Merging man & maths https://www.mathcity.org/ Thu, 04 Jun 2026 16:12:45 +0000 FeedCreator 1.8 https://www.mathcity.org/_media/logo.svg https://www.mathcity.org/ https://www.mathcity.org/fsc-part1-ptb/mcq-bank/ch04 MCQs: Ch 04 Quadratic Equations 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 $ax^2+bx+c=0$$ax^2+bx+c=0$$b \neq 0$$c \neq 0$$a \neq 0$$x$$ax^2+bx+c$$ax^2+bx+c=0$$\{a,b\}$$ax^2+bx+c=0$$a\neq 0$$x= \frac{b \pm \sqrt{b^2-4ac}}{a}$$x= \frac{-b \pm \sqrt{b^2+4ac}}{2a}$$x= \frac{-b \pm \sqrt{4ac-b^2}}{2a}$$x= \frac{-b \pm \sqrt{b^2-… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:50 +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/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/notes/ordinary-differential-equations-hamad-safi Ordinary Differential Equations (ODE) by Hammad Safi [Ordinary Differential Equations by Hammad Safi] An equation containing the derivatives of one or more dependent variables with respect to one or more independent variables is said to be a differential equation or a differential equation is an equation which contains one or more terms and derivatives of one or more dependent variables with respect to other variables (independent variables) or an equation that contains derivatives of dependent… anonymous@undisclosed.example.com (Anonymous) Sun, 09 Mar 2025 09:26:46 +0000 https://www.mathcity.org/fsc-part1-ptb/definitions Definitions: FSc Part 1 (Mathematics): PTB On this page, all the definitions of “Textbook of Algebra and Trigonometry Class XI, published by Punjab Textbook Board (PTB) Lahore, Pakistan are given. We are very thankful to Muhammad Waqas Sulaiman for his valuable contribution.$\frac{p}{q}$$p,q \in \mathbb{Z}$$q\neq 0$$\frac{p}{q}$$p,q \in \mathbb{Z}$$q\neq 0$$\mathbb{R}$$0.3333....,21.134134$$\pi = 3.1415...$$\divideontimes$$z=x+iy$$x,y \in \mathbb{R}, i = \sqrt{-1}$$x$$y$$z$$2, 3+\sqrt{3}i, \fra… anonymous@undisclosed.example.com (Anonymous) Sun, 22 Sep 2024 17:12:21 +0000 https://www.mathcity.org/notes/method-of-mathematical-physics-m-usman-hamid Method of Mathematical Physics by Mr. Muhammad Usman Hamid [Method of Mathematical Physics by Mr. Muhammad Usman Hamid] Method of Mathematical Physics is an area of mathematics concerned with the application of mathematical methods to physics problems. Mathematical techniques for statistical mechanics, quantum mechanics, and classical mechanics are all included in this large field. anonymous@undisclosed.example.com (Anonymous) Thu, 20 Apr 2023 11:54:01 +0000 https://www.mathcity.org/notes/partial-differential-equations-m-usman-hamid Partial Differential Equations (PDE) by M Usman Hamid The course provides a foundation to solve PDE’s with special emphasis on wave, heat and Laplace equations, formulation and some theory of these equations are also intended. We are really very thankful to Prof. Muhammad Usman Hamid for providing these notes and appreciates his effort to publish these notes on MathCity.org anonymous@undisclosed.example.com (Anonymous) Sun, 09 Mar 2025 09:27:42 +0000 https://www.mathcity.org/notes/partial-differential-equations-muzammil-tanveer Partial Differential Equations (PDE) by Muzammil Tanveer [Partial Differential Equations] These notes are provided and composed by Mr. Muzammil Tanveer. We are really very thankful to him for providing these notes and appreciates his effort to publish these notes on MathCity.org Name: Partial Differential Equations or PDEs anonymous@undisclosed.example.com (Anonymous) Sun, 09 Mar 2025 09:29:35 +0000 https://www.mathcity.org/khuram Khuram Ali Khan Khuram Ali Khan, PhD Associate Professor Department of Mathematics University of Sargodha Sargodha - PAKISTAN. Email: <khuram@MathCity.org> Field of Research: Difference and functional equations, Real functions, Mathematical inequalities involving convex functions, Time Scales Calculus, Soft Sets anonymous@undisclosed.example.com (Anonymous) Fri, 13 Jun 2025 12:03:59 +0000 https://www.mathcity.org/notes/mechanics-iii-analytic-dynamics-ii-dr-babar-ahmad Mechanics III (Analytic Dynamics II) by Dr Babar Ahmad [Mechanics III (Analytic Dynamics II) by Dr Babar Ahmad] We are very thankful to Dr Babar Ahmad for sharing his book on MathCity.org. This book is very helpful for undergraduate students of Science and Engineering Programs. This book is shared by the permission of the author and he keeps the copyright of the book. anonymous@undisclosed.example.com (Anonymous) Sat, 10 May 2025 17:33:01 +0000 https://www.mathcity.org/bsc/notes_of_mathematical_method/ch09_first_order_differential_equations Chapter 09: First Order Differential Equations Notes of the book Mathematical Method written by S.M. Yusuf, A. Majeed and M. Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN. Contents and summary * D.E and their classification * Formation of differential equation anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:45:50 +0000 https://www.mathcity.org/fsc-part2-ptb/definitions Definitions: FSc Part 2 (Mathematics): PTB On this page, all the definitions of “Calculus and Analytic Geometry, MATHEMATICS 12” (Mathematics FSc Part 2 or HSSC-II), Punjab Textbook Board (PTB) Lahore, Pakistan are given. We are very thankful to $A=x^2$$f:X\to Y$$X$$f:X\to Y$$y$$Y$$y=ax+b$$x$$y$$f(x)=2x-6$$p(x) = {a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + {a_{n - 2}}{x^{n - 2}} + ... + {a_1}x + {a_0}$${a_0},\,{a_1},\,{a_2},...,{a_n}$$f(x)=ax+b$$X$$I:X\to X$$X$$Y$$C:X \rightarrow Y$$C(x)=a$$x \in X$$… anonymous@undisclosed.example.com (Anonymous) Sun, 20 Oct 2024 18:16:52 +0000 https://www.mathcity.org/notes/mathematical-method-muzammil-tanveer Mathematical Method by Sir Muhammad Awais Aun [Mathematical Method by Muzammil Tanveer] Mathematical methods are the approaches employed by mathematicians to address issues in mathematics and science. Algebra, functions, relations and associated graphs, calculus, and statistics are examples of mathematical techniques. Through their usage in resolving practical issues, they are applied to modelling. anonymous@undisclosed.example.com (Anonymous) Mon, 17 Apr 2023 09:02:35 +0000 https://www.mathcity.org/fsc-part2-ptb/important-questions/unit-04-introduction-to-analytic-geometry Unit 04: Introduction to Analytic Geometry Here is the list of important questions. * Find the area between $x-axis$ and the curve $y=4x-x^2$ --- BSIC Gujranwala (2016) * Find $h$ if $A(-1,h)$, $B(3,2)$, $C(7,3)$ are collinear --- BSIC Gujranwala (2016) * Find the point three fifth of the way along the line segment from $A(-5,8)$$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/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$$\s… anonymous@undisclosed.example.com (Anonymous) Wed, 24 Jul 2024 18:33:10 +0000 https://www.mathcity.org/bsc/notes_of_mathematical_method/ch10_higher_order_linear_differential_equations Chapter 10: Higher Order Linear Differential Equations Notes of the book Mathematical Method written by S.M. Yusuf, A. Majeed and M. Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN. Contents and summary * Higher order linear differential equations anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:45:51 +0000 https://www.mathcity.org/fsc-part2-ptb/important-questions/unit-06-conic-section Unit 06: Conic section Here is the list of important questions. * Find the centre and radius of the circle given by the equation $4x^2+4y^2-8x+12y-25=0$ --- BSIC Gujranwala (2016) * Find equation of tangent to the circle $x^2+y^2=2$ parallel to the line $x-2y+1=0$ --- BSIC Gujranwala (2016)$x^2=-16y$$(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/bsc/notes_of_mathematical_method/ch09_first_order_differential_equations/viewer Chapter 09: Viewer Notes of Chapter 09: First Order Partial Differential Equations of Mathematical Method written by S.M. Yusuf, A. Majeed and M. Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN. PDF file of respective exercise can be downloaded from this page. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:54:45 +0000 https://www.mathcity.org/notes/differential-geometry-kaushef_salamat Differential Geometry (Notes) by Ms. Kaushef Salamat [Differential Geometry by Ms. Kaushef Salamat] Differential Geometry is an important field in the mathematical sciences. Usually, this is a compulsory or core subject for BS mathematics students. These notes are send by Ms. Kaushef Salamat. We are really very thankful to her for providing these notes and appreciates her effort to publish these notes on MathCity.org anonymous@undisclosed.example.com (Anonymous) Sat, 23 Aug 2025 11:29:10 +0000 https://www.mathcity.org/notes/theory-of-relativity-and-analytic-dynamics Theory of Relativity & Analytic Dynamics: Handwritten Notes [Theory of Relativity & Analytic Dynamics: Handwritten Notes] Theory of Relativity and Analytic Dynamics is a subject that encompasses two distinct topics: relativity theory and analytic dynamics. Albert Einstein's theory of relativity defines the fundamental principles that control how moving objects behave in relation to one another and to the observer. The motion of objects as a result of forces and torques is the subject of analyt… anonymous@undisclosed.example.com (Anonymous) Fri, 23 Aug 2024 07:42:56 +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/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/fsc/kpk_fsc_part_2 FSc Part 2 (KPK Boards) [A Textbook of Mathematics For Class XII] Notes of FSc Part 2 of “A Textbook of Mathematics For Class XII” published by Khyber Pakhtunkhwa Textbook Board, Peshawar. We are posting the notes chapter-wise. These notes are shared as open educational resources. This page will be continuously updated.$y=x^n$$y=(ax+b)^n$ anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:42:52 +0000 https://www.mathcity.org/fsc/fsc_part_1_solutions/ch04 Chapter 04: Quadratic Equations [Chapter 04: Quadratic Equations] Notes (Solutions) of Chapter 04: Quadratic Equations, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board, Lahore. Contents & summary * Introduction * Solutions of Quadratic Equations anonymous@undisclosed.example.com (Anonymous) Sun, 04 Jun 2023 16:13:15 +0000 https://www.mathcity.org/bsc/notes_of_mathematical_method/ch10_higher_order_linear_differential_equations/viewer Chapter 10: Viewer Notes of Chapter 10: Higher Order Linear Differential Equations of Mathematical Method written by S.M. Yusuf, A. Majeed and M. Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN. PDF file of respective exercise can be downloaded from this page. