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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…

Welcome to your new DokuWiki

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:45:13 +0000

Welcome to your new DokuWiki Congratulations, your wiki is now up and running. Here are a few more tips to get you started. Enjoy your work with DokuWiki, -- the developers Create your first pages Your wiki needs to have a start page. As long as it doesn't exist, this link will be red:

Question 7 & 8 Review Exercise 6

anonymous@undisclosed.example.com (Anonymous) — Thu, 01 Feb 2024 02:36:01 +0000

Question 7 & 8 Review Exercise 6 Solutions of Question 7 & 8 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$P(A)=0.8, P(B)=0.5$$P(B / A)=0.4$$P(A \cap B)$\begin{align} P(B \mid A)&=\dfrac{P(A \cap B)}{P(A)} \\ \Rightarrow P(A \cap B)&=P(B \mid A) \cdot P(A)\\ &=0.4 \times 0.8=0.32\end{align}$P(A)=0.8, P(B)=0.5$$P(B / A)=0.4$$P(A …

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 …

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)

Formatting Syntax

anonymous@undisclosed.example.com (Anonymous) — Sat, 10 Feb 2024 11:07:05 +0000

Formatting Syntax DokuWiki supports some simple markup language, which tries to make the datafiles to be as readable as possible. This page contains all possible syntax you may use when editing the pages. Simply have a look at the source of this page by pressing

Question 9 & 10 Review Exercise 6

anonymous@undisclosed.example.com (Anonymous) — Thu, 01 Feb 2024 02:36:03 +0000

Question 9 & 10 Review Exercise 6 Solutions of Question 9 & 10 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$2,3,0,3,4,2,3$$1$$=100,0000$$$=\dfrac{7 !}{3 ! \cdot 2 !}=420 $$$1$$0$$7$$0$$$=\dfrac{6 !}{2 ! 3 !}=60 $$$1$$420-50=360$$n$$n$$(n-1)$$(n - 1)$$(n-1)$$(n-2) !$$2$$2 !$$n$$$(n-2) ! \cdot 2 !=2(n-2) ! $$

Exercise 1.2 (Solutions)

anonymous@undisclosed.example.com (Anonymous) — Mon, 27 Oct 2025 18:50:09 +0000

Exercise 1.2 (Solutions) The solutions of the Exercise 1.2 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to real and imaginary part of complex numbers, modulus and conjugate of the complex numbers.$\operatorname{Re}(i z)=-\operatorname{Im}(z)$$\operatorname{Im}(i z)=\operatorname{Re}(z)$$$\left(z_{1} z_{2}\right)\left(z_{3} z_{4…

Question 18 and 19, Exercise 6.2

anonymous@undisclosed.example.com (Anonymous) — Sat, 08 Mar 2025 08:59:48 +0000

Question 18 and 19, Exercise 6.2 Solutions of Question 18 and 19 of Exercise 6.2 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $10,000$$0,2,3,5,6$$4$$10000$$3$$5$$0$$3={ }^{3} P=6$$0$$5=3 p_{6}=6$$10000$$5$$=6+6=12$$3$$4$$5$$={ }^{4} P_{3}=2$$4$$5$$4$$5={ }^{4} P_{3}=24$$3$$3$$3$$5=$${ }^{4} P_{2}=12$$3$$5$$3$$5=$${ }^{4} P_{2}=12$$2$$3$$2$$5={ }^{4} …

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}}$

MATH-510: Topology

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:25 +0000

MATH-510: Topology Topology is an important branch of mathematics that studies all the “qualitative” or “discrete” properties of continuous objects such as manifolds, i.e. all the properties that aren't changed by any continuous transformations except for the singular (infinitely extreme) ones.$(T_0, T_1, T_2)$$\mathbb{R}$$X=\{a\}$$X$$X$$X$$\tau$$\mathbb{N}$$\tau$$(\mathbb{Z}, \tau)$$\mathbb{N}$$\tau$$A=\{\pm 100,\pm 101, \pm 102, ... \}$$\tau$$E=\{0,\pm 2,\pm 4,...\}$$\tau$$\tau$$B=\{1,2,3,...…

MATH-510: Topology

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:40:24 +0000

MATH-510: Topology Objectives of the course This is an introductory course in topology, giving the basics of the theory. Course contents Topological spaces, bases and sub-bases, first and second axiom of countability, separability, continuous functions and homeomorphism, finite product space. Separation axioms $(T_0, T_1, T_2)$

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

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 Part1 KPK

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

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.

