Merging man & maths https://www.mathcity.org/ Thu, 04 Jun 2026 20:38:49 +0000 FeedCreator 1.8 https://www.mathcity.org/_media/logo.svg https://www.mathcity.org/ https://www.mathcity.org/fsc-part1-ptb/important-questions Important Questions: HSSC-I [Important Questions FSc/ICS Part 1] These are the important questions for “Textbook of Algebra and Trigonometry Class XI” published by Punjab Textbook Board (PTB) Lahore, Pakistan. These questions are taken from old papers. These are very helpful to understand the types of questions which may asked final paper of mathematics for FSc/ICS (HSSC) Part 1. Lot of energy has been put to collect and write these questions. These are taken from old papers of FBISE Islamabad,… anonymous@undisclosed.example.com (Anonymous) Thu, 18 Apr 2024 08:01:02 +0000 https://www.mathcity.org/fsc-part2-ptb/important-questions Important Questions: HSSC-II These questions are taken from old papers for the book Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Textbook Board (PTB) Lahore, Pakistan. * Unit 01: Functions and Limits * Unit 02: Differentiation * Unit 03: Integration * Unit 04: Introduction to Analytic Geometry * Unit 05: Linear Inequalities and Linear Programming * anonymous@undisclosed.example.com (Anonymous) Thu, 18 Apr 2024 08:02:52 +0000 https://www.mathcity.org/atiq/sp23-mth321 MTH321: Real Analysis I (Spring 2023) ~~DISCUSSION~~ [Photo-illustration of Zeno's Paradox] At the end of this course the students will be able to understand the basic set theoretic statements and emphasize the proofs’ development of various statements by induction. Define the limit of, a function at a value, a sequence and the Cauchy criterion. Prove various theorems about limits of sequences and functions and emphasize the proofs’ development. Define continuity of a function and uniform con… anonymous@undisclosed.example.com (Anonymous) Wed, 14 Jun 2023 14:47:57 +0000 https://www.mathcity.org/bsc/paper_pattern/punjab_university/b.sc._paper_pattern_for_general_mathematics_split_program Syllabus & Paper Pattern for General Mathematics (Split Program) There was one examination after two years for BA/BSc Program from University of Punjab (PU), Lahore but from this year (2016), PU has made changes in its examination policies for the said program. The BA/BSc Program has been split into two parts. Syllabus is break into two part year wise. After the each year of the program candidate has to appeared in examination instead of appearing after two year. In this regards syllabus of Gen… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:55:21 +0000 https://www.mathcity.org/fsc/fsc_part_2_mcqs/short_questions_by_mr._akhtar_abbas Short Questions by Mr. Akhtar Abbas * We are very thankful to Mr. Akhtar Abbas for sharing these short questions. * These short questions are selected from previous 5 years papers of different boards. Solve these at your own to perform well in annual exams.$\sqrt{x^2-4}$$f(x)=\frac{2x}{x-2}$$x=2$$x^{100}$ anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:46:47 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/ex6-4-p3 Question 3 Exercise 6.4 Solutions of Question 3 of Exercise 6.4 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$8$$8$$2^8$$$n(S)=256$$$$\dfrac{1}{256}$$$8$$$A=\{8\}$$$${ }^8 C_8=\dfrac{8 !}{(8-8) ! 8 !}=1$$$8$$$P(A)=\dfrac{1}{256}$$$7$$8$$2^8$$$n(S)=256$$$$\dfrac{1}{256}$$$7$$$B=\{7\}$$$7$$8$$$n(B)={ }^8 C_7=\dfrac{8 !}{(8-7) ! 7 !}=8$$$7$$8$$$P(B)=\d… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:51 +0000 https://www.mathcity.org/atiq/sp23-mth322 MTH322: Real Analysis II (Spring 2023) [MTH322: Real Analysis II (Spring 2023)] This course is offered to BS, Semester VI at Department of Mathematics, COMSATS University Islamabad, Attock campus. This course need rigorous knowledge of continuity, differentiation, integration, sequences and series of numbers, that is many notions included in $f\in \mathcal{R}[a,b]$$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/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$$p$ anonymous@undisclosed.example.com (Anonymous) Sun, 24 May 2026 17:45:10 +0000 https://www.mathcity.org/atiq/math-510 MATH-510: Topology Topology is an important branch of mathematics that studies all the “qualitative” or “discrete” properties of continuous objects such as manifolds, i.e. all the properties that aren't changed by any continuous transformations except for the singular (infinitely extreme) ones.$(T_0, T_1, T_2)$$\mathbb{R}$$X=\{a\}$$X$$X$$X$$\tau$$\mathbb{N}$$\tau$$(\mathbb{Z}, \tau)$$\mathbb{N}$$\tau$$A=\{\pm 100,\pm 101, \pm 102, ... \}$$\tau$$E=\{0,\pm 2,\pm 4,...\}$$\tau$$\tau$$B=\{1,2,3,...… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:25 +0000 https://www.mathcity.org/atiq/sp19-mth322 MTH322: Real Analysis II (Spring 2019) 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 anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:39 +0000 https://www.mathcity.org/msc/mcqs_short_questions/normed_spaces Normed Spaces: Short Questions and MCQs anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:48:16 +0000 https://www.mathcity.org/bsc/paper_pattern/punjab_university/b.sc._paper_pattern_for_a_and_b_course_of_mathematics Syllabus & Paper Pattern for A and B Course of Mathematics This is a new page created to discuss the syllabus or course outline of PU splitted into two part. It will take some time to complete this page. Please stay in touch with this page to be updated. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:55:18 +0000 https://www.mathcity.org/bsc/paper_pattern/sargodha_university/applied_mathematics_chapterwise Applied Mathematics Paper pattern for Applied Mathematics chapter-wise for University of Sargodha is given on this page. This pattern is extracted from syllabus, so use your own risk. Syllabus of Applied Mathematics can be seen here. Applied Mathematics is consists of two papers of 100 marks each. One is called “Paper A” and other is called “Paper B”. In every paper there are three sections with four questions each. A student have to attempt two questions from each section. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:55:54 +0000 https://www.mathcity.org/bsc/paper_pattern/sargodha_university/general_mathematics_chapterwise General Mathematics Paper pattern for General Mathematics chapter-wise for University of Sargodha is given on this page. This pattern is provided by Muhammad Siraj (+92-345-5365318). Syllabus of General Mathematics can be seen here. General Mathematics is consists of two papers of 100 marks each. One is called “Paper A” and other is called “Paper B”. In every paper there are three sections with four questions. A student have to attempt two questions from each section. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:55:58 +0000 https://www.mathcity.org/bsc/paper_pattern/sargodha_university/pure_mathematics_chapterwise Pure Mathematics Paper pattern for Pure Mathematics chapter-wise for University of Sargodha is given on this page. This pattern is extracted from syllabus, so use your own risk. Syllabus of Pure Mathematics can be seen here. Pure Mathematics is consists of two papers of 100 marks each. One is called “Paper A” and other is called “Paper B”. In every paper there are three sections with four questions each. A student have to attempt two questions from each section. