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- Symposium on “Computational Complexities, Innovations and Solutions (CCIS)", COMSATS, Abbottabad (10 - 11 May 2010)
- FSc Part 1 Mathematics Notes/Solutions
- Solution and Area of Oblique Triangle
- Solution & Area of Oblique Triangle
- FSc Part 1 Mathematics Notes/Solutions
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- Unit 1: Complex Numbers (Solutions)
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- Ch 14: Solutions of Trigonometric Equation
- Chapter 01: Number System
- Chapter 02: Sets, Functions and Groups
- Chapter 03: Matrices and Determinants
- Chapter 04: Quadratic Equations
- Chapter 05: Partial Fractions
- Chapter 06: Sequences and Series
- Chapter 07: Permutation, Combination and Probability
- Chapter 08: Mathematical Induction and Binomial Theorem
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- DOC Viewer: FSc Part 1 Solutions
- Unit 01: Functions and Limits
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- Chapter 12: Graph of Trigonometric and Inverse Trigonometric Functions and Solutions of Trigonometric Equations
- Unit 01: Complex Numbers (Solutions)
- Unit 02: Matrices and Determinants (Solutions)
- Unit 03: Vectors (Solutions)
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- Unit 06: Permutation, Combination and Probability (Solutions)
- Unit 07: Mathmatical Induction and Binomial Theorem (Solutions)
- Unit 10: Trigonometric Identities of Sum and Difference of Angles (Solutions)
- Unit 01: Complex Numbers (Solutions)
- Unit 02: Matrices and Determinants (Solutions)
- Mathematical Method by Khalid Latif Mir (Solutions)
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- Exercise 1.1 (Solutions)
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- Ch 01: Number System: Mathematics FSc Part 1
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- Ch 05: Partial Fractions: Mathematics FSc Part 1
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- Ch 07: Permutation, Combination and Probability: Mathematics FSc Part 1
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- Ch 10: Trigonometric Identities: Mathematics FSc Part 1
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- Ch 12: Application of Trigonometry: Mathematics FSc Part 1
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- Ch 14: Solutions of Trigonometric Equation
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- Exercise 6.1 (Solutions)
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- Exercise 11.1 (Solutions)
Fulltext results:
- Exercise 6.2 (Solutions) @math-11-nbf:sol:unit06
- }^n P_{n-2}$\\ [[math-11-nbf:sol:unit06:ex6-2-p1|Solution Question 1]] **Question 2.** Find $n$, if:\\ (i)... -1}=22: 7$ \\ [[math-11-nbf:sol:unit06:ex6-2-p2|Solution: Question 2 ]] **Question 3.** Find $r$, if:\\ (... +6}=1: 30800$\\ [[math-11-nbf:sol:unit06:ex6-2-p3|Solution: Question 3 ]] **Question 4.** How many 3 -digi... not repeated?\\ [[math-11-nbf:sol:unit06:ex6-2-p4|Solution: Question 4 & 5 ]] **Question 5.** How many 7 -d
- Question 7, Exercise 10.2 @fsc-part1-kpk:sol:unit10
