=diff Question 11 Exercise 7.1 Solutions of Question 11 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.\begin{align} & \left(\begin{array}{l} 2 \\ 2 \end{array}\right)+\left(\begin{array}{l} 3 \\ 2 \end{array}\right)+\left(\begin{array}{l} 4 \\ 2 \end{array}\right)+\ldots+\left(\begin{array}{l} n \\ 2 \end{array}\right)=\left(\begin{array}{c… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-8-p6?rev=1737476040&do=diff Question 11 and 12, Exercise 4.8 Solutions of Question 11 and 12 of Exercise 4.8 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sum_{k=1}^{n} \frac{1}{k(k+2)}$$T_k$$k$\begin{align*} T_k &= \frac{1}{k(k+2)}. \end{align*}\begin{align*} \frac{1}{k(k+2)} = \frac{A}{k} + \frac{B}{k+2} \ldots (1) \end{align*}$k(k+2)$\begin{align*} 1 = A(k+2) + Bk \ldots (2) \end{… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p2?rev=1737476040&do=diff Question 2, Exercise 8.1 Solutions of Question 2 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\cos 15^{\circ}$$\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/ex8-2-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 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-2-p5?rev=1737476036&do=diff Question 6, Exercise 1.2 Solutions of Question 6 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6(i) ${{z}_{1}}$${{z}_{2}}$$|{{z}_{1}}{{z}_{2}}|=|{{z}_{1}}||{{z}_{2}}|$${{z}_{1}}=a+bi$${{z}_{2}}=c+di$$|z_1=\sqrt{a^2+b^2}|$$|z_2=\sqrt{c^2+d^2}|$\begin{align} L.H.S.&=|{{z}_{1}}{{z}_{2}}|\\ &=|(a+bi)(c+di)|\\ &=|ac-bd+(ad+bc)i|\\ &=\sqrt{{{(ac-bd)}^{… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-3-p2?rev=1737476036&do=diff Question 2, Exercise 1.3 Solutions of Question 2 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2(i) $P(z)$$$P\left( z \right)={{z}^{3}}+6z+20$$$$p\left( z \right)={{z}^{3}}+6z+20$$$(z-a)$$P(z)$$P(a)=0$$z=-2$\begin{align} P(-2)&=(-2)^3+6(-2)+20\\ &=-8-12+20\\ &=0\end{align}$z+2$${{z}^{3}}+6z+20$$$\begin{array}{c|cccc} -2 & 1 & 0 & 6 & 20 \\ & \d… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-1-p3?rev=1737476036&do=diff Question 3, Exercise 10.1 Solutions of Question 3 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin u=\dfrac{3}{5}$$\sin v=\dfrac{4}{5}$$u$$v$$0$$\dfrac{\pi }{2}$$\cos \left( u+v \right)$$\sin u=\dfrac{3}{5},$$0\le u\le \dfrac{\pi }{2}.$$\sin v=\dfrac{4}{5},$$0\le v\le \dfrac{\pi }{2}.$$\cos u=\pm \sqrt{1-{{\sin }^… text/html 2025-01-21T16:13:57+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/ex1-3-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$$$\begin{array}{c|cccc} -2 & 1 & 0 & 6 & 20 \\ & \d… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p3?rev=1737476039&do=diff Question 3, Exercise 10.1 Solutions of Question 3 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin u=\dfrac{3}{5}$$\sin v=\dfrac{4}{5}$$u$$v$$0$$\dfrac{\pi }{2}$$\cos \left( u+v \right)$$\sin u=\dfrac{3}{5},$$0\le u\le \dfrac{\pi }{2}.$$\sin v=\dfrac{4}{5},$$0\le v\le \dfrac{\pi }{2}.$$\cos u=\pm \sqrt{1-{{\sin }^… text/html 2025-01-21T16:14:00+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 2025-01-21T16:13:59+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}.$$$n = 1$\begin{align}A^1 =\beg… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-8-p2?rev=1737476040&do=diff Question 3 and 4, Exercise 4.8 Solutions of Question 3 and 4 of Exercise 4.8 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $1+4+13+40+121+ \ldots$$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/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 $$-5=r\sin \varphi$\begin{align*} & (12)^2+(-5)^2=r^2 \cos^2\varphi+r^2 \sin^2 \varphi \\ \implies & 144+25={{r}^{2}}\left( {{\cos }^{2}}\varphi +{{\sin }^{2}}\varphi \r… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-ptb/definitions?rev=1737476037&do=diff Definitions: FSc Part 1 (Mathematics): PTB On this page, all the definitions of “Textbook of Algebra and Trigonometry Class XI, published by Punjab Textbook Board (PTB) Lahore, Pakistan are given. We are very thankful to Muhammad Waqas Sulaiman for his valuable contribution.$\frac{p}{q}$$p,q \in \mathbb{Z}$$q\neq 0$$\frac{p}{q}$$p,q \in \mathbb{Z}$$q\neq 0$$\mathbb{R}$$0.3333....,21.134134$$\pi = 3.1415...$$\divideontimes$$z=x+iy$$x,y \in \mathbb{R}, i = \sqrt{-1}$$x$$y$$z$$2, 3+\sqrt{3}i, \fra… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-1-p11?rev=1737476036&do=diff Question 13, Exercise 10.1 Solutions of Question 13 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$r\,\,\sin \left( \theta +\phi \right)$$\theta$$\phi$$4\sin \theta +3\cos \theta .$$4\sin \theta +3\cos \theta$$r\sin(\theta + \varphi)$$$4\sin \theta +3\cos \theta=r\cos\varphi\sin\theta+r\sin\varphi\cos\theta --- (1)$… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/re-ex10-p4?rev=1737476037&do=diff Question 6 & 7, Review Exercise 10 Solutions of Question 6 & 7 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\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/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$$$A=\left[ \begin{matrix} 1 & -3 & 4 \\ -3 & 2 & -5 \\ 4 & -5 & 0 \\ \end{ma… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p11?rev=1737476037&do=diff Question 12, Exercise 2.1 Solutions of Question 12 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 12(i) $A=\begin{bmatrix}3 & 2 & 1 \\4 & 5 & 6 \\-2 & 3 & 4\end{bmatrix}$$A+A^t$$$A=\left[ \begin{matrix} 3 & 2 & 1 \\ 4 & 5 & 6 \\ -2 & 3 & 4 \\ \end{matrix} \right]$$$$A^t=\left[ \begin{matrix} 3 & 4 & -2 \\ 2 & 5 & 3 \… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p1?rev=1737476037&do=diff Question 1, Exercise 2.2 Solutions of Question 1 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1 $A=\begin{bmatrix}1 & 3 & 1 \\-1 & 2 & 0 \\2 & 0 & -2 \end{bmatrix}$$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]$$$${{A}_{11}}={{\left(… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p11?rev=1737476039&do=diff Question 13, Exercise 10.1 Solutions of Question 13 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$r\,\,\sin \left( \theta +\phi \right)$$\theta$$\phi$$4\sin \theta +3\cos \theta .$$4\sin \theta +3\cos \theta$$r\sin(\theta + \varphi)$$$4\sin \theta +3\cos \theta=r\cos\varphi\sin\theta+r\sin\varphi\cos\theta --- (1)$… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/re-ex10-p1?rev=1737476039&do=diff Question 1, Review Exercise 10 Solutions of Question 1 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos {{50}^{\circ }}5{0}'\cos {{9}^{\circ }}1{0}'-\sin {{50}^{\circ }}5{0}'\sin {{9}^{\circ }}1{0}'=$$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-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 2025-01-21T16:14:00+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 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-ptb/solution-and-area-of-oblique-triangle?rev=1737476037&do=diff Solution and Area of Oblique Triangle These are the common formulas used in Chapter 12 of Textbook of Algebra and Trigonometry Class XI, Punjab Textbook Board Lahore. This handout is very helpful to remember the formulas. All these formulas are given for real valued and defined trigonometric functions. A PDF file can be downloaded for high quality printing and a word file is also given if you wish to modify the contents or credit as you need. <panel title=$a^2=b^2+c^2-2bc\cos \alpha$$b^2=c^2+a^… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-3-p4?rev=1737476037&do=diff Question 5, Exercise 1.3 Solutions of Question 5 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) ${{z}^{2}}+z+3=0$$${{z}^{2}}+z+3=0.$$$a=1$$b=1$$c=3$\begin{align}z&=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\\ &=\dfrac{-1\pm \sqrt{{{\left( 1 \right)}^{2}}-4\left( 1 \right)\left( 3 \right)}}{2\left( 1 \right)}\\ &=\dfrac{-1\pm \sqrt{1-12}}{2}\\ &=\… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p12?rev=1737476037&do=diff Question 14 & 15, Exercise 2.2 Solutions of Questions 14 & 15 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$A=\begin{bmatrix}0 & 2 & 2 \\-1 & 3 & 2 \\1 & 0 & 5\end{bmatrix}$$A^{-1}$$$A=\left[ \begin{matrix} 0 & 2 & 2 \\ -1 & 3 & 2 \\ 1 & 0 & 5 \\ \end{matrix} \right]$$$A^{-1}$$$A^{-1}=\dfrac{Adj\,\,A}{|A|}$$$$Adj\,\,A={{\left[ \begin… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p3?rev=1737476039&do=diff Question 3 Exercise 7.3 Solutions of Question 3 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sqrt{\frac{1-x}{1+x}}$$x^3$$\sqrt{\frac{1-x}{1+x}}$$$ =(1-x)^{\frac{1}{2}}(1+x)^{-\frac{1}{2}} \text {. } $$$$ \begin{aligned} & (1-x)^{\frac{1}{2}}(1+x)^{\frac{1}{2}} \\ & =\left[1-\frac{x}{2}+\frac{\frac{1}{2}\left(\frac{1}{2}-1\right)}{2… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p4?rev=1737476039&do=diff Question 7 and 8, Exercise 4.1 Solutions of Question 7 and 8 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$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 2025-01-21T16:14:00+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-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/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 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-2-p7?rev=1737476036&do=diff Question 8 and 9, Exercise 10.2 Solutions of Question 8 and 9 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${{\cos }^{4}}\theta $\begin{align}{{\cos}^{4}}\theta &={{\left( {{\cos }^{2}}\theta \right)}^{2}}\\ &={{\left( \dfrac{1+\cos 2\theta }{2} \right)}^{2}}\\ &=\dfrac{1+2\cos 2\theta +{{\cos }^{2}}2\theta }{4}\\… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/review-ex-1-p3?rev=1737476037&do=diff Question 4 & 5, Review Exercise 1 Solutions of Question 4 & 5 of Review Exercise 1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${{z}_{1}}=2-i$${{z}_{2}}=1+i,$$\left|\dfrac{{{z}_{1}}+{{z}_{2}}+1}{{{z}_{1}}-{{z}_{2}}+1}\right|$$z_1=2-i$$z_2=1+i$\begin{align} \dfrac{{{z}_{1}}+{{z}_{2}}+1}{{{z}_{1}}-{{z}_{2}}+1}&=\dfrac{\left( 2-i \right)+\left( 1+i \right)+1}{\left( 2-i \rig… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p4?rev=1737476037&do=diff Question 4, Exercise 2.1 Solutions of Question 4 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4 $A= \begin{bmatrix}1 & 4 & 4 \\ 4 & 1 & 4 \\ 4 & 4 & 1 \end{bmatrix}$$\dfrac{1}{3}A^2-2A-9I=0$$A=\begin{bmatrix} 1 & 4 & 4 \\ 4 & 1 & 4 \\ 4 & 4 & 1 \end{bmatrix}$\begin{align}\frac{1}{3}A^2&=\frac{1}{3}\left[ \begin{matrix} 1 & 4 & 4 … text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p6?rev=1737476037&do=diff Question 7, Exercise 2.1 Solutions of Question 7 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7 $ A=\begin{bmatrix}1 & 0 & -1 & 2 \\3 & 1 & 2 & \quad 5 \\0 & -2 & 1 & 6\end{bmatrix}$$ B=\begin{bmatrix} 2 & -1 & 3 & 1 \\1 & 3 & -1 & 4 \\3 & 1 & 2 & -1 \end{bmatrix}$$( A+B )^t=A^t+B^t$$A=\left[ \begin{matrix}1 & 0 & -1 & 2 \\3 & 1 &… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p7?rev=1737476037&do=diff Question 8, Exercise 2.1 Solutions of Question 8 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 8(i) $A=\begin{bmatrix}1 & 2 & 0 \\3 & -1 & 4 \end{bmatrix}$$( A^t )^t=A$$$A=\left[ \begin{matrix} 1 & 2 & 0 \\ 3 & -1 & 4 \\ \end{matrix} \right]$$$$A^t=\left[ \begin{matrix} 1 & 3 \\ 2 & -1 \\ 0 & 4 \\ \end{matrix} \rig… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p6?rev=1737476039&do=diff Question 7 and 8 Exercise 7.3 Solutions of Question 7 and 8 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$x^4$$(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/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 2025-01-21T16:14:00+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 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/playground/wrap?rev=1737476042&do=diff Examples for the Wrap Plugin Basic syntax An uppercase <WRAP> (or alternatively <block> or <div>) creates a div and should be used for “big” containers, surrounding paragraphs, lists, tables, etc. <WRAP classes #id width :language> "big" content </WRAP> or <block classes #id width :language> "big" content </block> or <div classes #id width :language> "big" content </div> text/html 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$$z_2=2+3i$\begin{align} {{z}_{1}}+{{z}_{2}}&=1+2i+2+3i\\ &=1+2+2i+3i\\ &=3+5i \end{align}\begin{align} |z_1+z_2|&=\sqrt{3^2+5^2}\\ &=\sqrt{9+25}\\ &=\sqrt{34}\end{align}${{z}_{1}}=1+2… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-1-p6?rev=1737476036&do=diff Question 6, Exercise 10.1 Solutions of Question 1 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos \alpha =2{{\cos }^{2}}\dfrac{\alpha }{2}-1=1-2{{\sin }^{2}}\dfrac{\alpha }{2}$\begin{align}L.H.S&=\cos \alpha \\ \cos \alpha &=\cos 2\dfrac{\alpha }{2}\\ &={{\cos }^{2}}\dfrac{\alpha }{2}-{{\sin }^{2}}\dfrac{\alpha }… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-1-p6?rev=1737476037&do=diff Question 7, Exercise 1.1 Solutions of Question 7 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7(i) ${{z}_{1}}=1+2i$${{z}_{2}}=2+3i$$|{{z}_{1}}+{{z}_{2}}|$$z_1=1+2i$$z_2=2+3i$\begin{align} {{z}_{1}}+{{z}_{2}}&=1+2i+2+3i\\ &=1+2+2i+3i\\ &=3+5i \end{align}\begin{align} |z_1+z_2|&=\sqrt{3^2+5^2}\\ &=\sqrt{9+25}\\ &=\sqrt{34}\end{align}${{z}_{1}}=1+2… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p6?rev=1737476039&do=diff Question 6, Exercise 10.1 Solutions of Question 1 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos \alpha =2{{\cos }^{2}}\dfrac{\alpha }{2}-1=1-2{{\sin }^{2}}\dfrac{\alpha }{2}$\begin{align}\cos \alpha &=\cos 2\dfrac{\alpha }{2}\\ &={{\cos }^{2}}\dfrac{\alpha }{2}-{{\sin }^{2}}\dfrac{\alpha }{2}\\ &={{\cos }^{2}}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-1-p1?rev=1737476039&do=diff Question 1, Exercise 1.1 Solutions of Question 1 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 1(i) ${{i}^{31}}$\begin{align}{{i}^{31}}&=i\cdot{{i}^{30}}\\ &=i\cdot{{\left( {{i}^{2}} \right)}^{15}}\\ &=i\cdot{{\left( -1 \right)}^{15}} \quad \because i^2=-1\\ &=i\cdot(-1)\\ &=-i.\end{align}${{\left( -i \right)}^{6}}$\begin{align} {{\left… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p3?rev=1737476039&do=diff Question 3, Exercise 2.6 Solutions of Question 3 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 x+3 y+4 z=2$$2 x+y+z=5$$3 x-2 y+z=-3$\begin{align*} \begin{aligned} 2x + 3y + 4z &= 2 \\ 2x + y + z &= 5 \\ 3x - 2y + z &= -3 \end{aligned}\end{align*}\begin{align*} A_{b} &=\quad \left[\begin{array}{cccc} 2 & 3 & 4 & 2 \\ 2 & 1 & 1 & 5 \\ 3… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p6?rev=1737476040&do=diff Question 7, Exercise 8.1 Solutions of Question 7 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\alpha$$\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 2025-01-21T16:13:54+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/atiq/fa22-mth604?rev=1737476034&do=diff ~~DISCUSSION~~ MTH604: Fixed Point Theory and Applications (Fall 2022) [FPTA] Course Objectives: This course is intended as a brief introduction to the subject with a focus on Banach Fixed Point theorems fixed point theorem and its application to nonlinear differential equations, nonlinear integral equations, real and complex implicit functions theorems and system of nonlinear equations. Some generalizations and similar results e. g. Kannan Fixed Point theorems, Banach Fixed Point theorem f… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/review-ex-1-p3?rev=1737476036&do=diff Question 4 & 5, Review Exercise 1 Solutions of Question 4 & 5 of Review Exercise 1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${{z}_{1}}=2-i$${{z}_{2}}=1+i,$$|\dfrac{{{z}_{1}}+{{z}_{2}}+1}{{{z}_{1}}-{{z}_{2}}+1}|$\begin{align}{{z}_{1}}&=2-i,\\ {{z}_{2}}&=1+i,\\ \dfrac{{{z}_{1}}+{{z}_{2}}+1}{{{z}_{1}}-{{z}_{2}}+1}&=\dfrac{\left( 2-i \right)+\left( 1+i \right)+1}{\left( 2-… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-2-p2?rev=1737476036&do=diff Question 2, Exercise 10.2 Solutions of Question 2 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin \theta =\dfrac{5}{13}$$\theta $$\sin 2\theta $$\sin \theta =\dfrac{5}{13}$$$\cos \theta =\pm \sqrt{1-{{\sin }^{2}}\theta }.