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:56:03 +0000 https://www.mathcity.org/fsc-part1-ptb/short-term-preparation-salman-sherazi Short Term Preparation FSc/ICS 1 fsc fsc_part1 m_salman_sherazi important_questions_fsc_1 [Short Term Preparation by M Salman Sherazi] This document contains all the important MCQs, Short Questions and Long Questions of Mathematics HSSC-I (FSc/ICS Part 1) from the Textbook of Algebra and Trigonometry for Class XI. It has been done to help the students and teachers at no cost by $\sqrt{2}$ anonymous@undisclosed.example.com (Anonymous) Thu, 02 May 2024 17:12:17 +0000 https://www.mathcity.org/fsc-part2-ptb/short-term-preparation-salman-sherazi Short Term Preparation FSc 2 fsc fsc_part2 m_salman_sherazi important_questions_fsc_2 [Short Term Preparation Guide FSc 2] This document contains all the important MCQs, Short Questions and Long Questions of Mathematics HSSC-II (FSc Part 2) from the Calculus and Analytic Geometry, MATHEMATICS 12. It has been done to help the students and teachers at no cost by M Salman Sherazi. This work (pdf) is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0. It has been done to… anonymous@undisclosed.example.com (Anonymous) Sun, 28 Nov 2021 18:17:44 +0000 https://www.mathcity.org/msc/mcqs_short_questions MCQs and Short Questions Topology: Short Questions and MCQs Topology is a compulsory subject in MSc Mathematics in most of the universities of Pakistan. Normed Spaces: Short Questions and MCQs Short questions and MCQs related to the normed spaces in a single PDF file. Real Analysis: Short Questions and MCQs It is very much similar to calculus but a little bit more abstract. anonymous@undisclosed.example.com (Anonymous) Thu, 05 Aug 2021 08:24:36 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/ex6-3-p6 Question 7 and 8 Exercise 6.3 Solutions of Question 7 and 8 of Exercise 6.3 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$20$\begin{align}{ }^{20} C_2&=\dfrac{20 !}{(20-2)2!}!\\ &=\dfrac{20!}{18!\cdot 2!}\\ &=190\end{align}$7$$10$$3$$7$$10$$${ }^{10} C_7=\dfrac{10 !}{(10-7) ! 7 !}=120$$$7$$4.$$4$$${ }^7 C_4=\dfrac{7 !}{(7-4) ! 4 !}=35.$$$35$$10.$ anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:47 +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/fa16-mth322 ~~DISCUSSION:off~~ MTH322: Real Analysis II (Fall 2016) Do you have questions or comments? Please use Discussion at the end of this page. This course is offered to MSc, Semester III at Department of Mathematics, COMSATS Institute of Information Technology, Attock campus. The is course need rigorous knowledge of continuity, differentiation, integration, sequences and series of numbers, that is many notion included in anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:09 +0000 https://www.mathcity.org/atiq/sp17-mth322 ~~DISCUSSION:closed~~ MTH322: Real Analysis II (Spring 2017) Do you have questions or comments? Please use Discussion at the end of this page. This course is offered to MSc, Semester III at Department of Mathematics, COMSATS Institute of Information Technology, Attock campus. The is course need rigorous knowledge of continuity, differentiation, integration, sequences and series of numbers, that is many notion included in anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:35 +0000 https://www.mathcity.org/fsc/fsc_part_2_mcqs MCQs/Objective: HSSC-II On this page, MCQ/Objective for FSc-II (HSSC-II) or FSc Part 2 are given. * Objective Mathematics 12th by Muhammad Shahbaz NEW * Short Questions without answers by Mr. Akhtar Abbas for FSc Part 2. * Short Questions by Mr. Akhtar Abbas * Short Questions without answers by Mr. Akhtar Abbas for FSc Part 2. anonymous@undisclosed.example.com (Anonymous) Fri, 21 Mar 2025 15:21:10 +0000 https://www.mathcity.org/fsc/fsc_part_1_mcqs/short_questions_by_mr._akhtar_abbas Short Questions by Mr. Akhtar Abbas * We are very thankful to Mr. Akhtar Abbas for sharing these short questions. * These short questions are selected from previous 5 years papers of different boards. Solve these at your own to perform well in annual exams. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:46:09 +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/msc/mcqs_short_questions/toplogy Topology: Short Questions and MCQs We are going to add short questions and MCQs for Topology. This is a compulsory subject in MSc and BS Mathematics in most of the universities of Pakistan. The author of this page is Dr. Atiq ur Rehman, PhD. This page will be updated periodically. $\mathbb{R}$$X=\{a\}$$X$$X$$X$$\tau$$\mathbb{N}$$\tau$$(\mathbb{Z}, \tau)$$\mathbb{N}$$\tau$$A=\{\pm 100,\pm 101, \pm 102, ... \}$$\tau$$E=\{0,\pm 2,\pm 4,...\}$$\tau$$\tau$$B=\{1,2,3,...,99\}$$\tau$$C=\{10^{10}+n : n … anonymous@undisclosed.example.com (Anonymous) Mon, 03 Apr 2023 06:51:14 +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}$${ }^{n} C_{r}+{ }^{n} C… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 09:00:35 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-3-p10 Question 13 and 14, Exercise 6.3 Solutions of Question 13 and 14 of Exercise 6.3 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $6$$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/fsc-part1-kpk FSc Part 1 (Mathematics): KPK [A Textbook of Mathematics for Class XI] A Textbook of Mathematics for Class XI is published by Khybar Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. The book has total of twelve (12) chapters. This book is written by Prof. Dr. Gulzar Ali Khan, Prof. Dr. Islam Noor and Prof. Dr. Muhammad Shah. anonymous@undisclosed.example.com (Anonymous) Tue, 12 Dec 2023 17:09:07 +0000 https://www.mathcity.org/math-11-kpk FSc/ICS Part 1 (Mathematics): KPK [A Textbook of Mathematics for Class XI] A Textbook of Mathematics for Class XI is published by Khybar Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. The book has total of twelve (12) chapters. This book is written by Prof. Dr. Gulzar Ali Khan, Prof. Dr. Islam Noor and Prof. Dr. Muhammad Shah. anonymous@undisclosed.example.com (Anonymous) Tue, 06 Feb 2024 13:20:19 +0000 https://www.mathcity.org/atiq/fa17-mth322 MTH322: Real Analysis II (Fall 2017) This course is offered to MSc, Semester III at Department of Mathematics, COMSATS Institute of Information Technology, Attock campus. The is course need rigorous knowledge of continuity, differentiation, integration, sequences and series of numbers, that is many notion included in anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:10 +0000 https://www.mathcity.org/atiq/fa21-mth321 MTH321: Real Analysis I (Fall 2021) Discussion is available at the end of this page. One is free to ask any question or comment. [Photo-illustration of Zeno's Paradox] At the end of this course the students will be able to understand the basic set theoretic statements and emphasize the proofs’ development of various statements by induction. Define the limit of, a function at a value, a sequence and the Cauchy criterion. Prove various theorems about limits of sequences and functions and emphas… anonymous@undisclosed.example.com (Anonymous) Fri, 28 Oct 2022 11:10:02 +0000 https://www.mathcity.org/atiq/math-510-s2012 MATH-510: Topology Objectives of the course This is an introductory course in topology, giving the basics of the theory. Course contents Topological spaces, bases and sub-bases, first and second axiom of countability, separability, continuous functions and homeomorphism, finite product space. Separation axioms $(T_0, T_1, T_2)$ anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:24 +0000 https://www.mathcity.org/atiq/sp16-mth322 MTH322: Real Analysis II (Spring 2016) This course was teach to MSc III and IV. Course Contents: Sequences of functions: convergence, uniform convergence, uniform convergence and continuity, uniform convergence and integration, uniform convergence and differentiation, the exponential and logarithmic function, the trigonometric functions. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:34 +0000 https://www.mathcity.org/atiq/sp20-mth321 MTH321: Real Analysis I (Spring 2020) Discussion is available at the end of this page. One is free to ask any question or comment. ~~DISCUSSION~~ [Photo-illustration of Zeno's Paradox] At the end of this course the students will be able to understand the basic set theoretic statements and emphasize the proofs’ development of various statements by induction. Define the limit of, a function at a value, a sequence and the Cauchy criterion. Prove various theorems about limits of sequences and fun… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:41 +0000 https://www.mathcity.org/fsc/fsc_part_1_mcqs MCQs/Objective: HSSC-I Short Questions by Mr. Akhtar Abbas NEW Short Questions without answers by Mr. Akhtar Abbas for FSc Part 1. MCQs-Short Questions by Mr Parvez Khan MCQs and Short Question by Mr. Parvez Khan composed by Momin Ali: Text Book of Algebra and Trigonometry Class XI (Punjab Textbook Board, Lahore) MCQs by Nauman Idrees anonymous@undisclosed.example.com (Anonymous) Thu, 02 May 2024 17:05:13 +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/fsc-part1-ptb/important-questions/ch02-functions-and-groups Ch 02: Functions and Groups The important questions of Chapter 2 of Textbook of Algebra and Trigonometry Class XI is published by Punjab Textbook Board (PTB) Lahore, Pakistan has been given on this page. These questions are selected from old papers.$(2,4)$$\{a,\{b,c\}\}$$A-B=A \cup B^c$$p \longrightarrow q$$\{(1,2),(2,5),(3,7),(4,9),(5,11)\}$$\{a,b \}$$\{\{a,b\}\}$$~(p \longrightarrow q) \longrightarrow p$$A \cap(B \cup C)=(A \cap B)\cup(A \cap C)$$A=\{1,2,3,4\}$$B=\{3,4,5,6,7,8\}$$C=\{5,6,7,9,… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:39 +0000 https://www.mathcity.org/fsc/fsc_part_1_mcqs/mcqs-short_questions_by_mr._parvez_khan MCQs-Short Questions by Mr Parvez Khan * MCQs and Short Question written by Mr. Parvez Khan, composed by Mr. Momin Ali from Text Book of Algebra and Trigonometry Class XI (Punjab Textbook Board, Lahore) * Key to the MCQs is given at page 57. Erratum $180^{\circ}$ anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:46:05 +0000 https://www.mathcity.org/fsc/fsc_part_1_old_papers/fbise_annual_2009 FBISE Annual 2009 * FSc part 1 (HSSC-I) mathematics paper conducted by Federal Board of Intermediate and Secondary Education (FBISE), Islamabad has been analyse on this web page with the help of chart. Three type of chart are given in which one includes bar chart between chapters and marks, 2nd one include relation between algebraic and trigonometric portion and 3rd one contains pie chart which show the portion of questions from exercises to non-exercise question from book a anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:46:16 +0000 https://www.mathcity.org/fsc/fsc_part_1_old_papers/fbise_annual_2011 FBISE Annual 2011 * FSc part 1 (HSSC-I) mathematics paper conducted by Federal Board of Intermediate and Secondary Education (FBISE), Islamabad has been analyse on this web page with the help of chart. Three type of chart are given in which one includes bar chart between chapters and marks, 2nd one include relation between algebraic and trigonometric portion and 3rd one contains pie chart which show the portion of questions from exercises to non-exercise question from book a anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:46:18 +0000 https://www.mathcity.org/fsc/fsc_part_1_old_papers/fbise_annual_2012 FBISE Annual 2012 * FSc part 1 (HSSC-I) mathematics paper conducted by Federal Board of Intermediate and Secondary Education (FBISE), Islamabad has been analyse on this web page with the help of chart. Three type of chart are given in which one includes bar chart between chapters and marks, 2nd one include relation between algebraic and trigonometric portion and 3rd one contains pie chart which show the portion of questions from exercises to non-exercise question from book a anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:46:18 +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_2_mcqs/mcqs-short_questions_by_mr._parvez_khan MCQs-Short Questions by Mr. Parvez Khan * MCQs and Short Question by Mr. Parvez Khan composed by Momin Ali: Calculus and Analytic Geometry, MATHEMATICS 12 (Punjab Textbook Board, Lahore). Answers are given at page 32. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:46:42 +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/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/fsc/fsc_part_1_mcqs/short_questions_by_mr._akhtar_abbas/view View * We are very thankful to Mr. Akhtar Abbas for sharing these short questions. These short questions are selected from previous five years papers of different boards. Solve these at your own to perform well in annual examination. Recommended book for these short questions is “Text Book of Algebra and Trigonometry Class XI (Punjab Textbook Board, Lahore)”. But any student Mathematics can get benefit from it. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:57:20 +0000 https://www.mathcity.org/fsc-part2-ptb FSc/ICS Part 2 (Mathematics): PTB [Calculus and Analytic Geometry, MATHEMATICS 12] 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. One this page we have posted Notes (Solutions), MCQs, short question, sample papers and old papers related to this subject. This book has wide scope and it is part of syllabus of Mathematics in FSc/ICS from all board (Board of Intermedi… anonymous@undisclosed.example.com (Anonymous) Sun, 22 Sep 2024 08:59:48 +0000 https://www.mathcity.org/math-11-nbf Mathematics 11 for FSc ICS (NBF) [A Textbook of Mathematics for Class XI] Model Textbook of Mathematics for Class XI is published by National Book Foundation (NBF), Islamabad, Pakistan. NBF can be considered as Federal Textbook Board Islamabad. The book has total of nine (9) chapters. This book is written by Dr. Khalid Mahmood, Dr. Saleem Ullah Satti, M Dabeer Mughal, Dr. Naveed Akmal and Dr. Shahzad Ahmad. anonymous@undisclosed.example.com (Anonymous) Sat, 14 Feb 2026 14:30:09 +0000 https://www.mathcity.org/bsc/tests_by_mobeen_munir Tests by Mobeen Munir Calculus Chapter 01 ARW Chapter 1,2 & 5 (Important Questions) NEW Download PDF (6KB) ARW Chapter 01, Test 01 Download PDF (51KB) Method Chapter 01 ARW Chapter 01, Test 03 Download PDF (52KB) Chapter 03 ARW Chapter 03, Test 01 Download PDF (67KB) Calculus & Methods (Mixed) anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:41:13 +0000 https://www.mathcity.org/fsc-part1-ptb/fbise-papers Old Question Papers/Model Papers HSSC-I (FSc-I): FBISE [FBISE Paper Papers HSSC-I] Old (past) question papers and model papers of mathematics for HSSC-I (FSc Part 1) conducted by Federal Board of Intermediate and Secondary Education (FBISE), Islamabad. Paper Pattern The recommended book for the mathematics paper is anonymous@undisclosed.example.com (Anonymous) Mon, 11 Dec 2023 13:01:04 +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$$a<b \wedge… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:43:03 +0000 https://www.mathcity.org/fsc-part2-ptb/fbise-papers Old Question Papers/Model Papers HSSC-II (FSc-II): FBISE Old (past) question papers and model papers of mathematics (math) for HSSC-II (FSc Part 2) conducted by Federal Board of Intermediate and Secondary Education (FBISE), Islamabad. Paper Pattern anonymous@undisclosed.example.com (Anonymous) Sun, 02 Jan 2022 18:31:48 +0000 https://www.mathcity.org/fsc/fsc_part_2_formulas_introduction_to_analytics_geometry Chap 04: Formulas Introduction to Analytics Geometry On these four pages, one can find all the formulas used in Chapter 04: Formulas Introduction to Analytics Geometry of FSc Part 2. There are five exercises in chapter 04 with lot of questions. These basic things help to solve the questions easily without going to the depth of each concept. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:42:43 +0000 https://www.mathcity.org/math-11-pectaa/mcqs Multiple Choice Questions (MCQs) [FSC ICS Part 1 Math MCQs (11th Class) – Punjab Textbook Board] FSC or ICS Part 1 Math Multiple Choice Questions (MCQs) (11th Class). Mathematics 11 (FSc or ICS Part 1, HSSC-I) is published by Punjab Education, Curriculum, Training and Assessment Authority (PECTAA) Lahore - Pakistan formally know as Punjab Textbook Board (PTB) anonymous@undisclosed.example.com (Anonymous) Tue, 26 Aug 2025 11:48:11 +0000 https://www.mathcity.org/atiq/sp15-mth321/mcqs MCQs or Short Questions On this page, MCQs or short questions with out answers are given. Students need to find the answer them self. This page will be updated occasionally and new MCQs or short question will be posted here. * A number which is neither even nor odd is$2n$$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/paper_pattern/sargodha_university Syllabus/Model Papers for Sargodha University Syllabus for the subjects General Mathematics, A-Course of Mathematics and B-Course of Mathematics for BSc (private and regular) from University of Sargodha, Sargodha - PAKISTAN. Every subject consists of two papers of 100 marks each. In every paper there are three sections with four questions. A student have to attempt two questions from each section. These papers are renamed from Pure and Applied Mathematics. For details, please see anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:46:03 +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$$\lim\limits_{\theta \to 0}\frac{\tan \theta-\sin \theta}{\sin^3\t… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:54 +0000 https://www.mathcity.org/fsc-part2-ptb/important-questions/unit-02-differentiation Unit 02: Differentiation Here is the list of important questions. * Differentiate $\frac{(x^2+1)^2}{x^2-1}$ $w.r.t.x$. --- BSIC Gujranwala (2016) * If $x=at^2$, $y=2at$. Find $\frac{dy}{dx}$ --- BSIC Gujranwala (2016) * Differentiate $x^2-\frac{1}{x^2}$ $w.r.t.x^2$. --- BSIC Gujranwala (2016) * Prove that $\frac{d}{dx}(tan^{-1}x)=\frac{1}{1+x^2}$ --- BSIC Gujranwala (2016)$\frac{d}{dx}(sinh^{-1}x)=\frac{1}{\sqrt{1+x^2}}$$y=x^2ln(\frac{1}{x})$$\frac{dy}{dx}$$x=sin\theta$$y=sin m\t… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:54 +0000 https://www.mathcity.org/fsc-part2-ptb/important-questions/unit-03-integration Unit 03: Integration Here is the list of important questions. * Evaluate $\int \frac{1}{\sqrt{x}(\sqrt{x}+1)}dx$ --- BSIC Gujranwala (2016) * Find $\int \frac{1}{1+ cosx}dx$ --- BSIC Gujranwala (2016) * Evaluate $\int \frac{1}{x \ln x}dx$ --- BSIC Gujranwala (2016) * Find $\int x \ln x dx$ --- BSIC Gujranwala (2016) * Evaluate $\int e^{2x}(-sinx+2cosx)dx$ --- BSIC Gujranwala (2016)$\int^2_1(x^2+1)dx$$\int^{\frac{\pi}{4}}_0 \sec x(\sec x+\tan x)dx$$\sin y cosec x \frac{dy}{d… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:55 +0000 https://www.mathcity.org/fsc-part2-ptb/important-questions/unit-04-introduction-to-analytic-geometry Unit 04: Introduction to Analytic Geometry Here is the list of important questions. * Find the area between $x-axis$ and the curve $y=4x-x^2$ --- BSIC Gujranwala (2016) * Find $h$ if $A(-1,h)$, $B(3,2)$, $C(7,3)$ are collinear --- BSIC Gujranwala (2016) * Find the point three fifth of the way along the line segment from $A(-5,8)$$B(5,3)$$2$$y-intercept$$5$$5x-12y+39=0$$2x^2+3xy-5y^2=0$$x-y-4=0$$7x+y+20=0$$6x+y-14=0$$5x-12y+39=0$$(4,6)$$(4,8)$$x-2y+1=0$$2x-y+2=0$$A(2,-5)$$B(-4,-3)$$C(-1… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:56 +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$$-y\leq… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:57 +0000 https://www.mathcity.org/fsc-part2-ptb/important-questions/unit-06-conic-section Unit 06: Conic section Here is the list of important questions. * Find the centre and radius of the circle given by the equation $4x^2+4y^2-8x+12y-25=0$ --- BSIC Gujranwala (2016) * Find equation of tangent to the circle $x^2+y^2=2$ parallel to the line $x-2y+1=0$ --- BSIC Gujranwala (2016)$x^2=-16y$$(0,\pm5)$$\frac{3}{5}$$ABC$$a^2=b^2+c^2-2bc \cos A$$A(4,5)$$B(-4,-3)$$C(8,-3)$$9x^2-18x+4y^2+8y-23=0$$x^2+y^2-6x+4y+13=0$$x^2+y^2=25$$(4,3)$$(-3,1)$$x=3$$(0,0)$$(6,0)$$(4,0)$$x^2-4x-8y+4=… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:57 +0000 https://www.mathcity.org/fsc-part2-ptb/important-questions/unit-07-vectors Unit 07: Vectors Here is the list of important questions. * Find position vector of a point which divide the join of $P$ and $Q$ with position vectors $2\underline i-3 \underline j$ and $3\underline i+2\underline j$ in ratio $4:3$. --- BSIC Gujranwala (2016) * Find $a$ and $b$ so that the vectors $3\underline i-\underline j+4\underline k$ and $a\underline i+b\underline j+2\underline k$ are parallel. $\cos$$u.v$$u=3\underline i+\underline j-\underline k$$v=2\underline i-\underline j-\un… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:57 +0000 https://www.mathcity.org/fsc/fsc_part_1_solutions/ch01 Chapter 01: Number System [Chapter 01: Number System] Notes (Solutions) of Chapter 01: Number System, Textbook of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary * Rational numbers and irrational numbers$\mathbb{C}$$(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$$\sqrt{-1}$ anonymous@undisclosed.example.com (Anonymous) Sat, 03 Jun 2023 16:30:31 +0000 https://www.mathcity.org/fsc/fsc_part_1_solutions/ch02 Chapter 02: Sets, Functions and Groups Notes (Solutions) of Chapter 02: Sets, Functions and Groups, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. [Chapter 02: Sets, Functions and Groups] Contents & summary * Introduction$p\leftrightarrow q$ anonymous@undisclosed.example.com (Anonymous) Wed, 05 Apr 2023 12:54:57 +0000 https://www.mathcity.org/fsc/fsc_part_1_solutions/ch04 Chapter 04: Quadratic Equations [Chapter 04: Quadratic Equations] Notes (Solutions) of Chapter 04: Quadratic Equations, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board, Lahore. Contents & summary * Introduction * Solutions of Quadratic Equations anonymous@undisclosed.example.com (Anonymous) Sun, 04 Jun 2023 16:13:15 +0000 https://www.mathcity.org/fsc/fsc_part_1_solutions/ch05 Chapter 05: Partial Fractions [Chapter 05: Partial Fractions] Notes (Solutions) of Chapter 05: Partial Fractions, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary * Introduction * Rational Fraction$\frac {P(x)}{Q(x)}$ anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:46:29 +0000 https://www.mathcity.org/fsc/fsc_part_1_solutions/ch06 Chapter 06: Sequences and Series [Chapter 06: Sequences and Series] Notes (Solutions) of Chapter 06: Sequences and Series, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary * Introduction * Types of Sequences$l,m,n$$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$$(a+x)^n$ anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:46:30 +0000 https://www.mathcity.org/fsc/fsc_part_1_solutions/ch09 Chapter 09: Fundamentals of Trigonometry [Chapter 09: Fundamentals of Trigonometry] Notes (Solutions) of Chapter 09: Fundamentals of Trigonometry, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. This chapter has four exercise and solutions of those exercises are given below which can be downloaded in PDF format or can be viewed online.