- \sin }^{4}}\theta =\dfrac{1}{\sec 2\theta }$. ====Solution==== \begin{align}L.H.S&={{\cos }^{4}}\theta -{{\s... }\dfrac{\theta }{2}=\dfrac{2}{\sin \theta }$. ====Solution==== \begin{align}L.H.S&=\tan \dfrac{\theta }{2}+c... eta }{1-\cos 2\theta }={{\cot }^{2}}\theta $. ====Solution==== \begin{align}L.H.S&=\dfrac{1+\cos 2\theta }{1... y $\cos ec2\theta -\cot 2\theta =tan\theta $. ====Solution==== \begin{align}L.H.S.&=\cos ec2\theta -\cot 2\t
- Question 7, Exercise 10.2 @math-11-kpk:sol:unit10
- \sin }^{4}}\theta =\dfrac{1}{\sec 2\theta }$. ====Solution==== \begin{align}L.H.S&={{\cos }^{4}}\theta -{{\s... }\dfrac{\theta }{2}=\dfrac{2}{\sin \theta }$. ====Solution==== \begin{align}L.H.S&=\tan \dfrac{\theta }{2}+c... eta }{1-\cos 2\theta }={{\cot }^{2}}\theta $. ====Solution==== \begin{align}L.H.S&=\dfrac{1+\cos 2\theta }{1... y $\cos ec2\theta -\cot 2\theta =tan\theta $. ====Solution==== \begin{align}L.H.S.&=\cos ec2\theta -\cot 2\t
- Exercise 6.3 (Solutions) @math-11-nbf:sol:unit06
- ^{n+1} C_{r}$\\ [[math-11-nbf:sol:unit06:ex6-3-p1|Solution: Question 1(i-v)]] **Question 1(vi-x).** Prove t... ible by $k$ !\\ [[math-11-nbf:sol:unit06:ex6-3-p2|Solution: Question 1(vi-x) ]] **Question 2.** Find $n$, i... C_{2}=12: 1$\\ [[math-11-nbf:sol:unit06:ex6-3-p3|Solution: Question 2 ]] **Question 3.** Find $r$, if:\\ (... {r}-1}=11: 5$\\ [[math-11-nbf:sol:unit06:ex6-3-p4|Solution: Question 3 ]] **Question 4.** Find $n$ and $r$,
- Exercise 6.2 @matric:9th_science
- .2 ====== On the following page we have given the solution of Exercise 6.2 of Mathematics 9 (Science) publis... x^2-x-6}{x^2-9}+\frac{x^2+2x-24}{x^2-x-12}$\\ **Solution:**\\ $\begin{align} \frac{x^2-x-6}{x^2-9}&+\fra... }-\frac{4x}{x^2+1}\right]+\frac{4x}{x^4-1}$\\ **Solution:**\\ $\begin{align} \left[\frac{x+1}{x-1}-\frac... +15}+\frac{1}{x^2-4x+3}-\frac{2}{x^2-6x+5}$\\ **Solution:**\\ $\begin{align} \frac{1}{x^2-8x+15}+\frac{1
- Exercise 2.8 (Solutions) @fsc-part1-ptb:sol:ch02
- 0 & 3 & 0 & 1 & 2 \\ \hline \end{array} \] **Solution** Suppose $G=\left\{ 0,1,2,3 \right\}$ i) The ... number form groups with respect to $\times$. **Solution** As $0\in \mathbb{Q}$, multiplicative inverse ... ional number form groups with respect to $+$. **Solution** a- Closure property holds in $\mathbb{Q}$ und... l number form groups with respect to $\times$. **Solution** a- Since for $a,b\in {{\mathbb{Q}}^{+}}$, $ab\
- Question 1, Exercise 2.6 @math-11-nbf:sol:unit02
- em of homogeneous linear equation for non-trivial solution if exists\\ $ 2 x_{1}-3 x_{2}+4 x_{3}=0$\\ $x_{1}... _{2}+3 x_{3}=0$\\ $4 x_{1}+x_{2}-6 x_{3}=0$\\ ** Solution. ** \begin{align*} &2 x_{1}-3 x_{2}+4 x_{3}=0\cdo... 4+36=0 \end{align*} So the system has non-trivial solution. \text{By}\quad(i)-2(ii), we have \begin{align*... es of $x_3$, there are infinite solutions. Hence solution is; \begin{align*} \left[ \begin{array}{c} x_3
- Exercise 4.1 @matric:9th_science
- .1 ====== On the following page we have given the solution of Exercise 4.1 of Mathematics 9 (Science) publis... ^2-3x+\sqrt{2}$\\ (iv) $\frac{3x}{2x-1}+8$\\ **Solution:** (i) $3x^2+\frac{1}{x}-5$\\ $No (Reason:\fr... \\ (iv) $\frac{2\sqrt{x}+3}{2\sqrt{x}-3}$\\ **Solution:**\\ (i) $3x^2+\frac{1}{x}-5$\\ $No (Reason:\... $\\ (viii) $\frac{9x^2-(x^2-4)^2}{4+3x-x^2}$\\ **Solution:**\\ (i) $\frac{120 x^2y^3z^5}{30x^3yz^2}$\\ $\be
- Exercise 6.1 @matric:9th_science