$$$\theta$$\cos$\begin{align}\cos\theta &=-\sqrt{1-{{\sin }^{2}}\theta }\\ &=-\sqrt{1-\left(\… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/re-ex10-p1?rev=1737476037&do=diff Question 1, Review Exercise 10 Solutions of Question 1 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\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 }}$$7+\sq… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p7?rev=1737476039&do=diff Question 9 Exercise 7.3 Solutions of Question 9 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$x^{\prime \prime}$$\left(\frac{1+x}{1-x}\right)^2$$$ \begin{aligned} & \left(\frac{1+x}{1-x}\right)^2=(1+x)^2(1-x)^{-2} \\ & =\left(x^2+2 x+1\right)(1-x)^2 \end{aligned} $$$$ \begin{aligned} & =\left(x^2+2 x+1\right)[1+2 x+ \\ & \frac{-2(-2-… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-2-p2?rev=1737476039&do=diff Question 2, Exercise 10.2 Solutions of Question 2 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin \theta =\dfrac{5}{13}$$\theta $$\sin 2\theta $$\sin \theta =\dfrac{5}{13}$$$\cos \theta =\pm \sqrt{1-{{\sin }^{2}}\theta }.$$$\theta$$\cos$\begin{align}\cos\theta &=-\sqrt{1-{{\sin }^{2}}\theta }\\ &=-\sqrt{1-\left(\… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p2?rev=1737476039&do=diff Question 2, Exercise 1.2 Solutions of Question 2 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 2 $$ \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} $$\begin{align} &(z_1 z_2)(z_3 z_4) \\ =&(z_1 z_2)z_5 \quad \text {Let }z_5=z_3 z_4 \\ =&z_1 (z_2 z_5) \… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p7?rev=1737476039&do=diff Question 6(x-xvii), Exercise 1.4 Solutions of Question 6(x-xvii) of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $7 \sqrt{2}\left(\cos \dfrac{5 \pi}{4}+i \sin \dfrac{5 \pi}{4}\right)$$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/unit02/ex2-3-p6?rev=1737476039&do=diff Question 6, Exercise 2.3 Solutions of Question 6 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{ccc}2 & 1 & -3 \\ 0 & 1 & 0 \\ 2 & 1 & 6\end{array}\right]$$A^{-1}$$A A^{-1}=A^{-1} A=I_{3}$\begin{align*} A &= \begin{bmatrix} 2 & 1 & -3 \\ 0 & 1 & 0 \\ 2 & 1 & 6 \end{bmatrix} \end{align*}$ A^{-1} $$ A $\begin{align*} … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p7?rev=1737476039&do=diff Question 7 and 8, Exercise 2.6 Solutions of Question 7 and 8 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{ccc}3 & 2 & 1 \\ 4 & -1 & 2 \\ 7 & 3 & -3\end{array}\right]$$A^{-1}$$3 x+4 y+7 z=14 ; 2 x-y+3 z=4 ; \quad x+2 y-3 z=0$\begin{align*} A &= \begin{bmatrix} 3 & 2 & 1 \\ 4 & -1 & 2 \\ 7 & 3 & -3 \end{bmatrix}\\ |… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-2-p4?rev=1737476040&do=diff Question 7 and 8, Exercise 5.2 Solutions of Question 7 and 8 of Exercise 5.2 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 x^{3}-15 x^{2}+27 x-10$$\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/re-ex-p3?rev=1737476040&do=diff Question 4 & 5, Review Exercise Solutions of Question 4 & 5 of Review Exercise of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $3 y-2$$6 y^{3}-y^{2}-5 y+2$\begin{align*}3y-2&=0\\ 3y&=2\\ y&=\frac{2}{3}\end{align*}\begin{align*} f(y) &= 6y^{3} - y^{2} - 5y + 2\\ f\left(\frac{2}{3}\right) &= 6\left(\frac{2}{3}\right)^{3} - \left(\frac{2}{3}\right)^{2} - 5\left(\frac{2}{3… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p3?rev=1737476040&do=diff Question 5 Exercise 8.2 Solutions of Question 5 of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin \theta$$\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$$\cos 2\theta$\begin{align*}\cos 2\theta & = - \sqrt{1-\sin^2 2\theta}\\… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-2-p2?rev=1737476036&do=diff Question 2, Exercise 1.2 Solutions of Question 2 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2 $z_1=-1+i$, $z_2=3-2i$${{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).$$\begin{align} {{z}_{1}}+{{z}_{2}}&=\left( -1+i \right)+\left( 3-2i \right)\\ &=2-i\end{align}\begin{align} \left( {{z}_{1}}+{{z}_{2}… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-1-p10?rev=1737476036&do=diff Question11 and 12, Exercise 10.1 Solutions of Question 11 and 12 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\alpha$$\beta$$\gamma$$ABC$$\cot \dfrac{\alpha }{2}+\cot \dfrac{\beta }{2}+\cot \dfrac{\gamma }{2}=\cot \dfrac{\alpha }{2}\cot \dfrac{\beta }{2}\cot \dfrac{\gamma }{2}$$\alpha$$\beta$$\gamma$\begin{align}&\… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-3-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.$$\dfrac{\cos {{75}^{\circ }}+\cos {{15}^{\circ }}}{\sin {{75}^{\circ }}-\sin {{15}^{\circ }}}=\sqrt{3}.$$$$\cos \alpha +\cos \beta =2\cos \left( \dfrac{\alpha +\beta }{2} \right)\cos \left( \dfrac{\alpha -\beta }{2} \righ… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-2-p2?rev=1737476037&do=diff Question 2, Exercise 1.2 Solutions of Question 2 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2 $z_1=-1+i$, $z_2=3-2i$${{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).$$\begin{align} {{z}_{1}}+{{z}_{2}}&=\left( -1+i \right)+\left( 3-2i \right)\\ &=2-i\end{align}\begin{align} \left( {{z}_{1}}+{{z}_{2}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p10?rev=1737476039&do=diff Question11 and 12, Exercise 10.1 Solutions of Question 11 and 12 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\alpha$$\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/ex10-3-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.$$\dfrac{\cos {{75}^{\circ }}+\cos {{15}^{\circ }}}{\sin {{75}^{\circ }}-\sin {{15}^{\circ }}}=\sqrt{3}.$$$$\cos \alpha +\cos \beta =2\cos \left( \dfrac{\alpha +\beta }{2} \right)\cos \left( \dfrac{\alpha -\beta }{2} \righ… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p8?rev=1737476039&do=diff Question 8, Exercise 1.2 Solutions of Question 8 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 8(i) $|2 z-i|=4$$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/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. $$ A=\left[\begin{array}{ccc} 2 & 0 \\ \sqrt{5} & 6 \\ 1 & 9 \end{array}\right]$$$$ A^t=\begin{bmatrix} 2 & \sqrt{5} & 1 \\ 0 & 6 & 9 \end{bmatrix}$$$$B=\left[\begin{array}{cccc} 1 & 6 & 2 & 0 \end{array}\right] $$$$B^t=\left[\begin{array}{c} 1… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p4?rev=1737476040&do=diff Question 7 and 8, Exercise 4.5 Solutions of Question 7 and 8 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=16, r=-\frac{1}{2}, n=10$$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.$$\begin{align*} S_{10} &= \frac{16 \left(1 - \left(-\frac{1}{2}\right)^{10}\right)}{1 - \left(-\frac{1}… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p7?rev=1737476040&do=diff Question 14, 15 and 16, Exercise 4.7 Solutions of Question 14, 15 and 16 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$n$$n+1$$T_n$$n$$$ T_{n} = n+1. $$\begin{align*}\sum_{n=1}^{\infty} T_{n} &= \sum_{n=1}^{\infty} (n+1)\\ & = \sum_{n=1}^{\infty} n + \sum_{n=1}^{\infty} 1 \\ & = \frac{n(n+1)}{2} + n \\ & = \frac{n(n+1)}{2} + \frac{2n}{2} \… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p11?rev=1737476040&do=diff Question 21 and 22, Exercise 4.7 Solutions of Question 21 and 22 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$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$$k$\begin{align*}T_k&=k(3k+1) \\ &=3k^2+k. \end{align*}\begin{align*}\sum_{k=1}^{n} T_{k} &= \sum_{k=1}^{n} (3k^2 +k)\… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p7?rev=1737476040&do=diff Question 8, Exercise 8.1 Solutions of Question 8 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin \alpha=\dfrac{3}{5}$$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/ex8-3-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/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$$\phi$\begin{align*} \cos^2 \theta &= 1-\sin^2\theta\\ &= 1-\left(\frac{3}{5}\right)^2\\ & =… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/msc/mcqs_short_questions/real_analysis?rev=1737476041&do=diff Real Analysis: Short Questions and MCQs <callout type=“info” icon=“true”> We are going to add short questions and MCQs for Real Analysis. The subject is similar to calculus but little bit more abstract. This is a compulsory subject in MSc and BS Mathematics in most of the universities of Pakistan. The author of this page is Dr. $\left\{\frac{1}{n+1} \right\}$$\left\{\frac{n+2}{n+1} \right\}$$\{x_n\}$$\{y_n\}$$\lim_{n\to\infty z_n}$$z_n=x_n-2y_n$$\{x_n\}$$\{y_n\}$$\lim_{n\to\infty z_n}$$x_n=2y_n… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-3-p1?rev=1737476036&do=diff Question 1, Exercise 10.3 Solutions of Question 1 of Exercise 10.3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. There are four parts in Question 1.$2\sin 6x\sin x$$$-2\sin \alpha \sin \beta =\cos (\alpha +\beta )-\cos (\alpha -\beta ).$$$\alpha =6x$$\beta =x$\begin{align}-\,2\sin 6x\sin x&=\cos (6x+x)-\cos (6x-x)\\ &=\cos 7x-\cos x… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p8?rev=1737476037&do=diff Question 8,9 & 10, Exercise 2.2 Solutions of Questions 8,9 & 10 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\left| \begin{matrix}1+x & y & z \\x & 1+y & z \\x & y & 1+z \end{matrix} \right|=1+x+y+z$$$L.H.S.=\left| \begin{matrix} 1+x & y & z \\ x & 1+y & z \\ x & y & 1+z \\ \end{matrix} \right|$$$$=\left| \begin{matrix} 1 & 0 & -… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-1-p3?rev=1737476038&do=diff Question 5 Exercise 4.1 Solutions of Question 5 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) $\sum_{j=1}^6(2 j-3)$\begin{align}\sum_{j=1}^6(2 j-3)&=(2.1-3)+(2.2-3)+(2.3-3)+(2.4-3)\\&+(2.5-3)+(2.6-3) \\ \implies \sum_{j=1}^6(2 j-3)&=-1+1+3+5+7+9 .\end{align}$\sum_{k=1}^5(-1)^k 2^{k-1}$\begin{align}\sum_{k=1}^5(-1)^k 2^{k-1}& =(-1)^1 2^{1-… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-4-p7?rev=1737476038&do=diff Question 7 Exercise 6.4 Solutions of Question 7 of Exercise 6.4 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.\begin{align}S&=\{(i, j) ; i, j=1,2,3,4,5,6\}\\ &=\left[\begin{array}{llllll} (1,1) & (1,2) & (1,3) & (1,4) & (1,5) & (1,6) \\ (2,1) & (2,2) & (2,3) & (2,4) & (2,5) & (2,6) \\ (3,1) & (3,2) & (3,3) & (3,4) & (3,5) & (3,6) \\ (4,1) & (4,2) & (… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p9?rev=1737476039&do=diff Question 9 Exercise 7.2 Solutions of Question 9 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$(x-y)="$$x=12$$y-4$$x=12$$$ \begin{aligned} & \left(x \quad y=20(12-y)^{20}\right. \\ & =12^{2 n}\left(\begin{array}{ll} 1 & \frac{y}{12} \end{array}\right)^{31} \end{aligned} $$$\frac{(n+1) \cdot x}{1+|x|}$$\left(\frac{1}{12}\right)^2 \cdot… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-3-p1?rev=1737476039&do=diff Question 1, Exercise 10.3 Solutions of Question 1 of Exercise 10.3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. There are four parts in Question 1.$2\sin 6x\sin x$$$-2\sin \alpha \sin \beta =\cos (\alpha +\beta )-\cos (\alpha -\beta ).$$$\alpha =6x$$\beta =x$\begin{align}-\,2\sin 6x\sin x&=\cos (6x+x)-\cos (6x-x)\\ &=\cos 7x-\cos x… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-3-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 2025-01-21T16:14:00+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}$$\begin{align*} & \alpha+\beta=180^{\circ}-\gamma \\ \implies & \tan(\alpha+\beta) = \tan(180^{\circ}-… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/re-ex-p1?rev=1737476040&do=diff Question 1, Review Exercise Solutions of Question 1 of Review Exercise of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin \left(45^{\circ}-30^{\circ}\right)=\ldots$$\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$$\frac{\sqrt{3}-1}… text/html 2025-01-21T16:13:54+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/atiq/sp23-mth322?rev=1737476034&do=diff MTH322: Real Analysis II (Spring 2023) [MTH322: Real Analysis II (Spring 2023)] This course is offered to BS, Semester VI at Department of Mathematics, COMSATS University Islamabad, Attock campus. This course need rigorous knowledge of continuity, differentiation, integration, sequences and series of numbers, that is many notions included in $f\in \mathcal{R}[a,b]$$b\ge a$$f(x)\ge 0$$x\ge a$$\int_{\,a}^{\,\infty }{f(x)\,dx}$$M>0$$\int\limits_{a}^{b}{f(x)\,dx}\leq M$$b\ge a$$f(x)$$g(x)$$x>a$$\li… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/real-analysis-notes-by-prof-syed-gul-shah?rev=1737476041&do=diff Real Analysis Notes by Prof Syed Gul Shah [Real Analysis Notes by Prof Syed Gul Shah] Real analysis, a discipline that explores the complexities of mathematical functions, limits, and sequences, can often be a difficult topic for students. Prof. Syed Gul Shah, as a true analyst, not only excelled in the subject but also gained fame for his extraordinary qualities as a human being.$s_n<u_n<t_n$$n\ge n_0$$\{s_n\}$$\{t_n\}$$\{u_n\}$$\{s_n\}$$\exists$$\left| {\,{s_n}}\right|>\frac{1}{2}s$$\{s_n\}$… text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/ppsc/ppsc-maths-2011?rev=1737476042&do=diff PPSC Paper 2011 (Lecturer in Mathematics) [PPSC Paper 2011 (Lecturer in Mathematics)] On this page, we have given question from old (past) paper of Lecturer in Mathematics conducted in year 2011. This is a MCQs paper and answers are given at the end of the paper. At the end of the PDF is also given to download. This paper is provided by Ms. $R$$x\in R$$x^2=x$$x^2=-x$$x^2=0$$x^2=1$$6$$8$$10$$4$$G$$H$$H$$G$$2$$4$$nZ$$Z$$n$$G$$24$$a$$a^{10}$$2$$12$$10$$V$$n$$V$$n+1$$n$$n-1$$v_1,v_2,v_3,....,v_r$$… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/msc/real_analysis_notes_by_syed_gul_shah/sequence_and_series?rev=1737476041&do=diff Chapter 02 - Sequence and Series Contents * Sequence, Subsequence, Increasing Sequence, Decreasing Sequence, Monotonic Sequence, Strictly Increasing or Decreasing * Bernoulli’s Inequality * Bounded Sequence * Convergence of the Sequence$s_n<u_n<t_n$$n\ge n_0$$\{s_n\}$$\{t_n\}$$\{u_n\}$$\{s_n\}$$\exists$$\left| {\,{s_n}}\right|>\frac{1}{2}s$$\{s_n\}$$\{t_n\}$$\left\{a{s_n}+b{t_n}\right\}$$as+bt$$\left\{{s_n}{t_n}\right\}$$\left\{\frac{{{s_n}}}{{{t_n}}} \right\}$$\frac{s}{t}$${t_n}\ne… 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/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 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-3-p1?rev=1737476037&do=diff Question 1, Exercise 1.3 Solutions of Question 1 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) \begin{align}&z-4w=3i\\ &2z+3w=11-5i\end{align}\begin{align}z-4w&=3i …(i)\\ 2z+3w&=11-5i …(ii)\end{align}$2$\begin{align}2z-8w&=6i …(iii)\end{align}\[\begin{array}{cccc} 2z&-8w&=6i \\ \mathop+\limits_{-}2z&\mathop+\limits_{-}3w&=\mathop-\limit… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/review-ex-1-p1?rev=1737476037&do=diff Question 1, Review Exercise 1 Solutions of Question 1 of Review Exercise 1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1 ${{\left( \dfrac{2i}{1+i} \right)}^{2}}$$i$$2i$$1-i$$i+1$$\dfrac{5+2i}{4-3i}$$-\dfrac{7}{25}+\dfrac{26}{25}i$$\dfrac{5}{4}-\dfrac{2}{3}i$$\dfrac{14}{25}+\dfrac{23}{25}i$$\dfrac{26}{7}+\dfrac{23}{7}i$${{i}^{57}}+\frac{1}{{{i}^{25}}}$$0$$2i$$-2… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p10?rev=1737476037&do=diff Question 11, Exercise 2.1 Solutions of Question 11 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 11 $A=\begin{bmatrix}0 & 1 & -2 \\-1 & 0 & 3 \\2 & -3 & 0 \end{bmatrix}$$B=\begin{bmatrix}0 & -6 & 11 \\6 & 0 & -7 \\-11 & 7 & 0 \end{bmatrix}$$A+B$$$A=\left[ \begin{matrix} 0 & 1 & -2 \\ -1 & 0 & 3 \\ 2 & -3 & 0 \\ \end{matri… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-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]$$$$A^t=\left[ \begin{matrix} 2 & -1 \\ 3 & 1 \\ \end{matrix} \right]$$$$|A^t|=5$$$$Ad… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p8?rev=1737476038&do=diff Question 12 Exercise 6.2 Solutions of Question 12 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$8.