Merging man & maths https://www.mathcity.org/ Thu, 04 Jun 2026 15:41:13 +0000 FeedCreator 1.8 https://www.mathcity.org/_media/logo.svg https://www.mathcity.org/ https://www.mathcity.org/math-11-nbf/sol/unit04 Unit 04: Sequences and Seeries This is a forth unit of the book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. On this page we have provided the solutions of the questions.$n$$n$$n$$n$$n$$n$$n$ anonymous@undisclosed.example.com (Anonymous) Sun, 06 Oct 2024 17:46:06 +0000 https://www.mathcity.org/math-11-nbf/sol/unit05 Unit 05: Polynomials [Unit 05: Polynomials] This is a fifth unit of the book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. On this page we have provided the solutions of the questions. anonymous@undisclosed.example.com (Anonymous) Sun, 13 Oct 2024 18:05:52 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08 Unit 08: Fundamental of Trigonometry [Unit 08: Fundamental of Trigonometry] This is a eight unit of the book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. On this page we have provided the solutions of the questions.$\cos(\alpha -\beta)=\cos \alpha \cos\beta+\sin\alpha \sin\beta$$\cos(\alpha +\beta)=\cos \alpha \cos\beta-\sin\alpha \sin\beta$$\sin(\alpha \pm \beta)=\sin \alpha \cos\beta \pm \sin\alpha \co… anonymous@undisclosed.example.com (Anonymous) Sat, 09 Nov 2024 18:51:30 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/rev-ex Review Exercise 2 (Solutions) The solutions of the Review Exercise 2 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the MCQs and question all topics included in this chapter.$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}=(-… anonymous@undisclosed.example.com (Anonymous) Sun, 08 Feb 2026 16:57:09 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01 Unit 01: Complex Numbers (Solutions) This is a first unit of the book Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. On this page we have provided the solutions of the questions.$z$$z^2+a^2$$z^3-3z^2+z=5$$pz^2+qz+r=0$$p,q,r$$z$ anonymous@undisclosed.example.com (Anonymous) Mon, 27 Oct 2025 18:47:40 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02 Unit 02: Matrices and Determinants (Solutions) This is a second unit of the book Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. On this page we have provided the solutions of the questions. anonymous@undisclosed.example.com (Anonymous) Sun, 08 Feb 2026 17:04:31 +0000 https://www.mathcity.org/math-11-nbf/sol/unit09 Unit 09: Trigonometric Functions This is a ninth unit of the book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. On this page we have provided the solutions of the questions.$a+b \sin \theta$$a+b \cos \theta$$a+b \sin(c \theta+d)$$a+b \cos(c \theta+d)$$a, b, c$$d$$\sin \theta$$\cos \theta$$\tan \theta$ anonymous@undisclosed.example.com (Anonymous) Wed, 27 Nov 2024 15:59:25 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p1 Question 1, Exercise 2.6 Solutions of Question 1 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $ 2 x_{1}-3 x_{2}+4 x_{3}=0$$x_{1}-2 x_{2}+3 x_{3}=0$$4 x_{1}+x_{2}-6 x_{3}=0$\begin{align*} &2 x_{1}-3 x_{2}+4 x_{3}=0\cdots (i)\\ &x_{1}-2 x_{2}+3 x_{3}=0\cdots (ii)\\ &4 x_{1}+x_{2}-6 x_{3}=0\cdots (iii)\\ \end{align*}\begin{align*} A &= \le… anonymous@undisclosed.example.com (Anonymous) Wed, 04 Sep 2024 03:03:49 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p7 Question 7 and 8, Exercise 2.6 Solutions of Question 7 and 8 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{ccc}3 & 2 & 1 \\ 4 & -1 & 2 \\ 7 & 3 & -3\end{array}\right]$$A^{-1}$$3 x+4 y+7 z=14 ; 2 x-y+3 z=4 ; \quad x+2 y-3 z=0$\begin{align*} A &= \begin{bmatrix} 3 & 2 & 1 \\ 4 & -1 & 2 \\ 7 & 3 & -3 \end{bmatrix}\\ |… anonymous@undisclosed.example.com (Anonymous) Wed, 04 Sep 2024 03:13:32 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-1 Exercise 1.1 (Solutions) The solutions of the Exercise 1.1 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to sum, product and division of the complex numbers.${{i}^{31}}$${{\left( -i \right)}^{6}}$${{\left( -1 \right)}^{\frac{-13}{2}}}$$\dfrac{2}{(-1)^{\frac{3}{2}}}$$i^{23}+i^{58}+i^{21}$$x+iy$$(3+2i)+(2+4i)$$(4+3i)-(2+5i)$$(4+7i… anonymous@undisclosed.example.com (Anonymous) Mon, 27 Oct 2025 18:50:29 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-2 Exercise 1.2 (Solutions) The solutions of the Exercise 1.2 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to real and imaginary part of complex numbers, modulus and conjugate of the complex numbers.$\operatorname{Re}(i z)=-\operatorname{Im}(z)$$\operatorname{Im}(i z)=\operatorname{Re}(z)$$$\left(z_{1} z_{2}\right)\left(z_{3} z_{4… anonymous@undisclosed.example.com (Anonymous) Mon, 27 Oct 2025 18:50:09 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-3 Exercise 1.3 (Solutions) The solutions of the Exercise 1.3 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to sum, product and division of the complex numbers.$z^{2}+169$$2 z^{2}+18$$3 z^{2}+363$$z^{2}+\dfrac{3}{25}$$2 z^{3}+3 z^{2}-10 z-15$$z^{3}-7 z+6$$z^{3}+2 z^{2}-23 z-60$$2 z^{3}+9 z^{2}-11 z-30$$z^{2}-7 z-8$$4 z^{2}-7 z-11$$… anonymous@undisclosed.example.com (Anonymous) Mon, 27 Oct 2025 18:51:15 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-4 Exercise 1.4 (Solutions) The solutions of the Exercise 1.4 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to polar form of the complex numbers.$2+i 2 \sqrt{3}$$3-i \sqrt{3}$$-2-i 2$$\dfrac{i-1}{\cos \dfrac{\pi}{3}+i \sin \dfrac{\pi}{3}}$$\left(\cos \dfrac{\pi}{6}+i \sin \dfrac{\pi}{6}\right)\left(\cos \dfrac{\pi}{3}+i \sin \dfrac… anonymous@undisclosed.example.com (Anonymous) Mon, 27 Oct 2025 18:51:55 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/re-ex-p7 Question 7, Review Exercise Solutions of Question 7 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 z^{2}-11 z+16=0$\begin{align*} &2 z^{2}-11 z+16=0\\ \implies&z^2 - \dfrac{11}{2}z + 8 = 0\\ \implies& z^2 - \dfrac{11}{2}z = -8\\ \implies& z^2 - 2z\dfrac{11}{4}z + \dfrac{121}{16} = -8 + \dfrac{121}{16}\\ \implies&\left(z-\dfrac{11}{4}\right)^2… anonymous@undisclosed.example.com (Anonymous) Wed, 17 Jul 2024 16:59:42 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/rev-ex Review Exercise 1 (Solutions) The solutions of the Review Exercise 1 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to polar form of the complex numbers. $i^{2}+i^{4}+i^{6}+\cdots+i^{100}$$\left|\dfrac{(3-2 i)(1+i)}{2-3 i}\right|$$|\overline{(3-2 i)(4-i)}|$$\left(\dfrac{3+5 i}{2-3 i}\right)^{-1}$$3 x^{2}+108$$4 x^{2}+40$$z=x+i y$… anonymous@undisclosed.example.com (Anonymous) Mon, 27 Oct 2025 18:52:39 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-1 Exercise 2.1 (Solutions) The solutions of the Exercise 2.1 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to order, different type of matrices and transpose of matrix.$A=\left[\begin{array}{lll}1 & 3 & 0 \\ 2 & 0 & 1\end{array}\right]$$B=\left[\begin{array}{ll}1 & 2 \\ 2 & 3 \\ 3 & 4\end{array}\right]$$C=\left[\begin{array}{l}1 \… anonymous@undisclosed.example.com (Anonymous) Sun, 08 Feb 2026 16:35:15 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-2 Exercise 2.2 (Solutions) The solutions of the Exercise 2.2 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to sum, product and different operations on matrices.$A=\left[a_{i j}\right]$$2 \times 2$$a_{i j}=\dfrac{i+3 j}{2}$$a_{i j}=\dfrac{i \times l}{2}$$a_{i j}=\dfrac{i}{j}$$a_{i j}=\dfrac{2 i-3 j}{3}$$B=\left[a_{\ell}\right]$$3 \… anonymous@undisclosed.example.com (Anonymous) Sun, 08 Feb 2026 16:40:19 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-3 Exercise 2.3 (Solutions) The solutions of the Exercise 2.3 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to determinant and inverse of the matrix.$\left[\begin{array}{ccc}2 & 3 & 1 \\ 1 & -1 & 2 \\ 4 & 1 & 2\end{array}\right]$$\left[\begin{array}{ccc}\cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1\en… anonymous@undisclosed.example.com (Anonymous) Sun, 08 Feb 2026 16:44:16 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-4 Exercise 2.4 (Solutions) The solutions of the Exercise 2.4 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to the determinant and properties of the determinant.$\left|\begin{array}{lll}9 & 27 & 36 \\ 18 & 54 & 24 \\ 27 & 81 & 28\end{array}\right|=0$$\left|\begin{array}{lll}1 / a & b c & b+c \\ 1 / b & a c & a+c \\ 1 / c & a b & a+… anonymous@undisclosed.example.com (Anonymous) Sun, 08 Feb 2026 16:46:40 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-5 Exercise 2.5 (Solutions) The solutions of the Exercise 2.5 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to solving system of the equation of three variables by using matrices.$\left[\begin{array}{ccc}1 & 3 & 5 \\ -6 & 8 & 3 \\ -4 & 6 & 5\end{array}\right]$$\left[\begin{array}{ll}2 & 1 \\ 3 & 2 \\ 1 & 9\end{array}\right]$$\left[… anonymous@undisclosed.example.com (Anonymous) Sun, 08 Feb 2026 16:50:14 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/re-ex-p1 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$$m \times p$$A$$1 \times n$$A^{t} A$$1 \times n$$n \times 1$$1 \times 1$$n \times n$$n \times n$$a_{i j}$$A$$a_{i j}=(-1)^{i+j} A_{i j}$$a_{i j}=(-1)^{i+j}… anonymous@undisclosed.example.com (Anonymous) Mon, 25 Nov 2024 17:51:01 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/re-ex-p3 Question 4 and 5, Review Exercise Solutions of Question 4 and 5 of Review Exercise of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left|\begin{array}{ccc}a+1 & l & l \\ l & a+1 & l \\ l & l & a+1\end{array}\right|=(a+1+2 l)(a+1-l)^{2}$\begin{align*} L.H.S &= \left|\begin{array}{ccc}a+1 & l & l \\ l & a+1 & l \\ l & l & a+1\end{array}\right|\\ &=\left|\b… anonymous@undisclosed.example.com (Anonymous) Wed, 04 Sep 2024 03:15:45 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-1 Exercise 6.1 (Solutions) The solutions of the Exercise 6.1 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to factorial function.$n!$$$ n!=\left\{\begin{array}{l} n(n-1)(n-2)\cdot \ldots \cdot 3 \cdot 2 \cdot 1 \text{ if } n\geq 1,\\ 1 \text{ if } n=0. \end{array} \right. $$$10!$$\dfrac{12!}{7! 3! 2!}$$\dfrac{4!-2!}{3!+5!}$$\dfrac… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:40 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-2 Exercise 6.2 (Solutions) The solutions of the Exercise 6.1 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to factorial function.$n!$$$ n!=\left\{\begin{array}{l} n(n-1)(n-2)\cdot \ldots \cdot 3 \cdot 2 \cdot 1 \text{ if } n\geq 1,\\ 1 \text{ if } n=0. \end{array} \right. $$$n \in \mathbb{N}$${ }^n P_r=\frac{n!}{(n-r)!}$$\quad{ }^… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:59 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-3 Exercise 6.3 (Solutions) The solutions of the Exercise 6.3 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to factorial function.$n \in \mathbb{N}$${ }^{n} C_{r}=\frac{n!}{r!(n-r)!}$$n,{ }^{n-1} C_{r-1}=(n-r+1){ }^{n} C_{r-1}$$r^{n} C_{r}=(n-r+1)^{n} C_{r-1}$${ }^{n-1} C_{r-1}+{ }^{n-1} C_{r}={ }^{n} C_{r}$${ }^{n} C_{r}+{ }^{n} C… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 09:00:35 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/rev-ex Review Exercise (Solutions) The solutions of the Review Exercise of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. Question 1.$4$$3$$0$$6$$0,2,3,4,5,7$$7$ anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 09:00:41 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06 Unit 06: Permutation and Combination This is a sixth unit of the book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. On this page we have provided the solutions of the questions.$n$$n!$$n$$r$$n$$r$ anonymous@undisclosed.example.com (Anonymous) Sun, 09 Mar 2025 10:56:17 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-1-p1 Question 1, Exercise 1.1 Solutions of Question 1 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 1(i) ${{i}^{31}}$\begin{align}{{i}^{31}}&=i\cdot{{i}^{30}}\\ &=i\cdot{{\left( {{i}^{2}} \right)}^{15}}\\ &=i\cdot{{\left( -1 \right)}^{15}} \quad \because i^2=-1\\ &=i\cdot(-1)\\ &=-i.\end{align}${{\left( -i \right)}^{6}}$\begin{align} {{\left… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Jul 2024 09:40:00 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-1-p2 Question 2, Exercise 1.1 Solutions of Question 2 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 2(i) $x+iy$$(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… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Jul 2024 12:10:43 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-1-p3 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}… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Jul 2024 12:24:16 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-1-p4 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$\begin{align}&(2+3i)x+(1+3i)y+2=0\\ \implies &(2x+y+2)+(3x+3y)i=0.