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Elementary Linear Algebra by Muhammad Usman Hamid

anonymous@undisclosed.example.com (Anonymous) — Sun, 10 Dec 2023 14:05:12 +0000

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…

General Mathematics (Paper A & B)

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:55:55 +0000

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.

MTH231: Linear Algebra

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:27 +0000

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…

Definitions: FSc Part 2 (Mathematics): PTB

anonymous@undisclosed.example.com (Anonymous) — Sun, 20 Oct 2024 18:16:52 +0000

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$$x \in X$$…

FSc Part 1 (KPK Boards)

anonymous@undisclosed.example.com (Anonymous) — Tue, 12 Dec 2023 17:30:33 +0000

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…

Chapter 10: Higher Order Linear Differential Equations

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:45:51 +0000

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

Syllabus & Paper Pattern for General Mathematics (Split Program)

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:55:21 +0000

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…

B-Course of Mathematics (Paper A & B)

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:55:55 +0000

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.$(\lambda ,\mu )$

Chapter 10: Viewer

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:54:54 +0000

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.

Unit 05: Linear Inequalities and Linear Programming

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:47:08 +0000

Unit 05: Linear Inequalities and Linear Programming [Unit 05: Linear Inequalities and Linear Programming] Notes (Solutions) of Unit 05: Linear Inequalities and Linear Programming, 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.

A-Course of Mathematics (Paper A & B)

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:55:54 +0000

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.

Question 1, Exercise 1.3

anonymous@undisclosed.example.com (Anonymous) — Sun, 14 Jul 2024 19:35:10 +0000

Question 1, Exercise 1.3 Solutions of Question 1 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 1(i) $z^{2}+169$\begin{align} & z^{2} + 169 \\ = & z^{2} - (13i)^2 \\ = &(z + 13i)(z - 13i). \end{align}$2 z^{2}+18$\begin{align} & 2z^2 + 18 \\ = &2(z^2 - (3i)^2)\\ = &2(z + 3i)(z - 3i) \end{align}$3 z^{2}+363$\begin{align} & 3z^2 + 363 \\ …

Linear Algebra: Important Definitions and Results

anonymous@undisclosed.example.com (Anonymous) — Wed, 10 Aug 2022 16:49:49 +0000

Linear Algebra: Important Definitions and Results [Linear Algebra: Important Definitions and Results] 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. These notes contains important definitions with examples and related theorem, which might be helpful to prepare interviews or any other written test after graduation like PPSC, FPSC or etc.$V$$n$$n$$V$$W$$V$$A$$A$$A…

Quantitative Reasoning II (QR2: Tools for Reasoning Skills)

anonymous@undisclosed.example.com (Anonymous) — Fri, 01 Aug 2025 18:31:56 +0000

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…

MCQs: Ch 04 Quadratic Equations

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:47:50 +0000

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-…

Definitions: Mathematics 12: PTB by Muzzammil Subhan

anonymous@undisclosed.example.com (Anonymous) — Sun, 20 Oct 2024 18:16:28 +0000

Definitions: Mathematics 12: PTB by Muzzammil Subhan Definitions from Calculus and Analytic Geometry, MATHEMATICS 12, published by Punjab Textbook Board (PTB) Lahore, Pakistan. We are very thankful to Muzzammil Subhan for his valuable contribution. Download or view PDF for all definitions. Samples is given below$P(x)=a_0 x^0+a_1 x^1+a_2 x^2+\ldots . .+a_{n-1} x^{n-1}+a_n x^n$$n \in W$$a_0, a_1, a_2, \ldots, a_n \in R$$f(x)=a x+b$$a, b \in R$$a \neq 0$$f(x)=x$$f(x)=c$$c \in R$$\frac{P(x)}{Q(x)}$…

Functional Analysis by Mr. Tahir Hussain Jaffery

anonymous@undisclosed.example.com (Anonymous) — Wed, 18 Sep 2024 18:13:33 +0000

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$

Unit 05: Linear Inequalities and Linear Programming

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:47:57 +0000

Unit 05: Linear Inequalities and Linear Programming Here is the list of important questions. * Graph the solution region of $2x+y \geq 2$ --- BSIC Gujranwala (2016) * Graph the feasible region subject to the following constraint: --- BSIC Gujranwala (2016)$2x-3y \leq 6$$2x+3y \leq 12$$x \geq 0$$y \geq 0$$2x+y\geq 2$$x+2y\leq10$$x\geq0,y\geq0$$2x+3y\leq 12$$z=x+3y$$2x+5y\leq30$$5x+4y\leq20$$x\geq0$$y\geq0$$x+2y\leq 14$$3x+4y\leq 36$$2x+y\leq 10$$x\geq0, y\geq0$$f(x)=2x+5y$$-x\leq8$$-y\leq…

Operation Research by Sir Haidar Ali

anonymous@undisclosed.example.com (Anonymous) — Fri, 21 Mar 2025 17:42:28 +0000

Operation Research by Sir Haidar Ali [Notes of Operation Research by Sir Haidar Ali] 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 are based on lectures delivered by Sir Haidar Ali at GC University Faisalabad.

