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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 - $4$ - $2$ - $8$
- PPSC Paper 2011 (Lecturer in Mathematics) @ppsc
- of these - If $H$ is a subgroup of index ------ then $H$ is a normal subgroup of $G$ \\ - $2$ ... be a cyclic group of order $24$ generated by $a$ then order of $a^{10}$ is ------ \\ - $2$ - $1... If a vector space $V$ has a basis of $n$ vectors, then every basis of $V$ must consist of exactly ----- ... ,\bar{4}$ - If $H$ is a normal subgroup of $G$, then $Na(H)=$ ------------ \\ - $H$ - $G$
- PPSC Paper 2021 (Lecturer in Mathematics) @ppsc
- - $10$ - Let \(X\) and \(Y\) be Banach spaces. Then the product space \(X\times Y\), with the norm de... \(\leq x;\) be integrable function on \([0,4]\), then \(\displaystyle \int_{0}^{4}[x]dx \) is equal to:... of a Hilbert space \(H\) such that \(X \perp Y\), then \(X+Y\) is : \\ - A closed subspace - \(X... , n\). If \(\lim\limits_{n \to \infty} a_n=l>0\), then \(\lim\limits_{n \to \infty}(a_n\ldots a_1)^{\fra
- Real Analysis: Short Questions and MCQs @msc:mcqs_short_questions
- n\}$ converges to 5 and $\{y_n\}$ converges to 2. Then find $\lim_{n\to\infty z_n}$, where $z_n=x_n-2y_n... n\}$ converges to 3 and $\{y_n\}$ converges to 4. Then find $\lim_{n\to\infty z_n}$, where $x_n=2y_n-3z_... anel><panel> 6. If a real number is not rational then it is ............... * (A) integer * (B)... hbb{N} \wedge x^2 \leq 7 \} \subset \mathbb{N}$. Then supremum of $A$ is * (A) 7 * (B) 3 *
- Definitions: FSc Part 1 (Mathematics): PTB @fsc-part1-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 2 (Mathematics): PTB @fsc-part2-ptb
- alue //x// determines exactly one value of //y//, then we say that //y// is function of //x//. The varia... ctions. * **Logarithmic Function:** If $x=a^y$, then $y=log_ax$, where $a<0$, $a\neq 1$. * **Explici... pressed in terms of the independent variable $x$, then $y$ is called an explicit function of $x$. =====... \in D_f$ and $f'(c)=0$ or $f'(c)$ does not exists then $c$ is called critical value or point. * ** St
- MTH322: Real Analysis II (Fall 2021) @atiq
- e a$. Assume that $f(x)\ge 0$ for each $x\ge a$. Then prove that $\int_{a}^{\infty }{f(x) dx}$ converg... ge a$ and $\int_{a}^{\infty }{g\,dx}$ converges, then $\int_{a}^{\infty }{f\,dx}$ converges and we hav... $\lim\limits_{x\to \infty }\frac{f(x)}{g(x)}=1$, then $\int\limits_{a}^{\infty }{f dx}$ and $\int\li... that $f\in \mathcal{R}[a,b]$ for every $b\ge a$. Then the integral $\int\limits_{a}^{\infty }{f dx}$ co
- MTH321: Real Analysis I (Spring 2023) @atiq
- }}<{{u}_{n}}<{{t}_{n}}$ for all $n\ge {{n}_{0}}$, then the sequence $\left\{ {{u}_{n}} \right\}$ also co... olimits_{n=1}^{\infty }{{{a}_{n}}}$ is convergent then $\underset{n\to \infty }{\mathop{\lim }}\,{{a}_{... \ge k$. - If $\sum{{{b}_{n}}}$ is convergent, then $\sum{{{a}_{n}}}$ is convergent. - If $\sum{{{a}_{n}}}$ is divergent, then $\sum{{{b}_{n}}}$ is divergent. - Prove that e
- MTH322: Real Analysis II (Spring 2023) @atiq
- e a$. Assume that $f(x)\ge 0$ for each $x\ge a$. Then $\int_{\,a}^{\,\infty }{f(x)\,dx}$ converges if,... a$. If $\lim_{x\to \infty } \frac{f(x)}{g(x)}=0$, then convergence of $\int_{a}^{\infty }{g(x)dx}$ impl... c}{f\,\,dx}\, \right|<\varepsilon \] for $b,c>B$, then $\int_{a}^{\infty }{f\,dx}$ is convergent. - If... ts_{a}^{\infty }{f\,dx}$ is absolutely converges, then it is convergent but the converse is not true in
- PPSC Paper 2015 (Lecturer in Mathematics) @ppsc
- surface and \(V\) is the volume enclosed by \(S\) then \(\displaystyle \iint_{s}\underline{r}\cdot \unde... 1]\cup (1,3]\) and \(R\) with usual metric space. Then \(A^\circ=\) ------------- \\ - $A\setminus ... t \(A\) be finite subset of a metric space \(X\). Then \(A^d=\) ------------ \\ - Singleton set \(0... A\) - Let \(A\) be a finite subset of \(X,d\) then \(A\) is ----- \\ - Open set - Open as
- Chapter 03 - Limits and Continuity @msc:real_analysis_notes_by_syed_gul_shah
- nto //X// (iv) //p// is the limit point of //E//. Then $\lim_{x\to p} f(x)=q$ iff $\lim_{n\to\infty}f(p_... ercies * Theorem: If $\lim_{x\to c}f(x)$ exists then it is unique. * Theorem: Suppose that a real va... open interval //G// except possibly at $c\in G$. Then $\lim_{x\to c}f(x)=l$ if and only if for every po... /G//. If $\lim_{x\to c}f(x)=\lim_{x\to c}g(x)=l$, then $\lim_{x\to c}h(x)=l$. * Theorem: (for sum, dif
- Question 9 Exercise 6.3 @math-11-kpk:sol:unit06
- 5,6$ or 7 men. If committee contains two men and then it will obviously contain $6$ women. so in this ... \\ &=21\end{align} If committee contains $3$ men then it will contain $5$ women, in this case the tota... \\\ &=210\end{align} If committee contain $4$ men then it will contain $4$ women, in this case total nu... \\ &=525\end{align} If committee contains $5$ men then it will contain $3$ women, in this case total num
- MTH424: Convex Analysis (Fall 2020) @atiq
- above. * If $f:[a,b]\to \mathbb{R}$ is convex, then $f$ is bounded above by $\max(f(a),f(b))$. ===... ive * If $f:I\rightarrow \mathbb{R}$ is convex, then $f_{-}'(x)$ and $f_{+}'(x)$ exist and are increas... .$$ * Suppose $f$ is differentiable on $(a,b)$. Then $f$ is convex [strictly convex] if, and only if, ... = * Let $f$ is twice differentiable on $(a,b)$. Then $f$ is convex on $(a,b)$ iff $f''(x)\geq 0$ for