- .1 ====== On the following page we have given the solution of Exercise 6.1 of Mathematics 9 (Science) publis... \ (ii) $102xy^2z$, $85x^2yz$ and $187xyz^2$ \\ **Solution:** (i) $39x^7y^3z=13\times 3\times x^7 y^3 z$\\ ... +2)$\\ (v) $36(3x^4+5x^3-2x^2)$, $54(27x^4-x)$ **Solution:**\\ (i) $\begin{align} x^2+5x+6&=x^2+3x+2x+6,... z^7$ \\ (ii)$102xy^2z$, $85x^2yz$ , $187xyz^2$ **Solution:**\\ (i) $\begin{align} 39x^7y^3z &=3 \times 13
- Exercise 6.3 @matric:9th_science
- .3 ====== On the following page we have given the solution of Exercise 6.3 of Mathematics 9 (Science) publis... -3)(2X^2+11x-21)$\\ (i) $4x^2-12xy +9y^2$\\ **Solution:**\\ $\begin{align}4x^2-12xy +9y^2\\&=4x^2-6x... gn}$ (ii) $x^2-1+\frac{1}{4x^2}, (x\neq 0)$\\ **Solution:**\\ $\begin{align}x^2-1+\frac{1}{4x^2}\\&=(x)... {1}{16}x^2-\frac{1}{12}xy+ \frac{1}{36}y^2$\\ **Solution:**\\ $\begin{align}\frac{1}{16}x^2-\frac{1}{12
- Question 7 Exercise 6.4 @math-11-kpk:sol:unit06
- obability of getting doublet of even numbers. ====Solution==== The sample space rolling a pair of dice is \b... probability of getting a sum less than $6$. ====Solution==== The sample space rolling a pair of dice is \b... e probability of getting a sum more than $7.$ ====Solution==== The sample space rolling a pair of dice is \b... bability of getting a sum greater than $10.$ ====Solution==== The sample space rolling a pair of dice is \b
- Question 1, Exercise 1.3 @math-11-nbf:sol:unit01
- polynomial into linear functions: $z^{2}+169$. **Solution.** \begin{align} & z^{2} + 169 \\ = & z^{2} - (... olynomial into linear functions: $2 z^{2}+18$. **Solution.** \begin{align} & 2z^2 + 18 \\ = &2(z^2 - (3i)^... nomial into linear functions: $3 z^{2}+363$. **Solution.** \begin{align} & 3z^2 + 363 \\ = & 3(z^2 - (1... into linear functions: $z^{2}+\dfrac{3}{25}$. **Solution.** \begin{align} & z^2 + \dfrac{3}{25} \\ = & z^
- Definitions: FSc Part 2 (Mathematics): PTB @fsc-part2-ptb
- problem constrain. * ** Feasible region:** The solution region of an inequality restricted to the first q... on. * ** Vertex or corner point:** A point of a solution region where two of its boundary lines intersects... constrain or decision variables. * ** Feasible Solution:** Each point of the feasible region is called the feasible solution of linear inequalities. * ** Feasible Solution
- Operation Research: Handwritten Notes @notes
- jective function 2 * Decision variables 2 * Solution space 2 * Optimum solution 2 * Standard form 2 * Conversion of maximization to minimization 3 * Linear programming formulation and graphical solution 4 <grid> <col sm="6"> * Simplex method (or the general solution technique) 22 * Algorithm 22 * Optimality c
- Unit 06: Conic Section: Mathematics FSc part 2 @fsc:fsc_part_2_solutions:ch06
- :view&cp=01&p=15&ch=06&fp=Ex-6-1-FSC-part2-ver3-1|Solution of Exercise 6.1]] * [[mdoku>fsc:fsc_part_2_s... 06:view&cp=01&p=10&ch=06&fp=Ex-6-2-FSC-part2-ver3|Solution of Exercise 6.2]] * [[mdoku>fsc:fsc_part_2_s... 06:view&cp=01&p=03&ch=06&fp=Ex-6-3-FSC-part2-ver3|Solution of Exercise 6.3]] * [[mdoku>fsc:fsc_part_2_s... 06:view&cp=01&p=13&ch=06&fp=Ex-6-4-FSC-part2-ver3|Solution of Exercise 6.4]] * [[mdoku>fsc:fsc_part_2_
- Chapter 04: Viewer @bsc:notes_of_calculus_with_analytic_geometry:ch04_techniques_of_integration_farooq
- Chapter 01: Viewer @bsc:notes_of_calculus_with_analytic_geometry:ch01_real_numbers_limits_and_continuity