$D^\circ M'S''$$45^\circ , 30^\circ , 60^\circ$$0^\circ , 90^\circ , 180^\circ , 270^\circ , 36… anonymous@undisclosed.example.com (Anonymous) Sun, 04 Jun 2023 16:03:04 +0000 https://www.mathcity.org/fsc/fsc_part_1_solutions/ch10 Chapter 10: Trigonometric Identities [Chapter 10: Trigonometric Identities] Notes (Solutions) of Chapter 10: Trigonometric Identities, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. There are four exercise in this chapter. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:46:32 +0000 https://www.mathcity.org/fsc/fsc_part_1_solutions/ch11 Chapter 11: Trigonometric Functions and their Graphs [Chapter 11: Trigonometric Functions and their Graphs] Notes (Solutions) of Chapter 11: Trigonometric Functions and their Graphs, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary $ y = \sin x$$-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_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/bsc/paper_pattern/sargodha_university/a-course_of_mathematics A-Course of Mathematics (Paper A & B) This subject is consists of two papers of 100 marks each. One is called “Paper A” and other is called “Paper B”. This page is updated on February 15, 2015. This syllabus is for 1st Annual 2015 and onward organized by University of Sargodha, Sargodha. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:55:54 +0000 https://www.mathcity.org/bsc/paper_pattern/sargodha_university/applied_mathematics Applied Mathematics (Paper A & B) This paper consista of two papers of 100 marks each. One paper is called “Paper A” and other is called “Paper B”. Paper A * NOTE: attempt two questions from each section. SECTION-I (4/12: 17,17,17,17) $(\lambda ,\mu )$ anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:55:46 +0000 https://www.mathcity.org/bsc/paper_pattern/sargodha_university/b-course_of_mathematics B-Course of Mathematics (Paper A & B) This subject is consists of two papers of 100 marks each. One is called “Paper A” and other is called “Paper B”. This page is updated on February 15, 2015. This syllabus is for 1st Annual 2015 and onward organized by University of Sargodha, Sargodha.$(\lambda ,\mu )$ anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:55:55 +0000 https://www.mathcity.org/bsc/paper_pattern/sargodha_university/general_mathematics General Mathematics (Paper A & B) This subject is consists of two papers of 100 marks each. One is called “Paper A” and other is called “Paper B”. This syllabus is for 1st Annual 2015 and onward organized by University of Sargodha (UoS), Sargodha. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:55:55 +0000 https://www.mathcity.org/bsc/paper_pattern/sargodha_university/pure_mathematics Pure Mathematics (Paper A & B) This paper consist of two papers of 100 marks each. One paper is called “Paper A” and the other is called “Paper B”. Paper A * NOTE: attempt two questions from each section. SECTION-I (4/12: 17,17,17,17) anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:55:59 +0000 https://www.mathcity.org/matric/9th_science/unit11/11-1 Exercise 11.1 (Solutions) On this page solutions of Exercise of Unit 11: Parallelograms and Triangles of Mathematics 9 written by Dr. Karamat H. Dar and Prof. Irfan-ul-Haq has been given. There are two questions in this exercise and solution of both the questions are given below.$ABCD$$m\angle B=130^\circ$$m\angle B=m\angle D$$m\angle B=m\angle D=130^\circ$\begin{align} & m\angle A +\,\,m\angle B=180^\circ \\ & m\angle A+\,{{130}^{\circ }}={{180}^{\circ }}\\ & m\angle A={{180}^{\circ }}-{{130}… anonymous@undisclosed.example.com (Anonymous) Mon, 20 Mar 2023 17:51:29 +0000 https://www.mathcity.org/fsc FSc Notes (Solutions), MCQs/Objective type questions, model papers and old/previous papers (of FBISE and BISE) given here, are useful for FSc Part 1 and Part 2 (HSSC). Text Book of Algebra and Trigonometry Class XI and Calculus and Analytic Geometry, MATHEMATICS 12, Punjab Text Book Board Lahore anonymous@undisclosed.example.com (Anonymous) Fri, 26 May 2023 06:29:22 +0000 https://www.mathcity.org/msc BS/MSc This section mainly divided into three parts. Please see below. Syllabus for M.Sc Mathematics Scheme of studies and syllabus for M.Sc Mathematics for University of Sargodha and University of the Punjab. Notes of Mathematics Selection of notes required to prepare different papers of MSc or BS Mathematics. It doesn't mean, these cannot be used for other purpose. anonymous@undisclosed.example.com (Anonymous) Tue, 09 Aug 2022 18:29:03 +0000 https://www.mathcity.org/notes Notes of Mathematics [Notes of Mathematics] Mathematics is a language of science and is a basic need for physical or natural sciences as well as social sciences. On this page, notes on different subjects related to mathematics are listed. These notes or resources might be helpful for ADS or BS or MSc or MPhil Mathematics. These notes are send by different students or teachers. We are very thankful to them for sending us these notes. These notes are provided as it is as open educational resource… anonymous@undisclosed.example.com (Anonymous) Mon, 25 May 2026 18:19:50 +0000 https://www.mathcity.org/participate Participate If you have written notes, then why not share on MathCity.org with others. This is your help to mathematics students and teachers (یہ صدقہ جاریہ ہے). For further details, please contact webmaster for more information. * The material given on MathCity.org is free and publish to help the students to learn Mathematics. It will take a lot of time to manage this website and prepare new material to publish on this website. anonymous@undisclosed.example.com (Anonymous) Sun, 31 May 2026 09:56:03 +0000 https://www.mathcity.org/atiq/fa19-mth321 MTH321: Real Analysis I (Fall 2019) [Photo-illustration of Zeno's Paradox by Juliana Jiménez Jaramillo. Photo by Twildlife/Thinkstock] At the end of this course the students will be able to understand the basic set theoretic statements and emphasize the proofs’ development of various statements by induction. Define the limit of, a function at a value, a sequence and the Cauchy criterion. Prove various theorems about limits of sequences and functions and emphasize the proofs’ development. Def… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:11 +0000 https://www.mathcity.org/atiq/fa19-mth611 MTH611: Integral Inequalities (Fall 2019) This course is offered to students of MS(Mathematics) at COMSATS University Islamabad. This is a three credit hour course. Contents Some Quadrature rules and their applications Ostrowski Inequality in L1-, Lp- and L∞ spaces and applications Grüss Inequality, its variants and applications Ostrowski- Grüss inequalities, their consequences and applications Purturbed results for Ostrowski and Ostrowski- Grüss type inequalities Inequalities for convex func… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:12 +0000 https://www.mathcity.org/atiq/fa20-mth424 MTH424: Convex Analysis (Fall 2020) [Convex Analysis] Objectives: At the end of this course the students will be able to understand the concept of Convex Analysis, convex sets, convex functions, Differential of the convex function. Developing ability to study the Hadamard-Hermite inequalities and their applications. Prepare students to be self independent and enhance their mathematical ability by giving them home work and projects.