Merging man & maths https://www.mathcity.org/ Thu, 04 Jun 2026 22:28:15 +0000 FeedCreator 1.8 https://www.mathcity.org/_media/logo.svg https://www.mathcity.org/ https://www.mathcity.org/ppsc/ppsc-maths-2015 PPSC Paper 2015 (Lecturer in Mathematics) [PPSC Paper 2011 (Lecturer in Mathematics)] On this page, we have given question from old (past) paper of Lecturer in Mathematics conducted in year 2011. This is a MCQs paper and answers are given at the end of the paper. At the end of the PDF is also given to download. This paper is provided by Kaushef Salamat. We are very thankful to her for providing this paper.\(\displaystyle \int_{-4}^{0}\frac{tdt}{\sqrt{16-t62}}\)$0$$-4$$4$\(A\cos wt+B\sin wt\)$\… anonymous@undisclosed.example.com (Anonymous) Tue, 18 Jan 2022 10:20:00 +0000 https://www.mathcity.org/math-11-pectaa/sol FSC/ICS Part 1 Math Solution (11th Class) – Punjab Board Solved Exercises [FSC ICS Part 1 Math Solution (11th Class) – Punjab Board Solved Exercises] FSC or ICS Part 1 Math Solution (11th Class), Punjab Board Solved Exercises. Mathematics 11 (FSc or ICS Part 1, HSSC-I) is published by Punjab Education, Curriculum, Training and Assessment Authority (PECTAA) Lahore - Pakistan formally know as Punjab Textbook Board (PTB) anonymous@undisclosed.example.com (Anonymous) Sun, 24 May 2026 10:06:29 +0000 https://www.mathcity.org/math-9th-pctb/sol Solutions: Math 9th PCTB [Solutions of Mathematics 9th PCTB] Notes (Solutions) of Mathematics 9 (Mathematics for Matric), Punjab Curriculum and Textbook Board (PCTB) Lahore. Our aim here is to provides clear and step-by-step solutions for mathematical concepts, including real numbers, logarithms, algebra, trigonometry, and probability. we will try our best to complete the solutions as soon as possible for teachers and students. To download a book, please anonymous@undisclosed.example.com (Anonymous) Mon, 04 May 2026 16:21:21 +0000 https://www.mathcity.org/software Software On this page, we have listed document viewers, graphing tools, calculators and other useful software for Mathematics. As most of the visitors of our websites are using android mobiles, therefore first of all we are giving some apps for Android.$\LaTeX$$\LaTeX$$\LaTeX$$\LaTeX$ anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:39:44 +0000 https://www.mathcity.org/matric/9th_science Mathematics 9 (Science Group) [Mathematics 9 (Science Group)] Mathematics 9 is written by Dr. Karamat H. Dar and Prof. Irfan-ul-Haq and this book is published by Carvan Book House, Lahore, Pakistan. This book consist of 302 pages and there are 17 units. Notes of Unit 1 and 3 are provided by $ka + kb + kc$$ac + ad + bc + bd$$a^2 + 2ab + b^2$$a^2 – b^2$$a^2 + 2ab + b^2 – c^2$$a^4 + a^2b^2 + b^4$$a^4 + 4b^4$$x^2 + px + q$$ax^2 + bx + c$$(ax^2 + bx + c) (ax2 + bx + d) + k$$(x + a) (x + b) (x + c) … anonymous@undisclosed.example.com (Anonymous) Wed, 08 Mar 2023 18:04:36 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-1-p2 Question 2, Exercise 1.1 Solutions of Question 2 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 2(i) $x+iy$$(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-12-nbf/sol Solutions: Math 12 NBF [Solutions of Textbook of Mathematics 12] Solutions of “Textbook of Mathematics 12 published by National Book Foundation (NBF), Islamabad, Pakistan”. NBF can be considered as Federal Textbook Board Islamabad. This comprehensive guide, Solutions for Mathematics 12, serves as a definitive resource for students mastering the advanced HSSC curriculum. Published by the National Book Foundation (NBF), it bridges the gap between complex theory and practical application. The tex… anonymous@undisclosed.example.com (Anonymous) Mon, 04 May 2026 16:11:21 +0000 https://www.mathcity.org/wiki/wrap Examples for the Wrap Plugin Basic syntax An uppercase <WRAP> (or alternatively <block> or <div>) creates a div and should be used for “big” containers, surrounding paragraphs, lists, tables, etc. <WRAP classes width :language> "big" content </WRAP> or <block classes width :language> "big" content </block> or <div classes width :language> "big" content </div> anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:45:13 +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/matric/10th_science Mathematics 10 (Science Group) [Matric Science 10th Book Cover] The notes/solutions, definitions, MCQs and important question for Mathematics 10 (Science Group), published by Ilmi Kitab Khana, Urdu Bazar, Lahore, Pakistan are available on this page. Whenever we found the notes we will update this page and will upload notes here. If you wish to contribute and send us the notes please contact us via our $(b^2-4ac)$$ax^2+bx+c$$\mathbb{N}$$\mathbb{W}$$\mathbb{Z}$$E$$O$$P$$\mathbb{Q}$$\cup$$\cap$$\s… anonymous@undisclosed.example.com (Anonymous) Wed, 24 Jul 2024 18:33:10 +0000 https://www.mathcity.org/wiki/syntax Formatting Syntax DokuWiki supports some simple markup language, which tries to make the datafiles to be as readable as possible. This page contains all possible syntax you may use when editing the pages. Simply have a look at the source of this page by pressing anonymous@undisclosed.example.com (Anonymous) Sat, 10 Feb 2024 11:07:05 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p8 Question 7, Exercise 1.4 Solutions of Question 7 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 7(i) $\arg (z-1)=-\dfrac{\pi}{4}$$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/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]$$D=\left[\begin{array}{lll}1 & 6 & 9 \\ 2 & 0 … anonymous@undisclosed.example.com (Anonymous) Sat, 27 Jul 2024 09:36:13 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-1-p4 Question 4, Exercise 2.1 Solutions of Question 4 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$ A=\left[\begin{array}{ccc} 2 & 0 \\ \sqrt{5} & 6 \\ 1 & 9 \end{array}\right]$$$$ A^t=\begin{bmatrix} 2 & \sqrt{5} & 1 \\ 0 & 6 & 9 \end{bmatrix}$$$$B=\left[\begin{array}{cccc} 1 & 6 & 2 & 0 \end{array}\right] $$$$B^t=\left[\begin{array}{c} 1… anonymous@undisclosed.example.com (Anonymous) Sat, 27 Jul 2024 09:36:55 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-1 Exercise 6.1 (Solutions) The solutions of the Exercise 6.1 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to factorial function.$n!$$$ 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/matric/9th_science/unit_02/exercise_2.1 Exercise 2.1 (Solutions) Question 1 Identify which of the following are rational and irrational numbers: (i) $\sqrt{3}$ (ii) $\frac{1}{6}$ (iii) $\pi$ (iv) $\frac{15}{2}$ (v) $7.25$ (vi)$\sqrt{29}$ Solution * Rational: $\frac{1}{6}$, $\frac{15}{2}$, $7.25$ * Irrational: $\sqrt{3}$, $\pi$, $\sqrt{29}$ Question 2 Convert the following fraction into decimal fraction.$\frac{17}{25}$$\frac{19}{4}$$\frac{57}{8}$$\frac{205}{18}$$\frac{5}{8}$$\frac{25}{38}$$\frac{2}{3}$$\pi$$\frac{1}{9}$$\fr… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 17:00:36 +0000 https://www.mathcity.org/matric/9th_science/ex-6-1 Exercise 6.1 On the following page we have given the solution of Exercise 6.1 of Mathematics 9 (Science) published by Caravan Book House, Lahore. We have created this page and it will be updated to add new solutions occasionally. Please stay in touch with this page.$39x^7y^3z$$91x^5y^6 z^7$$102xy^2z$$85x^2yz$$187xyz^2$$39x^7y^3z=13\times 3\times x^7 y^3 z$$91x^5y^6 z^7=13\times 7\times x^5 y^6 z^7$$13 x^5y^3z$$102xy^2z=2\times 3\times 17 xy^2z$$85x^2yz=3\times 17 x^2 y z$$187xyz^2 = 11\times 1… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:48:08 +0000 https://www.mathcity.org/msc/syllabus/pu Syllabus for PU Syllabus and scheme of studies for Regular/Private students doing MSc Mathematics from University of the Punjab, Lahore. 2 years M.Sc Mathematics programme consists of two parts namely Part-I and Part II. The regulation, Syllabi and Courses of Reading for the M.Sc. (Mathematics) Part-I and Part-II (Regular Scheme) are given below. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:50:16 +0000 https://www.mathcity.org/fsc-part1-ptb/sol/ch01/ex1-1 Exercise 1.1 (Solutions) Notes (Solutions) of Exercise 1.1: Textbook of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board (PTB) Lahore. The main topics of this exercise are properties of real numbers, binary operation, addition and multiplication law, properties of equality, properties of inequality (order properties), field, rule of fractions. These notes are based on the new Student Learning Outcomes (SLOs). Version: 4.0, Available at MathCity.org $\{0… anonymous@undisclosed.example.com (Anonymous) Thu, 13 Apr 2023 09:34:48 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-1 Exercise 1.1 (Solutions) The solutions of the Exercise 1.1 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to sum, product and division of the complex numbers.