MathCity.org

Quotes for the March

anonymous@undisclosed.example.com (Anonymous) — Fri, 28 Apr 2023 18:08:23 +0000

Quotes for the March “نئی ریاضی” کی اہمیت بنیادی طور پر اس حقیقت میں پنہاں ہے کہ اس نے ہمیں ڈسک اور دائرے کے درمیان فرق سکھایا ہے۔ [اسحاق ٹودھنٹر (1820-1884)]

Quotes for the May

anonymous@undisclosed.example.com (Anonymous) — Fri, 28 Apr 2023 18:45:11 +0000

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۔۔۔

Quotes for the April

anonymous@undisclosed.example.com (Anonymous) — Fri, 28 Apr 2023 18:32:46 +0000

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)

Quotes for the February

anonymous@undisclosed.example.com (Anonymous) — Wed, 22 Feb 2023 16:02:58 +0000

Quotes for the February An expert is someone who knows some of the worst mistakes that can be made in his subject, and how to avoid them [Werner Heisenberg (1901-1976)] ایک ماہر وہ ہوتا ہے جو اپنے مضمون میں ہونے والی کچھ بدترین غلطیوں اور ان سے بچنے کا طریقہ جانتا ہو [ورنر ہیزنبرگ (1901-1976)]

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…

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$

Chapter 04: Quadratic Equations

anonymous@undisclosed.example.com (Anonymous) — Sun, 04 Jun 2023 16:13:15 +0000

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

Unit 01: Functions and Limits

anonymous@undisclosed.example.com (Anonymous) — Sat, 03 Jun 2023 16:30:56 +0000

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{…

Unit 03: Integration

anonymous@undisclosed.example.com (Anonymous) — Sat, 03 Jun 2023 17:25:33 +0000

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)$

Chapter 03: General Theorem, Intermediate Forms

anonymous@undisclosed.example.com (Anonymous) — Fri, 21 Apr 2023 13:46:07 +0000

Chapter 03: General Theorem, Intermediate Forms [BSc Calculus 3rd Chapter] What is in the this chapter? * Rolle's theorem * Geometrical interpretation of Rolle's theorem * The mean value theorems * Another form of mean value theorem * Increasing and decreasing functions$\frac{0}{0}$$\frac{\infty}{\infty}$$0\times \infty$$\infty \times 0$$\infty-\infty$$0^\infty, 1^\infty, \infty^0$

Chapter 11: The Laplace Transform

anonymous@undisclosed.example.com (Anonymous) — Sun, 29 May 2022 17:43:26 +0000

Chapter 11: The Laplace Transform Notes of the book Mathematical Method written by S.M. Yusuf, A. Majeed and M. Amin. This book is published by Ilmi Kitab Khana, Lahore - PAKISTAN. Solutions of Chapter 11: The Laplace Transform are given here in pdf form. $f$$[0,\infty)$$f$$\mathcal{L}(f)$$F$$ provided the above improper integral converges. We have $

Chapter 01: Number System

anonymous@undisclosed.example.com (Anonymous) — Sat, 03 Jun 2023 16:30:31 +0000

Chapter 01: Number System [Chapter 01: Number System] Notes (Solutions) of Chapter 01: Number System, Textbook of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary * Rational numbers and irrational numbers$\mathbb{C}$$(x+iy)^n$$\left(\frac{x_1+iy_1}{x_2+iy_2}\right)^n, x_2+iy_2\neq 0$$\sqrt{-1}=i$$\sqrt{-1}$$i$$-i$$i$$-i$$-1$$i^2=-1$$\sqrt{-1}=i$$\sqrt{-1}$

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$

Chapter 06: Sequences and Series

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:46:29 +0000

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\\ &=…

Chapter 09: Fundamentals of Trigonometry

anonymous@undisclosed.example.com (Anonymous) — Sun, 04 Jun 2023 16:03:04 +0000

Chapter 09: Fundamentals of Trigonometry [Chapter 09: Fundamentals of Trigonometry] Notes (Solutions) of Chapter 09: Fundamentals of Trigonometry, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. This chapter has four exercise and solutions of those exercises are given below which can be downloaded in PDF format or can be viewed online.$D^\circ M'S''$$45^\circ , 30^\circ , 60^\circ$$0^\circ , 90^\circ , 180^\circ , 270^\circ , 36…

Unit 02: Differentiation

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:47:05 +0000

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…

Home

anonymous@undisclosed.example.com (Anonymous) — Tue, 24 Mar 2026 18:10:06 +0000

Home Welcome to MathCity.org. Please browse the website by using navigation bar or search the website. More Quotes :---: MathCity.org Updates * | FSc (PECTAA) | Solutions of the many exercises of Mathematics 11 (by PECTAA) has been add. read more for download NEW * | Notes | Mechanics III (Analytic Dynamics II) by Dr Babar Ahmad has been added in the notes section.

MTH321: Real Analysis I (Fall 2021)

anonymous@undisclosed.example.com (Anonymous) — Fri, 28 Oct 2022 11:10:02 +0000

MTH321: Real Analysis I (Fall 2021) Discussion is available at the end of this page. One is free to ask any question or comment. [Photo-illustration of Zeno's Paradox] At the end of this course the students will be able to understand the basic set theoretic statements and emphasize the proofs’ development of various statements by induction. Define the limit of, a function at a value, a sequence and the Cauchy criterion. Prove various theorems about limits of sequences and functions and emphas…

MTH321: Real Analysis I (Spring 2020)

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

MTH321: Real Analysis I (Spring 2020) Discussion is available at the end of this page. One is free to ask any question or comment. ~~DISCUSSION~~ [Photo-illustration of Zeno's Paradox] At the end of this course the students will be able to understand the basic set theoretic statements and emphasize the proofs’ development of various statements by induction. Define the limit of, a function at a value, a sequence and the Cauchy criterion. Prove various theorems about limits of sequences and fun…

Chapter 01: Real Numbers, Limits and Continuity

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:45:28 +0000

Chapter 01: Real Numbers, Limits and Continuity [Chapter 01 of Calculus with Analytic Geometry] 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. The notes of this chapter is written by Prof. $\mathbb{R}$$\mathbb{R}$$\mathbb{R}$

Chapter 02: The Derivative

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:45:28 +0000

Chapter 02: The Derivative [Chapter 02: The Derivative BSc Calculus] 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. Here are few online resource, which are very helpful to find derivative.

Chapter 07: Plane Curves II

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:45:32 +0000

Chapter 07: Plane Curves II Notes of the book Calculus with Analytic Geometry written by Dr. S. M. Yusuf and Prof. Muhammad Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN. [Asymptote] Contents and summary * Asymptotes: A straight line $l$ is called an asymptote for a curve $C$$l$$C$$l$$l$

Chapter 08: Analytic Geometry of Three Dimensions

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:45:33 +0000

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$\mathbb{R}^3$

Chapter 01: Complex Numbers

anonymous@undisclosed.example.com (Anonymous) — Mon, 11 Dec 2023 12:59:57 +0000

Chapter 01: Complex Numbers [Chapter 01 Complex Numbers Methods] Notes of the book Mathematical Method written by S.M. Yusuf, A. Majeed and M. Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN. A complex number is an element $(x,y)$ of the set $$ \mathbb{R}^2=\{(x,y): x,y \in \mathbb{R}\} $$ obeying the following rules of addition and multiplication.$z_1=(x_1,y_1)$$z_2=(x_2,y_2)$$z_1+z_2= (x_1+x_2, y_1+y_2)$$z_1 z_2 = (x_1 x_2 - y_1 y_2, x_1 y_2+y_1 x_2)$$\mathbb{R}^2$$\mathbb{C}$

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 ) *

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

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$p\leftrightarrow q$

Chapter 05: Partial Fractions

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:46:29 +0000

Chapter 05: Partial Fractions [Chapter 05: Partial Fractions] Notes (Solutions) of Chapter 05: Partial Fractions, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary * Introduction * Rational Fraction$\frac {P(x)}{Q(x)}$

Chapter 07: Permutation, Combination and Probability

anonymous@undisclosed.example.com (Anonymous) — Sat, 26 Mar 2022 16:28:40 +0000

Chapter 07: Permutation, Combination and Probability [Chapter 07: Permutation , Combination and Probability] Notes (Solutions) of Chapter 07: Permutation , Combination and Probability, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary

Chapter 08: Mathematical Induction and Binomial Theorem

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:46:30 +0000

Chapter 08: Mathematical Induction and Binomial Theorem [Chapter 08 Mathematical Induction and Binomial Theorem] Notes (Solutions) of Chapter 08: Mathematical Induction and Binomial Theorem, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore.$(a+x)^n$$(a+x)^n$

Chapter 10: Trigonometric Identities

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:46:32 +0000

Chapter 10: Trigonometric Identities [Chapter 10: Trigonometric Identities] Notes (Solutions) of Chapter 10: Trigonometric Identities, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. There are four exercise in this chapter.

