Merging man & maths https://www.mathcity.org/ Thu, 04 Jun 2026 18:44:50 +0000 FeedCreator 1.8 https://www.mathcity.org/_media/logo.svg https://www.mathcity.org/ https://www.mathcity.org/fsc-part1-ptb/mcq-bank/ch01 MCQs: Ch 01 Number Systems 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 * If $*$$A$$a, b \in A$$a+b \in A$$a-b \in A$$a \times b \in A$$a * b \in A$$z=(1,3)$$z^{-1}= $$(\displaystyle{\frac{1}{10}},\displaystyle{\frac{3}{10}})$$(-\displaystyle{\frac{1}{10}},\displaystyle{\frac{3}{10}})$$(\displaystyle{\frac{1}{10}},-\display… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:49 +0000 https://www.mathcity.org/fsc-part1-ptb/definitions Definitions: FSc Part 1 (Mathematics): PTB On this page, all the definitions of “Textbook of Algebra and Trigonometry Class XI, published by Punjab Textbook Board (PTB) Lahore, Pakistan are given. We are very thankful to Muhammad Waqas Sulaiman for his valuable contribution.$\frac{p}{q}$$p,q \in \mathbb{Z}$$q\neq 0$$\frac{p}{q}$$p,q \in \mathbb{Z}$$q\neq 0$$\mathbb{R}$$0.3333....,21.134134$$\pi = 3.1415...$$\divideontimes$$z=x+iy$$x,y \in \mathbb{R}, i = \sqrt{-1}$$x$$y$$z$$2, 3+\sqrt{3}i, \fra… anonymous@undisclosed.example.com (Anonymous) Sun, 22 Sep 2024 17:12:21 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/ex6-3-p7 Question 9 Exercise 6.3 Solutions of Question 9 of Exercise 6.3 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$8$$6$$7$$7$$6.$$=7+6=13$${ }^7 C_4$${ }^6 C_4$\begin{align}{ }^7 C_4 \cdot{ }^6 C_4&=\dfrac{7 !}{(7-4) ! 4 !} \cdot \dfrac{6 !}{(6-4)}\\\ &= 525\end{align}$8$$6$$7$$7$$6$$=7+6=13$$3,4,5,6$$6$\begin{align}{ }^7 C_2 \cdot{ }^6 C_6&=\dfrac{7 !}… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:48 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/ex6-4-p7 Question 7 Exercise 6.4 Solutions of Question 7 of Exercise 6.4 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.\begin{align}S&=\{(i, j) ; i, j=1,2,3,4,5,6\}\\ &=\left[\begin{array}{llllll} (1,1) & (1,2) & (1,3) & (1,4) & (1,5) & (1,6) \\ (2,1) & (2,2) & (2,3) & (2,4) & (2,5) & (2,6) \\ (3,1) & (3,2) & (3,3) & (3,4) & (3,5) & (3,6) \\ (4,1) & (4,2) & (… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:53 +0000 https://www.mathcity.org/fsc-part1-ptb/definitions-aurang-zaib 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 \( \dfrac{p}{q} \)\( p, q \in \mathbb{Z} \)\( q \neq 0 \)\( \dfrac{3}{4} \)\( \dfrac{7}{2} \)\( \sqrt{2} \)\( \pi \)\( \mathbb{R} \)\( 0.25 \)\( 3.75 \)\( 0.3333... \)\( 1.234234... \)\( \pi \)\( 3.1415... \)\( \sqrt{2} \)\(… anonymous@undisclosed.example.com (Anonymous) Sat, 30 Mar 2024 16:48:20 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p9 Question 13 Exercise 6.2 Solutions of Question 13 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\mathrm{E}$$n=10$$m_1=4$$E, m_2=2$$L$$m_3=2$$C$\begin{align}\text{total number of permutations are} &=\left(\begin{array}{c} n \\ m_1, m_2, m_3 \end{array}\right)\\&=\left(\begin{array}{c} 10 \\ 4,2,2 \end{array}\right) \\ & =\dfrac{10 !}… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:41 +0000 https://www.mathcity.org/matric/9th_science/unit_02/exercise_2.1 Exercise 2.1 (Solutions) Question 1 Identify which of the following are rational and irrational numbers: (i) $\sqrt{3}$ (ii) $\frac{1}{6}$ (iii) $\pi$ (iv) $\frac{15}{2}$ (v) $7.25$ (vi)$\sqrt{29}$ Solution * Rational: $\frac{1}{6}$, $\frac{15}{2}$, $7.25$ * Irrational: $\sqrt{3}$, $\pi$, $\sqrt{29}$ Question 2 Convert the following fraction into decimal fraction.$\frac{17}{25}$$\frac{19}{4}$$\frac{57}{8}$$\frac{205}{18}$$\frac{5}{8}$$\frac{25}{38}$$\frac{2}{3}$$\pi$$\frac{1}{9}$$\fr… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 17:00:36 +0000 https://www.mathcity.org/notes/algebraic-number-theory-notes-anwar-khan Algebraic Number Theory Notes by Anwar Khan [Algebraic Number Theory Notes by Anwar Khan] Algebraic number theory is a subfield of number theory that studies integers, rational numbers, and their generalisations using abstract algebra techniques. It covers Galois theory, ideals and units in rings of integers, unique factorization, and algebraic number fields and related rings of integers. It is a complex and in-depth subject with numerous linkages to other branches of mathematics.$\mathbb{R}$ anonymous@undisclosed.example.com (Anonymous) Sat, 05 Aug 2023 19:06:20 +0000 https://www.mathcity.org/notes/number-theory-handwritten-notes Number Theory: Handwritten Notes [Number Theory: Handwritten Notes] The study of the characteristics of the positive integers (1, 2, 3,...) is called number theory. It is significant because it has numerous uses in coding theory, combinatorics, cryptography, and other branches of mathematics and computer science. Some mathematicians also refer to number theory as the anonymous@undisclosed.example.com (Anonymous) Tue, 08 Aug 2023 11:20:45 +0000 https://www.mathcity.org/msc/mcqs_short_questions/real_analysis Real Analysis: Short Questions and MCQs We are going to add short questions and MCQs for Real Analysis. The subject is similar to calculus but little bit more abstract. This is a compulsory subject in MSc and BS Mathematics in most of the universities of Pakistan. The author of this page is Dr. $\left\{\frac{1}{n+1} \right\}$$\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… anonymous@undisclosed.example.com (Anonymous) Mon, 03 Apr 2023 04:06:26 +0000 https://www.mathcity.org/notes/number-theory-umer-shuaib Number Theory by Dr Muhammad Umer Shuaib [Number Theory Notes] A subfield of mathematics called number theory studies the characteristics of positive integers. Higher arithmetic is another name for it. The study of the relationships between various types of numbers, including prime numbers, rational numbers, and algebraic integers, is done using number theory, one of the oldest fields of mathematics.$\phi$ anonymous@undisclosed.example.com (Anonymous) Sat, 05 Aug 2023 18:11:04 +0000 https://www.mathcity.org/atiq/sp23-mth321 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… anonymous@undisclosed.example.com (Anonymous) Wed, 14 Jun 2023 14:47:57 +0000 https://www.mathcity.org/notes/number-theory-notes-anwar-khan Number Theory Notes by Anwar Khan [Number Theory Notes by Anwar Khan] Mathematicians who specialize in number theory examine the characteristics and connections between integers. “Higher arithmetic” and “the queen of mathematics” are some names for it. Because it examines the characteristics and connections between integers and arithmetic functions, number theory is interesting. It has numerous uses in coding theory, combinatorics, cryptography, and other branches of mathematics. Like the ones… anonymous@undisclosed.example.com (Anonymous) Sat, 05 Aug 2023 18:47:01 +0000 https://www.mathcity.org/bsc/number-theory-by-prof-asghar-ali Number Theory by Prof. Asghar Ali [Number Theory by M Asghar Ali] We are very thankful to Prof. Asghar Ali for send these notes. These notes are very helpful to prepare BSc or ADS mathematics portion of Number Theory. Number theory is a subject in which students learn different concepts created on the set of integers. For example, the concept of divisibilty exists in the set of integer. Let a and b be any two integers suhc that $a\neq 0$ anonymous@undisclosed.example.com (Anonymous) Mon, 21 Mar 2022 19:38:31 +0000 https://www.mathcity.org/ppsc/ppsc-maths-2011 PPSC Paper 2011 (Lecturer in Mathematics) [PPSC Paper 2011 (Lecturer in Mathematics)] On this page, we have given question from old (past) paper of Lecturer in Mathematics conducted in year 2011. This is a MCQs paper and answers are given at the end of the paper. At the end of the PDF is also given to download. This paper is provided by Ms. $R$$x\in R$$x^2=x$$x^2=-x$$x^2=0$$x^2=1$$6$$8$$10$$4$$G$$H$$H$$G$$2$$4$$nZ$$Z$$n$$G$$24$$a$$a^{10}$$2$$12$$10$$V$$n$$V$$n+1$$n$$n-1$$v_1,v_2,v_3,....,v_r$$… anonymous@undisclosed.example.com (Anonymous) Thu, 03 Feb 2022 11:54:32 +0000 https://www.mathcity.org/atiq/sp15-mth321/mcqs MCQs or Short Questions On this page, MCQs or short questions with out answers are given. Students need to find the answer them self. This page will be updated occasionally and new MCQs or short question will be posted here. * A number which is neither even nor odd is$2n$$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|<p$$n\in\mathbb{Z}$$s$$|s_n|<s$$n… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:45:18 +0000 https://www.mathcity.org/msc/notes/number-theory-iqra-liaqat Number Theory by Ms. Iqra Liaqat [Number Theory by Ms. Iqra Liaqat] Notes of number theory provided Ms. Iqra Liaqat is a very good addition in the MSc notes section. We are actually quite grateful to her for giving these notes and likes her encouragement to distribute these notes on MathCity.org Name anonymous@undisclosed.example.com (Anonymous) Tue, 11 May 2021 11:51:22 +0000 https://www.mathcity.org/fsc-part1-ptb/sol/ch01/ex1-2 Exercise 1.2 (Solutions) Notes (Solutions) of Exercise 1.2: Textbook of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board (PTB) Lahore. The main topics of this exercise are complex numbers, real part and imaginary part of complex numbers, properties of the fundamental operation on complex numbers, complex number as ordered pair of real numbers and special subset of complex numbers. These notes are based on the new Student Learning Outcomes (SLOs). Versio… anonymous@undisclosed.example.com (Anonymous) Sun, 09 Apr 2023 07:32:53 +0000 https://www.mathcity.org/atiq/fa21-mth321 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… anonymous@undisclosed.example.com (Anonymous) Fri, 28 Oct 2022 11:10:02 +0000 https://www.mathcity.org/fsc-part1-ptb/mcqs Multiple Choice Questions (MCQs) Textbook of Algebra and Trigonometry Class XI is published by Punjab Textbook Board (PTB) Lahore, Pakistan. The book has total of 14 chapters. Our plan is to give lot of Multiple Choice Questions (MCQs) for the above mentioned book. MCQs are very important because most of entry tests, admission tests and job tests consists of only MCQs.$\sqrt{3}$$n$$\sqrt{n}$$\forall a, b, c \in R$$a<b \wedge c>0\Rightarrow ac\geq bc$$a<b \wedge c>0\Rightarrow ac> bc$$a<b \wedge… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:43:03 +0000 https://www.mathcity.org/math-11-nbf/definitions 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… anonymous@undisclosed.example.com (Anonymous) Wed, 10 Jul 2024 18:51:53 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p1 Question 1, Exercise 1.4 Solutions of Question 1 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 1(i) $2+i 2 \sqrt{3}$$z=x+iy=2 + i 2 \sqrt{3}$\begin{align} r & = \sqrt{x^2 + y^2} = \sqrt{2^2 + (2\sqrt{3})^2} \\ & = \sqrt{4 + 12} = \sqrt{16} = 4. \end{align}\begin{align} \alpha & = \tan^{-1}\left|\frac{y}{x}\right| = \tan^{-1}\left|\fra… anonymous@undisclosed.example.com (Anonymous) Mon, 15 Jul 2024 12:24:40 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p6 Question 6(i-ix), Exercise 1.4 Solutions of Question 6(i-ix) of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\sqrt{2}\left(\cos 315^{\circ}+i \sin 315^{\circ}\right)$\begin{align} &\sqrt{2}\left(\cos 315^{\circ}+i \sin 315^{\circ}\right) \\ =& \sqrt{2} \left(\dfrac{1}{\sqrt{2}}-\dfrac{i}{\sqrt{2}} \right) \\ =& 1-i. \end{align}$5\left(\cos 210^{\ci… anonymous@undisclosed.example.com (Anonymous) Wed, 17 Jul 2024 12:40:31 +0000 https://www.mathcity.org/atiq/sp20-mth321 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… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:41 +0000 https://www.mathcity.org/notes/advanced-analysis-handwritten-notes Advanced Analysis: Handwritten Notes [Advanced Analysis: Handwritten Notes] These notes are provided by Mr. Anwar Khan. We are really very thankful to Mr. Anwar Khan for providing these notes and appreciates his effort to publish these notes on MathCity.org It covers the complete syllabus of Advanced Analysis paper of MSc Mathematics. See the contents of the notes given below to see the topics covered by these notes. anonymous@undisclosed.example.com (Anonymous) Fri, 14 Apr 2023 17:55:33 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p4 Question 7 and 8 Exercise 6.2 Solutions of Question 7 and 8 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$1,2,3,4$$E_1$$m_1=5$$E_2$$\cdot m_2=5$$E_3$$m_3=5$$$m_1 \cdot m_2 \cdot m_3=5.5 \cdot 5=125$$$1,2,3,4$$E_1$$m_1=5$$E_2$$m_2=4$$E_3$$m_3=3$$$m_1 \cdot m_2 \cdot m_3=5 \cdot 4 \cdot 3=60$$$8$$5$$=4$$=4$$=5$$=3$$4 ! \cdot 5 ! \cdot … anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:38 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p5 Question 9 Exercise 6.2 Solutions of Question 9 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$=^6 P_1=6$$s=^6 P_2=30$$=^6 P_3=120$$=^6 P_4=360$$=^6 P_5=720$$=^6 P_6=720$$6+30+120+360+720+720=1956$ anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:38 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p6 Question 10 Exercise 6.2 Solutions of Question 10 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$n=8$$r=5$\begin{align}^8 P_5&=\dfrac{8 !