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- Exercise 1.2 (Solutions) @fsc-part1-ptb:sol:ch01
- -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
- 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
- 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. '$\tim
- Ch 03: Matrices and Determinants @fsc-part1-ptb:important-questions
- {\begin{array}{c} x+3&1\\ -3& 3y-4 \end{array}} \right]= \left[ {\begin{array}{c} 2&1\\ -3&2 \end{array}} \right]$ --- // 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
- 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
- ble involved in it is called tautology.\\ e.g. $p\rightarrow q\leftrightarrow (\sim q \rightarrow \sim p)$ is a tautology. * **Contradiction:** A statement which is always f... n the truth values of variable. \\ e.g. $(p \longrightarrow q)\wedge (p \vee q)$ is the contingency.
- Solution and Area of Oblique Triangle
- +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 02 Sets, Functions and Groups @fsc-part1-ptb:mcq-bank
- itional - None of these - Statement $p \longrightarrow (q \longrightarrow r)$ is equivalent to - $(p \vee q)\longrightarrow r$ - $(p \wedge q)\longrightarrow r$ - $p \longrightarrow (q \wedge r)$ - $(r \longrigh
- Definitions: FSc Part 1 (Mathematics): PTB by Aurang Zaib
- th values of its variables. ===Example:=== \( p \rightarrow q \leftrightarrow (\neg q \rightarrow \neg p) \) is a tautology because its truth table shows that it is always true,... h values of its variables. ===Example:=== \( (p \rightarrow q) \land (p \lor q) \) is a contingency beca
- MCQs: Ch 01 Number Systems @fsc-part1-ptb:mcq-bank
- , 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
- Ch 02: Functions and Groups @fsc-part1-ptb:important-questions
- * Write converse and contra positive of $p \longrightarrow q$ --- // BISE Gujrawala(2015)// * Writ... // BISE Gujrawala(2017)// * Show that $~(p \longrightarrow q) \longrightarrow p$ --- // BISE Gujrawala(2017)// * Prove that $A \cap(B \cup C)=(A \cap B... * Write converse and contrapositive of $q \longrightarrow p$ --- // BISE Sargodha(2015)// * Wri
- Multiple Choice Questions (MCQs)
- 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
- 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