Chapter 11: Trigonometric Functions and their Graphs

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:46:34 +0000

Chapter 11: Trigonometric Functions and their Graphs [Chapter 11: Trigonometric Functions and their Graphs] Notes (Solutions) of Chapter 11: Trigonometric Functions and their Graphs, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary $ y = \sin x$$-2\pi \hbox{ to } 2\pi$$ y = \cos x$$-2\pi \hbox{ to } 2\pi$$ y = \tan x$$-\pi \hbox{ to } \pi$$ y = \cot x$$-2\pi \hbox{ to } \pi$$ y = \sec x$$-2\pi \hbox{ to } 2\pi…

Chapter 12: Application of Trigonometry

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:46:34 +0000

Chapter 12: Application of Trigonometry [Chapter 12: Application of Trigonometry] Notes (Solutions) of Chapter 12: Application of Trigonometry, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary * Introduction

Chapter 13: Inverse Trigonometric Functions

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:46:37 +0000

Chapter 13: Inverse Trigonometric Functions [Chapter 13: Inverse Trigonometric Functions] Notes (Solutions) of Chapter 13: Inverse Trigonometric Functions, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. Contents & summary * ${\sin ^{ - 1}}A + {\sin ^{ - 1}}B = {\sin ^{ - 1}}\left( {A\sqrt {1 - {B^2}} + B\sqrt {1 - {A^2}} } \right)$${\sin ^{ - 1}}A - {\sin ^{ - 1}}B = {\sin ^{ - 1}}\left( {A\sqrt {1 - {B^2}} - B\sqrt {1 - {…

Chapter 14: Solutions of Trigonometric Equation

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:46:37 +0000

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…

Unit 07: Vectors

anonymous@undisclosed.example.com (Anonymous) — Thu, 25 Mar 2021 17:27:32 +0000

Unit 07: Vectors [Unit 07: Vectors] Notes (Solutions) of Unit 07: Vectors, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. You can view online or download PDF. To view PDF, you must have PDF Reader installed on your system and it can be downloaded from Software section.$u\cdot v$$u\times v$$u\cdot(v\times w)$

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 PU

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

Syllabus for PU Syllabus and scheme of studies for Regular/Private students doing MSc Mathematics from University of the Punjab, Lahore. 2 years M.Sc Mathematics programme consists of two parts namely Part-I and Part II. The regulation, Syllabi and Courses of Reading for the M.Sc. (Mathematics) Part-I and Part-II (Regular Scheme) are given below.

Question 17 and 18, Exercise 4.7

anonymous@undisclosed.example.com (Anonymous) — Fri, 04 Oct 2024 19:06:41 +0000

Question 17 and 18, Exercise 4.7 Solutions of Question 17 and 18 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$2^{2}+5^{2}+8^{2}+\ldots$$2+5+8+\ldots$$a_k=2+(k-1)(3)=2+3k-3=3k-1$$T_k$$k$\begin{align*}T_k&=(3k-1)^2 \\ &=9k^2-6k+1. \end{align*}\begin{align*}\sum_{k=1}^{n} T_{k} &= \sum_{k=1}^{n} (9k^{2} - 6k + 1)\\ & = 9\sum_{k=1}^{n} k^{2} …

Question 21 and 22, Exercise 4.7

anonymous@undisclosed.example.com (Anonymous) — Fri, 04 Oct 2024 19:06:33 +0000

Question 21 and 22, Exercise 4.7 Solutions of Question 21 and 22 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$1 \times 4+2 \times 7+3 \times 10+\cdots$$4+7+10+\ldots$$a_k=4+(k-1)(3)=4+3k-3=3k+1$$1+2+3+...$$k$$k(3k+1)$$T_k$$k$\begin{align*}T_k&=k(3k+1) \\ &=3k^2+k. \end{align*}\begin{align*}\sum_{k=1}^{n} T_{k} &= \sum_{k=1}^{n} (3k^2 +k)\…

Dr Aamir Ali

anonymous@undisclosed.example.com (Anonymous) — Wed, 13 Dec 2023 07:11:53 +0000

Dr Aamir Ali This is a personal web page created for the students to provide academic resources Dr. Aamir Ali Associate Professor COMSATS University Islamabad, Attock Campus, Attock - PAKISTAN. CUI Profile: ResearchGate Profile: Google Scholar:

About Us

anonymous@undisclosed.example.com (Anonymous) — Thu, 03 Apr 2025 19:07:53 +0000

About Us The only purpose of this website is to help students to learn mathematics. This site contains material for the students of F.Sc., B.Sc., M.Sc., M.Phil. and Ph.D, in the subject of mathematics. The website division has been made according to the different classes for simple navigation. For example, student or teacher searching for notes of F.Sc may see

Dr. Muhammad Ahsan Binyamin

anonymous@undisclosed.example.com (Anonymous) — Sun, 19 Feb 2023 08:05:44 +0000

Dr. Muhammad Ahsan Binyamin This is a personal web page of Dr. Muhammad Ahsan Binyamin Associate Professor Government College University Faisalabad, Faisalabad - PAKISTAN. Google Scholar: ResearchGate Profile: Compute Codimension and Milnor number

Mathematics Conferences

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:39:21 +0000

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).

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

FSc Part 1 (Mathematics): KPK

anonymous@undisclosed.example.com (Anonymous) — Tue, 12 Dec 2023 17:09:07 +0000

FSc Part 1 (Mathematics): KPK [A Textbook of Mathematics for Class XI] A Textbook of Mathematics for Class XI is published by Khybar Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. The book has total of twelve (12) chapters. This book is written by Prof. Dr. Gulzar Ali Khan, Prof. Dr. Islam Noor and Prof. Dr. Muhammad Shah.

FSc/ICS Part 1 (Mathematics): PTB

anonymous@undisclosed.example.com (Anonymous) — Sat, 19 Jul 2025 17:19:20 +0000

FSc/ICS Part 1 (Mathematics): PTB This is an old book. Notes of new book are available at following URL: [Textbook of Algebra and Trigonometry Class XI] Textbook of Algebra and Trigonometry Class XI is published by Punjab Textbook Board (PTB) Lahore, Pakistan. The book has total of 14 chapters.

Dr. Junaid Alam Khan

anonymous@undisclosed.example.com (Anonymous) — Thu, 23 Feb 2023 17:55:59 +0000

Dr. Junaid Alam Khan This is a personal web page of Dr. Junaid Alam Khan Associate Professor Institute of Business Administration, Karachi - PAKISTAN. IBA Profile: ResearchGate Profile: Facebook: [Dr Junaid Alam Khan] Procedure to Compute Sasbi-Standard Bases

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).