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:54:54 +0000 https://www.mathcity.org/cui Mathematics CUI: LaTeX Resources [Department of Mathematics, COMSATS University Islamabad, Attock Campus] This page contains LaTeX template of CIIT Mathematics, MSc Project and MS Thesis templates. Templates Download a zip file given below and extract it by right clicking on the file. BS Project Template: (Version 1.5, Uploaded: Sep 29, 2022)$\$$I$$\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/fsc/kpk_fsc_part_1 FSc Part 1 (KPK Boards) These are the notes of old book. The notes of new book is AVAILABLE HERE [FSc Part 2 KPTP] Notes of FSc Part 1 of “A Textbook of Mathematics For Class XI” published by Khyber Pakhtunkhwa Textbook Board, Peshawar. We are posting the notes chapter-wise. These notes are shared as open educational resources. This page will be continuously updated.$P(z)$$(\sum)$$\sum n$$\sum n^2$$\sum n^3$$n$$n$$$\frac{a}{a(a+d)}+\frac{a}{(a+d)(a+2d)}+...$$$^nP_r$$^nC_r=\left(\begin{smallm… anonymous@undisclosed.example.com (Anonymous) Tue, 12 Dec 2023 17:30:33 +0000 https://www.mathcity.org/fsc-part1-ptb/important-questions/ch04-quadratic-equations Ch 04: Quadratic Equations * Reduce $x^{-2}-10=3x^{-1}$ to quadratic form --- BISE Gujrawala(2015) * Show that $x^3-y^3=(x-y)(x-wy)(x-w^2y)$ --- BISE Gujrawala(2015) * If $n$ is an odd integer, is $(x+a)$ factor of $(x^n+a^n)$? --- BISE Gujrawala(2015) * If the roots of $px^2+qx+q=0$ are $\alpha$, $\beta$,then prove that $$\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/fsc_part_2_solutions/ch04 Unit 04: Introduction to Analytic Geometry [Unit 01: Functions and Limits] Notes (Solutions) of Unit 04: Introduction to Analytic Geometry, 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.$ax^2+ 2hxy+by^2=0$ anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:07 +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/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/notes/special-functions-muzammil-tanveer Special Functions by Dr. Muhey-U-Din These notes are provided and composed by Mr. Muzammil Tanveer. We are really very thankful to him for providing these notes and appreciates his effort to publish these notes on MathCity.org. Thease notes are based on the lectures by Dr. Muhey-U-Din. anonymous@undisclosed.example.com (Anonymous) Sun, 23 Jul 2023 16:48:30 +0000 https://www.mathcity.org/math-11-kpk/sol/unit01/ex1-3-p4 Question 5, Exercise 1.3 Solutions of Question 5 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) ${{z}^{2}}+z+3=0$$${{z}^{2}}+z+3=0.$$$a=1$$b=1$$c=3$\begin{align}z&=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\\ &=\dfrac{-1\pm \sqrt{{{\left( 1 \right)}^{2}}-4\left( 1 \right)\left( 3 \right)}}{2\left( 1 \right)}\\ &=\dfrac{-1\pm \sqrt{1-12}}{2}\\ &=\… anonymous@undisclosed.example.com (Anonymous) Tue, 05 Dec 2023 02:45:04 +0000 https://www.mathcity.org/math-11-kpk/sol/unit01/ex1-3-p5 Question 6, Exercise 1.3 Solutions of Question 6 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6(i) ${{z}^{4}}+{{z}^{2}}+1=0$$$z^4+z^2+1=0$$$$z^4+2z^2+1-z^2=0$$$$( z^2+1 )^2-z^2=0$$$$( z^2+1+z)( z^2+1-z )=0$$$$( z^2+z+1 )( z^2-z+1 )=0$$$$(z^2+z+1 )=0$$$$z=\dfrac{-1\pm \sqrt{1-4}}{2}$$$$z=\dfrac{-1\pm \sqrt{3}i}{2}$$$$(z^2-z+1 )=0$$$$z=\dfrac{1\pm … anonymous@undisclosed.example.com (Anonymous) Tue, 05 Dec 2023 02:45:04 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p3 Question 3, Exercise 2.6 Solutions of Question 3 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 x+3 y+4 z=2$$2 x+y+z=5$$3 x-2 y+z=-3$\begin{align*} \begin{aligned} 2x + 3y + 4z &= 2 \\ 2x + y + z &= 5 \\ 3x - 2y + z &= -3 \end{aligned}\end{align*}\begin{align*} A_{b} &=\quad \left[\begin{array}{cccc} 2 & 3 & 4 & 2 \\ 2 & 1 & 1 & 5 \\ 3… anonymous@undisclosed.example.com (Anonymous) Wed, 04 Sep 2024 03:11:14 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p6 Question 6, Exercise 2.6 Solutions of Question 6 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $5 x+3 y+z=6$$2 x+y+3 z=19$$x+2 y+4 z=25$\begin{align*} A &= \begin{bmatrix} 5 & 3 & 1 \\ 2 & 1 & 3 \\ 1 & 2 & 4 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad B = \begin{bmatrix} 6 \\ 19 \\ 25 \end{bmatrix} \end{alig… anonymous@undisclosed.example.com (Anonymous) Wed, 04 Sep 2024 03:13:00 +0000 https://www.mathcity.org/notes/fluid-dynamics-i-m-usman-hamid Fluid Dynamics I by Muhammad Usman Hamid [Fluid Dynamics I by Muhammad Usman Hamid] Explore comprehensive notes on Fluid Dynamics by Muhammad Usman Hamid. Covers fundamental concepts (viscosity, stress fields, continuum), fluid statics, and differential analysis. Special thanks to Mr. Anwar Khan for contributing these resources to MathCity.org. anonymous@undisclosed.example.com (Anonymous) Sun, 03 May 2026 08:26:32 +0000 https://www.mathcity.org/notes/fluid-mechanics-ii-muzammil-tanveer Fluid Mechanics II by Dr Rao Muzamal Hussain [Fluid Mechanics I by Muzammil Tanveer] These notes are provided and composed by Mr. Muzammil Tanveer. We are really very thankful to him for providing these notes and appreciates his effort to publish these notes on MathCity.org. These notes are based on lectures delivered by Mr. Muzammil Hussain at GC University Faisalabad. anonymous@undisclosed.example.com (Anonymous) Tue, 29 Aug 2023 06:36:09 +0000 https://www.mathcity.org/ppsc/ppsc-maths-2011 PPSC Paper 2011 (Lecturer in Mathematics) [PPSC Paper 2011 (Lecturer in Mathematics)] On this page, we have given question from old (past) paper of Lecturer in Mathematics conducted in year 2011. This is a MCQs paper and answers are given at the end of the paper. At the end of the PDF is also given to download. This paper is provided by Ms. $R$$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/matric/9th_science/unit08 Unit 08: Linear Graph and their Application On this page notes of Unit 08 of Mathematics 9 written by Dr. Karamat H. Dar and Prof. Irfan-ul-Haq are given. [Unit 08: Linear Graph and their Application] After studying this unit the students will be able to: * Identity pair of real numbers as an ordered pair.$O$$\left( O \right)$$\left( a,b \right)$$a\,$$b$$y=c.$$x=a.$$y=mx.$$y=mx+c.$ anonymous@undisclosed.example.com (Anonymous) Sat, 28 May 2022 19:29:21 +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/math-11-kpk/sol/unit05/ex5-4-p1 Question 1 Exercise 5.3 Solutions of Question 1 of Exercise 5.4 of Unit 05: Mascellaneous series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) $\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+\ldots$$n$$$T_n=\dfrac{1}{n(n+1)}$$$T_n$$$\dfrac{1}{n(n+1)}=\dfrac{A}{n}+\dfrac{B}{(n+1)}$$$n(n+1)$$$1=A(n+1)+B n=(A+B) n+A$$$n$$$A+B=0 \text{and} A=1$$$A=1$\begin{align}1+B&=0\\ B&=-1\end{align}\beg… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:20 +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/notes/numerical-analysis-ii Numerical Analysis II [Numerical Analysis by Muzammil Tanveer] These notes are provided and composed by Mr. Muzammil Tanveer. We are really very thankful to him for providing these notes and appreciates his effort to publish these notes on MathCity.org. Name Numerical Analysis II Compiled by Muzammil Tanveer anonymous@undisclosed.example.com (Anonymous) Sat, 05 Aug 2023 17:56:32 +0000 https://www.mathcity.org/people/umer Umer Asghar We are very thankful to Mr. Umer Asghar for his contribution to the website. * Email: <umermth2016@gmail.com> * Skype ID: sp15mmth06678 * Cell: +92-307-4896454 Contribution: * Notes of Metric Spaces by Umer Asghar NEW * Notes of Number Theory by Umer Asghar * Exercise 9.2 (BSc Mathematical Method) | VIEW View online | [Download PDF] * anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:44:50 +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$$\sin y cosec x \frac{dy}{d… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:55 +0000 https://www.mathcity.org/fsc/fsc_part_2_solutions/ch06 Unit 06: Conic Section [Unit 06: Conic Section] Notes (Solutions) of Unit 06: Conic Section, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. You can view online or download PDF. To view PDF, you must have PDF Reader installed on your system and it can be downloaded from Software section. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:08 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-3-p4 Question 4, Exercise 1.3 Solutions of Question 4 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 4(i) $(1-i) z+(1+i) \omega=3 ; 2 z-(2+5 i) \omega=2+3 i$\begin{align} &(1-i) z+(1+i) \omega=3 \quad \cdots(1)\\ &2 z-(2+5 i) \omega=2+3i \quad\cdots(2) \end{align}$2$\begin{align} &(2-2i)z+(2+2i) \omega=6 \quad \cdots (3) \end{align}$(1-i)$\b… anonymous@undisclosed.example.com (Anonymous) Mon, 15 Jul 2024 12:19:22 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p8 Question 7, Exercise 1.4 Solutions of Question 7 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 7(i) $\arg (z-1)=-\dfrac{\pi}{4}$$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/ex2-6-p1 Question 1, Exercise 2.6 Solutions of Question 1 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $ 2 x_{1}-3 x_{2}+4 x_{3}=0$$x_{1}-2 x_{2}+3 x_{3}=0$$4 x_{1}+x_{2}-6 x_{3}=0$\begin{align*} &2 x_{1}-3 x_{2}+4 x_{3}=0\cdots (i)\\ &x_{1}-2 x_{2}+3 x_{3}=0\cdots (ii)\\ &4 x_{1}+x_{2}-6 x_{3}=0\cdots (iii)\\ \end{align*}\begin{align*} A &= \le… anonymous@undisclosed.example.com (Anonymous) Wed, 04 Sep 2024 03:03:49 +0000 https://www.mathcity.org/atiq/fa25-csc456 CSC456: Stochastic Processes (Fall 2025) [Stochastic Processes (Fall 2025), Image Courtesy: Gemini] Course Objectives: * To define basic concepts from the theory of Markov chains and present proofs for the most important theorems. * To compute probabilities of transition between states and return to the initial state after long time intervals in Markov chains. anonymous@undisclosed.example.com (Anonymous) Sun, 28 Dec 2025 14:20:45 +0000 https://www.mathcity.org/fsc/kpk_fsc_part_1/chapter_12_graph_of_trigonometric_and_inverse_trigonometric_functions_and_solutions_of_trigonometric_equations Chapter 12: Graph of Trigonometric and Inverse Trigonometric Functions and Solutions of Trigonometric Equations Notes of Chapter 12: Graph of Trigonometric and Inverse Trigonometric Functions and Solutions of Trigonometric Equations of “A Textbook of Mathematics for Class XI anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:26 +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/math-11-nbf/sol/unit02/ex2-6-p4 Question 4, Exercise 2.6 Solutions of Question 4 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 x_{1}-x_{2}-x_{3}=2$$3 x_{1}-4 x_{2}+3 x_{3}=7$$4 x_{1}+2 x_{2}-5 x_{3}=10$\begin{align*} 2x_1 - x_2 - x_3 &= 2, \\ 3x_1 - 4x_2 + 3x_3 &= 7, \\ 4x_1 + 2x_2 - 5x_3 &= 10, \end{align*}\begin{align*} A_b &= \begin{bmatrix} 2 & -1 & -1 & : & 2 … anonymous@undisclosed.example.com (Anonymous) Wed, 04 Sep 2024 03:11:42 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p5 Question 5, Exercise 2.6 Solutions of Question 5 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $x_{1}+x_{2}+2 x_{3}=8$$-x_{1}-2 x_{2}+3 x_{3}=1$$3 x_{1}-7 x_{2}+4 x_{3}=10$$A X=B$\begin{align*} &A = \begin{bmatrix} 1 & 1 & 2 \\ -1 & -2 & 3 \\ 3 & -7 & 4 \end{bmatrix}, \quad X = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}, \quad B = \… anonymous@undisclosed.example.com (Anonymous) Wed, 04 Sep 2024 03:12:33 +0000 https://www.mathcity.org/atiq Atiq ur Rehman, PhD Atiq ur Rehman, PhD Associate Professor (Tenured) Department of Mathematics COMSATS University Islamabad, Attock Campus Kamra Road, Attock - PAKISTAN. Email: <Atiq@MathCity.org>, <atiq@cuiatk.edu.pk> Field of Research: Difference and functional equations, Real functions, Inequalities in monotonic and convex functions anonymous@undisclosed.example.com (Anonymous) Sun, 01 Feb 2026 14:25:54 +0000 https://www.mathcity.org/atiq/sp26-csc456 CSC456: Stochastic Processes (Spring 2026) [Stochastic Processes (Spring 2026), Image Courtesy: Gemini] Course Learning Outcomes: * To define basic concepts from the theory of Markov chains and present proofs for the most important theorems. * To compute probabilities of transition between states and return to the initial state after long time intervals in Markov chains. anonymous@undisclosed.example.com (Anonymous) Wed, 03 Jun 2026 08:38:47 +0000 https://www.mathcity.org/fsc-part1-kpk/definitions Definitions: FSc Part1 KPK A Textbook of Mathematics for Class XI is published by Khybar Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. The book has total of twelve (12) chapters. Definition of the book provide the quick overview of the book.