Topology: Handwritten Notes

anonymous@undisclosed.example.com (Anonymous) — Sat, 01 Mar 2025 09:43:45 +0000

Topology: Handwritten Notes [House of Tau] A topological space is a collection of points with a topology-a structure that describes how close two points are to one another. It is a generalisation of Euclidean spaces that makes it possible to investigate boundaries, continuity, and connectivity. A topology is a group of open sets, or subsets, that adhere to certain principles.$T_0$$T_1$$T_2$$\varepsilon-$

PPSC Paper 2021 (Lecturer in Mathematics)

anonymous@undisclosed.example.com (Anonymous) — Tue, 23 Aug 2022 17:04:49 +0000

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(…

MCQs: Ch 02 Sets, Functions and Groups

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:47:49 +0000

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=…

Applied Mathematics (Paper A & B)

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:55:46 +0000

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) $(\lambda ,\mu )$

Pure Mathematics (Paper A & B)

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:55:59 +0000

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)

1st SIBAU-NU International Workshop on Matrix Analysis and Linear Algebra (15-17 October 2021)

anonymous@undisclosed.example.com (Anonymous) — Fri, 09 Jul 2021 18:34:07 +0000

1st SIBAU-NU International Workshop on Matrix Analysis and Linear Algebra (15-17 October 2021) [1st SIBAU-NU International Workshop on Matrix Analysis and Linear Algebra] * Conference Name: 1st SIBAU-NU International Workshop on Matrix Analysis and Linear Algebra * Venue: Online * Organizer: Department of Mathematics, Sukkur IBA University, (SIBAU)-Pakistan and Naresuan University, (NU)-Thailand

Mathematics 9 (Science Group)

anonymous@undisclosed.example.com (Anonymous) — Wed, 08 Mar 2023 18:04:36 +0000

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) …

Algebra II by Syed Sheraz Asghar

anonymous@undisclosed.example.com (Anonymous) — Sun, 10 Dec 2023 14:02:42 +0000

Algebra II by Syed Sheraz Asghar [Algebra II by Syed Sheraz Asghar] 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: Algebra II * Provider: Mr. Muzammil Tanveer

Functional Analysis by M Usman Hamid and Zeeshan Ahmad

anonymous@undisclosed.example.com (Anonymous) — Wed, 18 Sep 2024 18:09:17 +0000

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.

Chapter 09: Linear Programming

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:47:24 +0000

Chapter 09: Linear Programming Notes of Chapter 09: Linear Programming of “A Textbook of Mathematics for Class XI” published by Khyber Pakhtunkhwa (KPK) Textbook Board, Pesharwar. These notes are shared as open educational resources. [Download PDF ~ ch09-linear-programming-fsc1-kpk.pdf] fsc kpk_fsc_part1

Unit 08: Linear Graph and their Application

anonymous@undisclosed.example.com (Anonymous) — Sat, 28 May 2022 19:29:21 +0000

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.$

Khuram Ali Khan

anonymous@undisclosed.example.com (Anonymous) — Fri, 13 Jun 2025 12:03:59 +0000

Khuram Ali Khan Khuram Ali Khan, PhD Associate Professor Department of Mathematics University of Sargodha Sargodha - PAKISTAN. Email: Field of Research: Difference and functional equations, Real functions, Mathematical inequalities involving convex functions, Time Scales Calculus, Soft Sets

MATH-731: Convex Analysis

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:27 +0000

MATH-731: Convex Analysis Convex functions on the real line, Continuity and differentiability of convex functions, Characterizations, Differences of convex functions, Conjugate convex functions, Convex sets and affine sets, Convex functions on a normed linear space, Continuity of convex functions on normed linear space, Differentiable convex function on normed linear space, The support of convex functions, Differentiability of convex function on normed linear space.

Notes of Algebra

anonymous@undisclosed.example.com (Anonymous) — Wed, 26 Jun 2024 18:24:28 +0000

Notes of Algebra All the notes of algebra are given here. [All the notes of algebra] Algebra II by Syed Sheraz Asghar A notes of Algebra-II composed by Muzammil Tanveer. We are very thankful to him for sending these notes. Elementary Linear Algebra by Muhammad Usman Hamid Linear Algebra is the study of vectors and linear transformations. Best notes to prepare the subject written by

Method of Mathematical Physics by Mr. Muhammad Usman Hamid

anonymous@undisclosed.example.com (Anonymous) — Thu, 20 Apr 2023 11:54:01 +0000

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.

Vector Spaces (Handwritten notes)

anonymous@undisclosed.example.com (Anonymous) — Sat, 01 Apr 2023 15:18:56 +0000

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.

Chapter 04: System of Linear Equations

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:45:45 +0000

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 …

Chapter 06: Vector Spaces

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:45:47 +0000

Chapter 06: Vector Spaces Notes of Chapter 06 Vector Spaces 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 * Subspaces * Linear combinations and spanning sets

Chapter 03: Matrices and Determinants

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:46:27 +0000

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$

Question 2, Exercise 1.3

anonymous@undisclosed.example.com (Anonymous) — Sun, 01 Oct 2023 06:52:57 +0000

Question 2, Exercise 1.3 Solutions of Question 2 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 2(i) $P(z)$$$P\left( z \right)={{z}^{3}}+6z+20$$$$p\left( z \right)={{z}^{3}}+6z+20$$$(z-a)$$P(z)$$P(a)=0$$z=-2$\begin{align} P(-2)&=(-2)^3+6(-2)+20\\ &=-8-12+20\\ &=0\end{align}$z+2$${{z}^{3}}+6z+20$$$\begin{array}{c|cccc} -2 & 1 & 0 & 6 & 20 \\ & \d…