${{i}^{31}}$${{\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)$$(4+7i… anonymous@undisclosed.example.com (Anonymous) Mon, 27 Oct 2025 18:50:29 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-3 Exercise 2.3 (Solutions) The solutions of the Exercise 2.3 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to determinant and inverse of the matrix.$\left[\begin{array}{ccc}2 & 3 & 1 \\ 1 & -1 & 2 \\ 4 & 1 & 2\end{array}\right]$$\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/mathcraft/sample-01-latex MathCraft: PDF to LaTeX file: Sample-01 If the PDF file provided by you as follows: Then the output LaTeX file is as follows: \documentclass[10pt]{amsart} %\usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} %\usepackage[version=4]{mhchem} \usepackage{stmaryrd} \usepackage{bbold} \usepackage{hyperref} \usepackage{enumerate} \hypersetup{colorlinks=true, linkcolor=blue, filecolor=magenta, urlcolor=cyan,} \urlstyle{same} \title{… anonymous@undisclosed.example.com (Anonymous) Mon, 25 Mar 2024 16:29:59 +0000 https://www.mathcity.org/ppsc/ppsc-maths-2011 PPSC Paper 2011 (Lecturer in Mathematics) [PPSC Paper 2011 (Lecturer in Mathematics)] On this page, we have given question from old (past) paper of Lecturer in Mathematics conducted in year 2011. This is a MCQs paper and answers are given at the end of the paper. At the end of the PDF is also given to download. This paper is provided by Ms. $R$$x\in R$$x^2=x$$x^2=-x$$x^2=0$$x^2=1$$6$$8$$10$$4$$G$$H$$H$$G$$2$$4$$nZ$$Z$$n$$G$$24$$a$$a^{10}$$2$$12$$10$$V$$n$$V$$n+1$$n$$n-1$$v_1,v_2,v_3,....,v_r$$… anonymous@undisclosed.example.com (Anonymous) Thu, 03 Feb 2022 11:54:32 +0000 https://www.mathcity.org/fsc-part1-ptb/important-questions/ch05-partial-fractions Ch 05: Partial Fraction * Resolve $\frac{1}{(x^2+1)(x+1)}$ into partial fraction --- BISE Gujrawala(2015) * Resolve the following into partial fractions $\frac{2x^4}{(x-3)(x+2)^2}$ --- BISE Gujrawala(2017) * Resolve $\frac{x^2+1}{(x+1)(x-1)}$ into partial fraction --- BISE Sargodha(2015),BISE Sargodha(2017) * Resolve $\frac{9}{(x+2)^2(x-1)}$$\frac{1}{(x-1)^2+(x+1)}$$\frac{x^2+1}{(x^3+1)}$$\frac{1}{(x-1)^2(x^2+2)}$$\frac{1}{x^2-1}$$\frac{x^2}{(x-2)(x-1)^2}$$\frac{3x-1}{(x^2+1)(x+3)}… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:40 +0000 https://www.mathcity.org/fsc-part2-ptb/important-questions/unit-05-linear-inequalities-and-linear-programming Unit 05: Linear Inequalities and Linear Programming Here is the list of important questions. * Graph the solution region of $2x+y \geq 2$ --- BSIC Gujranwala (2016) * Graph the feasible region subject to the following constraint: --- BSIC Gujranwala (2016)$2x-3y \leq 6$$2x+3y \leq 12$$x \geq 0$$y \geq 0$$2x+y\geq 2$$x+2y\leq10$$x\geq0,y\geq0$$2x+3y\leq 12$$z=x+3y$$2x+5y\leq30$$5x+4y\leq20$$x\geq0$$y\geq0$$x+2y\leq 14$$3x+4y\leq 36$$2x+y\leq 10$$x\geq0, y\geq0$$f(x)=2x+5y$$-x\leq8$$-y\leq… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:57 +0000 https://www.mathcity.org/matric/9th_science/ex-4-1 Exercise 4.1 On the following page we have given the solution of Exercise 4.1 of Mathematics 9 (Science) published by Caravan Book House, Lahore. We have created this page and it will be updated to add new solutions occasionally. Please stay in touch with this page.$3x^2+\frac{1}{x}-5$$3x^3-4x^2-x\sqrt{x}+3$$x^2-3x+\sqrt{2}$$\frac{3x}{2x-1}+8$$3x^2+\frac{1}{x}-5$$No (Reason:\frac{1}{x})$$3x^3-4x^2-x\sqrt{x}+3$$No (Reasons \sqrt{x})$$x^2-3x+\sqrt{2}$$Yes$$\frac{3x}{2x-1}+8$$No (Reason:\frac{1}… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:48:06 +0000 https://www.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-1-p3 Question 3, Exercise 10.1 Solutions of Question 3 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin u=\dfrac{3}{5}$$\sin v=\dfrac{4}{5}$$u$$v$$0$$\dfrac{\pi }{2}$$\cos \left( u+v \right)$$\sin u=\dfrac{3}{5},$$0\le u\le \dfrac{\pi }{2}.$$\sin v=\dfrac{4}{5},$$0\le v\le \dfrac{\pi }{2}.$$\cos u=\pm \sqrt{1-{{\sin }^… anonymous@undisclosed.example.com (Anonymous) Thu, 24 Aug 2023 17:22:13 +0000 https://www.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-1-p11 Question 13, Exercise 10.1 Solutions of Question 13 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$r\,\,\sin \left( \theta +\phi \right)$$\theta$$\phi$$4\sin \theta +3\cos \theta .$$4\sin \theta +3\cos \theta$$r\sin(\theta + \varphi)$$$4\sin \theta +3\cos \theta=r\cos\varphi\sin\theta+r\sin\varphi\cos\theta --- (1)$… anonymous@undisclosed.example.com (Anonymous) Fri, 25 Aug 2023 02:54:27 +0000 https://www.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-2-p2 Question 2, Exercise 10.2 Solutions of Question 2 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin \theta =\dfrac{5}{13}$$\theta $$\sin 2\theta $$\sin \theta =\dfrac{5}{13}$$$\cos \theta =\pm \sqrt{1-{{\sin }^{2}}\theta }.$$$\theta$$\cos$\begin{align}\cos\theta &=-\sqrt{1-{{\sin }^{2}}\theta }\\ &=-\sqrt{1-\left(\… anonymous@undisclosed.example.com (Anonymous) Fri, 01 Sep 2023 18:35:56 +0000 https://www.mathcity.org/math-11-kpk/sol/unit04/ex4-1-p3 Question 5 Exercise 4.1 Solutions of Question 5 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) $\sum_{j=1}^6(2 j-3)$\begin{align}\sum_{j=1}^6(2 j-3)&=(2.1-3)+(2.2-3)+(2.3-3)+(2.4-3)\\&+(2.5-3)+(2.6-3) \\ \implies \sum_{j=1}^6(2 j-3)&=-1+1+3+5+7+9 .\end{align}$\sum_{k=1}^5(-1)^k 2^{k-1}$\begin{align}\sum_{k=1}^5(-1)^k 2^{k-1}& =(-1)^1 2^{1-… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 18:33:10 +0000 https://www.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p3 Question 3, Exercise 10.1 Solutions of Question 3 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin u=\dfrac{3}{5}$$\sin v=\dfrac{4}{5}$$u$$v$$0$$\dfrac{\pi }{2}$$\cos \left( u+v \right)$$\sin u=\dfrac{3}{5},$$0\le u\le \dfrac{\pi }{2}.$$\sin v=\dfrac{4}{5},$$0\le v\le \dfrac{\pi }{2}.$$\cos u=\pm \sqrt{1-{{\sin }^… anonymous@undisclosed.example.com (Anonymous) Tue, 05 Dec 2023 02:45:40 +0000 https://www.mathcity.org/math-11-kpk/sol/unit10/ex10-1-p11 Question 13, Exercise 10.1 Solutions of Question 13 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$r\,\,\sin \left( \theta +\phi \right)$$\theta$$\phi$$4\sin \theta +3\cos \theta .$$4\sin \theta +3\cos \theta$$r\sin(\theta + \varphi)$$$4\sin \theta +3\cos \theta=r\cos\varphi\sin\theta+r\sin\varphi\cos\theta --- (1)$… anonymous@undisclosed.example.com (Anonymous) Tue, 05 Dec 2023 02:45:37 +0000 https://www.mathcity.org/math-11-kpk/sol/unit10/ex10-2-p2 Question 2, Exercise 10.2 Solutions of Question 2 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin \theta =\dfrac{5}{13}$$\theta $$\sin 2\theta $$\sin \theta =\dfrac{5}{13}$$$\cos \theta =\pm \sqrt{1-{{\sin }^{2}}\theta }.$$$\theta$$\cos$\begin{align}\cos\theta &=-\sqrt{1-{{\sin }^{2}}\theta }\\ &=-\sqrt{1-\left(\… anonymous@undisclosed.example.com (Anonymous) Tue, 05 Dec 2023 02:45:52 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-1-p3 Question 3, Exercise 1.1 Solutions of Question 3 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 3(i) $\dfrac{(2+i)(3-2i)}{1+i}$\begin{align}&\dfrac{(2+i)(3-2i)}{1+i}\\ =&\dfrac{6-2i^2+3i-4i}{1+i}\\ =&\dfrac{8-i}{1+i}\\ =&\dfrac{8-i}{1+i}\times \dfrac{1-i}{1-i}\\ =&\dfrac{8+i^2-8i-i}{1^2-i^2}\\ =&\dfrac{7-9i}{2}\\ =&\dfrac{7}{2}-\dfrac{9}… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Jul 2024 12:24:16 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-1-p4 Question 4, Exercise 1.1 Solutions of Question 4 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 4(i) $x$$y$$(2+3i)x+(1+3i)y+2=0$\begin{align}&(2+3i)x+(1+3i)y+2=0\\ \implies &(2x+y+2)+(3x+3y)i=0.\end{align}\begin{align} 2x+y+2&=0 \quad \cdots(1)\\ 3x+3y&=0\quad \cdots (2) \end{align}\begin{align} &3x=-3y \\ x=-y \quad ... (3) \end{align}$… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Jul 2024 12:39:29 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p9 Question 9, Exercise 1.2 Solutions of Question 9 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 9(i) $(2+4 i)^{-1}$$z=2+4i$\begin{align} Re(2+4i)^{-1} & = Re(z^{-1}) = \dfrac{Re(z)}{|z|^2} \\ & =\dfrac{2}{2^2+4^2} = \dfrac{2}{20}\\ &= \dfrac{1}{10}. \end{align}\begin{align} Im(2+4i)^{-1} & = Im(z^{-1}) = -\dfrac{Im(z)}{|z|^2} \\ & =-\df… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Jul 2024 19:38:59 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-3 Exercise 1.3 (Solutions) The solutions of the Exercise 1.3 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to sum, product and division of the complex numbers.