$f(x)=x$$\mathbb{R}$$f(x)=x^2$$\mathbb{R}$$f:[a,b]\to \mathbb{… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:16 +0000 https://www.mathcity.org/atiq/fa22-mth321 MTH321: Real Analysis I (Fall 2022) ~~DISCUSSION~~ [Photo-illustration of Zeno's Paradox] At the end of this course the students will be able to understand the basic set theoretic statements and emphasize the proofs’ development of various statements by induction. Define the limit of, a function at a value, a sequence and the Cauchy criterion. Prove various theorems about limits of sequences and functions and emphasize the proofs’ development. Define continuity of a function and uniform conti… anonymous@undisclosed.example.com (Anonymous) Mon, 15 May 2023 07:16:43 +0000 https://www.mathcity.org/atiq/fa22-mth604 ~~DISCUSSION~~ MTH604: Fixed Point Theory and Applications (Fall 2022) [FPTA] Course Objectives: This course is intended as a brief introduction to the subject with a focus on Banach Fixed Point theorems fixed point theorem and its application to nonlinear differential equations, nonlinear integral equations, real and complex implicit functions theorems and system of nonlinear equations. Some generalizations and similar results e. g. Kannan Fixed Point theorems, Banach Fixed Point theorem f… anonymous@undisclosed.example.com (Anonymous) Fri, 06 Jan 2023 04:37:11 +0000 https://www.mathcity.org/atiq/fa23-mth480 MTH480: Introductory Quantum Mechanics Objective The physical principles and mathematical formalism of quantum theory, with emphasis on applications to atomic, molecular, and many-body physics; scattering phenomena; and electromagnetism (photon physics). $x(t)={{t}^{3}}+2\sin t$$t=\dfrac{\pi }{6}$$v(t)={{t}^{2}}+t{{e}^{t}}$ anonymous@undisclosed.example.com (Anonymous) Sat, 07 Oct 2023 18:27:32 +0000 https://www.mathcity.org/atiq/math-608 MATH-608: History of Mathematics Course contents History of Numerations: Egyptian, Babylonian, Hindu and Arabic contributions. Algebra: Including the contributions of Al-Khwarzmi and Ibn Kura. Geometry: the areas, the work of Al-Toussi on Euclud’s axioms, Analysis. The Calculus: Newton, Leibniz and Gauss, The concept of limit, Laplace. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:26 +0000 https://www.mathcity.org/atiq/sp18-mth604 MTH604: Fixed Point Theory and Applications 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 multi-valued mappings are also ed… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:39 +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/sp24-mth480 MTH480: Introductory Quantum Mechanics Objective The physical principles and mathematical formalism of quantum theory, with emphasis on applications to atomic, molecular, and many-body physics; scattering phenomena; and electromagnetism (photon physics). anonymous@undisclosed.example.com (Anonymous) Wed, 14 Feb 2024 09:26:16 +0000 https://www.mathcity.org/atiq/sp25-mth424 MTH424: Convex Analysis (Spring 2025) [Convex Analysis] Convex analysis is a branch of mathematics that studies convex sets and convex functions. A set is convex if a straight line between any two points in the set always stays inside it. This field is important in optimization, economics, and engineering. It helps in solving real-world problems like minimizing costs, maximizing profits, and designing efficient systems. Convex analysis is widely used in machine learning, finance, and physics. 😊… anonymous@undisclosed.example.com (Anonymous) Mon, 16 Jun 2025 18:51:28 +0000 https://www.mathcity.org/fsc-part1-kpk/important-questions Important Questions This page will be updated soon. * Prove that (without calculator) $\sin 10^{\circ}\sin 30^{\circ}\sin 50^{\circ}\sin 70^{\circ}=\frac{1}{16}$ --- BISE Gujrawala(2015) * Prove that $\sin(\frac{\pi}{4}-\theta)\sin(\frac{\pi}{4}+\theta)=\frac{1}{2}\csc^2\theta$ --- BISE Gujrawala(2017) * Prove that $\sin(\theta+\frac{\pi}{6})=\cos\theta$ --- BISE Gujrawala(2017) * Using without table or calculator find $tan(1110^{\circ})$ --- BISE Sargodha(2015), BISE Gujrawala(201… anonymous@undisclosed.example.com (Anonymous) Mon, 28 Aug 2023 17:06:07 +0000 https://www.mathcity.org/fsc-part1-kpk/mcqs Multiple Choice Questions (MCQs) Here are the sample MCQs at this time. Page will be updated periodically. SAMPLE MCQs * $i^{13}=$............. * (A) $i$ * (B) 1 * (C) -1 * (D) 2 * Set of all possible subsets of $S$ is called * (A) Equivalent sets$1, \omega, \omega^2$$-1, \omega, \omega^2$$-1, -\omega, -\omega^2$$1, -1, 2$$ax^2+bx+c=0$$a=0, b\neq 0$$a\neq 0$$a=b=0$$b=$$ax^2+bx+c=0$$a=0, b\neq 0$$a\neq 0$$a=b=0$$b=$$n!=n(n-1)(n-2)...3\cdot 2\cdot 1$$n$ anonymous@undisclosed.example.com (Anonymous) Mon, 28 Aug 2023 17:03:58 +0000 https://www.mathcity.org/math-11-kpk/important-questions Important Questions This page will be updated soon. * Prove that (without calculator) $\sin 10^{\circ}\sin 30^{\circ}\sin 50^{\circ}\sin 70^{\circ}=\frac{1}{16}$ --- BISE Gujrawala(2015) * Prove that $\sin(\frac{\pi}{4}-\theta)\sin(\frac{\pi}{4}+\theta)=\frac{1}{2}\csc^2\theta$ --- BISE Gujrawala(2017) * Prove that $\sin(\theta+\frac{\pi}{6})=\cos\theta$ --- BISE Gujrawala(2017) * Using without table or calculator find $tan(1110^{\circ})$ --- BISE Sargodha(2015), BISE Gujrawala(201… anonymous@undisclosed.example.com (Anonymous) Tue, 05 Dec 2023 02:44:42 +0000 https://www.mathcity.org/math-11-kpk/mcqs Multiple Choice Questions (MCQs) Here are the sample MCQs at this time. Page will be updated periodically. SAMPLE MCQs * $i^{13}=$............. * (A) $i$ * (B) 1 * (C) -1 * (D) 2 * Set of all possible subsets of $S$ is called * (A) Equivalent sets$1, \omega, \omega^2$$-1, \omega, \omega^2$$-1, -\omega, -\omega^2$$1, -1, 2$$ax^2+bx+c=0$$a=0, b\neq 0$$a\neq 0$$a=b=0$$b=$$ax^2+bx+c=0$$a=0, b\neq 0$$a\neq 0$$a=b=0$$b=$$n!=n(n-1)(n-2)...3\cdot 2\cdot 1$$n$ anonymous@undisclosed.example.com (Anonymous) Tue, 05 Dec 2023 02:44:44 +0000 https://www.mathcity.org/math-11-nbf/mcqs MCQs: Math 11 NBF Multiple Choice Questions (MCQs) of the Model Textbook of Mathematics for Class XI is published by National Book Foundation (NBF), Islamabad, Pakistan. NBF can be considered as Federal Textbook Board Islamabad. Unit 01: Complex Numbers $\operatorname{part}(\mathrm{s})$$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) Sun, 20 Oct 2024 18:55:54 +0000 https://www.mathcity.org/notes/computing-tools-for-mathematics-asif-arshad Computing Tools for Mathematics by Asif Arshad [omputing Tools for Mathematics by Asif Arshad] Computing tools for mathematics are algorithms that use computers to solve mathematical issues. They are employed in a number of scientific, technical, industrial, and technological domains where computer is crucial and central. They may assist in creating precise and effective numerical techniques, for instance, to solve physical or biological models. anonymous@undisclosed.example.com (Anonymous) Sat, 05 Aug 2023 20:30:34 +0000 https://www.mathcity.org/notes/fundamental-of-complex-analysis-prof-m-saleem Fundamental of Complex Analysis (Solutions of Some Exercises) [Fundamental of Complex Analysis, Solutions of Some Exercises] Complex analysis is the study of functions that exist in the complex plane, that is, functions with complex arguments and complex outputs. With roots in the 18th century and the years just before, it is one of the classical branches of mathematics. In the 20th century, significant figures in mathematics who are connected to complex numbers include Euler, Gauss, Riemann, … anonymous@undisclosed.example.com (Anonymous) Sat, 15 Apr 2023 18:26:12 +0000 https://www.mathcity.org/people/fiaz Muhammad Fiaz Hussain We are very thankful to Muhammad Fiaz Hussain for contributing to our website. * M.Sc, M.Phil. in Applied & Analytic Mathematics (COMSATS Institute of Information Technology Islamabad) * Email: <fiaz.hussain24@gmail.com> * Cell: +92-333-6639466; +92-300-504117 anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:44:41 +0000 https://www.mathcity.org/ppsc/ppsc-mock-interview-mathematics PPSC Mock Interview Lecturer Mathematics [PPSC Mock Interview Lecturer Mathematics] This handout is shared by Mr. Rashad Wattu and written by Sawaira Sikandar. We are really very thankful to him for providing this handout and appreciates his efforts to publish it on MathCity.org. This handout contains the questions collected from the different interviews of the Lecturer in Mathematics conducted by Public Service Commission (PSC). This is help full to prepare interview of all types of jobs whic… anonymous@undisclosed.example.com (Anonymous) Thu, 25 Mar 2021 16:31:54 +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/wiki/dokuwiki DokuWiki wiki:dokuwiki DokuWiki is a simple to use and highly versatile Open Source wiki software that doesn't require a database. It is loved by users for its clean and readable Formatting Syntax. The ease of maintenance, backup and integration makes it an administrator's favorite. Built in anonymous@undisclosed.example.com (Anonymous) Sat, 28 Jun 2025 07:28:13 +0000 https://www.mathcity.org/bsc/notes_of_mathematical_method/ch03_matrices Chapter 03: Matrices Notes of the book Mathematical Method written by S.M. Yusuf, A. Majeed and M. Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN. The difficulty level of this chapter is very low. Most of the questions involve calculations. This chapter is wide range of applications in Linear Algebra. In many universities teachers include this chapter in the syllabus of Linear Algebra for BS students of mathematics and other subjects. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:45:44 +0000 https://www.mathcity.org/bsc/notes_of_mathematical_method/ch04_system_of_linear_equations Chapter 04: System of Linear Equations Notes of the book Mathematical Method written by S.M. Yusuf, A. Majeed and M. Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN. The difficulty level of this chapter is low. Most of the questions involve calculations. This chapter is wide range of applications in Linear Algebra and Operations Research. In many universities teachers include this chapter in the syllabus of Linear Algebra and Operations Research for BS students of mathematics and other … anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:45:45 +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-kpk/sol/unit10 Unit 10: Trigonometric Identities of Sum and Difference of Angles (Solutions) This is a tenth unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions.$a\sin\theta + b\cos \theta$$r\sin(\theta +\psi )$$a = r\cos\psi$$b=r\sin\psi$ anonymous@undisclosed.example.com (Anonymous) Sat, 09 Sep 2023 07:00:54 +0000 https://www.mathcity.org/fsc-part1-ptb/important-questions/ch01-number-systems Ch 01: Number Systems * Simplify $(i)^{19}$ --- BISE Gujrawala(2015) * If $z$ be a complex number then prove that $\overline{z_1 + z_2}=\overline z_1 +\overline z_2$ --- BISE Sargodha(2015) * Simplify $\frac{2}{\sqrt{5}+\sqrt{-8}}$ in the form of $a+ib$ --- BISE Sargodha(2015) * Simplify by justify each step $\frac{\frac{1}{a}-\frac{1}{b}}{1-\frac{1}{a}\frac{1}{b}}$ --- BISE Sargodha(2015)$(\sqrt{2}, -\sqrt{5})$$\{0,-1\}$$a \div ib$$(-1)^\frac{-21}{2}$$(0,1)$$\{1,-1\}$$|z_… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:38 +0000 https://www.mathcity.org/fsc-part1-ptb/important-questions/ch03-matrices-and-determinants Ch 03: Matrices and Determinants * Fin $x$ and $y$ if $ \left[ {\begin{array}{c} x+3&1\\ -3& 3y-4 \end{array}} \right]= \left[ {\begin{array}{c} 2&1\\ -3&2 \end{array}} \right]$ --- BISE Gujrawala(2015) * Solve for matrix $A$ if $\left[ {\begin{array}{c}4&3\\ 2&2 \end{array}} \right]A-\left[ {\begin{array}{c} 2&3\\ -1&-2 \end{array}} \right]= \left[ {\begin{array}{c} -1&-4\\ 3&6 \end{array}} \right]$ --- BISE Gujrawala(2015) * Prove without expansion $ \left[ {\begin{array}{c} 6&7&… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:39 +0000 https://www.mathcity.org/fsc-part1-ptb/important-questions/ch04-quadratic-equations Ch 04: Quadratic Equations * Reduce $x^{-2}-10=3x^{-1}$ to quadratic form --- BISE Gujrawala(2015) * Show that $x^3-y^3=(x-y)(x-wy)(x-w^2y)$ --- BISE Gujrawala(2015) * If $n$ is an odd integer, is $(x+a)$ factor of $(x^n+a^n)$? --- BISE Gujrawala(2015) * If the roots of $px^2+qx+q=0$ are $\alpha$, $\beta$,then prove that $$\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/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-part1-ptb/important-questions/ch06-sequence-and-series Ch 06: Sequences and Series * If $\frac{1}{a}$, $\frac{1}{b}$ and $\frac{1}{c}$ are in $G.P$. Show that $r=\pm \sqrt{\frac{a}{c}}$ --- BISE Gujranwala(2015),BISE Sargodha(2015), BISE Sargodha(2017),BISE Lahore(2017) * With usual notation show that $AH=G^2$ --- BISE Gujrawala(2015) * Find $n$, so that $\frac{a^n+b^n}{a^{n-1}+b^{n-1}}$ maybe $A.M$ between $a$ and $b$$y=1+\frac{x}{2}+\frac{x^4}{4}+...$$x=2(\frac{y-1}{y})$$9th$$\frac{1}{3}, \frac{1}{5}, \frac{1}{7},...$$a=-2$$b=-6$$A.G$$\f… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:42 +0000 https://www.mathcity.org/fsc-part1-ptb/important-questions/ch07-permutation-combination-and-probablity Ch 07: Permutation, Combination and Probability * Find $n$ when ${^nC_{12}}={^nC_6}$ --- BISE Gujranwala(2015) * Evaluate ${^{20}C_{17}}$ without calculator --- BISE Gujranwala(2015) * How many $6-digit$ numbers can be formed from the digits $2,2,3,3,4,4$? How many of them with lie between $400,000$ and $430,000$? ---$``PLANE"$$^nC_4=^nC_{n-r}$$6-digits$$n^3-n$$6$$n=2,3$$n$$^nP_2=30$$6-dided$$n$$^nC_{12}=^nC_6$$^{n-1}C_r+^{n-1}C_{r-1}=^nC_r$$\frac{a_5}{a_3}=\frac{4}{9}$$a_2=\frac{4}{9}$… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:42 +0000 https://www.mathcity.org/fsc-part1-ptb/important-questions/ch08-mathematical-induction-and-binomial-theorem Ch 08: Mathematical Induction and Binomial Theorem * Using binomial theorem,expand $\left(\frac{x}{2}-\frac{2}{x^2}\right)$ --- BISE Gujranwala(2015) * Find the $6$th term in the expansion of $\left( x^2-\frac{3}{2x}\right)$ --- BISE Gujranwala(2015) * Expand $\left( 8-2x\right)^{-1}$ up to two terms. --- BISE Gujranwala(2015) * Use binomial theorem to show that $1+\frac{1}{4}+\frac{1.3}{4.8}+\frac{1.3.5}{4.8.12},...=\sqrt{2}$$(1.03)^{\frac{1}{3}}$$(a+x)$$n$$x$$(x-\frac{2}{x})^{10}$$… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:43 +0000 https://www.mathcity.org/fsc-part1-ptb/important-questions/ch09-fundamentals-of-trigonometry Ch 09: Fundamental of Trigonometry * Find the value of the remaining trigonometric functions of $\theta$, If $cos \theta=\frac{12}{13}$ and the terminal side of the angle is not in the $I$ Quadrant. --- BISE Gujrawala(2015) * Express in radian $120'40''$ --- BISE Gujrawala(2017) * Verify $2 $ $\sin 45^{\circ} +\frac{1}{2}\cos 45^{\circ}=\frac{3}{\sqrt{2}}$$cosce \theta+tan\theta sec \theta=cosec \theta sec^2 \theta$$(tan\theta+cot\theta)^2=sec^2\theta cosec^2\theta$$150^{\circ}$$\theta$$l… anonymous@undisclosed.example.com (Anonymous) Sun, 28 Nov 2021 18:19:36 +0000 https://www.mathcity.org/fsc-part1-ptb/important-questions/ch10-trigonometric-identities Ch 10: Trigonometric Identities * Prove that (without calculator) $\sin 10^{\circ}\sin 30^{\circ}\sin 50^{\circ}\sin 70^{\circ}=\frac{1}{16}$ --- BISE Gujrawala(2015) * Prove that $\sin(\frac{\pi}{4}-\theta)\sin(\frac{\pi}{4}+\theta)=\frac{1}{2}\csc^2\theta$ --- BISE Gujrawala(2017) * Prove that $\sin(\theta+\frac{\pi}{6})=\cos\theta$ --- BISE Gujrawala(2017) * Using without table or calculator find $tan(1110^{\circ})$ --- BISE Sargodha(2015), BISE Gujrawala(2017)$sin(180^{\circ}+\a… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:44 +0000 https://www.mathcity.org/fsc-part1-ptb/important-questions/ch11-trigonometric-functions-and-their-graphs Ch 11: Trigonometric Functions and Their Graphs * Find the period of $\sin 4x$ --- BISE Gujrawala(2015) * Find the period of $\tan 4x$ --- BISE Gujrawala(2017) * Find the period of $\sin\frac{x}{5}$ --- BISE Sargodha(2015), BISE Sargodha(2016) * Find the period of $cosec10x$ --- BISE Sargodha(2015)$\cot\frac{x}{2}$$\sin x$$2\pi$ anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:44 +0000 https://www.mathcity.org/fsc-part1-ptb/important-questions/ch12-application-of-trigonometry Ch 12: Applications of Trigonometry * Find the value of $tan\frac{\alpha}{2}$ in term of $s$ --- BISE Gujrawala(2015) * Solve $\triangle ABC$ if $b=125$, $r=53^{\circ}$, $\alpha=47^{\circ}$ --- BISE Gujrawala(2015) * Show that $r_1=stan\frac{\alpha}{2}$ --- BISE Gujrawala(2015) * Define an escribed circle.