\end{align}\begin{align} 2x+y+2&=0 \quad \cdots(1)\\ 3x+3y&=0\quad \cdots (2) \end{align}\begin{align} &3x=-3y \\ x=-y \quad ... (3) \end{align}$… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Jul 2024 12:39:29 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-1-p5 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$$\bar{z}=x-iy$\begin{align}&4z-3\bar{z}=\dfrac{1-18i}{2-i}\\ \implies &4(x+iy)-3(x-iy)=\dfrac{1-18i}{2-i}\times \dfrac{2+i}{2+i}\\ \implies &4x+4iy-3x+3iy=\dfrac{(1-18i)(2+i)}{2^2-i^2} \end{align}\b… anonymous@undisclosed.example.com (Anonymous) Tue, 02 Jul 2024 16:57:50 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-1-p6 Question 6, Exercise 1.1 Solutions of Question 6 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 6(i) $4-3 i$$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},$$\bar… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Jul 2024 12:40:49 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-1-p7 Question 7, Exercise 1.1 Solutions of Question 7 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 7(i) $11+12 i$$$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)$$z=(2+3i)−(2+6i)$\begin{align}z&=2+3i−2−6i\\ &=-3i \end{align}\begin{align} |z| &= \sqrt{0^2+(-3)^2} \\ &= \sqrt{… anonymous@undisclosed.example.com (Anonymous) Tue, 02 Jul 2024 16:57:27 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p1 Question 1, Exercise 1.2 Solutions of Question 1 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 1(i) $\operatorname{Re}(i z)=-\operatorname{Im}(z)$$$z=x+iy$$\begin{align} iz&=i(x+iy)\\ &=ix-y\end{align}\begin{align}Re(iz)&=-y\\ \implies Re(iz)&=-Im(z)\end{align}$\operatorname{Im}(i z)=\operatorname{Re}(z)$$$z=x+iy$$\begin{align}iz&=i(x+i… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Jul 2024 12:44:38 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p2 Question 2, Exercise 1.2 Solutions of Question 2 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 2 $$ \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} $$\begin{align} &(z_1 z_2)(z_3 z_4) \\ =&(z_1 z_2)z_5 \quad \text {Let }z_5=z_3 z_4 \\ =&z_1 (z_2 z_5) \… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Jul 2024 12:45:03 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p3 Question 3, Exercise 1.2 Solutions of Question 3 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 3(i) $z \in \mathbb{C}$$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$$z$\begin{align}& z=\bar{z}\\ \Righ… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Jul 2024 12:53:00 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p4 Question 4, Exercise 1.2 Solutions of Question 4 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 4 $z_{1}=2-3 i$$\left|z_{1} z_{2}\right|=16$$\left|z_{2}\right|$$$z_{1}=2-3i$$\begin{align}|z_1|&=\sqrt{(2)^2+(-3)^2}\\ &=\sqrt{13}\end{align}\begin{align}&|z_{1} z_{2}|=16\\ \Rightarrow \quad &|z_{1}|| z_{2}|=16\\ \Rightarrow \quad & \sqrt{13… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Jul 2024 12:54:23 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p5 Question 5, Exercise 1.2 Solutions of Question 5 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 5 $z_1$$z_2$$|z_1+z_2|^2-|z_1-z_2|^2=4Re(z_1)Re(z_2)$\begin{align}z_1&=x_1+iy_1 \text{ and } z_2&=x_2+iy_2\end{align}\begin{align}z_1+z_2&=x_1+iy_1+x_2+iy_2\\ &=x_1+x_2+i(y_1+y_2)\\ |z_1+z_2|^2&=(x_1+x_2)^2+(y_1+y_2)^2\\ &=x^2_1+x^2_2+2x_1x_… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Jul 2024 12:55:01 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p6 Question 6, Exercise 1.2 Solutions of Question 6 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 6 $\lambda$$\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)} \\ &=\… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Jul 2024 13:16:48 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p7 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$$\left(|x|-|y|)^{2} \geq 0\right)$\begin{align} &\left(|x|-|y|)^{2} \geq 0\right) \\ \implies & |x|^2+|y|^2-2|x||y| \geq 0 \\ \implies & |x|^2+|y|^2 \geq 2|x||y| \\ \implie… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Jul 2024 13:32:58 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p8 Question 8, Exercise 1.2 Solutions of Question 8 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 8(i) $|2 z-i|=4$$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… anonymous@undisclosed.example.com (Anonymous) Wed, 10 Jul 2024 19:37:43 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p9 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}$$z=2+4i$\begin{align} Re(2+4i)^{-1} & = Re(z^{-1}) = \dfrac{Re(z)}{|z|^2} \\ & =\dfrac{2}{2^2+4^2} = \dfrac{2}{20}\\ &= \dfrac{1}{10}. \end{align}\begin{align} Im(2+4i)^{-1} & = Im(z^{-1}) = -\dfrac{Im(z)}{|z|^2} \\ & =-\df… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Jul 2024 19:38:59 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p10 Question 10, Exercise 1.2 Solutions of Question 10 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 10(i) $z_{1}=-3+2 i$$$\left|z_{1}\right|=\left|-z_{1}\right|=\left|\overline{z_{!}}\right|=\left|-\overline{z_{!}}\right|.$$\begin{align} |z_1| &= \sqrt{(-3)^2 + (2)^2} \\ &= \sqrt{9 + 4} = \sqrt{13} \,\, -- (1) \end{align}\begin{align} -z_… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Jul 2024 17:34:55 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-3-p1 Question 1, Exercise 1.3 Solutions of Question 1 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 1(i) $z^{2}+169$\begin{align} & z^{2} + 169 \\ = & z^{2} - (13i)^2 \\ = &(z + 13i)(z - 13i). \end{align}$2 z^{2}+18$\begin{align} & 2z^2 + 18 \\ = &2(z^2 - (3i)^2)\\ = &2(z + 3i)(z - 3i) \end{align}$3 z^{2}+363$\begin{align} & 3z^2 + 363 \\ … anonymous@undisclosed.example.com (Anonymous) Sun, 14 Jul 2024 19:35:10 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-3-p2 Question 2, Exercise 1.3 Solutions of Question 2 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 2(i) $z^{2}-6 z+2=0$\begin{align} & z^2 - 6z + 2 = 0 \\ \implies & z^2 - 2(3)(z)+9-9+2=0 \\ \implies & (z - 3)^2+7= 0 \\ \implies & (z - 3)^2 = 7. \end{align}\begin{align} &z - 3 = \pm \sqrt{7} \\ \implies &z = 3 \pm \sqrt{7}\end{align}$\{3 … anonymous@undisclosed.example.com (Anonymous) Sun, 14 Jul 2024 19:41:53 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-3-p3 Question 3, Exercise 1.3 Solutions of Question 3 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 3(i) $\dfrac{1}{3} z^{2}+2 z-16=0$\begin{align}&\dfrac{1}{3}z^{2}+2 z-16=0\\ \implies &z^{2} + 6z - 48 = 0 \end{align}$$ z = \dfrac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a},$$$$a = 1,\quad b = 6,\quad \text{and}\quad c = -48.$$\begin{align} z& = \d… anonymous@undisclosed.example.com (Anonymous) Sun, 14 Jul 2024 19:45:23 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-3-p4 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… anonymous@undisclosed.example.com (Anonymous) Mon, 15 Jul 2024 12:19:22 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p1 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}$$z=x+iy=2 + i 2 \sqrt{3}$\begin{align} r & = \sqrt{x^2 + y^2} = \sqrt{2^2 + (2\sqrt{3})^2} \\ & = \sqrt{4 + 12} = \sqrt{16} = 4. \end{align}\begin{align} \alpha & = \tan^{-1}\left|\frac{y}{x}\right| = \tan^{-1}\left|\fra… anonymous@undisclosed.example.com (Anonymous) Mon, 15 Jul 2024 12:24:40 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p2 Question 2, Exercise 1.4 Solutions of Question 2 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 2(i) $\left(\cos \dfrac{\pi}{6}+i \sin \dfrac{\pi}{6}\right)\left(\cos \dfrac{\pi}{3}+i \sin \dfrac{\pi}{3}\right)$$z_1=\cos \dfrac{\pi}{6}+i \sin \dfrac{\pi}{6}=e^{i\frac{\pi}{6}}$$z_2=\cos \dfrac{\pi}{3}+i \sin \dfrac{\pi}{3}=e^{i\frac{\pi}{… anonymous@undisclosed.example.com (Anonymous) Thu, 05 Sep 2024 12:24:09 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p3 Question 3, Exercise 1.4 Solutions of Question 3 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 3(i) $\left(x_{1}+i y_{1}\right)\left(x_{2}+i y_{2}\right)\left(x_{3}+i y_{3}\right) \ldots\left(x_{n}+i y_{n}\right)=a+i b$$\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}… anonymous@undisclosed.example.com (Anonymous) Wed, 17 Jul 2024 16:39:59 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p4 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$$z=i \tan \theta$\begin{align}&\dfrac{1+z}{1-z}=\cos 2 \theta+i \sin 2 \theta\\ \implies &\dfrac{1+z}{1-z}=e^{i2\theta}\\ \implies &(1+z)=(1-z)e^{i2\theta}\\ \implies &z+z e^{i2\theta}=e^{i2\th… anonymous@undisclosed.example.com (Anonymous) Mon, 15 Jul 2024 12:40:46 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p5 Question 5, Exercise 1.4 Solutions of Question 5 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 5 $\cos \alpha+\cos \beta+\cos \gamma=\sin \alpha+\sin \beta+\sin \gamma=0$$\cos 3 \alpha+\cos 3 \beta+\cos 3 \gamma=3 \cos (\alpha+\beta+\gamma)$$\sin 3 \alpha+\sin 3 \beta+\sin 3 \gamma=3 \sin (\alpha+\beta+\gamma)$\begin{align} \cos \alpha … anonymous@undisclosed.example.com (Anonymous) Mon, 15 Jul 2024 12:41:11 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p6 Question 6(i-ix), Exercise 1.4 Solutions of Question 6(i-ix) of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sqrt{2}\left(\cos 315^{\circ}+i \sin 315^{\circ}\right)$\begin{align} &\sqrt{2}\left(\cos 315^{\circ}+i \sin 315^{\circ}\right) \\ =& \sqrt{2} \left(\dfrac{1}{\sqrt{2}}-\dfrac{i}{\sqrt{2}} \right) \\ =& 1-i. \end{align}$5\left(\cos 210^{\ci… anonymous@undisclosed.example.com (Anonymous) Wed, 17 Jul 2024 12:40:31 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p7 Question 6(x-xvii), Exercise 1.4 Solutions of Question 6(x-xvii) of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $7 \sqrt{2}\left(\cos \dfrac{5 \pi}{4}+i \sin \dfrac{5 \pi}{4}\right)$$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{\… anonymous@undisclosed.example.com (Anonymous) Wed, 17 Jul 2024 12:40:53 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p8 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}$$z=x+iy$\begin{align*} &\arg (z-1)=-\dfrac{\pi}{4} \\ \implies & \arg(x+iy-1) = -\dfrac{\pi}{4} \\ \implies & \arg(x-1+iy) = -\dfrac{\pi}{4} \\ \implies & \tan^{-1}\left(\dfrac{y}{x-1}\right) = -\dfrac{\pi}{4} … anonymous@undisclosed.example.com (Anonymous) Wed, 17 Jul 2024 12:41:24 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p9 Question 8, Exercise 1.4 Solutions of Question 8 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 8(i) $0.004 \mathrm{~mm}$$\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}… anonymous@undisclosed.example.com (Anonymous) Wed, 17 Jul 2024 12:41:56 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p10 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{… anonymous@undisclosed.example.com (Anonymous) Wed, 17 Jul 2024 12:46:32 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p11 Question 10, Exercise 1.4 Solutions of Question 10 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 10(i) $Z$$E=(-50+100 i)$$I=(-6-2 i)$$E=(-50+100 i)$$I=(-6-2 i)$$$ E = I \times Z $$$$(-50+100 i)= (-6-2 i) \times Z $$\begin{align} \implies Z & = \dfrac{-50+100 i}{-6-2 i} \\ & = \dfrac{(-50+100 i)(-6+2i)}{(-6-2 i)(-6+2i)}\\ & = \dfrac{300-… anonymous@undisclosed.example.com (Anonymous) Wed, 17 Jul 2024 12:47:07 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/re-ex-p1 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)$$(0,0)$$z$$|z|$$1 / z$$-z$$\bar{z}$$\bar{z}$$x$$y$$x y$$y$$z_{1}=3+2 i$$z_{2}=5+6 i$$z_{1}>z_{2}$$z_{1}<z_{2}$$\overline{z_{1}}=\overline{z_{2}}$$\overline{z_{1}}=-\overline{z_{2}}$… anonymous@undisclosed.example.com (Anonymous) Wed, 17 Jul 2024 12:53:04 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/re-ex-p2 Question 2, Review Exercise Solutions of Question 2 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $i^{2}+i^{4}+i^{6}+\cdots+i^{100}$\begin{align*} & i^{2}+i^{4}+i^{6}+\ldots+i^{100} \\ =& i^2 + (i^2)^2 + (i^2)^3 + (i^2)^4 + \ldots +(i^2)^{49} +(i^2)^{50} \\ =& -1 + (-1)^2 + (-1)^3 + (-1)^4 + \ldots + (-1)^{49}+(-1)^{50} \\ =& -1+1-1+1- \ldots -… anonymous@undisclosed.example.com (Anonymous) Wed, 17 Jul 2024 12:53:23 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/re-ex-p3 Question 3, Review Exercise Solutions of Question 3 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $3 x^{2}+108$\begin{align*} & 3 x^{2}+108\\ =&3 (x^{2}+36)\\ =&3 (x^{2}-(6i)^2)\\ =&3 (x+6i)(x-6i) \end{align*}$4 x^{2}+40$\begin{align*} &4 x^{2}+40\\ =&4 (x^{2}+10)\\ =&4 (x^{2}+(\sqrt{10}i)^2)\\ =&4 (x+\sqrt{10}i)(x-\sqrt{10}i) \end{align*} anonymous@undisclosed.example.com (Anonymous) Wed, 17 Jul 2024 12:53:46 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/re-ex-p4 Question 4, Review Exercise Solutions of Question 4 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $z=x+i y$$\left|\dfrac{z+2 i}{z-2 i}\right|=1$$z = x + iy$\begin{align*} & \left|\dfrac{z + 2i}{z - 2i}\right| = 1\\ \implies & |z + 2i| = |z - 2i|\\ \implies & |x + i(y + 2)| = |x + i(y - 2)|\\ \implies & \sqrt{x^2 + (y + 2)^2} = \sqrt{x^2 + (y -… anonymous@undisclosed.example.com (Anonymous) Wed, 17 Jul 2024 12:54:02 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/re-ex-p5 Question 5, Review Exercise Solutions of Question 5 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $z$$(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… anonymous@undisclosed.example.com (Anonymous) Wed, 17 Jul 2024 16:47:30 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/re-ex-p6 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]… anonymous@undisclosed.example.com (Anonymous) Wed, 17 Jul 2024 16:56:04 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/re-ex-p8 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}.