$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/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/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/fsc/fsc_part_1_solutions/ch04 Chapter 04: Quadratic Equations [Chapter 04: Quadratic Equations] Notes (Solutions) of Chapter 04: Quadratic Equations, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board, Lahore. Contents & summary * Introduction * Solutions of Quadratic Equations anonymous@undisclosed.example.com (Anonymous) Sun, 04 Jun 2023 16:13:15 +0000 https://www.mathcity.org/matric/9th_science/review_exercise Review exercise On the following page we have given the solution of Review exercise of Mathematics 9 (Science) published by Caravan Book House, Lahore. We have created this page and it will be updated to add new solutions occasionally. Please stay in touch with this page.$x-2$$x^2+x-6$$x^2+x-6$$x+3$$x-2$$x+2$$c$$a^3+b^3$$a^2-ab+b^2$$a+b$$a^2-ab+b^2$$(a-b)^2$$a^2+b^2$$c$$x^2-5x+6$$x^2-x-6$$x-3$$x+2$$x^2-4$$x-2$$a$$a^2-b^2$$a^3-b^3$$a-b$$a+b$$a^2+ab+b^2$$a^2-ab+b^2$$a$$x^2+3x+2$$x^2+4x+3$$x^2+5x… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:48:10 +0000 https://www.mathcity.org/math-11-kpk/sol/unit03/ex3-5-p6 Question 7 Exercise 3.5 Solutions of Question 7 of Exercise 3.5 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7(i) For what value of $c$$\vec{u}=\hat{i}+2 \hat{j}+3 \hat{k}$$\vec{v}=2 \hat{i}-3 \hat{j}+4 \hat{k} \cdot \vec{w}=3 \hat{i}+\hat{j}+c \hat{k}$\begin{align}\vec{u} \cdot \vec{v} \times \vec{w}&=0\\ \vec{u} \cdot \vec{v} \times \vec{w}&=0\\ \Rightarrow\left|\beg… anonymous@undisclosed.example.com (Anonymous) Wed, 03 Jan 2024 18:10:52 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-2 Exercise 1.2 (Solutions) The solutions of the Exercise 1.2 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to real and imaginary part of complex numbers, modulus and conjugate of the complex numbers.$\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/ex1-4-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}$$z=x+iy=2 + i 2 \sqrt{3}$\begin{align} r & = \sqrt{x^2 + y^2} = \sqrt{2^2 + (2\sqrt{3})^2} \\ & = \sqrt{4 + 12} = \sqrt{16} = 4. \end{align}\begin{align} \alpha & = \tan^{-1}\left|\frac{y}{x}\right| = \tan^{-1}\left|\fra… anonymous@undisclosed.example.com (Anonymous) Mon, 15 Jul 2024 12:24:40 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-1 Exercise 2.1 (Solutions) The solutions of the Exercise 2.1 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to order, different type of matrices and transpose of matrix.$A=\left[\begin{array}{lll}1 & 3 & 0 \\ 2 & 0 & 1\end{array}\right]$$B=\left[\begin{array}{ll}1 & 2 \\ 2 & 3 \\ 3 & 4\end{array}\right]$$C=\left[\begin{array}{l}1 \… anonymous@undisclosed.example.com (Anonymous) Sun, 08 Feb 2026 16:35:15 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-3-p5 Question 5, Exercise 2.3 Solutions of Question 5 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left[\begin{array}{ccc}1 & -1 & 1 \\ 2 & 1 & -1 \\ 1 & -2 & -1\end{array}\right]$\begin{align*} A &= \left[\begin{array}{ccc} 1 & -1 & 1 \\ 2 & 1 & -1 \\ 1 & -2 & -1 \end{array}\right]\\ |A|&= 1 [-1 - 2] + 1 [-2 + 1] + 1 [-4 - 1] \\ &= 1 \cd… anonymous@undisclosed.example.com (Anonymous) Fri, 23 Aug 2024 09:05:37 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-3-p7 Question 7, Exercise 2.3 Solutions of Question 7 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $(A B)^{-1}=B^{-1} A^{-1}$$A=\left[\begin{array}{ll}2 & 1 \\ 8 & 6\end{array}\right]$$B=\left[\begin{array}{ll}3 & 2 \\ 0 & 2\end{array}\right]$\begin{align*} A &= \left[\begin{array}{ll}2 & 1 \\ 8 & 6\end{array}\right] \\ |A|& = 12 - 8 = 4\\ … anonymous@undisclosed.example.com (Anonymous) Fri, 23 Aug 2024 09:48:00 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-5 Exercise 2.5 (Solutions) The solutions of the Exercise 2.5 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to solving system of the equation of three variables by using matrices.$\left[\begin{array}{ccc}1 & 3 & 5 \\ -6 & 8 & 3 \\ -4 & 6 & 5\end{array}\right]$$\left[\begin{array}{ll}2 & 1 \\ 3 & 2 \\ 1 & 9\end{array}\right]$$\left[… anonymous@undisclosed.example.com (Anonymous) Sun, 08 Feb 2026 16:50:14 +0000 https://www.mathcity.org/matric/9th_science/unit_02/exercise_2.2 Exercise 2.2 (Solutions) Question 1 Identify the property used in the following, * (i) $a + b = b + a$ ... ..... * (ii) $(ab)c = a(bc)$ ... ... ... * (iii) $7 \times 1 = 7$ ... ... ... * (iv) $x > y$ or $x = y$ or $x< y$ ... ... ... * (v) $ab = ba$ ... ... ... * (vi) $a + c = b + c \Rightarrow a = b$ ... ... ... * (vii) $5 + (-5) = 0$ ... ... ... * (viii) $7 \times \frac{1}{7} = 1$$a > b \Rightarrow ac > bc? (c >0)$$a + b = b + a$$(ab)c = a(bc)$$7 \times 1 = 7$$x > y$$x = y$$… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 17:00:36 +0000 https://www.mathcity.org/matric/9th_science/unit_02/exercise_2.6 Exercise 2.6 (Solutions) Question 1 Identify the following statements as true or false. (i) $\sqrt{-3}\cdot\sqrt{-3} = 3$ (ii) $i^{73}=-i$ (iii) $i^{10} = -1$ (iv) Complex conjugate of $(-6i + i^2) is (-1 + 6i)$ (v) Difference of complex numbers $z = a + ib$ and its conjugate is a real number. (vi) If $(a-1)-(b+3)i = 5+8i$, then a = 6 & b = -11 (vii) Product of complex number and its conjugate is always a non-negative real number.$a+ib$$(2+3i)+(7-2i)$$2(5+4i)+3(7-4i)$$-(-3+5i)-3(4+9i)$$… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 17:00:37 +0000 https://www.mathcity.org/cui Mathematics CUI: LaTeX Resources [Department of Mathematics, COMSATS University Islamabad, Attock Campus] This page contains LaTeX template of CIIT Mathematics, MSc Project and MS Thesis templates. Templates Download a zip file given below and extract it by right clicking on the file. BS Project Template: (Version 1.5, Uploaded: Sep 29, 2022)$\$$I$$\mathbb{R}$$f:I\to \mathbb{R}$$(\$$$\sin^2 \theta + \cos^2 \theta =1$$\begin{equation} \sin^2 \theta + \cos^2 \theta =1 \end{equation} anonymous@undisclosed.example.com (Anonymous) Fri, 02 Aug 2024 07:19:37 +0000 https://www.mathcity.org/junaid Dr. Junaid Alam Khan This is a personal web page of Dr. Junaid Alam Khan Associate Professor Institute of Business Administration, Karachi - PAKISTAN. IBA Profile: <https://oric.iba.edu.pk/profile.php?id=jakhan> ResearchGate Profile: <https://www.researchgate.net/scientific-contributions/2141923316_Junaid_Alam_Khan> Facebook: <https://www.facebook.com/junaid.a.khan.58511> [Dr Junaid Alam Khan] Procedure to Compute Sasbi-Standard Bases anonymous@undisclosed.example.com (Anonymous) Thu, 23 Feb 2023 17:55:59 +0000 https://www.mathcity.org/mathematician_of_the_day Mathematician of the day The complete list of the mathematician who were born or died today is given at the following URL * <https://mathshistory.st-andrews.ac.uk/OfTheDay/today/> It is worth mentioning that “The MacTutor History of Mathematics archive” is a database of famous mathematician and mathematics archive created by by John J O'Connor and Edmund F Robertson is available at the following anonymous@undisclosed.example.com (Anonymous) Mon, 11 Mar 2024 05:54:30 +0000 https://www.mathcity.org/atiq/s625-mth251 MTH251: Set Topology (Spring 25) [MTH251 Set Topology] Set topology is a branch of mathematics that studies the properties of shapes and spaces that remain unchanged even if they are stretched, twisted, or deformed (without tearing or gluing). It helps us understand concepts like continuity, connectedness, and boundaries.$\mathbb{R}$$T_1$$\mathbb{Z}$$A=\{1,2,3,...,20\}$$\mathbb{R}$$\mathbb{Q}$$\mathbb{R}$$A=\left\{1,\frac{1}{2},\frac{1}{3},... \right\}$$A$$\mathbb{R}$$A=\mathbb{N}$$B=\{1,2,3,.… anonymous@undisclosed.example.com (Anonymous) Sun, 01 Feb 2026 14:28:11 +0000 https://www.mathcity.org/atiq/sp14-mth231 MTH231: Linear Algebra Introduction Linear algebra is the branch of mathematics deals with algebraic equations, spaces (vector and scalar), linear mappings between such spaces etc. Combined with the theory of calculus, linear algebra ensures to have methodologies to compute the solutions of system of equations (algebraic and differential). Techniques from linear algebra are also used in analytically geometry, engineering, physics, natural sciences and computer sciences and particularly in econ… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:27 +0000 https://www.mathcity.org/atiq/sp18-mth251 MTH251: Set Topology (Spring 18) [Set Topology] Topology is an important branch of mathematics that studies all the “qualitative” or “discrete” properties of continuous objects such as manifolds, i.e. all the properties that aren't changed by any continuous transformations except for the singular (infinitely extreme) ones.