--- BISE Gujrawala(2015) * With usual notation prove that $r_1+r_2+r_3-r=4R$$\triangle ABC$$r=90^{\circ}$$\alpha=62^{\circ}40'$$b=796$$\beta$$a$$\triangle ABC$$a=18$$b=24$$c=30$$\fra… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:46 +0000 https://www.mathcity.org/fsc-part1-ptb/important-questions/ch13-inverse-trigonometry-functions Ch 13: Inverse Trigonometry Functions * Find the value of $cos^{-1}(\frac{1}{2})$ --- BISE Gujrawala(2015) * Prove that $2tan^{-1}(\frac{1}{3})+tan^{-1}(\frac{1}{7})=\frac{\pi}{4}$ --- BISE Gujrawala(2015), FBISE(2016) * Prove that $sin^{-1}(\frac{1}{\sqrt{5}})+cot^{-1}(3)=\frac{\pi}{4}$--- BISE Sargodha(2015), BISE Sargodha(2016), BISE Gujrawala(2017) * Prove that $cos^{-1}(-x)=\pi-cos^{-1}x$--- BISE Gujrawala(2017), FBISE(2017) $cos^{-1}(\frac{12}{13})=sin^{-1}(\frac{5}{13})$$cos(sin… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:46 +0000 https://www.mathcity.org/fsc-part1-ptb/important-questions/ch14-solutions-of-trigonometric-equation Ch 14: Solutions of Trigonometric Equation * Solve $cose^2\theta=\frac{4}{3}$ in $[0,2\pi]$--- BISE Gujrawala(2015), BISE Sargodha(2016), BISE Gujrawala(2017) * Solve $sinx=\frac{1}{2}$ in $[0,2\pi]$--- BISE Gujrawala(2015) * Solve $cot\theta = \frac{1}{\sqrt{3}}$, $\theta \in [0,2\pi]$--- BISE Gujrawala(2017), BISE Sargodha(2016) * Solve $sec^2\theta=\frac{4}{3}$ in $[0,2\pi]$--- BISE Sargodha(2015)$4cos^2x-3=0$$x \in [0,2\pi]$$secx=-2$$x \in [0,2\pi]$$cosec\theta=2$$[0,2\pi]$$tanx=-1… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:46 +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$$$ \begin{aligned} \frac{d}{dx}\left(\frac{x+1}{x-1}\right) &= \frac{(x-1)\frac{d}{dx}(x… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:05 +0000 https://www.mathcity.org/math-11-kpk/sol/unit01 Unit 01: Complex Numbers (Solutions) This is a first unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the students will be able to$z$$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/unit04 Unit 04: Sequence and Series (Solutions) This is a forth unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the students will be able to$n$$n$$n$$n$$n$ anonymous@undisclosed.example.com (Anonymous) Fri, 05 Jan 2024 17:30:16 +0000 https://www.mathcity.org/math-11-kpk/sol/unit05 Unit 05: Miscellaneous Series (Solutions) This is a fifth unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the students will be able to$n$$n$$n$$n$$n$$\dfrac{1}{a(a+d)}+\dfrac{1}{(a+d)(a+2 d)}+ \cdots $ anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:34:56 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06 Unit 06: Permutation, Combination and Probability (Solutions) This is a sixth unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the students will be able to$n$$n!$$n$$r$$^nP_r$$n$$r$$n$$r$ anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:34:51 +0000 https://www.mathcity.org/math-11-kpk/sol/unit07 Unit 07: Mathmatical Induction and Binomial Theorem (Solutions) This is a seventh unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the students will be able to$(x+y)^n$$n$$(x+y)^n$$(x+ y)^n.$$(1 +x)^n$$n$$n.$$(l +x)^n$$x$$|x| < 1$$n$ anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:34:59 +0000 https://www.mathcity.org/math-11-kpk/sol/unit10 Unit 10: Trigonometric Identities of Sum and Difference of Angles (Solutions) This is a tenth unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions.$a\sin\theta + b\cos \theta$$r\sin(\theta +\psi )$$a = r\cos\psi$$b=r\sin\psi$ anonymous@undisclosed.example.com (Anonymous) Tue, 05 Dec 2023 02:44:48 +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$$z$ anonymous@undisclosed.example.com (Anonymous) Mon, 27 Oct 2025 18:47:40 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02 Unit 02: Matrices and Determinants (Solutions) This is a second unit of the book Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. On this page we have provided the solutions of the questions. anonymous@undisclosed.example.com (Anonymous) Sun, 08 Feb 2026 17:04:31 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04 Unit 04: Sequences and Seeries This is a forth unit of the book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. On this page we have provided the solutions of the questions.$n$$n$$n$$n$$n$$n$$n$ anonymous@undisclosed.example.com (Anonymous) Sun, 06 Oct 2024 17:46:06 +0000 https://www.mathcity.org/math-11-nbf/sol/unit05 Unit 05: Polynomials [Unit 05: Polynomials] This is a fifth unit of the book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. On this page we have provided the solutions of the questions. anonymous@undisclosed.example.com (Anonymous) Sun, 13 Oct 2024 18:05:52 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06 Unit 06: Permutation and Combination This is a sixth unit of the book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. On this page we have provided the solutions of the questions.$n$$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/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/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/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/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/unit02/ex2-2-p8 Question 8,9 & 10, Exercise 2.2 Solutions of Questions 8,9 & 10 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\left| \begin{matrix}1+x & y & z \\x & 1+y & z \\x & y & 1+z \end{matrix} \right|=1+x+y+z$$$L.H.S.=\left| \begin{matrix} 1+x & y & z \\ x & 1+y & z \\ x & y & 1+z \\ \end{matrix} \right|$$$$=\left| \begin{matrix} 1 & 0 & -… anonymous@undisclosed.example.com (Anonymous) Tue, 05 Dec 2023 02:45:26 +0000 https://www.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p12 Question 14 & 15, Exercise 2.2 Solutions of Questions 14 & 15 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$A=\begin{bmatrix}0 & 2 & 2 \\-1 & 3 & 2 \\1 & 0 & 5\end{bmatrix}$$A^{-1}$$$A=\left[ \begin{matrix} 0 & 2 & 2 \\ -1 & 3 & 2 \\ 1 & 0 & 5 \\ \end{matrix} \right]$$$A^{-1}$$$A^{-1}=\dfrac{Adj\,\,A}{|A|}$$$$Adj\,\,A={{\left[ \begin… anonymous@undisclosed.example.com (Anonymous) Tue, 05 Dec 2023 02:45:20 +0000 https://www.mathcity.org/math-11-kpk/sol/unit02/ex2-2-p13 Question 16 & 17, Exercise 2.2 Solutions of Questions 16 & 17 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$A=\begin{bmatrix}3 & -1 \\4 & 2\end{bmatrix}$$|A^{-1}|=\dfrac{1}{|A|}$$$A=\left[ \begin{matrix} 3 & -1 \\ 4 & 2 \\ \end{matrix} \right]$$$$|A|=6+4$$$$\Rightarrow |A|=10\ldots (1)$$$$A^{-1}=\dfrac{1}{|A|}AdjA$$$$AdjA=\left[ \begin{ma… anonymous@undisclosed.example.com (Anonymous) Tue, 05 Dec 2023 02:45:21 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/re-ex6-p1 Question 1, Review Exercise 6 Solutions of Question 1 of Review Exercise 6 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $3\,\,^nP_3=^nP_4$$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