Unit 05: Linear Inequalities and Linear Programming: Mathematics FSc part 2

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 17:00:20 +0000

Unit 05: Linear Inequalities and Linear Programming: Mathematics FSc part 2 Notes (Solutions) of Unit 05: Linear Inequalities and Linear Programming, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. There are three exercises in this chapter. Please see the main page of this chapter for MCQs and important question

Question 2, Exercise 1.3

anonymous@undisclosed.example.com (Anonymous) — Tue, 05 Dec 2023 02:45:02 +0000

Question 4, Exercise 1.3

anonymous@undisclosed.example.com (Anonymous) — Mon, 15 Jul 2024 12:19:22 +0000

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…

Question 1, Exercise 2.6

anonymous@undisclosed.example.com (Anonymous) — Wed, 04 Sep 2024 03:03:49 +0000

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…

Question 3, Exercise 2.6

anonymous@undisclosed.example.com (Anonymous) — Wed, 04 Sep 2024 03:11:14 +0000

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…

Question 4, Exercise 2.6

anonymous@undisclosed.example.com (Anonymous) — Wed, 04 Sep 2024 03:11:42 +0000

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 …

Question 5, Exercise 2.6

anonymous@undisclosed.example.com (Anonymous) — Wed, 04 Sep 2024 03:12:33 +0000

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 = \…

Question 6, Exercise 2.6

anonymous@undisclosed.example.com (Anonymous) — Wed, 04 Sep 2024 03:13:00 +0000

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…

MTH103: Exploring Quantitative Skills

anonymous@undisclosed.example.com (Anonymous) — Wed, 27 Sep 2023 13:47:12 +0000

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…

Definitions: FSc Part 1 (Mathematics): PTB

anonymous@undisclosed.example.com (Anonymous) — Sun, 22 Sep 2024 17:12:21 +0000

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…

Definitions: FSc Part 1 (Mathematics): PTB by Aurang Zaib

anonymous@undisclosed.example.com (Anonymous) — Sat, 30 Mar 2024 16:48:20 +0000

Definitions: FSc Part 1 (Mathematics): PTB by Aurang Zaib Definitions from Textbook of Algebra and Trigonometry Class XI, published by Punjab Textbook Board (PTB) Lahore, Pakistan. We are very thankful to Aurang Zaib for his valuable contribution. Chapter 01: Number System $ \dfrac{p}{q} $$ p, q \in \mathbb{Z} $$ q \neq 0 $$ \dfrac{3}{4} $$ \dfrac{7}{2} $$ \sqrt{2} $$ \pi $$ \mathbb{R} $$ 0.25 $$ 3.75 $$ 0.3333... $$ 1.234234... $$ \pi $$ 3.1415... $$ \sqrt{2} $\(…

General Mathematics 9

anonymous@undisclosed.example.com (Anonymous) — Mon, 15 Dec 2025 16:49:07 +0000

General Mathematics 9 [General Mathematics 9th Class] There are ten chapters in General Mathematics 9 for Punjab Textbook Board, Lahore. Solutions of all the chapters are given below. One can download the PDF file of the notes. Please remember that, to view these notes one must have PDF reader installed in their system. We will try our best to add online view of the notes very soon.

Functional Analysis by Prof Mumtaz Ahmad

anonymous@undisclosed.example.com (Anonymous) — Wed, 18 Sep 2024 18:11:56 +0000

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$

Notes for Numerical Methods by M Usman Hamid

anonymous@undisclosed.example.com (Anonymous) — Sat, 05 Aug 2023 17:59:07 +0000

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)$

Partial Differential Equations (PDE) by M Usman Hamid

anonymous@undisclosed.example.com (Anonymous) — Sun, 09 Mar 2025 09:27:42 +0000

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

Umer Asghar

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:44:50 +0000

Umer Asghar We are very thankful to Mr. Umer Asghar for his contribution to the website. * Email: * 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] *

PPSC Paper 2011 (Lecturer in Mathematics)

anonymous@undisclosed.example.com (Anonymous) — Thu, 03 Feb 2022 11:54:32 +0000

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$$…

Unit 02: Matrices and Determinants (Solutions)

anonymous@undisclosed.example.com (Anonymous) — Sun, 08 Feb 2026 17:04:31 +0000

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.

Question 1, Exercise 1.3

anonymous@undisclosed.example.com (Anonymous) — Sat, 30 Sep 2023 18:43:14 +0000

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…

Question 1, Exercise 1.3

anonymous@undisclosed.example.com (Anonymous) — Tue, 05 Dec 2023 02:45:01 +0000

Notes of Mathematics

anonymous@undisclosed.example.com (Anonymous) — Mon, 25 May 2026 18:19:50 +0000

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…

PPSC General Information, Syllabus, Paper Pattern

anonymous@undisclosed.example.com (Anonymous) — Sun, 24 May 2026 17:45:57 +0000

~~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.