FSc/ICS Part 1 (Mathematics): KPK

anonymous@undisclosed.example.com (Anonymous) — Tue, 06 Feb 2024 13:20:19 +0000

FSc/ICS Part 1 (Mathematics): KPK [A Textbook of Mathematics for Class XI] A Textbook of Mathematics for Class XI is published by Khybar Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. The book has total of twelve (12) chapters. This book is written by Prof. Dr. Gulzar Ali Khan, Prof. Dr. Islam Noor and Prof. Dr. Muhammad Shah.

Mathematics 11 for FSc ICS (NBF)

anonymous@undisclosed.example.com (Anonymous) — Sat, 14 Feb 2026 14:30:09 +0000

Mathematics 11 for FSc ICS (NBF) [A Textbook of Mathematics for Class XI] Model Textbook of Mathematics for Class XI is published by National Book Foundation (NBF), Islamabad, Pakistan. NBF can be considered as Federal Textbook Board Islamabad. The book has total of nine (9) chapters. This book is written by Dr. Khalid Mahmood, Dr. Saleem Ullah Satti, M Dabeer Mughal, Dr. Naveed Akmal and Dr. Shahzad Ahmad.

Mathematics 11 (PECTAA)

anonymous@undisclosed.example.com (Anonymous) — Sun, 20 Jul 2025 06:50:20 +0000

Mathematics 11 (PECTAA) [Mathematics 11 for FSc and ICS (PECTAA)] Mathematics 11 is is based on Updated/Revised National Curriculum of Pakistan 2023 and has been approved by the Punjab Education, Curriculum, Training and Assessment Authority (PECTAA) previously known as Punjab Textbook Board (PTB). This is a textbook for all the boards of Punjab for 11th class (FSc or ICS Part 1 or HSSC-I). The book has total of fourteen (14) units.

Mathematics 12 for FSc ICS (NBF)

anonymous@undisclosed.example.com (Anonymous) — Mon, 16 Feb 2026 07:08:57 +0000

Mathematics 12 for FSc ICS (NBF) [Textbook of Mathematics for Class 12] Textbook of Mathematics for Class XII is published by National Book Foundation (NBF), Islamabad, Pakistan. NBF can be considered as Federal Textbook Board Islamabad. The book has total of ten (10) chapters. Dr, Khalid Mahmood is the Managing Author. Saleem Ullah Satti, Muhammad Dabeer Mughal, Dr. Naveed Akmal and Muhammad Anwar ul Haq are the co-authors of the book.

Matric

anonymous@undisclosed.example.com (Anonymous) — Sat, 08 Feb 2025 07:45:55 +0000

Matric This is a new section. We need to do lot of work to complete this section. We will add notes of mathematics for 9th and 10th classes (general and science). math-9th-pctb Mathematics 9th for Matric (PCTB) Mathematics (Science Group) * Notes for 9th Class (Science) * Notes for 10th Class (Science) General Mathematics * Notes for 9th Class (General)

Papers

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:39:38 +0000

Papers If you have any paper, which you think are also useful for others. you can send us (soft copy by email or hard copy by post) to publish on this page. Read more about here Old Papers of M.Phil. University of Sargodha Previous papers of MPhil (Mathematics) University of Sargodha, Sargodha.

Participate

anonymous@undisclosed.example.com (Anonymous) — Sun, 31 May 2026 09:56:03 +0000

Participate If you have written notes, then why not share on MathCity.org with others. This is your help to mathematics students and teachers (یہ صدقہ جاریہ ہے). For further details, please contact webmaster for more information. * The material given on MathCity.org is free and publish to help the students to learn Mathematics. It will take a lot of time to manage this website and prepare new material to publish on this website.

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.

Quote of the Day

anonymous@undisclosed.example.com (Anonymous) — Thu, 25 Jul 2024 06:21:51 +0000

Quote of the Day [Quote of the Day] The “quote of the day” is a useful tool for inspiration, motivation, and self-examination It offers a daily serving of knowledge from reputable and accomplished people in a range of sectors, including literature, politics, science, and entertainment. Everyday reading and reflection on a thought-provoking quote can help people learn more, see things from other people's perspectives, and advance their personal development. Sharing the

Dr. Shafiq Ur Rehman

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:39:43 +0000

Dr. Shafiq Ur Rehman This is personal webpage of Dr. Shafiq Ur Rehman, Assistant Professor, COMSATS Institute of information Technology, Attock, Pakistan. Email: shafiq@ciit-attock.edu.pk Spring2015 Fall2014 Spring2014 Fall2013 Spring2013

MTH322: Real Analysis II (Fall 2016)

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:09 +0000

~~DISCUSSION:off~~ MTH322: Real Analysis II (Fall 2016) Do you have questions or comments? Please use Discussion at the end of this page. This course is offered to MSc, Semester III at Department of Mathematics, COMSATS Institute of Information Technology, Attock campus. The is course need rigorous knowledge of continuity, differentiation, integration, sequences and series of numbers, that is many notion included in

MTH321: Real Analysis I (Fall 2018)

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:10 +0000

MTH321: Real Analysis I (Fall 2018) At the end of this course the students will be able to understand the basic set theoretic statements and emphasize the proofs’ development of various statements by induction. Define the limit of, a function at a value, a sequence and the Cauchy criterion. Prove various theorems about limits of sequences and functions and emphasize the proofs’ development. Define continuity of a function and uniform continuity of a function, prove various theorems about cont…

MTH321: Real Analysis I (Fall 2019)

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:11 +0000

MTH321: Real Analysis I (Fall 2019) [Photo-illustration of Zeno's Paradox by Juliana Jiménez Jaramillo. Photo by Twildlife/Thinkstock] At the end of this course the students will be able to understand the basic set theoretic statements and emphasize the proofs’ development of various statements by induction. Define the limit of, a function at a value, a sequence and the Cauchy criterion. Prove various theorems about limits of sequences and functions and emphasize the proofs’ development. Def…

MTH321: Real Analysis I (Fall 2022)

anonymous@undisclosed.example.com (Anonymous) — Mon, 15 May 2023 07:16:43 +0000

MTH321: Real Analysis I (Fall 2022) ~~DISCUSSION~~ [Photo-illustration of Zeno's Paradox] At the end of this course the students will be able to understand the basic set theoretic statements and emphasize the proofs’ development of various statements by induction. Define the limit of, a function at a value, a sequence and the Cauchy criterion. Prove various theorems about limits of sequences and functions and emphasize the proofs’ development. Define continuity of a function and uniform conti…

MTH322: Real Analysis II (Spring 2017)

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:35 +0000

~~DISCUSSION:closed~~ MTH322: Real Analysis II (Spring 2017) Do you have questions or comments? Please use Discussion at the end of this page. This course is offered to MSc, Semester III at Department of Mathematics, COMSATS Institute of Information Technology, Attock campus. The is course need rigorous knowledge of continuity, differentiation, integration, sequences and series of numbers, that is many notion included in

MTH321: Real Analysis I (Spring 2023)

anonymous@undisclosed.example.com (Anonymous) — Wed, 14 Jun 2023 14:47:57 +0000

MTH321: Real Analysis I (Spring 2023) ~~DISCUSSION~~ [Photo-illustration of Zeno's Paradox] At the end of this course the students will be able to understand the basic set theoretic statements and emphasize the proofs’ development of various statements by induction. Define the limit of, a function at a value, a sequence and the Cauchy criterion. Prove various theorems about limits of sequences and functions and emphasize the proofs’ development. Define continuity of a function and uniform con…

Notes of Metric Spaces

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:51 +0000

Notes of Metric Spaces These notes are related to Section IV of B Course of Mathematics, paper B. We are very thankful to Mr. Tahir Aziz for sending these notes. Name Notes of Metric Space Author Prof. Shahzad Ahmad Khan Send by Tahir Aziz