$360^\circ$$\theta$$90^{\circ} \pm \theta, 180^{\circ} \pm \theta, 270^{\circ} \pm \theta, 360^{\circ} \pm \theta$$16^\circ 13' 9''$$sin(\alpha+2\pi)=sin\alpha$$sin x=\frac{2}{7}$$cos x-tan x=0$ anonymous@undisclosed.example.com (Anonymous) Mon, 28 Aug 2023 16:59:59 +0000 https://www.mathcity.org/fsc/fsc_part_2_formulas_introduction_to_analytics_geometry Chap 04: Formulas Introduction to Analytics Geometry On these four pages, one can find all the formulas used in Chapter 04: Formulas Introduction to Analytics Geometry of FSc Part 2. There are five exercises in chapter 04 with lot of questions. These basic things help to solve the questions easily without going to the depth of each concept. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:42:43 +0000 https://www.mathcity.org/math-11-kpk/definitions Definitions: FSc Part1 KPK A Textbook of Mathematics for Class XI is published by Khybar Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. The book has total of twelve (12) chapters. Definition of the book provide the quick overview of the book.$360^\circ$$\theta$$90^{\circ} \pm \theta, 180^{\circ} \pm \theta, 270^{\circ} \pm \theta, 360^{\circ} \pm \theta$$16^\circ 13' 9''$$sin(\alpha+2\pi)=sin\alpha$$sin x=\frac{2}{7}$$cos x-tan x=0$ anonymous@undisclosed.example.com (Anonymous) Tue, 05 Dec 2023 02:44:42 +0000 https://www.mathcity.org/math-12-nbf/sol Solutions: Math 12 NBF [Solutions of Textbook of Mathematics 12] Solutions of “Textbook of Mathematics 12 published by National Book Foundation (NBF), Islamabad, Pakistan”. NBF can be considered as Federal Textbook Board Islamabad. This comprehensive guide, Solutions for Mathematics 12, serves as a definitive resource for students mastering the advanced HSSC curriculum. Published by the National Book Foundation (NBF), it bridges the gap between complex theory and practical application. The tex… anonymous@undisclosed.example.com (Anonymous) Mon, 04 May 2026 16:11:21 +0000 https://www.mathcity.org/mathcraft/sample-01-latex MathCraft: PDF to LaTeX file: Sample-01 If the PDF file provided by you as follows: Then the output LaTeX file is as follows: \documentclass[10pt]{amsart} %\usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} %\usepackage[version=4]{mhchem} \usepackage{stmaryrd} \usepackage{bbold} \usepackage{hyperref} \usepackage{enumerate} \hypersetup{colorlinks=true, linkcolor=blue, filecolor=magenta, urlcolor=cyan,} \urlstyle{same} \title{… anonymous@undisclosed.example.com (Anonymous) Mon, 25 Mar 2024 16:29:59 +0000 https://www.mathcity.org/notes/complex-analysis-m-usman-hamid Complex Analysis by M Usman Hamid [Complex Analysis by M Usman Hamid] We are really very thankful to Muhammad Usman Hamid for providing these notes and appreciates his effort to publish these notes on MathCity.org It covers the one part of the syllabus of Complex Analysis paper of MSc Mathematics. See the contents of the notes given below to see the topics covered by these notes.$$ x^2+4=0, x^2+x+1=0 \text{ and } x^2-2x+3=0 $$$i=\sqrt{-1}$$i^2=-1$ anonymous@undisclosed.example.com (Anonymous) Sat, 08 Feb 2025 08:11:42 +0000 https://www.mathcity.org/notes/differential_geometry_notes Differential Geometry: Handwritten Notes [Differential Geometry: Handwritten Notes] Differential geometry is a discipline of mathematics that investigates the geometry of smooth objects and spaces, sometimes known as smooth manifolds. It investigates the geometric properties of curves and surfaces using the methods of differential and integral calculus, linear algebra, and multilinear algebra. Mathematical analysis and differential geometry are related concepts. These are the lecture notes of … anonymous@undisclosed.example.com (Anonymous) Sat, 23 Aug 2025 11:27:57 +0000 https://www.mathcity.org/notes/differential-geometry-syed-hassan-waqas Differential Geometry by Syed Hassan Waqas [Differential Geometry by Syed Hassan Waqas] In the field of mathematics known as differential geometry, smooth manifolds—also known as smooth shapes and spaces—and their geometry are studied. Investigating the geometric characteristics of curves and surfaces involves the use of the tools of differential and integral calculus, linear algebra, and multilinear algebra. Differential calculus and mathematical analysis are related to differential geometry.… anonymous@undisclosed.example.com (Anonymous) Sat, 23 Aug 2025 11:28:41 +0000 https://www.mathcity.org/notes/fluid-mechanics-ali-raza Fluid Mechanics by Ali Raza [Fluid Mechanics by Ali Raza] Fluid mechanics is the branch of physics that studies how fluids (liquids, gases, and plasmas) behave and interact with forces and energy. Fluid mechanics has many applications in engineering, geophysics, biology, and other fields. anonymous@undisclosed.example.com (Anonymous) Sat, 27 Jul 2024 16:23:08 +0000 https://www.mathcity.org/notes/mathematical-method-usman-hamid Mathematical Method by Muhammad Usman Hamid [Mathematical Method by Muhammad Usman Hamid] These notes are send by Muhammad Usman Hamid. We acknowledged his efforts to published these notes on MathCity.org. The main objective of this course is to provide the students with a range of mathematical methods that are essential to the solution of advanced problems encountered in the fields of applied physics and engineering. In addition this course is intended to prepare the students with mathematic… anonymous@undisclosed.example.com (Anonymous) Mon, 17 Apr 2023 08:43:53 +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)$$\left(\frac{3}{8}\right)$ anonymous@undisclosed.example.com (Anonymous) Sat, 05 Aug 2023 17:59:07 +0000 https://www.mathcity.org/notes/qr2-tools-for-quantitative-reasoning-m-usman-hamid Quantitative Reasoning II (QR2: Tools for Reasoning Skills) [Quantitative Reasoning II (Tools for Reasoning Skills)] This handout provides a comprehensive exploration of fundamental mathematical and logical concepts, making it an essential read for students and professionals alike. It begins with an introduction to enumeration and its practical applications, followed by an in-depth discussion on quantitative reasoning, number systems, and arithmetic operations. The book highlights the contribut… anonymous@undisclosed.example.com (Anonymous) Fri, 01 Aug 2025 18:31:56 +0000 https://www.mathcity.org/fsc-part1-ptb/important-questions/ch03-matrices-and-determinants Ch 03: Matrices and Determinants * Fin $x$ and $y$ if $ \left[ {\begin{array}{c} x+3&1\\ -3& 3y-4 \end{array}} \right]= \left[ {\begin{array}{c} 2&1\\ -3&2 \end{array}} \right]$ --- BISE Gujrawala(2015) * Solve for matrix $A$ if $\left[ {\begin{array}{c}4&3\\ 2&2 \end{array}} \right]A-\left[ {\begin{array}{c} 2&3\\ -1&-2 \end{array}} \right]= \left[ {\begin{array}{c} -1&-4\\ 3&6 \end{array}} \right]$ --- BISE Gujrawala(2015) * Prove without expansion $ \left[ {\begin{array}{c} 6&7&… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:39 +0000 https://www.mathcity.org/fsc-part1-ptb/important-questions/ch14-solutions-of-trigonometric-equation Ch 14: Solutions of Trigonometric Equation * Solve $cose^2\theta=\frac{4}{3}$ in $[0,2\pi]$--- BISE Gujrawala(2015), BISE Sargodha(2016), BISE Gujrawala(2017) * Solve $sinx=\frac{1}{2}$ in $[0,2\pi]$--- BISE Gujrawala(2015) * Solve $cot\theta = \frac{1}{\sqrt{3}}$, $\theta \in [0,2\pi]$--- BISE Gujrawala(2017), BISE Sargodha(2016) * Solve $sec^2\theta=\frac{4}{3}$ in $[0,2\pi]$--- BISE Sargodha(2015)$4cos^2x-3=0$$x \in [0,2\pi]$$secx=-2$$x \in [0,2\pi]$$cosec\theta=2$$[0,2\pi]$$tanx=-1… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:46 +0000 https://www.mathcity.org/fsc/fsc_part_1_solutions/ch03 Chapter 03: Matrices and Determinants [Chapter 03: Matrices and Determinants] Notes (Solutions) of Chapter 03: Matrices and Determinants, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary * Introduction$2\times2$$2\times2$$2\times2$$n\geq 3$$n\geq 3$ anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:46:27 +0000 https://www.mathcity.org/fsc/fsc_part_1_solutions/ch14 Chapter 14: Solutions of Trigonometric Equation [Chapter 14: Solutions of Trigonometric Equation] Notes (Solutions) of Chapter 14: Solutions of Trigonometric Equation, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary ${\sin ^{ - 1}}A + {\sin ^{ - 1}}B = {\sin ^{ - 1}}\left( {A\sqrt {1 - {B^2}} + B\sqrt {1 - {A^2}} } \right)$${\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-part1-kpk/sol/unit01/ex1-3-p4 Question 5, Exercise 1.3 Solutions of Question 5 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) ${{z}^{2}}+z+3=0$${{z}^{2}}+z+3=0$$a=1,\,\,\,b=1$$c=3$\begin{align}z&=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\\ z&=\dfrac{-\left( 1 \right)\pm \sqrt{{{\left( 1 \right)}^{2}}-4\left( 1 \right)\left( 3 \right)}}{2\left( 1 \right)}\\ z&=\dfrac{-1\pm \s… anonymous@undisclosed.example.com (Anonymous) Mon, 25 Sep 2023 12:02:44 +0000 https://www.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-3-p5 Question 6, Exercise 1.3 Solutions of Question 6 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6(i) ${{z}^{4}}+{{z}^{2}}+1=0$\begin{align}{{z}^{4}}+{{z}^{2}}+1&=0\\ {{z}^{4}}+2\left( \dfrac{1}{2} \right){{z}^{2}}+\dfrac{1}{4}-\dfrac{1}{4}+1&=0\\ {{\left( {{z}^{2}}+\dfrac{1}{2} \right)}^{2}}+\dfrac{4-1}{4}&=0\\ {{\left( {{z}^{2}}+\dfrac{1}{2} \righ… anonymous@undisclosed.example.com (Anonymous) Mon, 25 Sep 2023 12:03:16 +0000 https://www.mathcity.org/math-11-kpk/sol/unit01/ex1-3-p3 Question 3 & 4, Exercise 1.3 Solutions of Question 3 & 4 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 ${{z}_{1}}=-1+i$${{z}_{2}}=-1-i$${{z}^{2}}+2z+2=0$$$z^2+2z_1+2=0\quad \ldots (i)$$$z_1=-1+i$\begin{align}L.H.S &= (-1+i)^2+2(-1+i)+2\\ &=1-2i-1-2+2i+2\\ &=0=R.H.S\end{align}$z_1=-1+i$$z_2=-1-i$\begin{align} L.H.S&=(-1-i)^2+2(-1-i)+2\\ &=1+2i-1-… anonymous@undisclosed.example.com (Anonymous) Tue, 05 Dec 2023 02:45:03 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/ex6-1-p3 Question 5 Exercise 6.1 Solutions of Question 5 of Exercise 6.1 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{(2 n) !