}{(8-5) !}\\ &=\dfrac{8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 !}{3 !}\\ &=6720\end{align}\begin{align}^2 P_2 \times^7 P_4&=2 \times \dfrac{7 !}{(7-4) !}\\ &=2 \times\dfrac{7.6 .5 .4 .3 !}{3 !}\\ &=… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:40 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p8 Question 12 Exercise 6.2 Solutions of Question 12 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$8.$$n=8$$\mathrm{O}$$m_1=3$\begin{align} \left(\begin{array}{c} n \\ m 1 \end{array}\right)&=\left(\begin{array}{l} 8 \\ 3 \end{array}\right) \\ & =\dfrac{8 !}{3 !}\\ &=\dfrac{8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 !}{3 !}\\ &=6,720 \e… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:41 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-1-p2 Question 2, Exercise 1.1 Solutions of Question 2 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 2(i) $x+iy$$(3+2i)+(2+4i)$\begin{align}&(3+i2)+(2+i4)\\ =&(3+2)+(i2+i4)\\ =&5+i6\end{align}$x+iy$$(4+3i)-(2+5i)$\begin{align}&(4+3i)-(2+5i)\\ =&(4-2)+(3i-5i)\\ =&2-2i\end{align}$x+iy$$(4+7i)+(4-7i)$\begin{align} &(4+7i)+(4-7i)\\ =&(4+4)+(7i-7i… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Jul 2024 12:10:43 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p7 Question 6(x-xvii), Exercise 1.4 Solutions of Question 6(x-xvii) of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $7 \sqrt{2}\left(\cos \dfrac{5 \pi}{4}+i \sin \dfrac{5 \pi}{4}\right)$$10 \sqrt{2}\left(\cos \dfrac{7 \pi}{4}+i \sin \dfrac{7 \pi}{4}\right)$$2\left(\cos\dfrac{5\pi}{2}+i \sin \dfrac{5\pi}{2}\right)$$\dfrac{1}{\sqrt{2}}\left(\cos \dfrac{\… anonymous@undisclosed.example.com (Anonymous) Wed, 17 Jul 2024 12:40:53 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-2-p11 Question 18 and 19, Exercise 6.2 Solutions of Question 18 and 19 of Exercise 6.2 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $10,000$$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$$5={ }^{4} … anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:48 +0000 https://www.mathcity.org/khuram Khuram Ali Khan Khuram Ali Khan, PhD Associate Professor Department of Mathematics University of Sargodha Sargodha - PAKISTAN. Email: <khuram@MathCity.org> Field of Research: Difference and functional equations, Real functions, Mathematical inequalities involving convex functions, Time Scales Calculus, Soft Sets anonymous@undisclosed.example.com (Anonymous) Fri, 13 Jun 2025 12:03:59 +0000 https://www.mathcity.org/math-11-kpk/sol/unit04/ex4-5-p10 Question 15 & 16 Exercise 4.5 Solutions of Question 15 & 16 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$2$$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$$a_1=1$\begi… anonymous@undisclosed.example.com (Anonymous) Fri, 05 Jan 2024 17:30:06 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/ex6-5-p7 Question 10 Exercise 6.5 Solutions of Question 10 of Exercise 6.5 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$20$$10$$5$$3$$2$$=20$$=10$$=5$$=3$$=15$$=5$$=10$$=3$$=22$$E$$a A$$B$$2$\begin{align}n(S)&={ }^{30} C_2\\ &=435\\ P(A)&=\dfrac{^{20} C_2}{^{30} C_2}\\ &=\dfrac{190}{435}=\dfrac{38}{87}\\ P(B)&=\dfrac{^{22} C_2}{^{30} C_2}\\ &=\dfrac{231}{43… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:58 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p6 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) ! $$ anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:36:03 +0000 https://www.mathcity.org/bsc ADS/BSc One this page we have listed notes/resources widely used in BSc or in Associated Degree of Science (ADS). This includes notes of some famous books and other resources. Notes of Famous Books Notes of Calculus with Analytic Geometry Notes of Calculus with Analytic Geometry Notes of Mathematical Method Notes of Mathematical Method Introduction to Mechanics Introduction to Mechanics Other notes Notes of Mechanics Notes of Mechanics written by different authors for BSc or BS… anonymous@undisclosed.example.com (Anonymous) Sat, 13 Jan 2024 11:00:49 +0000 https://www.mathcity.org/bsc/number-theory-by-prof-m-tanveer Number Theory by Prof. M. Tanveer [Number Theory by Prof. M. Tanveer] These notes are very helpful to prepare one of the sections of mathematics for BSc. Also these notes can be used for other classes. Author: Prof. M. Tanveer Type: Handwritten Format: PDF (972 kB) Pages: $a,b, \in \mathbb{Z}$$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$ anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:54 +0000 https://www.mathcity.org/dyk/2 What is the value of pi? What is the value of $\pi$? * (A) 3.1415 * (B) $\frac{22}{7}$ * (C) $\frac{333}{160}$ * (D) None of these ✔ Answer The number π is a mathematical constant, the ratio of a circle's circumference to its diameter. It is an irrational number, meaning that it cannot be written as the ratio of two integers or in terminating or repeated decimals.$\pi=3.14159...$$\sqrt{2}=1.4142...$$e=2.718...$$\pi$ anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:42:25 +0000 https://www.mathcity.org/dyk/3 Its about square root [DYK] The reason is not difficult if one knows about the definition of square root of real numbers. Definition: Let $x$ be a non-negative number. Then a non-negative number $r$ is called square root of $x$ iff $r^2=x$. Square root of $x$ is denoted by $\sqrt{x}$$2^2=4$$3^2=9$$x$$r$$x$$r^2=x$$2^2=4$$(-2)^2=4$$\sqrt{4}=\sqrt{2^2}=\sqrt{(-2)^2}=2$$\sqrt{4}=\sqrt{2^2}=\sqrt{(-2)^2}=\pm 2$$\sqrt{4}$$\sqrt{x}$$x\geq0$$\sqrt{x}=x^{\frac{1}{2}}$ anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:42:27 +0000 https://www.mathcity.org/math-11-nbf/mcqs MCQs: Math 11 NBF Multiple Choice Questions (MCQs) of the Model Textbook of Mathematics for Class XI is published by National Book Foundation (NBF), Islamabad, Pakistan. NBF can be considered as Federal Textbook Board Islamabad. Unit 01: Complex Numbers $\operatorname{part}(\mathrm{s})$$z$$z$$(0,0)$$(1,0)$$(0,1)$$(1,1)$$(0,0)$$z$$|z|$$1 / z$$-z$$\bar{z}$$\bar{z}$$x$$y$$x y$$y$$z_{1}=3+2 i$$z_{2}=5+6 i$$z_{1}>z_{2}$$z_{1}<z_{2}$$\overline{z_{1}}=\overline{z_{2}}$$\overline{z_{1}}=-\overline{z_{2… anonymous@undisclosed.example.com (Anonymous) Sun, 20 Oct 2024 18:55:54 +0000 https://www.mathcity.org/ppsc/ppsc-maths-2015 PPSC Paper 2015 (Lecturer in Mathematics) [PPSC Paper 2011 (Lecturer in Mathematics)] On this page, we have given question from old (past) paper of Lecturer in Mathematics conducted in year 2011. This is a MCQs paper and answers are given at the end of the paper. At the end of the PDF is also given to download. This paper is provided by Kaushef Salamat. We are very thankful to her for providing this paper.\(\displaystyle \int_{-4}^{0}\frac{tdt}{\sqrt{16-t62}}\)$0$$-4$$4$\(A\cos wt+B\sin wt\)$\… anonymous@undisclosed.example.com (Anonymous) Tue, 18 Jan 2022 10:20:00 +0000 https://www.mathcity.org/msc/real_analysis_notes_by_syed_gul_shah/real_number_system Chapter 01 - Real Number System Contents & Summary * Theorem: There is no rational p such that $p^2=2$. * Theorem: Let A be the set of all positive rationals p such that $p^2>2$ and let B consist of all positive rationals p such that $p^2<2$ then A contain no largest member and $x<y$$x<u<y$$x=\sup E$$x>0$$n>0$$y^n=x$$\underline x,\underline y\in \mathbb{R}^n$$\|\underline x^2\|=\underline x\cdot \underline x$$\|\underline x\cdot \underline y\|=\|\underline x\| \|\underline y\|$$\underline … anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:49:59 +0000 https://www.mathcity.org/fsc-part1-ptb/sol/ch02/ex2-8 Exercise 2.8 (Solutions) Notes (Solutions) of Exercise 2.8: Textbook of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board (PTB) Lahore. The main topic of this exercise are binary operation, semi-group, monoid, groups and abelian groups. These notes are based on the new Student Learning Outcomes (SLOs). Version: 4.1, Available at MathCity.org $\oplus$$G=\{0,1\}$\[ \begin{array}{|c|c|c|} \hline \oplus & 0 & 1 \\ \hline 0 & 1 & 1 \\ \hline 1 & 1 & … anonymous@undisclosed.example.com (Anonymous) Wed, 05 Apr 2023 12:55:15 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/ex6-4-p1 Question 1 Exercise 6.4 Solutions of Question 1 of Exercise 6.4 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.$S=\{1,2,3,4,5,6\}$$5$$5$\begin{align}A&=\{5\}\\ P(A)&=\dfrac{n(A)}{n(S)}\\ &=\dfrac{1}{6} \end{align}$S=\{1,2,3,4,5,6\}$$1$$1$\begin{align}B&=\{\}\\ &=\phi \text{then}\\ P(B)&=\dfrac{n(B)}{n(S)}\\ &=\dfrac{0}{6}\\ &=0\end{align}$S=\{1,2,3,4,… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:49 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/ex6-4-p2 Question 2 Exercise 6.4 Solutions of Question 2 of Exercise 6.4 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.$4$$5$$6$$3$$4+5+6=15$$${ }^{15} C_3=\dfrac{15 !}{(15-3) ! 3 !}=455 $$$${ }^6 C_4=\dfrac{6 !}{(6-4) ! 4 !}=15$$$$=\dfrac{15}{455}=\dfrac{3}{91}$$$4$$5$$6$$3$$4+5+6=15$$${ }^{15} C_3=\dfrac{15 !}{(15-3) ! 3 !}=455 $$$${ }^4 C_3=\dfrac{4 !}{(4-… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:50 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/ex6-4-p5 Question 5 Exercise 6.4 Solutions of Question 5 of Exercise 6.4 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$6$$4$$3$$2$$=6+4=10$$5$$10$\begin{align}{ }^{10)} C_5 &=\dfrac{10 !}{(10-5) ! 5 !}\\ &=252\\ n(S)&=252\end{align}$3$$2$$3$$2$\begin{align}{ }^6 \mathrm{C}_3\cdot{ }^{4} \mathrm{C}_2&=\dfrac{6 !}{(6-3) ! 3 !} \cdot \dfrac{4 !}{(4-2) ! 2 !}\\… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:52 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/re-ex-p1 Question 1, Review Exercise Solutions of Question 1 of Review Exercise 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. $\operatorname{part}(\mathrm{s})$$z$$z$$(0,0)$$(1,0)$$(0,1)$$(1,1)$$(0,0)$$z$$|z|$$1 / z$$-z$$\bar{z}$$\bar{z}$$x$$y$$x y$$y$$z_{1}=3+2 i$$z_{2}=5+6 i$$z_{1}>z_{2}$$z_{1}<z_{2}$$\overline{z_{1}}=\overline{z_{2}}$$\overline{z_{1}}=-\overline{z_{2}}$… anonymous@undisclosed.example.com (Anonymous) Wed, 17 Jul 2024 12:53:04 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-2-p4 Question 4 and 5, Exercise 6.2 Solutions of Question 4 and 5 of Exercise 6.2 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $3$$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_… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:51 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/re-ex6-p1 Question 1, Review Exercise 6 Solutions of Question 1 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. $3\,\,^nP_3=^nP_4$$n$$5$$6$$7$$8$$6$$ 480$$600$$720$$840$$720$$r$$r!