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Exercise 1.2 (Solutions) @fsc-part1-ptb:sol:ch01
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-z\in \mathbb{C} \mbox{ such that}\; z+\left( -z\right) =0 \\ \mbox{In fact if } z=a+bi, \mbox{ then } -... ber\] - Associative Law for Addition \[\left( z+w\right) +v= z +\left( w+v\right)\nonumber \] </panel> <panel> **Question 2** Verify the multiplication proper... $\quad (2,6)\div (3,7)=\dfrac{(2,6)}{\left( 3,7 \right)}$ $=\dfrac{2+6i}{3+7i}=\dfrac{2+6i}{3+7i}\times
Exercise 2.8 (Solutions) @fsc-part1-ptb:sol:ch02
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y} \] **Solution** Suppose $G=\left\{ 0,1,2,3 \right\}$ i) The given table show that each element of ... }$, $a+b\in \mathbb{Z}$. b- Since $a+\left( b+c \right)=\left( a+b \right)+c$, thus associative law holds in $\mathbb{Z}$. c- Since $0\in \mathbb{Z}$ such th... ts the sums of the elements of set $\left\{ E,O \right\}$. The identity element of the set is $E$ becau
Exercise 1.1 (Solutions) @fsc-part1-ptb:sol:ch01
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d{array} \] As $(-1)+(-1)=-2 \notin \{0,-1\}$. $\Rightarrow \{0,-1\}$ does not satisfy closure property ... y} \] As $(-1)\times (-1)= 1 \notin \{0,-1\}$. $\Rightarrow \{0,-1\}$ does not have closure property w.r... ine \end{array} \] As $1+1=2 \notin \{1,-1\}$. $\Rightarrow \{1,-1\}$ does not closure property w.r.t. '... he entries of the table belongs to $\{1,-1\}$. $\Rightarrow \{1,-1\}$ has closure property w.r.t. '$\tim
Ch 03: Matrices and Determinants @fsc-part1-ptb:important-questions
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{\begin{array}{c} x+3&1\\ -3& 3y-4 \end{array}} \right]= \left[ {\begin{array}{c} 2&1\\ -3&2 \end{array}} \right]$ --- // BISE Gujrawala(2015)// * Solve for m... $\left[ {\begin{array}{c}4&3\\ 2&2 \end{array}} \right]A-\left[ {\begin{array}{c} 2&3\\ -1&-2 \end{array}} \right]= \left[ {\begin{array}{c} -1&-4\\ 3&6 \end{array
Trigonometric Formulas
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rid><col sm="6"> * $\sin \left( \alpha +\beta \right)=\sin \alpha \cos \beta +\cos \alpha \sin \beta$ * $\sin \left( \alpha -\beta \right)=\sin \alpha \cos \beta -\cos \alpha \sin \beta$ * $\cos \left( \alpha +\beta \right)=\cos \alpha \cos \beta -\sin \alpha \sin \beta$ * $\cos \left( \alpha -\beta \right)=\cos \alpha \cos \beta +\sin \alpha \sin \beta$
Definitions: FSc Part 1 (Mathematics): PTB
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ble involved in it is called tautology.\\ e.g. $p\rightarrow q\leftrightarrow (\sim q \rightarrow \sim p)$ is a tautology. * **Contradiction:** A statement ... gin{array}{c} 1&2&3\\ 0&8&4\\ 2&1&1 \end{array}} \right]$ * **Order:** Order of Matrix tells us abou... ft[ {\begin{array}{c} 3&1&7\\ 0&5&4 \end{array}} \right]$. Order of $A$=$2 \times 3$ * **Row matrix:**
Solution and Area of Oblique Triangle
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+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"> * $\dfrac{c-a}{c+a}=\dfrac{\tan \le
MCQs: Ch 01 Number Systems @fsc-part1-ptb:mcq-bank
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, b, c \in \mathbb{R}$\\ (i) $a>b \wedge b>c \Rightarrow a>c$\\ (ii) $a<b \wedge b<c \Rightarrow a<c$\\ is called ----- property. - Translative ... - For all $ a, b, c \in \mathbb{R}$\\ (i) $a>b \Rightarrow a+c>b+c$\\ (ii) $a<b \Rightarrow a+c<b+c$\\ is called ----- property. - Additional - Adva
Definitions: FSc Part 1 (Mathematics): PTB by Aurang Zaib
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th values of its variables. ===Example:=== \( p \rightarrow q \leftrightarrow (\neg q \rightarrow \neg p) \) is a tautology because its truth table shows that ... h values of its variables. ===Example:=== \( (p \rightarrow q) \land (p \lor q) \) is a contingency beca... \) and \( B = \{a, b, c\} \). A function \( f: A \rightarrow B \) could be defined as \( f(1) = a, f(2) =
Multiple Choice Questions (MCQs)
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hat $\forall a, b, c \in R$ * $a<b \wedge c>0\Rightarrow ac\geq bc$ * $a<b \wedge c>0\Rightarrow ac> bc$ * $a<b \wedge c>0\Rightarrow ac< bc$ * $a>b \wedge c>0\Rightarrow ac= bc$ - Which of the following is an expression for $\
Ch 08: Mathematical Induction and Binomial Theorem @fsc-part1-ptb:important-questions
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l theorem,expand $\left(\frac{x}{2}-\frac{2}{x^2}\right)$ --- // BISE Gujranwala(2015)// * Find the $6$... 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 Gujranwala(20