Notes of Vector Analysis

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:54 +0000

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$

Notes of Fourier Series

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:46 +0000

Notes of Fourier Series These notes are provided by Mr. Muhammad Ashfaq. We are really very thankful to him for providing these notes and appreciates his effort to publish these notes on MathCity.org Name Notes of Fluid Mechanics Author Qayyum Ullah Khan

Notes of Metric Spaces by Umer Asghar

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

Notes of Metric Spaces by Umer Asghar These notes are related to Section IV of B Course of Mathematics, paper B. We are very thankful to Mr. Umer Asghar for sending these notes. Name Notes of Metric Space Author Mr. Umer Asghar Pages 19 pages

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

Number Theory by Prof. Asghar Ali

anonymous@undisclosed.example.com (Anonymous) — Mon, 21 Mar 2022 19:38:31 +0000

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 $a\neq 0$

Number Theory by Prof. M. Tanveer

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:54 +0000

Number Theory by Prof. M. Tanveer [Number Theory by Prof. M. Tanveer] These notes are very helpful to prepare one of the sections of mathematics for BSc. Also these notes can be used for other classes. Author: Prof. M. Tanveer Type: Handwritten Format: PDF (972 kB) Pages: $a,b, \in \mathbb{Z}$$a \neq 0$$a$$b$$c\in \mathbb{Z}$$b=ac$$a,b, \in \mathbb{Z}$$c \in \mathbb{Z}$$a$$b$$c | a$$c | b$

Vector Analysis by Hameed Ullah: Notes

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:41:13 +0000

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

International Conference on Computing and Mathematical Sciences, IBA Sukkur (February 25-26, 2017)

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:41:49 +0000

International Conference on Computing and Mathematical Sciences, IBA Sukkur (February 25-26, 2017) [IBA Shukar] The selected papers will be published in SIBA Journal of Computing and Mathematical Sciences (JCMS). * Name of conference: International Conference on Computing and Mathematical Sciences

Pakistan Mathematics Competitions (PMC) 2022 (1-3 April 2022)

anonymous@undisclosed.example.com (Anonymous) — Mon, 21 Mar 2022 13:15:06 +0000

Pakistan Mathematics Competitions (PMC) 2022 (1-3 April 2022) [Pakistan Mathematics Competitions (PMC) 2022] Mathematics Society – SIBAU is an active society, which has organized series of successfully events in the past. These events were Inter University Mathematics Olympiad 2014, 2015, National Mathematical Olympiad 2016, Pakistan National Mathematical Olympiad 2017, Calculus Contest 2017, Calculus Contest 2018, National Calculus Contest 2019, Pakistan National Mathematical Olympiad 2019 and…

Definitions: FSc Part1 KPK

anonymous@undisclosed.example.com (Anonymous) — Mon, 28 Aug 2023 16:59:59 +0000

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$

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} $\(…

Definitions: Mathematics 11: PTB by Muzzammil Subhan

anonymous@undisclosed.example.com (Anonymous) — Sun, 22 Sep 2024 17:24:14 +0000

Definitions: Mathematics 11: PTB by Muzzammil Subhan Definitions from Textbook of Algebra and Trigonometry Class XI, 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$y=2^x$$y=e^x$$f(x)=\log_a x$$f(x)=\log_e x$$y$$x$$y$$y=x^2+3x$$x^2+xy+y^2=4$$f(-x)=f(x)$$f(-x)=-f(x)$

Solution and Area of Oblique Triangle

anonymous@undisclosed.example.com (Anonymous) — Mon, 25 Sep 2023 15:48:59 +0000

Solution and Area of Oblique Triangle These are the common formulas used in Chapter 12 of Textbook of Algebra and Trigonometry Class XI, Punjab Textbook Board Lahore. This handout is very helpful to remember the formulas. All these formulas are given for real valued and defined trigonometric functions. A PDF file can be downloaded for high quality printing and a word file is also given if you wish to modify the contents or credit as you need.$a^2=b^2+c^2-2bc\cos \alpha$$b^2=c^2+a^2-2ca\cos \bet…

Trigonometric Formulas

anonymous@undisclosed.example.com (Anonymous) — Thu, 25 May 2023 02:59:50 +0000

Trigonometric Formulas These are the common formulas used in Chapter 9 to 14 of Textbook of Algebra and Trigonometry Class XI, Punjab Textbook Board Lahore. This handout is very helpful to remember the formulas. All these formulas are given for real valued and defined trigonometric functions. A PDF file can be downloaded for high quality printing and a word file is also given if you wish to modify the contents or credit as you need.${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$$1+{{\tan }^{2}}\t…

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)}$…

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 1 Mathematics Notes/Solutions

anonymous@undisclosed.example.com (Anonymous) — Sun, 20 Jul 2025 09:10:04 +0000

FSc Part 1 Mathematics Notes/Solutions This is an old book. Notes of new book are available at following link: [FSc Part1 PTB Book Cover] Notes (Solutions) of Textbook of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board (PTB) Lahore.

MCQS of Mathematics 9 (PCTB)

anonymous@undisclosed.example.com (Anonymous) — Tue, 18 Nov 2025 17:29:59 +0000

MCQS of Mathematics 9 (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).

Definitions: FSc Part1 KPK

anonymous@undisclosed.example.com (Anonymous) — Tue, 05 Dec 2023 02:44:42 +0000

Definitions: Mathematics 11 NBF

anonymous@undisclosed.example.com (Anonymous) — Wed, 10 Jul 2024 18:51:53 +0000

Definitions: Mathematics 11 NBF Model Textbook of Mathematics for Class XI is published by National Book Foundation (NBF), Islamabad, Pakistan. NBF can be considered as Federal Textbook Board Islamabad. The book has total of nine (9) chapters. Definition of the book provide the quick overview of the book.$x+iy$$x,y\in\mathbb{R}$$i^2=1$$\mathbb{C}$$x+i y$$x$$y$$x$$y$$Re(z)=x$$Im(z)=y$$z=x+i y$$\bar{z}$$\bar{z}=x-i y$$z=x+i y$$|z|$$|z|=\sqrt{x^{2}+y^{2}}$$z=x+iy$$Re(z)=x$$Im(z)=y$$Re(z^{-1})= \d…

Advance Functional Analysis by Waseem Akram

anonymous@undisclosed.example.com (Anonymous) — Sun, 31 May 2026 07:51:28 +0000

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…

Advance Functional Analysis by Waseem Akram

anonymous@undisclosed.example.com (Anonymous) — Mon, 25 May 2026 18:22:55 +0000

Advanced Analysis: Handwritten Notes

anonymous@undisclosed.example.com (Anonymous) — Fri, 14 Apr 2023 17:55:33 +0000

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.

Advance Analysis by Ms. Iqra Liaqat

anonymous@undisclosed.example.com (Anonymous) — Fri, 21 Mar 2025 15:08:17 +0000

Advance Analysis by Ms. Iqra Liaqat [Advance Analysis 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 Advance Analysis Sender Ms. Iqra Liaqat Author $\sigma$

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 $R^n$

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

Algebraic Number Theory Notes by Anwar Khan

anonymous@undisclosed.example.com (Anonymous) — Sat, 05 Aug 2023 19:06:20 +0000

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.$\mathbb{R}$

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.

Complex Analysis (Easy Notes of Complex Analysis)

anonymous@undisclosed.example.com (Anonymous) — Sun, 06 Aug 2023 11:17:51 +0000

Complex Analysis (Easy Notes of Complex Analysis) [Complex Analysis (Easy Notes of Complex Analysis) ] The study of complex numbers, together with their manipulation, derivatives, and other characteristics, is known as complex integration. Contour integration is a technique for analysing specific integrals along pathways in the complex plane in the field of complex analysis mathematics. The complex analytic technique known as the calculus of residues and contour integration are closely linked.