$$(r+1)!$$r!+1$$ 2r!$$r!$$6$$\dfrac{5}{2}\,\,6!$$6!$$\dfrac{1}{2}\,\,6!$$\dfrac{3}{2}\,\,6!$$\dfrac{5}{2}\,\,6!$$A=\{1,2,3,4,...,20\}. $$3$$5$$7$$8$$8$$A=\{1,3,5,7,… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 09:00:36 +0000 https://www.mathcity.org/atiq/fa14-mth321 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… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:06 +0000 https://www.mathcity.org/atiq/fa15-mth321 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… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:08 +0000 https://www.mathcity.org/atiq/fa18-mth321 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… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:10 +0000 https://www.mathcity.org/atiq/math-103 MATH 103: Number Theory Objectives of the course This course shall assume no experience of background in number theory of theoretical mathematics. The course introduces various strategies for composing mathematical proofs. Course contents Number systems: natural numbers, integers, rational numbers, real numbers, complex numbers, the equivalence and the difference of cardinality between them, de Morvie’s theorem with application, hyperbolic ad logarithmic functions, introduction to number the… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:17 +0000 https://www.mathcity.org/atiq/sp15-mth321 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… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:32 +0000 https://www.mathcity.org/bsc/notes-of-number-theory-by-umer-asghar 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 anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:48 +0000 https://www.mathcity.org/fsc-part1-kpk/mcqs Multiple Choice Questions (MCQs) Here are the sample MCQs at this time. Page will be updated periodically. SAMPLE MCQs * $i^{13}=$............. * (A) $i$ * (B) 1 * (C) -1 * (D) 2 * Set of all possible subsets of $S$ is called * (A) Equivalent sets$1, \omega, \omega^2$$-1, \omega, \omega^2$$-1, -\omega, -\omega^2$$1, -1, 2$$ax^2+bx+c=0$$a=0, b\neq 0$$a\neq 0$$a=b=0$$b=$$ax^2+bx+c=0$$a=0, b\neq 0$$a\neq 0$$a=b=0$$b=$$n!=n(n-1)(n-2)...3\cdot 2\cdot 1$$n$ anonymous@undisclosed.example.com (Anonymous) Mon, 28 Aug 2023 17:03:58 +0000 https://www.mathcity.org/math-11-kpk/mcqs Multiple Choice Questions (MCQs) Here are the sample MCQs at this time. Page will be updated periodically. SAMPLE MCQs * $i^{13}=$............. * (A) $i$ * (B) 1 * (C) -1 * (D) 2 * Set of all possible subsets of $S$ is called * (A) Equivalent sets$1, \omega, \omega^2$$-1, \omega, \omega^2$$-1, -\omega, -\omega^2$$1, -1, 2$$ax^2+bx+c=0$$a=0, b\neq 0$$a\neq 0$$a=b=0$$b=$$ax^2+bx+c=0$$a=0, b\neq 0$$a\neq 0$$a=b=0$$b=$$n!=n(n-1)(n-2)...3\cdot 2\cdot 1$$n$ anonymous@undisclosed.example.com (Anonymous) Tue, 05 Dec 2023 02:44:44 +0000 https://www.mathcity.org/notes/real-analysis-notes-by-prof-syed-gul-shah 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<u_n<t_n$$n\ge n_0$$\{s_n\}$$\{t_n\}$$\{u_n\}$$\{s_n\}$$\exists$$\left| {\,{s_n}}\right|>\frac{1}{2}s$$\{s_n\}$… anonymous@undisclosed.example.com (Anonymous) Mon, 17 Apr 2023 04:04:23 +0000 https://www.mathcity.org/fsc/fsc_part_1_mcqs/mcqs_with_answers MCQs with Answers (FSc/ICS Part 1) [MCQs Choice] In this one PDF, MCQs of all chapters of FSc/ICS Part1 are given. There are seven chapters. Answers of MCQs is starting from page 71. SAMPLE MCQs * $i^{13}=$............. * (A) $i$ * (B) 1 * (C) -1 * (D) 2 * $S$$1, \omega, \omega^2$$-1, \omega, \omega^2$$-1, -\omega, -\omega^2$$1, -1, 2$$ax^2+bx+c=0$$a=0, b\neq 0$$a\neq 0$$a=b=0$$b=$$ax^2+bx+c=0$$a=0, b\neq 0$$a\neq 0$$a=b=0$$b=$$n!=n(n-1)(n-2)...3\cdot 2\cdot 1$$n$ anonymous@undisclosed.example.com (Anonymous) Thu, 02 May 2024 17:03:11 +0000 https://www.mathcity.org/fsc/fsc_part_1_solutions/ch01 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}$ anonymous@undisclosed.example.com (Anonymous) Sat, 03 Jun 2023 16:30:31 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p3 Question 5 and 6 Exercise 6.2 Solutions of Question 5 and 6 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$7.$$7$$7$\begin{align}^7 P_7&=\dfrac{7 !}{(7-7) !}\\ & =7 !\\ &=5,040 \end{align}$2,4,5,7,9$$2,4,5,7,9$$\mathrm{n} . \mathrm{m}$$e$$$=5.4 .3 .2=120\quad \text{or}$$$$^5 P_4=\dfrac{5 !}{5-4} !=120$$$2$$4$$3$$E_1$$m_1=2$$E_2$$m_2=3… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:36 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p4 Question 5 & 6 Review Exercise 6 Solutions of Question 5 & 6 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.$$n=6$$$$(n-1) !=(6-1) !=5 !=120$$$120-24=96$$n=6$$(n-1) !=(6-1) !=5 !=120$$$(n-1) !=(5-1) !=4 !=24$$$$(n-1) !=(6-1) !=5 !=120$$$$4 ! \cdot 2 !=48$$$(5-1) !$ anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:36:01 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-1-p6 Question 6(vi-ix), Exercise 6.1 Solutions of Question 6(vi-ix) of Exercise 6.1 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n\in N$$\quad 33!$$2^{15}$$$33!=33.32.31\cdots4.3.2.1$$$2,4,8,16,32$$2^1,2^2,2^3,2^4,2^5$\begin{align*}33!&=2^5.2^4.2^3.2^2.2(33\times31\times\cdots6\times5\times3\times1)\\ &=2^{15}(33\times31\times\cdots\times3\times1)\end{al… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:34 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-3-p7 Question 7 and 8, Exercise 6.3 Solutions of Question 7 and 8 of Exercise 6.3 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $5$$6$$4$$2$$2$$3$$={ }^{4} C_{2} \times{ }^{6} C_{3}=6 \times 20=120$$5$$6$$4$$2$$2$$2$$2,3$$2$$120$$3$$2$$={ }^{4} C_{3} \cdot{ }^{6} C_{2}=4 \times 15=60$$4$$={ }^{4} C_{4} \times{ }^{6} C_{1}=1 \times 6=6$$=120+60+6=186$$5$$6$… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 09:00:29 +0000 https://www.mathcity.org/notes Notes of Mathematics [Notes of Mathematics] Mathematics is a language of science and is a basic need for physical or natural sciences as well as social sciences. On this page, notes on different subjects related to mathematics are listed. These notes or resources might be helpful for ADS or BS or MSc or MPhil Mathematics. These notes are send by different students or teachers. We are very thankful to them for sending us these notes. These notes are provided as it is as open educational resource… anonymous@undisclosed.example.com (Anonymous) Mon, 25 May 2026 18:19:50 +0000 https://www.mathcity.org/atiq/fa19-mth321 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… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:11 +0000 https://www.mathcity.org/atiq/fa22-mth321 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… anonymous@undisclosed.example.com (Anonymous) Mon, 15 May 2023 07:16:43 +0000 https://www.mathcity.org/atiq/sp14-mth321 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… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:29 +0000 https://www.mathcity.org/fsc-part2-ptb/definitions Definitions: FSc Part 2 (Mathematics): PTB On this page, all the definitions of “Calculus and Analytic Geometry, MATHEMATICS 12” (Mathematics FSc Part 2 or HSSC-II), Punjab Textbook Board (PTB) Lahore, Pakistan are given. We are very thankful to $A=x^2$$f:X\to Y$$X$$f:X\to Y$$y$$Y$$y=ax+b$$x$$y$$f(x)=2x-6$$p(x) = {a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + {a_{n - 2}}{x^{n - 2}} + ... + {a_1}x + {a_0}$${a_0},\,{a_1},\,{a_2},...,{a_n}$$f(x)=ax+b$$X$$I:X\to X$$X$$Y$$C:X \rightarrow Y$$C(x)=a$$x \in X$$… anonymous@undisclosed.example.com (Anonymous) Sun, 20 Oct 2024 18:16:52 +0000 https://www.mathcity.org/notes/real-analysis-hand-written-kaushef 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. $\infty$ anonymous@undisclosed.example.com (Anonymous) Fri, 20 Jan 2023 17:26:27 +0000 https://www.mathcity.org/fsc-part1-ptb/mcq-bank/ch02 MCQs: Ch 02 Sets, Functions and Groups High quality MCQs of Chapter 02 Sets, Functions and Groups of Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. The answers are given at the end of the page.$\forall$$\wedge$$<$$\in$$A$$B$$A\cap B=\phi$$A=B$$B\subseteq A$$A \subseteq B$$A$$B$$A-B \neq \phi$$A=B$$A \subseteq B$$B\subseteq A$$A$$B$$A\cap B=A$$B \subseteq A$$A\cap B=\phi$$A\subseteq B$$B\subseteq A$$A=\phi$$A \cup B=A$$A \cap B=… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:49 +0000 https://www.mathcity.org/msc/real_analysis_notes_by_syed_gul_shah/sequence_and_series Chapter 02 - Sequence and Series Contents * Sequence, Subsequence, Increasing Sequence, Decreasing Sequence, Monotonic Sequence, Strictly Increasing or Decreasing * Bernoulli’s Inequality * Bounded Sequence * Convergence of the Sequence$s_n<u_n<t_n$$n\ge n_0$$\{s_n\}$$\{t_n\}$$\{u_n\}$$\{s_n\}$$\exists$$\left| {\,{s_n}}\right|>\frac{1}{2}s$$\{s_n\}$$\{t_n\}$$\left\{a{s_n}+b{t_n}\right\}$$as+bt$$\left\{{s_n}{t_n}\right\}$$\left\{\frac{{{s_n}}}{{{t_n}}} \right\}$$\frac{s}{t}$${t_n}\ne… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:49:59 +0000 https://www.mathcity.org/fsc-part1-kpk/sol/unit01/review-ex-1-p2 Question 2 & 3, Review Exercise 1 Solutions of Question 2 & 3 of Review Exercise 1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${{i}^{n}}+{{i}^{n+1}}+{{i}^{n+2}}+{{i}^{n+3}}=0$$\forall n\in N$\begin{align}{{i}^{n}}+{{i}^{n+1}}+{{i}^{n+2}}+{{i}^{n+3}}&=0\\ L.H.S.&={{i}^{n}}+{{i}^{n}}\cdot i+{{i}^{n}}\cdot {{i}^{2}}+{{i}^{n}}\cdot {{i}^{3}}\\ &={{i}^{n}}\left( 1+i+{{i}^{2}}… anonymous@undisclosed.example.com (Anonymous) Mon, 25 Sep 2023 12:09:19 +0000 https://www.mathcity.org/math-11-kpk/sol/unit01/review-ex-1-p2 Question 2 & 3, Review Exercise 1 Solutions of Question 2 & 3 of Review Exercise 1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${{i}^{n}}+{{i}^{n+1}}+{{i}^{n+2}}+{{i}^{n+3}}=0$$\forall n\in N$\begin{align}{{i}^{n}}+{{i}^{n+1}}+{{i}^{n+2}}+{{i}^{n+3}}&=0\\ L.H.S.&={{i}^{n}}+{{i}^{n}}\cdot i+{{i}^{n}}\cdot {{i}^{2}}+{{i}^{n}}\cdot {{i}^{3}}\\ &={{i}^{n}}\left( 1+i+{{i}^{2}}… anonymous@undisclosed.example.com (Anonymous) Tue, 05 Dec 2023 02:45:05 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/ex6-3-p5 Question 5 and 6 Exercise 6.3 Solutions of Question 5 and 6 of Exercise 6.3 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$12$$n=12$${ }^{12} C_2=66$$12$$n=12$${ }^{12} C_3=220$$${ }^6 C_2=\dfrac{6 !}{(6-2) ! 2 !}=15 $$$6$$\quad 15-6=9$ anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:46 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/ex6-4-p6 Question 6 Exercise 6.4 Solutions of Question 6 of Exercise 6.4 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.$52$$$=52$$$=4$$$=\dfrac{4}{52}=\dfrac{1}{13}$$$52$$=52$$13$$13$$$\dfrac{13}{52}+ \dfrac{13}{52}=\dfrac{1}{4}+\dfrac{1}{4}=\dfrac{2}{4}=\dfrac{1}{2}$$$52$$=52$$13.$$$=\dfrac{13}{52}=\dfrac{1}{4}$$$52$$=52$$12.$$$=\dfrac{12}{52}=\dfrac{3}{13}$… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:53 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-1 Exercise 1.1 (Solutions) The solutions of the Exercise 1.1 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to sum, product and division of the complex numbers.${{i}^{31}}$${{\left( -i \right)}^{6}}$${{\left( -1 \right)}^{\frac{-13}{2}}}$$\dfrac{2}{(-1)^{\frac{3}{2}}}$$i^{23}+i^{58}+i^{21}$$x+iy$$(3+2i)+(2+4i)$$(4+3i)-(2+5i)$$(4+7i… anonymous@undisclosed.example.com (Anonymous) Mon, 27 Oct 2025 18:50:29 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-1-p4 Question 4, Exercise 1.1 Solutions of Question 4 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 4(i) $x$$y$$(2+3i)x+(1+3i)y+2=0$\begin{align}&(2+3i)x+(1+3i)y+2=0\\ \implies &(2x+y+2)+(3x+3y)i=0.