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)]

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$

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 15 & 16 Exercise 4.5

anonymous@undisclosed.example.com (Anonymous) — Fri, 05 Jan 2024 17:30:06 +0000

Question 15 & 16 Exercise 4.5 Solutions of Question 15 & 16 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$2$$4$$15^{\text {th }}$$a_1=R s .1$$a_2=R s .2$$a_3=R s .4$$1,2,4,8, \ldots$$a_1=1 . \quad r=2 . \quad n=15$$a_n=a_1 r^{n-1}$$15^{1 / 2}$$$a_{15}=a_1 r^{14} $$$$a_{15}=1 .(2)^{1 4}=R s .16384 $$$$S_{30}=\dfrac{a_1(r^{30}-1)}{r-1} $$$r-2$$a_1=1$\begi…

Question 6, Exercise 10.1

anonymous@undisclosed.example.com (Anonymous) — Tue, 05 Dec 2023 02:45:42 +0000

Question 6, 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.$\cos \alpha =2{{\cos }^{2}}\dfrac{\alpha }{2}-1=1-2{{\sin }^{2}}\dfrac{\alpha }{2}$\begin{align}\cos \alpha &=\cos 2\dfrac{\alpha }{2}\\ &={{\cos }^{2}}\dfrac{\alpha }{2}-{{\sin }^{2}}\dfrac{\alpha }{2}\\ &={{\cos }^{2}}…

Question 7, Exercise 1.2

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Jul 2024 13:32:58 +0000

Question 7, Exercise 1.2 Solutions of Question 7 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 7 $\sqrt{2}|z| \geq|\operatorname{Re}(z)|+|\operatorname{Im}(z)| \quad$$\left(|x|-|y|)^{2} \geq 0\right)$\begin{align} &\left(|x|-|y|)^{2} \geq 0\right) \\ \implies & |x|^2+|y|^2-2|x||y| \geq 0 \\ \implies & |x|^2+|y|^2 \geq 2|x||y| \\ \implie…

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 4 and 5, Exercise 6.2

anonymous@undisclosed.example.com (Anonymous) — Sat, 08 Mar 2025 08:59:51 +0000

Question 4 and 5, Exercise 6.2 Solutions of Question 4 and 5 of Exercise 6.2 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $3$$1,2,3,4,5,6,$$3$$6$$\mathrm{I}:$$2$$$\underline{ },\underline{ },\underline{2}$$$={ }^{5} P_{2}=20$$4$$$\underline{ },\underline{ },\underline{4}$$$={ }^{5} P_{2}=20$$6$$$\underline{ },\underline{ },\underline{6}$$$={ }^{5} P_…

Question 8 and 9, Exercise 6.2

anonymous@undisclosed.example.com (Anonymous) — Sat, 08 Mar 2025 08:59:54 +0000

Question 8 and 9, Exercise 6.2 Solutions of Question 8 and 9 of Exercise 6.2 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $4$$5$$5$$4$$41$$=41 \times 41=576$$2$$3$$4$$$\text{Total flags} =9$$$$\text{Repetition of blue }=2$$$$\text{Repetition of yellow}=3$$$$\text{Repetition of green}=4$$\begin{align*}\text{Total signals }&=\dfrac{9!}{2!3!4!}\\ &=\dfr…

Question 20 and 21, Exercise 6.2

anonymous@undisclosed.example.com (Anonymous) — Sat, 08 Mar 2025 08:59:45 +0000

Question 20 and 21, Exercise 6.2 Solutions of Question 20 and 21 of Exercise 6.2 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $6$$6$$5!$$6$$6!$$=5! \times 6!=86400$$= n = 4$$= {}^4P_4 = 24.$

Question 4, 5 and 6, Review Exercise 6

anonymous@undisclosed.example.com (Anonymous) — Sat, 08 Mar 2025 09:00:38 +0000

Question 4, 5 and 6, Review Exercise 6 Solutions of Question 4, 5 and 6 of Review Exercise 6 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $0,2,3,4,5,7$$6$$$=6!=720$$$0$$5$$$5!=120$$$6-$$$=720-120=600$$$7$$$(7-1)!=720$$$2$$$\dfrac{720}{2}=360$$$11!$$10!$$$=11!-10!=36288000$$