MathCity.org

Definitions: FSc Part 2 (Mathematics): PTB

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

Definitions: FSc Part 2 (Mathematics): PTB On this page, all the definitions of “Calculus and Analytic Geometry, MATHEMATICS 12” (Mathematics FSc Part 2 or HSSC-II), Punjab Textbook Board (PTB) Lahore, Pakistan are given. We are very thankful to $A=x^2$$f:X\to Y$$X$$f:X\to Y$$y$$Y$$y=ax+b$$x$$y$$f(x)=2x-6$$p(x) = {a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + {a_{n - 2}}{x^{n - 2}} + ... + {a_1}x + {a_0}$${a_0},\,{a_1},\,{a_2},...,{a_n}$$f(x)=ax+b$$X$$I:X\to X$$X$$Y$$C:X \rightarrow Y$$C(x)=a$$x \in X$$…

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)}$…

Khuram Ali Khan

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

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

MCQs: Ch 02 Sets, Functions and Groups

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

MCQs: Ch 02 Sets, Functions and Groups High quality MCQs of Chapter 02 Sets, Functions and Groups of Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. The answers are given at the end of the page.$\forall$$\wedge$$<$$\in$$A$$B$$A\cap B=\phi$$A=B$$B\subseteq A$$A \subseteq B$$A$$B$$A-B \neq \phi$$A=B$$A \subseteq B$$B\subseteq A$$A$$B$$A\cap B=A$$B \subseteq A$$A\cap B=\phi$$A\subseteq B$$B\subseteq A$$A=\phi$$A \cup B=A$$A \cap B=…

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{…

Definitions: FSc Part 1 (Mathematics): PTB

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

Definitions: FSc Part 1 (Mathematics): PTB On this page, all the definitions of “Textbook of Algebra and Trigonometry Class XI, published by Punjab Textbook Board (PTB) Lahore, Pakistan are given. We are very thankful to Muhammad Waqas Sulaiman for his valuable contribution.$\frac{p}{q}$$p,q \in \mathbb{Z}$$q\neq 0$$\frac{p}{q}$$p,q \in \mathbb{Z}$$q\neq 0$$\mathbb{R}$$0.3333....,21.134134$$\pi = 3.1415...$$\divideontimes$$z=x+iy$$x,y \in \mathbb{R}, i = \sqrt{-1}$$x$$y$$z$$2, 3+\sqrt{3}i, \fra…

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

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

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

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.

MTH322: Real Analysis II (Fall 2021)

anonymous@undisclosed.example.com (Anonymous) — Thu, 30 Dec 2021 19:16:17 +0000

MTH322: Real Analysis II (Fall 2021) This course is offered to MSc, Semester II at Department of Mathematics, COMSATS University Islamabad, Attock campus. This course need rigorous knowledge of continuity, differentiation, integration, sequences and series of numbers, that is many notion included in $\int_{1}^{\infty }{{{x}^{-p}} dx}$$p$$f\in \mathcal{R}[a,b]$$b\ge a$$f(x)\ge 0$$x\ge a$$\int_{a}^{\infty }{f(x) dx}$$M>0$$\int\limits_{a}^{b}{f(x)\,dx} \le M$$b\ge a$$f\in \mathcal{R}[a,b]$$b\ge a$…

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.

MTH424: Convex Analysis (Fall 2020)

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

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

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$…

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)$

Are the functions are same?

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:42:25 +0000

Are the functions are same? [Are the functions are same?] Consider two functions $f(x)=x+3$ and $\displaystyle g(x)=\frac{x^2-9}{x-3}$. Is $f=g$? Answer A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Set of inputs is usually know as $f:A \to B$$f = A$$B$$A$$A$$B$$A=\mathbb{N}$$B=\mathbb{R}$$f(x)=x+1$$1 \mapsto 2$$\quad 2 \mapsto 3$$\quad 3 \mapsto 4$$f = \mathbb{N}$$A=\mathbb{R}$$B=\mathbb…

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$

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(…

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.