}{n !}=2^n(1.3 .5 \ldots(2 n-1))$\begin{align}\dfrac{(2 n) !}{n !}&=\dfrac{1}{n !}[(2 n)(2 n-1)(2 n-2) \\ &=(2 n-3)(2 n-4)(2 n-5) \ldots(2 n-(2 n-4))\\ &(2 n-(2 n-3))(2 n-(2 n-2))(2 n-(2 n-1))]\end{align}$2 n$\begin{align}\dfra… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:32 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-3-p2 Question 2, Exercise 1.3 Solutions of Question 2 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 2(i) $z^{2}-6 z+2=0$\begin{align} & z^2 - 6z + 2 = 0 \\ \implies & z^2 - 2(3)(z)+9-9+2=0 \\ \implies & (z - 3)^2+7= 0 \\ \implies & (z - 3)^2 = 7. \end{align}\begin{align} &z - 3 = \pm \sqrt{7} \\ \implies &z = 3 \pm \sqrt{7}\end{align}$\{3 … anonymous@undisclosed.example.com (Anonymous) Sun, 14 Jul 2024 19:41:53 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-3-p3 Question 3, Exercise 1.3 Solutions of Question 3 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 3(i) $\dfrac{1}{3} z^{2}+2 z-16=0$\begin{align}&\dfrac{1}{3}z^{2}+2 z-16=0\\ \implies &z^{2} + 6z - 48 = 0 \end{align}$$ z = \dfrac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a},$$$$a = 1,\quad b = 6,\quad \text{and}\quad c = -48.$$\begin{align} z& = \d… anonymous@undisclosed.example.com (Anonymous) Sun, 14 Jul 2024 19:45:23 +0000 https://www.mathcity.org/math-11-nbf/sol/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/re-ex-p1 Question 1, Review Exercise Solutions of Question 1 of Review Exercise of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A$$m \times n$$B$$n \times p$$A B$$n \times p$$m \times p$$p \times m$$n \times n$$m \times p$$A$$1 \times n$$A^{t} A$$1 \times n$$n \times 1$$1 \times 1$$n \times n$$n \times n$$a_{i j}$$A$$a_{i j}=(-1)^{i+j} A_{i j}$$a_{i j}=(-1)^{i+j}… anonymous@undisclosed.example.com (Anonymous) Mon, 25 Nov 2024 17:51:01 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/rev-ex Review Exercise 2 (Solutions) The solutions of the Review Exercise 2 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the MCQs and question all topics included in this chapter.$A$$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$$a_{i j}=(-… anonymous@undisclosed.example.com (Anonymous) Sun, 08 Feb 2026 16:57:09 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p4 Question 5 and 6, Exercise 4.2 Solutions of Question 5 and 6 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $a_{17}=-40$$a_{28}=-73$$a_{1}$$d$$$a_n=a_1+(n-1)d$$\begin{align*} & a_{17} = -40 \\ \implies &a_1 + 16d = -40 \quad \cdots (1) \end{align*}\begin{align*} &a_{28}=-73\\ \implies &a_1 + 27d = -73 \quad \cdots (2) \end{align*}\begin{align*}… anonymous@undisclosed.example.com (Anonymous) Fri, 20 Sep 2024 17:08:12 +0000 https://www.mathcity.org/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/atiq/fa14-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 16:40:08 +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-mth103 MTH103: Exploring Quantitative Skills Course Objectives This course aims to develop the basic mathematical skills which ultimately enhance problem-solving skills using inductive and deductive reasoning, Polya's strategy, and sets. The basic concepts will be develop with applications form the real world such as algebraic models with equations, rates, ratios, and percentages will be discussed. Students will also explore linear models, including rectangular coordinates, functions, empowering them… anonymous@undisclosed.example.com (Anonymous) Wed, 27 Sep 2023 13:47:12 +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/bsc/notes_of_mathematical_method Notes of Mathematical Method [BSc Mathematical Method] Notes of the Mathematical Method written by by S.M. Yusuf, A. Majeed and M. Amin and published by Ilmi Kitab Khana, Lahore. This is an old and good book of mathematical method. The notes given here are provided by awesome peoples, who dare to help others. Some of the notes are send by the authors of these notes and other are send by people who didn't write but share these notes as Open Educational Resources (OER). We are thankful to anonymous@undisclosed.example.com (Anonymous) Sun, 02 Jul 2023 07:49:56 +0000 https://www.mathcity.org/dyk/3 Its about square root [DYK] The reason is not difficult if one knows about the definition of square root of real numbers. Definition: Let $x$ be a non-negative number. Then a non-negative number $r$ is called square root of $x$ iff $r^2=x$. Square root of $x$ is denoted by $\sqrt{x}$$2^2=4$$3^2=9$$x$$r$$x$$r^2=x$$2^2=4$$(-2)^2=4$$\sqrt{4}=\sqrt{2^2}=\sqrt{(-2)^2}=2$$\sqrt{4}=\sqrt{2^2}=\sqrt{(-2)^2}=\pm 2$$\sqrt{4}$$\sqrt{x}$$x\geq0$$\sqrt{x}=x^{\frac{1}{2}}$ anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:42:27 +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$$n$ anonymous@undisclosed.example.com (Anonymous) Mon, 28 Aug 2023 17:03:58 +0000 https://www.mathcity.org/math-11-kpk/mcqs Multiple Choice Questions (MCQs) Here are the sample MCQs at this time. Page will be updated periodically. SAMPLE MCQs * $i^{13}=$............. * (A) $i$ * (B) 1 * (C) -1 * (D) 2 * Set of all possible subsets of $S$ is called * (A) Equivalent sets$1, \omega, \omega^2$$-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/notes/classical-mechanics-muhammad-usman-hamid Classical Mechanics by Muhammad Usman Hamid [Classical Mechanics by Muhammad Usman Hamid] Objectives of the course: To provide solid understanding of classical mechanics and enable the students to use this understanding while studying courses on quantum mechanics, statistical mechanics, electromagnetism, fluid dynamics, space-flight dynamics, astrodynamics and continuum mechanics. anonymous@undisclosed.example.com (Anonymous) Mon, 29 Jul 2024 18:38:28 +0000 https://www.mathcity.org/notes/functional-analysis-m-usman-hamid-and-zeeshan-ahmad Functional Analysis by M Usman Hamid and Zeeshan Ahmad [Functional Analysis by M Usman Hamid and Zeeshan Ahmad] These notes are send by Muhammad Usman Hamid and written by Muhammad Usman Hamid and Zeeshan Ahmad. We are really very thankful to him for providing these notes and appreciate his effort to publish these notes on MathCity.org. Usman is dedicated and committed mathematician, who is working very hard for better understanding of mathematics to it readers. anonymous@undisclosed.example.com (Anonymous) Wed, 18 Sep 2024 18:09:17 +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-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$$A \cap B=… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:49 +0000 https://www.mathcity.org/fsc/fsc_part_1_mcqs/mcqs_with_answers MCQs with Answers (FSc/ICS Part 1) [MCQs Choice] In this one PDF, MCQs of all chapters of FSc/ICS Part1 are given. There are seven chapters. Answers of MCQs is starting from page 71. SAMPLE MCQs * $i^{13}=$............. * (A) $i$ * (B) 1 * (C) -1 * (D) 2 * $S$$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$$n$ anonymous@undisclosed.example.com (Anonymous) Thu, 02 May 2024 17:03:11 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01 Unit 01: Complex Numbers (Solutions) This is a first unit of the book Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. On this page we have provided the solutions of the questions.$z$$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/fsc-part1-kpk/sol/unit01/ex1-3-p1 Question 1, Exercise 1.3 Solutions of Question 1 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) \begin{align}&z-4w=3i\\ &2z+3w=11-5i\end{align}\begin{align}z-4w&=3i …(i)\\ 2z+3w&=11-5i …(ii)\end{align}$2$\begin{align}2z-8w&=6i …(iii)\end{align}\[\begin{array}{cccc} 2z&-8w&=6i \\ \mathop+\limits_{-}2z&\mathop+\limits_{-}3w&=\mathop-\limit… anonymous@undisclosed.example.com (Anonymous) Sat, 30 Sep 2023 18:43:14 +0000 https://www.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-3-p3 Question 3 & 4, Exercise 1.3 Solutions of Question 3 & 4 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 ${{z}_{1}}=-1+i$${{z}_{2}}=-1-i$${{z}^{2}}+2z+2=0$$$z^2+2z_1+2=0\quad \ldots (i)$$$z_1=-1+i$\begin{align}L.H.S &= (-1+i)^2+2(-1+i)+2\\ &=1-2i-1-2+2i+2\\ &=0=R.H.S\end{align}$z_1=-1+i$$z_2=-1-i$\begin{align} L.H.S&=(-1-i)^2+2(-1-i)+2\\ &=1+2i-1-… anonymous@undisclosed.example.com (Anonymous) Tue, 03 Oct 2023 03:15:42 +0000 https://www.mathcity.org/math-11-kpk/sol/unit01/ex1-3-p1 Question 1, Exercise 1.3 Solutions of Question 1 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) \begin{align}&z-4w=3i\\ &2z+3w=11-5i\end{align}\begin{align}z-4w&=3i …(i)\\ 2z+3w&=11-5i …(ii)\end{align}$2$\begin{align}2z-8w&=6i …(iii)\end{align}\[\begin{array}{cccc} 2z&-8w&=6i \\ \mathop+\limits_{-}2z&\mathop+\limits_{-}3w&=\mathop-\limit… anonymous@undisclosed.example.com (Anonymous) Tue, 05 Dec 2023 02:45:01 +0000 https://www.mathcity.org/math-11-kpk/sol/unit05/ex5-4-p2 Question 2 & 3 Exercise 5.4 Solutions of Question 2 & 3 of Exercise 5.4 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2 $\sum_{k=1}^n \dfrac{1}{9 k^2+3 k-2}$\begin{align}\text { Let } S_n&=\sum_{k=1}^n \dfrac{1}{9 k^2+3 k-2} \\ S_n&=\sum_{k=1}^n \dfrac{1}{9 k^2+6 k-3 k-2} \\ & =\sum_{k=1}^n \dfrac{1}{3 k(3 k+2)-1(3 k+2)} \\ S_n&=\sum_{k=1}^n \dfrac{1}{(3 k-1… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:21 +0000 https://www.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p6 Question 7 and 8 Exercise 7.3 Solutions of Question 7 and 8 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$x^4$$(1-x)^{\frac{1}{4}}+(1-x)^{\frac{1}{4}}=a-b x^2$$a$$b$$$ \begin{aligned} & (1+x)^{\frac{1}{4}}+(1-x)^{\frac{1}{4}} \\ & =\left[1+\frac{x}{4}+\frac{\frac{1}{4}\left(\frac{1}{4}-1\right)}{2 !} x^2+\right. \\ & \left.