Define \(f: \mathbb{R} \rightarrow \mathbb{R}\) by \[f(x)= \begin{cases}(1-\cos x) \sin \left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x=0\end{cases}\] Then,\(\qquad\) (a) \(f\) is discontinuous. \(\qquad\) (b) \(f\) is continuous but not differentiable. \(\qquad\) \(\qquad\) \(\qquad\) (c) \(f\) is differentiable and its derivative is discontinuous. \(\qquad\) \(\qquad\) \(\qquad\)\(\qquad\)\(\qquad\) \(\qquad\) (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 \(f: \mathbb{R} \rightarrow[0, \infty)\) 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 \[\left.\frac{d f(x)}{d x}\right|_{x=1}=2\] Then, the value of \(f(1) \log _{e} f(1)\) is

For \(0 \leq x<2 \pi\), the number of solutions of the equation \[\sin ^{2} x+2 \cos ^{2} x+3 \sin x \cos x=0\] is

Let \[\begin{gathered} p(x)=x^{3}-3 x^{2}+2 x, x \in \mathbb{R}, \\ f_{0}(x)= \begin{cases}\int_{0}^{x} p(t) d t, & x \geq 0, \\ -\int_{x}^{0} p(t) d t, & x<0,\end{cases} \\ f_{1}(x)=e^{f_{0}(x)}, \quad f_{2}(x)=e^{f_{1}(x)}, \quad \ldots \quad, f_{n}(x)=e^{f_{n-1}(x)} \end{gathered}\] How many roots does the equation \(\frac{d f_{n}(x)}{d x}=0\) have in the interval \((-\infty, \infty)?\)

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, \[\lim _{n \rightarrow \infty}\left\{(3+2 \sqrt{2})^{n}\right\}=\]

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be any twice differentiable function such that its second derivative is continuous and \[\frac{d f(x)}{d x} \neq 0 \text { for all } x \neq 0 \text { . }\] If then prove that \[\lim _{x \rightarrow 0} \frac{f(x)}{x^{2}}=\pi\] for all \(x \neq 0, \quad f(x)>f(0).\)

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

Consider the following two subsets of \(\mathbb{C}\) : \[A=\left\{\frac{1}{z}:|z|=2\right\} \text { and } B=\left\{\frac{1}{z}:|z-1|=2\right\} \text { . }\] Then \(\qquad\) (a) \(A\) is a circle, but \(B\) is not a circle. \(\qquad\) (b) \(B\) is a circle, but \(A\) is not a circle. \(\qquad\qquad\) (c) \(A\) and \(B\) are both circles. \(\qquad\qquad\quad\) (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 \[x^{2} \frac{d^{2} f(x)}{d x^{2}}+4 x \frac{d f(x)}{d x}+2 f(x)>0 \text { for all } x \in(a, b) \text { . }\] Then,\(\qquad\) (a) \(f\) is negative for all \(x \in(a, b).\) \(\qquad\) (b) \(f\) is positive for all \(x \in(a, b).\) \(\qquad\quad\) (c) \(f(x)=0\) for exactly one \(x \in(a, b).\) \(\qquad\) (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 \(x^{4}+4 x+c=0\) has at least one real root if and only if \(\qquad\) (a) \(c<2;\)\(\qquad\) (b) \(c \leq 2;\)\(\qquad\) (c) \(3>c\)\(\qquad\) (d) \(c \leq 3.\)

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 \(d=\operatorname{gcd}(a, b).\) 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 \[g(\theta)=\int_{-1}^{1} f\left(\frac{x}{\theta}\right) d x, \theta \neq 0\] Then, \[\lim _{\theta \rightarrow 0} \frac{g(\theta)}{\theta}=\]

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) 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 \(S=\{(x, y, z):|x|+2|y|+3|z| \leq 6\}\) 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 \(\alpha \geq 1\) or \(\alpha \leq-1.\)

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

Let \(f \colon \mathbb{R} \to \mathbb{R} \) be a continuous function such that \[f(x+1)=\frac{1}{2}{f(x)}\space \forall x \in \mathbb{R},\] and let \(a_n=\int_{0}^{n} {f(x)}{dx}\) for all integers \(n\geq {1}.\) Then \[\lim_{n\to \infty} {a_n}\space \text{exists and equals}\space 2\int _{0}^{1} {f(x)}{dx}.\]

The sum of all the solutions of \(2+\log_{2}(x-2)=\log_{(x-2)}{8}\) in the interval \((2, \infty)\) 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 \[\left[\frac{\frac{1}{a}}{\frac{1}{a}-\frac{1}{b}}-\frac{\frac{1}{b}}{\frac{1}{a}+\frac{1}{b}}\right]\left({\frac{1}{a}-\frac{1}{b}}\right)\cdot\frac{1}{\frac{1}{a^2}+\frac{1}{b^2}}=\frac{2}{3},\]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 \[\frac{15}{x^2y}+\frac{3}{xy}-\frac{2}{x}=2.\]

Let \(x, y, z\) and \(w\) represent four \(\textit{distinct}\) 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 \(a.\)