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Definitions: FSc Part 1 (Mathematics): PTB by Aurang Zaib @fsc-part1-ptb
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alent if there exists a one-to-one correspondence between their elements. ===Example:=== \( A = \{2, 4, 6,... thers. ====Function==== A function is a relation between two non-empty sets \( A \) and \( B \), where eac
Definitions: FSc Part 1 (Mathematics): PTB @fsc-part1-ptb
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t if one to one correspondence can be established between them.\\ e.g. $A=\{2,4,6,8\}$, $B=\{a,b,c,d\}$ ... etic Mean:** A number $A$ is said to be the $A.M$ between the two numbers $a$ and $b$. If $a,A,b$ are in $A... ic Mean:** A number is said to be geometric means between two numbers $a$ and $b$. If $a,G,b$ are in $G.P$.... mber $H$ is said to be the harmonic means ($H.M$) between two numbers $a$ and $b$, if $a, H, b$ are in $H.P
Mathematics CUI: LaTeX Resources
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$ $ (dollar) sign to write equation or symbols in between statements or sentences, e.g.\\ Let $I$ be an int
MathCraft: PDF to LaTeX file: Sample-02 @mathcraft
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eq f(x) \leq s(x)$$ Integrating both inequalities between $a$ and $b$ \begin{equation*} \int_{a}^{b} r(x)
MTH480: Introductory Quantum Mechanics @atiq
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underlying physical principles. The relationship between classical and quantum mechanics is explored to il
Question 14 Exercise 4.2 @math-11-kpk:sol:unit04
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Question 14(i)===== Insert three arithmetic means between 6 and 41. GOOD ====Solution==== Let $A_1, A_2, A_3$ be three arithmetic means between 6 and 41. Then $6, A_1, A_2, A_3, 41$ are in A.P.... ac{1}{4}.\end{align} Hence three arithmetic means between 6 and 41 are $$14\dfrac{3}{4},23\dfrac{1}{2},32\d... Question 14(ii)===== Insert four arithmetic means between 17 and 32. GOOD ====Solution==== Let $A_1, A_2, A
Question 17 Exercise 4.2 @math-11-kpk:sol:unit04
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==Question 17===== There are $n$ arithmetic means between 5 and 32 such that the ratio of the 3rd and 7th m... 1, A_2, A_3, \ldots, A_n$ be $n$ arithmetic means between 5 and 32. Then $5, A_1, A_2, A_3, \ldots, A_n, 32
Question 16 Exercise 4.2 @math-11-kpk:sol:unit04
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====Question 16===== Insert five arithmetic means between $5$ and $8$ and show that their sum is five times the arithmetic mean between $5$ and $8$. GOOD ====Solution==== Let $A_1, A_2, A_3, A_4, A_5$ be five arithmetic means between $5$ and $8$. Then $5, A_1, A_2, A_3, A_4, A_5, 8$... },6,\dfrac{13}{2},7, \dfrac{15}{2}$ are five A.Ms between $5$ & $8$. Now \begin{align}A_1&+A_2+A_3+A_4+A_5
Question 15 Exercise 4.2 @math-11-kpk:sol:unit04
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a^{n+1}+b^{n+1}}{a^n+b^n}$ is the arithmetic mean between $a$ and $b$. Where $a$ and $b$ are not zero simul... on==== Suppose $A$ represents the arithmetic mean between $a$ and $b$, then $$ A=\dfrac{a+b}{2}. --- (1) $$
Question 12 & 13 Exercise 4.2 @math-11-kpk:sol:unit04
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=====Question 13(i)===== Find the arithmetic mean between $12$ and $18$. GOOD ====Solution==== Here $a=12, ... }\\&=\dfrac{30}{2}=15.\end{align} Hence 15 is A.M between 12 and 18. GOOD =====Question 13(ii)===== Find the arithmetic mean between $\dfrac{1}{3}$ and $\dfrac{1}{4}$. ====Solution==... ===Question 13(iii)===== Find the arithmetic mean between $-6,-216$. GOOD ====Solution==== Here $a=-6, b=-2
Question 5 and 6 Exercise 6.3 @math-11-kpk:sol:unit06
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total number lines. We know one line can be drawn between each two points, so total number of lines are: $$
Question 11 Exercise 6.2 @math-11-kpk:sol:unit06
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=====Question 11===== How many numbers each lying between $10$ and $1000$ can be formed with digits $2.3,4,
Question 14 and 15 Exercise 6.2 @math-11-kpk:sol:unit06
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!$ And the two particular people can be arranged between them selves in $2 !=2$ ways. Hence, number of wa
Mathematics 10 (Science Group) @matric
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angle in circular system) and prove the relation between radians and degree. * establish the rule $l=r\t
Unit 04: Sequence and Series (Solutions) @math-11-kpk:sol
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ing arithmetic sequence. * Know arithmetic mean between two * Insert n arithmetic means tEtween two num... series. * Show that sum of $n$ arithmetic means between two numbers is equal to n times their arithmetic ... lving geometric sequence. * Know geometric mean between two numbers. * Insert $n$ geometric means between two numbers. * Define a geometric series. * Find t
Question 12 Exercise 4.4 @math-11-kpk:sol:unit04
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Question 11 Exercise 4.4 @math-11-kpk:sol:unit04
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Question 10 Exercise 4.4 @math-11-kpk:sol:unit04
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Question 9 Exercise 4.4 @math-11-kpk:sol:unit04
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Question 13 & 14 Exercise 4.3 @math-11-kpk:sol:unit04
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Question 9 & 10 Exercise 4.3 @math-11-kpk:sol:unit04
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Unit 03: Vectors (Solutions) @math-11-kpk:sol
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Question 6 & 7 Review Exercise 3 @math-11-kpk:sol:unit03
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Question 1 Review Exercise 3 @math-11-kpk:sol:unit03
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Question 7 & 8 Exercise 3.3 @math-11-kpk:sol:unit03
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Question 2 and 3 Exercise 3.3 @math-11-kpk:sol:unit03