\frac{\fra… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:36:35 +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/unit02/ex2-6-p2 Question 2, Exercise 2.6 Solutions of Question 2 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\lambda$$\lambda$$2 x_{1}-\lambda x_{2}+x_{3}=0$$2 x_{1}+3 x_{2}-x_{3}=0$$3 x_{1}-2 x_{2}+4 x_{3}=0$\begin{align*} &2 x_{1}-\lambda x_{2}+x_{3}=0 \cdots(i)\\ &2 x_{1}+3 x_{2}-x_{3}=0\cdots(ii)\\ &3 x_{1}-2 x_{2}+4 x_{3}=0\cdots(iii)\\ \end{ali… anonymous@undisclosed.example.com (Anonymous) Wed, 04 Sep 2024 03:04:14 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-8-p5 Question 9 and 10, Exercise 4.8 Solutions of Question 9 and 10 of Exercise 4.8 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$\frac{1}{1 \cdot 3}+\frac{1}{2 \cdot 5}+\frac{1}{3 \cdot 7}+\ldots \ldots \text{ up to } \infty$$$\sum_{k=3}^{n} \dfrac{1}{(k+1)(k+2)}$\begin{align*} T_k &= \frac{1}{(k+1)(k+2)}. \end{align*}\begin{align*} \frac{1}{(k+1)(k+2)} = \frac… anonymous@undisclosed.example.com (Anonymous) Sun, 06 Oct 2024 17:48:05 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-8-p7 Question 13, 14 and 15, Exercise 4.8 Solutions of Question 13, 14 and 15 of Exercise 4.8 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\frac{1}{5 \cdot 11}+\frac{1}{7 \cdot 13}+\frac{1}{9 \cdot 15}+\ldots \ldots$$n$$T_k$$k$\begin{align*} T_k &= \frac{1}{(2k+3)(2k+9)}. \end{align*}\begin{align*} \frac{1}{(2k+3)(2k+9)} = \frac{A}{2k+3} + \frac{B}{2k+9} \ldots … anonymous@undisclosed.example.com (Anonymous) Sun, 06 Oct 2024 17:49:14 +0000 https://www.mathcity.org/math-9th-pctb Mathematics 9th for Matric (PCTB) [Mathematics 9th for Matric (PCTB)] Mathematics 9 is published by Punjab Curriculum and Textbook Board (PCTB), Lahore, Pakistan. This is a textbook for all the boards of Punjab for matriculation. The book has total of thirteen (13) chapters. The book is written by Muhammad Akhtar Shirani (Senior Subject Specialist, Mathematics), Madiha Mehmood, (Subject Specialist, Statistics) and Ghulam Murtaza (Subject Specialist, Mathematics). anonymous@undisclosed.example.com (Anonymous) Mon, 04 May 2026 16:19:23 +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/chem-501 CHEM-501: Basic Mathematics for Chemist Course contents Introdtuction; Review of basic algebra, Graphs and their significance in chemistry. Trigonometric, logarithmic and exponential functions. Differentiation, partial differentiation, differential equations and their use in chemical problems. Concept of maxima and minima. integration, Determinants and Matrices, their properties and use in chemical problems. solutions of linear equations (simple, determinant and matrices methods), operator the… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:04 +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}$$v(t)={{t}^{2}}+t{{e}^{t}}$ anonymous@undisclosed.example.com (Anonymous) Sat, 07 Oct 2023 18:27:32 +0000 https://www.mathcity.org/atiq/math-300 MATH-300: Basic Mathematics for Chemist Without mathematics the sciences cannot be understood, nor made clear, nor taught, nor learned. (Roger Bacon, 1214–1292) Course contents Introdtuction; Review of basic algebra, Graphs and their significance in chemistry. Trigonometric, logarithmic and exponential functions. Differentiation, partial differentiation, differential equations and their use in chemical problems. Concept of maxima and minima. integration, Determinants and Matrices, their prope… anonymous@undisclosed.example.com (Anonymous) Wed, 31 May 2023 05:38:37 +0000 https://www.mathcity.org/atiq/sp23-mth322 MTH322: Real Analysis II (Spring 2023) [MTH322: Real Analysis II (Spring 2023)] This course is offered to BS, Semester VI at Department of Mathematics, COMSATS University Islamabad, Attock campus. This course need rigorous knowledge of continuity, differentiation, integration, sequences and series of numbers, that is many notions included in $f\in \mathcal{R}[a,b]$$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/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/bsc/notes_of_vector_analysis Notes of Vector Analysis [Vector Ananlysis] Notes of the vector analysis are given on this page. These notes are helpful for BSc or equivalent classes. These notes are written by Amir Taimur Mohmand of University of Peshawar. The books of these notes is not known. If you know about the book, please inform us.$f$$P$ anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:54 +0000 https://www.mathcity.org/conferences/differential_equations_and_applications_may_26-28_2016_lums_lahore International Conference on Differential Equations and Applications, LUMS Lahore (May 26-28 2016) [Main Building LUMS, Lahore] * Name of conference: Differential Equations and Applications * Palace: Lahore University of Management Sciences (LUMS), Lahore - PAKISTAN. * Date: anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:41:47 +0000 https://www.mathcity.org/events/mathematics-olympiad-2019-sukkur-iba Mathematics Olympiad 2019 Sukkur IBA (11-13 November 2019) [Mathematics Olympiad 2019] Mathematical Olympiad is a contest of mathematics among the students. It is a very healthy activity to promote and learn mathematics. Contents for the test are as follows: * Qudratic equations and expressions anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:42:33 +0000 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/mathcraft/sample-01-word MathCraft: PDF/Image to Word: Sample 01 If the PDF file provided by you as follows: Then the output Word file is as follows. It contains equation, which can be editable with MS Office built-in Equation editor. anonymous@undisclosed.example.com (Anonymous) Mon, 25 Mar 2024 17:29:19 +0000 https://www.mathcity.org/mathcraft/sample-02-latex MathCraft: PDF to LaTeX file: Sample-02 If the PDF file provided by you as follows: Then the output LaTeX file is as follows: \documentclass[4pt]{article} \usepackage[utf8]{inputenc} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage[version=4]{mhchem} \usepackage{stmaryrd} \usepackage{bbold} \usepackage[a4paper]{geometry} \linespread{1.3} % double spaces lines \textwidth 6.3truein % These 4 commands define more efficient margins \textheight 9.9truein \oddsidemargi… anonymous@undisclosed.example.com (Anonymous) Mon, 25 Mar 2024 16:32:04 +0000 https://www.mathcity.org/mathcraft/sample-02-word MathCraft: PDF/Image to Word: Sample 02 If the PDF file provided by you as follows: Then the output Word file is as follows. It contains equation, which can be editable with MS Office built-in Equation editor. anonymous@undisclosed.example.com (Anonymous) Mon, 25 Mar 2024 17:30:02 +0000 https://www.mathcity.org/notes/advanced-analysis-handwritten-notes Advanced Analysis: Handwritten Notes [Advanced Analysis: Handwritten Notes] These notes are provided by Mr. Anwar Khan. We are really very thankful to Mr. Anwar Khan for providing these notes and appreciates his effort to publish these notes on MathCity.org It covers the complete syllabus of Advanced Analysis paper of MSc Mathematics. See the contents of the notes given below to see the topics covered by these notes. anonymous@undisclosed.example.com (Anonymous) Fri, 14 Apr 2023 17:55:33 +0000 https://www.mathcity.org/notes/mechanics-by-sir-nouman-siddique Mechanics by Sir Nouman Siddique These notes are provided and composed by Mr. Muzammil Tanveer. We are really very thankful to him for providing these notes and appreciates his effort to publish these notes on MathCity.org * Name: Mechanics * Provider: Mr. Muzammil Tanveer anonymous@undisclosed.example.com (Anonymous) Wed, 26 Jun 2024 18:35:58 +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/multivariable-calculus-m-usman-hamid-and-m-zeeshan-ahmad Multivariable Calculus by M Usman Hamid and M Zeeshan Ahmad [Multivariable Calculus by M Usman Hamid and M Zeeshan Ahmad] Multivariable calculus is a fundamental subject that extends the concepts of single-variable calculus to higher-dimensional spaces. It provides a powerful framework for analyzing and modeling complex phenomena in fields such as physics, engineering, economics, and computer science. This textbook is designed to provide a comprehensive introduction to multivariable calculus, c… anonymous@undisclosed.example.com (Anonymous) Sun, 15 Jun 2025 16:25:59 +0000 https://www.mathcity.org/notes/number-theory-umer-shuaib Number Theory by Dr Muhammad Umer Shuaib [Number Theory Notes] A subfield of mathematics called number theory studies the characteristics of positive integers. Higher arithmetic is another name for it. The study of the relationships between various types of numbers, including prime numbers, rational numbers, and algebraic integers, is done using number theory, one of the oldest fields of mathematics.$\phi$ anonymous@undisclosed.example.com (Anonymous) Sat, 05 Aug 2023 18:11:04 +0000 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$\(A\cos wt+B\sin wt\)$\… anonymous@undisclosed.example.com (Anonymous) Tue, 18 Jan 2022 10:20:00 +0000 https://www.mathcity.org/ppsc/ppsc-mock-interview-mathematics PPSC Mock Interview Lecturer Mathematics [PPSC Mock Interview Lecturer Mathematics] This handout is shared by Mr. Rashad Wattu and written by Sawaira Sikandar. We are really very thankful to him for providing this handout and appreciates his efforts to publish it on MathCity.org. This handout contains the questions collected from the different interviews of the Lecturer in Mathematics conducted by Public Service Commission (PSC). This is help full to prepare interview of all types of jobs whic… anonymous@undisclosed.example.com (Anonymous) Thu, 25 Mar 2021 16:31:54 +0000 https://www.mathcity.org/bsc/notes_of_calculus_with_analytic_geometry/ch06_plane_curves_i Chapter 06: Plane Curves I Notes of the book Calculus with Analytic Geometry written by Dr. S. M. Yusuf and Prof. Muhammad Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN. [Conic section] Contents and summary * Conic sections * The parabola * The ellipse anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:45:31 +0000 https://www.mathcity.org/fsc-part1-kpk/sol/unit01 Unit 1: Complex Numbers (Solutions) This is a first unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the students will be able to$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/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/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/matric/10th_science/unit01-view Unit 01: Quadratic Equations: Online View On this page the solutions of Unit 01: Quadratic Equations, Mathematics 10 (Science Group), published by Ilmi Kitab Khana, Urdu Bazar, Lahore, Pakistan. List of all exercise of Unit 01 * Exercise 1.1 * Exercise 1.2 * Exercise 1.3 anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:48:01 +0000 https://www.mathcity.org/matric/10th_science/unit02-view Unit 02: Theory of Quadratic Equations: Online View On this page the solutions of Unit 02: Theory of Quadratic Equations, Mathematics 10 (Science Group), published by Ilmi Kitab Khana, Urdu Bazar, Lahore, Pakistan. List of all exercise of Unit 02 anonymous@undisclosed.example.com (Anonymous) Wed, 24 Jul 2024 18:33:41 +0000 https://www.mathcity.org/bsc/notes_of_mathematical_method/ch04_system_of_linear_equations/viewer Chapter 04: Viewer Notes of Chapter 04: System of linear equations of Mathematical Method written by S.M. Yusuf, A. Majeed and M. Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN. PDF file of respective exercise can be downloaded from this page. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:53:51 +0000 https://www.mathcity.org/bsc/notes_of_mathematical_method/ch04system_of_linear_equations/viewer Chapter 04: Viewer Notes of Chapter 04: System of linear equations of Mathematical Method written by S.M. Yusuf, A. Majeed and M. Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN. PDF file of respective exercise can be downloaded from this page. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:53:27 +0000 https://www.mathcity.org/fsc/fsc_part_1_solutions/ch14/view Ch 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 Textbook Board (PTB), Lahore. There are four exercises in this chapter. Please see the main page of this chapter for MCQs and important question at anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:59:08 +0000 https://www.mathcity.org/math-11-kpk/sol/unit02/ex2-1-p5 Question 5 & 6, Exercise 2.1 Solutions of Question 5 & 6 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$A= \begin{bmatrix} 0 & 2b & -2 \\ 3 & 1 & 3 \\ 3a & 3 & -1 \end{bmatrix}$$a$$b$$A=\begin{bmatrix} 0 & 2b & -2 \\ 3 & 1 & 3 \\ 3a & 3 & -1 \end{bmatrix}$$$A^t=\left[ \begin{matrix} 0 & 3 & 3a \\ 2b & 1 & 3 \\ -2 & 3 & -1 \\ \end{ma… anonymous@undisclosed.example.com (Anonymous) Tue, 05 Dec 2023 02:45:14 +0000 https://www.mathcity.org/math-11-kpk/sol/unit03/ex3-2-p3 Question 3 & 4, Exercise 3.2 Solutions of Question 3 & 4 of Exercise 3.2 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 If $\vec{r}=\hat{i}-9\hat{j}$$\vec{a}=\hat{i}+2\hat{j}$$\vec{b}=5\hat{i}-\hat{j}$$p$$q$$\vec{r}=p\vec{a}+q\vec{b}$$$\vec{r}=p\vec{a}+q\vec{b}.$$$\vec{r},\vec{a}$$\vec{b}$$$\hat{i}-9\hat{j}=p(\hat{i}+2\hat{j})+q(5\hat{i}-\hat{j})$$$$\implies \hat{i}-9\… anonymous@undisclosed.example.com (Anonymous) Wed, 03 Jan 2024 18:10:31 +0000 https://www.mathcity.org/math-11-kpk/sol/unit05/ex5-1-p1 Question 1 Exercise 5.1 Solutions of Question 1 of Exercise 5.1 of Unit 05: Mascellaneous series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) $1^2+3^2+5^2+7^2+\ldots$$n$$1+3+5+\ldots$$n^{\text {th }}$$2 n-1$$n^{t h}$$$T_j=(2 j-1)^2$$\begin{align}& \sum_{j=1}^n T_j=\sum_{j=1}^n(2 j-1)^2 \\ & =\sum_{j=1}^n(4 j^2-4 j+1)\\ & =4 \sum_{j=1}^n j^2-4 \sum_{j=1}^n j+\sum_{j=1}^n 1 \\ & =4 \dfr… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:10 +0000 https://www.mathcity.org/math-11-kpk/sol/unit05/ex5-1-p3 Question 4 & 5 Exercise 5.1 Solutions of Question 4 & 5 of Exercise 5.1 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4 $2+(2+5)+(2+5+8)+\ldots$$n$\begin{align}& T_j=\dfrac{j}{2}[2(2)+3(j-1)]\\ &=\dfrac{j(3 j+1)}{2} \\ & =\dfrac{1}{2}(3 j^2+j)\end{align}\begin{align}& \sum_{j=1}^n T_i=\dfrac{1}{2}[3 \sum_{j=1}^n j^2+\sum_{j=1}^n j] \\ & =\dfrac{1}{2}[3 \dfra… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:11 +0000 https://www.mathcity.org/math-11-kpk/sol/unit05/ex5-1-p6 Question 9 Exercise 5.1 Solutions of Question 9 of Exercise 5.1 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9(i) $n$$n$$n$\begin{align} & T_n=n^2(2 n+3)=2 n^3+3 n^2 \\ & \Rightarrow T_j=2 j^3+3 j^2\end{align}\begin{align} & \sum_{j=1}^n T_j=2 \sum_{j=1}^n j^3+3 \sum_{j=1}^n j^2 \\ & =2(\dfrac{n(n+1)}{2})^2+3 \dfrac{n(n+1)(2 n+1)}{6} \\ & =\dfrac{n(n+1)}{2}… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:13 +0000 https://www.mathcity.org/math-11-kpk/sol/unit05/re-ex5-p3 Question 4 Review Exercise Solutions of Question 4 of Review Exercise of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4 $\dfrac{1}{1.4 .7}+\dfrac{1}{4.7 .10}+\dfrac{1}{7.10 .13}+\ldots$$1,4,7, \ldots$$$a_n=\dfrac{1}{(3 n-2)(3 n+1)(3 n+4)}$$\begin{align} \dfrac{1}{(3 n-2)(3 n+1)(3 n+4)}&=\dfrac{A}{3 n-2}+\dfrac{B}{3 n+1}+\dfrac{C}{3 n+4}\end{align}$(3 n-2)(3 n+… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:24 +0000 https://www.mathcity.org/math-11-kpk/sol/unit05/re-ex5-p4 Question 5 & 6 Review Exercise Solutions of Question 5 & 6 of Review Exercise of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$5+12 x+19 x^2+26 x^3+\ldots$$n$\begin{align}S_n&=5+12 x+19 x^2+26 x^3+\cdots+(7 n-2) x^{n-1}...(i)\\ x S_n&=5 x+12 x^2+19 x^3+\cdots+(7 n-9) x^{n-1}+(7 n-1) x^n....(ii)\end{align}\begin{align}(1-x) S_n&=5+(12-5) x+(19-12) x^2+\cdots\\ &+[7 n-2-(… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:24 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/ex6-1-p2 Question 3 & 4 Exercise 6.1 Solutions of Question 3 & 4 of Exercise 6.1 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{1}{6 !}+\dfrac{2}{7 !}+\dfrac{3}{8 !}=\dfrac{75}{8 !}$\begin{align}\dfrac{1}{6 !}+\dfrac{2}{7 !}+\dfrac{3}{8 !}&=\dfrac{1}{6 !}+\dfrac{2}{7.6 !}+\dfrac{3}{8.7 .6 !} \\ & =\dfrac{56+16+3}{8 !}\\ &=\dfrac{75}{8 !}\end{align}$\df… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:30 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p2 Question 3 and 4 Exercise 6.2 Solutions of Question 3 and 4 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$^n P_r=n(^{n-1} P_{r-1})$$$^n P_r=n({ }^{n-1} P_{r-1})$$\begin{align}n(^{n-1} P_{r-1})&=n \dfrac{(n-1) !}{((n-1)-(r-1)) !} \\ & =\dfrac{n(n-1) !}{(n-r) !}\\ &=\dfrac{n !}{(n-r) !}\\ &=^n P_r\end{align}$^n P_r=^{n-1} P_r+r(^{n-1} … anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:37 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p4 Question 7 and 8 Exercise 6.2 Solutions of Question 7 and 8 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$1,2,3,4$$E_1$$m_1=5$$E_2$$\cdot m_2=5$$E_3$$m_3=5$$$m_1 \cdot m_2 \cdot m_3=5.5 \cdot 5=125$$$1,2,3,4$$E_1$$m_1=5$$E_2$$m_2=4$$E_3$$m_3=3$$$m_1 \cdot m_2 \cdot m_3=5 \cdot 4 \cdot 3=60$$$8$$5$$=4$$=4$$=5$$=3$$4 ! \cdot 5 ! \cdot … anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:38 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-2-p3 Question 3 and 4, Exercise 4.2 Solutions of Question 3 and 4 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $0.07,0.12,0.7, \ldots$$$0.07,0.12,0.7, \ldots$$$a_1 = 0.07$$d=0.05$$a_{11}=?$\begin{align*} a_n&=a_1+(n-1)d \\ \implies a_{11}&= 0.07+(11-1)(0.05)\\ &=0.07+(10)(0.05)\\ &=0.57 \end{align*}$a_{11}=0.57.$$a_3 = 14$$a_9 = -1$$$a_n = a_1 + (… anonymous@undisclosed.example.com (Anonymous) Fri, 20 Sep 2024 16:59:34 +0000 https://www.mathcity.org/matric/9th_science/unit07/viewer Unit 07: Linear Equations and Inequalities: Online View On this page the solutions of Unit 07: Linear Equations and Inequalities, Mathematics 9 (Science Group), are given List of all exercise of Unit 07 * Exercise 7.1 * Exercise 7.2 * Exercise 7.3 anonymous@undisclosed.example.com (Anonymous) Sat, 30 Apr 2022 18:35:29 +0000 https://www.mathcity.org/matric/10th_science/unit01/viewer Unit 01: Quadratic Equations: Online View On this page the solutions of Unit 01: Quadratic Equations, Mathematics 10 (Science Group), published by Ilmi Kitab Khana, Urdu Bazar, Lahore, Pakistan. List of all exercise of Unit 01 * Exercise 1.1 * Exercise 1.2 * Exercise 1.3 anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 17:00:28 +0000 https://www.mathcity.org/matric/10th_science/unit02/viewer Unit 02: Theory of Quadratic Equations: Online View On this page the solutions of Unit 02: Theory of Quadratic Equations, Mathematics 10 (Science Group), published by Ilmi Kitab Khana, Urdu Bazar, Lahore, Pakistan. List of all exercise of Unit 02 anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 17:00:28 +0000 https://www.mathcity.org/conferences Mathematics Conferences This page will no longer be updated. To spread the scope of this page, it has been merged with Mathematics Events. Conferences can be ideal places to meet with professionals and present our work to live audience to get engage with them. It might be best place to find collaborators from with in country and outside of the country. On this page we have posted mathematics conferences occurring all over the country (PAKISTAN). anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:39:21 +0000 https://www.mathcity.org/mathcraft MathCraft Introducing “MathCraft”: Your Solution for Document Transformation! [MathCraft] We are thrilled to unveil our latest service, MathCraft, tailored exclusively for the mathematics community. With MathCraft, you can easily get code from PDFs and pictures into LaTeX or Word files without spending too much money. Whether you're a student, researcher, or teacher, MathCraft can help you create your math documents in the format you want. anonymous@undisclosed.example.com (Anonymous) Sun, 14 Apr 2024 10:56:57 +0000 https://www.mathcity.org/atiq/cal2 MATH 102: Calculus II Course outline * Techniques of integration * Further applications of integration * Parametric equations and polar coordinates * Conic sections * Sequence and series * Power series representation of functions Assignments anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:05 +0000 https://www.mathcity.org/atiq/fa25-mth324 MTH324: Complex Analysis (Fall 2025) [MTH324: Complex Analysis (Fall 2025) Courtesy: Copilot] Course Objectives: At the end of this course the students will be able to understand the basic properties of functions of a complex variable with the theory of analytic functions and its applications. Course contents: anonymous@undisclosed.example.com (Anonymous) Sun, 28 Sep 2025 07:23:19 +0000 https://www.mathcity.org/atiq/math-305 MATH-305: Real Analysis-I Objectives of the course: This is the first rigorous course in analysis and has a theoretical emphasis. It tegorously develops the fundamental ideas of calculus and is aimed to develop the students’ ability to deal with abstract mathematics and mathematical proofs. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:21 +0000 https://www.mathcity.org/bsc/number-theory-by-prof-asghar-ali Number Theory by Prof. Asghar Ali [Number Theory by M Asghar Ali] We are very thankful to Prof. Asghar Ali for send these notes. These notes are very helpful to prepare BSc or ADS mathematics portion of Number Theory. Number theory is a subject in which students learn different concepts created on the set of integers. For example, the concept of divisibilty exists in the set of integer. Let a and b be any two integers suhc that $a\neq 0$ anonymous@undisclosed.example.com (Anonymous) Mon, 21 Mar 2022 19:38:31 +0000 https://www.mathcity.org/bsc/vector_analysis_by_hameed_ullah Vector Analysis by Hameed Ullah: Notes [right triangle in semi circle] Note of vector analysis by Hammed Ullah. These notes are send by Umer Asghar, we are very thankful to him for providing these notes. These notes are for helpful for undergraduate level (BSc or BS). Name Notes of vector analysis anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:41:13 +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/notes/advance-functional-analysis-waseem_akram Advance Functional Analysis by Waseem Akram [Advance Functional Analysis by Waseem Akram] These notes provide a concise yet dense overview of key theorems in functional analysis, including the Hahn-Banach Theorem, Baire Category Theorem, Open Mapping Theorem, Closed Graph Theorem, and Banach Fixed Point Theorem. The presentation is somewhat informal, with occasional typographical errors, incomplete proofs, and missing equation references. However, the core logical flow is preserved, making it u… anonymous@undisclosed.example.com (Anonymous) Sun, 31 May 2026 07:51:28 +0000 https://www.mathcity.org/notes/advance-functional-analysis-waseem-akram Advance Functional Analysis by Waseem Akram [Advance Functional Analysis by Waseem Akram] These notes provide a concise yet dense overview of key theorems in functional analysis, including the Hahn-Banach Theorem, Baire Category Theorem, Open Mapping Theorem, Closed Graph Theorem, and Banach Fixed Point Theorem. The presentation is somewhat informal, with occasional typographical errors, incomplete proofs, and missing equation references. However, the core logical flow is preserved, making it u… anonymous@undisclosed.example.com (Anonymous) Mon, 25 May 2026 18:22:55 +0000 https://www.mathcity.org/notes/algebraic-number-theory-notes-anwar-khan Algebraic Number Theory Notes by Anwar Khan [Algebraic Number Theory Notes by Anwar Khan] Algebraic number theory is a subfield of number theory that studies integers, rational numbers, and their generalisations using abstract algebra techniques. It covers Galois theory, ideals and units in rings of integers, unique factorization, and algebraic number fields and related rings of integers. It is a complex and in-depth subject with numerous linkages to other branches of mathematics.