$$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 2025-01-21T16:13:59+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/ex2-6-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 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch13?rev=1737476036&do=diff Chapter 13: Inverse Trigonometric Functions [Chapter 13: Inverse Trigonometric Functions] Notes (Solutions) of Chapter 13: Inverse Trigonometric Functions, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary * ${\sin ^{ - 1}}A + {\sin ^{ - 1}}B = {\sin ^{ - 1}}\left( {A\sqrt {1 - {B^2}} + B\sqrt {1 - {A^2}} } \right)$${\sin ^{ - 1}}A - {\sin ^{ - 1}}B = {\sin ^{ - 1}}\left( {A\sqrt {1 - {B^2}} - B\sqrt {1 - {… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch14?rev=1737476036&do=diff Chapter 14: Solutions of Trigonometric Equation [Chapter 14: Solutions of Trigonometric Equation] Notes (Solutions) of Chapter 14: Solutions of Trigonometric Equation, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary ${\sin ^{ - 1}}A + {\sin ^{ - 1}}B = {\sin ^{ - 1}}\left( {A\sqrt {1 - {B^2}} + B\sqrt {1 - {A^2}} } \right)$${\sin ^{ - 1}}A - {\sin ^{ - 1}}B = {\sin ^{ - 1}}\left( {A\sqrt {1 - {B^2}} - B\sqr… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-2-p5?rev=1737476036&do=diff Question 6, Exercise 10.2 Solutions of Question 6 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos {{15}^{\circ }}$${{15}^{\circ }}=\dfrac{{{30}^{\circ }}}{2}$$\dfrac{\theta }{2}=\dfrac{{{30}^{\circ }}}{2}$$\cos {{15}^{\circ }}$\begin{align}\cos {{15}^{\circ }}&=\cos \dfrac{{{30}^{\circ }}}{2}=\sqrt{\dfrac{1+\cos … text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p3?rev=1737476037&do=diff Question 3, Exercise 2.2 Solutions of Question 3 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 $A$$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}$$\begin{align}|A|&=a_{11} \left( a_{22} a_{33}-a_{23} a_{32} \right)-a_{12}\left( a_{21}a_{33}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-2-p5?rev=1737476039&do=diff Question 6, Exercise 10.2 Solutions of Question 6 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos {{15}^{\circ }}$${{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-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/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$$\left[\begin{array}{lll}\lambda & 1 & 3 \\ 2 & 1 & 8 \\ 0 & 3 & 1\end{array}\right]$\begin{align*} A &= \left[\begin{array}{ccc} \lambda & 1 & 3 \\ 2 & 1 & 8 \\ 0 & 3 & 1 \end{array}\right]\\ |A| &= \lambda \cdot (-23) - 1 \cdot 2 + 3… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p1?rev=1737476039&do=diff Question 1, Exercise 2.6 Solutions of Question 1 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $ 2 x_{1}-3 x_{2}+4 x_{3}=0$$x_{1}-2 x_{2}+3 x_{3}=0$$4 x_{1}+x_{2}-6 x_{3}=0$\begin{align*} &2 x_{1}-3 x_{2}+4 x_{3}=0\cdots (i)\\ &x_{1}-2 x_{2}+3 x_{3}=0\cdots (ii)\\ &4 x_{1}+x_{2}-6 x_{3}=0\cdots (iii)\\ \end{align*}\begin{align*} A &= \le… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p3?rev=1737476039&do=diff Question 5, 6 and 7, Exercise 4.4 Solutions of Question 5, 6 and 7 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=3, r=-2$$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 2025-01-21T16:14:00+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/ex4-7-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 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/ppsc/ppsc-maths-2015?rev=1737476042&do=diff PPSC Paper 2015 (Lecturer in Mathematics) [PPSC Paper 2011 (Lecturer in Mathematics)] On this page, we have given question from old (past) paper of Lecturer in Mathematics conducted in year 2011. This is a MCQs paper and answers are given at the end of the paper. At the end of the PDF is also given to download. This paper is provided by Kaushef Salamat. We are very thankful to her for providing this paper.\(\displaystyle \int_{-4}^{0}\frac{tdt}{\sqrt{16-t62}}\)$0$$-4$$4$\(A\cos wt+B\sin wt\)$\… text/html 2024-08-05T06:27:23+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/wiki/syntax?rev=1722839243&do=diff Formatting Syntax DokuWiki supports some simple markup language, which tries to make the datafiles to be as readable as possible. This page contains all possible syntax you may use when editing the pages. Simply have a look at the source of this page by pressing text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-1-p1?rev=1737476036&do=diff Question 1, Exercise 10.1 Solutions of Question 1 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. There are four parts in Question 1.$\sin {{37}^{\circ }}\cos {{22}^{\circ }}+\cos {{37}^{\circ }}\sin {{22}^{\circ }}$\begin{align} \sin (\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta, \end{align}\begin{a… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-2-p4?rev=1737476036&do=diff Question 4 and 5, Exercise 10.2 Solutions of Question 4 and 5 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos \theta =-\dfrac{3}{7}$$\theta $$\sin \dfrac{\theta }{2}$$\cos \theta =-\dfrac{3}{7}$$\theta$\begin{align}&\pi < \theta < \dfrac{3\pi}{2} \\ \implies &\frac{\pi}{2} < \frac{\theta}{2} < \dfrac{3\pi}{4}\end… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/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$$\… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p11?rev=1737476037&do=diff Question 13, Exercise 2.2 Solutions of Question 13 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 13(i) $x,$$\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,$$\left… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-3-p7?rev=1737476037&do=diff Question 11, Exercise 3.3 Solutions of Question 11 of Exercise 3.3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 11 (i) Show that the vectors $3 \hat{i}-2 \hat{j}+$$\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}$\begin{align}|\vec{a}|&… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p8?rev=1737476038&do=diff Question 8 Exercise 7.2 Solutions of Question 8 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$(3-2 x)^{10}$$x=\frac{3}{4}$$\left(3-2,1^{10}=3^{10}\left(1-\frac{3 x}{2}\right)^{10}\right.$$\left(1-\frac{3 x}{2}\right)^{10}$$p+1$$: 3-\mathbf{2}_1 1^{10}$$T_{5} !=\left(\begin{array}{c}10 \\ 5\end{array}\right) 3^{10} 5-2 \gamma^{15}$$x=… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p10?rev=1737476039&do=diff Question 12 Exercise 7.3 Solutions of Question 12 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$2 y=\frac{1}{2^2}+\frac{1.3}{2 !} \cdot \frac{1}{2^4}+\frac{1.3 \cdot 5}{3 !} \cdot \frac{1}{2^6}+\ldots$$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/re-ex7-p1?rev=1737476039&do=diff Question 1 Review Exercise 7 Solutions of Question 1 of Review Exercise 7 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Chose the correct option. <panel>$O, P, Q, R, S, T, U$$2520$$9040$$5140$$4880$$\{1,2,3,4,5,6,7\}$$14$$42$$28$$21$$\{1,2,3,4,6,7,8\}$$3$$7$$120$$180$$144$$96$$\dfrac{(n+2) !(n-2) !}{(n+1) !(n-1) !}$$(n-3)$$(\dot{n}-1)$$\dfrac{n+1}{n… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p1?rev=1737476039&do=diff Question 1, Exercise 10.1 Solutions of Question 1 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. There are four parts in Question 1.$\sin {{37}^{\circ }}\cos {{22}^{\circ }}+\cos {{37}^{\circ }}\sin {{22}^{\circ }}$\begin{align} \sin (\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta, \end{align}\begin{a… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-2-p4?rev=1737476039&do=diff Question 4 and 5, Exercise 10.2 Solutions of Question 4 and 5 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos \theta =-\dfrac{3}{7}$$\theta $$\sin \dfrac{\theta }{2}$$\cos \theta =-\dfrac{3}{7}$$\theta$\begin{align}&\pi < \theta < \dfrac{3\pi}{2} \\ \implies &\frac{\pi}{2} < \frac{\theta}{2} < \dfrac{3\pi}{4}\end… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/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-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/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/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}$$|A|$\begin{align*} A&=\left[\begin{array}{ccc}1 & 2 & 0 \\ -3 & 4 & 9 \\ 2 & 1 & 6\end{array}\right]\\ A_{13} &= (-1)^{… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p2?rev=1737476040&do=diff Question 3 and 4, Exercise 4.5 Solutions of Question 3 and 4 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=5$$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/ex4-7-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/ex4-8-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/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-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 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/matric/9th_science/unit_02/exercise_2.2?rev=1737476041&do=diff Exercise 2.2 (Solutions) Question 1 Identify the property used in the following, * (i) $a + b = b + a$ ... ..... * (ii) $(ab)c = a(bc)$ ... ... ... * (iii) $7 \times 1 = 7$ ... ... ... * (iv) $x > y$ or $x = y$ or $x< y$ ... ... ... * (v) $ab = ba$ ... ... ... * (vi) $a + c = b + c \Rightarrow a = b$ ... ... ... * (vii) $5 + (-5) = 0$ ... ... ... * (viii) $7 \times \frac{1}{7} = 1$$a > b \Rightarrow ac > bc? (c >0)$$a + b = b + a$$(ab)c = a(bc)$$7 \times 1 = 7$$x > y$$x = y$$… text/html 2025-01-21T16:13:54+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/atiq/fa21-mth322?rev=1737476034&do=diff MTH322: Real Analysis II (Fall 2021) This course is offered to MSc, Semester II at Department of Mathematics, COMSATS University Islamabad, Attock campus. This course need rigorous knowledge of continuity, differentiation, integration, sequences and series of numbers, that is many notion included in $\int_{1}^{\infty }{{{x}^{-p}} dx}$$p$$f\in \mathcal{R}[a,b]$$b\ge a$$f(x)\ge 0$$x\ge a$$\int_{a}^{\infty }{f(x) dx}$$M>0$$\int\limits_{a}^{b}{f(x)\,dx} \le M$$b\ge a$$f\in \mathcal{R}[a,b]$$b\ge a$… text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/bsc/series_of_important_functions?rev=1737476035&do=diff Some important series of functions On this page we are going to post some series of functions, which are used in mathematics at undergraduate level. * $\sinh x = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} +... \qquad -\infty <x< \infty$ * $\cosh x = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!} +... \qquad -\infty <x< \infty$ * $\tanh x = x - \frac{x^3}{3} + 2\frac{x^5}{15} - 17\frac{x^7}{315} +... \qquad \left |x\right |< \frac{\pi}{2}$ * $\coth x = \frac{1}{x} … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/mcqs?rev=1737476039&do=diff MCQs: Math 11 NBF <lead>Multiple Choice Questions (MCQs) of the Model Textbook of Mathematics for Class XI is published by National Book Foundation (NBF), Islamabad, Pakistan. NBF can be considered as Federal Textbook Board Islamabad. </lead> Unit 01: Complex Numbers $\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}}$$… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/matric/9th_science?rev=1737476041&do=diff Mathematics 9 (Science Group) [Mathematics 9 (Science Group)] Mathematics 9 is written by Dr. Karamat H. Dar and Prof. Irfan-ul-Haq and this book is published by Carvan Book House, Lahore, Pakistan. This book consist of 302 pages and there are 17 units. Notes of Unit 1 and 3 are provided by $ka + kb + kc$$ac + ad + bc + bd$$a^2 + 2ab + b^2$$a^2 – b^2$$a^2 + 2ab + b^2 – c^2$$a^4 + a^2b^2 + b^4$$a^4 + 4b^4$$x^2 + px + q$$ax^2 + bx + c$$(ax^2 + bx + c) (ax2 + bx + d) + k$$(x + a) (x + b) (x + c) … text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/metric-spaces-notes?rev=1737476041&do=diff Metric Spaces (Notes) [Metric Spaces (Notes)] These are updated version of previous notes. Many mistakes and errors have been removed. These notes are collected, composed and corrected by Atiq ur Rehman, PhD. These are actually based on the lectures delivered by Prof. Muhammad Ashfaq (Ex HoD, Department of Mathematics, Government College Sargodha). $(X,d)$$x,y\in X$$$\left| {\,d(x,\,A)\, - \,d(y,\,A)\,} \right|\,\, \le \,\,d(x,\,y).$$$A^c$$A\subset X$$x \in X$$B(x;r)$$A \subset X$$f:(X,d)\to (Y… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/msc/real_analysis_notes_by_syed_gul_shah/real_number_system?rev=1737476041&do=diff Chapter 01 - Real Number System Contents & Summary * Theorem: There is no rational p such that $p^2=2$. * Theorem: Let A be the set of all positive rationals p such that $p^2>2$ and let B consist of all positive rationals p such that $p^2<2$ then A contain no largest member and $x<y$$x<u<y$$x=\sup E$$x>0$$n>0$$y^n=x$$\underline x,\underline y\in \mathbb{R}^n$$\|\underline x^2\|=\underline x\cdot \underline x$$\|\underline x\cdot \underline y\|=\|\underline x\| \|\underline y\|$$\underline … 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$$z_2=-1-i$\begin{align} L.H.S&=(-1-i)^2+2(-1-i)+2\\ &=1+2i-1-… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-1-p7?rev=1737476036&do=diff Question 7, Exercise 10.1 Solutions of Question 1 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cot \left( \alpha +\beta \right)=\dfrac{\cot \alpha \cot \beta -1}{\cot \alpha +\cot \beta }$\begin{align}L.H.S.&=\cot (\alpha +\beta )\\ &=\dfrac{1}{\tan (\alpha +\beta )}\\ &=\dfrac{1}{\,\dfrac{\tan \alpha +\tan \beta… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-4-p2?rev=1737476037&do=diff Question 2 Exercise 3.4 Solutions of Question 2 of Exercise 3.4 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2(i) Show in two different ways that the vectors $\vec{a}$$\vec{b}$$\vec{a}=-\hat{i}+2 \hat{j}-3 \hat{k}, \quad \vec{b}=2 \hat{i}-4 \hat{j}+$$6 \hat{k}$\begin{align}\vec{a} \times \vec{b}&=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ -1 & 2 & -3 \\ 2 … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-4-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$$E$\begin{align}\overrightarrow{A E}&=\overrightarrow{E C}\\ &=\dfrac{1}{2} \vec{a}\\ &=2 \hat{i}+… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-5-p2?rev=1737476038&do=diff Question 3 & 4 Exercise 3.5 Solutions of Question 3 & 4 of Exercise 3.5 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 For the vectors $\vec{a}=3 \hat{i}+2 \hat{k}$$\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}$$\vec{a} \cdot \vec{b}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p8?rev=1737476039&do=diff Question 10 Exercise 7.3 Solutions of Question 10 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$1-\frac{1}{2^2}+\frac{1.3}{2 !