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…

MTH322: Real Analysis II (Spring 2023)

anonymous@undisclosed.example.com (Anonymous) — Thu, 15 Jun 2023 01:08:47 +0000

MTH322: Real Analysis II (Spring 2023) [MTH322: Real Analysis II (Spring 2023)] This course is offered to BS, Semester VI at Department of Mathematics, COMSATS University Islamabad, Attock campus. This course need rigorous knowledge of continuity, differentiation, integration, sequences and series of numbers, that is many notions included in $f\in \mathcal{R}[a,b]$$b\ge a$$f(x)\ge 0$$x\ge a$$\int_{\,a}^{\,\infty }{f(x)\,dx}$$M>0$$\int\limits_{a}^{b}{f(x)\,dx}\leq M$$b\ge a$$f(x)$$g(x)$$x>a$$\li…

Atiq ur Rehman, PhD

anonymous@undisclosed.example.com (Anonymous) — Sun, 01 Feb 2026 14:25:54 +0000

Atiq ur Rehman, PhD Atiq ur Rehman, PhD Associate Professor (Tenured) Department of Mathematics COMSATS University Islamabad, Attock Campus Kamra Road, Attock - PAKISTAN. Email: , Field of Research: Difference and functional equations, Real functions, Inequalities in monotonic and convex functions

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…

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 $\sigma$

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…

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. $\infty$

Question 4(i-iv), Exercise 9.1

anonymous@undisclosed.example.com (Anonymous) — Wed, 27 Nov 2024 15:59:11 +0000

Question 4(i-iv), Exercise 9.1 Solutions of Question 4(i-iv) of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y=\sin x+x \cdot \cos x$$f(x)=\sin x+x \cdot \cos x$\begin{align*} f(-x) = \sin (-x) + (-x)\cdot \cos (-x) \end{align*}$\sin(-x)=-\sin x$$\cos (-x) = \cos x$\begin{align*} f(x) & = -\sin x - x \cdot \cos x \\ & = -(\sin x + x \cdot …

Question 4(v-viii), Exercise 9.1

anonymous@undisclosed.example.com (Anonymous) — Wed, 27 Nov 2024 15:59:12 +0000

Question 4(v-viii), Exercise 9.1 Solutions of Question 4(v-viii) of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y=\dfrac{\sin ^{2} x}{x+\tan x}$\[y = \frac{\sin^2 x}{x + \tan x}\]\begin{align*} y(-x) &= \frac{\big(-\sin x\big)^2}{-x - \tan x} \\ &= \frac{\sin^2 x}{-x - \tan x}\\ & = \frac{\sin^2 x}{-(x + \tan x)}\\ & = -\frac{\sin^2 x}{x +…

Question 1,Review Exercise

anonymous@undisclosed.example.com (Anonymous) — Wed, 27 Nov 2024 15:59:18 +0000

Question 1,Review Exercise Solutions of Question 1 of Review Exercise of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\cos \theta=\frac{\sqrt{3}}{2}$$\sin \theta=$$\frac{1}{2}$$-\frac{1}{2}$$\sqrt{3}$$-\frac{2}{\sqrt{3}}$$-\frac{1}{2}$$\tan (-15 \pi)=$$ 0$$-1$$1$$0$$2 \sin \theta+\frac{1}{2}cosec \theta \theta $$\theta=45^{\circ}$$\frac{1}{\sqrt{2}}$$\frac…

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$

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 Part1 KPK

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

MathCraft: PDF to LaTeX file: Sample-01

anonymous@undisclosed.example.com (Anonymous) — Mon, 25 Mar 2024 16:29:59 +0000

MathCraft: PDF to LaTeX file: Sample-01 If the PDF file provided by you as follows: Then the output LaTeX file is as follows: \documentclass[10pt]{amsart} %\usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} %\usepackage[version=4]{mhchem} \usepackage{stmaryrd} \usepackage{bbold} \usepackage{hyperref} \usepackage{enumerate} \hypersetup{colorlinks=true, linkcolor=blue, filecolor=magenta, urlcolor=cyan,} \urlstyle{same} \title{…

