Define $f:\mathbb{R}\to \mathbb{R}$ by $$f(x)=\{\begin{array}{ll}(1-\cos x)\sin \left(\frac{1}{x}\right),& x\ne 0\\ 0,& x=0\end{array}$$ Then,$$ (a) $f$ is discontinuous. $$ (b) $f$ is continuous but not differentiable. $$ $$ $$ (c) $f$ is differentiable and its derivative is discontinuous. $$ $$ $$$$$$ $$ (d) $f$ is differentiable and its derivative is continuous.

If two real numbers $x$ and $y$ satisfy $(x+5{)}^2+(y-10{)}^2=196$, then the minimum possible value of $x^2+2x+y^2-4y$ is

If the maximum and minimum values of ${\sin }^{6}x+{\cos }^{6}x,$ as $x$ takes all real values, are $a$ and $b$, respectively, then $a-b$ equals

The expression $$\sum _{k=0}^{10}{2}^k\tan \left(2^k\right)$$ equals

Let $f:\mathbb{R}\to [0,\mathrm{\infty })$ be a continuous function such that $$f(x+y)=f(x)f(y)$$ for all $x,y\in \mathbb{R}$. Suppose that $f$ is differentiable at $x=1$ and $${\frac{df(x)}{dx}|}_{x=1}=2$$ Then, the value of $f(1){\log }_{e}f(1)$ is

For $0\le x<2\pi$, the number of solutions of the equation $${\sin }^{2}x+2{\cos }^{2}x+3\sin x\cos x=0$$ is

Let $$\begin{array}{c}p(x)=x^3-3x^2+2x,x\in \mathbb{R},\\ f_0(x)=\{\begin{array}{ll}{\int }_0^{x}p(t)dt,& x\ge 0,\\ -{\int }_x^{0}p(t)dt,& x<0,\end{array}\\ f_1(x)=e^{f_0(x)},f_2(x)=e^{f_1(x)},\dots ,f_n(x)=e^{f_{n-1}(x)}\end{array}$$ How many roots does the equation $\frac{d{f}_n(x)}{dx}=0$ have in the interval $(-\mathrm{\infty },\mathrm{\infty })?$

Let us denote the fractional part of a real number $x$ by $\{x\}$ (note: $\{x\}=x-[x]$ where $[x]$ is the integer part of $x$ ). Then, $$\underset{n\to \mathrm{\infty }}{lim}\{(3+2\sqrt{2}{)}^n\}=$$

Let $f:\mathbb{R}\to \mathbb{R}$ be any twice differentiable function such that its second derivative is continuous and $$\frac{df(x)}{dx}\ne 0\text{ for all }x\ne 0\text{ . }$$ If then prove that $$\underset{x\to 0}{lim}\frac{f(x)}{x^2}=\pi$$ for all $x\ne 0,f(x)>f(0).$

For a positive integer $n,$ the equation $$x^2=n+y^2,x,y\text{ integers }$$ does not have a solution if and only if $$ (a) $n=2.$ $$ (b) $n$ is a prime number. $$ (c) $n$ is an odd number. $$ (d) $n$ is an even number not divisible by $4.$

Consider the following two subsets of $\mathbb{C}$ : $$A=\{\frac{1}{z}:|z|=2\}\text{ and }B=\{\frac{1}{z}:|z-1|=2\}\text{ . }$$ Then $$ (a) $A$ is a circle, but $B$ is not a circle. $$ (b) $B$ is a circle, but $A$ is not a circle. $$ (c) $A$ and $B$ are both circles. $$ (d) Neither $A$ nor $B$ is a circle.

Suppose $f(x)$ is a twice differentiable function on $[a,b]$ such that $$f(a)=0=f(b)$$ and $$x^2\frac{d^{2}f(x)}{d{x}^2}+4x\frac{df(x)}{dx}+2f(x)>0\text{ for all }x\in (a,b)\text{ . }$$ Then,$$ (a) $f$ is negative for all $x\in (a,b).$ $$ (b) $f$ is positive for all $x\in (a,b).$ $$ (c) $f(x)=0$ for exactly one $x\in (a,b).$ $$ (d) $f(x)=0$ for atleast two $x\in (a,b).$

The number of different values of $a$for which the equation $x^3-x+a=0$ has two identical real roots is

The number of all integer solutions of the equation $x^2+y^2+x-y=2021$ is

The polynomial $x^4+4x+c=0$ has at least one real root if and only if $$ (a) $c<2;$$$ (b) $c\le 2;$$$ (c) $3>c$$$ (d) $c\le 3.$

Let $a,b,c$ and $d$ be four non-negative real numbers where $a+b+c+d=1.$ The number of different ways one can choose these numbers such that $a^2+b^2+c^2+d^2=max\{a,b,c,d\}$ is