MATH-301: Complex Analysis

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:18 +0000

MATH-301: Complex Analysis Objectives of the course This is an introductory course in complex analysis, giving the basics of the theory along with applications, with an emphasis on applications of complex analysis and especially conformal mappings. Students should have a background in real analysis (as in the course Real Analysis I), including the ability to write a simple proof in an analysis context. $\cot 2z$

MATH-505: Complex Analysis

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:23 +0000

MATH-505: Complex Analysis Provisional Results MMAF13E101 = 65 MMAF13E102 = 65 MMAF13E103 = 58 MMAF13E104 = 58 MMAF13E105 = 78 MMAF13E106 = 62 MMAF13E107 = 50 MMAF13E108 = 75 MMAF13E109 = 61 MMAF13E110 = 50 MMAF13E111 = 50 MMAF13E112 = 85 $\cot 2z$

MTH604: Fixed Point Theory and Applications (Spring 2020)

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:41 +0000

~~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…

Notes of Mathematical Method

anonymous@undisclosed.example.com (Anonymous) — Sun, 02 Jul 2023 07:49:56 +0000

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

Important Questions: HSSC-II

anonymous@undisclosed.example.com (Anonymous) — Thu, 18 Apr 2024 08:02:52 +0000

Important Questions: HSSC-II These questions are taken from old papers for the book Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Textbook Board (PTB) Lahore, Pakistan. * Unit 01: Functions and Limits * Unit 02: Differentiation * Unit 03: Integration * Unit 04: Introduction to Analytic Geometry * Unit 05: Linear Inequalities and Linear Programming *

Subjective Mathematics 12th by Muhammad Shahbaz

anonymous@undisclosed.example.com (Anonymous) — Tue, 23 Sep 2025 19:59:04 +0000

Subjective Mathematics 12th by Muhammad Shahbaz [Subjective Mathematics 12th by Muhammad Shahbaz] These handwritten notes are sent by Muhammad Shahbaz. We are very thankful to him for sending these notes. These notes are based on the book “Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc/ICS Part 2 or HSSC-II), Punjab Textbook Board (PTB) Lahore, Pakistan.'' There are total seven (7) units in this book.

FSc Part 2 (KPK Boards)

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:42:52 +0000

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$

Mathematics 10 (Science Group)

anonymous@undisclosed.example.com (Anonymous) — Wed, 24 Jul 2024 18:33:10 +0000

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…

Affine and Euclidean Geometry by Shahzad Idress

anonymous@undisclosed.example.com (Anonymous) — Mon, 24 Jul 2023 10:30:52 +0000

Affine and Euclidean Geometry by Shahzad Idress [Affine and Euclidean Geometry by Shahzad Idress] These notes are sent by shahzad-idress. We acknowledged his efforts to published these notes on MathCity.org. These are short notes containing topics related to Affine and Euclidean Geometry. The main sections includes $R^n$

Classical Mechanics by Muhammad Usman Hamid

anonymous@undisclosed.example.com (Anonymous) — Mon, 29 Jul 2024 18:38:28 +0000

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.

Mechanics I (Statics) by Dr Babar Ahmad

anonymous@undisclosed.example.com (Anonymous) — Mon, 24 Jul 2023 12:49:29 +0000

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.

Multivariable Calculus by M Usman Hamid and M Zeeshan Ahmad

anonymous@undisclosed.example.com (Anonymous) — Sun, 15 Jun 2025 16:25:59 +0000

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…

Number Theory by Dr Muhammad Umer Shuaib

anonymous@undisclosed.example.com (Anonymous) — Sat, 05 Aug 2023 18:11:04 +0000

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.$\phi$

Numerical Analysis II

anonymous@undisclosed.example.com (Anonymous) — Sat, 05 Aug 2023 17:56:32 +0000

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

Operation Research: Handwritten Notes

anonymous@undisclosed.example.com (Anonymous) — Sat, 05 Aug 2023 16:53:29 +0000

Operation Research: Handwritten Notes [Operation Research: Handwritten Notes] Operation research (OR) is a scientific field that enhances organisational decision-making and problem-solving via the application of mathematical and analytical techniques. The management and administration of many processes, including military, governmental, economic, and industrial ones, include the use of scientific principles. OR is carried out by a group of professionals from various linked fields, depending on …

Ordinary Differential Equations (ODE) by Hammad Safi

anonymous@undisclosed.example.com (Anonymous) — Sun, 09 Mar 2025 09:26:46 +0000

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…

Theory of Optimization by Ma'am Iqra Razzaq

anonymous@undisclosed.example.com (Anonymous) — Sun, 23 Jul 2023 16:28:59 +0000

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.

Vector Space (Review) by Rashad Wattu

anonymous@undisclosed.example.com (Anonymous) — Sat, 01 Apr 2023 14:51:01 +0000

Vector Space (Review) by Rashad Wattu Mathematical vector space is a crucial and fundamental idea. It depends on the concept of field. These notes were primarily written to help maths students understand vector space, which is a concept that they all need to be familiar with.