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$

MTH424: Convex Analysis (Spring 2025)

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

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

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

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$$…

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}$

Unit 02: Functions and Limits

anonymous@undisclosed.example.com (Anonymous) — Fri, 24 Jan 2025 18:15:12 +0000

Unit 02: Functions and Limits [Unit 02: Functions and Limits] We know that function is a rule or correspondence between two non-empty sets X and Y in such a way that, each element of X corresponds to one and only one element of Y. Here X is called the domain of the function and the set of corresponding elements of Y is called the range of the function.

Question 1, Exercise 10.1

anonymous@undisclosed.example.com (Anonymous) — Thu, 24 Aug 2023 17:24:38 +0000

Question 1, Exercise 10.1 Solutions of Question 1 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. There are four parts in Question 1.$\sin {{37}^{\circ }}\cos {{22}^{\circ }}+\cos {{37}^{\circ }}\sin {{22}^{\circ }}$\begin{align} \sin (\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta, \end{align}\begin{a…

Question 1, Exercise 10.1

anonymous@undisclosed.example.com (Anonymous) — Tue, 05 Dec 2023 02:45:35 +0000

Question 5(vi-x), Exercise 9.1

anonymous@undisclosed.example.com (Anonymous) — Wed, 27 Nov 2024 15:59:14 +0000

Question 5(vi-x), Exercise 9.1 Solutions of Question 5(vi-x) of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y=2 \operatorname{Sin} 3 x$$y=3 \operatorname{Cos} x$$y=\operatorname{Cos}^{2} x$$y=\operatorname{Sin}^{2} x$$y=\operatorname{Tan}^{2} x$$y=\operatorname{Sin} \frac{x}{2}$

MTH321: Real Analysis 1

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

MTH321: Real Analysis 1 At the end of this course the students will be able to uunderstand 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 continuous func…

MTH321: Real Analysis I (Fall 2015)

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

MTH321: Real Analysis I (Fall 2015) At the end of this course the students will be able to uunderstand 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 con…

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 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 (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…

MTH321: Real Analysis 1

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:29 +0000

MTH321: Real Analysis 1 (Spring 2015)

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:32 +0000

MTH321: Real Analysis 1 (Spring 2015) At the end of this course the students will be able to uunderstand 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 c…

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…

MTH604: Fixed Point Theory and Applications (Spring 2021)

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

~~DISCUSSION~~ MTH604: Fixed Point Theory and Applications (Spring 2021) Course Objectives: This course is intended as a brief introduction to the subject with a focus on Banach Fixed Point theorems fixed point theorem and its application to nonlinear differential equations, nonlinear integral equations, real and complex implicit functions theorems and system of nonlinear equations. Some generalizations and similar results e. g. Kannan Fixed Point theorems, Banach Fixed Point theorem for mul…

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

Unit 01: Functions and Limits

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:47:54 +0000

Unit 01: Functions and Limits Here is the list of important questions. * Evaluate $\lim\limits_{\theta \to 0}\frac{1-\cos \theta}{\sin^3\theta}$ --- FBSIC (2016) * Graph the curve of the following parametric equations $x=\sec \theta$, $y=\tan\theta$ where $\theta$ is a parameter.--- FBSIC (2016) * Evaluate $\lim\limits_{x \to 2}\frac{\sqrt{x}-\sqrt{2}}{x-2}$ --- BSIC Rawalpendi(2016), BSIC Rawalpendi(2017)$f(x)=x^3+x$$\lim\limits_{\theta \to 0}\frac{\tan \theta-\sin \theta}{\sin^3\t…

Question 2(i, ii, iii, iv and v) Exercise 8.3

anonymous@undisclosed.example.com (Anonymous) — Mon, 28 Oct 2024 17:14:31 +0000

Question 2(i, ii, iii, iv and v) Exercise 8.3 Solutions of Question 2(i, ii, iii, iv and v) of Exercise 8.3 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sin 70^{\circ} + \sin 30^{\circ}$\begin{align*} & \quad \sin 70^{\circ} + \sin 30^{\circ} \\ & = 2 \sin \left(\frac{70+30}{2} \right) \cos \left(\frac{70-30}{2} \right) \\ & = 2 \sin \left(\frac{1…