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FSc Part 1 (KPK Boards) @fsc
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Fluid Mechanics by Ali Raza @notes
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Question 3, Exercise 10.1 @math-11-kpk:sol:unit10
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MTH480: Introductory Quantum Mechanics @atiq
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Fluid Mechanics II by Dr Rao Muzamal Hussain @notes
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Question 3, Exercise 10.1 @fsc-part1-kpk:sol:unit10
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Metric Spaces (Notes) @notes
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Number Theory Notes by Anwar Khan @notes
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Number Theory by Dr Muhammad Umer Shuaib @notes
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Operation Research: Handwritten Notes @notes
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Affine and Euclidean Geometry by Shahzad Idress @notes
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MTH322: Real Analysis II (Spring 2023) @atiq
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MTH321: Real Analysis I (Spring 2023) @atiq
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Chapter 04: Quadratic Equations @fsc:fsc_part_1_solutions
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Unit 03: Integration @fsc:fsc_part_2_solutions
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MATH-300: Basic Mathematics for Chemist @atiq
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MTH321: Real Analysis I (Fall 2022) @atiq
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Quotes for the April @quote-of-the-day
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Quotes for the March @quote-of-the-day
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Functional Analysis by Mr. Tahir Hussain Jaffery @notes
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Functional Analysis by Prof Mumtaz Ahmad @notes
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Advanced Analysis: Handwritten Notes @notes
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Mathematics 9 (Science Group) @matric
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Quotes for the February @quote-of-the-day
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Measure Theory Handwritten Notes by Asim Marwat @notes
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MTH321: Real Analysis I (Fall 2021) @atiq
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MTH211: Discrete Mathematics (Spring 2022) @atiq
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Groups (Handwritten Notes) @notes
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Group Theory by Mr. Muhammad Iftikhar @notes
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MTH322: Real Analysis II (Spring 2022) @atiq
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MTH322: Real Analysis II (Fall 2021) @atiq
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MTH211: Discrete Mathematics (Fall 2020) @atiq
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Number Theory by Ms. Iqra Liaqat @msc:notes
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Unit 07: Vectors @fsc:fsc_part_2_solutions
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Exercise 2.1 (Solutions) @matric:9th_science:unit_02
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Unit 07: Vectors @fsc-part2-ptb:important-questions
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Unit 03: Integration @fsc-part2-ptb:important-questions
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FBISE Annual 2011 @fsc:fsc_part_1_old_papers
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FBISE Annual 2012 @fsc:fsc_part_1_old_papers
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FBISE Annual 2009 @fsc:fsc_part_1_old_papers
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What is Mathematics? @atiq:math-608
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MCQs or Short Questions @atiq:sp15-mth321
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FSc Part 2 (KPK Boards) @fsc
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Are the functions are same? @dyk
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MTH321: Real Analysis I (Spring 2020) @atiq
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MTH211: Discrete Mathematics (Spring 2020) @atiq
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MTH322: Real Analysis II (Spring 2019) @atiq
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MTH322: Real Analysis II (Spring 2017) @atiq
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MTH321: Real Analysis 1 (Spring 2015) @atiq
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MTH321: Real Analysis 1 @atiq
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MTH231: Linear Algebra @atiq
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MATH-608: History of Mathematics @atiq
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MATH-510: Topology @atiq
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MATH 103: Number Theory @atiq
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MTH322: Real Analysis II (Fall 2020) @atiq
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MTH211: Discrete Mathematics (Fall 2020) @atiq
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MTH322: Real Analysis II (Fall 2019) @atiq
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MTH322: Real Analysis II (Fall 2018) @atiq
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MTH321: Real Analysis I (Fall 2019) @atiq
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MTH321: Real Analysis I (Fall 2018) @atiq
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MTH322: Real Analysis II (Fall 2017) @atiq
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MTH321: Real Analysis I (Fall 2015) @atiq
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MTH321: Real Analysis 1 @atiq
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CHEM-501: Basic Mathematics for Chemist @atiq
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