\end{align}\begin{align} 2x+y+2&=0 \quad \cdots(1)\\ 3x+3y&=0\quad \cdots (2) \end{align}\begin{align} &3x=-3y \\ x=-y \quad ... (3) \end{align}$… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Jul 2024 12:39:29 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-1-p6 Question 6, Exercise 1.1 Solutions of Question 6 of Exercise 1.1 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 6(i) $4-3 i$$z=4-3 i$$\bar{z}=4+3i$$3 i+8$$2+\sqrt{\dfrac{-1}{5}}$\begin{align}z=&2+\sqrt{\dfrac{-1}{5}}\\ =&2+\sqrt{\dfrac{1}{5}}i,\end{align}$$\bar{z}=2-\sqrt{\dfrac{1}{5}}i$$$\dfrac{5 }{2}i-\dfrac{7}{8}$$z=\dfrac{5 }{2}i-\dfrac{7}{8},$$\bar… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Jul 2024 12:40:49 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-4 Exercise 1.4 (Solutions) The solutions of the Exercise 1.4 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to polar form of the complex numbers.$2+i 2 \sqrt{3}$$3-i \sqrt{3}$$-2-i 2$$\dfrac{i-1}{\cos \dfrac{\pi}{3}+i \sin \dfrac{\pi}{3}}$$\left(\cos \dfrac{\pi}{6}+i \sin \dfrac{\pi}{6}\right)\left(\cos \dfrac{\pi}{3}+i \sin \dfrac… anonymous@undisclosed.example.com (Anonymous) Mon, 27 Oct 2025 18:51:55 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-2 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{ }^… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:59 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-3 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}$${ }^{n} C_{r}+{ }^{n} C… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 09:00:35 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/re-ex6-p3 Question 4, 5 and 6, Review Exercise 6 Solutions of Question 4, 5 and 6 of Review Exercise 6 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $0,2,3,4,5,7$$6$$$=6!=720$$$0$$5$$$5!=120$$$6-$$$=720-120=600$$$7$$$(7-1)!=720$$$2$$$\dfrac{720}{2}=360$$$11!$$10!$$$=11!-10!=36288000$$ anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 09:00:38 +0000 https://www.mathcity.org/matric/9th_science/unit_02/exercise_2.6 Exercise 2.6 (Solutions) Question 1 Identify the following statements as true or false. (i) $\sqrt{-3}\cdot\sqrt{-3} = 3$ (ii) $i^{73}=-i$ (iii) $i^{10} = -1$ (iv) Complex conjugate of $(-6i + i^2) is (-1 + 6i)$ (v) Difference of complex numbers $z = a + ib$ and its conjugate is a real number. (vi) If $(a-1)-(b+3)i = 5+8i$, then a = 6 & b = -11 (vii) Product of complex number and its conjugate is always a non-negative real number.$a+ib$$(2+3i)+(7-2i)$$2(5+4i)+3(7-4i)$$-(-3+5i)-3(4+9i)$$… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 17:00:37 +0000 https://www.mathcity.org/msc/syllabus/uos/preparation_guide 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 $z= f(x,y)$ anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 17:00:44 +0000 https://www.mathcity.org/mathcraft MathCraft Introducing “MathCraft”: Your Solution for Document Transformation! [MathCraft] We are thrilled to unveil our latest service, MathCraft, tailored exclusively for the mathematics community. With MathCraft, you can easily get code from PDFs and pictures into LaTeX or Word files without spending too much money. Whether you're a student, researcher, or teacher, MathCraft can help you create your math documents in the format you want. anonymous@undisclosed.example.com (Anonymous) Sun, 14 Apr 2024 10:56:57 +0000 https://www.mathcity.org/atiq/chem-501 CHEM-501: Basic Mathematics for Chemist Course contents Introdtuction; Review of basic algebra, Graphs and their significance in chemistry. Trigonometric, logarithmic and exponential functions. Differentiation, partial differentiation, differential equations and their use in chemical problems. Concept of maxima and minima. integration, Determinants and Matrices, their properties and use in chemical problems. solutions of linear equations (simple, determinant and matrices methods), operator the… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:04 +0000 https://www.mathcity.org/atiq/math-300 MATH-300: Basic Mathematics for Chemist Without mathematics the sciences cannot be understood, nor made clear, nor taught, nor learned. (Roger Bacon, 1214–1292) Course contents Introdtuction; Review of basic algebra, Graphs and their significance in chemistry. Trigonometric, logarithmic and exponential functions. Differentiation, partial differentiation, differential equations and their use in chemical problems. Concept of maxima and minima. integration, Determinants and Matrices, their prope… anonymous@undisclosed.example.com (Anonymous) Wed, 31 May 2023 05:38:37 +0000 https://www.mathcity.org/fsc/kpk_fsc_part_2 FSc Part 2 (KPK Boards) [A Textbook of Mathematics For Class XII] Notes of FSc Part 2 of “A Textbook of Mathematics For Class XII” published by Khyber Pakhtunkhwa Textbook Board, Peshawar. We are posting the notes chapter-wise. These notes are shared as open educational resources. This page will be continuously updated.$y=x^n$$y=(ax+b)^n$ anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:42:52 +0000 https://www.mathcity.org/fsc-part1-kpk/sol/unit01 Unit 1: Complex Numbers (Solutions) This is a first unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the students will be able to$z$$z=a+ib$$(a,b)$$a$$b$$i=\sqrt{-1}$$a$$z$$b$$z$$\bar{z} = a —ib$$z=a+ib$$|z| = \sqrt{a^2+b^2}$$z=a+ib$$'+'$$'\times'$$z$$|z|=|-z|=|\bar{z}=|-\bar{z}|$$pz^2+ qz+ r = 0$$p,q,r$$z$ anonymous@undisclosed.example.com (Anonymous) Mon, 25 Sep 2023 12:08:05 +0000 https://www.mathcity.org/fsc-part1-ptb/mcq-bank/ch04 MCQs: Ch 04 Quadratic Equations High quality MCQs of Chapter 01 Number System of Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. The answers are given at the end of the page. MCQs $ax^2+bx+c=0$$ax^2+bx+c=0$$b \neq 0$$c \neq 0$$a \neq 0$$x$$ax^2+bx+c$$ax^2+bx+c=0$$\{a,b\}$$ax^2+bx+c=0$$a\neq 0$$x= \frac{b \pm \sqrt{b^2-4ac}}{a}$$x= \frac{-b \pm \sqrt{b^2+4ac}}{2a}$$x= \frac{-b \pm \sqrt{4ac-b^2}}{2a}$$x= \frac{-b \pm \sqrt{b^2-… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:50 +0000 https://www.mathcity.org/math-11-kpk/sol/unit01 Unit 01: Complex Numbers (Solutions) This is a first unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the students will be able to$z$$z=a+ib$$(a,b)$$a$$b$$i=\sqrt{-1}$$a$$z$$b$$z$$\bar{z} = a —ib$$z=a+ib$$|z| = \sqrt{a^2+b^2}$$z=a+ib$$'+'$$'\times'$$z$$|z|=|-z|=|\bar{z}=|-\bar{z}|$$pz^2+ qz+ r = 0$$p,q,r$$z$ anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:28:42 +0000 https://www.mathcity.org/math-11-kpk/sol/unit02 Unit 02: Matrices and Determinants (Solutions) This is a second unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the students will be able to$z$$z=a+ib$$(a,b)$$a$$b$$i=\sqrt{-1}$$a$$z$$b$$z$$\bar{z} = a —ib$$z=a+ib$$|z| = \sqrt{a^2+b^2}$$z=a+ib$$'+'$$'\times'$$z$$|z|=|-z|=|\bar{z}=|-\bar{z}|$$pz^2+ qz+ r = 0$$p,q,r$$z$ anonymous@undisclosed.example.com (Anonymous) Tue, 05 Dec 2023 02:44:47 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01 Unit 01: Complex Numbers (Solutions) This is a first unit of the book Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. On this page we have provided the solutions of the questions.$z$$z^2+a^2$$z^3-3z^2+z=5$$pz^2+qz+r=0$$p,q,r$$z$ anonymous@undisclosed.example.com (Anonymous) Mon, 27 Oct 2025 18:47:40 +0000 https://www.mathcity.org/msc/real_analysis_notes_by_syed_gul_shah/limits_and_continuity 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|… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:49:57 +0000 https://www.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-1-p3 Question 4, Exercise 1.1 Solutions of Question 4 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4(i) $\left( a,0 \right)\left( 2,-b \right)$\begin{align}&\left( a,0 \right)-\left( 2,-b \right)\\ &=\left( a+0i \right)-\left( 2-bi \right)\\ &=\left( a-2 \right)+\left( 0+b \right)i\\ &=\left( a-2 \right)+bi\end{align}$\left( -3,\dfrac{1}{2} \right)\le… anonymous@undisclosed.example.com (Anonymous) Sat, 30 Sep 2023 12:31:57 +0000 https://www.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-1-p4 Question 5, Exercise 1.1 Solutions of Question 5 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) $8i+11,-7+5i$\begin{align}&(8i+11)\times (-7+5i)\\ &=\left( 11+8i \right)\times \left( -7+5i \right)\\ &=\left( -77+40{{i}^{2}} \right)+\left( 55-56 \right)i\\ &=\left( -77+40\left( -1 \right) \right)+\left( 55-56 \right)i\\ &=\left( -77-40 \right)+… anonymous@undisclosed.example.com (Anonymous) Sat, 30 Sep 2023 13:35:32 +0000 https://www.mathcity.org/math-11-kpk/sol/unit01/ex1-1-p3 Question 4, Exercise 1.1 Solutions of Question 4 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4(i) $\left( a,0 \right)\left( 2,-b \right)$\begin{align}&\left( a,0 \right)-\left( 2,-b \right)\\ &=\left( a+0i \right)-\left( 2-bi \right)\\ &=\left( a-2 \right)+\left( 0+b \right)i\\ &=\left( a-2 \right)+bi\end{align}$\left( -3,\dfrac{1}{2} \right)\le… anonymous@undisclosed.example.com (Anonymous) Tue, 05 Dec 2023 02:44:52 +0000 https://www.mathcity.org/math-11-kpk/sol/unit01/ex1-1-p4 Question 5, Exercise 1.1 Solutions of Question 5 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 5(i) $8i+11,-7+5i$\begin{align}&(8i+11)\times (-7+5i)\\ &=\left( 11+8i \right)\times \left( -7+5i \right)\\ &=\left( -77+40{{i}^{2}} \right)+\left( 55-56 \right)i\\ &=\left( -77+40\left( -1 \right) \right)+\left( 55-56 \right)i\\ &=\left( -77-40 \right)+… anonymous@undisclosed.example.com (Anonymous) Tue, 05 Dec 2023 02:44:52 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/ex6-5-p2 Question 3 and 4 Exercise 6.5 Solutions of Question 3 and 4 of Exercise 6.5 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)=0.5$$P(A \cup B)=0.6$$P(B)$$A$$B$$\mathrm{A}$$B$$A \cap B=\emptyset$\begin{align}P(A \cup B)&=P(A)+P(B)\\ \Rightarrow P(B)&=P(A \cup B)-P(A)\\ &=0.6-.0 .5=0.1 \end{align}$30$$1$$30.$\begin{align}S&=\{1,2,3, \ldots, 50\} \tex… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:55 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/ex6-5-p3 Question 5 and 6 Exercise 6.5 Solutions of Question 5 and 6 of Exercise 6.5 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{8}{9}$$$E=\{ event\, passing\, the\, test \}$$$$E^{\prime}=\{ event\, failing\, the\, test \}$$$E$$E^{\prime}$$P(E)=\dfrac{8}{9}$\begin{align}P(E^{\prime})&=1-P(E)=1-\dfrac{8}{9}=\dfrac{1}{9}\end{align}$4$$4$\begin{align}S… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:56 +0000 https://www.mathcity.org/math-11-kpk/sol/unit07/ex7-2-p5 Question 5 Exercise 7.2 Solutions of Question 5 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$(\dfrac{a}{x}+b x)^8$$a=\dfrac{a}{x}$$b=b x$$n=8$$n-8$$8+1=9$$$(\dfrac{8+2}{2})^{t h}=5^{t h}$$T_{r+1}$$$T_{r+1}=\dfrac{8 !}{(8-r) ! r !}(\dfrac{a}{x})^{8-r}(b x)^r$$$T_5$$r=4$\begin{align}T_5&=\dfrac{8 !}{(8-4) ! 4 !}(\dfrac{a}{x})^{8-4}(b … anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:36:25 +0000 https://www.mathcity.org/math-11-kpk/sol/unit07/re-ex7-p1 Question 1 Review Exercise 7 Solutions of Question 1 of Review Exercise 7 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Chose the correct option.$O, P, Q, R, S, T, U$$2520$$9040$$5140$$4880$$2520$$\{1,2,3,4,5,6,7\}$$14$$42$$28$$21$$28$$\{1,2,3,4,6,7,8\}$$3$$7$$120$$180$$144$$96$$120$$\dfrac{(n+2) !(n-2) !}{(n+1) !(n-1) !}$$(n-3)$$(\dot{n}-1)$$\dfrac… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:36:38 +0000 https://www.mathcity.org/math-11-nbf/sol/unit05/ex5-3-p2 Question 2, Exercise 5.3 Solutions of Question 2 of Exercise 5.3 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 2 $t(x)=x^{3}-12 x^{2}+48 x+74$$x$$$t(x)=x^{3}-12 x^{2}+48 x+74.