Consider all $2\times 2$ matrices whose entries are distinct and taken from the set $\{1,2,3,4\}.$ The sum of determinants of all such matrices is

Define $a=p^3+p^2+p+11$ and $b=p^2+1,$ where $p$ is any prime number. Let $d=\gcd (a,b).$ Then the set of possible values of $d$ is

The number of different ways to colour the vertices of a square $PQRS$ using one or more colours from the set $\{Red,Blue,Green,Yellow\},$ such that no two adjacent vertices have the same colour is

Let $$f(x)=e^{-|x|},x\in \mathbb{R}$$ and $$g(\theta )={\int }_{-1}^{1}f\left(\frac{x}{\theta }\right)dx,\theta \ne 0$$ Then, $$\underset{\theta \to 0}{lim}\frac{g(\theta )}{\theta }=$$

Let $f:\mathbb{R}\to \mathbb{R}$ be a twice differentiable function such that $\frac{d^{2}f(x)}{d{x}^2}$ is positive for all $x\in \mathbb{R},$ and suppose $f(0)=1,f(1)=4.$ Find the value of $f(2)?$

The volume of the region $S=\{(x,y,z):|x|+2|y|+3|z|\le 6\}$ is

The value of $$1+\frac{1}{1+2}+\frac{1}{1+2+3}+\cdots +\frac{1}{1+2+3+\cdots 2021}$$ is

Let $f(x)=\sin x+\alpha x,x\in \mathbb{R}$, where $\alpha$ is a fixed real number. Prove the function $f$ is one-to-one if and only if $\alpha \ge 1$ or $\alpha \le -1.$

Let $a,b,c,d>0$, be any real numbers. Then the maximum possible value of $cx+dy$, over all points on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ must be

A box has $13$ distinct pairs of socks. Let $p_r$ denote the probability of having at least one matching pair among a bunch of $r$ socks drawn at random from the box. If $r_0$ is the maximum possible value of $r$ such that $p_r<1$, then the value of $p_{r_0}$ is

Consider the curves $x^2+y^2-4x-6y-12=0,9x^2+4y^2-900=0$ and $y^2-6y-6x+51=0.$ The maximum number of disjoint regions into which these curves divide the $XY$-plane (excluding the curves themselves), is

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function such that $$f(x+1)=\frac{1}{2}f(x)\mathrm{\forall }x\in \mathbb{R},$$ and let $a_n={\int }_0^{n}f(x)dx$ for all integers $n\ge 1.$ Then $$\underset{n\to \mathrm{\infty }}{lim}{a}_n\text{exists and equals}2{\int }_0^{1}f(x)dx.$$

The sum of all the solutions of $2+{\log }_2(x-2)={\log }_{(x-2)}8$ in the interval $(2,\mathrm{\infty })$ is

The number of ways one can express $2^2{3}^3{5}^5{7}^7$ as a product of two numbers $a$ and $b$, where $(a,b)=1$ and $b>a>1$, is

Given that the integers $a,b$ satisfy the equation $$[\frac{\frac{1}{a}}{\frac{1}{a}-\frac{1}{b}}-\frac{\frac{1}{b}}{\frac{1}{a}+\frac{1}{b}}]\left(\frac{1}{a}-\frac{1}{b}\right)\cdot \frac{1}{\frac{1}{a^2}+\frac{1}{b^2}}=\frac{2}{3},$$find the value of $a+b.$

$p,q$ are two integers, and the two roots of the equation in $x$ $$x^2-\frac{p^2+11}{9}x+\frac{15}{4}(p+q)+16=0$$ are $p$ and $q$ also. Find the values of $p$ and $q.$

Let $p$ be a positive prime number such that the equation $$x^2-px-580p=0$$ has two integer solutions. Find the value of $p.$

How many ordered pairs of integers $(x,y)$ satisfy the equation $$x^2+y^2=2(x+y)+xy?$$

Find the number of positive integer solutions of the equation $$\frac{2}{x}-\frac{3}{y}=\frac{1}{4}.$$

Find the number of positive integer solutions to the equation $$\frac{x}{3}+\frac{14}{y}=3.$$

Find the number of non-zero integer solutions $x,y$ to the equation $$\frac{15}{x^{2}y}+\frac{3}{xy}-\frac{2}{x}=2.$$

Let $x,y,z$ and $w$ represent four $\mathit{\text{distinct}}$ positive integers such that $$x^2-y^2=z^2-w^2=81.$$ Find the value of $xz+yw+xw+yz.$

How many number of pairs $(x,y)$ of two integers satisfy the equation $$x^2-y^2=12?$$

Given that $\frac{1260}{a^2+a-6}$ is a positive integer, where $a$ is a positive integer. Find the value of $a.$