Nawab Ali

anonymous@undisclosed.example.com (Anonymous) — Fri, 20 May 2022 11:38:18 +0000

Nawab Ali We are very thankful to Mr. Nawab Ali for providing notes of functional analysis. His notes are attached below, which might be helpful in BS Mathematics. * Name: Nawab Ali * Father Name: Muhammad Ayar * Email: * University: Abdul wali Khan University Mardan

Chapter 03: Matrices

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:45:44 +0000

Chapter 03: Matrices Notes of the book Mathematical Method written by S.M. Yusuf, A. Majeed and M. Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN. The difficulty level of this chapter is very low. Most of the questions involve calculations. This chapter is wide range of applications in Linear Algebra. In many universities teachers include this chapter in the syllabus of Linear Algebra for BS students of mathematics and other subjects.

Unit 1: Complex Numbers (Solutions)

anonymous@undisclosed.example.com (Anonymous) — Mon, 25 Sep 2023 12:08:05 +0000

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$

Chapter 02: Sets, Functions and Groups

anonymous@undisclosed.example.com (Anonymous) — Wed, 05 Apr 2023 12:54:57 +0000

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$p\leftrightarrow q$

Objective Mathematics 12th by Muhammad Shahbaz

anonymous@undisclosed.example.com (Anonymous) — Fri, 21 Mar 2025 15:20:30 +0000

Objective Mathematics 12th by Muhammad Shahbaz [Objective Mathematics 12th by Muhammad Shahbaz] These notes are sent by Muhammad Shahbaz. We are very thankful to him for send this booklet. This booklet is very helpful to cover the following things for Mathematics 12 (Mathematics for Intermediate) * Summary

Assessment Scheme

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:46:51 +0000

Assessment Scheme Chapter wise marks distribution is given below. Mathematics Class 12 Time: 3 Hours Marks: 100 Ch # Chapter name Weightage % Distribution of marks 1 Functions and limits 7% 11 2 Differentiation 21% 30

Unit 04: Introduction to Analytic Geometry

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:47:07 +0000

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.$ax^2+ 2hxy+by^2=0$

Unit 01: Complex Numbers (Solutions)

anonymous@undisclosed.example.com (Anonymous) — Thu, 01 Feb 2024 02:28:42 +0000

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$

Unit 02: Matrices and Determinants (Solutions)

anonymous@undisclosed.example.com (Anonymous) — Tue, 05 Dec 2023 02:44:47 +0000

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$

Matric 9th (General): Online View

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:48:06 +0000

Matric 9th (General): Online View List of all chapters * VIEW Unit 01: Percentage Ratio and Proportion * VIEW Unit 02: Zakat, Ushr and Inheritance * VIEW Unit 03: Business Mathematics * VIEW Unit 04: Financial Mathematics * VIEW Unit 05: Consumer Mathematics * VIEW Unit 06: Exponents and Logarithms * VIEW Unit 07: Arithmatic and Geometric Sequence * VIEW Unit 08: Sets and Functions * VIEW Unit 09: Linear Graphs * VIEW Unit 10: Basic Statistic matric 9th_general

Number Theory by Ms. Iqra Liaqat

anonymous@undisclosed.example.com (Anonymous) — Tue, 11 May 2021 11:51:22 +0000

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

How to prepare admission test (A short guide)

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:50:25 +0000

How to prepare admission test (A short guide) MathCity.org does not represent any official or government/semi-government/private educational institute or board or university. The resources given on the site holds no official position in government/semi-government/private educational institute or board or university. While using a resources given on this site you agreed to the term that we (MathCity.org or person related to MathCity.org) do not take any responsibility for these resources. The sug…

Chapter 04: Viewer

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:53:51 +0000

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.

Chapter 04: Viewer

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:53:27 +0000

Exercise 1.3 (Solutions)

anonymous@undisclosed.example.com (Anonymous) — Mon, 27 Oct 2025 18:51:15 +0000

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$$4 z^{2}-7 z-11$$…

Question 2, Exercise 2.6

anonymous@undisclosed.example.com (Anonymous) — Wed, 04 Sep 2024 03:04:14 +0000

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…

Unit 07: Linear Equations and Inequalities: Online View

anonymous@undisclosed.example.com (Anonymous) — Sat, 30 Apr 2022 18:35:29 +0000

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

Unit 08: Linear Graph and their Application: Online View

anonymous@undisclosed.example.com (Anonymous) — Sat, 28 May 2022 19:00:24 +0000

Unit 08: Linear Graph and their Application: Online View On this page the solutions of Unit 08: Linear Graph and their Application, Mathematics 9 (Science Group), are given List of all exercise of Unit 08 * Exercise 8.1 (Solutions) * Exercise 8.2 (Solutions) * Exercise 8.3 (Solutions)

Mathematical Events

anonymous@undisclosed.example.com (Anonymous) — Sun, 23 Mar 2025 06:48:21 +0000

Mathematical Events This page is dedicated to events related to field of mathematics, which includes conferences, seminars, workshops, games, occurring all over the country (PAKISTAN). If you wish to add upcoming mathematical event on this page, please contact to

Mathematics 9th for Matric (PCTB)

anonymous@undisclosed.example.com (Anonymous) — Mon, 04 May 2026 16:19:23 +0000

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).

CHEM-501: Basic Mathematics for Chemist

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:04 +0000

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…

MTH424: Convex Analysis

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:06 +0000

MTH424: Convex Analysis Objectives: At the end of this course the students will be able to understand the concept of Convex Analysis, convex sets, convex functions, Differential of the convex function. Developing ability to study the Hadamard-Hermite inequalities and their applications. Prepare students to be self independent and enhance their mathematical ability by giving them home work and projects.