$$$x_{\max}$\begin{align} &x=x_{\max} e^{i\theta} \\ \implies & \sqrt{2} + i \sqrt{2}=x_{\max} e^{i\dfrac{\pi}{4}} \\ \implies & x_{\max} \left(\cos\dfr… anonymous@undisclosed.example.com (Anonymous) Wed, 17 Jul 2024 17:13:41 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-1-p1 Question 1, Exercise 2.1 Solutions of Question 1 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{lll}1 & 3 & 0 \\ 2 & 0 & 1\end{array}\right]$\begin{align}\text{Order of A}&= 2\times 3\end{align}$B=\left[\begin{array}{ll}1 & 2 \\ 2 & 3 \\ 3 & 4\end{array}\right]$\begin{align}\text{Order of B}&= 3\times 2\end{align}$C… anonymous@undisclosed.example.com (Anonymous) Sat, 27 Jul 2024 09:35:52 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-1-p2 Question 2, Exercise 2.1 Solutions of Question 2 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\quad A=\left[\begin{array}{lll}3 & 6 & 2 \\ 2 & 1 & 9\end{array}\right]$$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 … anonymous@undisclosed.example.com (Anonymous) Sat, 27 Jul 2024 09:36:13 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-1-p3 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. $$A=\begin{bmatrix} 3 & 0 & 0 \\ 0 & 1 & 0 \\ 2 & 6 & 0 \end{bmatrix}$$$$B=\begin{bmatrix} -6 & 0 & 0 \\ 0 & -6 & 0 \\ 0 & 0 & -6 \end{bmatrix}$$$$C=\begin{bmatrix} 1 & 0 \\ 2 & 0 \end{bmatrix}$$$$D=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 &… anonymous@undisclosed.example.com (Anonymous) Sat, 27 Jul 2024 09:36:33 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-1-p4 Question 4, Exercise 2.1 Solutions of Question 4 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$ A=\left[\begin{array}{ccc} 2 & 0 \\ \sqrt{5} & 6 \\ 1 & 9 \end{array}\right]$$$$ A^t=\begin{bmatrix} 2 & \sqrt{5} & 1 \\ 0 & 6 & 9 \end{bmatrix}$$$$B=\left[\begin{array}{cccc} 1 & 6 & 2 & 0 \end{array}\right] $$$$B^t=\left[\begin{array}{c} 1… anonymous@undisclosed.example.com (Anonymous) Sat, 27 Jul 2024 09:36:55 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p1 Question 1, Exercise 2.2 Solutions of Question 1 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[a_{i j}\right]$$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} … anonymous@undisclosed.example.com (Anonymous) Tue, 30 Jul 2024 07:26:46 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p2 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 &… anonymous@undisclosed.example.com (Anonymous) Tue, 30 Jul 2024 07:29:12 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p3 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 &… anonymous@undisclosed.example.com (Anonymous) Tue, 30 Jul 2024 07:30:34 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p4 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… anonymous@undisclosed.example.com (Anonymous) Tue, 30 Jul 2024 07:29:39 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p5 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]$$X^{2}-4 X-5 I=0$\begin{align}L.H.S. & =X^{2}-4 X-5 I \\ &=\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2… anonymous@undisclosed.example.com (Anonymous) Mon, 19 Aug 2024 16:11:12 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p6 Question 6, Exercise 2.2 Solutions of Question 6 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{cc}2 & 1 \\ 3 & -3\end{array}\right]$$\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… anonymous@undisclosed.example.com (Anonymous) Mon, 19 Aug 2024 16:24:24 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p7 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}.$$$n = 1$\begin{align}A^1 =\beg… anonymous@undisclosed.example.com (Anonymous) Mon, 19 Aug 2024 16:42:50 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p8 Question 8, Exercise 2.2 Solutions of Question 8 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A$$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}… anonymous@undisclosed.example.com (Anonymous) Mon, 19 Aug 2024 16:43:59 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p9 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$$(A+B)^{t}=A^{t}+B^{t}$\begin{align*} A &= \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} \\ B &= \begin{pmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22… anonymous@undisclosed.example.com (Anonymous) Mon, 19 Aug 2024 16:44:43 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p10 Question 10, Exercise 2.2 Solutions of Question 10 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A$$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… anonymous@undisclosed.example.com (Anonymous) Mon, 19 Aug 2024 16:47:00 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p11 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}$$a_{i j}=i^{2}-j^{2}$\begin{align} a_{ji} & = j^2 -i^2 \\ &= - (i^2 -j^2) \\ &= - a_{ij} \end{align}$a_{ij}=-a_{ji}$ anonymous@undisclosed.example.com (Anonymous) Mon, 19 Aug 2024 16:55:20 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p12 Question 12, Exercise 2.2 Solutions of Question 12 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A$$\left(A^{n}\right)^{t}=\left(A^{t}\right)^{n}$ anonymous@undisclosed.example.com (Anonymous) Mon, 19 Aug 2024 16:58:16 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-2-p13 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)\\… anonymous@undisclosed.example.com (Anonymous) Mon, 19 Aug 2024 16:58:35 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-3-p1 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 |… anonymous@undisclosed.example.com (Anonymous) Fri, 23 Aug 2024 09:03:34 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-3-p2 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… anonymous@undisclosed.example.com (Anonymous) Fri, 23 Aug 2024 09:04:07 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-3-p3 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) … anonymous@undisclosed.example.com (Anonymous) Fri, 23 Aug 2024 09:04:42 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-3-p4 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$$\left[\begin{array}{lll}\lambda & 1 & 3 \\ 2 & 1 & 8 \\ 0 & 3 & 1\end{array}\right]$\begin{align*} A &= \left[\begin{array}{ccc} \lambda & 1 & 3 \\ 2 & 1 & 8 \\ 0 & 3 & 1 \end{array}\right]\\ |A| &= \lambda \cdot (-23) - 1 \cdot 2 + 3… anonymous@undisclosed.example.com (Anonymous) Fri, 23 Aug 2024 09:05:13 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-3-p5 Question 5, Exercise 2.3 Solutions of Question 5 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left[\begin{array}{ccc}1 & -1 & 1 \\ 2 & 1 & -1 \\ 1 & -2 & -1\end{array}\right]$\begin{align*} A &= \left[\begin{array}{ccc} 1 & -1 & 1 \\ 2 & 1 & -1 \\ 1 & -2 & -1 \end{array}\right]\\ |A|&= 1 [-1 - 2] + 1 [-2 + 1] + 1 [-4 - 1] \\ &= 1 \cd… anonymous@undisclosed.example.com (Anonymous) Fri, 23 Aug 2024 09:05:37 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-3-p6 Question 6, Exercise 2.3 Solutions of Question 6 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{ccc}2 & 1 & -3 \\ 0 & 1 & 0 \\ 2 & 1 & 6\end{array}\right]$$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*} … anonymous@undisclosed.example.com (Anonymous) Fri, 23 Aug 2024 09:41:19 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-3-p7 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\\ … anonymous@undisclosed.example.com (Anonymous) Fri, 23 Aug 2024 09:48:00 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-5-p1 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 & … anonymous@undisclosed.example.com (Anonymous) Wed, 04 Sep 2024 03:01:24 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-5-p2 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}{… anonymous@undisclosed.example.com (Anonymous) Wed, 04 Sep 2024 03:01:56 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-5-p3 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]$$A A^{-1}=A^{-1} A=I$\begin{align*} A&=\left[ \begin{array}{ccc} 0 & -1 & -1 \\ -1 & 3 & 0 \\ 1 & -1 & 4 \end{array} \right]\\ |A|&=0+1(-4)-1(1-3)\\ &=-4+3\… anonymous@undisclosed.example.com (Anonymous) Wed, 04 Sep 2024 03:02:35 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p2 Question 2, Exercise 2.6 Solutions of Question 2 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\lambda$$\lambda$$2 x_{1}-\lambda x_{2}+x_{3}=0$$2 x_{1}+3 x_{2}-x_{3}=0$$3 x_{1}-2 x_{2}+4 x_{3}=0$\begin{align*} &2 x_{1}-\lambda x_{2}+x_{3}=0 \cdots(i)\\ &2 x_{1}+3 x_{2}-x_{3}=0\cdots(ii)\\ &3 x_{1}-2 x_{2}+4 x_{3}=0\cdots(iii)\\ \end{ali… anonymous@undisclosed.example.com (Anonymous) Wed, 04 Sep 2024 03:04:14 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p3 Question 3, Exercise 2.6 Solutions of Question 3 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 x+3 y+4 z=2$$2 x+y+z=5$$3 x-2 y+z=-3$\begin{align*} \begin{aligned} 2x + 3y + 4z &= 2 \\ 2x + y + z &= 5 \\ 3x - 2y + z &= -3 \end{aligned}\end{align*}\begin{align*} A_{b} &=\quad \left[\begin{array}{cccc} 2 & 3 & 4 & 2 \\ 2 & 1 & 1 & 5 \\ 3… anonymous@undisclosed.example.com (Anonymous) Wed, 04 Sep 2024 03:11:14 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p4 Question 4, Exercise 2.6 Solutions of Question 4 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 x_{1}-x_{2}-x_{3}=2$$3 x_{1}-4 x_{2}+3 x_{3}=7$$4 x_{1}+2 x_{2}-5 x_{3}=10$\begin{align*} 2x_1 - x_2 - x_3 &= 2, \\ 3x_1 - 4x_2 + 3x_3 &= 7, \\ 4x_1 + 2x_2 - 5x_3 &= 10, \end{align*}\begin{align*} A_b &= \begin{bmatrix} 2 & -1 & -1 & : & 2 … anonymous@undisclosed.example.com (Anonymous) Wed, 04 Sep 2024 03:11:42 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p5 Question 5, Exercise 2.6 Solutions of Question 5 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $x_{1}+x_{2}+2 x_{3}=8$$-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 = \… anonymous@undisclosed.example.com (Anonymous) Wed, 04 Sep 2024 03:12:33 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p6 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… anonymous@undisclosed.example.com (Anonymous) Wed, 04 Sep 2024 03:13:00 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p8 Question 9 and 10, Exercise 2.6 Solutions of Question 9 and 10 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 x-y+3 z=\alpha ; 3 x+y-5 z=\beta ;-5 x-5 y+21 z=\gamma$$\gamma \neq 2 \alpha-3 \beta$$2$$2$$3$$3$ anonymous@undisclosed.example.com (Anonymous) Wed, 04 Sep 2024 03:13:56 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/re-ex-p2 Question 2 and 3, Review Exercise Solutions of Question 2 and 3 of Review Exercise of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{ccc}1 & 2 & 0 \\ -3 & 4 & 9 \\ 2 & 1 & 6\end{array}\right]$$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)^{… anonymous@undisclosed.example.com (Anonymous) Wed, 04 Sep 2024 03:15:23 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p1 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… anonymous@undisclosed.example.com (Anonymous) Sat, 14 Sep 2024 16:29:10 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p2 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… anonymous@undisclosed.example.com (Anonymous) Sat, 14 Sep 2024 16:33:03 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p3 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.