$\mathbb{R}$$T_1$$\mathbb{Z}$$A=\{1,2,3,...,20\}$$\mathbb{R}$$\mathbb{Q}$$\mathbb{R}$$A=\left\{1,\frac{1}{2},\frac{1}{3},... \right\}$$A$$\mathbb{R}$$A=\mathbb{N}$$B=\{1,2,3,.… anonymous@undisclosed.example.com (Anonymous) Fri, 07 Feb 2025 11:25:49 +0000 https://www.mathcity.org/atiq/sp25-mth251 MTH251: Set Topology (Spring 25) [MTH251 Set Topology] Set topology is a branch of mathematics that studies the properties of shapes and spaces that remain unchanged even if they are stretched, twisted, or deformed (without tearing or gluing). It helps us understand concepts like continuity, connectedness, and boundaries.$\mathbb{R}$$T_1$$\mathbb{Z}$$A=\{1,2,3,...,20\}$$\mathbb{R}$$\mathbb{Q}$$\mathbb{R}$$A=\left\{1,\frac{1}{2},\frac{1}{3},... \right\}$$A$$\mathbb{R}$$A=\mathbb{N}$$B=\{1,2,3,.… anonymous@undisclosed.example.com (Anonymous) Tue, 29 Apr 2025 10:10:47 +0000 https://www.mathcity.org/bsc/notes_of_calculus_with_analytic_geometry/ch07_plane_curves_ii Chapter 07: Plane Curves II Notes of the book Calculus with Analytic Geometry written by Dr. S. M. Yusuf and Prof. Muhammad Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN. [Asymptote] Contents and summary * Asymptotes: A straight line $l$ is called an asymptote for a curve $C$$l$$C$$l$$l$ anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:45:32 +0000 https://www.mathcity.org/fsc-part1-ptb/important-questions/ch03-matrices-and-determinants Ch 03: Matrices and Determinants * Fin $x$ and $y$ if $ \left[ {\begin{array}{c} x+3&1\\ -3& 3y-4 \end{array}} \right]= \left[ {\begin{array}{c} 2&1\\ -3&2 \end{array}} \right]$ --- BISE Gujrawala(2015) * Solve for matrix $A$ if $\left[ {\begin{array}{c}4&3\\ 2&2 \end{array}} \right]A-\left[ {\begin{array}{c} 2&3\\ -1&-2 \end{array}} \right]= \left[ {\begin{array}{c} -1&-4\\ 3&6 \end{array}} \right]$ --- BISE Gujrawala(2015) * Prove without expansion $ \left[ {\begin{array}{c} 6&7&… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:39 +0000 https://www.mathcity.org/fsc-part1-ptb/important-questions/ch08-mathematical-induction-and-binomial-theorem Ch 08: Mathematical Induction and Binomial Theorem * Using binomial theorem,expand $\left(\frac{x}{2}-\frac{2}{x^2}\right)$ --- BISE Gujranwala(2015) * Find the $6$th term in the expansion of $\left( x^2-\frac{3}{2x}\right)$ --- BISE Gujranwala(2015) * Expand $\left( 8-2x\right)^{-1}$ up to two terms. --- BISE Gujranwala(2015) * Use binomial theorem to show that $1+\frac{1}{4}+\frac{1.3}{4.8}+\frac{1.3.5}{4.8.12},...=\sqrt{2}$$(1.03)^{\frac{1}{3}}$$(a+x)$$n$$x$$(x-\frac{2}{x})^{10}$$… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:43 +0000 https://www.mathcity.org/fsc/fsc_part_1_solutions/ch03 Chapter 03: Matrices and Determinants [Chapter 03: Matrices and Determinants] Notes (Solutions) of Chapter 03: Matrices and Determinants, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary * Introduction$2\times2$$2\times2$$2\times2$$n\geq 3$$n\geq 3$ anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:46:27 +0000 https://www.mathcity.org/fsc/fsc_part_1_solutions/ch09 Chapter 09: Fundamentals of Trigonometry [Chapter 09: Fundamentals of Trigonometry] Notes (Solutions) of Chapter 09: Fundamentals of Trigonometry, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. This chapter has four exercise and solutions of those exercises are given below which can be downloaded in PDF format or can be viewed online.$D^\circ M'S''$$45^\circ , 30^\circ , 60^\circ$$0^\circ , 90^\circ , 180^\circ , 270^\circ , 36… anonymous@undisclosed.example.com (Anonymous) Sun, 04 Jun 2023 16:03:04 +0000 https://www.mathcity.org/fsc/fsc_part_2_mcqs/objective-mathematics-12th-by-m-shahbaz Objective Mathematics 12th by Muhammad Shahbaz [Objective Mathematics 12th by Muhammad Shahbaz] These notes are sent by Muhammad Shahbaz. We are very thankful to him for send this booklet. This booklet is very helpful to cover the following things for Mathematics 12 (Mathematics for Intermediate) * Summary anonymous@undisclosed.example.com (Anonymous) Fri, 21 Mar 2025 15:20:30 +0000 https://www.mathcity.org/fsc/fsc_part_2_solutions/ch01 Unit 01: Functions and Limits [Unit 01: Functions and Limits] Notes (Solutions) of Unit 01: Functions and Limits, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. There are five exercises in this chapter. You can view online or download PDF. To view PDF, you must have PDF Reader installed on your system and it can be downloaded from $\lim_{x\to a}\frac{x^n-a^n}{x-a} = na^{n-1}$$\lim_{x\to0}\frac{\sqrt{x+a} - \sqrt{a}}{x} = \frac{… anonymous@undisclosed.example.com (Anonymous) Sat, 03 Jun 2023 16:30:56 +0000 https://www.mathcity.org/fsc/fsc_part_2_solutions/ch03 Unit 03: Integration [Unit 03: Integration] Notes (Solutions) of Unit 03: Integration, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. You can view online or download PDF. To view PDF, you must have PDF Reader installed on your system and it can be downloaded from Software section.$dy$$\delta{y}$$[f(x)]^n f'(x)$$[f(x)]^{-1}f'(x)$ anonymous@undisclosed.example.com (Anonymous) Sat, 03 Jun 2023 17:25:33 +0000 https://www.mathcity.org/matric/9th_science/ex-6-3 Exercise 6.3 On the following page we have given the solution of Exercise 6.3 of Mathematics 9 (Science) published by Caravan Book House, Lahore. We have created this page and it will be updated to add new solutions occasionally. Please stay in touch with this page.$4x^2-12xy +9y^2$$x^2-1+\frac{1}{4x^2}, (x\neq 0)$$\frac{1}{16}x^2-\frac{1}{12}xy+ \frac{1}{36}y^2$$4(a+b)^2-12(a^2-b^2)+9(a-b)^2$$\frac{4x^6-12x^3y^3+9y^6}{9x^4-24x^2y^2+16y^4},(x \neq 0)$$\left( x+\frac{1}{x}\right)^2-4\left( x-\f… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:48:09 +0000 https://www.mathcity.org/msc/syllabus/uos Syllabus for UoS (Private only) Syllabus and scheme of studies for private students doing MSc Mathematics from University of Sargodha, Sargodha. The syllabus has been changed and few optional subjects has been dropped. Please be alert --- 2017/08/25 17:05 anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:50:15 +0000 https://www.mathcity.org/bsc/notes_of_calculus_with_analytic_geometry/ch07_plane_curves_ii/viewer Chapter 07: Viewer Notes of “Chapter 07: Plane Curve II” of Calculus with Analytic Geometry written by Dr. S. M. Yusuf and Prof. Muhammad Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN. You can also download a PDF file of respective exercise from this page. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:52:14 +0000 https://www.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-2-p3 Question 3, Exercise 10.2 Solutions of Question 3 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin \theta =\dfrac{4}{5}$$\theta$$\sin2\theta$$\sin \theta =\dfrac{4}{5}$$\theta$$\cos \theta =-\dfrac{3}{5}$\begin{align}\sin 2\theta &=2\sin \theta \cos \theta \\ &=2\left( \dfrac{4}{5} \right)\left( -\dfrac{3}{5} \rig… anonymous@undisclosed.example.com (Anonymous) Fri, 01 Sep 2023 18:50:54 +0000 https://www.mathcity.org/math-11-kpk/sol/unit10/ex10-2-p3 Question 3, Exercise 10.2 Solutions of Question 3 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\sin \theta =\dfrac{4}{5}$$\theta$$\sin2\theta$$\sin \theta =\dfrac{4}{5}$$\theta$$\cos \theta =-\dfrac{3}{5}$\begin{align}\sin 2\theta &=2\sin \theta \cos \theta \\ &=2\left( \dfrac{4}{5} \right)\left( -\dfrac{3}{5} \rig… anonymous@undisclosed.example.com (Anonymous) Tue, 05 Dec 2023 02:45:52 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p11 Question 10, Exercise 1.4 Solutions of Question 10 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 10(i) $Z$$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-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/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/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$$S_n$\begin{align} S_n&=\frac{n}{2}[2a_1+(n-1)d]\\ \implies S_{20}& =\frac{20}{2}[2(6,000)+(20-1)(64,000)]\\ & =10 \times [12,000+1,216,000]\\ & =12,280,000. \end{ali… anonymous@undisclosed.example.com (Anonymous) Sat, 14 Sep 2024 16:25:19 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-2 Exercise 6.2 (Solutions) The solutions of the Exercise 6.1 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to factorial function.