Complex Analysis (Notes) by Ms. Iqra Liaqat

anonymous@undisclosed.example.com (Anonymous) — Sun, 06 Aug 2023 09:33:02 +0000

Complex Analysis (Notes) by Ms. Iqra Liaqat [Notes of Complex Analysis 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 Complex analysis is the study of functions of complex numbers. It is useful in a variety of mathematical fields, such as algebraic geometry, number theory, analytic combinatorics, and applied mathematics, as well as in physics fields like…

Complex Analysis by M Usman Hamid

anonymous@undisclosed.example.com (Anonymous) — Sat, 08 Feb 2025 08:11:42 +0000

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$

Complex Analysis (Quick Review)

anonymous@undisclosed.example.com (Anonymous) — Wed, 26 Jun 2024 18:33:42 +0000

Complex Analysis (Quick Review) [Complex Analysis: Quick Review] 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. Important definitions and important results are the part of these notes, these might be helpful to prepare interviews or any other written test after graduation like PPSC, FPSC or etc.$z_1, z_2 \in S$$S$$v(x,y)$$u(x,y)$$f(z)=u(x,y)+iv(x,y)$$f$$D$$C$$D$…

Computing Tools for Mathematics by Asif Arshad

anonymous@undisclosed.example.com (Anonymous) — Sat, 05 Aug 2023 20:30:34 +0000

Computing Tools for Mathematics by Asif Arshad [omputing Tools for Mathematics by Asif Arshad] Computing tools for mathematics are algorithms that use computers to solve mathematical issues. They are employed in a number of scientific, technical, industrial, and technological domains where computer is crucial and central. They may assist in creating precise and effective numerical techniques, for instance, to solve physical or biological models.

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 (Notes) by Ms. Kaushef Salamat

anonymous@undisclosed.example.com (Anonymous) — Sat, 23 Aug 2025 11:29:10 +0000

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

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.…

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…

Fluid Dynamics I by Muhammad Usman Hamid

anonymous@undisclosed.example.com (Anonymous) — Sun, 03 May 2026 08:26:32 +0000

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.

Fluid Mechanics by Ali Raza

anonymous@undisclosed.example.com (Anonymous) — Sat, 27 Jul 2024 16:23:08 +0000

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.

Fluid Mechanics I by Dr Rao Muzamal Hussain

anonymous@undisclosed.example.com (Anonymous) — Mon, 24 Jul 2023 12:45:33 +0000

Fluid Mechanics I 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.

Fluid Mechanics II by Dr Rao Muzamal Hussain

anonymous@undisclosed.example.com (Anonymous) — Tue, 29 Aug 2023 06:36:09 +0000

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.

Fluid Mechanics by Muhammad Usman Hamid

anonymous@undisclosed.example.com (Anonymous) — Sun, 16 Mar 2025 12:46:08 +0000

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.

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$

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$

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.

Fundamental of Complex Analysis (Solutions of Some Exercises)

anonymous@undisclosed.example.com (Anonymous) — Sat, 15 Apr 2023 18:26:12 +0000

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, …

Fuzzy Sets Theory by Umar Saeed Bunarai

anonymous@undisclosed.example.com (Anonymous) — Tue, 08 Aug 2023 11:56:34 +0000

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.

General Topology by Raheel Ahmad

anonymous@undisclosed.example.com (Anonymous) — Tue, 08 Aug 2023 11:51:09 +0000

General Topology by Raheel Ahmad A handwritten notes of Topology by Mr. Raheel Ahmad. We are very thankful to him for sending these notes. Name General Topology Author Mr. Raheel Ahmad Pages 87 pages Format Mobile Scanned PDF Size 8.0 MB What is in the notes?

General Topology by Azhar Hussain

anonymous@undisclosed.example.com (Anonymous) — Sun, 06 Aug 2023 19:18:17 +0000

General Topology by Azhar Hussain [Topology Notes by Azhar Hussain] The area of topology known as general topology (also known as point set topology) is concerned with the fundamental concepts and constructs of set theory utilised in topology. Most other fields of topology, such as differential topology, geometric topology, and algebraic topology, are built upon it. The three key ideas of point-set topology are connectedness, compactness, and continuity. Continuous functions move points from on…

Group Theory: Important Definitions and Results

anonymous@undisclosed.example.com (Anonymous) — Wed, 10 Aug 2022 15:31:46 +0000

Group Theory: Important Definitions and Results [Group Theory: 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 effort 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.$G$$a\in G$$\exists$$e\in G$$a\…

Group Theory by Mr. Muhammad Iftikhar

anonymous@undisclosed.example.com (Anonymous) — Tue, 09 Aug 2022 18:28:27 +0000

Group Theory by Mr. Muhammad Iftikhar [Group Theory by Mr. Muhammad Iftikhar] These notes are send by Mr. Muhammad Iftikhar. We are really very thankful to him for providing these notes and appreciates his effort to publish these notes on MathCity.org. Name Lecture Notes on Group Theory Author Mr. Muhammad Iftikhar

Handwritten Notes of Real Analysis by Asim Marwat

anonymous@undisclosed.example.com (Anonymous) — Sat, 15 Apr 2023 17:26:13 +0000

Handwritten Notes of Real Analysis by Asim Marwat [Handwritten Notes of Real Analysis by Asim Marwat] Real analysis is a branch of mathematics that analyses how real numbers, sequences and series, and real functions behave. It focuses on real numbers and frequently extends the real line by including positive and negative infinity. Real analysis investigates a number of the properties of real-valued sequences and functions, including convergence, limits, continuity, smoothness, differentiabilit…

History of Mathematics by Muhammad Usman Hamid

anonymous@undisclosed.example.com (Anonymous) — Tue, 08 Aug 2023 12:01:51 +0000

History of Mathematics by Muhammad Usman Hamid [History of Mathematics by Muhammad Usman Hamid] Mathematics is a unique aspect of human thought, and its history differs in essence from all other histories. These notes are written by Muhammad Usman Hamid. We are very thankful to him for sharing these notes on our website.

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…

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 Method by Muhammad Usman Hamid

anonymous@undisclosed.example.com (Anonymous) — Mon, 17 Apr 2023 08:43:53 +0000

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…

Mathematical Statistics I by Muzammil Tanveer

anonymous@undisclosed.example.com (Anonymous) — Thu, 20 Apr 2023 16:11:58 +0000

Mathematical Statistics I by Muzammil Tanveer [Mathematical Statistics I] 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 Mathematical Statistics

Mathematical Statistics II by Sir Haidar Ali

anonymous@undisclosed.example.com (Anonymous) — Thu, 20 Apr 2023 12:16:43 +0000

Mathematical Statistics II by Sir Haidar Ali [Mathematical Statistics II] A subfield of mathematics called mathematical statistics is concerned with using mathematical techniques to solve statistical problems. It involves using mathematical analysis and probability theory to the study of statistical issues like estimate, hypothesis testing, and confidence intervals. Financial, engineering, and scientific fields all benefit from the use of mathematical statistics, which is a significant area of…

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 Notes by Anwar Khan

anonymous@undisclosed.example.com (Anonymous) — Mon, 01 May 2023 13:54:40 +0000

Measure Theory Notes by Anwar Khan [Measure Theory Notes by Anwar Khan] Measure theory is a branch of mathematics concerned with the concept of “measure,” which is a method of assigning a numerical value to specific sets. The concepts of length, area, and volume are generalised via measurements to more abstract environments, such as infinite-dimensional spaces and areas that cannot be seen.$X$$\sigma-$$X$$\sigma-$$\sigma-$$\lim\limits_{k\to \infty} \sup A_k$$\lim\limits_{k\to \infty} \inf A_k$…

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$

Measure Theory by M Usman Hamid & Saima Akram

anonymous@undisclosed.example.com (Anonymous) — Wed, 26 Jun 2024 17:55:59 +0000

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).