$$$t=12$\begin{align*} t(12)&=(12)^3-12(12)^2+48(12)+74 \\ &=650. \end{align*} anonymous@undisclosed.example.com (Anonymous) Sun, 13 Oct 2024 17:52:37 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-2-p12 Question 20 and 21, Exercise 6.2 Solutions of Question 20 and 21 of Exercise 6.2 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $6$$6$$5!$$6$$6!$$=5! \times 6!=86400$$= n = 4$$= {}^4P_4 = 24.$ anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:45 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-3-p8 Question 9 and 10, Exercise 6.3 Solutions of Question 9 and 10 of Exercise 6.3 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$n$$n$$n$$C_{2}$$n$$n$$={ }^{n} C_{2}-n$$=\frac{n!}{2!(n-2)!}-n$$10$$10$$3$$7$$10$${ }^{10} C_{7}$$7$$3$$7={ }^{10} C_{7}=120$ anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 09:00:33 +0000 https://www.mathcity.org/mathematician_of_the_day Mathematician of the day The complete list of the mathematician who were born or died today is given at the following URL * <https://mathshistory.st-andrews.ac.uk/OfTheDay/today/> It is worth mentioning that “The MacTutor History of Mathematics archive” is a database of famous mathematician and mathematics archive created by by John J O'Connor and Edmund F Robertson is available at the following anonymous@undisclosed.example.com (Anonymous) Mon, 11 Mar 2024 05:54:30 +0000 https://www.mathcity.org/atiq/math-305 MATH-305: Real Analysis-I Objectives of the course: This is the first rigorous course in analysis and has a theoretical emphasis. It tegorously develops the fundamental ideas of calculus and is aimed to develop the students’ ability to deal with abstract mathematics and mathematical proofs. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:21 +0000 https://www.mathcity.org/dyk/1 Are the functions are same? [Are the functions are same?] Consider two functions $f(x)=x+3$ and $\displaystyle g(x)=\frac{x^2-9}{x-3}$. Is $f=g$? Answer A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Set of inputs is usually know as $f:A \to B$$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}$$B=\mathbb… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:42:25 +0000 https://www.mathcity.org/fsc-part1-kpk/definitions Definitions: FSc Part1 KPK A Textbook of Mathematics for Class XI is published by Khybar Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. The book has total of twelve (12) chapters. Definition of the book provide the quick overview of the book.$360^\circ$$\theta$$90^{\circ} \pm \theta, 180^{\circ} \pm \theta, 270^{\circ} \pm \theta, 360^{\circ} \pm \theta$$16^\circ 13' 9''$$sin(\alpha+2\pi)=sin\alpha$$sin x=\frac{2}{7}$$cos x-tan x=0$ anonymous@undisclosed.example.com (Anonymous) Mon, 28 Aug 2023 16:59:59 +0000 https://www.mathcity.org/math-11-kpk/definitions Definitions: FSc Part1 KPK A Textbook of Mathematics for Class XI is published by Khybar Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. The book has total of twelve (12) chapters. Definition of the book provide the quick overview of the book.$360^\circ$$\theta$$90^{\circ} \pm \theta, 180^{\circ} \pm \theta, 270^{\circ} \pm \theta, 360^{\circ} \pm \theta$$16^\circ 13' 9''$$sin(\alpha+2\pi)=sin\alpha$$sin x=\frac{2}{7}$$cos x-tan x=0$ anonymous@undisclosed.example.com (Anonymous) Tue, 05 Dec 2023 02:44:42 +0000 https://www.mathcity.org/notes/advanced-analysis-iqra-liaqat Advance Analysis by Ms. Iqra Liaqat [Advance Analysis by Ms. Iqra Liaqat] These notes are send by Ms. Iqra Liaqat. We are really very thankful to her for providing these notes and appreciates her effort to publish these notes on MathCity.org Name Advance Analysis Sender Ms. Iqra Liaqat Author $\sigma$ anonymous@undisclosed.example.com (Anonymous) Fri, 21 Mar 2025 15:08:17 +0000 https://www.mathcity.org/notes/complex-analysis-quick-review Complex Analysis (Quick Review) [Complex Analysis: Quick Review] These notes are made and shared by Mr. Akhtar Abbas. We are really very thankful to him for providing these notes and appreciates his efforts to publish these notes on MathCity.org. Important definitions and important results are the part of these notes, these might be helpful to prepare interviews or any other written test after graduation like PPSC, FPSC or etc.$z_1, z_2 \in S$$S$$v(x,y)$$u(x,y)$$f(z)=u(x,y)+iv(x,y)$$f$$D$$C$$D$… anonymous@undisclosed.example.com (Anonymous) Wed, 26 Jun 2024 18:33:42 +0000 https://www.mathcity.org/notes/metric-spaces-notes Metric Spaces (Notes) [Metric Spaces (Notes)] These are updated version of previous notes. Many mistakes and errors have been removed. These notes are collected, composed and corrected by Atiq ur Rehman, PhD. These are actually based on the lectures delivered by Prof. Muhammad Ashfaq (Ex HoD, Department of Mathematics, Government College Sargodha). $(X,d)$$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… anonymous@undisclosed.example.com (Anonymous) Fri, 18 Aug 2023 18:05:58 +0000 https://www.mathcity.org/people/mformathodology 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. anonymous@undisclosed.example.com (Anonymous) Fri, 20 Jun 2025 11:23:21 +0000 https://www.mathcity.org/ppsc/ppsc-maths-2021 PPSC Paper 2021 (Lecturer in Mathematics) [PPSC Paper 2011 (Lecturer in Mathematics)] On this page, we have given question from old (past) paper of Lecturer in Mathematics conducted in year 2021. This is a MCQs paper and answers are given at the end of the paper. At the end of the PDF is also given to download. This paper is provided by Ms. \(2018\)$4$\(6\)$8$$10$\(X\)\(Y\)\(X\times Y\)\(\parallel (x,y) \parallel=\parallel x\parallel+\parallel y\parallel, \,\forall \, (x,y)\in X \times Y\)\(f(… anonymous@undisclosed.example.com (Anonymous) Tue, 23 Aug 2022 17:04:49 +0000 https://www.mathcity.org/ppsc/ppsc-mock-interview-mathematics PPSC Mock Interview Lecturer Mathematics [PPSC Mock Interview Lecturer Mathematics] This handout is shared by Mr. Rashad Wattu and written by Sawaira Sikandar. We are really very thankful to him for providing this handout and appreciates his efforts to publish it on MathCity.org. This handout contains the questions collected from the different interviews of the Lecturer in Mathematics conducted by Public Service Commission (PSC). This is help full to prepare interview of all types of jobs whic… anonymous@undisclosed.example.com (Anonymous) Thu, 25 Mar 2021 16:31:54 +0000 https://www.mathcity.org/wiki/syntax 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 anonymous@undisclosed.example.com (Anonymous) Sat, 10 Feb 2024 11:07:05 +0000 https://www.mathcity.org/fsc-part1-ptb/important-questions/ch01-number-systems Ch 01: Number Systems * Simplify $(i)^{19}$ --- BISE Gujrawala(2015) * If $z$ be a complex number then prove that $\overline{z_1 + z_2}=\overline z_1 +\overline z_2$ --- BISE Sargodha(2015) * Simplify $\frac{2}{\sqrt{5}+\sqrt{-8}}$ in the form of $a+ib$ --- BISE Sargodha(2015) * Simplify by justify each step $\frac{\frac{1}{a}-\frac{1}{b}}{1-\frac{1}{a}\frac{1}{b}}$ --- BISE Sargodha(2015)$(\sqrt{2}, -\sqrt{5})$$\{0,-1\}$$a \div ib$$(-1)^\frac{-21}{2}$$(0,1)$$\{1,-1\}$$|z_… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:38 +0000 https://www.mathcity.org/fsc-part1-ptb/important-questions/ch07-permutation-combination-and-probablity Ch 07: Permutation, Combination and Probability * Find $n$ when ${^nC_{12}}={^nC_6}$ --- BISE Gujranwala(2015) * Evaluate ${^{20}C_{17}}$ without calculator --- BISE Gujranwala(2015) * How many $6-digit$ numbers can be formed from the digits $2,2,3,3,4,4$? How many of them with lie between $400,000$ and $430,000$? ---$``PLANE"$$^nC_4=^nC_{n-r}$$6-digits$$n^3-n$$6$$n=2,3$$n$$^nP_2=30$$6-dided$$n$$^nC_{12}=^nC_6$$^{n-1}C_r+^{n-1}C_{r-1}=^nC_r$$\frac{a_5}{a_3}=\frac{4}{9}$$a_2=\frac{4}{9}$… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:42 +0000 https://www.mathcity.org/msc/notes/targets Targets Here we have listed the notes for MSc or BS Mathematics, which will be published on MathCity.org. We are working hard to find these notes. Whenever we found these notes we will put them on our website. Here are our targets. * Fluid Mechanics anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:49:43 +0000 https://www.mathcity.org/msc/syllabus/uos Syllabus for UoS (Private only) Syllabus and scheme of studies for private students doing MSc Mathematics from University of Sargodha, Sargodha. The syllabus has been changed and few optional subjects has been dropped. Please be alert --- 2017/08/25 17:05 anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:50:15 +0000 https://www.mathcity.org/papers/old_papers_for_msc_mathematics/sargodha_university University of Sargodha, Sargodha (Old Papers) * To open or print a DjVu file, you must have some DjVu file viewer, e.g. WinDjVu. It can be downloaded from here * From 1st Annual 2013, the paper pattern has been changed. Check the complete syllabus anonymous@undisclosed.example.com (Anonymous) Thu, 28 Mar 2024 16:26:23 +0000 https://www.mathcity.org/fsc-part1-kpk/sol/unit01/ex1-2-p3 Question 3 & 4, Exercise 1.2 Solutions of Question 3 & 4 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 ${{z}_{1}}=\sqrt{3}+\sqrt{2}i$${{z}_{2}}=\sqrt{2}-\sqrt{3}i$${{z}_{3}}=2+3i$${{z}_{1}}=\sqrt{3}+\sqrt{2}i$${{z}_{2}}=\sqrt{2}-\sqrt{3}i$${{z}_{3}}=2+3i$\begin{align}{{z}_{1}}\left( {{z}_{2}}+{{z}_{3}} \right)&={{z}_{1}}{{z}_{2}}+{{z}_{1}}{{z}_{… anonymous@undisclosed.example.com (Anonymous) Sat, 30 Sep 2023 17:09:17 +0000 https://www.mathcity.org/fsc/fsc_part_1_solutions/ch01/view Ch 01: Number System: Mathematics FSc Part 1 Notes (Solutions) of Chapter 01: Number System, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board (PTB), Lahore. There are three exercises in this chapter. Please see the main page of this chapter for MCQs and important question at anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:57:21 +0000 https://www.mathcity.org/math-11-kpk/sol/unit01/ex1-2-p3 Question 3 & 4, Exercise 1.2 Solutions of Question 3 & 4 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 ${{z}_{1}}=\sqrt{3}+\sqrt{2}i$${{z}_{2}}=\sqrt{2}-\sqrt{3}i$${{z}_{3}}=2+3i$${{z}_{1}}=\sqrt{3}+\sqrt{2}i$${{z}_{2}}=\sqrt{2}-\sqrt{3}i$${{z}_{3}}=2+3i$\begin{align}{{z}_{1}}\left( {{z}_{2}}+{{z}_{3}} \right)&={{z}_{1}}{{z}_{2}}+{{z}_{1}}{{z}_{… anonymous@undisclosed.example.com (Anonymous) Tue, 05 Dec 2023 02:44:58 +0000 https://www.mathcity.org/math-11-kpk/sol/unit04/ex4-3-p1 Question 1 Exercise 4.3 Solutions of Question 1 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 1(i) $9,7,5,3, \ldots$$a_1$$d$\begin{align}&a_1=9 \\ &d=7-9=-2 \\ &n=20. \end{align}\begin{align}&a_n=a_1+(n-1)d \\ \implies &a_20=9+(20-1)(-2)=-29. \end{align}$S_n$$n$\begin{align} S_n&=\dfrac{n}{2}[a_1+a_n], \\ \implies S_{20}&=\dfrac{20}{2}[9-29] … anonymous@undisclosed.example.com (Anonymous) Sun, 11 Feb 2024 11:01:45 +0000 https://www.mathcity.org/math-11-kpk/sol/unit04/ex4-3-p8 Question 13 & 14 Exercise 4.3 Solutions of Question 13 & 14 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.\begin{align}\text{Total number of rows}& n=40,\\ \text{Seats in a first row} a_1&=20\\ \text{Seat in a second row} a_2&=23\\ \text{Seats in third row} a_3&=26\end{align}$20,23,26, \ldots$$S_{40}$$$S_n=\dfrac{n}{2} [{2} a_1+(n-1) d] \text {.}$$\begin… anonymous@undisclosed.example.com (Anonymous) Fri, 05 Jan 2024 17:29:58 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p2 Question 3 and 4 Exercise 6.2 Solutions of Question 3 and 4 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$^n P_r=n(^{n-1} P_{r-1})$$$^n P_r=n({ }^{n-1} P_{r-1})$$\begin{align}n(^{n-1} P_{r-1})&=n \dfrac{(n-1) !}{((n-1)-(r-1)) !} \\ & =\dfrac{n(n-1) !}{(n-r) !}\\ &=\dfrac{n !}{(n-r) !