$$\begin{align*} a_1 &= (1)^2 - 2(1) = 1 - 2 = -1\\ a_2 &= (2)^2 - 2(2) = 4 - 4 = 0\\ a_3 &= (3)^2 - 2(3) = 9 - 6 = 3\\ a_4 &= (4)^2 - 2(4) = 16 - 8 = 8\\ \end{align*}\begin{align*} a_{… anonymous@undisclosed.example.com (Anonymous) Sat, 14 Sep 2024 16:37:26 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p4 Question 7 and 8, Exercise 4.1 Solutions of Question 7 and 8 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$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} =… anonymous@undisclosed.example.com (Anonymous) Sat, 14 Sep 2024 16:39:31 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p5 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 &=… anonymous@undisclosed.example.com (Anonymous) Sat, 14 Sep 2024 16:42:36 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p6 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$$a_{n}=5 n+11 ; a_{9}$ anonymous@undisclosed.example.com (Anonymous) Sat, 14 Sep 2024 16:45:13 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p7 Question 13 and 14, Exercise 4.1 Solutions of Question 13 and 14 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{n}=(3 n+4)(2 n-5) ; a_{7}$$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… anonymous@undisclosed.example.com (Anonymous) Sat, 14 Sep 2024 16:46:57 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p8 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_{… anonymous@undisclosed.example.com (Anonymous) Sat, 14 Sep 2024 16:48:53 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p9 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.$$\begin{align*} a_{67} &= \ln e^… anonymous@undisclosed.example.com (Anonymous) Sat, 14 Sep 2024 17:38:01 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p10 Question 19 and 20, Exercise 4.1 Solutions of Question 19 and 20 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{n}$$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}$$3,9,27,81,243, \ldots$\begin… anonymous@undisclosed.example.com (Anonymous) Sat, 14 Sep 2024 17:59:38 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p11 Question 21 and 22, Exercise 4.1 Solutions of Question 21 and 22 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{n}$$\sqrt{2}, \sqrt{4}, \sqrt{6}, \sqrt{8}, \sqrt{10}, \ldots$$$\sqrt{2}, \sqrt{4}, \sqrt{6}, \sqrt{8}, \sqrt{10}, \ldots$$\begin{align*} &a_1=\sqrt{2 \cdot 1}, \\ &a_2=\sqrt{4}=\sqrt{2 \cdot 2} \\ &a_3=\sqrt{6}=\sqrt{2 \cdot 3}\\… anonymous@undisclosed.example.com (Anonymous) Sat, 14 Sep 2024 18:01:00 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p1 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.$$\begin{align*} … anonymous@undisclosed.example.com (Anonymous) Sun, 15 Sep 2024 12:26:08 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p2 Question 2, Exercise 4.2 Solutions of Question 2 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $5,9,13, \ldots$$$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$$… anonymous@undisclosed.example.com (Anonymous) Sun, 15 Sep 2024 12:29:31 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p3 Question 3 and 4, Exercise 4.2 Solutions of Question 3 and 4 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $0.07,0.12,0.7, \ldots$$$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 + (… anonymous@undisclosed.example.com (Anonymous) Fri, 20 Sep 2024 16:59:34 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p4 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$$\begin{align*} & a_{17} = -40 \\ \implies &a_1 + 16d = -40 \quad \cdots (1) \end{align*}\begin{align*} &a_{28}=-73\\ \implies &a_1 + 27d = -73 \quad \cdots (2) \end{align*}\begin{align*}… anonymous@undisclosed.example.com (Anonymous) Fri, 20 Sep 2024 17:08:12 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p5 Question 7 and 8, Exercise 4.2 Solutions of Question 7 and 8 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $-6,-2,2, \ldots$$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… anonymous@undisclosed.example.com (Anonymous) Fri, 20 Sep 2024 17:20:28 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p6 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}-… anonymous@undisclosed.example.com (Anonymous) Fri, 20 Sep 2024 17:28:31 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p7 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],$$\begin{align*} S_{20} &= \frac{20}{2}[2(1000)+(20-1)2000]\\ &= 10 [2000+(19)2000] \… anonymous@undisclosed.example.com (Anonymous) Fri, 20 Sep 2024 17:46:01 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p8 Question 13, Exercise 4.2 Solutions of Question 13 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $7$$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… anonymous@undisclosed.example.com (Anonymous) Fri, 20 Sep 2024 18:05:05 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p9 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$$y$… anonymous@undisclosed.example.com (Anonymous) Fri, 20 Sep 2024 18:31:40 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p10 Question 16 and 17, Exercise 4.2 Solutions of Question 16 and 17 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $5$$17$$A_1$$A_2$$5$$17$$5$$A_1$$A_2$$17$$a_1=5$$a_4=17$$$a_n=a_1+(n-1)d.$$\begin{align*} &a_4 = a_1 + 3d \\ \implies & 17=5+3d\\ \implies & 3d=12\\ \implies & \boxed{d=4}.\end{align*}\begin{align*} A_1 &= a_2= a_1+d \\ &=5+4=9 \end{a… anonymous@undisclosed.example.com (Anonymous) Fri, 20 Sep 2024 18:45:04 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p1 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$$… anonymous@undisclosed.example.com (Anonymous) Sat, 14 Sep 2024 16:15:56 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p2 Question 3 and 4, Exercise 4.3 Solutions of Question 3 and 4 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=5$$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$$n=1… anonymous@undisclosed.example.com (Anonymous) Sat, 14 Sep 2024 16:17:02 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p3 Question 5 and 6, Exercise 4.3 Solutions of Question 5 and 6 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=50$$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$$-3+(-7)+(-11)+\cd… anonymous@undisclosed.example.com (Anonymous) Sat, 14 Sep 2024 16:17:27 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p4 Question 7 and 8, Exercise 4.3 Solutions of Question 7 and 8 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $9+11+13+15+\cdots$$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$$$2+4+6+...+100 (50 \tex… anonymous@undisclosed.example.com (Anonymous) Sat, 14 Sep 2024 16:17:56 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p5 Question 9 and 10, Exercise 4.3 Solutions of Question 9 and 10 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $1$$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$$523$$… anonymous@undisclosed.example.com (Anonymous) Sat, 14 Sep 2024 16:19:25 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p6 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$$a_{1… anonymous@undisclosed.example.com (Anonymous) Sat, 14 Sep 2024 16:19:47 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p7 Question 13 and 14, Exercise 4.3 Solutions of Question 13 and 14 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $S_s$$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… anonymous@undisclosed.example.com (Anonymous) Sat, 14 Sep 2024 16:23:14 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p8 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$$n=?$\begin{align} & a_n=a_1+(n-1)d \\ \implies & 15=91+(n-1)(-4) \\ \implies & 15=91-4n+4 \\ \implies & 4n=95-15 \\ \implies & 4n = 80\\ \implies & n = 20. \end{align}\begin{… anonymous@undisclosed.example.com (Anonymous) Sat, 14 Sep 2024 16:23:39 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p9 Question 17, 18 and 19, Exercise 4.3 Solutions of Question 17, 18 and 19 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $6+12+18+\ldots+96$$$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}… anonymous@undisclosed.example.com (Anonymous) Sat, 14 Sep 2024 16:23:56 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p10 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$$d$\begin{align} &S_n=\frac{n}{2}[a_1+a_n]\\ \implies & 876=\frac{n}{2}[7+139]\\ \implies & 1752=146n\\ \implies & n=\frac{1752}{146}=12. \end{align}\begin{align… anonymous@undisclosed.example.com (Anonymous) Sat, 14 Sep 2024 16:24:21 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p11 Question 23 and 24, Exercise 4.3 Solutions of Question 23 and 24 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$ 14+16+18+...+a_{25}.$$$a_1=14$$d=16-14=2$$n=25$$a_25$$S_25$\begin{align} a_n&=a_1+(n-1)d\\ \implies a_{25}&= 14+(25-1)(2)\\ &=62. \end{align}\begin{align} S_n&=\frac{n}{2}[a_1+a_n]\\ \implies S_{25}& =\frac{25}{2}[14+62]\\ & =25 \t… anonymous@undisclosed.example.com (Anonymous) Sat, 14 Sep 2024 16:24:52 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-3-p12 Question 25 and 26, Exercise 4.3 Solutions of Question 25 and 26 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$ 6000+70,000+...+a_{20}.$$$a_1=6,000$$d=70,000-6,000=64,000$$n=20$$S_n$\begin{align} S_n&=\frac{n}{2}[2a_1+(n-1)d]\\ \implies S_{20}& =\frac{20}{2}[2(6,000)+(20-1)(64,000)]\\ & =10 \times [12,000+1,216,000]\\ & =12,280,000. \end{ali… anonymous@undisclosed.example.com (Anonymous) Sat, 14 Sep 2024 16:25:19 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p1 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{… anonymous@undisclosed.example.com (Anonymous) Sun, 22 Sep 2024 18:26:03 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p2 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} = … anonymous@undisclosed.example.com (Anonymous) Sun, 22 Sep 2024 18:26:06 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p3 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$$a_4=-… anonymous@undisclosed.example.com (Anonymous) Sun, 22 Sep 2024 18:26:06 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p4 Question 8 and 9, Exercise 4.4 Solutions of Question 8 and 9 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$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=… anonymous@undisclosed.example.com (Anonymous) Sun, 22 Sep 2024 18:26:06 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p5 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}… anonymous@undisclosed.example.com (Anonymous) Sun, 22 Sep 2024 18:26:07 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p6 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}… anonymous@undisclosed.example.com (Anonymous) Sun, 22 Sep 2024 18:26:07 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p7 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^{… anonymous@undisclosed.example.com (Anonymous) Sun, 22 Sep 2024 18:26:08 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p8 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}$$a_{n}=… anonymous@undisclosed.example.com (Anonymous) Sun, 22 Sep 2024 18:26:08 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p9 Question 18 and 19, Exercise 4.4 Solutions of Question 18 and 19 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{1}=32, n=6, r=-\frac{1}{2}$$a_{1}=32$$n=6$$r=-\frac{1}{2}$$a_{n}=a_{1} r^{n-1}.$\begin{align*} a_6 &= 32 \times \left(-\frac{1}{2}\right)^{6-1} \\ &= 32 \times \left(-\frac{1}{2}\right)^{5} \\ &= 32 \times \left(-\frac{1}{32}\ri… anonymous@undisclosed.example.com (Anonymous) Sun, 22 Sep 2024 18:26:09 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p10 Question 20 and 21, Exercise 4.4 Solutions of Question 20 and 21 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$3 , \_\_\_ , \_\_\_ , \_\_\_ , 48$$$a_1=3$$a_5=48$$r$$$ a_n=ar^{n-1}. $$\begin{align*} &a_5=a_1 r^4 \\ \implies & 48=3r^4 \\ \implies & r^4 = 16 \\ \implies & r^4 = 2^4 \\ \implies & r = 2. \end{align*}\begin{align*} & a_2=a_1 r= (3… anonymous@undisclosed.example.com (Anonymous) Sun, 22 Sep 2024 18:26:02 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p11 Question 22 and 23, Exercise 4.4 Solutions of Question 22 and 23 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$8 , \_\_\_, \_\_\_, \_\_\_, \_\_\_, \dfrac{1}{4}$$$a_1=8$$a_6=\frac{1}{4}$$r$$n$$a_n = a_1 r^{n-1}.