$n!$$$ 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/matric/9th_science/unit_02/exercise_2.3 Exercise 2.3 (Solutions) Question 1 Write each radical expression in exponential notation and each exponential expression in radical notation, Do not simplify. * (i) $\sqrt[3]{-64}$ *(ii) $2^{35}$ * (iii) $-7^\frac{1}{3}$ * (iv) $y^\frac{-2}{3}$$\sqrt[3]{-64} = -64^\frac{1}{3}$$2^\frac{3}{5} = \sqrt[5]{2}^{3}$$-7^\frac{1}{3} = -\sqrt[3]{7}$$y^\frac{-2}{3} = \sqrt[3]{y}^{-2}$$ 5^\frac{1}{5} = \sqrt{5}$$2^\frac{2}{3} = \sq… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 17:00:36 +0000 https://www.mathcity.org/matric/9th_science/unit_02/exercise_2.5 Exercise 2.5 (Solutions) Question 1 * Evaluate (i) $i^7$ (ii) $i^{50}$ (iii) $i^{12}$ (iv) $\left(-i\right)^8$ (v) $\left(-i\right)^5$ (vi) $i^{27}$ Solution $$\begin{array}{cl} i^7 &= {i^6}\cdot i\\ &= (i^2)^3\cdot i\\ &= {-1}^3 \cdot i\\ &= -i \end{array}$$$$\begin{array}{cl} i^{50} &= (i^2 )^{25}\\ &= {-1}^{25}\\ … anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 17:00:37 +0000 https://www.mathcity.org/matric/9th_science/unit_05/viewer Unit 05: Factorization: Online View On this page, online view of the notes of unit 05 are given. After studying this unit , the students will be able to: * Recall factorization of expressions of the following types. * $ka + kb + kc$ * $ac + ad + bc + bd$ * $a^2 + 2ab + b^2$ * $a^2 – b^2$ * $a^2 + 2ab + b^2 – c^2$ * Factorize the expressions of the following types.$a^4 + a^2b^2 + b^4$$a^4 + 4b^4$$x^2 + px + q$$ax^2 + bx + c$$(ax^2 + bx + c) (ax2 + bx + d) + k$$(x + a) (x + b) (x +… anonymous@undisclosed.example.com (Anonymous) Wed, 30 Mar 2022 17:00:19 +0000 https://www.mathcity.org/fsc-part1-ptb FSc/ICS Part 1 (Mathematics): PTB This is an old book. Notes of new book are available at following URL: <https://www.mathcity.org/math-11-pectaa> [Textbook of Algebra and Trigonometry Class XI] Textbook of Algebra and Trigonometry Class XI is published by Punjab Textbook Board (PTB) Lahore, Pakistan. The book has total of 14 chapters. anonymous@undisclosed.example.com (Anonymous) Sat, 19 Jul 2025 17:19:20 +0000 https://www.mathcity.org/ppsc ~~DISCUSSION~~ PPSC General Information, Syllabus, Paper Pattern [PPSC] Our aim is to give general information, syllabus and paper pattern of paper couducted by Punjab Public Service Commission (PPSC) for the post of Lecturer in Mathematics. This page might be helpful for other jobs as subject specialist or for public service commission of other provinces. anonymous@undisclosed.example.com (Anonymous) Sun, 24 May 2026 17:45:57 +0000 https://www.mathcity.org/atiq/fa21-mth322 MTH322: Real Analysis II (Fall 2021) This course is offered to MSc, Semester II at Department of Mathematics, COMSATS University Islamabad, Attock campus. This course need rigorous knowledge of continuity, differentiation, integration, sequences and series of numbers, that is many notion included in $\int_{1}^{\infty }{{{x}^{-p}} dx}$$p$$f\in \mathcal{R}[a,b]$$b\ge a$$f(x)\ge 0$$x\ge a$$\int_{a}^{\infty }{f(x) dx}$$M>0$$\int\limits_{a}^{b}{f(x)\,dx} \le M$$b\ge a$$f\in \mathcal{R}[a,b]$$b\ge a$… anonymous@undisclosed.example.com (Anonymous) Thu, 30 Dec 2021 19:16:17 +0000 https://www.mathcity.org/atiq/fa22-mth604 ~~DISCUSSION~~ MTH604: Fixed Point Theory and Applications (Fall 2022) [FPTA] Course Objectives: This course is intended as a brief introduction to the subject with a focus on Banach Fixed Point theorems fixed point theorem and its application to nonlinear differential equations, nonlinear integral equations, real and complex implicit functions theorems and system of nonlinear equations. Some generalizations and similar results e. g. Kannan Fixed Point theorems, Banach Fixed Point theorem f… anonymous@undisclosed.example.com (Anonymous) Fri, 06 Jan 2023 04:37:11 +0000 https://www.mathcity.org/atiq/sp20-mth604 ~~DISCUSSION~~ MTH604: Fixed Point Theory and Applications (Spring 2020) Course Objectives: This course is intended as a brief introduction to the subject with a focus on Banach Fixed Point theorems fixed point theorem and its application to nonlinear differential equations, nonlinear integral equations, real and complex implicit functions theorems and system of nonlinear equations. Some generalizations and similar results e. g. Kannan Fixed Point theorems, Banach Fixed Point theorem for mul… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:41 +0000 https://www.mathcity.org/atiq/sp21-mth604 ~~DISCUSSION~~ MTH604: Fixed Point Theory and Applications (Spring 2021) Course Objectives: This course is intended as a brief introduction to the subject with a focus on Banach Fixed Point theorems fixed point theorem and its application to nonlinear differential equations, nonlinear integral equations, real and complex implicit functions theorems and system of nonlinear equations. Some generalizations and similar results e. g. Kannan Fixed Point theorems, Banach Fixed Point theorem for mul… anonymous@undisclosed.example.com (Anonymous) Mon, 22 Feb 2021 15:12:31 +0000 https://www.mathcity.org/atiq/sp23-mth322 MTH322: Real Analysis II (Spring 2023) [MTH322: Real Analysis II (Spring 2023)] This course is offered to BS, Semester VI at Department of Mathematics, COMSATS University Islamabad, Attock campus. This course need rigorous knowledge of continuity, differentiation, integration, sequences and series of numbers, that is many notions included in $f\in \mathcal{R}[a,b]$$b\ge a$$f(x)\ge 0$$x\ge a$$\int_{\,a}^{\,\infty }{f(x)\,dx}$$M>0$$\int\limits_{a}^{b}{f(x)\,dx}\leq M$$b\ge a$$f(x)$$g(x)$$x>a$$\li… anonymous@undisclosed.example.com (Anonymous) Thu, 15 Jun 2023 01:08:47 +0000 https://www.mathcity.org/bsc/formula-pages Formula Pages On this page, formula pages for BSc or BS level are given. * Some Important Derivative | [Download PDF] | View Online * Some Important Integrals | [Download PDF] | View Online The following pages has been send by Mansoor Tahir. * Differentiation | [Download PDF] | View Online * Integration Formulas anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:46 +0000 https://www.mathcity.org/fsc-part1-ptb/mcqs Multiple Choice Questions (MCQs) Textbook of Algebra and Trigonometry Class XI is published by Punjab Textbook Board (PTB) Lahore, Pakistan. The book has total of 14 chapters. Our plan is to give lot of Multiple Choice Questions (MCQs) for the above mentioned book. MCQs are very important because most of entry tests, admission tests and job tests consists of only MCQs.$\sqrt{3}$$n$$\sqrt{n}$$\forall a, b, c \in R$$a<b \wedge c>0\Rightarrow ac\geq bc$$a<b \wedge c>0\Rightarrow ac> bc$$a<b \wedge… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:43:03 +0000 https://www.mathcity.org/fsc-part2-ptb/subjective-mathematics-m-shahbaz Subjective Mathematics 12th by Muhammad Shahbaz [Subjective Mathematics 12th by Muhammad Shahbaz] These handwritten notes are sent by Muhammad Shahbaz. We are very thankful to him for sending these notes. These notes are based on the book “Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc/ICS Part 2 or HSSC-II), Punjab Textbook Board (PTB) Lahore, Pakistan.'' There are total seven (7) units in this book. anonymous@undisclosed.example.com (Anonymous) Tue, 23 Sep 2025 19:59:04 +0000 https://www.mathcity.org/fsc/fsc_part_1_solutions FSc Part 1 Mathematics Notes/Solutions This is an old book. Notes of new book are available at following link: <https://www.mathcity.org/math-11-pectaa> [FSc Part1 PTB Book Cover] Notes (Solutions) of Textbook of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board (PTB) Lahore. anonymous@undisclosed.example.com (Anonymous) Sun, 20 Jul 2025 09:10:04 +0000 https://www.mathcity.org/fsc/fsc_part_2_formulas_introduction_to_analytics_geometry Chap 04: Formulas Introduction to Analytics Geometry On these four pages, one can find all the formulas used in Chapter 04: Formulas Introduction to Analytics Geometry of FSc Part 2. There are five exercises in chapter 04 with lot of questions. These basic things help to solve the questions easily without going to the depth of each concept. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:42:43 +0000 https://www.mathcity.org/notes/multiple-choice-questions-bsc-bs-ppsc-akhtar-abbas Multiple Choice Questions (BSc/BS/PPSC) by Akhtar Abbas [Multiple Choice Questions (BSc/BS/PPSC)] These notes are made and shared by Mr. Akhtar Abbas. We are really very thankful to him for providing these notes and appreciates his efforts to publish these notes on MathCity.org. Multiple Choice Questions (MCQs) are given in these notes, which might be helpful in BSc, BS or Punjab Public Service Commission (PPSC) exams.