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- Exercise 1.2 (Solutions) @fsc-part1-ptb:sol:ch01
- exists}\; -z\in \mathbb{C} \mbox{ such that}\; z+\left( -z\right) =0 \\ \mbox{In fact if } z=a+bi, \mbox... ray}\nonumber\] - Associative Law for Addition \[\left( z+w\right) +v= z +\left( w+v\right)\nonumber \] </panel> <panel> **Question 2** Verify the multiplic... olutions** $\quad (2,6)\div (3,7)=\dfrac{(2,6)}{\left( 3,7 \right)}$ $=\dfrac{2+6i}{3+7i}=\dfrac{2+6i}
- Exercise 2.8 (Solutions) @fsc-part1-ptb:sol:ch02
- \hline \end{array} \] **Solution** Suppose $G=\left\{ 0,1,2,3 \right\}$ i) The given table show that... n \mathbb{Z}$, $a+b\in \mathbb{Z}$. b- Since $a+\left( b+c \right)=\left( a+b \right)+c$, thus associative law holds in $\mathbb{Z}$. c- Since $0\in \mathb... ble represents the sums of the elements of set $\left\{ E,O \right\}$. The identity element of the set
- Ch 03: Matrices and Determinants @fsc-part1-ptb:important-questions
- nts ====== <list-group> * Fin $x$ and $y$ if $ \left[ {\begin{array}{c} x+3&1\\ -3& 3y-4 \end{array}} \right]= \left[ {\begin{array}{c} 2&1\\ -3&2 \end{array}} \right... E Gujrawala(2015)// * Solve for matrix $A$ if $\left[ {\begin{array}{c}4&3\\ 2&2 \end{array}} \right]A-\left[ {\begin{array}{c} 2&3\\ -1&-2 \end{array}} \righ
- Trigonometric Formulas
- id> </panel> <panel><grid><col sm="6"> * $\sin \left( \alpha +\beta \right)=\sin \alpha \cos \beta +\cos \alpha \sin \beta$ * $\sin \left( \alpha -\beta \right)=\sin \alpha \cos \beta -\cos \alpha \sin \beta$ * $\cos \left( \alpha +\beta \right)=\cos \alpha \cos \beta -\sin \alpha \sin \beta$ * $\cos \left( \alpha -\beta \right)=\cos \alpha \cos \beta +\
- Exercise 1.1 (Solutions) @fsc-part1-ptb:sol:ch01
- imes$'. ---- (ix) $a(b-c)= ab-ac$ **Property:** Left distributive property. ---- (x) $(x-y)z= xz-yz$ ... . ---- (xii) $a(b+c-d)= ab+ac-ad$ **Property:** Left distributive property. </panel> <panel> **Quest... + 1 \times \frac{c}{d}\\ &= \frac{a}{b} \times \left(d \times \frac{1}{d}\right)+\left(b \times \frac{1}{b}\right)\times \frac{c}{d}\\ &= \frac{a}{b} \time
- Solution and Area of Oblique Triangle
- d><col sm="4"> * $\dfrac{a-b}{a+b}=\dfrac{\tan \left( \tfrac{\alpha -\beta }{2} \right)}{\tan \left( \tfrac{\alpha +\beta }{2} \right)}$ </col><col sm="4"> * $\dfrac{b-c}{b+c}=\dfrac{\tan \left( \tfrac{\beta-\gamma}{2} \right)}{\tan \left( \tfrac{\beta+\gamma}{2} \right)}$ </col><col sm="4"> *
- Definitions: FSc Part 1 (Mathematics): PTB
- n square brackets is called Matrix. \\ e.g. $A = \left[ {\begin{array}{c} 1&2&3\\ 0&8&4\\ 2&1&1 \end{arr... = no. of rows $\times$ no. of column. e.g. $A = \left[ {\begin{array}{c} 3&1&7\\ 0&5&4 \end{array}} \ri... aving single row is called Row Matrix. e.g. $B = \left[ {\begin{array}{c} 1&4&6 \end{array}} \right]$ ... single column is called column Matrix. e.g. $B = \left[ {\begin{array}{c} 1\\3\\5 \end{array}} \right]$
- Ch 08: Mathematical Induction and Binomial Theorem @fsc-part1-ptb:important-questions
- <list-group> * Using binomial theorem,expand $\left(\frac{x}{2}-\frac{2}{x^2}\right)$ --- // BISE Guj... )// * Find the $6$th term in the expansion of $\left( x^2-\frac{3}{2x}\right)$ --- // BISE Gujranwala(2015)// * Expand $\left( 8-2x\right)^{-1}$ up to two terms. --- // BISE G