$\begin{align*} a_6 &= a_1 r^5 \\ \implies \frac{1}{4} &= 8 \cdot r^5 \\ \implies r^5 &= \frac{1}{4 \cdot 8} \\ \implies r^5 &= \frac… anonymous@undisclosed.example.com (Anonymous) Sun, 22 Sep 2024 18:26:03 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p12 Question 24 and 25, Exercise 4.4 Solutions of Question 24 and 25 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$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… anonymous@undisclosed.example.com (Anonymous) Sun, 22 Sep 2024 18:26:04 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p13 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*}… anonymous@undisclosed.example.com (Anonymous) Sun, 22 Sep 2024 18:26:04 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p14 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. $$\begin{align*} S_6&=\frac{(1)\left(1-2^7 \right)}{1-2} \\ &=\frac{1-128}{-2}\\ &=127 \end{align… anonymous@undisclosed.example.com (Anonymous) Sun, 22 Sep 2024 18:26:05 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-4-p15 Question 30, Exercise 4.4 Solutions of Question 30 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $=a_1= 1$$=a_2= 3$$=a_3=3\times 3 = 9$$=a_4=3\times 9 = 27$$=a_5=3\times 27 = 81$$81$$a_1=1$$r=3$$a_5=?$$$a_n=a_1 r^{n-1}.$$\begin{align*} a_5&=a_1 r^4 \\ &=(1)(3)^4 = 81 \end{align*}$$S_n=a_1+a_2+a_3+a_4+a_5.$$ anonymous@undisclosed.example.com (Anonymous) Sun, 22 Sep 2024 18:26:05 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p1 Question 1 and 2, Exercise 4.5 Solutions of Question 1 and 2 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $16+16+16+\ldots$$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 \… anonymous@undisclosed.example.com (Anonymous) Sun, 22 Sep 2024 18:34:37 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p2 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)}… anonymous@undisclosed.example.com (Anonymous) Sun, 22 Sep 2024 18:26:10 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p3 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… anonymous@undisclosed.example.com (Anonymous) Sun, 22 Sep 2024 18:26:10 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p4 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}… anonymous@undisclosed.example.com (Anonymous) Sun, 22 Sep 2024 18:26:11 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p5 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… anonymous@undisclosed.example.com (Anonymous) Sun, 22 Sep 2024 18:26:11 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p6 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 & … anonymous@undisclosed.example.com (Anonymous) Sun, 22 Sep 2024 18:26:12 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p7 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}$$9.99999 ...$$… anonymous@undisclosed.example.com (Anonymous) Sun, 22 Sep 2024 18:26:12 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p8 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$$$D=30+2\left(12+\frac{24}{5}+\frac{24}{5}+... \right)$$$$ 12+\frac{24}{5}+\frac{24}{5… anonymous@undisclosed.example.com (Anonymous) Sun, 22 Sep 2024 18:26:13 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p9 Question 16, Exercise 4.5 Solutions of Question 16 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $80 ft$$90\%$$a_1$$a_1 r$$a_1 r^2$$=a_1= 80 ft$$r=90% = \frac{90}{100} =0.9$$A$\begin{align} A &= a_1+a_1r+a_1r^2+... \\ & = \frac{a_1}{1-r} \\ & = \frac{80}{1-0.9}\\ &= 800 \end{align}$800 ft$ anonymous@undisclosed.example.com (Anonymous) Sun, 22 Sep 2024 18:42:06 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-6-p1 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) … anonymous@undisclosed.example.com (Anonymous) Tue, 24 Sep 2024 18:04:32 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-6-p2 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}.… anonymous@undisclosed.example.com (Anonymous) Tue, 24 Sep 2024 18:04:55 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-6-p3 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=… anonymous@undisclosed.example.com (Anonymous) Tue, 24 Sep 2024 18:08:03 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-6-p4 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*} … anonymous@undisclosed.example.com (Anonymous) Tue, 24 Sep 2024 18:07:44 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-6-p5 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_… anonymous@undisclosed.example.com (Anonymous) Tue, 24 Sep 2024 18:08:26 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-6-p6 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}$$b=\dfrac{4}{7}$\begin{align*} \text{H.M.}&=\frac{2ab}{a+b} \\ &=\frac{2\times\frac{2}{3}\times\frac{4}{7}}{\frac{2}{3}+\frac{4}{7}} \\ &=\frac{16/21}{26/21} \\ &=\frac{8}{13} \\ \end{align*}$\dfrac{8}{13… anonymous@undisclosed.example.com (Anonymous) Tue, 24 Sep 2024 18:09:11 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-6-p7 Question 12, Exercise 4.6 Solutions of Question 12 of Exercise 4.6 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\dfrac{1}{3}$$\dfrac{1}{11}$$H_1, H_2, H_3, H_4$$H.Ms$$\dfrac{1}{3}$$\dfrac{1}{11}$$$\dfrac{1}{3},H_1, H_2, H_3, H_4, \dfrac{1}{11} \text{ are in H.P.}$$$$\quad 3,\dfrac{1}{H_1},\dfrac{1}{H_2}, \dfrac{1}{H_3}, \dfrac{1}{H_4},11 \text{ are in A.P.}… anonymous@undisclosed.example.com (Anonymous) Tue, 24 Sep 2024 18:09:33 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p1 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}\\ &= … anonymous@undisclosed.example.com (Anonymous) Fri, 04 Oct 2024 19:06:32 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p2 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(… anonymous@undisclosed.example.com (Anonymous) Fri, 04 Oct 2024 19:06:38 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p3 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.… anonymous@undisclosed.example.com (Anonymous) Fri, 04 Oct 2024 19:06:39 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p4 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… anonymous@undisclosed.example.com (Anonymous) Fri, 04 Oct 2024 19:06:39 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p5 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$$ anonymous@undisclosed.example.com (Anonymous) Fri, 04 Oct 2024 19:06:40 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p6 Question 11, 12 and 13, Exercise 4.7 Solutions of Question 11, 12 and 13 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $-2+4-8+16-32+64$$$ -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} +… anonymous@undisclosed.example.com (Anonymous) Fri, 04 Oct 2024 19:06:40 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p7 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} \… anonymous@undisclosed.example.com (Anonymous) Fri, 04 Oct 2024 19:06:41 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p8 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$$k$\begin{align*}T_k&=(3k-1)^2 \\ &=9k^2-6k+1. \end{align*}\begin{align*}\sum_{k=1}^{n} T_{k} &= \sum_{k=1}^{n} (9k^{2} - 6k + 1)\\ & = 9\sum_{k=1}^{n} k^{2} … anonymous@undisclosed.example.com (Anonymous) Fri, 04 Oct 2024 19:06:41 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p9 Question 19 and 20, Exercise 4.7 Solutions of Question 19 and 20 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$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^… anonymous@undisclosed.example.com (Anonymous) Fri, 04 Oct 2024 19:06:43 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p10 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$$k$\begin{align*}T_k&=(2k-1) \\ &=9k^2-6k+1. \end{align*}\begin{align*}\sum_{k=1}^{n} T_{k} &= \sum_{k=1}^{n} (2k - 1)\\ & = 2 \sum_{k=1}^{n} k - \sum_{k=1}^{n} 1 \… anonymous@undisclosed.example.com (Anonymous) Fri, 04 Oct 2024 19:06:32 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p11 Question 21 and 22, Exercise 4.7 Solutions of Question 21 and 22 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$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$$k$\begin{align*}T_k&=k(3k+1) \\ &=3k^2+k. \end{align*}\begin{align*}\sum_{k=1}^{n} T_{k} &= \sum_{k=1}^{n} (3k^2 +k)\… anonymous@undisclosed.example.com (Anonymous) Fri, 04 Oct 2024 19:06:33 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p12 Question 23 and 24, Exercise 4.7 Solutions of Question 23 and 24 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$$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}… anonymous@undisclosed.example.com (Anonymous) Fri, 04 Oct 2024 19:06:34 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p13 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… anonymous@undisclosed.example.com (Anonymous) Fri, 04 Oct 2024 19:06:36 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p14 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$$1, \dfrac{1}{3}, \d… anonymous@undisclosed.example.com (Anonymous) Fri, 04 Oct 2024 19:06:36 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-7-p15 Question 29 and 30, Exercise 4.7 Solutions of Question 29 and 30 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$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_{\… anonymous@undisclosed.example.com (Anonymous) Fri, 04 Oct 2024 19:06:37 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-8-p1 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*} \… anonymous@undisclosed.example.com (Anonymous) Sun, 06 Oct 2024 17:46:32 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-8-p2 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… anonymous@undisclosed.example.com (Anonymous) Sun, 06 Oct 2024 17:46:48 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-8-p3 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… anonymous@undisclosed.example.com (Anonymous) Sun, 06 Oct 2024 17:47:09 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-8-p4 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… anonymous@undisclosed.example.com (Anonymous) Sun, 06 Oct 2024 17:47:35 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-8-p5 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… anonymous@undisclosed.example.com (Anonymous) Sun, 06 Oct 2024 17:48:05 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-8-p6 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{… anonymous@undisclosed.example.com (Anonymous) Sun, 06 Oct 2024 17:48:33 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-8-p7 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$$k$\begin{align*} T_k &= \frac{1}{(2k+3)(2k+9)}. \end{align*}\begin{align*} \frac{1}{(2k+3)(2k+9)} = \frac{A}{2k+3} + \frac{B}{2k+9} \ldots … anonymous@undisclosed.example.com (Anonymous) Sun, 06 Oct 2024 17:49:14 +0000 https://www.mathcity.org/math-11-nbf/sol/unit05/ex5-1-p1 Question 1, Exercise 5.1 Solutions of Question 1 of Exercise 5.1 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 1(i) $2 x^{3}+3 x^{2}-4 x+1$$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… anonymous@undisclosed.example.com (Anonymous) Thu, 10 Oct 2024 09:44:24 +0000 https://www.mathcity.org/math-11-nbf/sol/unit05/ex5-1-p2 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$$… anonymous@undisclosed.example.com (Anonymous) Thu, 10 Oct 2024 09:44:46 +0000 https://www.mathcity.org/math-11-nbf/sol/unit05/ex5-1-p3 Question 4 and 5, Exercise 5.1 Solutions of Question 4 and 5 of Exercise 5.1 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $4 y^{3}-4 y^{2}+10+2 y$$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)$$p(-1)=0$\begin{align*} &(-1)^3+q(-1)^2-7(-1)+6=0 \\ -&1+q+7+6=0\\ &q+12=0\\ &q=-12 \end{align*} anonymous@undisclosed.example.com (Anonymous) Thu, 10 Oct 2024 09:45:10 +0000 https://www.mathcity.org/math-11-nbf/sol/unit05/ex5-1-p4 Question 6 and 7, Exercise 5.1 Solutions of Question 6 and 7 of Exercise 5.1 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $m$$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… anonymous@undisclosed.example.com (Anonymous) Thu, 10 Oct 2024 09:45:38 +0000 https://www.mathcity.org/math-11-nbf/sol/unit05/ex5-1-p5 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$$p(x)=2x^3+3x^2-11x-6$\begin{align} p(2) &= 2(2)^3+3(2)^2-11(2)-6 \\ &=16+12-22-6 = 0 \end{align}$p(x)$\begin{align} \begin{array}{r|rrrr} 2 & 2 & 3 & -11 & -6 \\ & \downarrow & 4 & 14 & 6 \\ \hline & 2 & 7 & 3 & 0 \\ \… anonymous@undisclosed.example.com (Anonymous) Thu, 10 Oct 2024 09:45:57 +0000 https://www.mathcity.org/math-11-nbf/sol/unit05/ex5-1-p6 