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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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- Solutions: Math 11 KPK
- Fundamental of Complex Analysis (Solutions of Some Exercises)
- Vector & Tensor Analysis by Dr Nawazish Ali (Solutions)
- Unit 1: Complex Numbers (Solutions)
- Unit 10: Trigonometric Identities of Sum and Difference of Angles (Solutions)
- 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
- Chapter 09: Fundamentals of Trigonometry
- Chapter 10: Trigonometric Identities
- Chapter 11: Trigonometric Functions and their Graphs
- Chapter 12: Application of Trigonometry
- Chapter 13: Inverse Trigonometric Functions
- Chapter 14: Solutions of Trigonometric Equation
- DOC Viewer: FSc Part 1 Solutions
- Unit 01: Functions and Limits
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- Unit 04: Introduction to Analytic Geometry
- Unit 05: Linear Inequalities and Linear Programming
- Unit 06: Conic Section
- Unit 07: Vectors
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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)
- Unit 04: Sequence and Series (Solutions)
- Unit 05: Miscellaneous Series (Solutions)
- 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)
- Mathematical Method by Khalid Latif Mir (Solutions)
- Theoretical Mechanics by Khalid Latif Mir (Solutions)
- Exercise 1.1 (Solutions)
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- Exercise 2.8 (Solutions)
- 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 06: Sequences and Series: Mathematics FSc Part 1
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- Ch 07: Permutation, Combination and Probability: Mathematics FSc Part 1
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- Ch 08: Mathematical Induction and Binomial Theorem: Mathematics FSc Part 1
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- Ch 09: Fundamentals of Trigonometry: 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 13: Inverse Trigonometric Functions: Mathematics FSc Part 1
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- Ch 14: Solutions of Trigonometric Equation
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- Unit 02: Differentiation: Mathematics FSc part 2
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- Unit 06: Conic Section: Mathematics FSc part 2
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- Exercise 2.1 (Solutions)
- Exercise 2.2 (Solutions)
- Exercise 2.3 (Solutions)
- Exercise 2.4 (Solutions)
- Exercise 2.5 (Solutions)
- Exercise 2.6 (Solutions)
- Exercise 11.1 (Solutions)
Fulltext results:
- 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 $. ====Solution==== \begin{align}L.H.S.&=\cos ec2\theta -\cot 2\t
- Exercise 6.2 @matric:9th_science
- .2 ====== On the following page we have given the solution of Exercise 6.2 of Mathematics 9 (Science) publis... x^2-x-6}{x^2-9}+\frac{x^2+2x-24}{x^2-x-12}$\\ **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 $a,b\in {{\mathbb{Q}}^{+}}$, $ab\
- Exercise 4.1 @matric:9th_science
- .1 ====== On the following page we have given the solution of Exercise 4.1 of Mathematics 9 (Science) publis... ^2-3x+\sqrt{2}$\\ (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
- 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_
- View Online (Solutions of Unit 06) @fsc:fsc_part_2_solutions:ch06
- er&cp=01&p=15&ch=06&fp=Ex-6-1-FSc-part2-ver-2-0-0|Solution of Exercise 6.1]] * [[mdoku>fsc:fsc_part_2_solu... er&cp=01&p=09&ch=06&fp=Ex-6-2-FSc-part2-ver-2-1-0|Solution of Exercise 6.2]] * [[mdoku>fsc:fsc_part_2_solu... er&cp=01&p=03&ch=06&fp=Ex-6-3-FSc-part2-ver-2-0-0|Solution of Exercise 6.3]] * [[mdoku>fsc:fsc_part_2_solu... er&cp=01&p=13&ch=06&fp=Ex-6-4-FSc-part2-ver-2-1-1|Solution of Exercise 6.4]] * [[mdoku>fsc:fsc_part_2_sol
- B-Course of Mathematics (Paper A & B) @bsc:paper_pattern:sargodha_university
- f higher order (homogeneous and non-homogeneous). Solution by: D-operator and undetermined co-officients Met... er linear DE. Cauchy-Euler equation. Power series solution about an arbitrary point. ==== SECTION-II (4/12: 16,16,16,16) ==== Laplace Transformation, solution of ODEs. Error analysis. Solution of non-linear (algebraic and transcendental) equation in one variable usi
- Exercise 1.1 (Solutions) @fsc-part1-ptb:sol:ch01
- properties used in the following inequalities: **Solution** (i) $-3<-2 \quad \Rightarrow 0<1$. **Property... $$\frac{a}{c}+\frac{b}{c} = \frac{a+b}{c}$$. **Solution** \begin{align} L.H.S &= \frac{a}{c}+\frac{b}{c}... $$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$$. **Solution** \begin{align} L.H.S &= \frac{a}{b}+\frac{c}{d}... \frac{7}{12}-\frac{5}{18}=\frac{-21-10}{36}$$. **Solution** \begin{align} L.H.S &= -\frac{7}{12}-\frac{5}{
- Question 1 and 2 Exercise 4.1 @math-11-kpk:sol:unit04
- nd infinite sequences\\ $2,4,6,8, \ldots ,50$ ====Solution==== It is finite sequence whose last term is $50... and infinite sequences. $1,0,1,0,1, \ldots$. ====Solution==== It is infinite sequence, the last term may be... nfinite sequences: $...,-4,0,4,8, \ldots, 60$ ====Solution==== This is infinite sequence. =====Question 1(i... }{9},-\dfrac{1}{27}, \ldots,-\dfrac{1}{2187}$ ====Solution==== This finite sequence. =====Question 2(i)====
- 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