MTH104: Calculus & Analytical Geometry

anonymous@undisclosed.example.com (Anonymous) — Wed, 25 Dec 2024 17:21:25 +0000

MTH104: Calculus & Analytical Geometry [MTH104: Calculus & Analytical Geometry] Course Objectives The main objective of Calculus and Analytical Geometry for students is to continue learning the basics of the calculus of functions of one variable. They will study functions, their types, limit and continuity of a function, derivatives, rate of change, chain rule, the concepts and techniques of integration, maxima and minima for the function of one variable, power series sequence and series, Tay…

MTH604: Fixed Point Theory and Applications

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

MTH604: Fixed Point Theory and Applications Course Objectives: This course is intended as a brief introduction to the subject with a focus on Banach Fixed Point theorems fixed point theorem and its application to nonlinear differential equations, nonlinear integral equations, real and complex implicit functions theorems and system of nonlinear equations. Some generalizations and similar results e. g. Kannan Fixed Point theorems, Banach Fixed Point theorem for multi-valued mappings are also ed…

MTH604: Fixed Point Theory and Applications (Spring 2020)

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

~~DISCUSSION~~ MTH604: Fixed Point Theory and Applications (Spring 2020) Course Objectives: This course is intended as a brief introduction to the subject with a focus on Banach Fixed Point theorems fixed point theorem and its application to nonlinear differential equations, nonlinear integral equations, real and complex implicit functions theorems and system of nonlinear equations. Some generalizations and similar results e. g. Kannan Fixed Point theorems, Banach Fixed Point theorem for mul…

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

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…

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$

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.

MCQs: Ch 04 Quadratic Equations

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

MCQs: Ch 04 Quadratic Equations High quality MCQs of Chapter 01 Number System of Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. The answers are given at the end of the page. MCQs $ax^2+bx+c=0$$ax^2+bx+c=0$$b \neq 0$$c \neq 0$$a \neq 0$$x$$ax^2+bx+c$$ax^2+bx+c=0$$\{a,b\}$$ax^2+bx+c=0$$a\neq 0$$x= \frac{b \pm \sqrt{b^2-4ac}}{a}$$x= \frac{-b \pm \sqrt{b^2+4ac}}{2a}$$x= \frac{-b \pm \sqrt{4ac-b^2}}{2a}$$x= \frac{-b \pm \sqrt{b^2-…

Unit 02: Differentiation

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:47:54 +0000

Unit 02: Differentiation Here is the list of important questions. * Differentiate $\frac{(x^2+1)^2}{x^2-1}$ $w.r.t.x$. --- BSIC Gujranwala (2016) * If $x=at^2$, $y=2at$. Find $\frac{dy}{dx}$ --- BSIC Gujranwala (2016) * Differentiate $x^2-\frac{1}{x^2}$ $w.r.t.x^2$. --- BSIC Gujranwala (2016) * Prove that $\frac{d}{dx}(tan^{-1}x)=\frac{1}{1+x^2}$ --- BSIC Gujranwala (2016)$\frac{d}{dx}(sinh^{-1}x)=\frac{1}{\sqrt{1+x^2}}$$y=x^2ln(\frac{1}{x})$$\frac{dy}{dx}$$x=sin\theta$$y=sin m\t…

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 - {…

MCQs with key

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

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

Question 1, Exercise 9.1

anonymous@undisclosed.example.com (Anonymous) — Wed, 27 Nov 2024 15:59:09 +0000

Question 1, Exercise 9.1 Solutions of Question 1 of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y=2-2 \operatorname{Cos} \theta$\begin{align*} -1 \leq \operatorname{Cos} \theta \leq 1 \end{align*}$-2$\begin{align*} & 2 \geq -2 \operatorname{Cos} \theta \geq -2 \end{align*}$2$\begin{align*} & 4 \geq 2-2 \operatorname{Cos} \theta \geq 0 \\ …