$a$$b$$n$$na > b$$(p − 1)! \equiv −1(mod p)$$p$$p$$p$ anonymous@undisclosed.example.com (Anonymous) Sun, 24 May 2026 17:45:10 +0000 https://www.mathcity.org/notes/note-for-numerical-methods-m-usman-hamid Notes for Numerical Methods by M Usman Hamid [Notes for Numerical Methods by M Usman Hamid] These notes are initially provided by Mr. Anwar Khan. Later the updated version is send by Muhammad Tahir. We are really very thankful to Mr. Anwar Khan and Muhammad Tahir for providing these notes and appreciates their effort to publish these notes on MathCity.org$\left(\frac{1}{3}\right)$$\left(\frac{3}{8}\right)$ anonymous@undisclosed.example.com (Anonymous) Sat, 05 Aug 2023 17:59:07 +0000 https://www.mathcity.org/notes/number-theory-handwritten-notes Number Theory: Handwritten Notes [Number Theory: Handwritten Notes] The study of the characteristics of the positive integers (1, 2, 3,...) is called number theory. It is significant because it has numerous uses in coding theory, combinatorics, cryptography, and other branches of mathematics and computer science. Some mathematicians also refer to number theory as the anonymous@undisclosed.example.com (Anonymous) Tue, 08 Aug 2023 11:20:45 +0000 https://www.mathcity.org/open-notes/real-analysis-i Open Notes on Real Analysis I [Open Notes on Real Analysis I] This is an initial release of the notes. The beautiful thing for these notes is that you can edit these notes. If you feel that your version of notes is better than given here, we can share your version here. Please send an email to first author Dr. Atiq ur Rehman at anonymous@undisclosed.example.com (Anonymous) Tue, 26 Aug 2025 18:16:07 +0000 https://www.mathcity.org/ppsc/ppsc-maths-2021 PPSC Paper 2021 (Lecturer in Mathematics) [PPSC Paper 2011 (Lecturer in Mathematics)] On this page, we have given question from old (past) paper of Lecturer in Mathematics conducted in year 2021. This is a MCQs paper and answers are given at the end of the paper. At the end of the PDF is also given to download. This paper is provided by Ms. \(2018\)$4$\(6\)$8$$10$\(X\)\(Y\)\(X\times Y\)\(\parallel (x,y) \parallel=\parallel x\parallel+\parallel y\parallel, \,\forall \, (x,y)\in X \times Y\)\(f(… anonymous@undisclosed.example.com (Anonymous) Tue, 23 Aug 2022 17:04:49 +0000 https://www.mathcity.org/quote-of-the-day/may Quotes for the May مختصراً، پوری دنیا خلا اور وقت میں اشیا کی ریاضیاتی طور پر ظاہر کی جانے والی حرکات کا مجموعہ ہے، اور پوری کائنات ایک عظیم، ہم آہنگ اور ریاضیاتی طور پر تیار کی گئی مشین ہے۔۔۔ anonymous@undisclosed.example.com (Anonymous) Fri, 28 Apr 2023 18:45:11 +0000 https://www.mathcity.org/atiq/sp15-mth321/mcqs MCQs or Short Questions On this page, MCQs or short questions with out answers are given. Students need to find the answer them self. This page will be updated occasionally and new MCQs or short question will be posted here. * A number which is neither even nor odd is$2n$$n \in \mathbb{Z}$$2\pi$$\pi$$\pi$$\sqrt{2}$$\sqrt{3}$$A$$f:A\to \mathbb{N}$$f$$f$$f$$A=\{x| x\in \mathbb{N} \wedge x^2 \leq 7 \}$$A$$\{s_n\}$$\lambda$$|s_n|<\lambda$$n\in\mathbb{Z}$$p$$|s_n|<p$$n\in\mathbb{Z}$$s$$|s_n|<s$$n… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:45:18 +0000 https://www.mathcity.org/bsc/notes_of_mathematical_method/ch01_complex_numbers Chapter 01: Complex Numbers [Chapter 01 Complex Numbers Methods] Notes of the book Mathematical Method written by S.M. Yusuf, A. Majeed and M. Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN. A complex number is an element $(x,y)$ of the set $$ \mathbb{R}^2=\{(x,y): x,y \in \mathbb{R}\} $$ obeying the following rules of addition and multiplication.$z_1=(x_1,y_1)$$z_2=(x_2,y_2)$$z_1+z_2= (x_1+x_2, y_1+y_2)$$z_1 z_2 = (x_1 x_2 - y_1 y_2, x_1 y_2+y_1 x_2)$$\mathbb{R}^2$$\mathbb{C}$ anonymous@undisclosed.example.com (Anonymous) Mon, 11 Dec 2023 12:59:57 +0000 https://www.mathcity.org/fsc-part1-kpk/sol/unit01 Unit 1: Complex Numbers (Solutions) This is a first unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the students will be able to$z$$z=a+ib$$(a,b)$$a$$b$$i=\sqrt{-1}$$a$$z$$b$$z$$\bar{z} = a —ib$$z=a+ib$$|z| = \sqrt{a^2+b^2}$$z=a+ib$$'+'$$'\times'$$z$$|z|=|-z|=|\bar{z}=|-\bar{z}|$$pz^2+ qz+ r = 0$$p,q,r$$z$ anonymous@undisclosed.example.com (Anonymous) Mon, 25 Sep 2023 12:08:05 +0000 https://www.mathcity.org/fsc-part1-ptb/important-questions/ch04-quadratic-equations Ch 04: Quadratic Equations * Reduce $x^{-2}-10=3x^{-1}$ to quadratic form --- BISE Gujrawala(2015) * Show that $x^3-y^3=(x-y)(x-wy)(x-w^2y)$ --- BISE Gujrawala(2015) * If $n$ is an odd integer, is $(x+a)$ factor of $(x^n+a^n)$? --- BISE Gujrawala(2015) * If the roots of $px^2+qx+q=0$ are $\alpha$, $\beta$,then prove that $$\sqrt {\frac{\alpha}{\beta}}+\sqrt {\frac{\beta}{\alpha}}+\sqrt{\frac{p}{q}}=0$$ --- BISE Gujrawala(2017),BISE Sagodha(2017$${\begin{array}{c} x^2-5xy+6y^2=0\\x^2… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:40 +0000 https://www.mathcity.org/fsc-part1-ptb/mcq-bank/ch01 MCQs: Ch 01 Number Systems High quality MCQs of Chapter 01 Number System of Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. The answers are given at the end of the page. MCQs * If $*$$A$$a, b \in A$$a+b \in A$$a-b \in A$$a \times b \in A$$a * b \in A$$z=(1,3)$$z^{-1}= $$(\displaystyle{\frac{1}{10}},\displaystyle{\frac{3}{10}})$$(-\displaystyle{\frac{1}{10}},\displaystyle{\frac{3}{10}})$$(\displaystyle{\frac{1}{10}},-\display… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:49 +0000 https://www.mathcity.org/fsc-part1-ptb/mcq-bank/ch02 MCQs: Ch 02 Sets, Functions and Groups High quality MCQs of Chapter 02 Sets, Functions and Groups of Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. The answers are given at the end of the page.$\forall$$\wedge$$<$$\in$$A$$B$$A\cap B=\phi$$A=B$$B\subseteq A$$A \subseteq B$$A$$B$$A-B \neq \phi$$A=B$$A \subseteq B$$B\subseteq A$$A$$B$$A\cap B=A$$B \subseteq A$$A\cap B=\phi$$A\subseteq B$$B\subseteq A$$A=\phi$$A \cup B=A$$A \cap B=… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:49 +0000 https://www.mathcity.org/fsc-part2-ptb/important-questions/unit-01-functions-and-limits Unit 01: Functions and Limits Here is the list of important questions. * Evaluate $\lim\limits_{\theta \to 0}\frac{1-\cos \theta}{\sin^3\theta}$ --- FBSIC (2016) * Graph the curve of the following parametric equations $x=\sec \theta$, $y=\tan\theta$ where $\theta$ is a parameter.--- FBSIC (2016) * Evaluate $\lim\limits_{x \to 2}\frac{\sqrt{x}-\sqrt{2}}{x-2}$ --- BSIC Rawalpendi(2016), BSIC Rawalpendi(2017)$f(x)=x^3+x$$\lim\limits_{\theta \to 0}\frac{\tan \theta-\sin \theta}{\sin^3\t… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:54 +0000 https://www.mathcity.org/fsc/fsc_part_1_solutions/ch01 Chapter 01: Number System [Chapter 01: Number System] Notes (Solutions) of Chapter 01: Number System, Textbook of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary * Rational numbers and irrational numbers$\mathbb{C}$$(x+iy)^n$$\left(\frac{x_1+iy_1}{x_2+iy_2}\right)^n, x_2+iy_2\neq 0$$\sqrt{-1}=i$$\sqrt{-1}$$i$$-i$$i$$-i$$-1$$i^2=-1$$\sqrt{-1}=i$$\sqrt{-1}$ anonymous@undisclosed.example.com (Anonymous) Sat, 03 Jun 2023 16:30:31 +0000 https://www.mathcity.org/fsc/fsc_part_1_solutions/ch02 Chapter 02: Sets, Functions and Groups Notes (Solutions) of Chapter 02: Sets, Functions and Groups, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. [Chapter 02: Sets, Functions and Groups] Contents & summary * Introduction$p\leftrightarrow q$ anonymous@undisclosed.example.com (Anonymous) Wed, 05 Apr 2023 12:54:57 +0000 https://www.mathcity.org/fsc/fsc_part_1_solutions/ch05 Chapter 05: Partial Fractions [Chapter 05: Partial Fractions] Notes (Solutions) of Chapter 05: Partial Fractions, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary * Introduction * Rational Fraction$\frac {P(x)}{Q(x)}$ anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:46:29 +0000 https://www.mathcity.org/fsc/fsc_part_1_solutions/ch06 Chapter 06: Sequences and Series [Chapter 06: Sequences and Series] Notes (Solutions) of Chapter 06: Sequences and Series, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary * Introduction * Types of Sequences$l,m,n$$p$$q$$r$$$l(q-r)+m(r-p)+n(p-q)=0$$$a_1$$d$$$\begin{align}l=a_1+(p-1)d,\\ m=a_1+(q-1)d,\\ n=a_1+(r-1)d.