Question 10, Exercise 5.1 Solutions of Question 10 of Exercise 5.1 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 10 $\left(x^{3}+11 x^{2}+34 x+24\right)$$(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… anonymous@undisclosed.example.com (Anonymous) Thu, 10 Oct 2024 09:46:15 +0000 https://www.mathcity.org/math-11-nbf/sol/unit05/ex5-2-p1 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… anonymous@undisclosed.example.com (Anonymous) Sun, 13 Oct 2024 17:52:33 +0000 https://www.mathcity.org/math-11-nbf/sol/unit05/ex5-2-p2 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 &… anonymous@undisclosed.example.com (Anonymous) Sun, 13 Oct 2024 17:52:34 +0000 https://www.mathcity.org/math-11-nbf/sol/unit05/ex5-2-p3 ; 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 \… anonymous@undisclosed.example.com (Anonymous) Sun, 13 Oct 2024 17:52:35 +0000 https://www.mathcity.org/math-11-nbf/sol/unit05/ex5-2-p4 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… anonymous@undisclosed.example.com (Anonymous) Sun, 13 Oct 2024 17:52:35 +0000 https://www.mathcity.org/math-11-nbf/sol/unit05/ex5-3-p1 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$$\begin{align*} p(2)&=2^3+13(2)^2+30(2)-120 \\ &=8+52+60-120 =0 \end{ali… anonymous@undisclosed.example.com (Anonymous) Sun, 13 Oct 2024 17:52:36 +0000 https://www.mathcity.org/math-11-nbf/sol/unit05/ex5-3-p2 Question 2, Exercise 5.3 Solutions of Question 2 of Exercise 5.3 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 2 $t(x)=x^{3}-12 x^{2}+48 x+74$$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*} anonymous@undisclosed.example.com (Anonymous) Sun, 13 Oct 2024 17:52:37 +0000 https://www.mathcity.org/math-11-nbf/sol/unit05/ex5-3-p3 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)$$2(3)+… anonymous@undisclosed.example.com (Anonymous) Sun, 13 Oct 2024 17:52:37 +0000 https://www.mathcity.org/math-11-nbf/sol/unit05/ex5-3-p4 Question 4, Exercise 5.3 Solutions of Question 4 of Exercise 5.3 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 4 $x$$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.$$\begin{align*} p(11)&=2(11)^3-11^2-6(11)-2475 \\ &=2662… anonymous@undisclosed.example.com (Anonymous) Sun, 13 Oct 2024 17:52:37 +0000 https://www.mathcity.org/math-11-nbf/sol/unit05/ex5-3-p5 Question 5, Exercise 5.3 Solutions of Question 5 of Exercise 5.3 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 5 $6 x^{2}+38 x+56$$2 x+8$$ACED$$ABFG$$ACED$$6 x^{2}+38 x+56$$2 x+8$\begin{align*} & 6 x^{2}+38 x+56 \\ = & 2(3x^2+19x+28) \\ = & 2(3x^2+12x+7x+28) \\ = & 2(3x(x+4)+7(x+4)) \\ =& 2(x+4)(3x+7) \\ =& (2x+8)(3x+7) \end{align*}\begin{align*} & Length … anonymous@undisclosed.example.com (Anonymous) Sun, 13 Oct 2024 17:52:38 +0000 https://www.mathcity.org/math-11-nbf/sol/unit05/ex5-3-p6 Question 6, Exercise 5.3 Solutions of Question 6 of Exercise 5.3 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 6 $y^3-2y^2-y+2$$y-2$$=p(y)=y^3-2y^2-y+2$$y-2$$y-2$$p(y)$$2$$p(y)$\[ \begin{array}{r|rrrr} 2 & 1 & -2 & -1 & 2 \\ & \downarrow & 2 & 0 & -2 \\ \hline & 1 & 0 & -1 & 0 \\ \end{array} \]\begin{align*} p(y) & = (y-2)(y^2-1) \\ & = (y-2)(y+1)(y-1)… anonymous@undisclosed.example.com (Anonymous) Sun, 13 Oct 2024 17:52:38 +0000 https://www.mathcity.org/math-11-nbf/sol/unit05/ex5-4-p2 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 anonymous@undisclosed.example.com (Anonymous) Sun, 13 Oct 2024 17:52:38 +0000 https://www.mathcity.org/math-11-nbf/sol/unit05/re-ex-p1 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)$$(x+1)(x-2)$$9 y^{2}+9 y-10$$3 y-2$$ 0$$1$$2$$3$$ 0$$\frac{x^{2}-x-9}{x-3}=x+2+\frac{?}{x-3}$$-27$$-3$$\frac{3}{x-3}+x+2$$ 3$$-3$$3 x^{3}-2 x^{2}+5$$x+1$$x+1$$x^{3}+5 x^{2}-4 x+k$… anonymous@undisclosed.example.com (Anonymous) Sun, 13 Oct 2024 17:52:39 +0000 https://www.mathcity.org/math-11-nbf/sol/unit05/re-ex-p2 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 … anonymous@undisclosed.example.com (Anonymous) Sun, 13 Oct 2024 17:52:39 +0000 https://www.mathcity.org/math-11-nbf/sol/unit05/re-ex-p3 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$$6 y^{3}-y^{2}-5 y+2$\begin{align*}3y-2&=0\\ 3y&=2\\ y&=\frac{2}{3}\end{align*}\begin{align*} f(y) &= 6y^{3} - y^{2} - 5y + 2\\ f\left(\frac{2}{3}\right) &= 6\left(\frac{2}{3}\right)^{3} - \left(\frac{2}{3}\right)^{2} - 5\left(\frac{2}{3… anonymous@undisclosed.example.com (Anonymous) Sun, 13 Oct 2024 17:52:40 +0000 https://www.mathcity.org/math-11-nbf/sol/unit05/re-ex-p4 Question 6 & 7, Review Exercise Solutions of Question 6 & 7 of Review Exercise of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $k$$\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… anonymous@undisclosed.example.com (Anonymous) Sun, 13 Oct 2024 17:52:40 +0000 https://www.mathcity.org/math-11-nbf/sol/unit05/re-ex-p5 Question 8, Review Exercise Solutions of Question 8 of Review Exercise of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 8 $y^{3}+6 y^{2}-y-30$$(y-2)$$(y+3)$$p(y)=y^{3}+6 y^{2}-y-30$$(y-2)$$(y+3)$$p(y)$$2$$-3$$p(y)$\[ \begin{array}{r|rrrr} 2 & 1 & 6 & -1 & -30 \\ & \downarrow & 2 & 16 & 30 \\ \hline -3 & 1 & 8 & 15 & 0 \\ & \downarrow & -3 & -15 & … anonymous@undisclosed.example.com (Anonymous) Sun, 13 Oct 2024 17:52:41 +0000 https://www.mathcity.org/math-11-nbf/sol/unit05/re-ex-p6 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)$ anonymous@undisclosed.example.com (Anonymous) Sun, 13 Oct 2024 17:52:42 +0000 https://www.mathcity.org/math-11-nbf/sol/unit05/re-ex-p7 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 $3 x^{2}-x+32-\frac{121}{x+4}$ anonymous@undisclosed.example.com (Anonymous) Sun, 13 Oct 2024 17:52:42 +0000 https://www.mathcity.org/math-11-nbf/sol/unit05/re-ex-p8 Question 8, Review Exercise Solutions of Question 8 of Review Exercise of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 8 $y^{3}+6 y^{2}-y-30$$(y-2)$$(y+3)$ anonymous@undisclosed.example.com (Anonymous) Sun, 13 Oct 2024 17:52:43 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-1-p1 Question 1, Exercise 6.1 Solutions of Question 1 of Exercise 6.1 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $10!$\begin{align*} 10! &= 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \\ &= 3628800 \end{align*}$\dfrac{12!}{7! 3! 2!}$\begin{align*} \dfrac{12!}{7! \, 3! \, 2!} &= \dfrac{12 \times 11 \times 10 \times 9 \times 8 \times 7!}{7! … anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:29 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-1-p2 Question 2, Exercise 6.1 Solutions of Question 2 of Exercise 6.1 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\quad 14\cdot 13\cdot 12\cdot 11$\begin{align*} &14\cdot 13\cdot 12\cdot 11\\ = & \dfrac{14\cdot 13\cdot 12\cdot 11\cdot 10!}{10!} \\ = & \dfrac{14!}{10!} \end{align*}$\quad 1\cdot 3\cdot 5 \cdot 7 \cdot 9$$$1 \times 3\times 5\times 7 \time… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:29 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-1-p3 Question 3 and 4, Exercise 6.1 Solutions of Question 3 and 4 of Exercise 6.1 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\quad \dfrac{1}{5!}+\dfrac{3}{6!}+\dfrac{1}{7!}=\dfrac{4}{315}$\begin{align*} LHS = & \dfrac{1}{5!}+\dfrac{3}{6!}+\dfrac{1}{7!} \\ = & \dfrac{1}{5!}+\dfrac{3}{6\cdot 5!}+\dfrac{1}{7\cdot 6\cdot 5!}\\ = & \dfrac{1}{5!}\left(1+\dfr… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:31 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-1-p4 Question 5, Exercise 6.1 Solutions of Question 5 of Exercise 6.1 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$\quad \dfrac{n}{(n-4)!}=\dfrac{3.3!}{(n-3)!}$\begin{align*} \dfrac{n}{(n-4)!}&=\dfrac{3.3!}{(n-3)!}\\ \dfrac{n}{(n-4)!}&=\dfrac{3.3!}{(n-3)(n-4)!}\\ n&=\dfrac{3\times 6}{n-3}\\ n(n-3)&=18\\ n^2-3n&=18\\ n^2-2n-18&=0\\ n^2+3n-6n-18&=0\\ n(… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:34 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-1-p5 Question 6(i-v), Exercise 6.1 Solutions of Question 6(i-v) of Exercise 6.1 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n\in N$$\quad (2n)!=2^n(n!)[1\cdot3\cdot5 \cdots (2n-1)]$\begin{align*} (2n)!&= (2n)(2n-1)(2n-2)(2n-3)\cdots (2n-n)(2n-(n+1))\cdots4.3.2.1\\ &= (2n)(2n-1)(2n-2)(2n-3)\cdots (n)(n-1)\cdots4.3.2.1\\ &= (2n)(2n-1)(2n-2)(2n-3)\cdots (n… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:32 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-1-p6 Question 6(vi-ix), Exercise 6.1 Solutions of Question 6(vi-ix) of Exercise 6.1 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n\in N$$\quad 33!$$2^{15}$$$33!=33.32.31\cdots4.3.2.1$$$2,4,8,16,32$$2^1,2^2,2^3,2^4,2^5$\begin{align*}33!&=2^5.2^4.2^3.2^2.2(33\times31\times\cdots6\times5\times3\times1)\\ &=2^{15}(33\times31\times\cdots\times3\times1)\end{al… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:34 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-1-p7 Question 7(i-vi), Exercise 6.1 Solutions of Question 7(i-vi) of Exercise 6.1 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$\quad \dfrac{n!}{(n-2)!}=930,\quad n \geq 2$\begin{align*} \dfrac{n!}{(n-2)!}&=930\\ \dfrac{n(n-1)(n-2)!}{(n-2)!}&=930\\ n(n-1)&=930\\ n^2-n-930&=0 \end{align*}\begin{align*} n&=\dfrac{1\pm \sqrt{1+4(930)}}{2}\\ &=\dfrac{1\pm … anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:37 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-1-p8 Question 7(vii-xi), Exercise 6.1 Solutions of Question 7(vii-xi) of Exercise 6.1 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$\quad n!=990 \cdot (n-3)!$\begin{align*} n!&=990 (n-3)!\\ n(n-1)(n-1)(n-3)!&=990 (n-3)!\\ n(n-1)(n-1)&=990 \\ n^3-3n^2+2n-990 &=0\\ \end{align*}\[\begin{array}{c|cccc} & 1 & -3 & 2 & -990 \\ 11 & 0 & 11 & 88 & 990 \\… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:36 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-2-p1 Question 1, Exercise 6.2 Solutions of Question 1 of Exercise 6.2 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n \in N$$\quad^nP_r=\dfrac{n!}{(n-r)!}$$n$$r$$\dot{n}$$2^{\text {nd }}$$(n-1)$$3^{\text {rd }}$$(n-2)$$r^{\text {th }}$$(n-(r-1)$$$ { }^{n} P_{r}=n(n-1)(n-2) \ldots(n-(r-1)) $$$(n-r)(n-r-1) \ldots 3,2.1$${ }^{n} P_{r}=\frac{n(n-1)(n-2)\ldots… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:40 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-2-p2 Question 2, Exercise 6.2 Solutions of Question 2 of Exercise 6.2 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$\quad ^nP_4=20\, ^nP_2$\begin{align*} \dfrac{m}{(n-4)!}&=20 \cdot \dfrac{m}{(n-2)!}\\ \dfrac{1}{(n-4)!}&=\dfrac{20}{(n-2)(n-3)(n-4)!}\\ (n-2)(n-3)&=20\\ n^{2}-5 n+6&=20\\ n^{2}-5 n-14&=0\\ n^{2}+2 n-7 n-14&=0\\ n(n+2)-7(n+2)&=0\\ (n+2)(n-… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:49 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-2-p3 Question 3, Exercise 6.2 Solutions of Question 3 of Exercise 6.2 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $r$$^6P_{r-1}=^5P_4$\begin{align*}{ }^{6} P_{r-1}&={ }^{5} P_{4}\\ \dfrac{6!}{(6-(r-1))!}&=\dfrac{5!}{(5-4)!}\\ \dfrac{6!}{(7-r)!}&=\dfrac{5!}{1!} \\ \frac{6 \times 5!}{(7-r)!}=\dfrac{5!}{1} 6&=(7-r)!\\ 3!&=(7-r)!\\ 3&=7-r \\ r&=7-3\\ r&=4\en… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:50 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-2-p4 Question 4 and 5, Exercise 6.2 Solutions of Question 4 and 5 of Exercise 6.2 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $3$$1,2,3,4,5,6,$$3$$6$$\mathrm{I}:$$2$$$\underline{ },\underline{ },\underline{2}$$$={ }^{5} P_{2}=20$$4$$$\underline{ },\underline{ },\underline{4}$$$={ }^{5} P_{2}=20$$6$$$\underline{ },\underline{ },\underline{6}$$$={ }^{5} P_… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:51 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-2-p5 Question 6 and 7, Exercise 6.2 Solutions of Question 6 and 7 of Exercise 6.2 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $4$$$1,2,3,4,5,6$$6$$6$$6$$6$$$\text{Total possibilities }6 \times 6 \times 6 \times 6=6^4=1296$$$1,1,2,2,3,3,4$$=7$$2^{\text {nd }}$$6^{\text {th }}$$1^{\text {st }}, 3^{\text {rd }}, 5^{\text {th }}$$7^{\text {th }}$$2,2,4$$1,1,… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:54 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-2-p6 Question 8 and 9, Exercise 6.2 Solutions of Question 8 and 9 of Exercise 6.2 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $4$$5$$5$$4$$41$$=41 \times 41=576$$2$$3$$4$$$\text{Total flags} =9$$$$\text{Repetition of blue }=2$$$$\text{Repetition of yellow}=3$$$$\text{Repetition of green}=4$$\begin{align*}\text{Total signals }&=\dfrac{9!