} \cdot \frac{1}{2^4}+\ldots$$(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}$$\frac{n(n-1)}{2 !} x^2=\frac{1.3}{2 !} \cdot … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p7?rev=1737476039&do=diff Question 7, Exercise 10.1 Solutions of Question 1 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cot \left( \alpha +\beta \right)=\dfrac{\cot \alpha \cot \beta -1}{\cot \alpha +\cot \beta }$\begin{align}L.H.S.&=\cot (\alpha +\beta )\\ &=\dfrac{1}{\tan (\alpha +\beta )}\\ &=\dfrac{1}{\,\dfrac{\tan \alpha +\tan \beta… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-1-p2?rev=1737476039&do=diff Question 2, Exercise 2.1 Solutions of Question 2 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\quad A=\left[\begin{array}{lll}3 & 6 & 2 \\ 2 & 1 & 9\end{array}\right]$$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/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/ex4-4-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 2025-01-21T16:14:00+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/ex4-8-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-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/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 2025-01-21T16:13:56+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/ex10-2-p1?rev=1737476036&do=diff Question 1, Exercise 10.2 Solutions of Question 1 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. There are four parts in Question 1.$\sin 2\theta ,\,\,\cos 2\theta$$\tan 2\theta$$\tan \theta =-\dfrac{1}{5}$$\theta$$\sin \theta =\dfrac{1}{\sqrt{26}}$$\cos \theta =\dfrac{-5}{\sqrt{26}}$\begin{align}\sin 2\theta &=2\sin… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/review-ex-1-p4?rev=1737476037&do=diff Question 6, 7 & 8, Review Exercise 1 Solutions of Question 6, 7 & 8 of Review Exercise 1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{1}{3+4i}$$$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.$$$\dfrac{3i+2}{3-2… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p7?rev=1737476037&do=diff Question 7, Exercise 2.2 Solutions of Question 7 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7(i) $\left| \begin{matrix}3860 & 3861 \\3862 & 3863 \end{matrix} \right|$$$\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/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/ex3-4-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}$$\vec{b}$\begin{align}\hat{n}&=\dfrac{\vec{a} \times \vec{b}}{\mid \vec{a} \times \vec{b}} \\ \text { … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/ex10-2-p1?rev=1737476039&do=diff Question 1, Exercise 10.2 Solutions of Question 1 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. There are four parts in Question 1.$\sin 2\theta ,\,\,\cos 2\theta$$\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-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$$\left(|x|-|y|)^{2} \geq 0\right)$\begin{align} &\left(|x|-|y|)^{2} \geq 0\right) \\ \implies & |x|^2+|y|^2-2|x||y| \geq 0 \\ \implies & |x|^2+|y|^2 \geq 2|x||y| \\ \implie… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-3-p3?rev=1737476039&do=diff Question 3, Exercise 1.3 Solutions of Question 3 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 3(i) $\dfrac{1}{3} z^{2}+2 z-16=0$\begin{align}&\dfrac{1}{3}z^{2}+2 z-16=0\\ \implies &z^{2} + 6z - 48 = 0 \end{align}$$ z = \dfrac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a},$$$$a = 1,\quad b = 6,\quad \text{and}\quad c = -48.$$\begin{align} z& = \d… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-4-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}.$$\begin{align} x&=x_{\max} e^{i\theta} \\ &=0.004 e^{i\dfrac{\pi}{4}} \\ &=\frac{4}{1000} \left(\cos\left(\dfrac{\pi}{4}\right) +i \sin\left(\dfrac{\pi}{4}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/re-ex-p2?rev=1737476039&do=diff Question 2, Review Exercise Solutions of Question 2 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $i^{2}+i^{4}+i^{6}+\cdots+i^{100}$\begin{align*} & i^{2}+i^{4}+i^{6}+\ldots+i^{100} \\ =& i^2 + (i^2)^2 + (i^2)^3 + (i^2)^4 + \ldots +(i^2)^{49} +(i^2)^{50} \\ =& -1 + (-1)^2 + (-1)^3 + (-1)^4 + \ldots + (-1)^{49}+(-1)^{50} \\ =& -1+1-1+1- \ldots -… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-1-p1?rev=1737476039&do=diff Question 1, Exercise 2.1 Solutions of Question 1 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{lll}1 & 3 & 0 \\ 2 & 0 & 1\end{array}\right]$\begin{align}\text{Order of A}&= 2\times 3\end{align}$B=\left[\begin{array}{ll}1 & 2 \\ 2 & 3 \\ 3 & 4\end{array}\right]$\begin{align}\text{Order of B}&= 3\times 2\end{align}$C… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p4?rev=1737476039&do=diff Question 8 and 9, Exercise 4.4 Solutions of Question 8 and 9 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$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-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 2025-01-21T16:14:00+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}.$$\begin{align… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-3-p3?rev=1737476040&do=diff Question 1(ix, x & xi) Exercise 8.3 Solutions of Question 1(ix, x & xi) of Exercise 8.3 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 \sin 75{\circ} \sin 15{\circ}$\begin{align*} &\quad2 \sin 75^{\circ} \sin 15^{\circ} \\ &= \cos(75^{\circ} - 15^{\circ}) - \cos(75^{\circ} + 15^{\circ}) \\ &= \cos 60^{\circ} - \cos 90^{\circ} \\ \end{align*}$4 \sin … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/re-ex-p6?rev=1737476040&do=diff Question 7, Review Exercise Solutions of Question 7 of Review Exercise of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\dfrac{4 \sin ^{2} \theta \cos \theta}{\cos 3 \theta+\cos \theta}=\tan 2 \theta \tan \theta$\begin{align*} LHS&=\frac{4 \sin^2 \theta \cos \theta}{\cos 3 \theta + \cos \theta}\\ &=\frac{4 \sin \theta\sin \theta \cos \theta}{4\cos^ 3 \t… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/matric/9th_science/unit_02/exercise_2.3?rev=1737476041&do=diff Exercise 2.3 (Solutions) Question 1 Write each radical expression in exponential notation and each exponential expression in radical notation, Do not simplify. * (i) $\sqrt[3]{-64}$ *(ii) $2^{35}$ * (iii) $-7^\frac{1}{3}$ * (iv) $y^\frac{-2}{3}$$\sqrt[3]{-64} = -64^\frac{1}{3}$$2^\frac{3}{5} = \sqrt[5]{2}^{3}$$-7^\frac{1}{3} = -\sqrt[3]{7}$$y^\frac{-2}{3} = \sqrt[3]{y}^{-2}$$ 5^\frac{1}{5} = \sqrt{5}$$2^\frac{2}{3} = \sq… text/html 2025-01-21T16:13:54+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/atiq/sp20-mth604?rev=1737476034&do=diff ~~DISCUSSION~~ MTH604: Fixed Point Theory and Applications (Spring 2020) Course Objectives: This course is intended as a brief introduction to the subject with a focus on Banach Fixed Point theorems fixed point theorem and its application to nonlinear differential equations, nonlinear integral equations, real and complex implicit functions theorems and system of nonlinear equations. Some generalizations and similar results e. g. Kannan Fixed Point theorems, Banach Fixed Point theorem for mul… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-1-p9?rev=1737476036&do=diff Question 11, Exercise 1.1 Solutions of Question 11 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 11(i) ${{z}_{1}}=2-i$${{z}_{2}}=-2+i$${\rm Re}\left( \dfrac{{{z}_{1}}{{z}_{2}}}{\overline{{{z}_{1}}}} \right)$$z_1=2-i$$z_2=-2+i$$\overline{z_1}=2+i$\begin{align} z_1 z_2&=(2-i)(-2+i)\\ &=-4+1+2i+2i\\ &=-3+4i \end{align}\begin{align} \dfrac{z_1 z_2}{\… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-2-p4?rev=1737476036&do=diff Question 5, Exercise 1.2 Solutions of Question 5 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) ${{z}_{1}}=2+4i$${{z}_{2}}=1-3i$$\overline{{{z}_{1}}+{{z}_{2}}}=\overline{{{z}_{1}}}+\overline{{{z}_{2}}}$${{z}_{1}}=2+4i$${{z}_{2}}=1-3i$$\overline{{{z}_{1}}}=2-4i$$\overline{{{z}_{2}}}=1+3i$\begin{align}z_1+z_2&=2+4i+1-3i\\ &=3+i \end{align}\begin… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-1-p9?rev=1737476037&do=diff Question 11, Exercise 1.1 Solutions of Question 11 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 11(i) ${{z}_{1}}=2-i$${{z}_{2}}=-2+i$${\rm Re}\left( \dfrac{{{z}_{1}}{{z}_{2}}}{\overline{{{z}_{1}}}} \right)$$z_1=2-i$$z_2=-2+i$$\overline{z_1}=2+i$\begin{align} z_1 z_2&=(2-i)(-2+i)\\ &=-4+1+2i+2i\\ &=-3+4i \end{align}\begin{align} \dfrac{z_1 z_2}{\… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-2-p4?rev=1737476037&do=diff Question 5, Exercise 1.2 Solutions of Question 5 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) ${{z}_{1}}=2+4i$${{z}_{2}}=1-3i$$\overline{{{z}_{1}}+{{z}_{2}}}=\overline{{{z}_{1}}}+\overline{{{z}_{2}}}$${{z}_{1}}=2+4i$${{z}_{2}}=1-3i$$\overline{{{z}_{1}}}=2-4i$$\overline{{{z}_{2}}}=1+3i$\begin{align}z_1+z_2&=2+4i+1-3i\\ &=3+i \end{align}\begin… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p5?rev=1737476037&do=diff Question 5, Exercise 2.2 Solutions of Question 5 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) $\begin{vmatrix}a & b & c\\l & m & n\\x & y & z \end{vmatrix}=\begin{vmatrix}a & l & x\\b & m & y\\c & n & z \end{vmatrix}$\begin{align}L.H.S.&=\begin{vmatrix} a & b & c \\ l & m & n \\ x & y & z \end{vmatrix}\\ &=\begin{vmatrix} a & b &… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-3-p2?rev=1737476037&do=diff Question 2 and 3 Exercise 3.3 Solutions of Question 2 and 3 of Exercise 3.3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2 $$\vec{a}=2 \hat{i} + 2 \hat{j}-5 \hat{k}, \quad \vec{b}=2 \hat{i}+\hat{j}-7 \hat{k}$$\begin{align}\vec{a}+\vec{b}&=(2 \hat{i}+2 \hat{j}-5 \hat{k})+(2 \hat{i}+\hat{j}-7 \hat{k}) \\ \Rightarrow &=4 \hat{i}+3 \hat{j}-12 \hat{k}\\ \Rightarrow|\vec{a}+\… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-4-p1?rev=1737476037&do=diff Question 1 Exercise 3.4 Solutions of Question 1 of Exercise 3.4 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) Find the cross product $\hat{j} \times(2 \hat{j}+3 \hat{k})$\begin{align}\vec{a}=\hat{j}&=0 \hat{i}+\hat{j}+0 \hat{k}\\ \vec{b}&=0 \hat{i}+2 \hat{j}-3 \hat{k}\\ \vec{a} \times \vec{b}&=\hat{j} \times(2 \hat{j}+3 \hat{k})\\ &=\left|\begin{array}{lll}\hat{i}… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-4-p4?rev=1737476037&do=diff Question 4 Exercise 3.4 Solutions of Question 4 of Exercise 3.4 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4(i) If $\vec{a}=3 \hat{i}-6 \hat{j}+5 \hat{k},\quad\vec{b}=2\hat{i}-\hat{j}+4 \hat{k} \quad$ and $\quad \vec{c}=\hat{i}+\hat{j} \quad \hat{k},\quad$$\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-p7?rev=1737476037&do=diff Question 7 & 8 Exercise 3.4 Solutions of Question 7 & 8 of Exercise 3.4 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7 If $\vec{A}+\vec{B}+\vec{C}=\vec{O}$$$\vec{A} \times \vec{B}=\vec{B} \times \vec{C}=\vec{C} \times \vec{A}.$$$$\vec{A}+\vec{B}+\vec{C}=\vec{O} \text {. }$$$\vec{A}$$$\vec{A} \times(\vec{A}+\vec{B}+\vec{C})=0$$\begin{align}\Rightarrow \vec{A} \times \ve… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-5-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}$$\vec{a} \times \v… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-5-p6?rev=1737476038&do=diff Question 7 Exercise 3.5 Solutions of Question 7 of Exercise 3.5 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7(i) For what value of $c$$\vec{u}=\hat{i}+2 \hat{j}+3 \hat{k}$$\vec{v}=2 \hat{i}-3 \hat{j}+4 \hat{k} \cdot \vec{w}=3 \hat{i}+\hat{j}+c \hat{k}$\begin{align}\vec{u} \cdot \vec{v} \times \vec{w}&=0\\ \vec{u} \cdot \vec{v} \times \vec{w}&=0\\ \Rightarrow\left|\beg… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/review-ex3-p1?rev=1737476038&do=diff Question 1 Review Exercise 3 Solutions of Question 1 of Review Exercise 3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1 $\hat{i} \cdot(\hat{j} \times \hat{k})+\hat{j} \cdot(\hat{i} \times \hat{k})+\hat{k} \cdot(\hat{i} \times \hat{j})$$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}$$\hat{i}-2 \hat{i}+\hat{j}+3 \… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p3?rev=1737476038&do=diff Question 5 and 6 Exercise 4.2 Solutions of Question 5 and 6 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$\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}).$$\begin{align}a_n&=\log(a b^{n-1}). \end{align}\begin{align} d&=a_{n+1}-a_n \\ &=\log (a b^n)-\log (a b^{n-1}) \\ &=\log \left(\… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p5?rev=1737476038&do=diff Question 5 Exercise 7.1 Solutions of Question 5 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$1^3+2^3+3^3+\ldots+n^3=\left[\dfrac{n(n+1)}{2}\right]^2$$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 $$ term of the given series on l… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p4?rev=1737476039&do=diff Question 4 Exercise 7.3 Solutions of Question 4 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$x$$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-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$$$ S=y+1=1+\f… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/re-ex7-p4?rev=1737476039&do=diff Question 5 & 6 Review Exercise 7 Solutions of Question 5 & 6 of Review Exercise 7 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\left(\frac{2}{x^2}+\frac{x^2}{2}\right)^{10}$$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-nbf/sol/unit02/ex2-6-p5?rev=1737476039&do=diff Question 5, Exercise 2.6 Solutions of Question 5 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $x_{1}+x_{2}+2 x_{3}=8$$-x_{1}-2 x_{2}+3 x_{3}=1$$3 x_{1}-7 x_{2}+4 x_{3}=10$$A X=B$\begin{align*} &A = \begin{bmatrix} 1 & 1 & 2 \\ -1 & -2 & 3 \\ 3 & -7 & 4 \end{bmatrix}, \quad X = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}, \quad B = \… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/re-ex-p3?rev=1737476039&do=diff Question 4 and 5, Review Exercise Solutions of Question 4 and 5 of Review Exercise of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left|\begin{array}{ccc}a+1 & l & l \\ l & a+1 & l \\ l & l & a+1\end{array}\right|=(a+1+2 l)(a+1-l)^{2}$\begin{align*} L.H.S &= \left|\begin{array}{ccc}a+1 & l & l \\ l & a+1 & l \\ l & l & a+1\end{array}\right|\\ &=\left|\b… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p1?rev=1737476039&do=diff Question 1 and 2, Exercise 4.5 Solutions of Question 1 and 2 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $16+16+16+\ldots$$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/ex4-7-p14?rev=1737476040&do=diff Question 27 and 28, Exercise 4.7 Solutions of Question 27 and 28 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$5+\frac{7}{3}+\frac{9}{9}+\frac{11}{27}+\ldots$$$$5+\frac{7}{3}+\frac{9}{9}+\frac{11}{27}+\ldots$$$$ 5\times 1+7\times\frac{1}{3}+9\times\frac{1}{9}+11\times\frac{1}{27}+\ldots $$$5,7,9,11,4,\ldots$$a=5$$d=7-5=2$$1, \dfrac{1}{3}, \d… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-8-p7?rev=1737476040&do=diff Question 13, 14 and 15, Exercise 4.8 Solutions of Question 13, 14 and 15 of Exercise 4.8 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\frac{1}{5 \cdot 11}+\frac{1}{7 \cdot 13}+\frac{1}{9 \cdot 15}+\ldots \ldots$$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/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 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/ppsc/ppsc-maths-2021?rev=1737476042&do=diff PPSC Paper 2021 (Lecturer in Mathematics) [PPSC Paper 2011 (Lecturer in Mathematics)] On this page, we have given question from old (past) paper of Lecturer in Mathematics conducted in year 2021. This is a MCQs paper and answers are given at the end of the paper. At the end of the PDF is also given to download. This paper is provided by Ms. \(2018\)$4$\(6\)$8$$10$\(X\)\(Y\)\(X\times Y\)\(\parallel (x,y) \parallel=\parallel x\parallel+\parallel y\parallel, \,\forall \, (x,y)\in X \times Y\)\(f(… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-ptb/important-questions/ch08-mathematical-induction-and-binomial-theorem?rev=1737476037&do=diff Ch 08: Mathematical Induction and Binomial Theorem <list-group> * Using binomial theorem,expand $\left(\frac{x}{2}-\frac{2}{x^2}\right)$ --- BISE Gujranwala(2015) * Find the $6$th term in the expansion of $\left( x^2-\frac{3}{2x}\right)$ --- BISE Gujranwala(2015) * Expand $\left( 8-2x\right)^{-1}$ up to two terms. --- BISE Gujranwala(2015)$1+\frac{1}{4}+\frac{1.3}{4.8}+\frac{1.3.5}{4.8.12},...