\end{align}$$ Now $$\begin{align}L.H.S &= l(q-r)+m(r-p)+n(p-q)\\ &= lq-lr+mr-mp+np-nq\\ &=… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:46:29 +0000 https://www.mathcity.org/fsc/fsc_part_1_solutions/ch07 Chapter 07: Permutation, Combination and Probability [Chapter 07: Permutation , Combination and Probability] Notes (Solutions) of Chapter 07: Permutation , Combination and Probability, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary anonymous@undisclosed.example.com (Anonymous) Sat, 26 Mar 2022 16:28:40 +0000 https://www.mathcity.org/fsc/fsc_part_1_solutions/ch08 Chapter 08: Mathematical Induction and Binomial Theorem [Chapter 08 Mathematical Induction and Binomial Theorem] Notes (Solutions) of Chapter 08: Mathematical Induction and Binomial Theorem, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore.$(a+x)^n$$(a+x)^n$ anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:46:30 +0000 https://www.mathcity.org/fsc/fsc_part_1_solutions/ch10 Chapter 10: Trigonometric Identities [Chapter 10: Trigonometric Identities] Notes (Solutions) of Chapter 10: Trigonometric Identities, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. There are four exercise in this chapter. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:46:32 +0000 https://www.mathcity.org/fsc/fsc_part_1_solutions/ch11 Chapter 11: Trigonometric Functions and their Graphs [Chapter 11: Trigonometric Functions and their Graphs] Notes (Solutions) of Chapter 11: Trigonometric Functions and their Graphs, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary $ y = \sin x$$-2\pi \hbox{ to } 2\pi$$ y = \cos x$$-2\pi \hbox{ to } 2\pi$$ y = \tan x$$-\pi \hbox{ to } \pi$$ y = \cot x$$-2\pi \hbox{ to } \pi$$ y = \sec x$$-2\pi \hbox{ to } 2\pi… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:46:34 +0000 https://www.mathcity.org/fsc/fsc_part_1_solutions/ch12 Chapter 12: Application of Trigonometry [Chapter 12: Application of Trigonometry] Notes (Solutions) of Chapter 12: Application of Trigonometry, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary * Introduction anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:46:34 +0000 https://www.mathcity.org/fsc/fsc_part_1_solutions/ch13 Chapter 13: Inverse Trigonometric Functions [Chapter 13: Inverse Trigonometric Functions] Notes (Solutions) of Chapter 13: Inverse Trigonometric Functions, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary * ${\sin ^{ - 1}}A + {\sin ^{ - 1}}B = {\sin ^{ - 1}}\left( {A\sqrt {1 - {B^2}} + B\sqrt {1 - {A^2}} } \right)$${\sin ^{ - 1}}A - {\sin ^{ - 1}}B = {\sin ^{ - 1}}\left( {A\sqrt {1 - {B^2}} - B\sqrt {1 - {… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:46:37 +0000 https://www.mathcity.org/fsc/fsc_part_1_solutions/ch14 Chapter 14: Solutions of Trigonometric Equation [Chapter 14: Solutions of Trigonometric Equation] Notes (Solutions) of Chapter 14: Solutions of Trigonometric Equation, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary ${\sin ^{ - 1}}A + {\sin ^{ - 1}}B = {\sin ^{ - 1}}\left( {A\sqrt {1 - {B^2}} + B\sqrt {1 - {A^2}} } \right)$${\sin ^{ - 1}}A - {\sin ^{ - 1}}B = {\sin ^{ - 1}}\left( {A\sqrt {1 - {B^2}} - B\sqr… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:46:37 +0000 https://www.mathcity.org/fsc/fsc_part_2_solutions/ch02 Unit 02: Differentiation [Unit 02: Differentiation] Notes (Solutions) of Unit 02: Differentiation, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. You can view online or download PDF. To view PDF, you must have PDF Reader installed on your system and it can be downloaded from Software section.$f'(x)$$x^n$$n \in \mathbb{Z}$$\frac{x+1}{x-1}$$x$$$ \begin{aligned} \frac{d}{dx}\left(\frac{x+1}{x-1}\right) &= \frac{(x-1)\frac{d}{dx}(x… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:05 +0000 https://www.mathcity.org/fsc/fsc_part_2_solutions/ch07 Unit 07: Vectors [Unit 07: Vectors] Notes (Solutions) of Unit 07: Vectors, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. You can view online or download PDF. To view PDF, you must have PDF Reader installed on your system and it can be downloaded from Software section.$u\cdot v$$u\times v$$u\cdot(v\times w)$ anonymous@undisclosed.example.com (Anonymous) Thu, 25 Mar 2021 17:27:32 +0000 https://www.mathcity.org/math-11-kpk/sol/unit01 Unit 01: Complex Numbers (Solutions) This is a first unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the students will be able to$z$$z=a+ib$$(a,b)$$a$$b$$i=\sqrt{-1}$$a$$z$$b$$z$$\bar{z} = a —ib$$z=a+ib$$|z| = \sqrt{a^2+b^2}$$z=a+ib$$'+'$$'\times'$$z$$|z|=|-z|=|\bar{z}=|-\bar{z}|$$pz^2+ qz+ r = 0$$p,q,r$$z$ anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:28:42 +0000 https://www.mathcity.org/math-11-kpk/sol/unit02 Unit 02: Matrices and Determinants (Solutions) This is a second unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the students will be able to$z$$z=a+ib$$(a,b)$$a$$b$$i=\sqrt{-1}$$a$$z$$b$$z$$\bar{z} = a —ib$$z=a+ib$$|z| = \sqrt{a^2+b^2}$$z=a+ib$$'+'$$'\times'$$z$$|z|=|-z|=|\bar{z}=|-\bar{z}|$$pz^2+ qz+ r = 0$$p,q,r$$z$ anonymous@undisclosed.example.com (Anonymous) Tue, 05 Dec 2023 02:44:47 +0000 https://www.mathcity.org/math-11-kpk/sol/unit03 Unit 03: Vectors (Solutions) This is a third unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the students will be able to$i$$j$$i$$j$$k$$O$$-A$$A$$i.i=j.j=k.k=1$$i.j=j.k=k.i=0$$i\times i =j\times j =k\times k=0$$i\times j = k$$j\times k =k\times j = i$$A \times B$$A$$B$$i.j\times k =j.k\times i=k.i\times j=1$$i.k\times j = J.i\times k=k.j\times i=-1$ anonymous@undisclosed.example.com (Anonymous) Wed, 03 Jan 2024 18:11:32 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06 Unit 06: Permutation, Combination and Probability (Solutions) This is a sixth unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the students will be able to$n$$n!$$n$$r$$^nP_r$$n$$r$$n$$r$ anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:34:51 +0000 https://www.mathcity.org/matric/9th_science/ex-6-2 Exercise 6.2 On the following page we have given the solution of Exercise 6.2 of Mathematics 9 (Science) published by Caravan Book House, Lahore. We have created this page and it will be updated to add new solutions occasionally. Please stay in touch with this page.$\frac{x^2-x-6}{x^2-9}+\frac{x^2+2x-24}{x^2-x-12}$\begin{align} \frac{x^2-x-6}{x^2-9}&+\frac{x^2+2x-24}{x^2-x-12}\\ &=\frac{x^2-3x+2x-6}{(x)^2-(3)^2}+\frac{x^2+6x-4x-24}{x^2-4x+3x-12}\\&= \frac{x(x-3)+2(x-3)}{(x-3)(x+3)}+\frac{x(x+6… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:48:08 +0000 https://www.mathcity.org/msc/mcqs_short_questions/real_analysis Real Analysis: Short Questions and MCQs We are going to add short questions and MCQs for Real Analysis. The subject is similar to calculus but little bit more abstract. This is a compulsory subject in MSc and BS Mathematics in most of the universities of Pakistan. The author of this page is Dr. $\left\{\frac{1}{n+1} \right\}$$\left\{\frac{n+2}{n+1} \right\}$$\{x_n\}$$\{y_n\}$$\lim_{n\to\infty z_n}$$z_n=x_n-2y_n$$\{x_n\}$$\{y_n\}$$\lim_{n\to\infty z_n}$$x_n=2y_n-3z_n$$(1,2)$$\left(\frac{1}{2},\fr… anonymous@undisclosed.example.com (Anonymous) Mon, 03 Apr 2023 04:06:26 +0000 https://www.mathcity.org/papers/old_admission_test_of_assms_for_ph.d._mathematics/how_to_prepare_admission_test_a_short_guide How to prepare admission test (A short guide) MathCity.org does not represent any official or government/semi-government/private educational institute or board or university. The resources given on the site holds no official position in government/semi-government/private educational institute or board or university. While using a resources given on this site you agreed to the term that we (MathCity.org or person related to MathCity.org) do not take any responsibility for these resources. The sug… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:50:25 +0000 https://www.mathcity.org/papers/old_papers_for_bsc_mathematics/sargodha_university University of Sargodha (Old Papers): BSc (Mathematics only) Old/previous papers of BSc (Mathematics), University of Sargodha, Sargodha are posted on this page. There are three type of papers in BSc: General Mathematics, A-Course of Mathematics and B-Course of Mathematics. The A-Course of Mathematics is renamed from anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:50:33 +0000 https://www.mathcity.org/bsc/notes_of_mechanics/introduction_to_mechanics/ch04-centres-of-mass-and-gravity Chapter 04: Centres of Mass and Gravity Notes of Introduction to Mechanics by Q. K. Ghori published by West Pak Publishing Company (Pvt) Ltd. These notes are written by Prof. Muhammad Tanveer. Following are the main topics of this chapter. * Linear Moment anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:55:04 +0000 https://www.mathcity.org/bsc/notes_of_mechanics/introduction_to_mechanics/ch05-friction Chapter 05: Friction Notes of Introduction to Mechanics by Q. K. Ghori published by West Pak Publishing Company (Pvt) Ltd. These notes are written by Prof. Muhammad Tanveer. Following are the main topics of this chapter. * Friction * Laws of friction anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:55:06 +0000 https://www.mathcity.org/fsc-part1-kpk/sol/unit10/ex10-2-p1 Question 1, Exercise 10.2 Solutions of Question 1 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. There are four parts in Question 1.