Mechanics (Easy Notes of Mechanics)

anonymous@undisclosed.example.com (Anonymous) — Tue, 08 Aug 2023 11:46:58 +0000

Mechanics (Easy Notes of Mechanics) [Easy Notes of Mechanics] These notes are provided and prepared by Dr. Amir Mahmood. We are really very thankful to him for providing these notes and appreciates his effort to publish these notes on MathCity.org If there is error or mistake in the notes, then please directly contact to Dr. Amir Mahmood for correction. He can be accessed via his email address at

Mechanics by Sir Nouman Siddique

anonymous@undisclosed.example.com (Anonymous) — Wed, 26 Jun 2024 18:35:58 +0000

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

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.

Mechanics II (Analytic Dynamics I) by Dr Babar Ahmad

anonymous@undisclosed.example.com (Anonymous) — Sat, 24 May 2025 18:13:54 +0000

Mechanics II (Analytic Dynamics I) by Dr Babar Ahmad [Mechanics II (Analytic Dynamics I) 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.

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.

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.

Metric Spaces (Notes)

anonymous@undisclosed.example.com (Anonymous) — Fri, 18 Aug 2023 18:05:58 +0000

Metric Spaces (Notes) [Metric Spaces (Notes)] These are updated version of previous notes. Many mistakes and errors have been removed. These notes are collected, composed and corrected by Atiq ur Rehman, PhD. These are actually based on the lectures delivered by Prof. Muhammad Ashfaq (Ex HoD, Department of Mathematics, Government College Sargodha). $(X,d)$$x,y\in X$$$\left| {\,d(x,\,A)\, - \,d(y,\,A)\,} \right|\,\, \le \,\,d(x,\,y).$$$A^c$$A\subset X$$x \in X$$B(x;r)$$A \subset X$$f:(X,d)\to (Y…

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$

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…

Multivariable Calculus by Sheikh Muhammad Saleem Shahzad

anonymous@undisclosed.example.com (Anonymous) — Sun, 17 Dec 2023 17:13:50 +0000

Multivariable Calculus by Sheikh Muhammad Saleem Shahzad [Multivariable Calculus by Sheikh Muhammad Saleem Shahzad] * Have you ever wondered how we can understand the speed of a moving object at any instant of time? * Did you know that Calculus can help us predict future trends by analyzing patterns in data?

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)$

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…

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 by M Usman Hamid

anonymous@undisclosed.example.com (Anonymous) — Sat, 05 Aug 2023 17:01:51 +0000

Numerical Analysis 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

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 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.

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…

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

Partial Differential Equations (PDE) by Muzammil Tanveer

anonymous@undisclosed.example.com (Anonymous) — Sun, 09 Mar 2025 09:29:35 +0000

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

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…

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…

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. $\infty$

Real Analysis Notes by Prof Syed Gul Shah

anonymous@undisclosed.example.com (Anonymous) — Mon, 17 Apr 2023 04:04:23 +0000

Real Analysis Notes by Prof Syed Gul Shah [Real Analysis Notes by Prof Syed Gul Shah] Real analysis, a discipline that explores the complexities of mathematical functions, limits, and sequences, can often be a difficult topic for students. Prof. Syed Gul Shah, as a true analyst, not only excelled in the subject but also gained fame for his extraordinary qualities as a human being.$s_n\frac{1}{2}s$$\{s_n\}$…

Ring (Notes) by Prof. M. Dabeer Mughal

anonymous@undisclosed.example.com (Anonymous) — Sat, 01 Apr 2023 06:56:21 +0000

Ring (Notes) by Prof. M. Dabeer Mughal [Ring (Notes) by Prof. M. Dabeer Mughal] A handwritten notes on Ring (Algebra) by Prof. M. Dabeer Mughal (Federal Directorate of Education, Islamabad, Pakistan). It is best to prepare a “Rings and Vector Spaces” section of your algebra paper or Algebra II for BS or MS Mathematics.$\phi$

Rings (Handwritten notes) by Atiq ur Rehman

anonymous@undisclosed.example.com (Anonymous) — Wed, 26 Jun 2024 18:30:03 +0000

Rings (Handwritten notes) by Atiq ur Rehman [Rings (Handwritten notes) by Atiq ur Rehman] Ring is a two-operation mathematical structure. It is an abelian group with one operation and a semi-group with another operation, and the distributive law is true for the first operation relative to the second operation. This mathematical idea is fundamentally pure. These notes offer a very simple method for learning the concept of rings and other ideas that are closely related to rings. These are lecture…

Rings & Modules by Ms. Iqra Liaqat

anonymous@undisclosed.example.com (Anonymous) — Sat, 01 Apr 2023 14:30:33 +0000

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.

Special Functions by Dr. Muhey-U-Din

anonymous@undisclosed.example.com (Anonymous) — Sun, 23 Jul 2023 16:48:30 +0000

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.

Special Theory of Relativity by M Usman Hamid and M Zeeshan Ahmad

anonymous@undisclosed.example.com (Anonymous) — Tue, 23 Aug 2022 12:32:09 +0000

Special Theory of Relativity by M Usman Hamid and M Zeeshan Ahmad [Special Theory of Relativity by M Usman Hamid and M Zeeshan Ahmad] Special theory of relativity developed by Einstein in 1905, after the failure of Newton‘s laws, when he was dealing with the relative motion in space. The theory of relativity deals with relations which exist between physical quantities (such as mass of a particle, length of a rod, electric field at a point etc.) as they appear to different observers in relative …

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.

Theory of Relativity & Analytic Dynamics: Handwritten Notes

anonymous@undisclosed.example.com (Anonymous) — Fri, 23 Aug 2024 07:42:56 +0000

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…

Topology and Functional Analysis Solved Paper by Noman Khalid

anonymous@undisclosed.example.com (Anonymous) — Sun, 11 Feb 2024 12:38:09 +0000

Topology and Functional Analysis Solved Paper by Noman Khalid These solved papers are written and provided by Mr. noman-khalid. We are very thankful to him for providing these notes to publish on MathCity.org. Topology and Functional Analysis is considered as one the tough subject in BS/MSc Mathematics in different universities. Here solved paper for the said subject is given for the University of the Punjab (PU), Lahore.

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-$

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.

Vector & Tensor Analysis by Dr Nawazish Ali (Solutions)

anonymous@undisclosed.example.com (Anonymous) — Sun, 06 Aug 2023 18:20:39 +0000

Vector & Tensor Analysis by Dr Nawazish Ali (Solutions) [Vector & Tensor Analysis by Dr Nawazish Ali (Solutions)] We are very thankful to Prof. Fazal Abbas Sajid for sharing these solutions. Vector & Tensor Analysis for Scientists and Engineers, by Prof. Dr. Nawazish Ali Shah is a famous book taught in different universities of the Pakistan. On this page, we have added the solutions of the exercises of the book. Solutions of Chapter 5 is written by

Vector & Tensor Analysis by Prof Fazal Abbas

anonymous@undisclosed.example.com (Anonymous) — Wed, 24 Aug 2022 07:12:53 +0000

Vector & Tensor Analysis by Prof Fazal Abbas [Vector & Tensor Analysis by Prof Fazal Abbas] Solution of Chapter 6: Curvilinear Coordinates of the book Vector & Tensor Analysis by Prof. Dr. Nawazish Ali Shah written by Prof. Fazal Abbas Sajid. Here solutions of chapter 6 are provided by the author of the book, for the solutions of all the chapters of the book, please buy the solution manual from the market.

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.

Old PPSC Written tests for Lecturer (Mathematics)

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:44:09 +0000

Old PPSC Written tests for Lecturer (Mathematics) * Previous Written tests (papers) of Punjab Public Service Commission, Lahore for Lecturer in Mathematics. * From now the PPSC will conduct a MCQs test for this post. * We have created a page to give information to students. This page is

Muhammad Idrees

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:44:42 +0000

Muhammad Idrees Muhammad Idrees; M.Sc, MSC, M.Phil Department of Mathematics Govt: Boys Degree College, Nushki-Balochistan. Emails: , We are very thankful to Muhammad Idrees for contributing to the resources of MathCity.org Contribution * Exercise 3.3 |

Muzammil Tanveer

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

Muzammil Tanveer We are very thankful to Mr. Muzammil Tanveer for his contribution to the website. Mr. Muzammil Tanveer is also an expert to compose mathematics notes, books or papers. You may contact him via his email or cell number given below for professional paid composing.