=\sqrt{2}$$(1.03)^{\frac{1}{3}}$$(a+x)$$n$$x$$(x-\frac{2}{x})^{10}$$n^3-n$$6$$n=2,3$$4^n>3^n+… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_2_solutions/ch02?rev=1737476036&do=diff Unit 02: Differentiation [Unit 02: Differentiation] Notes (Solutions) of Unit 02: Differentiation, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. You can view online or download PDF. To view PDF, you must have PDF Reader installed on your system and it can be downloaded from Software section.$f'(x)$$x^n$$n \in \mathbb{Z}$$\frac{x+1}{x-1}$$x$$$ \begin{aligned} \frac{d}{dx}\left(\frac{x+1}{x-1}\right) &= \frac{(x-1)\frac{d}{dx}(x… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/matric/9th_science/unit08?rev=1737476041&do=diff Unit 08: Linear Graph and their Application On this page notes of Unit 08 of Mathematics 9 written by Dr. Karamat H. Dar and Prof. Irfan-ul-Haq are given. [Unit 08: Linear Graph and their Application] After studying this unit the students will be able to: * Identity pair of real numbers as an ordered pair.$O$$\left( O \right)$$\left( a,b \right)$$a\,$$b$$y=c.$$x=a.$$y=mx.$$y=mx+c.$ text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/msc/real_analysis_notes_by_syed_gul_shah/limits_and_continuity?rev=1737476041&do=diff Chapter 03 - Limits and Continuity * Limit of the function, examples and definition * Theorem: Suppose (i) $(X,{d_x})$ and $(Y,{d_y})$ be two metric spaces (ii) $E\subset X$ (iii) $f:E\to Y$ i.e. f maps E into X (iv) p is the limit point of E. Then $\lim_{x\to p} f(x)=q$ iff $\lim_{n\to\infty}f(p_n)=q$ for every sequence {$p_n$} in E such that ${p_n}\ne p$$\lim_{n\to\infty}{p_n}=p$$\lim_{x\to c}f(x)$$c\in G$$\lim_{x\to c}f(x)=l$$\varepsilon$$\delta>0$$|f(t)-f(s)|<\varepsilon$$\left\{x:|x-c|… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-1-p5?rev=1737476036&do=diff Question 6, Exercise 1.1 Solutions of Question 6 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6(i) $\dfrac{4+i}{3+5i}$$a+ib$\begin{align}\dfrac{4+i}{3+5i}&=\dfrac{4+i}{3+5i}\times \dfrac{3-5i}{3-5i}\\ &=\dfrac{\left( 12+5 \right)+\left( 3-20 \right)i}{9-25{{i}^{2}}}\\ &=\dfrac{17-17i}{9+25}\\ &=\dfrac{17}{34}-\dfrac{17}{34}i\\ &=\dfrac{1}{2}-\dfr… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-2-p8?rev=1737476036&do=diff Question 9, Exercise 1.2 Solutions of Question 9 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9(i) $z=3+2i,$$-|z|\leq \operatorname{Re}\left( z \right)\leq |z|$$z=3+2i$$|z|=\sqrt{9+4}=\sqrt{13}$${\rm Re}z=3=\sqrt{9}$\begin{align} &-\sqrt{13} \leq \sqrt{9} \leq \sqrt{13}\\ \implies &-|z|\leq \operatorname{Re}\left( z \right)\leq |z|\end{align}$z=3… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-2-p3?rev=1737476036&do=diff Question 3, Exercise 10.2 Solutions of Question 3 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin \theta =\dfrac{4}{5}$$\theta$$\sin2\theta$$\sin \theta =\dfrac{4}{5}$$\theta$$\cos \theta =-\dfrac{3}{5}$\begin{align}\sin 2\theta &=2\sin \theta \cos \theta \\ &=2\left( \dfrac{4}{5} \right)\left( -\dfrac{3}{5} \rig… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit10/re-ex10-p2?rev=1737476037&do=diff Question 2 and 3, Review Exercise 10 Solutions of Question 2 and 3 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\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/math-11-kpk/sol/unit01/ex1-1-p5?rev=1737476037&do=diff Question 6, Exercise 1.1 Solutions of Question 6 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6(i) $\dfrac{4+i}{3+5i}$$a+ib$\begin{align}\dfrac{4+i}{3+5i}&=\dfrac{4+i}{3+5i}\times \dfrac{3-5i}{3-5i}\\ &=\dfrac{\left( 12+5 \right)+\left( 3-20 \right)i}{9-25{{i}^{2}}}\\ &=\dfrac{17-17i}{9+25}\\ &=\dfrac{17}{34}-\dfrac{17}{34}i\\ &=\dfrac{1}{2}-\dfr… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-2-p8?rev=1737476037&do=diff Question 9, Exercise 1.2 Solutions of Question 9 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9(i) $z=3+2i,$$-|z|\leq \operatorname{Re}\left( z \right)\leq |z|$$z=3+2i$$|z|=\sqrt{9+4}=\sqrt{13}$${\rm Re}z=3=\sqrt{9}$\begin{align} &-\sqrt{13} \leq \sqrt{9} \leq \sqrt{13}\\ \implies &-|z|\leq \operatorname{Re}\left( z \right)\leq |z|\end{align}$z=3… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit02/ex2-3-p3?rev=1737476037&do=diff Question 3, Exercise 2.3 Solutions of Question 3 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3(i) $$\left[ \begin{matrix} 1 & 0 & -2 \\ 2 & 2 & 1 \\ -1 & 2 & 3 \\ \end{matrix} \right]$$\begin{align}&\begin{bmatrix} 1 & 0 & -2 \\ 2 & 2 & 1 \\ -1 & 2 & 3 \end{bmatrix}\\ \underset{\sim}{R}& \begin{bmatrix} 1 & 0 & -2 \\ 0 &… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-4-p5?rev=1737476037&do=diff Question 5 Exercise 3.4 Solutions of Question 5 of Exercise 3.4 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) Use the vector product to compute the area of the triangle with the given vertices $P(-2,-3), \quad Q(3,2)\quad$$\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 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-5-p1?rev=1737476038&do=diff Question 1 & 2 Exercise 3.5 Solutions of Question 1 & 2 of Exercise 3.5 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1 Find $\vec{a} \cdot \vec{b} \times \vec{c}$$\vec{a}=2 \hat{i}+\hat{j}+3 \hat{k}$$\vec{b}=-\hat{i}+2 \hat{j}+\hat{k} \quad \text { and }\quad \vec{c}=3 \hat{i}+\hat{j}+2 \hat{k} \text {. }$\begin{align}V&=\vec{a} \cdot \vec{b} \times \vec{c}\\ &=\left|\… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-5-p7?rev=1737476038&do=diff Question 8 Exercise 3.5 Solutions of Question 8 of Exercise 3.5 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 8(i) Find the volume of tetrahedron with the Vectors as coterminous edges \begin{align}\vec{a}&=\hat{i}+2 \hat{j}+3 \hat{k},\\ \vec{b}&=4 \hat{i}+5 \hat{j}+6 \hat{k}, \\ \vec{c}&=7 \hat{j}+8 \hat{k}\end{align}\begin{align}V&=\dfrac{1}{6}[\vec{u} \cdot \vec{v} \… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/review-ex3-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/unit04/ex4-2-p11?rev=1737476038&do=diff Question 15 Exercise 4.2 Solutions of Question 15 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 15 $n, \dfrac{a^{n+1}+b^{n+1}}{a^n+b^n}$$a$$b$$a$$b$$A$$a$$b$$$ A=\dfrac{a+b}{2}. --- (1) $$$$ A=\dfrac{a^{n+1}+b^{n+1}}{a^n+b^n}. --- (2) $$\begin{align}&\dfrac{a+b}{2}=\dfrac{a^{n+1}+b^{n+1}}{a^n+b^n}, --- (3) \\ \implies &(a^n+b^n)(a+b)=2(a^{n+1… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-3-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$$n$\begin{align} S_n&=\dfrac{n}{2}[a_1+a_n], \\ \implies S_{20}&=\dfrac{20}{2}[9-29] … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p3?rev=1737476038&do=diff Question 3 & 4 Review Exercise 6 Solutions of Question 3 & 4 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${ }^{56} P_{r+6}:{ }^{54} P_{r+3}=30800: 1$$r$\begin{align} { }^{56} P_{r+6}:{ }^{54} P_r+3&=30800: 1 \\ \Rightarrow \dfrac{\dfrac{56 !}{[56-(r+6)] !}}{\dfrac{54 !}{[54-(r+3)] !}}&=\dfrac{30800}{1} \\ \Rightarrow \dfrac{56… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p1?rev=1737476038&do=diff Question 1 Exercise 7.2 Solutions of Question 1 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$(x^2-\dfrac{1}{y})^4$\begin{align}(x^2-\dfrac{1}{y})^4&=(x^2)^4+{ }^4 C_1(x^2)^3(-\dfrac{1}{y})+ \\ & { }^4 C_2(x^2)^2(-\dfrac{1}{y})^2+{ }^4 C_3(x^2)(-\dfrac{1}{y})^3 + { }^4 C_4(-\dfrac{1}{y})^4 \\ & =x^8- \dfrac{4x^6}{y}+\dfrac{6x^4}{y^2}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/re-ex7-p2?rev=1737476039&do=diff Question 2 Review Exercise 7 Solutions of Question 2 of Review Exercise 7 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\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/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$$\cos \theta =-\dfrac{3}{5}$\begin{align}\sin 2\theta &=2\sin \theta \cos \theta \\ &=2\left( \dfrac{4}{5} \right)\left( -\dfrac{3}{5} \rig… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/re-ex10-p2?rev=1737476039&do=diff Question 2 and 3, Review Exercise 10 Solutions of Question 2 and 3 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{2\sin \theta \sin 2\theta }{\cos \theta +\cos 3\theta }=\tan 2\theta \tan \theta $\begin{align}L.H.S.&=\dfrac{2\sin \theta \sin 2\theta }{\cos \theta +\cos 3\theta }\\ &=\dfrac{2\sin \theta \s… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p3?rev=1737476039&do=diff Question 3, Exercise 1.2 Solutions of Question 3 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 3(i) $z \in \mathbb{C}$$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$$\begin{align}|z_1|&=\sqrt{(2)^2+(-3)^2}\\ &=\sqrt{13}\end{align}\begin{align}&|z_{1} z_{2}|=16\\ \Rightarrow \quad &|z_{1}|| z_{2}|=16\\ \Rightarrow \quad & \sqrt{13… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p6?rev=1737476039&do=diff Question 6, Exercise 1.2 Solutions of Question 6 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 6 $\lambda$$\left|\dfrac{z_{1}}{z_{2}}+\lambda\right|=\sqrt{\lambda+2}$$z_{1}=3+i$$z_{2}=1+i$\begin{align} &z_{1}=3+i\text{ and } z_{2}=1+i.\end{align}\begin{align} \dfrac{z_1}{z_2} &= \dfrac{3+i}{1+i}\\ &=\dfrac{(3+i)(1-i)}{(1+i)(1-i)} \\ &=\… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-3-p2?rev=1737476039&do=diff Question 2, Exercise 1.3 Solutions of Question 2 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 2(i) $z^{2}-6 z+2=0$\begin{align} & z^2 - 6z + 2 = 0 \\ \implies & z^2 - 2(3)(z)+9-9+2=0 \\ \implies & (z - 3)^2+7= 0 \\ \implies & (z - 3)^2 = 7. \end{align}\begin{align} &z - 3 = \pm \sqrt{7} \\ \implies &z = 3 \pm \sqrt{7}\end{align}$\{3 … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-4-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$$z=i \tan \theta$\begin{align}&\dfrac{1+z}{1-z}=\cos 2 \theta+i \sin 2 \theta\\ \implies &\dfrac{1+z}{1-z}=e^{i2\theta}\\ \implies &(1+z)=(1-z)e^{i2\theta}\\ \implies &z+z e^{i2\theta}=e^{i2\th… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/re-ex-p4?rev=1737476039&do=diff Question 4, Review Exercise Solutions of Question 4 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $z=x+i y$$\left|\dfrac{z+2 i}{z-2 i}\right|=1$$z = x + iy$\begin{align*} & \left|\dfrac{z + 2i}{z - 2i}\right| = 1\\ \implies & |z + 2i| = |z - 2i|\\ \implies & |x + i(y + 2)| = |x + i(y - 2)|\\ \implies & \sqrt{x^2 + (y + 2)^2} = \sqrt{x^2 + (y -… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/re-ex-p7?rev=1737476039&do=diff Question 7, Review Exercise Solutions of Question 7 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 z^{2}-11 z+16=0$\begin{align*} &2 z^{2}-11 z+16=0\\ \implies&z^2 - \dfrac{11}{2}z + 8 = 0\\ \implies& z^2 - \dfrac{11}{2}z = -8\\ \implies& z^2 - 2z\dfrac{11}{4}z + \dfrac{121}{16} = -8 + \dfrac{121}{16}\\ \implies&\left(z-\dfrac{11}{4}\right)^2… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/re-ex-p8?rev=1737476039&do=diff Question 8, Review Exercise Solutions of Question 8 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sqrt{2}+i \sqrt{2}$$\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/ex2-2-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-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]$$X+3 Y=\left[\begin{array}{ccc}4 & 3 & 2 \\ 1 & -3 & 0\end{array}\right]$\begin{align*} 2X - Y = \begin{pmatrix} 1 & 6 & -3 \\ 2 & 1 & 7 \end{pmatrix} \cdots (i)\\… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p7?rev=1737476039&do=diff Question 13 and 14, Exercise 4.3 Solutions of Question 13 and 14 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $S_s$$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/ex4-4-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}$$a_{n}=… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p14?rev=1737476039&do=diff Question 28 and 29, Exercise 4.4 Solutions of Question 28 and 29 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $=a_1= 1$$= a_2 = 2$$= a_3 = 2(2)=4$$= a_7$$$ 1+2+4+...+a_7 $$$a_1=1$$r=2$$n=7$$$ S_n=\frac{a_1\left(1-r^n \right)}{1-r}, \quad r\neq 1. $$\begin{align*} S_6&=\frac{(1)\left(1-2^7 \right)}{1-2} \\ &=\frac{1-128}{-2}\\ &=127 \end{align… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-6-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/ex4-7-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/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/re-ex-p2?rev=1737476040&do=diff Question 2 and 3, Review Exercise Solutions of Question 2 and 3 of Review Exercise of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\cos \theta -\sin \theta=\sqrt{2}\sin \theta,$$\cos \theta+ \sin \theta=\sqrt{2} \cos \theta$$$\cos \theta -\sin \theta=\sqrt{2}\sin \theta$$\begin{align*} & \cos \theta=\sqrt{2}\sin \theta + \sin \theta \\ \implies & \cos \the… text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/junaid?rev=1737476042&do=diff Dr. Junaid Alam Khan <callout type=“info” icon=“true”> This is a personal web page of Dr. Junaid Alam Khan Associate Professor Institute of Business Administration, Karachi - PAKISTAN. IBA Profile: <https://oric.iba.edu.pk/profile.php?id=jakhan> ResearchGate Profile: <https://www.researchgate.net/scientific-contributions/2141923316_Junaid_Alam_Khan> Facebook: text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/quote-of-the-day?rev=1737476042&do=diff Quote of the Day [Quote of the Day] The “quote of the day” is a useful tool for inspiration, motivation, and self-examination It offers a daily serving of knowledge from reputable and accomplished people in a range of sectors, including literature, politics, science, and entertainment. Everyday reading and reflection on a thought-provoking quote can help people learn more, see things from other people's perspectives, and advance their personal development. Sharing the text/html 2025-01-21T16:13:54+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/atiq/fa20-mth424?rev=1737476034&do=diff MTH424: Convex Analysis (Fall 2020) [Convex Analysis] Objectives: At the end of this course the students will be able to understand the concept of Convex Analysis, convex sets, convex functions, Differential of the convex function. Developing ability to study the Hadamard-Hermite inequalities and their applications. Prepare students to be self independent and enhance their mathematical ability by giving them home work and projects.$f(x)=x$$\mathbb{R}$$f(x)=x^2$$\mathbb{R}$$f:[a,b]\to \mathbb{… text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/bsc/vector_analysis_by_hameed_ullah?rev=1737476035&do=diff Vector Analysis by Hameed Ullah: Notes [right triangle in semi circle] Note of vector analysis by Hammed Ullah. These notes are send by Umer Asghar, we are very thankful to him for providing these notes. These notes are for helpful for undergraduate level (BSc or BS). <div> <center> </div> Name Notes of vector analysis <div> </center> </div> text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/definitions?rev=1737476036&do=diff Definitions: FSc Part1 KPK A Textbook of Mathematics for Class XI is published by Khybar Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. The book has total of twelve (12) chapters. Definition of the book provide the quick overview of the book.