MTH604: Fixed Point Theory and Applications

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:08 +0000

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…

MTH424: Convex Analysis (Fall 2020)

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:16 +0000

MTH424: Convex Analysis (Fall 2020) [Convex Analysis] Objectives: At the end of this course the students will be able to understand the concept of Convex Analysis, convex sets, convex functions, Differential of the convex function. Developing ability to study the Hadamard-Hermite inequalities and their applications. Prepare students to be self independent and enhance their mathematical ability by giving them home work and projects.$f(x)=x$$\mathbb{R}$$f(x)=x^2$$\mathbb{R}$$f:[a,b]\to \mathbb{…

MTH604: Fixed Point Theory and Applications (Fall 2022)

anonymous@undisclosed.example.com (Anonymous) — Fri, 06 Jan 2023 04:37:11 +0000

~~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…

MATH-300: Basic Mathematics for Chemist

anonymous@undisclosed.example.com (Anonymous) — Wed, 31 May 2023 05:38:37 +0000

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…

MTH633: Advanced Convex Analysis

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:31 +0000

MTH633: Advanced Convex Analysis Convex sets, convex hull, their properties, separation theorems, hyperplane, Best approximation theorem and its applications, Farkas and Gordan Theorems, Extreme points and Polyhedral. Convex functions, Basic Definitions, properties, various generalizations, differentiable convex functions, subgradient, characterization and applications in linear and nonlinear optimization, complementarity problems and its equivalent formulations.

MTH633: Advanced Convex Analysis (Spring 2015)

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:33 +0000

MTH633: Advanced Convex Analysis (Spring 2015) Convex sets, convex hull, their properties, separation theorems, hyperplane, Best approximation theorem and its applications, Farkas and Gordan Theorems, Extreme points and Polyhedral. Convex functions, Basic Definitions, properties, various generalizations, differentiable convex functions, subgradient, characterization and applications in linear and nonlinear optimization, complementarity problems and its equivalent formulations.

MTH633: Advanced Convex Analysis (Spring 2017)

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:36 +0000

MTH633: Advanced Convex Analysis (Spring 2017) Convex sets, convex hull, their properties, separation theorems, hyperplane, Best approximation theorem and its applications, Farkas and Gordan Theorems, Extreme points and Polyhedral. Convex functions, Basic Definitions, properties, various generalizations, differentiable convex functions, subgradient, characterization and applications in linear and nonlinear optimization, complementarity problems and its equivalent formulations.

MTH604: Fixed Point Theory and Applications

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:39 +0000

MTH633: Advanced Convex Analysis (Spring 2019)

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:40 +0000

MTH633: Advanced Convex Analysis (Spring 2019) Convex sets, convex hull, their properties, separation theorems, hyperplane, Best approximation theorem and its applications, Farkas and Gordan Theorems, Extreme points and Polyhedral. Convex functions, Basic Definitions, properties, various generalizations, differentiable convex functions, subgradient, characterization and applications in linear and nonlinear optimization, complementarity problems and its equivalent formulations.$\mathbb{R}$$\math…

MTH604: Fixed Point Theory and Applications (Spring 2021)

anonymous@undisclosed.example.com (Anonymous) — Mon, 22 Feb 2021 15:12:31 +0000

~~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…

MTH424: Convex Analysis (Spring 2024)

anonymous@undisclosed.example.com (Anonymous) — Fri, 29 Mar 2024 03:48:00 +0000

MTH424: Convex Analysis (Spring 2024) [Convex Analysis] Objectives: At the end of this course the students will be able to understand the concept of Convex Analysis, convex sets, convex functions, Differential of the convex function. Developing ability to study the Hadamard-Hermite inequalities and their applications. Prepare students to be self independent and enhance their mathematical ability by giving them home work and projects.

MTH424: Convex Analysis (Spring 2025)

anonymous@undisclosed.example.com (Anonymous) — Mon, 16 Jun 2025 18:51:28 +0000

MTH424: Convex Analysis (Spring 2025) [Convex Analysis] Convex analysis is a branch of mathematics that studies convex sets and convex functions. A set is convex if a straight line between any two points in the set always stays inside it. This field is important in optimization, economics, and engineering. It helps in solving real-world problems like minimizing costs, maximizing profits, and designing efficient systems. Convex analysis is widely used in machine learning, finance, and physics. 😊…

Notes of Number Theory by Umer Asghar

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:48 +0000

Notes of Number Theory by Umer Asghar These notes are very helpful to prepare one of the sections paper of mathematics for BSc. Author: Umer Asghar Type: Composed Format: PDF (1.14 mB) Pages: 24 Contents and Summary * Divisibility

Solutions: Math 9th PCTB

anonymous@undisclosed.example.com (Anonymous) — Mon, 04 May 2026 16:21:21 +0000

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

Differential Geometry: Handwritten Notes

anonymous@undisclosed.example.com (Anonymous) — Sat, 23 Aug 2025 11:27:57 +0000

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 …

Differential Geometry by M Usman Hamid

anonymous@undisclosed.example.com (Anonymous) — Sat, 23 Aug 2025 11:29:44 +0000

Differential Geometry by M Usman Hamid [Differential Geometry by M Usman Hamid] These notes are initially provided by Mr. Anwar Khan. Later send by many persons. We are really very thankful to all for providing these notes and appreciates their effort to publish these notes on MathCity.org

Differential Geometry by Syed Hassan Waqas

anonymous@undisclosed.example.com (Anonymous) — Sat, 23 Aug 2025 11:28:41 +0000

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.…

Mathematical Method by Sir Muhammad Awais Aun

anonymous@undisclosed.example.com (Anonymous) — Mon, 17 Apr 2023 09:02:35 +0000

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.

