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, \[\lim _{\theta \rightarrow 0} \frac{g(\theta)}{\theta}=\]