$360^\circ$$\theta$$90^{\circ} \pm \theta, 180^{\circ} \pm \theta, 270^{\circ} \pm \theta, 360^{\circ} \pm \theta$$16^\circ 13' 9''$$sin(\alpha+2\pi)=sin\alpha$$sin x=\frac{2}{7}$$cos x-tan x=0$ text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/kpk_fsc_part_1?rev=1737476036&do=diff FSc Part 1 (KPK Boards) AVAILABLE HERE [FSc Part 2 KPTP] Notes of FSc Part 1 of “A Textbook of Mathematics For Class XI” published by Khyber Pakhtunkhwa Textbook Board, Peshawar. We are posting the notes chapter-wise. These notes are shared as open educational resources. This page will be continuously updated.$P(z)$$(\sum)$$\sum n$$\sum n^2$$\sum n^3$$n$$n$$$\frac{a}{a(a+d)}+\frac{a}{(a+d)(a+2d)}+...$$$^nP_r$$^nC_r=\left(\begin{smallmatrix}n\\ r\end{smallmatrix} \right)=\frac{n!}{r!(n-r)!}$$P(… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/definitions?rev=1737476037&do=diff Definitions: FSc Part1 KPK A Textbook of Mathematics for Class XI is published by Khybar Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. The book has total of twelve (12) chapters. Definition of the book provide the quick overview of the book.$360^\circ$$\theta$$90^{\circ} \pm \theta, 180^{\circ} \pm \theta, 270^{\circ} \pm \theta, 360^{\circ} \pm \theta$$16^\circ 13' 9''$$sin(\alpha+2\pi)=sin\alpha$$sin x=\frac{2}{7}$$cos x-tan x=0$ text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/note-for-numerical-methods-m-usman-hamid?rev=1737476041&do=diff Notes for Numerical Methods by M Usman Hamid [Notes for Numerical Methods by M Usman Hamid] These notes are initially provided by Mr. Anwar Khan. Later the updated version is send by Muhammad Tahir. We are really very thankful to Mr. Anwar Khan and Muhammad Tahir for providing these notes and appreciates their effort to publish these notes on MathCity.org$\left(\frac{1}{3}\right)$$\left(\frac{3}{8}\right)$ text/html 2025-01-21T16:13:54+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/atiq/math-608/what_is_mathematics?rev=1737476034&do=diff What is Mathematics? Different people would gave different answers of the above title. A student in elementary school would probably say it was about adding, subtracting, multiplying and dividing. Oh yes--- about functions and decimals too. A student in high school would probably say that it is about learning rules and formulas to solve equations. Oh yes text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part2-ptb/important-questions/unit-04-introduction-to-analytic-geometry?rev=1737476037&do=diff Unit 04: Introduction to Analytic Geometry Here is the list of important questions. <list-group> * Find the area between $x-axis$ and the curve $y=4x-x^2$ --- BSIC Gujranwala (2016) * Find $h$ if $A(-1,h)$, $B(3,2)$, $C(7,3)$ are collinear --- BSIC Gujranwala (2016) * Find the point three fifth of the way along the line segment from $A(-5,8)$$B(5,3)$$2$$y-intercept$$5$$5x-12y+39=0$$2x^2+3xy-5y^2=0$$x-y-4=0$$7x+y+20=0$$6x+y-14=0$$5x-12y+39=0$$(4,6)$$(4,8)$$x-2y+1=0$$2x-y+2=0$$A(2,-5)$$B… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/msc/real_analysis_notes_by_syed_gul_shah/differentiation?rev=1737476041&do=diff Chapter 04 - Differentiation * Derivative of a function * Theorem: Let f be defined on [a,b], if f is differentiable at a point $x\in [a,b]$, then f is continuous at x. (Differentiability implies continuity) * Theorem (derivative of sum, product and quotient of two functions)$x\in [a,b]$$f'(x)$$f'(x)=0$$\mathbb{R}^k$$\underline{f}$$x\in (a,b)$$\left|\underline{f}(b)-\underline{f}(a)\right|\le (b-a)\left|\underline{f'}(x)\right|$ 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}$$$A=\left[ \begin{matrix} -\lambda & 1 & 0 \\ 1 & -\lambda & 1 \\ 0 & 1 & -\lambda \\ \end{matrix} \right]$$$$|A|=-\lamb… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-3-p3?rev=1737476037&do=diff Question 4 and 5 Exercise 3.3 Solutions of Question 4 and 5 of Exercise 3.3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4 $\hat{i}+7 \hat{j} + 3 \hat{k}$$\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}$$\vec{c} = 2 \hat{i}-\hat{j}-3 \hat{k}$\begin{align}\vec{a} \cdot \vec{b}&=(\hat{i}+7 \h… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-3-p8?rev=1737476037&do=diff Question 12 & 13, Exercise 3.3 Solutions of Question 12 & 13 of Exercise 3.3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 12 $\overrightarrow{B A} \cdot \overrightarrow{A C}=0$$|\vec{a}|=\vec{b}|=| \vec{c} \mid=$$\vec{b}=-\vec{c}$$\triangle A B O$\begin{align}\overrightarrow{O B}+\overrightarrow{A B}&=\overrightarrow{O A}\\ \Rightarrow \overrightarrow{B A}&=\overrightar… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-5-p5?rev=1737476038&do=diff Question 6 Exercise 3.5 Solutions of Question 6 of Exercise 3.5 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6 Do the points $(4. 2.1)$$(5,1,6)$$(2.2,-5)$$(3.5 .0)$$A(4,-2,1), B(5,1,6)$$C(2,2,-5)$$D(3,5.0)$$A, \overrightarrow{O A}=4 \hat{i}-2 \hat{j}+\hat{k}$$B, \overrightarrow{O B}=5 \hat{i}+\hat{j}+6 \hat{k}$$C, \overrightarrow{O C}=2 \hat{i}+2 \hat{i}-5 \hat{k}$$D, … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-5-p8?rev=1737476038&do=diff Question 9 Exercise 3.5 Solutions of Question 9 of Exercise 3.5 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9 (i) Write the value of $(\hat{i} \times \hat{j}). \hat{k}+\hat{i}. \hat{j}$\begin{align} (\hat{i} \times \hat{j}) \cdot \hat{k}&=\left|\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right|&=1 ....(1)\\ \text { and } \hat{i} \cdot \hat{j}&=0… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/review-ex3-p2?rev=1737476038&do=diff Question 2 & 3 Review Exercise 3 Solutions of Question 2 & 3 of Review Exercise 3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2 $\lambda$$\mu$$$(\hat{i}+3 \hat{j}+9 \hat{k}) \times(3 \hat{i}-\lambda \hat{j}+\mu \hat{k})=\overrightarrow{0} \text {. }$$\begin{align}(\hat{i}+3 \hat{j}+9 \hat{k}) \times(3 \hat{i}-\lambda \hat{j}+\mu \hat{k})&=\vec{O} \\ \Rightarrow\left|\b… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p4?rev=1737476038&do=diff Question 7 Exercise 4.2 Solutions of Question 7 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7 $a_6+a_4=6$$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/unit05/re-ex5-p1?rev=1737476038&do=diff Question 1 Review Exercise 5 Solutions of Question 1 of Review Exercise 5 of Unit 05: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1 $t_n=6 n+5$$t_{n+1}=$$6 n-1$$6 n+11$$6 n+6$$6 n-5$$1+\dfrac{2}{3}+\dfrac{6}{3^2}+\dfrac{10}{3^3}+\dfrac{14}{3^4}+\ldots$$6$$2$$3$$4$$1+2.2+3.2^2+\cdots+100.2^{\prime \prime}$$99.2^{100}$$100.2^{100}$$99.2^{100}+1$$1000.2^{100}$$n^{t h}$$1.2+2.3+3.4+\… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p2?rev=1737476038&do=diff Question 3 and 4 Exercise 6.2 Solutions of Question 3 and 4 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$^n P_r=n(^{n-1} P_{r-1})$$$^n P_r=n({ }^{n-1} P_{r-1})$$\begin{align}n(^{n-1} P_{r-1})&=n \dfrac{(n-1) !}{((n-1)-(r-1)) !} \\ & =\dfrac{n(n-1) !}{(n-r) !}\\ &=\dfrac{n !}{(n-r) !}\\ &=^n P_r\end{align}$^n P_r=^{n-1} P_r+r(^{n-1} … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-5-p5?rev=1737476038&do=diff Question 8 Exercise 6.5 Solutions of Question 8 of Exercise 6.5 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$7$$11.$\begin{align}s&=(i i, j): i, j-1,2,3,4,5,6\}\\ &=\left[\begin{array}{llllll} (1.1) & (1.2) & (1.3) & (1.4) & (1.5) & (1.6) \\ (2.1) & (2.2) & (2.3) & (2.4) & (2.5) & (2.6) \\ (3.1) & (3.2) & (3.3) & (3.4) & (3.5) & (3.6) \\ (4.1) & (4… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p9?rev=1737476038&do=diff Question 9 Exercise 7.1 Solutions of Question 9 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\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$$$\dfrac{1}{3}+\dfrac{1}{9}+\dfrac{1}{27}+\ldots… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/re-ex7-p3?rev=1737476039&do=diff Question 3 & 4 Review Exercise 7 Solutions of Question 3 & 4 of Review Exercise 7 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$(2 x-4 y)^7$$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-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} $$$$ \begin{aligned} & =4.7^k+3\left[7^k-3^k\… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-1-p7?rev=1737476039&do=diff Question 7, Exercise 1.1 Solutions of Question 7 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 7(i) $11+12 i$$$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)$$z=(2+3i)−(2+6i)$\begin{align}z&=2+3i−2−6i\\ &=-3i \end{align}\begin{align} |z| &= \sqrt{0^2+(-3)^2} \\ &= \sqrt{… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p10?rev=1737476039&do=diff Question 9, Exercise 1.4 Solutions of Question 9 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 9(i) $x=2+3 i$$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/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/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$$$a_n = a_1 + (… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p13?rev=1737476039&do=diff Question 26 and 27, Exercise 4.4 Solutions of Question 26 and 27 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $16\,\, ft$$6$$16\,\,ft$$a_1$$a_2$$a_3,...$$$a_1 = 16\times \dfrac{1}{4} = 4\,\, ft.$$$r=\dfrac{1}{4}$$a_6$$$a_{n}=a_{1} r^{n-1}.$$\begin{align*} a_{6}&=a_{1} r^5 \\ &=(4)\left(\dfrac{1}{4} \right)^5 \\ & = \dfrac{1}{256} \end{align*}… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p8?rev=1737476040&do=diff Question 15, Exercise 4.5 Solutions of Question 15 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $30 ft$$\frac{2}{5}$$= 30 ft$$= 30 \times \frac{2}{5} = 12 ft$$= 12 \times \frac{2}{5} = \frac{24}{5} ft$$= \frac{24}{5} \times \frac{2}{5} = \frac{48}{25} ft$$D$$$D=30+2\left(12+\frac{24}{5}+\frac{24}{5}+... \right)$$$$ 12+\frac{24}{5}+\frac{24}{5… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-6-p3?rev=1737476040&do=diff Question 5 & 6, Exercise 4.6 Solutions of Question 5 & 6 of Exercise 4.6 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\dfrac{1}{27}, \dfrac{1}{20}, \dfrac{1}{13}, \ldots \quad$\begin{align*} &\frac{1}{27}, \frac{1}{20}, \frac{1}{13}, \ldots \quad \text{ is in H.P.} \\ &27, 20, 13, \ldots \quad \text{ is in A.P.} \end{align*}$a_1 = 27$$d = 20 - 27 = -7$$a_n=… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p4?rev=1737476040&do=diff Question 7 and 8, Exercise 4.7 Solutions of Question 7 and 8 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sum_{k=0}^{5}\left(k^{2}-2 k+3\right)$\begin{align*} \sum_{k=0}^{5} (k^{2} - 2k + 3) &= (0^{2} - 2(0) + 3) + (1^{2} - 2(1) + 3) + (2^{2} - 2 (2) + 3) \\ &+ (3^{2} - 2 (3) + 3) + (4^{2} - 2 (4) + 3) + (5^{2} - 2 (5) + 3) \\ &= (0 - 0 + 3… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p6?rev=1737476040&do=diff Question 11, 12 and 13, Exercise 4.7 Solutions of Question 11, 12 and 13 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $-2+4-8+16-32+64$$$ -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-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/unit05/re-ex-p4?rev=1737476040&do=diff Question 6 & 7, Review Exercise Solutions of Question 6 & 7 of Review Exercise of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $k$$\left(x^{2}+8 x+k\right)$$(x-4)$\( p(x) = x^{2} + 8x + k \)\( p(x) \)\( (x - 4) \)\( p(4) \)\( p(4) = 0 \)\begin{align*} p(4) &= (4)^2 + 8(4) + k \\ &= 16 + 32 + k \\ &= 48 + k. \end{align*}\[ 48 + k = 0. \]\[ k = -48. \]$3 x^{2}-x+32-\frac… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/re-ex-p6?rev=1737476040&do=diff Question 6, Review Exercise Solutions of Question 6 of Review Exercise of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 6 $k$$\left(x^{2}+8 x+k\right)$$(x-4)$ text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/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/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-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$$\tan \left(\theta-45^{\circ}\right)=\frac{1}{3}$\begin{align*} & \frac{\tan \theta - \tan 45^{\circ}}{1 + \tan \theta \cdot \tan 45^{\circ}} =\frac{1}{3}\\ \implies & \frac{\tan \theta - 1}{1 + \tan \theta}= \f… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/matric/9th_science/unit_02/exercise_2.1?rev=1737476041&do=diff Exercise 2.1 (Solutions) Question 1 Identify which of the following are rational and irrational numbers: (i) $\sqrt{3}$ (ii) $\frac{1}{6}$ (iii) $\pi$ (iv) $\frac{15}{2}$ (v) $7.25$ (vi)$\sqrt{29}$ Solution * Rational: $\frac{1}{6}$, $\frac{15}{2}$, $7.25$ * Irrational: $\sqrt{3}$, $\pi$, $\sqrt{29}$ Question 2 Convert the following fraction into decimal fraction.$\frac{17}{25}$$\frac{19}{4}$$\frac{57}{8}$$\frac{205}{18}$$\frac{5}{8}$$\frac{25}{38}$$\frac{2}{3}$$\pi$$\frac{1}{9}$$\fr… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/matric/9th_science/unit_02/exercise_2.5?rev=1737476041&do=diff Exercise 2.5 (Solutions) Question 1 * Evaluate (i) $i^7$ (ii) $i^{50}$ (iii) $i^{12}$ (iv) $\left(-i\right)^8$ (v) $\left(-i\right)^5$ (vi) $i^{27}$ Solution $$\begin{array}{cl} i^7 &= {i^6}\cdot i\\ &= (i^2)^3\cdot i\\ &= {-1}^3 \cdot i\\ &= -i \end{array}$$$$\begin{array}{cl} i^{50} &= (i^2 )^{25}\\ &= {-1}^{25}\\ … text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/ahsan?rev=1737476042&do=diff Dr. Muhammad Ahsan Binyamin <callout type=“info” icon=“true”> This is a personal web page of Dr. Muhammad Ahsan Binyamin Associate Professor Government College University Faisalabad, Faisalabad - PAKISTAN. Google Scholar: <https://scholar.google.com.pk/citations?user=PJQDQy0AAAAJ&hl=en> ResearchGate Profile: text/html 2025-01-21T16:14:02+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/cui?rev=1737476042&do=diff Mathematics CUI: LaTeX Resources [Department of Mathematics, COMSATS University Islamabad, Attock Campus] This page contains LaTeX template of CIIT Mathematics, MSc Project and MS Thesis templates. Templates Download a zip file given below and extract it by right clicking on the file. BS Project Template: (Version 1.5, Uploaded: Sep 29, 2022)$\$$I$$\mathbb{R}$$f:I\to \mathbb{R}$$(\$$$\sin^2 \theta + \cos^2 \theta =1$$\begin{equation} \sin^2 \theta + \cos^2 \theta =1 \end{equation} text/html 2025-01-21T18:32:03+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/home?rev=1737484323&do=diff Home Welcome to MathCity.org. Please browse the website by using navigation bar or search the website. “”“”“” More Quotes--- “”“”“” “” Updates * | Math 11 (NBF) | Notes of unit 08 for FSc/ICS part 1 mathematics by NBF has been added text/html 2025-01-21T16:13:54+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/atiq/math-510?rev=1737476034&do=diff MATH-510: Topology <div> <img src="../images/Mug_and_Torus_morph.gif" alt="A continuous deformation (a type of homeomorphism) of a mug into a doughnut (torus) and back." title="Topologically equivalence figures" class="mediaright" /><br> </div> Topology is an important branch of mathematics that studies all the “qualitative” or “discrete” properties of continuous objects such as manifolds, i.e. all the properties that aren't changed by any continuous transformations except for the singular (in… text/html 2025-01-21T16:13:54+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/atiq/sp18-mth251?rev=1737476034&do=diff MTH251: Set Topology [Set Topology] Topology is an important branch of mathematics that studies all the “qualitative” or “discrete” properties of continuous objects such as manifolds, i.e. all the properties that aren't changed by any continuous transformations except for the singular (infinitely extreme) ones.