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$$…

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$$\…

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(…

PPSC Mock Interview Lecturer Mathematics

anonymous@undisclosed.example.com (Anonymous) — Thu, 25 Mar 2021 16:31:54 +0000

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…

Chapter 04: Techniques of Integration

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:45:29 +0000

Chapter 04: Techniques of Integration These notes are written by Mr. Aqeel Nawaz. We are very thankful to him for providing these notes. * Anti-derivative * Table of integrals * Integration by substitution * Integration by parts * Column (or tabular) integration

Chapter 04: Techniques of Integration

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:45:30 +0000

Chapter 04: Techniques of Integration These notes are written by Prof. Muhammad Farooq. We are very thankful to him for providing these notes. * Anti-derivative * Table of integrals * Integration by substitution * Integration by parts * Column (or tabular) integration

Chapter 05: The Definite Integral

anonymous@undisclosed.example.com (Anonymous) — Tue, 05 Dec 2023 03:01:56 +0000

Chapter 05: The Definite Integral These notes are written by Mr. Abrar Mustafa. We are very thankful to him for providing these notes. * The definite integral as limit of a sum * Evaluation of limit of a sum * Evaluation of definite integral

Chapter 06: Plane Curves I

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:45:31 +0000

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

Chapter 09: Functions of Several Variables

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:45:33 +0000

Chapter 09: Functions of Several Variables * Homogeneous Functions * Differentials * Change of variables, the chain rule * Implicit functions * Directional derivative * Tangent planes and normal lines * Extrema of functions of two variables

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.

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 05: Determinants

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

Chapter 05: Determinants * Determinant of a square matrix * Axiomatic definition of a determinant * Determinant as sum of products of elements * Determinant of the transpose * An algorithm to evaluate Det A * Determinants and inverse of matrices

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 07: Inner Product Spaces

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:45:48 +0000

Chapter 07: Inner Product Spaces Notes of the book Mathematical Method written by S.M. Yusuf, A. Majeed and M. Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN. Inner product spaces form and important topic of Functional Analysis. These are simply vector space over the field of real or complex numbers and with an inner product defined on them.

Chapter 08: Infinite Series

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:45:49 +0000

Chapter 08: Infinite Series Notes of the book Mathematical Method written by S.M. Yusuf, A. Majeed and M. Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN. Infinite series are of great importance in both pure and applied mathematics. They play a significant role in Physics and engineering. In fact many functions can be represented by infinite series. The theory of infinite series is developed through the use of special kind of function called sequence.

Chapter 09: First Order Differential Equations

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:45:50 +0000

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

Introduction to Mechanics

anonymous@undisclosed.example.com (Anonymous) — Sat, 13 Jan 2024 10:50:44 +0000

Introduction to Mechanics [Introduction to Mechanics by Q.K Ghori] Notes of Introduction to Mechanics by Q. K. Ghori published by West Pak Publishing Company (Pvt) Ltd. There are thirteen chapters in this book. We don't have all the notes of this book but the notes which we have are listed below. Please choose your require chapter to see the notes.

MCQs with Answers (FSc/ICS Part 1)

anonymous@undisclosed.example.com (Anonymous) — Thu, 02 May 2024 17:03:11 +0000

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$

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

Short Questions by Mr. Akhtar Abbas

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:46:47 +0000

Short Questions by Mr. Akhtar Abbas * We are very thankful to Mr. Akhtar Abbas for sharing these short questions. * These short questions are selected from previous 5 years papers of different boards. Solve these at your own to perform well in annual exams.$\sqrt{x^2-4}$$f(x)=\frac{2x}{x-2}$$x=2$$x^{100}$

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 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.

Unit 06: Conic Section

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

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.

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.$

Real Analysis: Short Questions and MCQs

anonymous@undisclosed.example.com (Anonymous) — Mon, 03 Apr 2023 04:06:26 +0000

Real Analysis: Short Questions and MCQs We are going to add short questions and MCQs for Real Analysis. The subject is similar to calculus but little bit more abstract. This is a compulsory subject in MSc and BS Mathematics in most of the universities of Pakistan. The author of this page is Dr. $\left\{\frac{1}{n+1} \right\}$$\left\{\frac{n+2}{n+1} \right\}$$\{x_n\}$$\{y_n\}$$\lim_{n\to\infty z_n}$$z_n=x_n-2y_n$$\{x_n\}$$\{y_n\}$$\lim_{n\to\infty z_n}$$x_n=2y_n-3z_n$$(1,2)$$\left(\frac{1}{2},\fr…

Topology: Short Questions and MCQs

anonymous@undisclosed.example.com (Anonymous) — Mon, 03 Apr 2023 06:51:14 +0000

Topology: Short Questions and MCQs We are going to add short questions and MCQs for Topology. This is a compulsory subject in MSc and BS Mathematics in most of the universities of Pakistan. The author of this page is Dr. Atiq ur Rehman, PhD. This page will be updated periodically. $\mathbb{R}$$X=\{a\}$$X$$X$$X$$\tau$$\mathbb{N}$$\tau$$(\mathbb{Z}, \tau)$$\mathbb{N}$$\tau$$A=\{\pm 100,\pm 101, \pm 102, ... \}$$\tau$$E=\{0,\pm 2,\pm 4,...\}$$\tau$$\tau$$B=\{1,2,3,...,99\}$$\tau$$C=\{10^{10}+n : n …

Mathematical Method by Khalid Latif Mir (Solutions)

anonymous@undisclosed.example.com (Anonymous) — Sat, 23 Dec 2023 16:48:39 +0000

Mathematical Method by Khalid Latif Mir (Solutions) [Mathematical Method by Khalid Latif Mir (Solutions)] We are very thankful to Prof. Fazal Abbas Sajid for sharing these solutions. Problems & Methods Mathematical Method by Khalid Latif Mir is a famous book taught in different universities of the Pakistan at BS and Master level. On this page, we have added the solutions of the exercises of the book. The solutions of Chapter 06: The Laplace Transform and its Applications is written by Marrium …

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

Theoretical Mechanics by Khalid Latif Mir (Solutions)

anonymous@undisclosed.example.com (Anonymous) — Sat, 27 Feb 2021 08:58:47 +0000

Theoretical Mechanics by Khalid Latif Mir (Solutions) [Theoretical Mechanics by Khalid Latif Mir (Solutions)] We are very thankful to Prof. Fazal Abbas Sajid for sharing these solutions. An Intermediate Course in Theoretical Mechanics By Khalid Latif Mir is a famous book taught in different universities of the Pakistan. On this page, we have added the solutions of the exercises of the book.

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…

University of Sargodha, Sargodha (Old Papers)

anonymous@undisclosed.example.com (Anonymous) — Thu, 28 Mar 2024 16:26:23 +0000

University of Sargodha, Sargodha (Old Papers) * To open or print a DjVu file, you must have some DjVu file viewer, e.g. WinDjVu. It can be downloaded from here * From 1st Annual 2013, the paper pattern has been changed. Check the complete syllabus

Chapter 02: Composition of Forces

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:55:05 +0000

Chapter 02: Composition of Forces Notes of Chapter 02: Composition of Forces: Introduction to Mechanics by Q. K. Ghori published by West Pak Publishing Company (Pvt) Ltd. Notes of this chapter is send by Mr. Tahir Aziz. We are very thankful for his such effort.$(\lambda,\mu)$

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

Chapter 05: Friction

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:55:06 +0000

Chapter 05: Friction Notes of Introduction to Mechanics by Q. K. Ghori published by West Pak Publishing Company (Pvt) Ltd. These notes are written by Prof. Muhammad Tanveer. Following are the main topics of this chapter. * Friction * Laws of friction

Chapter 06: Virtual Work

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

Chapter 06: Virtual Work Notes of Chapter 06: Virtual Work: Introduction to Mechanics by Q. K. Ghori published by West Pak Publishing Company (Pvt) Ltd. Notes of this chapter is provide by *Prof. M. Tanveer*. We are very thankful to him for such a kind effort.