$\mathbb{R}$ anonymous@undisclosed.example.com (Anonymous) Sat, 05 Aug 2023 19:06:20 +0000 https://www.mathcity.org/notes/elementary-linear-algebra-m-usman-hamid Elementary Linear Algebra by Muhammad Usman Hamid [Elementary Linear Algebra by Muhammad Usman Hamid] Linear Algebra is the study of vectors and linear transformations. The main objective of this course is to help students learn in rigorous manner, the tools and methods essential for studying the solution spaces of problems in mathematics, engineering, the natural sciences and social sciences and develop mathematical skills needed to apply these to the problems arising within their field of st… anonymous@undisclosed.example.com (Anonymous) Sun, 10 Dec 2023 14:05:12 +0000 https://www.mathcity.org/notes/fluid-mechanics-m-usman-hamid Fluid Mechanics by Muhammad Usman Hamid [Fluid Mechanics by Muhammad Usman Hamid] We casually look around most things seem to be solids but when one thinks of the oceans, the atmosphere and on out into space it becomes rather obvious that a large portion of the earth surface and of the entire universe is in a fluid state. Therefore, it becomes essential for sciences and engineers to know something about fluid mechanics. anonymous@undisclosed.example.com (Anonymous) Sun, 16 Mar 2025 12:46:08 +0000 https://www.mathcity.org/notes/functional-analysis-by-prof-mumtaz-ahmad Functional Analysis by Prof Mumtaz Ahmad [Functional Analysis by Prof Mumtaz Ahmad] Functional analysis is a subfield of mathematics that deals with vector space theory and linear algebra. It entails researching the connections between roles, things, incidents, actions, and outcomes. The word $l^\infty$$l^\infty$ anonymous@undisclosed.example.com (Anonymous) Wed, 18 Sep 2024 18:11:56 +0000 https://www.mathcity.org/notes/functional-analysis-by-tahir-hussain-jaffery Functional Analysis by Mr. Tahir Hussain Jaffery [Functional Analysis by Prof Mumtaz Ahmad] Functional analysis is a branch of mathematics concerned with vector space theory and linear algebra. It requires looking into the relationships between various roles, objects, incidents, actions, and results. The term $M$$M+N$ anonymous@undisclosed.example.com (Anonymous) Wed, 18 Sep 2024 18:13:33 +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/notes/fuzzy-sets-theory-umar-saeed-bunarai Fuzzy Sets Theory by Umar Saeed Bunarai [Fuzzy Sets Theory by Umar Saeed Bunarai] In the field of mathematics known as fuzzy set theory, sets having varying degrees of membership are studied. This implies that an element can be a member of a set only partially or not at all. In several disciplines, including logic, control, data analysis, and decision-making, fuzzy set theory is used to model uncertainty, vagueness, and imprecision. anonymous@undisclosed.example.com (Anonymous) Tue, 08 Aug 2023 11:56:34 +0000 https://www.mathcity.org/notes/groups-handwritten-notes Groups (Handwritten Notes) by Atiq ur Rehman Groups are a basic idea in algebra, introduced in high school. It's a very interesting topic in math. [Cube root of unity group] * Name: Groups (Handwritten notes)- Lecture Notes * Author: Atiq ur Rehman * Pages: 82 pages anonymous@undisclosed.example.com (Anonymous) Wed, 26 Jun 2024 18:07:44 +0000 https://www.mathcity.org/notes/measure-thoery-muhsa Measure Theory by M Usman Hamid & Saima Akram [Measure Theory by M Usman Hamid & Saima Akram] The study of measures on sets is the focus of the mathematical field known as measure theory. A measure is a function that gives specific subsets of a given set a non-negative real integer, indicating their size inferentially. The concept of measure is a formalisation and generalisation of common concepts like mass and event probability as well as geometrical measurements (length, area, and volume). anonymous@undisclosed.example.com (Anonymous) Wed, 26 Jun 2024 17:55:59 +0000 https://www.mathcity.org/notes/mechanics-i-statics-dr-babar-ahmad Mechanics I (Statics) by Dr Babar Ahmad [Mechanics I (Statics) by Dr Babar Ahmad] We are very thankful to Dr Babar Ahmad for sharing his book on MathCity.org. This book is very helpful for undergraduate students of Science and Engineering Programs. This book is shared by the permission of the author and he keeps the copyright of the book. anonymous@undisclosed.example.com (Anonymous) Mon, 24 Jul 2023 12:49:29 +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/notes/number-theory-notes-anwar-khan Number Theory Notes by Anwar Khan [Number Theory Notes by Anwar Khan] Mathematicians who specialize in number theory examine the characteristics and connections between integers. “Higher arithmetic” and “the queen of mathematics” are some names for it. Because it examines the characteristics and connections between integers and arithmetic functions, number theory is interesting. It has numerous uses in coding theory, combinatorics, cryptography, and other branches of mathematics. Like the ones… anonymous@undisclosed.example.com (Anonymous) Sat, 05 Aug 2023 18:47:01 +0000 https://www.mathcity.org/notes/rings-and-modules-iqra-liaqat Rings & Modules by Ms. Iqra Liaqat Ring is a mathematical structure with two operations. With one operation it is abelian group and with other operation it is semi-group with distributive law holds with first operation w.r.t the other operation. This concept is of pure in nature in mathematics. These advanced level notes are typically taken as an elective in a mathematics undergraduate degree. anonymous@undisclosed.example.com (Anonymous) Sat, 01 Apr 2023 14:30:33 +0000 https://www.mathcity.org/notes/theory-of-optimization-muzammil-tanveer Theory of Optimization by Ma'am Iqra Razzaq [Special Theory of Optimization by Ma'am Iqra Razzaq] These notes are provided and composed by Mr. Muzammil Tanveer. We are really very thankful to him for providing these notes and appreciates his effort to publish these notes on MathCity.org. These notes are based on the lectures by Ma'am Iqra Razzaq. anonymous@undisclosed.example.com (Anonymous) Sun, 23 Jul 2023 16:28:59 +0000 https://www.mathcity.org/notes/vector_spaces_handwritten_notes Vector Spaces (Handwritten notes) [Vector Spaces (Handwritten notes) by Atiq ur Rehman] Vector space is a fundamental subject in mathematics. At the undergraduate and upper secondary levels, the concept of vector space is regarded as basic and fundamental. These are lecture notes of Prof. Dr. Muhammad Khalid of University of Sargodha, Sargodha written by Atiq ur Rehman. anonymous@undisclosed.example.com (Anonymous) Sat, 01 Apr 2023 15:18:56 +0000 https://www.mathcity.org/quote-of-the-day/apr Quotes for the April کاسمولوجسٹ اکثر غلط ہوتے ہیں، لیکن کبھی شک میں نہیں۔۔۔ لیو لینڈاؤ (1908-1968) Cosmologists are often wrong, but never in doubt. --- Lev Landau (1908-1968) (Courtesy: MacTutor) anonymous@undisclosed.example.com (Anonymous) Fri, 28 Apr 2023 18:32:46 +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/math-608/what_is_mathematics What is Mathematics? Different people would gave different answers of the above title. A student in elementary school would probably say it was about adding, subtracting, multiplying and dividing. Oh yes--- about functions and decimals too. A student in high school would probably say that it is about learning rules and formulas to solve equations. Oh yes anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:45:18 +0000 https://www.mathcity.org/bsc/notes_of_calculus_with_analytic_geometry/ch08_analytic_geometry_of_three_dimensions Chapter 08: Analytic Geometry of Three Dimensions Notes of the book Calculus with Analytic Geometry written by Dr. S. M. Yusuf and Prof. Muhammad Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN. Contents & Summary * Distance between two points$\mathbb{R}^3$ anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:45:33 +0000 https://www.mathcity.org/bsc/notes_of_mathematical_method/ch02_groups Chapter 02: Groups [Chapter 02: Groups] Notes of the book Mathematical Method written by S.M. Yusuf, A. Majeed and M. Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN. Contents and summary * Definition (axioms of group) * Definition ( commutative group ) * anonymous@undisclosed.example.com (Anonymous) Mon, 11 Dec 2023 13:00:23 +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\}$$C=\{5,6,7,9,… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:39 +0000 https://www.mathcity.org/fsc-part2-ptb/important-questions/unit-01-functions-and-limits Unit 01: Functions and Limits Here is the list of important questions. * Evaluate $\lim\limits_{\theta \to 0}\frac{1-\cos \theta}{\sin^3\theta}$ --- FBSIC (2016) * Graph the curve of the following parametric equations $x=\sec \theta$, $y=\tan\theta$ where $\theta$ is a parameter.--- FBSIC (2016) * Evaluate $\lim\limits_{x \to 2}\frac{\sqrt{x}-\sqrt{2}}{x-2}$ --- BSIC Rawalpendi(2016), BSIC Rawalpendi(2017)$f(x)=x^3+x$$\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/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/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 $$\begin{align}L.H.S &= l(q-r)+m(r-p)+n(p-q)\\ &= lq-lr+mr-mp+np-nq\\ &=… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:46:29 +0000 https://www.mathcity.org/fsc/fsc_part_2_solutions/ch02 Unit 02: Differentiation [Unit 02: Differentiation] Notes (Solutions) of Unit 02: Differentiation, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. You can view online or download PDF. To view PDF, you must have PDF Reader installed on your system and it can be downloaded from Software section.