$\sin 2\theta ,\,\,\cos 2\theta$$\tan 2\theta$$\tan \theta =-\dfrac{1}{5}$$\theta$$\sin \theta =\dfrac{1}{\sqrt{26}}$$\cos \theta =\dfrac{-5}{\sqrt{26}}$\begin{align}\sin 2\theta &=2\sin… anonymous@undisclosed.example.com (Anonymous) Fri, 01 Sep 2023 18:26:41 +0000 https://www.mathcity.org/fsc-part1-kpk/sol/unit10/re-ex10-p1 Question 1, Review Exercise 10 Solutions of Question 1 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos {{50}^{\circ }}5{0}'\cos {{9}^{\circ }}1{0}'-\sin {{50}^{\circ }}5{0}'\sin {{9}^{\circ }}1{0}'=$$0$$\dfrac{1}{2}$$1$$\dfrac{\sqrt{3}}{2}$$\dfrac{1}{2}$$\tan {{15}^{\circ }}=2-\sqrt{3}$${{\cot }^{2}}{{75}^{\… anonymous@undisclosed.example.com (Anonymous) Sat, 09 Sep 2023 07:17:17 +0000 https://www.mathcity.org/fsc-part1-ptb/sol/ch01/ex1-2 Exercise 1.2 (Solutions) Notes (Solutions) of Exercise 1.2: Textbook of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board (PTB) Lahore. The main topics of this exercise are complex numbers, real part and imaginary part of complex numbers, properties of the fundamental operation on complex numbers, complex number as ordered pair of real numbers and special subset of complex numbers. These notes are based on the new Student Learning Outcomes (SLOs). Versio… anonymous@undisclosed.example.com (Anonymous) Sun, 09 Apr 2023 07:32:53 +0000 https://www.mathcity.org/fsc-part1-ptb/sol/ch02/ex2-8 Exercise 2.8 (Solutions) Notes (Solutions) of Exercise 2.8: Textbook of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board (PTB) Lahore. The main topic of this exercise are binary operation, semi-group, monoid, groups and abelian groups. These notes are based on the new Student Learning Outcomes (SLOs). Version: 4.1, Available at MathCity.org $\oplus$$G=\{0,1\}$\[ \begin{array}{|c|c|c|} \hline \oplus & 0 & 1 \\ \hline 0 & 1 & 1 \\ \hline 1 & 1 & … anonymous@undisclosed.example.com (Anonymous) Wed, 05 Apr 2023 12:55:15 +0000 https://www.mathcity.org/math-11-kpk/sol/unit04/ex4-5-p8 Question 11 & 12 Exercise 4.5 Solutions of Question 11 & 12 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$p^{t h}, q^{t h}$$r^{t h}$$a, b, c$$a^{q-r} b^{r-p} c^{p-q}=1$$a_n=a_1 r^{n-1}$$a_p=a_1 r^{p-1}=a \quad a_q=a_1 r^{q-1}=b$$a_r=a_1 r^{r-1}$\begin{align}a^{q-r}&=(a_1 r^{p-1})^{q-r} . \\ b^{r-p}&=(a_1 r^{q-1})^{r-p}, \text { and } \\ c^{p-q}&=(a_1 r^… anonymous@undisclosed.example.com (Anonymous) Fri, 05 Jan 2024 17:30:14 +0000 https://www.mathcity.org/math-11-kpk/sol/unit05/ex5-2-p2 Question 2 & 3 Exercise 5.2 Solutions of Question 2 & 3 of Exercise 5.2 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2 $1+3^2 x+5^2 x^2+7^2 x^3+\ldots, x<1$\begin{align} & S_{\infty}=1+3^2 x+5^2 x^2+7^2 x^3+\ldots ..(1)\\ & x S_{\infty}=x+3^2 x^2+5^2 x^3+7^2 x^4+\ldots..(2)\end{align}\begin{align}& (1-x) S_{\infty}=1^2+(3^2-1^2) x+(5^2-3^2) x^2+(7^2-5^2) x^… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:14 +0000 https://www.mathcity.org/math-11-kpk/sol/unit05/ex5-2-p3 Question 4 & 5 Exercise 5.2 Solutions of Question 4 & 5 of Exercise 5.2 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4 $5+\dfrac{7}{3}+\dfrac{9}{3^2}+\dfrac{11}{3^3}+\ldots$\begin{align} & S_{\infty}=5+\dfrac{7}{3}+\dfrac{9}{3^2}+\dfrac{11}{3^3}+\ldots.(i) \\ & \dfrac{1}{3} S_{\infty}=\dfrac{5}{3}+\dfrac{7}{3^2}+\dfrac{9}{3^2}+\dfrac{11}{3^3}+\ldots.(ii) \e… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:15 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p7 Question 11 Review Exercise 6 Solutions of Question 11 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$n(S)=4$$$\dfrac{1}{4}$$$\quad P( orange )=\dfrac{1}{4}$$$\dfrac{1}{4}$$\dfrac{1}{4}$\begin{align}P(\operatorname{Red})&=\dfrac{1}{4}\\ P( Green )&=\dfrac{1}{4}\end{align}$P(R \cap G)=\phi$$R$$G$\begin{align}\boldsymbol{P}( Red o… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:36:03 +0000 https://www.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p8 Question 10 Exercise 7.3 Solutions of Question 10 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$1-\frac{1}{2^2}+\frac{1.3}{2 !} \cdot \frac{1}{2^4}+\ldots$$(1+x)^n$$$ \begin{aligned} & 1+n x+\frac{n(n-1)}{2 !} x^2 \\ & +\frac{n(n-1(n-2))}{3 !} x^3+\ldots \end{aligned} $$$n x=-\frac{1}{4}$$\frac{n(n-1)}{2 !} x^2=\frac{1.3}{2 !} \cdot … anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:36:37 +0000 https://www.mathcity.org/math-11-kpk/sol/unit10/ex10-2-p1 Question 1, Exercise 10.2 Solutions of Question 1 of Exercise 10.2 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. There are four parts in Question 1.$\sin 2\theta ,\,\,\cos 2\theta$$\tan 2\theta$$\tan \theta =-\dfrac{1}{5}$$\theta$$\sin \theta =\dfrac{1}{\sqrt{26}}$$\cos \theta =\dfrac{-5}{\sqrt{26}}$\begin{align}\sin 2\theta &=2\sin… anonymous@undisclosed.example.com (Anonymous) Tue, 05 Dec 2023 02:45:47 +0000 https://www.mathcity.org/math-11-kpk/sol/unit10/re-ex10-p1 Question 1, Review Exercise 10 Solutions of Question 1 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\cos {{50}^{\circ }}5{0}'\cos {{9}^{\circ }}1{0}'-\sin {{50}^{\circ }}5{0}'\sin {{9}^{\circ }}1{0}'=$$0$$\dfrac{1}{2}$$1$$\dfrac{\sqrt{3}}{2}$$\dfrac{1}{2}$$\tan {{15}^{\circ }}=2-\sqrt{3}$${{\cot }^{2}}{{75}^{\… anonymous@undisclosed.example.com (Anonymous) Sat, 16 Mar 2024 13:12:33 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-4 Exercise 2.4 (Solutions) The solutions of the Exercise 2.4 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to the determinant and properties of the determinant.$\left|\begin{array}{lll}9 & 27 & 36 \\ 18 & 54 & 24 \\ 27 & 81 & 28\end{array}\right|=0$$\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/ex2-6-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/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$$3$ anonymous@undisclosed.example.com (Anonymous) Wed, 04 Sep 2024 03:13:56 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/re-ex-p3 Question 4 and 5, Review Exercise Solutions of Question 4 and 5 of Review Exercise of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left|\begin{array}{ccc}a+1 & l & l \\ l & a+1 & l \\ l & l & a+1\end{array}\right|=(a+1+2 l)(a+1-l)^{2}$\begin{align*} L.H.S &= \left|\begin{array}{ccc}a+1 & l & l \\ l & a+1 & l \\ l & l & a+1\end{array}\right|\\ &=\left|\b… anonymous@undisclosed.example.com (Anonymous) Wed, 04 Sep 2024 03:15:45 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/rev-ex Review Exercise 2 (Solutions) The solutions of the Review Exercise 2 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the MCQs and question all topics included in this chapter.$A$$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$$a_{i j}=(-… anonymous@undisclosed.example.com (Anonymous) Sun, 08 Feb 2026 16:57:09 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-1-p5 Question 9 and 10, Exercise 4.1 Solutions of Question 9 and 10 of Exercise 4.1 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$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/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$$= {}^4P_4 = 24.$ anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:45 +0000 https://www.mathcity.org/math-11-nbf/sol/unit08/re-ex-p1 Question 1, Review Exercise Solutions of Question 1 of Review Exercise of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin \left(45^{\circ}-30^{\circ}\right)=\ldots$$\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}$$\tan \left(\frac{\pi}{6}+\frac{\pi}{4}\rig… anonymous@undisclosed.example.com (Anonymous) Sat, 09 Nov 2024 17:48:41 +0000 https://www.mathcity.org/math-11-nbf/sol/unit09/ex9-1-p11 Question 10, Exercise 9.1 Solutions of Question 10 of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $ V(t)=a \operatorname{Sin}(k(t-d))+c$$56 \mathrm{~Hz} A C$$k$ anonymous@undisclosed.example.com (Anonymous) Wed, 27 Nov 2024 15:59:10 +0000