$\mathbb{R}$$T_1$$\mathbb{Z}$$A=\{1,2,3,...,20\}$$\mathbb{R}$$\mathbb{Q}$$\mathbb{R}$$A=\left\{1,\frac{1}{2},\frac{1}{3},... \right\}$$A$$\mathbb{R}$$A=\mathbb{N}$$B=\{1,2,3,...,100\}$$C=… text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/dyk/3?rev=1737476035&do=diff Its about square root [DYK] The reason is not difficult if one knows about the definition of square root of real numbers. Definition: Let $x$ be a non-negative number. Then a non-negative number $r$ is called square root of $x$ iff $r^2=x$. Square root of $x$ is denoted by $\sqrt{x}$$2^2=4$$3^2=9$$x$$r$$x$$r^2=x$$2^2=4$$(-2)^2=4$$\sqrt{4}=\sqrt{2^2}=\sqrt{(-2)^2}=2$$\sqrt{4}=\sqrt{2^2}=\sqrt{(-2)^2}=\pm 2$$\sqrt{4}$$\sqrt{x}$$x\geq0$$\sqrt{x}=x^{\frac{1}{2}}$ text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/kpk_fsc_part_2?rev=1737476036&do=diff FSc Part 2 (KPK Boards) [A Textbook of Mathematics For Class XII] Notes of FSc Part 2 of “A Textbook of Mathematics For Class XII” published by Khyber Pakhtunkhwa Textbook Board, Peshawar. We are posting the notes chapter-wise. These notes are shared as open educational resources. This page will be continuously updated.$y=x^n$$y=(ax+b)^n$ text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/handwritten-notes-real-analysis-asim-marwat?rev=1737476041&do=diff Handwritten Notes of Real Analysis by Asim Marwat [Handwritten Notes of Real Analysis by Asim Marwat] Real analysis is a branch of mathematics that analyses how real numbers, sequences and series, and real functions behave. It focuses on real numbers and frequently extends the real line by including positive and negative infinity. Real analysis investigates a number of the properties of real-valued sequences and functions, including convergence, limits, continuity, smoothness, differentiabilit… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/notes/ring_notes?rev=1737476041&do=diff Ring (Notes) by Prof. M. Dabeer Mughal [Ring (Notes) by Prof. M. Dabeer Mughal] A handwritten notes on Ring (Algebra) by Prof. M. Dabeer Mughal (Federal Directorate of Education, Islamabad, Pakistan). It is best to prepare a “Rings and Vector Spaces” section of your algebra paper or Algebra II for BS or MS Mathematics.$\phi$ text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/bsc/notes_of_calculus_with_analytic_geometry/ch01_real_numbers_limits_and_continuity?rev=1737476035&do=diff Chapter 01: Real Numbers, Limits and Continuity [Chapter 01 of Calculus with Analytic Geometry] Notes of the book Calculus with Analytic Geometry written by Dr. S. M. Yusuf and Prof. Muhammad Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN. The notes of this chapter is written by Prof. $\mathbb{R}$$\mathbb{R}$$\mathbb{R}$ text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/bsc/notes_of_calculus_with_analytic_geometry/ch08_analytic_geometry_of_three_dimensions?rev=1737476035&do=diff Chapter 08: Analytic Geometry of Three Dimensions Notes of the book Calculus with Analytic Geometry written by Dr. S. M. Yusuf and Prof. Muhammad Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN. Contents & Summary * Distance between two points$\mathbb{R}^3$ text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/bsc/paper_pattern/sargodha_university?rev=1737476035&do=diff Syllabus/Model Papers for Sargodha University <div> <img src=http://www.mathcity.org/images/UoS_Gate.jpg class="mediaright" align="right" /> </div> Syllabus for the subjects General Mathematics, A-Course of Mathematics and B-Course of Mathematics for BSc (private and regular) from University of Sargodha, Sargodha - PAKISTAN. Every subject consists of two papers of 100 marks each. In every paper there are three sections with four questions. A student have to attempt two questions from each se… text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch01?rev=1737476035&do=diff Chapter 01: Number System [Chapter 01: Number System] Notes (Solutions) of Chapter 01: Number System, Textbook of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary * Rational numbers and irrational numbers$\mathbb{C}$$(x+iy)^n$$\left(\frac{x_1+iy_1}{x_2+iy_2}\right)^n, x_2+iy_2\neq 0$$\sqrt{-1}=i$$\sqrt{-1}$$i$$-i$$i$$-i$$-1$$i^2=-1$$\sqrt{-1}=i$$\sqrt{-1}$ text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc/fsc_part_1_solutions/ch12?rev=1737476036&do=diff Chapter 12: Application of Trigonometry [Chapter 12: Application of Trigonometry] Notes (Solutions) of Chapter 12: Application of Trigonometry, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary * Introduction text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/msc/mcqs_short_questions/toplogy?rev=1737476041&do=diff Topology: Short Questions and MCQs <callout type=“info” icon=“true”> We are going to add short questions and MCQs for Topology. This is a compulsory subject in MSc and BS Mathematics in most of the universities of Pakistan. The author of this page is Dr. $\mathbb{R}$$X=\{a\}$$X$$X$$X$$\tau$$\mathbb{N}$$\tau$$(\mathbb{Z}, \tau)$$\mathbb{N}$$\tau$$A=\{\pm 100,\pm 101, \pm 102, ... \}$$\tau$$E=\{0,\pm 2,\pm 4,...\}$$\tau$$\tau$$B=\{1,2,3,...,99\}$$\tau$$C=\{10^{10}+n : n \in \mathbb{Z} \}$$\tau$$S… text/html 2025-01-21T16:13:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/bsc/paper_pattern/punjab_university/b.sc._paper_pattern_for_general_mathematics_split_program?rev=1737476035&do=diff Syllabus & Paper Pattern for General Mathematics (Split Program) There was one examination after two years for BA/BSc Program from University of Punjab (PU), Lahore but from this year (2016), PU has made changes in its examination policies for the said program. The BA/BSc Program has been split into two parts. Syllabus is break into two part year wise. After the each year of the program candidate has to appeared in examination instead of appearing after two year. In this regards syllabus of Gen… text/html 2025-01-21T16:13:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-2-p1?rev=1737476036&do=diff Question 1, Exercise 1.2 Solutions of Question 1 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1 If ${{z}_{1}}=2+i$${{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 2025-01-21T16:13:57+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/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.$$\begin{align}z_1 z_2 … text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit01/ex1-3-p3?rev=1737476037&do=diff Question 3 & 4, Exercise 1.3 Solutions of Question 3 & 4 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 ${{z}_{1}}=-1+i$${{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/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}$$B=\begin… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-2-p2?rev=1737476037&do=diff Question 2, Exercise 3.2 Solutions of Question 2 of Exercise 3.2 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2(i) Find unit vector having the same direction as the vector $3\hat{i}.$$$\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$$H$\begin{align}\overrightarrow… text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-3-p4?rev=1737476037&do=diff Question 6 Exercise 3.3 Solutions of Question 6 of Exercise 3.3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6(i) Let $\vec{a}=\hat{i}+3 \hat{j}-4 \hat{k}$ and $\vec{b}=2 \hat{i}-3 \hat{j}-5 \hat{k}$$m$$\vec{a}+m \vec{b}$$\vec{a}$\begin{align} \vec{a}+m \vec{b}& =\hat{i}+3 \hat{j}-4 \hat{k}+m(2 \hat{i}-3 \hat{j}+5 \hat{k}) \\ & =(1+2 m) \hat{i}+(3-3 m) \hat{j}+(5 m-4) … text/html 2025-01-21T16:13:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-3-p6?rev=1737476037&do=diff Question 9 & 10, Exercise 3.3 Solutions of Question 9 & 10 of Exercise 3.3 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9 $\vec{k}-2 \hat{i}+3 \hat{j}+\hat{k}$$\vec{S}=2 \hat{i}+\hat{j}-\hat{k}$\begin{align}W &=\vec{F} \cdot s \\ \Rightarrow W &=(2 \hat{i}+3 \hat{j}+\hat{k}) \cdot(2 \hat{i}+\hat{j}-\hat{k}) \\ \Rightarrow W &=2(2) \div 3(1)+1(-1) \\ \Rightarrow W &=4+3 … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit03/ex3-5-p4?rev=1737476038&do=diff Question 5(iii) & 5(iv) Exercise 3.5 Solutions of Question 5(iii) & 5(iv) of Exercise 3.5 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\vec{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}\quad$$\quad\vec{b}=b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k}\quad$$(\vec{a}. \vec{b})^2,\quad|a|^2,\quad|b|^2$\begin{align}\vec{a} \cdot \vec{b}&=(a_1 \hat{i}+a_2 \hat{j} + a_3 \hat{k}) \cdot(b_1 \hat{i}+b_2 \… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-1-p1?rev=1737476038&do=diff Question 1 and 2 Exercise 4.1 Solutions of Question 1 and 2 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$2,4,6,8, \ldots ,50$$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/ex4-2-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)$$\begin{align}a_{n+1}=2(n+1)+7=2… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p5?rev=1737476038&do=diff Question 8 Exercise 4.2 Solutions of Question 8 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 8 $\dfrac{b+c-a}{a}, \dfrac{c+a-b}{b}, \dfrac{a+b-c}{c}$$\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-p12?rev=1737476038&do=diff Question 16 Exercise 4.2 Solutions of Question 16 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 16 $5$$8$$5$$8$$A_1, A_2, A_3, A_4, A_5$$5$$8$$5, A_1, A_2, A_3, A_4, A_5, 8$$$a_1=5 \text{ and } a_7=8.$$\begin{align}&a_7=a+6d\\ \implies &8=5+6d\\ \implies &6d=8-5\\ \implies &d=\dfrac{3}{6}=\dfrac{1}{2}. \end{align}\begin{align} A_1&=a+d=5+\dfra… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-3-p2?rev=1737476038&do=diff Question 2 Exercise 4.3 Solutions of Question 2 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2(i) $a_1, a_n, n, d$$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$$S_6$\begin{align}S_n&=\dfrac{n}{2}[2 a_1+(n-1) d] \\ \therefore S_6&=\dfrac{6}{2}(2.16+5… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-4-p1?rev=1737476038&do=diff Question 1 Exercise 4.4 Solutions of Question 1 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question(i) $a_1=5, \quad r=3$$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$$n+2$\begin{align}a_{n+2}&=a_1 r^{n i 1}=a r^{n+1}=b \\ \because a_1&=a \\ \Rightarrow \quad r^{n+1}&=\dfrac{b}{a} .… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit04/ex4-5-p1?rev=1737476038&do=diff Question 1 Exercise 4.5 Solutions of Question 1 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) $3+6+12+\ldots+3.2^9$$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/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$$1.2+2.2^2+3.2^3+4.2^4+\ldots$\begin{align} & S_n=1.2+2.2^2+3 \cdot 2^3+4 \cdot 2^4+\ldots +n \cdot 2^n....(i) \\ & 2 S_n=1.2^2+2.2^3+3.2^4+4.2^5+\ldots +n \cdot 2^n....(ii)\end{align}\begin{align} (1-2) S_n&=1 \cdot 2+(2-1) 2^2+(3-2) 2^2+(4-… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-2-p2?rev=1737476038&do=diff Question 2 & 3 Exercise 5.2 Solutions of Question 2 & 3 of Exercise 5.2 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2 $1+3^2 x+5^2 x^2+7^2 x^3+\ldots, x<1$\begin{align} & S_{\infty}=1+3^2 x+5^2 x^2+7^2 x^3+\ldots ..(1)\\ & x S_{\infty}=x+3^2 x^2+5^2 x^3+7^2 x^4+\ldots..(2)\end{align}\begin{align}& (1-x) S_{\infty}=1^2+(3^2-1^2) x+(5^2-3^2) x^2+(7^2-5^2) x^… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-3-p1?rev=1737476038&do=diff Question 1 Exercise 5.3 Solutions of Question 1 of Exercise 5.3 of Unit 05: Mascellaneous series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1 $n$$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$$3+9+21+45+93+189+\ldots$\begin{align} & a_2-a_1=9-3=6 \\ & a_3-a_2=21-9=12 \\ & a_4-a_3=45-21=24\\ & \text {... ... ... } \\ & \text {... ... ... } \\ &a_n-a_{n-1}=(\mathrm{n}-1)\quad \text{ term of the sequence}\quad 6,12,24, \ldots\end{ali… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit05/ex5-4-p1?rev=1737476038&do=diff Question 1 Exercise 5.3 Solutions of Question 1 of Exercise 5.4 of Unit 05: Mascellaneous series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) $\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+\ldots$$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/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$\begin{align} & a_2-a_1=7-3=4 \\ & a_3-a_2=13-7=6 \\ & a_4-a_3=21-13=8 \\ & \ldots \quad \ldots \quad \ldots \\ & \ldots \quad \cdots \quad \ldots \\ & a_n-a_{n-1}=(n-1) \text { term of the series } \\ & 4,6,8, \ldo… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-1-p1?rev=1737476038&do=diff Question 1 and 2 Exercise 6.1 Solutions of Question 1 and 2 of Exercise 6.1 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\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}$\df… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-1-p4?rev=1737476038&do=diff Question 4 Exercise 6.1 Solutions of Question 4 of Exercise 6.1 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p1?rev=1737476038&do=diff Question 1 and 2 Exercise 6.2 Solutions of Question 1 and 2 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$^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-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$$$100 + 25=125… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-3-p1?rev=1737476038&do=diff Question 1 Exercise 6.3 Solutions of Question 1 of Exercise 6.3 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$^n C_2=36$$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$$10.$ text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-4-p1?rev=1737476038&do=diff Question 1 Exercise 6.4 Solutions of Question 1 of Exercise 6.4 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$S=\{1,2,3,4,5,6\}$$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.$$$=\dfrac{12}{52}=\dfrac{3}{13}$… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-5-p1?rev=1737476038&do=diff Question 1 and 2 Exercise 6.5 Solutions of Question 1 and 2 of Exercise 6.5 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$A$$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}$$=1-\dfrac{17}{52}=\dfr… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-5-p6?rev=1737476038&do=diff Question 9 Exercise 6.5 Solutions of Question 9 of Exercise 6.5 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$2$$\dfrac{1}{7}$$\dfrac{1}{5}$\begin{align} P(\text { Ajmal scicction })&=\dfrac{1}{7} \\ \Rightarrow P(\text { Ajmal not selected })&=\dfrac{6}{7} \\ P(\text { Bushra selection })&=\dfrac{1}{5} \\ \Rightarrow P(\text { Bushra not selected }… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p1?rev=1737476038&do=diff Question 1 Review Exercise 6 Solutions of Question 1 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$O, P, Q, R, S, T, U$$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-p4?rev=1737476038&do=diff Question 5 & 6 Review Exercise 6 Solutions of Question 5 & 6 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$n=6$$$$(n-1) !=(6-1) !=5 !=120$$$120-24=96$$n=6$$(n-1) !=(6-1) !=5 !=120$$$(n-1) !=(5-1) !=4 !=24$$$$(n-1) !=(6-1) !=5 !=120$$$$4 ! \cdot 2 !=48$$$(5-1) !