Question 2, Exercise 9.1

anonymous@undisclosed.example.com (Anonymous) — Wed, 27 Nov 2024 15:59:10 +0000

Question 2, Exercise 9.1 Solutions of Question 2 of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y=\dfrac{1}{4+3 \operatorname{Sin} \theta}$\begin{align*} -1 \leq \operatorname{Sin} \theta \leq 1 \end{align*}$3$\begin{align*} -3 \leq 3 \operatorname{Sin} \theta \leq 3 \end{align*}$4$\begin{align*} & 1 \leq 4+3 \operatorname{Sin} \theta \l…

Question 5(i-v), Exercise 9.1

anonymous@undisclosed.example.com (Anonymous) — Wed, 27 Nov 2024 15:59:13 +0000

Question 5(i-v), Exercise 9.1 Solutions of Question 5(i-v) of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $y=2 \operatorname{Sin} x$$y=2 \operatorname{Cos} 3 x$$y=2 \operatorname{Tan} 2 x$$\mathrm{y}=\operatorname{Cos} \frac{\mathrm{x}}{2}$

MTH604: Fixed Point Theory and Applications (Fall 2022)

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

~~DISCUSSION~~ MTH604: Fixed Point Theory and Applications (Fall 2022) [FPTA] Course Objectives: This course is intended as a brief introduction to the subject with a focus on Banach Fixed Point theorems fixed point theorem and its application to nonlinear differential equations, nonlinear integral equations, real and complex implicit functions theorems and system of nonlinear equations. Some generalizations and similar results e. g. Kannan Fixed Point theorems, Banach Fixed Point theorem f…

MATH-300: Basic Mathematics for Chemist

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

MATH-300: Basic Mathematics for Chemist Without mathematics the sciences cannot be understood, nor made clear, nor taught, nor learned. (Roger Bacon, 1214–1292) Course contents Introdtuction; Review of basic algebra, Graphs and their significance in chemistry. Trigonometric, logarithmic and exponential functions. Differentiation, partial differentiation, differential equations and their use in chemical problems. Concept of maxima and minima. integration, Determinants and Matrices, their prope…

MATH-305: Real Analysis-I

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:21 +0000

MATH-305: Real Analysis-I Objectives of the course: This is the first rigorous course in analysis and has a theoretical emphasis. It tegorously develops the fundamental ideas of calculus and is aimed to develop the students’ ability to deal with abstract mathematics and mathematical proofs.

MATH-731: Convex Analysis

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

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

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…

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

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…

Short Questions by Mr. Akhtar Abbas

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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}$

Chapter 03 - Limits and Continuity

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:49:57 +0000

Chapter 03 - Limits and Continuity * Limit of the function, examples and definition * Theorem: Suppose (i) $(X,{d_x})$ and $(Y,{d_y})$ be two metric spaces (ii) $E\subset X$ (iii) $f:E\to Y$ i.e. f maps E into X (iv) p is the limit point of E. Then $\lim_{x\to p} f(x)=q$ iff $\lim_{n\to\infty}f(p_n)=q$ for every sequence {$p_n$} in E such that ${p_n}\ne p$$\lim_{n\to\infty}{p_n}=p$$\lim_{x\to c}f(x)$$c\in G$$\lim_{x\to c}f(x)=l$$\varepsilon$$\delta>0$$|f(t)-f(s)|<\varepsilon$$\left\{x:|x-c|…

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…

Mathematics CUI: LaTeX Resources

anonymous@undisclosed.example.com (Anonymous) — Fri, 02 Aug 2024 07:19:37 +0000

Mathematics CUI: LaTeX Resources [Department of Mathematics, COMSATS University Islamabad, Attock Campus] This page contains LaTeX template of CIIT Mathematics, MSc Project and MS Thesis templates. Templates Download a zip file given below and extract it by right clicking on the file. BS Project Template: (Version 1.5, Uploaded: Sep 29, 2022)$\$$I$$\mathbb{R}$$f:I\to \mathbb{R}$$(\$$$\sin^2 \theta + \cos^2 \theta =1$$\begin{equation} \sin^2 \theta + \cos^2 \theta =1 \end{equation}

