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- MCQs: Ch 02 Sets, Functions and Groups @fsc-part1-ptb:mcq-bank
- element of a set $A$ is also element of set $B$, then - $A\cap B=\phi$ - $A=B$ - $B\subsete... ement of a set $A$ is also as element of set $B$, then - $A\cap B=A$ - $B \subseteq A$ - $A\... f these - If $A\subseteq B$ and $B\subseteq A$, then - $A=\phi$ - $A \cup B=A$ - $A \cap B... - None of these - If $A$ is a subset of $B$ then $A=B$, then we say that $A$ is an - Proper su
- MCQs: Ch 04 Quadratic Equations @fsc-part1-ptb:mcq-bank
- dentity - None of these - If $ax^2+bx+c=0$, then $\{a,b\}$ is called - Factors - Solution ... {3i}}{2}$ - If $w$ is the complex root of unity then its conjugate is - $-w$ - $-w^2$ - $w... $f(x)$ of degree $x \geq 1$ is divided by $(x-a)$ then reminder is - $a$ - $f(a)$ - $n$ ... nomial $f(x)=x^3+4x^2-2x+5$ is divided by $(x-1)$ then the reminder is - $4$ - $2$ - $8$
- Definitions: FSc Part 1 (Mathematics): PTB
- in a set is called its order.\\ e.g. $A=\{3,4\}$ then order of $A$ is $2$. * **Equal set:** Two... ch element of set $A$ is also an element set $B$. Then $A$ is called subset of $B$ written as $A \subset... contains at least one element which is not in $A$ then $A$ is called proper subset of $B$ denoted by $A ... * **Improper subset:** If a set of $B$ and $A+B$ then $A$ is improper subset of $B$ its follow that eve
- MCQs: Ch 01 Number Systems @fsc-part1-ptb:mcq-bank
- == - If $*$ is a binary operation in a set $A$, then for all $a, b \in A$ - $a+b \in A$ - $a-b... mes b \in A$ - $a * b \in A$ - If $z=(1,3)$ then $z^{-1}= $ - $(\displaystyle{\frac{1}{10}},\d... 2$ - $>z_1+z_2$ - If $z_1=2+i$, $z_2=1+3i$, then $z_1-z_2=$ - $1-7i$ - $-1+7i$ - $1-2i$ - $3+4i$ - If $z_1=2+i$, $z_2=1+3i$, then $-i lm (z_1-z_2)=$ - $2i$ - $-2i$ - $
- Definitions: FSc Part 1 (Mathematics): PTB by Aurang Zaib
- in a set. ===Example:=== If \( A = \{3, 4\} \), then the order of \( A \) is 2. ====Equal Set==== Two... If \( A = \{1, 2\} \) and \( B = \{1, 2, 3\} \), then \( A \subseteq B \). ====Proper Subset==== A pro... == * If \( C = \{1\} \) and \( D = \{1, 2\} \), then \( C \subset D \). ====Improper Subset==== An im... tself. ===Example:=== * If \( E = \{1, 2\} \), then \( E \subseteq E \). ====Power Set==== The power
- Definitions: Mathematics 11: PTB by Muzzammil Subhan
- tion:** If $y$ is easily expressed in term of $x$ then $y$ is called an explicit function. E.g. $y=x^2+3... it Function:** If y is not expressed in term of x then y is called an implicit function. E.g. $x^2+xy+y^... int from a fixed point bear a constant ratio less then one to the distance from a fixed line. * **Hype... from a fixed point bear a constant ratio greater then one to the distance from a fixed line. <callout
- Multiple Choice Questions (MCQs)
- (C) integer * (D) prime - If $n$ is prime, then $\sqrt{n}$ is * rational number * whole n... nate plane correspond to the ordered pair $(a,b)$ then $a,b$ are called the ------- of $A$. * ordina... * elements * none of these - If $z=-3-5i$ then $z^{-1}=------$ * $\frac{-3}{34}+\frac{5}{34}... dinate plane correspond to the order pair $(a,b)$ then $b$ is called * abscissa * $x$-coordinate
- Ch 02: Functions and Groups @fsc-part1-ptb:important-questions
- ,4\}$, $B=\{3,4,5,6,7,8\}$ and $C=\{5,6,7,9,10\}$ then verify associativity of union --- // BISE Sarg... (2015)// * If $A,B$ are elements of a group $G$ then show that $(ab)^{-1}=b^{-1}a^{-1}$ --- // BI... 015)// * If $a,b$ being elements of a group $G$ then solve (a) $ax=b$ (b) $x a=b$ --- // BI
- Ch 04: Quadratic Equations @fsc-part1-ptb:important-questions
- the roots of $px^2+qx+q=0$ are $\alpha$, $\beta$,then prove that $$\sqrt {\frac{\alpha}{\beta}}+\sqrt {... e polynomial $x^3+2x^2+kx+4$ is divided by $x-2$, then the reminder is $14$, find $k$. --- //BISE Gujraw... \beta$ are the roots of the equation $5x^2-x-2=0$ then form the equation whose roots are $\frac{3}{\alph
- Ch 06: Sequences and Series @fsc-part1-ptb:important-questions
- 1}{a}$, $\frac{1}{b}$, $\frac{1}{c}$ are in $A.P$ then show that common difference is $\frac{a-c}{2ac}$ ... {9}x^2+\frac{8}{27}x^3+...$ and $0<x<\frac{3}{2}$ then show that $x=\frac{3y}{2(1+y)}$ --- //BISE Sargod... }+\frac{1}{4}x^2+\frac{1}{8}x^3+...$ and $0<x<2$, then prove that $x=\frac{2y}{1+y}$ --- //BISE Lahore(2
- Exercise 1.2 (Solutions) @fsc-part1-ptb:sol:ch01
- tion** Let \(z,w,\) and \(v\) be complex numbers. Then the following properties hold. - Commutative Law... -z\right) =0 \\ \mbox{In fact if } z=a+bi, \mbox{ then } -z=-a-bi. \end{array}\nonumber\] - Associative ... **Solutions** Suppose $z=\left( -4,7 \right)$, then multiplicative inverse of $z$ $=z^{-1} =\dfrac{1
- Trigonometric Formulas
- $n$ is even or odd. **Step II:** If $n$ is even then the answer will be in $\sin$ and if the $n$ is odd then $\sin$ will be converted to $\cos$ and vice versa
- Exercise 2.8 (Solutions) @fsc-part1-ptb:sol:ch02
- is an identity element. d- If $a\in \mathbb{Q}$ then additive inverse $-a\in \mathbb{Q}$ such that $a... ces over the real field. (i) Suppose $A,B\in G$, then $A_{2\times 2} \times B_{2\times 2} = C_{2\times
- Ch 01: Number Systems @fsc-part1-ptb:important-questions
- Gujrawala(2015)// * If $z$ be a complex number then prove that $\overline{z_1 + z_2}=\overline z_1 +\
- Ch 08: Mathematical Induction and Binomial Theorem @fsc-part1-ptb:important-questions
- at bits square and higher powers can be neglected then show that $\frac{1-x}{\sqrt{1+x}} \approx {1-\fra