}\\ &=^n P_r\end{align}$^n P_r=^{n-1} P_r+r(^{n-1} … anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:37 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p7 Question 11 Exercise 6.2 Solutions of Question 11 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$10$$1000$$2.3,4,0,8,9$$10$$1000$$10$$100$$E_1$$m_1=5$$E_2$$m_2=5$$10$$100$$$m_1 \cdot m_2=5.5=25$$$100$$1000$$0$$E_1$$m_1=5$$E_2$$\boldsymbol{m}_2=5$$E_3$$m_3=4$$100$$1000$$$m_1 \cdot m_2 \cdot m_3=5.5 \cdot 4=100$$$10$$1000$$$100 + 25=125… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:40 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p10 Question 14 and 15 Exercise 6.2 Solutions of Question 14 and 15 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$\dfrac{(n-1) !}{2}=\dfrac{(5-1) !}{2}=\dfrac{24}{2}=12 $$$7$$7$$6 !$$6$$5!$$2 !=2$$7$$$2 \times 5 !=240$$ anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:35 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/ex6-3-p6 Question 7 and 8 Exercise 6.3 Solutions of Question 7 and 8 of Exercise 6.3 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$20$\begin{align}{ }^{20} C_2&=\dfrac{20 !}{(20-2)2!}!\\ &=\dfrac{20!}{18!\cdot 2!}\\ &=190\end{align}$7$$10$$3$$7$$10$$${ }^{10} C_7=\dfrac{10 !}{(10-7) ! 7 !}=120$$$7$$4.$$4$$${ }^7 C_4=\dfrac{7 !}{(7-4) ! 4 !}=35.$$$35$$10.$ anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:47 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/ex6-5-p5 Question 8 Exercise 6.5 Solutions of Question 8 of Exercise 6.5 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.$7$$11.$\begin{align}s&=(i i, j): i, j-1,2,3,4,5,6\}\\ &=\left[\begin{array}{llllll} (1.1) & (1.2) & (1.3) & (1.4) & (1.5) & (1.6) \\ (2.1) & (2.2) & (2.3) & (2.4) & (2.5) & (2.6) \\ (3.1) & (3.2) & (3.3) & (3.4) & (3.5) & (3.6) \\ (4.1) & (4… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:57 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p1 Question 1 Review Exercise 6 Solutions of Question 1 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$O, P, Q, R, S, T, U$$2520$$9040$$5140$$4880$$2520$$\{1,2,3,4,5,6,7\}$$14$$42$$28$$21$$28$$\{1,2,3,4,6,7,8\}$$3$$7$$120$$180$$144$$96$$120$$\dfrac{(n+2) !(n-2) !}{(n+1) !(n-1) !}$$(n-3)$$(\dot{n}-1)$$\dfrac{n+1}{n+2}$$\dfrac{n+2}{n-… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:59 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p3 Question 3 & 4 Review Exercise 6 Solutions of Question 3 & 4 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${ }^{56} P_{r+6}:{ }^{54} P_{r+3}=30800: 1$$r$\begin{align} { }^{56} P_{r+6}:{ }^{54} P_r+3&=30800: 1 \\ \Rightarrow \dfrac{\dfrac{56 !}{[56-(r+6)] !}}{\dfrac{54 !}{[54-(r+3)] !}}&=\dfrac{30800}{1} \\ \Rightarrow \dfrac{56… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:36:00 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p5 Question 7 & 8 Review Exercise 6 Solutions of Question 7 & 8 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$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 … anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:36:01 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p1 Question 1, Exercise 1.2 Solutions of Question 1 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 1(i) $\operatorname{Re}(i z)=-\operatorname{Im}(z)$$$z=x+iy$$\begin{align} iz&=i(x+iy)\\ &=ix-y\end{align}\begin{align}Re(iz)&=-y\\ \implies Re(iz)&=-Im(z)\end{align}$\operatorname{Im}(i z)=\operatorname{Re}(z)$$$z=x+iy$$\begin{align}iz&=i(x+i… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Jul 2024 12:44:38 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-4-p2 Question 2, Exercise 1.4 Solutions of Question 2 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 2(i) $\left(\cos \dfrac{\pi}{6}+i \sin \dfrac{\pi}{6}\right)\left(\cos \dfrac{\pi}{3}+i \sin \dfrac{\pi}{3}\right)$$z_1=\cos \dfrac{\pi}{6}+i \sin \dfrac{\pi}{6}=e^{i\frac{\pi}{6}}$$z_2=\cos \dfrac{\pi}{3}+i \sin \dfrac{\pi}{3}=e^{i\frac{\pi}{… anonymous@undisclosed.example.com (Anonymous) Thu, 05 Sep 2024 12:24:09 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-2-p10 Question 16 and 17, Exercise 6.2 Solutions of Question 16 and 17 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. $1,2,3,4,5,6$$1, 3$$5$$=3 \times{ }^{5} P_{5}=360$$4$$1,2,3,4$$5$$1$$={ }^{4} P_{3}=24$$3$$24$$5$$24$$24+24+24=72$$4$ anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:43 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-3-p9 Question 11 and 12, Exercise 6.3 Solutions of Question 11 and 12 of Exercise 6.3 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $n$$35$$n$\begin{align*}{ }^{n} C_{2}-n&=35\\ \text{or} \quad \dfrac{n!}{2!(n-2)!}-n&=35\\ \dfrac{n(n-1)(n-2)!}{2(n-2)!}-n&=35\\ \dfrac{n(n-1)-2 n}{2}&=35\\ n^{2}-n-2 n&=70\\ n^{2}-3 n-70&=0\\ n^{2}+7 n-10 n-70 & =0 \\ n(n+7)-… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 09:00:31 +0000 https://www.mathcity.org/ahsan Dr. Muhammad Ahsan Binyamin This is a personal web page of Dr. Muhammad Ahsan Binyamin Associate Professor Government College University Faisalabad, Faisalabad - PAKISTAN. Google Scholar: <https://scholar.google.com.pk/citations?user=PJQDQy0AAAAJ&hl=en> ResearchGate Profile: <https://www.researchgate.net/profile/Muhammad_Banyamin> Compute Codimension and Milnor number anonymous@undisclosed.example.com (Anonymous) Sun, 19 Feb 2023 08:05:44 +0000 https://www.mathcity.org/cui Mathematics CUI: LaTeX Resources [Department of Mathematics, COMSATS University Islamabad, Attock Campus] This page contains LaTeX template of CIIT Mathematics, MSc Project and MS Thesis templates. Templates Download a zip file given below and extract it by right clicking on the file. BS Project Template: (Version 1.5, Uploaded: Sep 29, 2022)$\$$I$$\mathbb{R}$$f:I\to \mathbb{R}$$(\$$$\sin^2 \theta + \cos^2 \theta =1$$\begin{equation} \sin^2 \theta + \cos^2 \theta =1 \end{equation} anonymous@undisclosed.example.com (Anonymous) Fri, 02 Aug 2024 07:19:37 +0000 https://www.mathcity.org/atiq/fa21-mth322 MTH322: Real Analysis II (Fall 2021) This course is offered to MSc, Semester II at Department of Mathematics, COMSATS University Islamabad, Attock campus. This course need rigorous knowledge of continuity, differentiation, integration, sequences and series of numbers, that is many notion included in $\int_{1}^{\infty }{{{x}^{-p}} dx}$$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$… anonymous@undisclosed.example.com (Anonymous) Thu, 30 Dec 2021 19:16:17 +0000 https://www.mathcity.org/atiq/fa23-mth103 MTH103: Exploring Quantitative Skills Course Objectives This course aims to develop the basic mathematical skills which ultimately enhance problem-solving skills using inductive and deductive reasoning, Polya's strategy, and sets. The basic concepts will be develop with applications form the real world such as algebraic models with equations, rates, ratios, and percentages will be discussed. Students will also explore linear models, including rectangular coordinates, functions, empowering them… anonymous@undisclosed.example.com (Anonymous) Wed, 27 Sep 2023 13:47:12 +0000 https://www.mathcity.org/atiq/math-301 MATH-301: Complex Analysis Objectives of the course This is an introductory course in complex analysis, giving the basics of the theory along with applications, with an emphasis on applications of complex analysis and especially conformal mappings. Students should have a background in real analysis (as in the course Real Analysis I), including the ability to write a simple proof in an analysis context. $\cot 2z$ anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:18 +0000 https://www.mathcity.org/atiq/math-505 MATH-505: Complex Analysis Provisional Results MMAF13E101 = 65 MMAF13E102 = 65 MMAF13E103 = 58 MMAF13E104 = 58 MMAF13E105 = 78 MMAF13E106 = 62 MMAF13E107 = 50 MMAF13E108 = 75 MMAF13E109 = 61 MMAF13E110 = 50 MMAF13E111 = 50 MMAF13E112 = 85 $\cot 2z$ anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:23 +0000 https://www.mathcity.org/atiq/sp19-mth633 MTH633: Advanced Convex Analysis (Spring 2019) Convex sets, convex hull, their properties, separation theorems, hyperplane, Best approximation theorem and its applications, Farkas and Gordan Theorems, Extreme points and Polyhedral. Convex functions, Basic Definitions, properties, various generalizations, differentiable convex functions, subgradient, characterization and applications in linear and nonlinear optimization, complementarity problems and its equivalent formulations.$\mathbb{R}$$\math… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:40 +0000 https://www.mathcity.org/atiq/sp20-mth604 ~~DISCUSSION~~ MTH604: Fixed Point Theory and Applications (Spring 2020) Course Objectives: This course is intended as a brief introduction to the subject with a focus on Banach Fixed Point theorems fixed point theorem and its application to nonlinear differential equations, nonlinear integral equations, real and complex implicit functions theorems and system of nonlinear equations. Some generalizations and similar results e. g. Kannan Fixed Point theorems, Banach Fixed Point theorem for mul… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:40:41 +0000 https://www.mathcity.org/conferences/14th-rammma-2019-lahore Recent Advances in Mathematical Methods, Models & Applications, LSC Lahore, Pakistan (April 13-14, 2019) [RAMMMA LSE Lahore] New mathematical ideas may help in improving modeling of real problems, in deriving innovative and efficient numerical methods, and in developing approximate models which are amenable to mathematical analysis. There exists a non-trivial interplay between mathematics, mathematical modeling of complex systems and mathematical and computer methods oriented towards the quali… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:41:39 +0000 https://www.mathcity.org/events/mathematics-olympiad-2019-sukkur-iba Mathematics Olympiad 2019 Sukkur IBA (11-13 November 2019) [Mathematics Olympiad 2019] Mathematical Olympiad is a contest of mathematics among the students. It is a very healthy activity to promote and learn mathematics. Contents for the test are as follows: * Qudratic equations and expressions anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:42:33 +0000 https://www.mathcity.org/fsc-part1-ptb/important-questions Important Questions: HSSC-I [Important Questions FSc/ICS Part 1] These are the important questions for “Textbook of Algebra and Trigonometry Class XI” published by Punjab Textbook Board (PTB) Lahore, Pakistan. These questions are taken from old papers. These are very helpful to understand the types of questions which may asked final paper of mathematics for FSc/ICS (HSSC) Part 1. Lot of energy has been put to collect and write these questions. These are taken from old papers of FBISE Islamabad,… anonymous@undisclosed.example.com (Anonymous) Thu, 18 Apr 2024 08:01:02 +0000 https://www.mathcity.org/fsc-part1-ptb/short-term-preparation-salman-sherazi Short Term Preparation FSc/ICS 1 fsc fsc_part1 m_salman_sherazi important_questions_fsc_1 [Short Term Preparation by M Salman Sherazi] This document contains all the important MCQs, Short Questions and Long Questions of Mathematics HSSC-I (FSc/ICS Part 1) from the Textbook of Algebra and Trigonometry for Class XI. It has been done to help the students and teachers at no cost by $\sqrt{2}$ anonymous@undisclosed.example.com (Anonymous) Thu, 02 May 2024 17:12:17 +0000 https://www.mathcity.org/fsc-part2-ptb/definitions-muzzammil-subhan 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)}$… anonymous@undisclosed.example.com (Anonymous) Sun, 20 Oct 2024 18:16:28 +0000 https://www.mathcity.org/fsc/kpk_fsc_part_1 FSc Part 1 (KPK Boards) These are the notes of old book. The notes of new book is AVAILABLE HERE [FSc Part 2 KPTP] Notes of FSc Part 1 of “A Textbook of Mathematics For Class XI” published by Khyber Pakhtunkhwa Textbook Board, Peshawar. We are posting the notes chapter-wise. These notes are shared as open educational resources. This page will be continuously updated.