CHEM-501: Basic Mathematics for Chemist

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

CHEM-501: Basic Mathematics for Chemist Course contents Introdtuction; Review of basic algebra, Graphs and their significance in chemistry. Trigonometric, logarithmic and exponential functions. Differentiation, partial differentiation, differential equations and their use in chemical problems. Concept of maxima and minima. integration, Determinants and Matrices, their properties and use in chemical problems. solutions of linear equations (simple, determinant and matrices methods), operator the…

MTH424: Convex Analysis

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

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

MTH322: Real Analysis II (Fall 2017)

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

MTH322: Real Analysis II (Fall 2017) This course is offered to MSc, Semester III at Department of Mathematics, COMSATS Institute of Information Technology, Attock campus. The is course need rigorous knowledge of continuity, differentiation, integration, sequences and series of numbers, that is many notion included in

MTH322: Real Analysis II (Fall 2018)

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

MTH322: Real Analysis II (Fall 2018) This course is offered to MSc, Semester II at Department of Mathematics, COMSATS University Islamabad, Attock campus. This course need rigorous knowledge of continuity, differentiation, integration, sequences and series of numbers, that is many notion included in

MTH322: Real Analysis II (Fall 2019)

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:12 +0000

MTH322: Real Analysis II (Fall 2019) This course is offered to MSc, Semester II at Department of Mathematics, COMSATS University Islamabad, Attock campus. This course need rigorous knowledge of continuity, differentiation, integration, sequences and series of numbers. these notions included in

MTH211: Discrete Mathematics (Fall 2020)

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:14 +0000

MTH211: Discrete Mathematics (Fall 2020) Course Objectives: Discrete Mathematics is branch of Mathematics which deals with discrete structures like logic. sequences, graphs, relations in contrast to Calculus. where we enjoy the continuity of functions and the set of real numbers. This course is introduction to discrete structures which are not the part of main stream courses. Discrete Mathematics has applications in Computer Science. Economics and Decision Making etc. This course will help t…

MTH322: Real Analysis II (Fall 2020)

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:14 +0000

MTH322: Real Analysis II (Fall 2020) This course is offered to MSc, Semester II at Department of Mathematics, COMSATS University Islamabad, Attock campus. This course need rigorous knowledge of continuity, differentiation, integration, sequences and series of numbers, that is many notion included in

MTH103: Exploring Quantitative Skills

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

MTH103: Exploring Quantitative Skills Course Objectives This course aims to develop the basic mathematical skills which ultimately enhance problem-solving skills using inductive and deductive reasoning, Polya's strategy, and sets. The basic concepts will be develop with applications form the real world such as algebraic models with equations, rates, ratios, and percentages will be discussed. Students will also explore linear models, including rectangular coordinates, functions, empowering them…

MTH480: Introductory Quantum Mechanics

anonymous@undisclosed.example.com (Anonymous) — Sat, 07 Oct 2023 18:27:32 +0000

MTH480: Introductory Quantum Mechanics Objective The physical principles and mathematical formalism of quantum theory, with emphasis on applications to atomic, molecular, and many-body physics; scattering phenomena; and electromagnetism (photon physics). $x(t)={{t}^{3}}+2\sin t$$t=\dfrac{\pi }{6}$$v(t)={{t}^{2}}+t{{e}^{t}}$

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

MTH322: Real Analysis II (Spring 2019)

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

MTH322: Real Analysis II (Spring 2019) This course is offered to MSc, Semester II at Department of Mathematics, COMSATS University Islamabad, Attock campus. This course need rigorous knowledge of continuity, differentiation, integration, sequences and series of numbers, that is many notion included in