Chapter 07: Kinematics

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:55:09 +0000

Chapter 07: Kinematics Notes of Chapter 07: Kinematics: Introduction to Mechanics by Q. K. Ghori published by West Pak Publishing Company (Pvt) Ltd. Notes of this chapter is send by Mr. Tahir Aziz and Mr. Umair Sabi Ullah. We are very thankful to them for such a kind effort.

Chapter 08: Rectilinear Motion

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:55:09 +0000

Chapter 08: Rectilinear Motion Notes of Chapter 08: Rectilinear Motion: Introduction to Mechanics by Q. K. Ghori published by West Pak Publishing Company (Pvt) Ltd. Thanks to Mr. Tahir Aziz and Atiq ur Rehman for sending these notes. * Motion with constant acceleration

Chapter 10: Motion of a Projectile

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:55:09 +0000

Chapter 10: Motion of a Projectile Notes of Chapter 10: Motion of a Projectile: Introduction to Mechanics by Q. K. Ghori published by West Pak Publishing Company (Pvt) Ltd. These notes are written by Prof. Shariq Mehtab Syed * Trajectory of a Projectile

Chapter 12: Orbital Motion

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:55:09 +0000

Chapter 12: Orbital Motion Notes of Chapter 12: Orbital Motion: Introduction to Mechanics by Q. K. Ghori published by West Pak Publishing Company (Pvt) Ltd. * Motion under the Central * Elliptic Orbit under a central Force * Polar form of the Orbit

Paper Pattern for General Mathematics

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:55:17 +0000

Paper Pattern for General Mathematics Paper pattern for General Mathematics chapter-wise for University of the Punjab, Lahore is given on this page. This pattern is extracted from syllabus and old papers, so use at your own risk. This pattern is made by Prof. Muhammad Farooq.

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…

Paper Pattern for A-Course of Mathematics

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:55:23 +0000

Paper Pattern for A-Course of Mathematics Paper pattern for A-Course of Mathematics, chapter-wise for University of the Punjab, Lahore is given on this page. This pattern is extracted from syllabus and old papers, so use at your own risk. * Download PDF ~ BSC_maths_paper_pattern_farooq.pdf bsc punjab_university paper_pattern syllabus

Paper Pattern for B-Course of Mathematics

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:55:29 +0000

Paper Pattern for B-Course of Mathematics Paper pattern for B-Course of Mathematics chapter-wise for University of the Punjab, Lahore is given on this page. This pattern is extracted from syllabus and old papers, so use at your own risk. This pattern is made by Prof. Muhammad Farooq.

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 14 and 15, Exercise 4.2

anonymous@undisclosed.example.com (Anonymous) — Fri, 20 Sep 2024 18:31:40 +0000

Question 14 and 15, Exercise 4.2 Solutions of Question 14 and 15 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $b$$10$$b$$20$$a= b$$b=20$\begin{align*} &\text{A.M.} = \frac{a + b}{2} \\ \implies & 10 = \frac{b + 20}{2} \\ \implies & 20 = b + 20 \\ \implies & b = 20 - 20 \\ \implies & b = 0 \end{align*}$b = 0$$b$$25$$b$$20$$b$$10$$b$$-10$$x$$y$…

Question 20 and 21, Exercise 4.4

anonymous@undisclosed.example.com (Anonymous) — Sun, 22 Sep 2024 18:26:02 +0000

Question 20 and 21, Exercise 4.4 Solutions of Question 20 and 21 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$3 , \_\_\_ , \_\_\_ , \_\_\_ , 48$$$a_1=3$$a_5=48$$r$$$ a_n=ar^{n-1}. $$\begin{align*} &a_5=a_1 r^4 \\ \implies & 48=3r^4 \\ \implies & r^4 = 16 \\ \implies & r^4 = 2^4 \\ \implies & r = 2. \end{align*}\begin{align*} & a_2=a_1 r= (3…

Question 30, Exercise 4.4

anonymous@undisclosed.example.com (Anonymous) — Sun, 22 Sep 2024 18:26:05 +0000

Question 30, Exercise 4.4 Solutions of Question 30 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $=a_1= 1$$=a_2= 3$$=a_3=3\times 3 = 9$$=a_4=3\times 9 = 27$$=a_5=3\times 27 = 81$$81$$a_1=1$$r=3$$a_5=?$$$a_n=a_1 r^{n-1}.$$\begin{align*} a_5&=a_1 r^4 \\ &=(1)(3)^4 = 81 \end{align*}$$S_n=a_1+a_2+a_3+a_4+a_5.$$

Question 19 and 20, Exercise 4.7

anonymous@undisclosed.example.com (Anonymous) — Fri, 04 Oct 2024 19:06:43 +0000

Question 19 and 20, Exercise 4.7 Solutions of Question 19 and 20 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$1^{3}+3^{3}+5^{3}+$$1+3+5+\ldots$$a_k=1+(k-1)(2)=1+2k-2=2k-1$$T_k$$k$\begin{align*}T_k&=(2k-1)^3 \\ &=(2k)^3+3(2k)^2(-1)+3(2k)(-1)^2+(-1)^3 \\ &=8k^3-12k^2+6k+1 \end{align*}\begin{align*}\sum_{k=1}^{n} T_{k} &= \sum_{k=1}^{n} (8k^…

Question 19 and 20, Exercise 4.7

anonymous@undisclosed.example.com (Anonymous) — Fri, 04 Oct 2024 19:06:32 +0000

Question 19 and 20, Exercise 4.7 Solutions of Question 19 and 20 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$1^{3}+3^{3}+5^{3}+$$1+3+5+\ldots$$a_k=1+(k-1)(2)=1+2k-2=2k-1$$T_k$$k$\begin{align*}T_k&=(2k-1) \\ &=9k^2-6k+1. \end{align*}\begin{align*}\sum_{k=1}^{n} T_{k} &= \sum_{k=1}^{n} (2k - 1)\\ & = 2 \sum_{k=1}^{n} k - \sum_{k=1}^{n} 1 \…

Exercise 6.1 (Solutions)

anonymous@undisclosed.example.com (Anonymous) — Sat, 08 Mar 2025 08:59:40 +0000

Exercise 6.1 (Solutions) The solutions of the Exercise 6.1 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to factorial function.$n!$$$ n!=\left\{\begin{array}{l} n(n-1)(n-2)\cdot \ldots \cdot 3 \cdot 2 \cdot 1 \text{ if } n\geq 1,\\ 1 \text{ if } n=0. \end{array} \right. $$$10!$$\dfrac{12!}{7! 3! 2!}$$\dfrac{4!-2!}{3!+5!}$$\dfrac…

Exercise 6.2 (Solutions)

anonymous@undisclosed.example.com (Anonymous) — Sat, 08 Mar 2025 08:59:59 +0000

Exercise 6.2 (Solutions) The solutions of the Exercise 6.1 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to factorial function.$n!$$$ n!=\left\{\begin{array}{l} n(n-1)(n-2)\cdot \ldots \cdot 3 \cdot 2 \cdot 1 \text{ if } n\geq 1,\\ 1 \text{ if } n=0. \end{array} \right. $$$n \in \mathbb{N}$${ }^n P_r=\frac{n!}{(n-r)!}$$\quad{ }^…