}{2!3!4!}\\ &=\dfr… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:54 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-2-p7 Question 10 and 11, Exercise 6.2 Solutions of Question 10 and 11 of Exercise 6.2 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $=6$$={ }^{6}{ }^{1} P_{5}=6$$720$$F$$F$$5$$5!=120$$=9$$S=2$$T=2$$A=2$\begin{align*}\text{possible permutation}&=\dfrac{9!}{2!2!2!}\\ &=\dfrac{362880}{8}\\ &=45360\end{align*} anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:56 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-2-p8 Question 12 and 13, Exercise 6.2 Solutions of Question 12 and 13 of Exercise 6.2 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $=5$$5$$=5!=120$$O$$E$$21=2$$3$$3!= 6$$O$$E$$=6 \times 2=12$$0$$E$$=120-12=108$$=7$$7$$=71=5040$$3$$A, I$$E$$3$$4$$$5!=120$$$3$$=6$$$6 \times 120=\mathbf{7 2 0}$$$5040$$720$$3$$5040-720=4320$ anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:55 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-2-p9 Question 14 and 15, Exercise 6.2 Solutions of Question 14 and 15 of Exercise 6.2 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $3$$=7$$=3$$={ }^{7} P_{3}=\dfrac{7!}{4!}=210$$5$$3$$2$$3$$=31=6$$=(5!\times 3!\times 2!) \times 31=8640$ anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:57 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-2-p10 Question 16 and 17, Exercise 6.2 Solutions of Question 16 and 17 of Exercise 6.2 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $1,2,3,4,5,6$$1, 3$$5$$=3 \times{ }^{5} P_{5}=360$$4$$1,2,3,4$$5$$1$$={ }^{4} P_{3}=24$$3$$24$$5$$24$$24+24+24=72$$4$ anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:43 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-2-p11 Question 18 and 19, Exercise 6.2 Solutions of Question 18 and 19 of Exercise 6.2 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $10,000$$0,2,3,5,6$$4$$10000$$3$$5$$0$$3={ }^{3} P=6$$0$$5=3 p_{6}=6$$10000$$5$$=6+6=12$$3$$4$$5$$={ }^{4} P_{3}=2$$4$$5$$4$$5={ }^{4} P_{3}=24$$3$$3$$3$$5=$${ }^{4} P_{2}=12$$3$$5$$3$$5=$${ }^{4} P_{2}=12$$2$$3$$2$$5={ }^{4} … anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:48 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-2-p12 Question 20 and 21, Exercise 6.2 Solutions of Question 20 and 21 of Exercise 6.2 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $6$$6$$5!$$6$$6!$$=5! \times 6!=86400$$= n = 4$$= {}^4P_4 = 24.$ anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:45 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-2-p13 Question 22 and 23, Exercise 6.2 Solutions of Question 22 and 23 of Exercise 6.2 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $(3\quad 4\quad 6\quad 1\quad 5\quad2)$$1$$2$$L=1, A=2, H=3, O=4, R=5, E=6$$(3,4,6,1,5,2)=$$(4\quad 6\quad 3\quad 2\quad 1\quad5)$$(4,6,3,2,1,5)$$4^{\text {th }}$$2^{\text {nd }}$$6^{\text {th }}$$3^{\text {rd }}$$3^{\text {rd… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:46 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-3-p1 Question 1(i-v), Exercise 6.3 Solutions of Question 1(i-v) of Exercise 6.3 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n\in N$$\quad^nC_r=\dfrac{n!}{r!(n-r)!}$$n$$r$$0<r<n$$X$$r$$r$$r$$$ \begin{array}{ll} & r!X={ }^{n} P_{r} \\ \Rightarrow & r!X=\frac{n!}{(n-r)!} \\ \Rightarrow & X=\frac{n!}{r!(n-r)!}={ }^{n}{C}_{r} \end{array} $$$n\in N$$n\cdot^{… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 09:00:24 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-3-p2 Question 1(vi-x), Exercise 6.3 Solutions of Question 1(vi-x) of Exercise 6.3 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n\in N$$\quad^{2n}C_n=\dfrac{2^n[1.3.5.\cdots(2n-1)]}{n!}$\begin{align*}L.H.S &=\quad^{2n}C_n \\ &=\dfrac{(2 n)!}{n!(2 n-n)!}\\ &=\dfrac{(2 n)(2 n-1)(2 n-2)(2 n-n)(2 n-(n+1)) ..2 .1}{n!\cdot n!}\\ &=\dfrac{(2 n)(2 n-1)(2 n-2) ..(… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 09:00:26 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-3-p3 Question 2, Exercise 6.3 Solutions of Question 2 of Exercise 6.3 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$\,\, ^nC_5=\,\, ^nC_8$\begin{align*}\dfrac{n!}{5!(n-5)!}=\dfrac{n!}{8!(n-8)!}&\\ \dfrac{1}{8!(n-5)(n-6)(n-7)(n-8)!}&\\ =\dfrac{1}{8 \times 7 \times 6 \times 8!(n-8)!}&\\ 336=(n-5)(n-6)(n-7)&\\ (n-5)\left(n^{2}-13 n+42\right)&=336\\ n^{3}-… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 09:00:26 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-3-p4 Question 3, Exercise 6.3 Solutions of Question 3 of Exercise 6.3 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $r$$\quad^{15}C_{3r}= ^{15}C_{3+r}$$${ }^{n} C_{r}={ }^{n} C_{n-r}$$\begin{align*}n=15\quad&\text{put} \quad r=3 r\\ n-3 r&=r+3\\ 15-3 r&=r+3 \\ 15-3 & =3 r+r \\ 12 & =4 r\\ r&=3\end{align*}$r$$\quad^{8}C_{r}-\,^7C_3= ^{7}C_{2}$\begin{align*}… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 09:00:27 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-3-p5 Question 4, Exercise 6.3 Solutions of Question 4 of Exercise 6.3 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$r$$\,\,^nC_{r-1}:\,^nC_{r}:\,^nC_{r+1}=6:14:21$\begin{align*}\dfrac{n!}{(r-1)!(n-(r-1))!}&: \dfrac{n!}{r!(n-r)!}\\ : \dfrac{n!}{(r+1)!(n-(r+1))!} &= 6:14:21\\ \dfrac{1}{(r-1)!(n-r+1)!}: \dfrac{1}{r!(n-r)!}&\\ : \dfrac{1}{(r+1)!(n-r-1)!}&=… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 09:00:27 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-3-p6 Question 5 and 6, Exercise 6.3 Solutions of Question 5 and 6 of Exercise 6.3 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $11$$16$$11$$16$${ }^{16} C_{11}$$4368$$11$$11$$16$$10$$15$$$ { }^{15} C_{10}=3003 $$$5$$3$$3$$1$${ }^{5} C_{1}$$2$${ }^{3} C_{2}$$={ }^{5} C_{1} \times{ }^{3} C_{2}=5 \times 3=15$$2$$={ }^{5} C_{2} \times{ }^{3} C_{1}=10 \times 3… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 09:00:29 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-3-p7 Question 7 and 8, Exercise 6.3 Solutions of Question 7 and 8 of Exercise 6.3 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $5$$6$$4$$2$$2$$3$$={ }^{4} C_{2} \times{ }^{6} C_{3}=6 \times 20=120$$5$$6$$4$$2$$2$$2$$2,3$$2$$120$$3$$2$$={ }^{4} C_{3} \cdot{ }^{6} C_{2}=4 \times 15=60$$4$$={ }^{4} C_{4} \times{ }^{6} C_{1}=1 \times 6=6$$=120+60+6=186$$5$$6$… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 09:00:29 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-3-p8 Question 9 and 10, Exercise 6.3 Solutions of Question 9 and 10 of Exercise 6.3 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$n$$n$$n$$C_{2}$$n$$n$$={ }^{n} C_{2}-n$$=\frac{n!}{2!(n-2)!}-n$$10$$10$$3$$7$$10$${ }^{10} C_{7}$$7$$3$$7={ }^{10} C_{7}=120$ anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 09:00:33 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-3-p9 Question 11 and 12, Exercise 6.3 Solutions of Question 11 and 12 of Exercise 6.3 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$35$$n$\begin{align*}{ }^{n} C_{2}-n&=35\\ \text{or} \quad \dfrac{n!}{2!(n-2)!}-n&=35\\ \dfrac{n(n-1)(n-2)!}{2(n-2)!}-n&=35\\ \dfrac{n(n-1)-2 n}{2}&=35\\ n^{2}-n-2 n&=70\\ n^{2}-3 n-70&=0\\ n^{2}+7 n-10 n-70 & =0 \\ n(n+7)-… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 09:00:31 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-3-p10 Question 13 and 14, Exercise 6.3 Solutions of Question 13 and 14 of Exercise 6.3 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $6$$1$$1$$2$$6$$1$$6={ }^{6} C_{1}=6$$2$$6={ }^{6} C_{2}=15$$3$$6={ }^{6} C_{3}=20$$4$$6={ }^{6} C_{4}=15$$5$$6={ }^{6} C_{5}=6$$6$$6={ }^{6} C_{6}=1$$$\text{Total}\quad =6+15+20+15+6+1=63$$$A$$B$$C$$8$$5$$A$$3$$B$$C$$5$$8$$A$… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 09:00:22 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/re-ex6-p1 Question 1, Review Exercise 6 Solutions of Question 1 of Review Exercise 6 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $3\,\,^nP_3=^nP_4$$n$$5$$6$$7$$8$$6$$ 480$$600$$720$$840$$720$$r$$r!$$(r+1)!$$r!+1$$ 2r!$$r!$$6$$\dfrac{5}{2}\,\,6!$$6!$$\dfrac{1}{2}\,\,6!$$\dfrac{3}{2}\,\,6!$$\dfrac{5}{2}\,\,6!$$A=\{1,2,3,4,...,20\}. $$3$$5$$7$$8$$8$$A=\{1,3,5,7,… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 09:00:36 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/re-ex6-p2 Question 2 and 3, Review Exercise 6 Solutions of Question 2 and 3 of Review Exercise 6 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $4$$26$$$^{26}P_4=358800$$$3$$0$$3$$100's$$9$$3$$$=10\times 10\times10=1000$$$0$$10$$3-$$0$$$=10\times 10=100$$ anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 09:00:43 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/re-ex6-p3 Question 4, 5 and 6, Review Exercise 6 Solutions of Question 4, 5 and 6 of Review Exercise 6 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $0,2,3,4,5,7$$6$$$=6!=720$$$0$$5$$$5!=120$$$6-$$$=720-120=600$$$7$$$(7-1)!=720$$$2$$$\dfrac{720}{2}=360$$$11!$$10!$$$=11!-10!=36288000$$ anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 09:00:38 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p1 Question 1, Exercise 8.1 Solutions of Question 1 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\cos (\alpha \pm \beta), \sin (\alpha \pm \beta)$$\tan (\alpha \pm \beta)$$\alpha=180^{\circ}, \beta=60^{\circ}$$\alpha=180^{\circ}$$\beta=60^{\circ}$\begin{align*} \cos (\alpha + \beta) & = \cos \alpha \cos \beta - \sin \alpha \sin \beta \… anonymous@undisclosed.example.com (Anonymous) Sun, 20 Oct 2024 17:53:30 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p2 Question 2, Exercise 8.1 Solutions of Question 2 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\cos 15^{\circ}$$\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… anonymous@undisclosed.example.com (Anonymous) Sun, 20 Oct 2024 17:53:35 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p3 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{… anonymous@undisclosed.example.com (Anonymous) Sun, 20 Oct 2024 17:53:37 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p4 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… anonymous@undisclosed.example.com (Anonymous) Sun, 20 Oct 2024 17:53:38 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p5 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$$\cos$\begin{alig… anonymous@undisclosed.example.com (Anonymous) Sun, 20 Oct 2024 17:53:39 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p6 Question 7, Exercise 8.1 Solutions of Question 7 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\alpha$$\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… anonymous@undisclosed.example.com (Anonymous) Sun, 20 Oct 2024 17:53:39 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p7 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}$$\alpha$\begin{align… anonymous@undisclosed.example.com (Anonymous) Sun, 20 Oct 2024 17:53:41 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p8 Question 9, Exercise 8.1 Solutions of Question 9 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\alpha$$\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*… anonymous@undisclosed.example.com (Anonymous) Sun, 20 Oct 2024 17:53:41 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p9 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 \\ & =… anonymous@undisclosed.example.com (Anonymous) Sun, 20 Oct 2024 17:53:42 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p10 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 \… anonymous@undisclosed.example.com (Anonymous) Sun, 20 Oct 2024 17:53:31 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p11 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}$$\begin{align*} & \alpha+\beta=180^{\circ}-\gamma \\ \implies & \tan(\alpha+\beta) = \tan(180^{\circ}-… anonymous@undisclosed.example.com (Anonymous) Sun, 20 Oct 2024 17:53:31 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p12 Question 13, Exercise 8.1 Solutions of Question 13 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $r \sin (\theta+\phi)$$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… anonymous@undisclosed.example.com (Anonymous) Sun, 20 Oct 2024 17:53:32 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-1-p13 Question 14, Exercise 8.1 Solutions of Question 14 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\theta$$\sin \theta$$\cos \theta$$\alpha$\begin{align*} &\tan\alpha = \frac{\overline{BC}}{\overline{AB}} \\ \implies &\tan\alpha = \frac{3}{3} = 1 \\ \implies &\alpha = \tan^{-1}(1) = 45^\circ \end{align*}$45^\circ$$\theta$\begin{align*} … anonymous@undisclosed.example.com (Anonymous) Sun, 20 Oct 2024 17:54:47 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p1 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}.