MTH211: Discrete Mathematics (Spring 2020)

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

MTH211: Discrete Mathematics (Spring 2020) Course Objectives: Discrete Mathematics is branch of Mathematics which deals with discrete structures like logic. sequences, graphs, relations in contrast to Calculus. where we enjoy the continuity of functions and the set of real numbers. This course is introduction to discrete structures which are not the part of main stream courses. Discrete Mathematics has applications in Computer Science. Economics and Decision Making etc. This course will help…

MTH211: Discrete Mathematics (Fall 2020)

anonymous@undisclosed.example.com (Anonymous) — Thu, 10 Jun 2021 09:02:12 +0000

MTH211: Discrete Mathematics (Spring 2022)

anonymous@undisclosed.example.com (Anonymous) — Mon, 12 Sep 2022 04:56:59 +0000

MTH211: Discrete Mathematics (Spring 2022) Course Objectives: Discrete Mathematics is branch of Mathematics which deals with discrete structures like logic. sequences, graphs, relations in contrast to Calculus. where we enjoy the continuity of functions and the set of real numbers. This course is introduction to discrete structures which are not the part of main stream courses. Discrete Mathematics has applications in Computer Science. Economics and Decision Making etc. This course will help…

MTH322: Real Analysis II (Spring 2022)

anonymous@undisclosed.example.com (Anonymous) — Wed, 13 Apr 2022 05:45:43 +0000

MTH322: Real Analysis II (Spring 2022) This course is offered to BS, Semester VI at Department of Mathematics, COMSATS University Islamabad, Attock campus. This course need rigorous knowledge of continuity, differentiation, integration, sequences and series of numbers, that is many notion included in

MTH424: Convex Analysis (Spring 2024)

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

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

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$

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

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).

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.

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\}$…

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: 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-$

M for Mathodology

anonymous@undisclosed.example.com (Anonymous) — Fri, 20 Jun 2025 11:23:21 +0000

M for Mathodology [M for Mathodology] We are very thankful to M for Mathodology for providing us notes. I’m Rabbia Abid – a pure math enthusiast on a mission to transform the way we experience mathematics! Currently pursuing a bachelor’s degree in Mathematics, I’m captivated by the elegance and challenge of pure mathematics. My academic journey is punctuated by exciting math olympiads and a never-ending curiosity for proofs, theorems, and abstract reasoning.

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 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 $

Ch 11: Trigonometric Functions and Their Graphs

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:47:44 +0000

Ch 11: Trigonometric Functions and Their Graphs * Find the period of $\sin 4x$ --- BISE Gujrawala(2015) * Find the period of $\tan 4x$ --- BISE Gujrawala(2017) * Find the period of $\sin\frac{x}{5}$ --- BISE Sargodha(2015), BISE Sargodha(2016) * Find the period of $cosec10x$ --- BISE Sargodha(2015)$\cot\frac{x}{2}$$\sin x$$2\pi$

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 08: Function and Graphs

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:47:23 +0000

Chapter 08: Function and Graphs Notes of Chapter 08: Function and Graphs of “A Textbook of Mathematics for Class XI” published by Khyber Pakhtunkhwa (KPK) Textbook Board, Pesharwar. These notes are shared as open educational resources. [Download PDF ~ ch08-functions-and-graphs-fsc1-kpk.pdf] fsc kpk_fsc_part1

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…

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

Chapter 04 - Differentiation

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:49:57 +0000

Chapter 04 - Differentiation * Derivative of a function * Theorem: Let f be defined on [a,b], if f is differentiable at a point $x\in [a,b]$, then f is continuous at x. (Differentiability implies continuity) * Theorem (derivative of sum, product and quotient of two functions)$x\in [a,b]$$f'(x)$$f'(x)=0$$\mathbb{R}^k$$\underline{f}$$x\in (a,b)$$\left|\underline{f}(b)-\underline{f}(a)\right|\le (b-a)\left|\underline{f'}(x)\right|$

General Mathematics (Paper A & B)

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

General Mathematics (Paper A & B) This subject is consists of two papers of 100 marks each. One is called “Paper A” and other is called “Paper B”. This syllabus is for 1st Annual 2015 and onward organized by University of Sargodha (UoS), Sargodha.