$ text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p5?rev=1737476038&do=diff Question 7 & 8 Review Exercise 6 Solutions of Question 7 & 8 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$P(A)=0.8, P(B)=0.5$$P(B / A)=0.4$$P(A \cap B)$\begin{align} P(B \mid A)&=\dfrac{P(A \cap B)}{P(A)} \\ \Rightarrow P(A \cap B)&=P(B \mid A) \cdot P(A)\\ &=0.4 \times 0.8=0.32\end{align}$P(A)=0.8, P(B)=0.5$$P(B / A)=0.4$$P(A … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p6?rev=1737476038&do=diff Question 9 & 10 Review Exercise 6 Solutions of Question 9 & 10 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$2,3,0,3,4,2,3$$1$$=100,0000$$$=\dfrac{7 !}{3 ! \cdot 2 !}=420 $$$1$$0$$7$$0$$$=\dfrac{6 !}{2 ! 3 !}=60 $$$1$$420-50=360$$n$$n$$(n-1)$$(n - 1)$$(n-1)$$(n-2) !$$2$$2 !$$n$$$(n-2) ! \cdot 2 !=2(n-2) ! $$ text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p1?rev=1737476038&do=diff Question 1 Exercise 7.1 Solutions of Question 1 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$2+4+6+\cdots+2 n=n(n+1)$$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$$(k+1)^{t h}$\begin{align} 3+7+11+\cdots+(4 k-1)+[4(k+1)-1] & =k(2 k+1)+4(k+1)-1 \… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p6?rev=1737476038&do=diff Question 6 Exercise 7.1 Solutions of Question 6 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$1(1 !)+2(2 !)+3(3 !)+\ldots+n(n !)= -(n+1) !-1$$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}$\begin{align}1+2+2^2+2^3+\ldots+2^{k-1}-2^k & =2^k-12^k \\ & =2^k+2^k-… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-1-p12?rev=1737476038&do=diff Question 12 Exercise 7.1 Solutions of Question 12 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\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$$n=k+1$\begin{align} 3^{2(k+1)-1}+2^{2(k+1)-1} & =3^{2 k+2-1}+2^{2… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p2?rev=1737476038&do=diff Question 2 Exercise 7.2 Solutions of Question 2 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$4^{th}$$(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$\begin{align} T_{r+1}&=\dfrac{23 !}{(23-r) ! r !}(2 \sqrt{x})^{23-r}(-\dfrac{3}{x \sqrt{x}})^r \\ & =\dfrac{23 !}{(23-r) ! r !} \cdot 2^{23-r} \cdot(-3)… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p11?rev=1737476038&do=diff Question 11 Exercise 7.2 Solutions of Question 11 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$(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-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$$$ p x^p-q x^q=p(1+h)^p-q(1+h)^q $$$$ \begin{aligned} & p x^p-q x^q \\ & =p(1+p h+\text { higher powers h) } \\ & -q(1+q h+\text { higher powcrs } h) \\ & \Rightarrow p x^p-q x^q=… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit07/re-ex7-p6?rev=1737476039&do=diff Question 9 and 10 Review Exercise 7 Solutions of Question 9 and 10 of Review Exercise 7 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit10/re-ex10-p3?rev=1737476039&do=diff Question 4 & 5, Review Exercise 10 Solutions of Question 4 & 5 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${{\sin }^{2}}\dfrac{\theta }{2}=\dfrac{\sin \theta \tan \dfrac{\theta }{2}}{2}$\begin{align}R.H.S.&=\dfrac{\sin \theta \tan \dfrac{\theta }{2}}{2}\\ &=\dfrac{\sin \theta \sin \dfrac{\theta }{2}}{2\cos \d… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-1-p2?rev=1737476039&do=diff Question 2, Exercise 1.1 Solutions of Question 2 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 2(i) $x+iy$$(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},$$\bar… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p5?rev=1737476039&do=diff Question 5, Exercise 1.2 Solutions of Question 5 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 5 $z_1$$z_2$$|z_1+z_2|^2-|z_1-z_2|^2=4Re(z_1)Re(z_2)$\begin{align}z_1&=x_1+iy_1 \text{ and } z_2&=x_2+iy_2\end{align}\begin{align}z_1+z_2&=x_1+iy_1+x_2+iy_2\\ &=x_1+x_2+i(y_1+y_2)\\ |z_1+z_2|^2&=(x_1+x_2)^2+(y_1+y_2)^2\\ &=x^2_1+x^2_2+2x_1x_… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p5?rev=1737476039&do=diff Question 5, Exercise 1.4 Solutions of Question 5 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 5 $\cos \alpha+\cos \beta+\cos \gamma=\sin \alpha+\sin \beta+\sin \gamma=0$$\cos 3 \alpha+\cos 3 \beta+\cos 3 \gamma=3 \cos (\alpha+\beta+\gamma)$$\sin 3 \alpha+\sin 3 \beta+\sin 3 \gamma=3 \sin (\alpha+\beta+\gamma)$\begin{align} \cos \alpha … text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/re-ex-p3?rev=1737476039&do=diff Question 3, Review Exercise Solutions of Question 3 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $3 x^{2}+108$\begin{align*} & 3 x^{2}+108\\ =&3 (x^{2}+36)\\ =&3 (x^{2}-(6i)^2)\\ =&3 (x+6i)(x-6i) \end{align*}$4 x^{2}+40$\begin{align*} &4 x^{2}+40\\ =&4 (x^{2}+10)\\ =&4 (x^{2}+(\sqrt{10}i)^2)\\ =&4 (x+\sqrt{10}i)(x-\sqrt{10}i) \end{align*} text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit01/re-ex-p5?rev=1737476039&do=diff Question 5, Review Exercise Solutions of Question 5 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $z$$(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/unit02/ex2-1-p3?rev=1737476039&do=diff Question 3, Exercise 2.1 Solutions of Question 3 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$A=\begin{bmatrix} 3 & 0 & 0 \\ 0 & 1 & 0 \\ 2 & 6 & 0 \end{bmatrix}$$$$B=\begin{bmatrix} -6 & 0 & 0 \\ 0 & -6 & 0 \\ 0 & 0 & -6 \end{bmatrix}$$$$C=\begin{bmatrix} 1 & 0 \\ 2 & 0 \end{bmatrix}$$$$D=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 &… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p8?rev=1737476039&do=diff Question 8, Exercise 2.2 Solutions of Question 8 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A$$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/ex2-6-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/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/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.$$\begin{align*} a_1 &= (1)^2 - 2(1) = 1 - 2 = -1\\ a_2 &= (2)^2 - 2(2) = 4 - 4 = 0\\ a_3 &= (3)^2 - 2(3) = 9 - 6 = 3\\ a_4 &= (4)^2 - 2(4) = 16 - 8 = 8\\ \end{align*}\begin{align*} a_{… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p5?rev=1737476039&do=diff Question 9 and 10, Exercise 4.1 Solutions of Question 9 and 10 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$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-p9?rev=1737476039&do=diff Question 17 and 18, Exercise 4.1 Solutions of Question 17 and 18 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{n}=\log 10^{n} ; a_{43}$$$a_n = \log 10^n.$$\begin{align*} a_{43} &= \log 10^{43} \\ &= 43 \cdot \log 10 \\ &= 43 \cdot 1 \\ &= 43 \end{align*}$a_{43}= 43$$a_{n}=\ln e^{n} ; a_{67}$$$a_n = \ln e^n.$$\begin{align*} a_{67} &= \ln e^… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p10?rev=1737476039&do=diff Question 19 and 20, Exercise 4.1 Solutions of Question 19 and 20 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{n}$$1,3,5,7,9, \ldots$$$1, 3, 5, 7, 9, \ldots$$$a_1=1$$d=3-1=2$$$a_n = a_1 + (n - 1) d$$\begin{align*} \implies a_n &= 1 + (n - 1) \cdot 2\\ &= 1 + 2n - 2\\ &= 2n - 1 \end{align*}$a_n = 2n - 1$$a_{n}$$3,9,27,81,243, \ldots$\begin… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p1?rev=1737476039&do=diff Question 1, Exercise 4.2 Solutions of Question 1 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=4, d=3$$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-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$$y$… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p1?rev=1737476039&do=diff Question 1 and 2, Exercise 4.3 Solutions of Question 1 and 2 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $4+7+10+13+16+19+22+25$$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-p8?rev=1737476039&do=diff Question 15 and 16, Exercise 4.3 Solutions of Question 15 and 16 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $S_n$$a_{1}=91$$d=-4$$a_{n}=15$$a_{1}=91$$d=-4$$a_{n}=15$$n=?$\begin{align} & a_n=a_1+(n-1)d \\ \implies & 15=91+(n-1)(-4) \\ \implies & 15=91-4n+4 \\ \implies & 4n=95-15 \\ \implies & 4n = 80\\ \implies & n = 20. \end{align}\begin{… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p9?rev=1737476039&do=diff Question 17, 18 and 19, Exercise 4.3 Solutions of Question 17, 18 and 19 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $6+12+18+\ldots+96$$$6+12+18+\ldots+96.$$$a_{1}=6$$d=12-6=6$$a_{n}=96$$n=?$\begin{align} & a_n=a_1+(n-1)d \\ \implies & 96=6+(n-1)(6) \\ \implies & 96=6+6n-6 \\ \implies & 6n=96 \\ \implies & n = 24. \end{align}\begin{align}… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p10?rev=1737476039&do=diff Question 20, 21 and 22, Exercise 4.3 Solutions of Question 20, 21 and 22 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=7$$a_{n}=139$$S_{n}=876$$a_{1}=7$$a_{n}=139$$S_{n}=876$$n$$d$\begin{align} &S_n=\frac{n}{2}[a_1+a_n]\\ \implies & 876=\frac{n}{2}[7+139]\\ \implies & 1752=146n\\ \implies & n=\frac{1752}{146}=12. \end{align}\begin{align… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p11?rev=1737476039&do=diff Question 23 and 24, Exercise 4.3 Solutions of Question 23 and 24 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$ 14+16+18+...+a_{25}.$$$a_1=14$$d=16-14=2$$n=25$$a_25$$S_25$\begin{align} a_n&=a_1+(n-1)d\\ \implies a_{25}&= 14+(25-1)(2)\\ &=62. \end{align}\begin{align} S_n&=\frac{n}{2}[a_1+a_n]\\ \implies S_{25}& =\frac{25}{2}[14+62]\\ & =25 \t… text/html 2025-01-21T16:13:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p1?rev=1737476039&do=diff Question 1 and 2, Exercise 4.4 Solutions of Question 1 and 2 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $5,20,100,500, \ldots$$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-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-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-p12?rev=1737476039&do=diff Question 24 and 25, Exercise 4.4 Solutions of Question 24 and 25 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$5 , \_\_\_, \_\_\_, \_\_\_, 80$$$a_1=5$$a_5=80$$r$$n$$$a_n = a_1 r^{n-1}.$$\begin{align*} a_5 &= a_1 r^4 \\ \implies 80 &= 5 \cdot r^4 \\ \implies r^4 &= \frac{80}{5} \\ \implies r^4 &= 16 \\ \implies r &= 2. \end{align*}\begin{alig… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p6?rev=1737476040&do=diff Question 11, 12 and 13, Exercise 4.5 Solutions of Question 11, 12 and 13 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}$$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 ...$$… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit04/ex4-6-p1?rev=1737476040&do=diff Question 1 and 2, Exercise 4.6 Solutions of Question 1 and 2 of Exercise 4.6 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\frac{1}{9}, \frac{1}{12}, \frac{1}{15}, \cdots \quad 7$$$\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-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/ex4-7-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-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$$ text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-1-p1?rev=1737476040&do=diff Question 1, Exercise 5.1 Solutions of Question 1 of Exercise 5.1 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 1(i) $2 x^{3}+3 x^{2}-4 x+1$$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/ex5-3-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$$2 x+8$\begin{align*} & 6 x^{2}+38 x+56 \\ = & 2(3x^2+19x+28) \\ = & 2(3x^2+12x+7x+28) \\ = & 2(3x(x+4)+7(x+4)) \\ =& 2(x+4)(3x+7) \\ =& (2x+8)(3x+7) \end{align*}\begin{align*} & Length … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/ex5-4-p2?rev=1737476040&do=diff Question 2, Review Exercise Solutions of Question 2 of Review Exercise of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Go to text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit05/re-ex-p1?rev=1737476040&do=diff Question 1, Review Exercise Solutions of Question 1 of Review Exercise of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 1 $-2-x+x^{2}$$(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-p7?rev=1737476040&do=diff Question 7, Review Exercise Solutions of Question 7 of Review Exercise of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 7 $3 x^{2}-x+32-\frac{121}{x+4}$ text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit08/ex8-2-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-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/ex8-3-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/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/unit09/ex9-1-p2?rev=1737476040&do=diff Question 2, Exercise 9.1 Solutions of Question 2 of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y=\dfrac{1}{4+3 \operatorname{Sin} \theta}$\begin{align*} -1 \leq \operatorname{Sin} \theta \leq 1 \end{align*}$3$\begin{align*} -3 \leq 3 \operatorname{Sin} \theta \leq 3 \end{align*}$4$\begin{align*} & 1 \leq 4+3 \operatorname{Sin} \theta \l… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p4?rev=1737476040&do=diff Question 4(i-iv), Exercise 9.1 Solutions of Question 4(i-iv) of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y=\sin x+x \cdot \cos x$$f(x)=\sin x+x \cdot \cos x$\begin{align*} f(-x) = \sin (-x) + (-x)\cdot \cos (-x) \end{align*}$\sin(-x)=-\sin x$$\cos (-x) = \cos x$\begin{align*} f(x) & = -\sin x - x \cdot \cos x \\ & = -(\sin x + x \cdot … text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p5?rev=1737476040&do=diff Question 4(v-viii), Exercise 9.1 Solutions of Question 4(v-viii) of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y=\dfrac{\sin ^{2} x}{x+\tan x}$\[y = \frac{\sin^2 x}{x + \tan x}\]\begin{align*} y(-x) &= \frac{\big(-\sin x\big)^2}{-x - \tan x} \\ &= \frac{\sin^2 x}{-x - \tan x}\\ & = \frac{\sin^2 x}{-(x + \tan x)}\\ & = -\frac{\sin^2 x}{x +… text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p6?rev=1737476040&do=diff Question 5(i-v), Exercise 9.1 Solutions of Question 5(i-v) of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y=2 \operatorname{Sin} x$$y=2 \operatorname{Cos} 3 x$$y=2 \operatorname{Tan} 2 x$$\mathrm{y}=\operatorname{Cos} \frac{\mathrm{x}}{2}$ text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p7?rev=1737476040&do=diff Question 5(vi-x), Exercise 9.1 Solutions of Question 5(vi-x) of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y=2 \operatorname{Sin} 3 x$$y=3 \operatorname{Cos} x$$y=\operatorname{Cos}^{2} x$$y=\operatorname{Sin}^{2} x$$y=\operatorname{Tan}^{2} x$$y=\operatorname{Sin} \frac{x}{2}$ text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p9?rev=1737476040&do=diff Question 7 & 8, Exercise 9.1 Solutions of Question 7 & 8 of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y=\operatorname{Sin} x$$y=\operatorname{Sin} 2 x$$[0,2 \pi]$$y=\operatorname{Cos} x$$y=\operatorname{Cos} 2 x$$[0,2 \pi]$ text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p10?rev=1737476040&do=diff Question 9, Exercise 9.1 Solutions of Question 9 of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin x=\cos x$$\cos x=x$$\sin x=x$$\tan x=x$ text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/re-ex-1-p2?rev=1737476040&do=diff Question 2 and 3,Review Exercise Solutions of Question 2 and 3 of Review Exercise of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/re-ex-1-p3?rev=1737476040&do=diff Question 4, Review Exercise Solutions of Question 4 of Review Exercise of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. text/html 2025-01-21T16:14:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-nbf/sol/unit09/re-ex-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-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.