$P(z)$$(\sum)$$\sum n$$\sum n^2$$\sum n^3$$n$$n$$$\frac{a}{a(a+d)}+\frac{a}{(a+d)(a+2d)}+...$$$^nP_r$$^nC_r=\left(\begin{smallm… anonymous@undisclosed.example.com (Anonymous) Tue, 12 Dec 2023 17:30:33 +0000 https://www.mathcity.org/matric/9th_science Mathematics 9 (Science Group) [Mathematics 9 (Science Group)] Mathematics 9 is written by Dr. Karamat H. Dar and Prof. Irfan-ul-Haq and this book is published by Carvan Book House, Lahore, Pakistan. This book consist of 302 pages and there are 17 units. Notes of Unit 1 and 3 are provided by $ka + kb + kc$$ac + ad + bc + bd$$a^2 + 2ab + b^2$$a^2 – b^2$$a^2 + 2ab + b^2 – c^2$$a^4 + a^2b^2 + b^4$$a^4 + 4b^4$$x^2 + px + q$$ax^2 + bx + c$$(ax^2 + bx + c) (ax2 + bx + d) + k$$(x + a) (x + b) (x + c) … anonymous@undisclosed.example.com (Anonymous) Wed, 08 Mar 2023 18:04:36 +0000 https://www.mathcity.org/notes/complex-analysis-iqra-liaqat Complex Analysis (Notes) by Ms. Iqra Liaqat [Notes of Complex Analysis by Ms. Iqra Liaqat] These notes are send by Ms. Iqra Liaqat. We are really very thankful to her for providing these notes and appreciates her effort to publish these notes on MathCity.org Complex analysis is the study of functions of complex numbers. It is useful in a variety of mathematical fields, such as algebraic geometry, number theory, analytic combinatorics, and applied mathematics, as well as in physics fields like… anonymous@undisclosed.example.com (Anonymous) Sun, 06 Aug 2023 09:33:02 +0000 https://www.mathcity.org/notes/complex-analysis-m-usman-hamid Complex Analysis by M Usman Hamid [Complex Analysis by M Usman Hamid] We are really very thankful to Muhammad Usman Hamid for providing these notes and appreciates his effort to publish these notes on MathCity.org It covers the one part of the syllabus of Complex Analysis paper of MSc Mathematics. See the contents of the notes given below to see the topics covered by these notes.$$ x^2+4=0, x^2+x+1=0 \text{ and } x^2-2x+3=0 $$$i=\sqrt{-1}$$i^2=-1$ anonymous@undisclosed.example.com (Anonymous) Sat, 08 Feb 2025 08:11:42 +0000 https://www.mathcity.org/notes/computing-tools-for-mathematics-asif-arshad 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. anonymous@undisclosed.example.com (Anonymous) Sat, 05 Aug 2023 20:30:34 +0000 https://www.mathcity.org/notes/general-topology-azhar-hussain 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… anonymous@undisclosed.example.com (Anonymous) Sun, 06 Aug 2023 19:18:17 +0000 https://www.mathcity.org/notes/handwritten-notes-real-analysis-asim-marwat 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… anonymous@undisclosed.example.com (Anonymous) Sat, 15 Apr 2023 17:26:13 +0000 https://www.mathcity.org/notes/measure-theory-by-anwar-khan 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$… anonymous@undisclosed.example.com (Anonymous) Mon, 01 May 2023 13:54:40 +0000 https://www.mathcity.org/notes/mechanics-iii-analytic-dynamics-ii-dr-babar-ahmad Mechanics III (Analytic Dynamics II) by Dr Babar Ahmad [Mechanics III (Analytic Dynamics II) by Dr Babar Ahmad] We are very thankful to Dr Babar Ahmad for sharing his book on MathCity.org. This book is very helpful for undergraduate students of Science and Engineering Programs. This book is shared by the permission of the author and he keeps the copyright of the book. anonymous@undisclosed.example.com (Anonymous) Sat, 10 May 2025 17:33:01 +0000 https://www.mathcity.org/notes/multiple-choice-questions-bsc-bs-ppsc-akhtar-abbas Multiple Choice Questions (BSc/BS/PPSC) by Akhtar Abbas [Multiple Choice Questions (BSc/BS/PPSC)] These notes are made and shared by Mr. Akhtar Abbas. We are really very thankful to him for providing these notes and appreciates his efforts to publish these notes on MathCity.org. Multiple Choice Questions (MCQs) are given in these notes, which might be helpful in BSc, BS or Punjab Public Service Commission (PPSC) exams.$a$$b$$n$$na > b$$(p − 1)! \equiv −1(mod p)$$p$$p$$p$ anonymous@undisclosed.example.com (Anonymous) Sun, 24 May 2026 17:45:10 +0000 https://www.mathcity.org/notes/note-for-numerical-methods-m-usman-hamid Notes for Numerical Methods by M Usman Hamid [Notes for Numerical Methods by M Usman Hamid] These notes are initially provided by Mr. Anwar Khan. Later the updated version is send by Muhammad Tahir. We are really very thankful to Mr. Anwar Khan and Muhammad Tahir for providing these notes and appreciates their effort to publish these notes on MathCity.org$\left(\frac{1}{3}\right)$$\left(\frac{3}{8}\right)$ anonymous@undisclosed.example.com (Anonymous) Sat, 05 Aug 2023 17:59:07 +0000 https://www.mathcity.org/notes/qr2-tools-for-quantitative-reasoning-m-usman-hamid Quantitative Reasoning II (QR2: Tools for Reasoning Skills) [Quantitative Reasoning II (Tools for Reasoning Skills)] This handout provides a comprehensive exploration of fundamental mathematical and logical concepts, making it an essential read for students and professionals alike. It begins with an introduction to enumeration and its practical applications, followed by an in-depth discussion on quantitative reasoning, number systems, and arithmetic operations. The book highlights the contribut… anonymous@undisclosed.example.com (Anonymous) Fri, 01 Aug 2025 18:31:56 +0000 https://www.mathcity.org/notes/topology-handwritten-notes 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-$ anonymous@undisclosed.example.com (Anonymous) Sat, 01 Mar 2025 09:43:45 +0000 https://www.mathcity.org/people/ammar-mujahid Ammar Mujahid [Ammar Mujahid] Ammar Mujahid holds a Master’s degree in Mathematics (MS Mathematics) and is a dedicated Mathematics Instructor with extensive experience in teaching F.Sc (Intermediate) and Undergraduate level Mathematics. He specializes in delivering concept-oriented lectures in subjects such as Calculus, Algebra, and other core areas of Pure Mathematics. anonymous@undisclosed.example.com (Anonymous) Sat, 28 Feb 2026 17:24:55 +0000 https://www.mathcity.org/people/muzammil 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. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:44:45 +0000 https://www.mathcity.org/people/umer Umer Asghar We are very thankful to Mr. Umer Asghar for his contribution to the website. * Email: <umermth2016@gmail.com> * Skype ID: sp15mmth06678 * Cell: +92-307-4896454 Contribution: * Notes of Metric Spaces by Umer Asghar NEW * Notes of Number Theory by Umer Asghar * Exercise 9.2 (BSc Mathematical Method) | VIEW View online | [Download PDF] * anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:44:50 +0000 https://www.mathcity.org/quote-of-the-day/apr Quotes for the April کاسمولوجسٹ اکثر غلط ہوتے ہیں، لیکن کبھی شک میں نہیں۔۔۔ لیو لینڈاؤ (1908-1968) Cosmologists are often wrong, but never in doubt. --- Lev Landau (1908-1968) (Courtesy: MacTutor) anonymous@undisclosed.example.com (Anonymous) Fri, 28 Apr 2023 18:32:46 +0000 https://www.mathcity.org/quote-of-the-day/feb 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)] anonymous@undisclosed.example.com (Anonymous) Wed, 22 Feb 2023 16:02:58 +0000 https://www.mathcity.org/quote-of-the-day/may Quotes for the May مختصراً، پوری دنیا خلا اور وقت میں اشیا کی ریاضیاتی طور پر ظاہر کی جانے والی حرکات کا مجموعہ ہے، اور پوری کائنات ایک عظیم، ہم آہنگ اور ریاضیاتی طور پر تیار کی گئی مشین ہے۔۔۔ anonymous@undisclosed.example.com (Anonymous) Fri, 28 Apr 2023 18:45:11 +0000 https://www.mathcity.org/wiki/dokuwiki DokuWiki wiki:dokuwiki DokuWiki is a simple to use and highly versatile Open Source wiki software that doesn't require a database. It is loved by users for its clean and readable Formatting Syntax. The ease of maintenance, backup and integration makes it an administrator's favorite. Built in anonymous@undisclosed.example.com (Anonymous) Sat, 28 Jun 2025 07:28:13 +0000 https://www.mathcity.org/bsc/notes_of_mathematical_method/ch01_complex_numbers 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}$ anonymous@undisclosed.example.com (Anonymous) Mon, 11 Dec 2023 12:59:57 +0000 https://www.mathcity.org/fsc-part2-ptb/important-questions/unit-06-conic-section Unit 06: Conic section Here is the list of important questions. * Find the centre and radius of the circle given by the equation $4x^2+4y^2-8x+12y-25=0$ --- BSIC Gujranwala (2016) * Find equation of tangent to the circle $x^2+y^2=2$ parallel to the line $x-2y+1=0$ --- BSIC Gujranwala (2016)$x^2=-16y$$(0,\pm5)$$\frac{3}{5}$$ABC$$a^2=b^2+c^2-2bc \cos A$$A(4,5)$$B(-4,-3)$$C(8,-3)$$9x^2-18x+4y^2+8y-23=0$$x^2+y^2-6x+4y+13=0$$x^2+y^2=25$$(4,3)$$(-3,1)$$x=3$$(0,0)$$(6,0)$$(4,0)$$x^2-4x-8y+4=… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:57 +0000 https://www.mathcity.org/fsc/fsc_part_2_solutions/ch01 Unit 01: Functions and Limits [Unit 01: Functions and Limits] Notes (Solutions) of Unit 01: Functions and Limits, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. There are five exercises in this chapter. You can view online or download PDF. To view PDF, you must have PDF Reader installed on your system and it can be downloaded from $\lim_{x\to a}\frac{x^n-a^n}{x-a} = na^{n-1}$$\lim_{x\to0}\frac{\sqrt{x+a} - \sqrt{a}}{x} = \frac{… anonymous@undisclosed.example.com (Anonymous) Sat, 03 Jun 2023 16:30:56 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06 Unit 06: Permutation, Combination and Probability (Solutions) This is a sixth unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the students will be able to$n$$n!$$n$$r$$^nP_r$$n$$r$$n$$r$ anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:34:51 +0000 https://www.mathcity.org/math-11-kpk/sol/unit07 Unit 07: Mathmatical Induction and Binomial Theorem (Solutions) This is a seventh unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the students will be able to$(x+y)^n$$n$$(x+y)^n$$(x+ y)^n.$$(1 +x)^n$$n$$n.$$(l +x)^n$$x$$|x| < 1$$n$ anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:34:59 +0000 https://www.mathcity.org/matric/9th_science/unit08 Unit 08: Linear Graph and their Application On this page notes of Unit 08 of Mathematics 9 written by Dr. Karamat H. Dar and Prof. Irfan-ul-Haq are given. [Unit 08: Linear Graph and their Application] After studying this unit the students will be able to: * Identity pair of real numbers as an ordered pair.$O$$\left( O \right)$$\left( a,b \right)$$a\,$$b$$y=c.$$x=a.$$y=mx.$$y=mx+c.$ anonymous@undisclosed.example.com (Anonymous) Sat, 28 May 2022 19:29:21 +0000 https://www.mathcity.org/msc/syllabus/pu 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. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:50:16 +0000 https://www.mathcity.org/bsc/paper_pattern/sargodha_university/a-course_of_mathematics A-Course of Mathematics (Paper A & B) This subject is consists of two papers of 100 marks each. One is called “Paper A” and other is called “Paper B”. This page is updated on February 15, 2015. This syllabus is for 1st Annual 2015 and onward organized by University of Sargodha, Sargodha. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:55:54 +0000 https://www.mathcity.org/bsc/paper_pattern/sargodha_university/general_mathematics General Mathematics (Paper A & B) This subject is consists of two papers of 100 marks each. One is called “Paper A” and other is called “Paper B”. This syllabus is for 1st Annual 2015 and onward organized by University of Sargodha (UoS), Sargodha. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:55:55 +0000 https://www.mathcity.org/bsc/paper_pattern/sargodha_university/pure_mathematics Pure Mathematics (Paper A & B) This paper consist of two papers of 100 marks each. One paper is called “Paper A” and the other is called “Paper B”. Paper A * NOTE: attempt two questions from each section. SECTION-I (4/12: 17,17,17,17) anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:55:59 +0000 https://www.mathcity.org/bsc/paper_pattern/sargodha_university/pure_mathematics_chapterwise Pure Mathematics Paper pattern for Pure Mathematics chapter-wise for University of Sargodha is given on this page. This pattern is extracted from syllabus, so use your own risk. Syllabus of Pure Mathematics can be seen here. Pure Mathematics is consists of two papers of 100 marks each. One is called “Paper A” and other is called “Paper B”. In every paper there are three sections with four questions each. A student have to attempt two questions from each section. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:56:03 +0000 https://www.mathcity.org/math-11-kpk/sol/unit03/ex3-2-p3 Question 3 & 4, Exercise 3.2 Solutions of Question 3 & 4 of Exercise 3.2 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 3 If $\vec{r}=\hat{i}-9\hat{j}$$\vec{a}=\hat{i}+2\hat{j}$$\vec{b}=5\hat{i}-\hat{j}$$p$$q$$\vec{r}=p\vec{a}+q\vec{b}$$$\vec{r}=p\vec{a}+q\vec{b}.$$$\vec{r},\vec{a}$$\vec{b}$$$\hat{i}-9\hat{j}=p(\hat{i}+2\hat{j})+q(5\hat{i}-\hat{j})$$$$\implies \hat{i}-9\… anonymous@undisclosed.example.com (Anonymous) Wed, 03 Jan 2024 18:10:31 +0000 https://www.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p2 Question 3 and 4 Exercise 4.2 Solutions of Question 3 and 4 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$6,9,12, \ldots, 78$$a_1=6$$d=9-6=3$$a_n=78$$$a_n=a_1+(n-1) d$$\begin{align}&78=6+(n-1) 3 \\ \implies &3(n-1)=78-6 \\ \implies &n-1=\dfrac{72}{3} \\ \implies &n=24+1=25.\end{align}$25$$n$$a_n=2n+7$$$a_n=2 n+7. --- (1)$$\begin{align}a_{n+1}=2(n+1)+7=2… anonymous@undisclosed.example.com (Anonymous) Sun, 04 Feb 2024 03:20:11 +0000 https://www.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p3 Question 5 and 6 Exercise 4.2 Solutions of Question 5 and 6 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$\log a, \log (a b), \log \left(a b^2\right), \log \left(a b^3\right), \ldots$$$n$$\log$$a$$b$$b$$$a_n=\log (a b^{n-1}).$$\begin{align}a_n&=\log(a b^{n-1}). \end{align}\begin{align} d&=a_{n+1}-a_n \\ &=\log (a b^n)-\log (a b^{n-1}) \\ &=\log \left(\… anonymous@undisclosed.example.com (Anonymous) Sat, 10 Feb 2024 16:55:28 +0000 https://www.mathcity.org/math-11-kpk/sol/unit04/ex4-2-p8 Question 11 Exercise 4.2 Solutions of Question 11 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 11 $a_1$$$a_1=1000.$$$= d=100$$a_n=5400$$n$\begin{align} &a_n=a_1+(n-1)d \\ \implies &5400=1000+(n-1)100\\ \implies &5400=900+100n \\ \implies &100n=5400-900\\ \implies &100n=4500\\ \implies &n=45.\end{align} anonymous@undisclosed.example.com (Anonymous) Sat, 10 Feb 2024 18:43:43 +0000 https://www.mathcity.org/math-11-kpk/sol/unit04/ex4-3-p6 Question 9 & 10 Exercise 4.3 Solutions of Question 9 & 10 of Exercise 4.3 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 9 $$306,315,324,333, \ldots, 693$$$a=306$$$d=(315-306) = 9 \text { and } a_n=693 .$$$n$\begin{align}a_n&=a_1+(n-1) d \text { becomes } \\ \Rightarrow a_1+(n-1) d&=693 \\ \Rightarrow 306+(n-1) \cdot 9&=693 \\ \Rightarrow 9 n&=396 \\ \Rightarr… anonymous@undisclosed.example.com (Anonymous) Fri, 05 Jan 2024 17:29:57 +0000 https://www.mathcity.org/math-11-kpk/sol/unit04/ex4-4-p3 Question 4 & 5 Exercise 4.4 Solutions of Question 4 & 5 of Exercise 4.4 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 4 $\dfrac{1}{64}$$r=\dfrac{1}{2}$$a_1=16$$a_n=\dfrac{1}{64}$$r=\dfrac{1}{2}$$n$$$a_n=a_1 r^{n-1} \quad \text{then}$$\begin{align}\dfrac{1}{64}&=16(\dfrac{1}{2})^{n-1} \\ \Rightarrow(\dfrac{1}{2})^{n-1}&=\dfrac{1}{64 \times 16}=\dfrac{1}{1024} … anonymous@undisclosed.example.com (Anonymous) Fri, 05 Jan 2024 17:30:00 +0000 https://www.mathcity.org/math-11-kpk/sol/unit04/ex4-5-p6 Question 7 & 8 Exercise 4.5 Solutions of Question 7 & 8 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Question 7 $\operatorname{sum} S_n$$n$$\{(\dfrac{1}{2})^n\}$$$\{(\dfrac{1}{2})^n\}=\dfrac{1}{2}, \dfrac{1}{2^2}, \dfrac{1}{2^3}, \ldots$$$$a_1=\dfrac{1}{2}$$$$r=\dfrac{\dfrac{1}{2^2}}{\dfrac{1}{2}}=\dfrac{1}{2}$$\begin{align}S_n&=\dfrac{a_1(1-r^n)}{1-r… anonymous@undisclosed.example.com (Anonymous) Fri, 05 Jan 2024 17:30:12 +0000 https://www.mathcity.org/math-11-kpk/sol/unit05/ex5-1-p2 Question 2 & 3 Exercise 5.1 Solutions of Question 2 & 3 of Exercise 5.1 of Unit 05: Miscullaneous Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan. Q2 Find the sum $1.2+2.3+3.4+\ldots+99.100$$1+2+3+\ldots+99$$2+3+4+\ldots+100$$n^{\text {th }}$$n(n+1)$$n^{\text {th }}$$\quad T_j=j(j+1)=j^2+j$$j=1$$j=99$$$ \begin{aligned} & \sum_{j=1}^{99} \tau_j=\sum_{j=1}^{99} j^2+\sum_{j=1}^{99} j \\ & =\frac{99… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:35:10 +0000 https://www.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p7 Question 11 Review Exercise 6 Solutions of Question 11 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$n(S)=4$$$\dfrac{1}{4}$$$\quad P( orange )=\dfrac{1}{4}$$$\dfrac{1}{4}$$\dfrac{1}{4}$\begin{align}P(\operatorname{Red})&=\dfrac{1}{4}\\ P( Green )&=\dfrac{1}{4}\end{align}$P(R \cap G)=\phi$$R$$G$\begin{align}\boldsymbol{P}( Red o… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:36:03 +0000 https://www.mathcity.org/math-11-kpk/sol/unit07/re-ex7-p2 Question 2 Review Exercise 7 Solutions of Question 2 of Review Exercise 7 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\left(2 x^3+3 y\right)^8$$a=2 x^3$$b=3 y$$n=8$$n=8$$\frac{8+2}{2}=5$$$ \begin{aligned} & T_5=\frac{8 !}{(8-4) ! 4 !}\left(2 x^3\right)^{8-4}(3 y)^4 \\ & T_5=70.2^4 \cdot 3^4 \cdot x^{12} \cdot y^4 \\ & =90720 x^{12} y^4 \end{aligne… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:36:40 +0000 https://www.mathcity.org/math-11-kpk/sol/unit07/re-ex7-p5 Question 7 & 8 Review Exercise 7 Solutions of Question 7 & 8 of Review Exercise 7 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$7^n-3^n$$n=1$$7^k-3^k=7-4=4$$n=1$$n=k>1$$7^n-3^n=4 Q$$Q$$n=k+1$$$ \begin{aligned} & 7^{k+1}-3^{k+1}=7.7^k-3.3^k \\ & =(4+3) \cdot 7^k-3.3^k \\ & =4.7^k+3.7^k-3.3^k \end{aligned} $$$$ \begin{aligned} & =4.7^k+3\left[7^k-3^k\… anonymous@undisclosed.example.com (Anonymous) Thu, 01 Feb 2024 02:36:41 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-1-p5 Question 5, Exercise 1.1 Solutions of Question 5 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. Question 5 $z$$4z-3\bar{z}=\dfrac{1-18i}{2-i}$$z=x+iy$$\bar{z}=x-iy$\begin{align}&4z-3\bar{z}=\dfrac{1-18i}{2-i}\\ \implies &4(x+iy)-3(x-iy)=\dfrac{1-18i}{2-i}\times \dfrac{2+i}{2+i}\\ \implies &4x+4iy-3x+3iy=\dfrac{(1-18i)(2+i)}{2^2-i^2} \end{align}\b… anonymous@undisclosed.example.com (Anonymous) Tue, 02 Jul 2024 16:57:50 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-2 Exercise 1.2 (Solutions) The solutions of the Exercise 1.2 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to real and imaginary part of complex numbers, modulus and conjugate of the complex numbers.$\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… anonymous@undisclosed.example.com (Anonymous) Mon, 27 Oct 2025 18:50:09 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/ex1-2-p2 Question 2, Exercise 1.2 Solutions of Question 2 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 2 $$ \left(z_{1} z_{2}\right)\left(z_{3} z_{4}\right)=\left(z_{1} z_{3}\right)\left(z_{2} z_{4}\right)=z_{3}\left(z_{1} z_{2}\right) z_{4} $$\begin{align} &(z_1 z_2)(z_3 z_4) \\ =&(z_1 z_2)z_5 \quad \text {Let }z_5=z_3 z_4 \\ =&z_1 (z_2 z_5) \… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Jul 2024 12:45:03 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/re-ex-p4 Question 4, Review Exercise Solutions of Question 4 of Review Exercise 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. $z=x+i y$$\left|\dfrac{z+2 i}{z-2 i}\right|=1$$z = x + iy$\begin{align*} & \left|\dfrac{z + 2i}{z - 2i}\right| = 1\\ \implies & |z + 2i| = |z - 2i|\\ \implies & |x + i(y + 2)| = |x + i(y - 2)|\\ \implies & \sqrt{x^2 + (y + 2)^2} = \sqrt{x^2 + (y -… anonymous@undisclosed.example.com (Anonymous) Wed, 17 Jul 2024 12:54:02 +0000 https://www.mathcity.org/math-11-nbf/sol/unit01/rev-ex Review Exercise 1 (Solutions) The solutions of the Review Exercise 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 polar form of the complex numbers. $i^{2}+i^{4}+i^{6}+\cdots+i^{100}$$\left|\dfrac{(3-2 i)(1+i)}{2-3 i}\right|$$|\overline{(3-2 i)(4-i)}|$$\left(\dfrac{3+5 i}{2-3 i}\right)^{-1}$$3 x^{2}+108$$4 x^{2}+40$$z=x+i y$… anonymous@undisclosed.example.com (Anonymous) Mon, 27 Oct 2025 18:52:39 +0000 https://www.mathcity.org/math-11-nbf/sol/unit04/ex4-5-p3 Question 5 and 6, Exercise 4.5 Solutions of Question 5 and 6 of Exercise 4.5 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}=7, r=2, n=14$$a_1 = 7$$r = 2$$n = 14$$n$$$S_n = \frac{a_1 \left(1 - r^n\right)}{1 - r}, \quad r \neq 1.$$\begin{align*} S_{14} &= \frac{7 \left(1 - 2^{14}\right)}{1 - 2} \\ &= \frac{7 \left(1 - 16384\right)}{-1} \\ &= \frac{7 \time… anonymous@undisclosed.example.com (Anonymous) Sun, 22 Sep 2024 18:26:10 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-1 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!}$$\dfrac… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:40 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-2-p5 Question 6 and 7, Exercise 6.2 Solutions of Question 6 and 7 of Exercise 6.2 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $4$$$1,2,3,4,5,6$$6$$6$$6$$6$$$\text{Total possibilities }6 \times 6 \times 6 \times 6=6^4=1296$$$1,1,2,2,3,3,4$$=7$$2^{\text {nd }}$$6^{\text {th }}$$1^{\text {st }}, 3^{\text {rd }}, 5^{\text {th }}$$7^{\text {th }}$$2,2,4$$1,1,… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 08:59:54 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/ex6-3-p6 Question 5 and 6, Exercise 6.3 Solutions of Question 5 and 6 of Exercise 6.3 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $11$$16$$11$$16$${ }^{16} C_{11}$$4368$$11$$11$$16$$10$$15$$$ { }^{15} C_{10}=3003 $$$5$$3$$3$$1$${ }^{5} C_{1}$$2$${ }^{3} C_{2}$$={ }^{5} C_{1} \times{ }^{3} C_{2}=5 \times 3=15$$2$$={ }^{5} C_{2} \times{ }^{3} C_{1}=10 \times 3… anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 09:00:29 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/re-ex6-p2 Question 2 and 3, Review Exercise 6 Solutions of Question 2 and 3 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. $4$$26$$$^{26}P_4=358800$$$3$$0$$3$$100's$$9$$3$$$=10\times 10\times10=1000$$$0$$10$$3-$$0$$$=10\times 10=100$$ anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 09:00:43 +0000 https://www.mathcity.org/math-11-nbf/sol/unit06/rev-ex Review Exercise (Solutions) The solutions of the Review Exercise of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. Question 1.$4$$3$$0$$6$$0,2,3,4,5,7$$7$ anonymous@undisclosed.example.com (Anonymous) Sat, 08 Mar 2025 09:00:41 +0000 https://www.mathcity.org/matric/9th_science/unit_02/exercise_2.2 Exercise 2.2 (Solutions) Question 1 Identify the property used in the following, * (i) $a + b = b + a$ ... ..... * (ii) $(ab)c = a(bc)$ ... ... ... * (iii) $7 \times 1 = 7$ ... ... ... * (iv) $x > y$ or $x = y$ or $x< y$ ... ... ... * (v) $ab = ba$ ... ... ... * (vi) $a + c = b + c \Rightarrow a = b$ ... ... ... * (vii) $5 + (-5) = 0$ ... ... ... * (viii) $7 \times \frac{1}{7} = 1$$a > b \Rightarrow ac > bc? (c >0)$$a + b = b + a$$(ab)c = a(bc)$$7 \times 1 = 7$$x > y$$x = y$$… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 17:00:36 +0000