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{ }^…

Question 10, Exercise 9.1

anonymous@undisclosed.example.com (Anonymous) — Wed, 27 Nov 2024 15:59:10 +0000

Question 10, Exercise 9.1 Solutions of Question 10 of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $ V(t)=a \operatorname{Sin}(k(t-d))+c$$56 \mathrm{~Hz} A C$$k$

MTH322: Real Analysis II (Fall 2015)

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

MTH322: Real Analysis II (Fall 2015) Course Contents: Sequences of functions: convergence, uniform convergence, uniform convergence and continuity, uniform convergence and integration, uniform convergence and differentiation, the exponential and logarithmic function, the trigonometric functions.

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

MTH322: Real Analysis II (Spring 2016)

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:34 +0000

MTH322: Real Analysis II (Spring 2016) This course was teach to MSc III and IV. Course Contents: Sequences of functions: convergence, uniform convergence, uniform convergence and continuity, uniform convergence and integration, uniform convergence and differentiation, the exponential and logarithmic function, the trigonometric functions.

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 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

MathCraft: PDF to LaTeX file: Sample-02

anonymous@undisclosed.example.com (Anonymous) — Mon, 25 Mar 2024 16:32:04 +0000

MathCraft: PDF to LaTeX file: Sample-02 If the PDF file provided by you as follows: Then the output LaTeX file is as follows: \documentclass[4pt]{article} \usepackage[utf8]{inputenc} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage[version=4]{mhchem} \usepackage{stmaryrd} \usepackage{bbold} \usepackage[a4paper]{geometry} \linespread{1.3} % double spaces lines \textwidth 6.3truein % These 4 commands define more efficient margins \textheight 9.9truein \oddsidemargi…

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$…

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.

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, …

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?

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.

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?

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.$\phi$

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…

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 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…

MCQs or Short Questions

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:45:18 +0000

MCQs or Short Questions On this page, MCQs or short questions with out answers are given. Students need to find the answer them self. This page will be updated occasionally and new MCQs or short question will be posted here. * A number which is neither even nor odd is$2n$$n \in \mathbb{Z}$$2\pi$$\pi$$\pi$$\sqrt{2}$$\sqrt{3}$$A$$f:A\to \mathbb{N}$$f$$f$$f$$A=\{x| x\in \mathbb{N} \wedge x^2 \leq 7 \}$$A$$\{s_n\}$$\lambda$$|s_n|<\lambda$$n\in\mathbb{Z}$$p$$|s_n|

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.

Unit 05: Linear Inequalities and Linear Programming

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

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

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 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 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)$

Unit 09: Trigonometric Functions

anonymous@undisclosed.example.com (Anonymous) — Wed, 27 Nov 2024 15:59:25 +0000

Unit 09: Trigonometric Functions This is a ninth unit of the book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. On this page we have provided the solutions of the questions.$a+b \sin \theta$$a+b \cos \theta$$a+b \sin(c \theta+d)$$a+b \cos(c \theta+d)$$a, b, c$$d$$\sin \theta$$\cos \theta$$\tan \theta$

Exercise 6.3 (Solutions)

anonymous@undisclosed.example.com (Anonymous) — Sat, 08 Mar 2025 09:00:35 +0000

Exercise 6.3 (Solutions) The solutions of the Exercise 6.3 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to factorial function.$n \in \mathbb{N}$${ }^{n} C_{r}=\frac{n!}{r!(n-r)!}$$n,{ }^{n-1} C_{r-1}=(n-r+1){ }^{n} C_{r-1}$$r^{n} C_{r}=(n-r+1)^{n} C_{r-1}$${ }^{n-1} C_{r-1}+{ }^{n-1} C_{r}={ }^{n} C_{r}$${ }^{n} C_{r}+{ }^{n} C…

Preparation Guide

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

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