$f'(x)$$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/msc/notes/number-theory-iqra-liaqat Number Theory by Ms. Iqra Liaqat [Number Theory by Ms. Iqra Liaqat] Notes of number theory provided Ms. Iqra Liaqat is a very good addition in the MSc notes section. We are actually quite grateful to her for giving these notes and likes her encouragement to distribute these notes on MathCity.org Name anonymous@undisclosed.example.com (Anonymous) Tue, 11 May 2021 11:51:22 +0000 https://www.mathcity.org/fsc-part1-kpk/sol/unit01/review-ex-1-p4 Question 6, 7 & 8, Review Exercise 1 Solutions of Question 6, 7 & 8 of Review Exercise 1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{1}{3+4i}$\begin{align}\dfrac{1}{3+4i}&=\dfrac{1}{3+4i}\times \dfrac{3-4i}{3-4i}\\ &=\dfrac{3-4i}{9+16}\\ &=\dfrac{3-4i}{25}\\ &=\dfrac{3}{25}-\dfrac{4i}{25}\end{align}$\dfrac{3i+2}{3-2i}$\begin{align}\dfrac{3i+2}{3-2i}\\ \dfrac{3i+2}… anonymous@undisclosed.example.com (Anonymous) Mon, 25 Sep 2023 12:10:41 +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/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/fsc/fsc_part_1_solutions/ch04/view View Online (Notes of Chapter 04) 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. These notes are provided by M. Shahid Nadeem, Lecturer in Mathematics, Punjab College Wah Cantt. One can also download PDF of the notes from this page. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:57:41 +0000 https://www.mathcity.org/fsc/fsc_part_1_solutions/ch14/viewer View Online (Solutions of Chapter 14) Notes (Solutions) of Chapter 14: Solutions of Trigonometric Equation of Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. In this chapter there is only one exercise. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:59:08 +0000 https://www.mathcity.org/math-11-kpk/sol/unit01/review-ex-1-p4 Question 6, 7 & 8, Review Exercise 1 Solutions of Question 6, 7 & 8 of Review Exercise 1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{1}{3+4i}$$$z=\dfrac{1}{3+4i}.$$\begin{align}z&=\dfrac{1}{3+4i}\times \dfrac{3-4i}{3-4i}\\ &=\dfrac{3-4i}{9+16}\\ &=\dfrac{3-4i}{25}\\ &=\dfrac{3}{25}-\dfrac{4}{25}i\end{align}$$\bar{z}=\dfrac{3}{25}+\dfrac{4}{25}i.$$$\dfrac{3i+2}{3-2… anonymous@undisclosed.example.com (Anonymous) Tue, 05 Dec 2023 02:45:06 +0000 https://www.mathcity.org/math-11-kpk/sol/unit03/ex3-2-p9 Question 12, 13 & 14, Exercise 3.2 Solutions of Question 12, 13 & 14 of Exercise 3.2 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 12 $\alpha ,$$|\alpha \hat{i}+(\alpha +1)\hat{j}+2\hat{k}|=3$\begin{align}|\alpha \hat{i}+(\alpha +1)\hat{j}+2\hat{k}|&=3.\end{align}\begin{align}\sqrt{(\alpha )^2+(\alpha +1)^2+(2)^2}&=3.\end{align}\begin{align}&{\alpha ^2+(\alpha +1)^2}+4=9… anonymous@undisclosed.example.com (Anonymous) Wed, 03 Jan 2024 18:10:36 +0000 https://www.mathcity.org/math-11-kpk/sol/unit04/ex4-3-p2 Question 2 Exercise 4.3 Solutions of Question 2 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2(i) $a_1, a_n, n, d$$S_n$$a_1=2, n=17, d=3$$a_1=2, n=17, d=3$$a_{17}$$S_{17}$$$a_{n}=a_1+(n-1)d.$$$$a_{17}=2+(17-1)(3)=50.$$$$S_n=\dfrac{n}{2}[a_1+a_n]$$\begin{align}S_{17}&=\dfrac{17}{2}(a_1+a_17) \\ &=\dfrac{17}{2}(2+50)=442.\end{align}$a_{17}=50$$… anonymous@undisclosed.example.com (Anonymous) Sun, 11 Feb 2024 11:48:37 +0000 https://www.mathcity.org/math-11-kpk/sol/unit04/ex4-5-p5 Question 5 & 6 Exercise 4.5 Solutions of Question 5 & 6 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5 $r$$S_{10}=244 S_5$$$S_n=\dfrac{a_1(r^n-1)}{r-1}$$$$S_{10}=\dfrac{a_1(r^{10}-1)}{r-1} \quad \text{and}\quad S_5=\dfrac{a_1(r^5-1)}{r-1}$$$S_{10}$$S_S$\begin{align}\dfrac{a_1(r^{10}-1)}{r-1}&=244 \dfrac{a_1(r^5-1)}{r-1} \\ \Rightarrow r^{10}-… anonymous@undisclosed.example.com (Anonymous) Fri, 05 Jan 2024 17:30:10 +0000 https://www.mathcity.org/math-11-kpk/sol/unit04/ex4-5-p8 Question 11 & 12 Exercise 4.5 Solutions of Question 11 & 12 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$p^{t h}, q^{t h}$$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$$a_r=a_1 r^{r-1}$\begin{align}a^{q-r}&=(a_1 r^{p-1})^{q-r} . \\ b^{r-p}&=(a_1 r^{q-1})^{r-p}, \text { and } \\ c^{p-q}&=(a_1 r^… anonymous@undisclosed.example.com (Anonymous) Fri, 05 Jan 2024 17:30:14 +0000 https://www.mathcity.org/math-11-kpk/sol/unit05/ex5-1-p4 Question 6 Exercise 5.1 Solutions of Question 6 of Exercise 5.1 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 6 $1.2 \cdot 3+2 \cdot 3.4+3.4 .5+\ldots$$n$$1+2+3+\ldots, \quad 2+3+4+5+\ldots$$3+4+5+6+7+\ldots$$n^{t h}$$j, j+1$$j+2$$n^{t h}$\begin{align} & T_j=j(j+1)(j+2)-j(j^2+3 j+2) \\ & =j^3+3 j^2+2 j\end{align}\begin{align} & \sum_{j=1}^n T_j=\sum_{j=1}^n … anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:12 +0000 https://www.mathcity.org/math-11-kpk/sol/unit05/ex5-1-p5 Question 7 & 8 Exercise 5.1 Solutions of Question 7 & 8 of Exercise 5.1 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7 $n$$1.5 .9+2.6 .10+3.7 .11+\ldots$$T_j=j(j+4)(j+8)$\begin{align} & =j(j^2+12 j+32) \\ & =j^3+12 j^2+32 j\end{align}\begin{align} & \sum_{j=1}^n T_j=\sum_{j=1}^n j^3+12 \sum_{j=1}^n j^2+32 \sum_{j=1}^n j \\ & =(\dfrac{n(n+1)}{2})^2+12 \dfrac… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:13 +0000 https://www.mathcity.org/math-11-kpk/sol/unit05/ex5-4-p3 Question 4 Exercise 5.4 Solutions of Question 4 of Exercise 5.4 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4 $\sum_{k=1}^n \dfrac{1}{k^2+7 k+12}$\begin{align}S_n &=\sum_{k=1}^n \dfrac{1}{k^2+7 k+12} \\ & =\sum_{k=1}^n \dfrac{1}{(k+3)(k+4)}\end{align}$n^{\text {th }}$$$u_n=\dfrac{1}{(n+3)(n+4)}$$$$\dfrac{1}{(n+3)(n+4)}=\dfrac{A}{n+3}+\dfrac{B}{n+4}$$$A$$B$… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:21 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/ex6-3-p2 Question 2 Exercise 6.3 Solutions of Question 2 of Exercise 6.3 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$n$$r$${ }^n P_r=840$${ }^n C_r=35$\begin{align} &^n P_r=\dfrac{n !}{(n-r) !}=840 ....(i)\\ &^n C_r=\dfrac{n !}{(n-r) ! r !}=35....(ii)\end{align}\begin{align}\dfrac{n !}{(n-r) !} \cdot \dfrac{(n-r) ! r !}{n !}&=\dfrac{840}{35}\\ r!&=24\\ \te… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:43 +0000 https://www.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p10 Question 10 Exercise 7.2 Solutions of Question 10 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$n=2 ;$$s=2^{n-1}$$$ \left.(1+x)^n=\left(\begin{array}{l} n \\ \vdots \end{array}\right)+\left(\begin{array}{l} m \\ 1 \end{array}\right) x+\left(\begin{array}{l} n \\ 2 \end{array}\right) x^2-\ldots+i_n^*\right) x^n \cdot $$$x=1$$(1 \div 1… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:36:22 +0000 https://www.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p9 Question 11 Exercise 7.3 Solutions of Question 11 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$y=\frac{1}{2^2}+\frac{1.3}{2 !} \cdot \frac{1}{2^4}+\frac{1 \cdot 3 \cdot 5}{3 !} \cdot \frac{1}{2^6}+\ldots$$y^2+2 y-1=0$$y=\frac{1}{2^2}+\frac{1.3}{2 !} \cdot \frac{1}{2^4}+\frac{1.3 \cdot 5}{3 !} \cdot \frac{1}{2^6}+\ldots$$$ S=y+1=1+\f… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:36:38 +0000 https://www.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p10 Question 12 Exercise 7.3 Solutions of Question 12 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$2 y=\frac{1}{2^2}+\frac{1.3}{2 !} \cdot \frac{1}{2^4}+\frac{1.3 \cdot 5}{3 !} \cdot \frac{1}{2^6}+\ldots$$4 y^2+4 y-1=0$$$ 2 y=\frac{1}{2^2}+\frac{1.3}{2 !} \cdot \frac{1}{2^4}-\frac{1.3 \cdot 5}{3 !} \cdot \frac{1}{2^6}+\ldots $$$S=2 y+1=… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:36:30 +0000 https://www.mathcity.org/math-11-kpk/sol/unit07/ex7-3-p12 Question 14 Exercise 7.3 Solutions of Question 14 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$x$$p x^p-q x^q=(p-q) x^{p+q}$$x$$x=1+h$$h \longrightarrow 0$$$ p x^p-q x^q=p(1+h)^p-q(1+h)^q $$$$ \begin{aligned} & p x^p-q x^q \\ & =p(1+p h+\text { higher powers h) } \\ & -q(1+q h+\text { higher powcrs } h) \\ & \Rightarrow p x^p-q x^q=… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:36:31 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-4 Exercise 1.4 (Solutions) The solutions of the Exercise 1.4 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to polar form of the complex numbers.$2+i 2 \sqrt{3}$$3-i \sqrt{3}$$-2-i 2$$\dfrac{i-1}{\cos \dfrac{\pi}{3}+i \sin \dfrac{\pi}{3}}$$\left(\cos \dfrac{\pi}{6}+i \sin \dfrac{\pi}{6}\right)\left(\cos \dfrac{\pi}{3}+i \sin \dfrac… anonymous@undisclosed.example.com (Anonymous) Mon, 27 Oct 2025 18:51:55 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-5 Exercise 2.5 (Solutions) The solutions of the Exercise 2.5 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to solving system of the equation of three variables by using matrices.$\left[\begin{array}{ccc}1 & 3 & 5 \\ -6 & 8 & 3 \\ -4 & 6 & 5\end{array}\right]$$\left[\begin{array}{ll}2 & 1 \\ 3 & 2 \\ 1 & 9\end{array}\right]$$\left[… anonymous@undisclosed.example.com (Anonymous) Sun, 08 Feb 2026 16:50:14 +0000 https://www.mathcity.org/math-11-nbf/sol/unit02/ex2-6-p8 Question 9 and 10, Exercise 2.6 Solutions of Question 9 and 10 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 x-y+3 z=\alpha ; 3 x+y-5 z=\beta ;-5 x-5 y+21 z=\gamma$$\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/unit04/ex4-8-p6 Question 11 and 12, Exercise 4.8 Solutions of Question 11 and 12 of Exercise 4.8 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sum_{k=1}^{n} \frac{1}{k(k+2)}$$T_k$$k$\begin{align*} T_k &= \frac{1}{k(k+2)}. \end{align*}\begin{align*} \frac{1}{k(k+2)} = \frac{A}{k} + \frac{B}{k+2} \ldots (1) \end{align*}$k(k+2)$\begin{align*} 1 = A(k+2) + Bk \ldots (2) \end{… anonymous@undisclosed.example.com (Anonymous) Sun, 06 Oct 2024 17:48:33 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-1-p8 Question 7(vii-xi), Exercise 6.1 Solutions of Question 7(vii-xi) of Exercise 6.1 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$\quad n!=990 \cdot (n-3)!$\begin{align*} n!&=990 (n-3)!\\ n(n-1)(n-1)(n-3)!&=990 (n-3)!\\ n(n-1)(n-1)&=990 \\ n^3-3n^2+2n-990 &=0\\ \end{align*}\[\begin{array}{c|cccc} & 1 & -3 & 2 & -990 \\ 11 & 0 & 11 & 88 & 990 \\… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:36 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-3-p5 Question 4, Exercise 6.3 Solutions of Question 4 of Exercise 6.3 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$r$$\,\,^nC_{r-1}:\,^nC_{r}:\,^nC_{r+1}=6:14:21$\begin{align*}\dfrac{n!}{(r-1)!(n-(r-1))!}&: \dfrac{n!}{r!(n-r)!}\\ : \dfrac{n!}{(r+1)!(n-(r+1))!} &= 6:14:21\\ \dfrac{1}{(r-1)!(n-r+1)!}: \dfrac{1}{r!(n-r)!}&\\ : \dfrac{1}{(r+1)!(n-r-1)!}&=… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 09:00:27 +0000 https://www.mathcity.org/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