Mathematical Statistics by Ms. Iqra Liaqat

anonymous@undisclosed.example.com (Anonymous) — Thu, 20 Apr 2023 12:30:11 +0000

Mathematical Statistics by Ms. Iqra Liaqat [Mathematical Statistics by Ms. Iqra Liaqat] These notes are send by Ms. Iqra Liaqat. We are really very thankful to her for providing these notes and appreciates her effort to publish these notes on MathCity.org Name Mathematical Statistics Provided by

Measure Theory Handwritten Notes by Asim Marwat

anonymous@undisclosed.example.com (Anonymous) — Fri, 10 Feb 2023 17:43:56 +0000

Measure Theory Handwritten Notes by Asim Marwat [Measure Theory Notes by Asim Marwat] These notes are made and shared by Mr. Asim Marwat. He has our sincere gratitude for supplying these notes, and we value his effort in having them published on MathCity.org. Measure Theory is an important subject in BS Mathematics. These notes contain topic from base level, like equivalence set, to advance level, like convergent in measure.$L_p$$L_p$$L^\infty$$L_p$

Mechanics III (Analytic Dynamics II) by Dr Babar Ahmad

anonymous@undisclosed.example.com (Anonymous) — Sat, 10 May 2025 17:33:01 +0000

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.

Multiple Choice Questions (BSc/BS/PPSC) by Akhtar Abbas

anonymous@undisclosed.example.com (Anonymous) — Sun, 24 May 2026 17:45:10 +0000

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$

Number Theory: Handwritten Notes

anonymous@undisclosed.example.com (Anonymous) — Tue, 08 Aug 2023 11:20:45 +0000

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

Number Theory Notes by Anwar Khan

anonymous@undisclosed.example.com (Anonymous) — Sat, 05 Aug 2023 18:47:01 +0000

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…

Quantitative Reasoning-I (QR1: Exploring Quantitative Skills)

anonymous@undisclosed.example.com (Anonymous) — Mon, 15 Dec 2025 16:47:25 +0000

Quantitative Reasoning-I (QR1: Exploring Quantitative Skills) [Quantitative Reasoning-I (Exploring Quantitative Skills) by M Usman Hamid] This handout offers a comprehensive introduction to numeracy and quantitative reasoning, laying a solid foundation for analytical thinking. It explores fundamental concepts, from basic arithmetic and unit analysis to more advanced topics like probability and statistics. A unique highlight is its exploration of the contributions of mathematicians and statis…

Real Analysis Handwritten Notes by Kaushef Salamat

anonymous@undisclosed.example.com (Anonymous) — Fri, 20 Jan 2023 17:26:27 +0000

Real Analysis Handwritten Notes by Kaushef Salamat [Real Analysis Handwritten Notes by Kaushef Salamat] We are very thankful to Ms. Kaushef Salamat for providing these notes. Real Analysis is a core subject in BS or MSc Mathematics. Without a fundamental grasp of Real Analysis, one cannot claim to be a mathematician. These notes are very comprehensive containing almost all the notions of Real Analysis. For providing these notes, Ms. $\infty$

Akhtar Abbas

anonymous@undisclosed.example.com (Anonymous) — Fri, 25 Aug 2023 08:59:45 +0000

Akhtar Abbas Mr. Akhtar Abbas is Lecture in Mathematics at University of Jhang, Jhang, Punjab, Pakistan. He is a dedicated and hardworking teacher. We are very thankful to him for his great contribution to our website. * Email: * YouTube Channel:

Sheikh Muhammad Saleem Shahzad

anonymous@undisclosed.example.com (Anonymous) — Sat, 23 Dec 2023 13:41:25 +0000

Sheikh Muhammad Saleem Shahzad “Mr. Saleem, an accomplished mathematician with an M.Phil in Processor Optimization through Memoryless Computations from the University of Sargodha, currently serves as a dedicated Mathematics Lecturer at Government Graduate College Jauharabad. His teaching experience includes a role as a Visiting Lecturer at the University of Education, Lahore, where he instructed courses such as Calculus, Set Theory, Real Analysis, Group Theory, and Linear Algebra. Additionally,…

PPSC Paper 2015 (Lecturer in Mathematics)

anonymous@undisclosed.example.com (Anonymous) — Tue, 18 Jan 2022 10:20:00 +0000

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$$\…

Chapter 02: Groups

anonymous@undisclosed.example.com (Anonymous) — Mon, 11 Dec 2023 13:00:23 +0000

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 ) *

Unit 03: Integration

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:47:55 +0000

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…

Unit 04: Introduction to Analytic Geometry

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:47:56 +0000

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…

MCQs with key

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:46:44 +0000

MCQs with key [MCQs Choice] In this one PDF, MCQs of all chapters of FSc Part2 are given. There are seven chapters. Keys of MCQs is starting from page 51. SAMPLE MCQs * A function $I(x)=x$ is called * (A) A linear function * (B) An identity function * (C) A quadratic function$\frac{d}{dx} \tan 3x =$$3\sec^2 3x$$\frac{1}{3}\sec^2 3x$$\cot 3x$$\sec^2 x$$y=f(x)$$y$$dy=f'(x)$$dy=f'(x) dx$$dy=f(x)$$\frac{dy}{dx}$$x<0$$y<0$$P(x,y)$$ax+by