$$\begin{align… anonymous@undisclosed.example.com (Anonymous) Sat, 26 Oct 2024 18:54:35 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p2 Question 4 Exercise 8.2 Solutions of Question 4 of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin 2 \theta$$\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… anonymous@undisclosed.example.com (Anonymous) Sat, 26 Oct 2024 18:54:40 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p3 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}\\… anonymous@undisclosed.example.com (Anonymous) Sat, 26 Oct 2024 18:54:42 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p4 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… anonymous@undisclosed.example.com (Anonymous) Sat, 26 Oct 2024 18:54:45 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p5 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… anonymous@undisclosed.example.com (Anonymous) Sat, 26 Oct 2024 18:54:47 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p6 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… anonymous@undisclosed.example.com (Anonymous) Sat, 26 Oct 2024 18:54:48 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p7 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 \… anonymous@undisclosed.example.com (Anonymous) Sat, 26 Oct 2024 18:54:50 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p8 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… anonymous@undisclosed.example.com (Anonymous) Sat, 26 Oct 2024 18:54:53 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p9 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+\… anonymous@undisclosed.example.com (Anonymous) Sat, 26 Oct 2024 18:54:54 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p10 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… anonymous@undisclosed.example.com (Anonymous) Sat, 26 Oct 2024 18:56:27 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p11 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… anonymous@undisclosed.example.com (Anonymous) Sat, 26 Oct 2024 18:56:02 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-2-p12 Question 8(xix, xx, xxi & xxii) Exercise 8.2 Solutions of Question 8(xix, xx, xxi & xxii) of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$\frac{\sin 2 \alpha}{\sin \alpha}-\frac{\cos 2 \alpha}{\cos \alpha}=\sec \alpha$$\begin{align*} LHS &= \dfrac{\sin 2 \alpha}{\sin \alpha}-\frac{\cos 2 \alpha}{\cos \alpha}\\ &= \dfrac{\sin 2 \alpha … anonymous@undisclosed.example.com (Anonymous) Sat, 26 Oct 2024 18:54:46 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-3-p1 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 \… anonymous@undisclosed.example.com (Anonymous) Mon, 28 Oct 2024 17:14:30 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-3-p2 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… anonymous@undisclosed.example.com (Anonymous) Mon, 28 Oct 2024 17:14:29 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-3-p3 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 … anonymous@undisclosed.example.com (Anonymous) Mon, 28 Oct 2024 17:14:30 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-3-p4 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… anonymous@undisclosed.example.com (Anonymous) Mon, 28 Oct 2024 17:14:31 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-3-p5 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} \\ & … anonymous@undisclosed.example.com (Anonymous) Mon, 28 Oct 2024 17:14:31 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-3-p6 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 (… anonymous@undisclosed.example.com (Anonymous) Mon, 28 Oct 2024 17:14:32 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-3-p7 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 \\ & =… anonymous@undisclosed.example.com (Anonymous) Mon, 28 Oct 2024 17:14:32 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/ex8-3-p8 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… anonymous@undisclosed.example.com (Anonymous) Mon, 28 Oct 2024 17:14:33 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/re-ex-p1 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}$$\frac{\sqrt{6}-\sqrt{2}}{4}$$\tan \left(\frac{\pi}{6}+\frac{\pi}{4}\rig… anonymous@undisclosed.example.com (Anonymous) Sat, 09 Nov 2024 17:48:41 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/re-ex-p2 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$$\phi$\begin{align*} \cos^2 \theta &= 1-\sin^2\theta\\ &= 1-\left(\frac{3}{5}\right)^2\\ & =… anonymous@undisclosed.example.com (Anonymous) Sat, 09 Nov 2024 18:31:57 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/re-ex-p3 Question 3, Review Exercise Solutions of Question 3 of Review Exercise of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\frac{1}{\sqrt{2}}(\sin \beta+\cos \beta)$\begin{align*} &\frac{1}{\sqrt{2}}(\sin \beta+\cos \beta)\\ =& \sin \frac{\pi}{4}\sin \beta+\cos \frac{\pi}{4}\cos \beta\\ =& \cos(\beta -\frac{\pi}{4}) \end{align*}$\frac{1}{\sqrt{2}} \sin 75^… anonymous@undisclosed.example.com (Anonymous) Sat, 09 Nov 2024 18:32:33 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/re-ex-p4 Question 4, Review Exercise Solutions of Question 4 of Review Exercise of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\dfrac{1+\tan 15^{\circ}}{1-\tan 15^{\circ}}$\begin{align*} &\frac{1+\tan 15^{\circ}}{1-\tan 15^{\circ}}\\ =&\frac{1+\tan 15^{\circ}}{1-1 \cdot \tan 15^{\circ}}\\ =&\frac{\tan 45^{\circ} + \tan 15^{\circ}}{1 - \tan 45^{\circ} \tan 15^{… anonymous@undisclosed.example.com (Anonymous) Sat, 09 Nov 2024 18:33:14 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/re-ex-p5 Question 5 and 6, Review Exercise Solutions of Question 5 and 6 of Review Exercise of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\tan \theta$$\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… anonymous@undisclosed.example.com (Anonymous) Sat, 09 Nov 2024 18:44:48 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/re-ex-p6 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… anonymous@undisclosed.example.com (Anonymous) Sat, 09 Nov 2024 18:47:22 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/re-ex-p7 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… anonymous@undisclosed.example.com (Anonymous) Sat, 09 Nov 2024 18:47:40 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/re-ex-p8 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… anonymous@undisclosed.example.com (Anonymous) Sat, 09 Nov 2024 18:49:13 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/re-ex-p9 Question 10, Review Exercise Solutions of Question 10 of Review Exercise of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin (16 x)=16 \sin (x) \cos (x) \cos (2 x) \cos (4 x) \cos (8 x)$\begin{align*} RHS&=16 \sin (x) \cos (x) \cos (2 x) \cos (4 x) \cos (8 x) \\ &= 8(2 \sin (x) \cos (x) )\cos (2 x) \cos (4 x) \cos (8 x) \\ &= 8 \sin2 (x) \cos (2 x) \… anonymous@undisclosed.example.com (Anonymous) Sat, 09 Nov 2024 18:51:06 +0000 https://www.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p1 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 \\ … anonymous@undisclosed.example.com (Anonymous) Wed, 27 Nov 2024 15:59:09 +0000 https://www.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p2 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… anonymous@undisclosed.example.com (Anonymous) Wed, 27 Nov 2024 15:59:10 +0000 https://www.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p3 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} \… anonymous@undisclosed.example.com (Anonymous) Wed, 27 Nov 2024 15:59:11 +0000 https://www.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p4 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 … anonymous@undisclosed.example.com (Anonymous) Wed, 27 Nov 2024 15:59:11 +0000 https://www.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p5 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 +… anonymous@undisclosed.example.com (Anonymous) Wed, 27 Nov 2024 15:59:12 +0000 https://www.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p6 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$$\mathrm{y}=\operatorname{Cos} \frac{\mathrm{x}}{2}$ anonymous@undisclosed.example.com (Anonymous) Wed, 27 Nov 2024 15:59:13 +0000 https://www.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p7 Question 5(vi-x), Exercise 9.1 Solutions of Question 5(vi-x) of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y=2 \operatorname{Sin} 3 x$$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}$ anonymous@undisclosed.example.com (Anonymous) Wed, 27 Nov 2024 15:59:14 +0000 https://www.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p8 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… anonymous@undisclosed.example.com (Anonymous) Wed, 27 Nov 2024 15:59:15 +0000 https://www.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p9 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$$[0,2 \pi]$ anonymous@undisclosed.example.com (Anonymous) Wed, 27 Nov 2024 15:59:15 +0000 https://www.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p10 Question 9, Exercise 9.1 Solutions of Question 9 of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin x=\cos x$$\cos x=x$$\sin x=x$$\tan x=x$ anonymous@undisclosed.example.com (Anonymous) Wed, 27 Nov 2024 15:59:09 +0000 https://www.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p11 Question 10, Exercise 9.1 Solutions of Question 10 of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $ V(t)=a \operatorname{Sin}(k(t-d))+c$$56 \mathrm{~Hz} A C$$k$ anonymous@undisclosed.example.com (Anonymous) Wed, 27 Nov 2024 15:59:10 +0000 https://www.mathcity.org/math-11-nbf/sol/unit09/re-ex-1-p2 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. anonymous@undisclosed.example.com (Anonymous) Wed, 27 Nov 2024 15:59:17 +0000 https://www.mathcity.org/math-11-nbf/sol/unit09/re-ex-1-p3 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. anonymous@undisclosed.example.com (Anonymous) Wed, 27 Nov 2024 15:59:18 +0000 https://www.mathcity.org/math-11-nbf/sol/unit09/re-ex-p1 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}$$-\frac{2}{\sqrt{3}}$$-\frac{1}{2}$$\tan (-15 \pi)=$$ 0$$-1$$1$$0$$2 \sin \theta+\frac{1}{2}cosec \theta \theta $$\theta=45^{\circ}$$\frac{1}{\sqrt{2}}$$\frac… anonymous@undisclosed.example.com (Anonymous) Wed, 27 Nov 2024 15:59:18 +0000 https://www.mathcity.org/math-11-nbf/sol/unit09/re-ex-p2 Question 2 and 3, Review Exercise Solutions of Question 2 and 3 of Review Exercise of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\cos \theta -\sin \theta=\sqrt{2}\sin \theta,$$\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… anonymous@undisclosed.example.com (Anonymous) Wed, 27 Nov 2024 16:04:15 +0000 https://www.mathcity.org/math-11-nbf/sol/unit09/re-ex-p3 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. anonymous@undisclosed.example.com (Anonymous) Wed, 27 Nov 2024 15:59:20 +0000 https://www.mathcity.org/math-11-nbf/sol/unit09/re-ex-p4 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. anonymous@undisclosed.example.com (Anonymous) Wed, 27 Nov 2024 15:59:22 +0000 https://www.mathcity.org/math-11-nbf/sol/unit09/re-ex-p5 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. anonymous@undisclosed.example.com (Anonymous) Wed, 27 Nov 2024 15:59:22 +0000 https://www.mathcity.org/math-11-nbf/sol/unit09/re-ex-p6 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. anonymous@undisclosed.example.com (Anonymous) Wed, 27 Nov 2024 15:59:23 +0000 https://www.mathcity.org/math-11-nbf/sol/unit09/re-ex-p7 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. anonymous@undisclosed.example.com (Anonymous) Wed, 27 Nov 2024 15:59:23 +0000 https://www.mathcity.org/math-11-nbf/sol/unit09/re-ex-p8 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. anonymous@undisclosed.example.com (Anonymous) Wed, 27 Nov 2024 15:59:24 +0000 https://www.mathcity.org/math-11-nbf/sol/unit09/re-ex-p9 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. anonymous@undisclosed.example.com (Anonymous) Wed, 27 Nov 2024 15:59:24 +0000 https://www.mathcity.org/math-11-nbf/sol/unit09/re-ex-p10 Question 10(xi-xv), Review Exercise Solutions of Question 10(xi-xv) of Review Exercise of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. anonymous@undisclosed.example.com (Anonymous) Wed, 27 Nov 2024 15:59:19 +0000