Unit 01: Complex Numbers (Solutions)

anonymous@undisclosed.example.com (Anonymous) — Mon, 27 Oct 2025 18:47:40 +0000

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$

Unit 05: Polynomials

anonymous@undisclosed.example.com (Anonymous) — Sun, 13 Oct 2024 18:05:52 +0000

Unit 05: Polynomials [Unit 05: Polynomials] This is a fifth unit of the book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. On this page we have provided the solutions of the questions.

Unit 11: Parallelograms and Triangles

anonymous@undisclosed.example.com (Anonymous) — Mon, 20 Mar 2023 15:25:59 +0000

Unit 11: Parallelograms and Triangles On this page notes of Unit 11 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: * prove that in a parallelogram * the opposite sides are congruent,

Syllabus for UoS (Private only)

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:50:15 +0000

Syllabus for UoS (Private only) Syllabus and scheme of studies for private students doing MSc Mathematics from University of Sargodha, Sargodha. The syllabus has been changed and few optional subjects has been dropped. Please be alert --- 2017/08/25 17:05

Chapter 04: Centres of Mass and Gravity

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:55:04 +0000

Chapter 04: Centres of Mass and Gravity Notes of Introduction to Mechanics by Q. K. Ghori published by West Pak Publishing Company (Pvt) Ltd. These notes are written by Prof. Muhammad Tanveer. Following are the main topics of this chapter. * Linear Moment

Applied Mathematics

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:55:54 +0000

Applied Mathematics Paper pattern for Applied Mathematics chapter-wise for University of Sargodha is given on this page. This pattern is extracted from syllabus, so use your own risk. Syllabus of Applied Mathematics can be seen here. Applied Mathematics is consists of two papers of 100 marks each. One is called “Paper A” and other is called “Paper B”. In every paper there are three sections with four questions each. A student have to attempt two questions from each section.

General Mathematics

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:55:58 +0000

General Mathematics Paper pattern for General Mathematics chapter-wise for University of Sargodha is given on this page. This pattern is provided by Muhammad Siraj (+92-345-5365318). Syllabus of General Mathematics can be seen here. General Mathematics is consists of two papers of 100 marks each. One is called “Paper A” and other is called “Paper B”. In every paper there are three sections with four questions. A student have to attempt two questions from each section.

Question 1, Review Exercise

anonymous@undisclosed.example.com (Anonymous) — Mon, 25 Nov 2024 17:51:01 +0000

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}…

Review Exercise 2 (Solutions)

anonymous@undisclosed.example.com (Anonymous) — Sun, 08 Feb 2026 16:57:09 +0000

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}=(-…

Question 8, Review Exercise

anonymous@undisclosed.example.com (Anonymous) — Sun, 13 Oct 2024 17:52:41 +0000

Question 8, Review Exercise Solutions of Question 8 of Review Exercise of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 8 $y^{3}+6 y^{2}-y-30$$(y-2)$$(y+3)$$p(y)=y^{3}+6 y^{2}-y-30$$(y-2)$$(y+3)$$p(y)$$2$$-3$$p(y)$\[ \begin{array}{r|rrrr} 2 & 1 & 6 & -1 & -30 \\ & \downarrow & 2 & 16 & 30 \\ \hline -3 & 1 & 8 & 15 & 0 \\ & \downarrow & -3 & -15 & …

Question 8, Review Exercise

anonymous@undisclosed.example.com (Anonymous) — Sun, 13 Oct 2024 17:52:43 +0000

Unit 05: Factorization: Online View

anonymous@undisclosed.example.com (Anonymous) — Wed, 30 Mar 2022 17:00:19 +0000

Unit 05: Factorization: Online View On this page, online view of the notes of unit 05 are given. After studying this unit , the students will be able to: * Recall factorization of expressions of the following types. * $ka + kb + kc$ * $ac + ad + bc + bd$ * $a^2 + 2ab + b^2$ * $a^2 – b^2$ * $a^2 + 2ab + b^2 – c^2$ * Factorize the expressions of the following types.$a^4 + a^2b^2 + b^4$$a^4 + 4b^4$$x^2 + px + q$$ax^2 + bx + c$$(ax^2 + bx + c) (ax2 + bx + d) + k$$(x + a) (x + b) (x +…

Preparation Guide

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 17:00:44 +0000

Preparation Guide This guide is made by Mr. Anwar Khan, PhD. We are very thankful to him for sharing. This guide is helpful to prepare papers for MSc Mathematics (annual system) from University of Sargodha. Part 1 1. REAL ANAYSIS * Real Analysis (Notes by Syed Gul Shah) * Chapter # 08 sequences and series of Mathematical Method by SM Yousaf (solutions are available $z= f(x,y)$