# Continuity and differentiability JEE Main Previous year Questions with solution

On this page you will get Continuity and differentiability JEE Main Previous year Questions with complete and detailed solution. These solutions are arranged in chronological order.

## Continuity and differentiability JEE Main Previous year Questions with solution

### 2020

Q. Let $f(x)=x .\left[\frac{x}{2}\right]$, for $-10<x<10$, where $[t]$ denotes the greatest integer function. Then the number of points of discontinuity of $f$ is equal to

[NA Sep. 05, 2020 (I)]

Sol.

Q. If a function $f(x)$ defined by

$f(x)=\left\{\begin{array}{ll}a e^{x}+b e^{-x}, & -1 \leq x<1 \\ c x^{2} & , 1 \leq x \leq 3 \\ a x^{2}+2 c x, & 3<x \leq 4\end{array}\right.$ be continuous for some $a, b, c \in \mathbf{R}$ and $f^{\prime}(0)+f^{\prime}(2)=e$, then the value of $a$ is :

(a) $\frac{1}{e^{2}-3 e+13}$

(b) $\frac{e}{e^{2}-3 e-13}$

(c) $\frac{e}{e^{2}+3 e+13}$

(d) $\frac{e}{e^{2}-3 e+13}$

[Sep. 02, 2020 (I)]

Sol.

Q. Let $[t]$ denote the greatest integer $\leq t$ and $\lim _{x \rightarrow 0} x\left[\frac{4}{x}\right]=\mathrm{A}$. Then the function, $f(x)=\left[x^{2}\right] \sin (\pi x)$ is discontinuous, when $x$ is equal to :

(a) $\sqrt{\mathrm{A}+1}$

(b) $\sqrt{\mathrm{A}+5}$

(c) $\sqrt{\mathrm{A}+21}$

(d) $\sqrt{\mathrm{A}}$

[Jan. 9, 2020 (II)]

Sol.

Q. If the function $f$ defined on $\left(-\frac{1}{3}, \frac{1}{3}\right)$ by

$f(x)=\left\{\begin{array}{l}\frac{1}{x} \log _{e}\left(\frac{1+3 x}{1-2 x}\right), \text { when } x \neq 0 \\ k, \text { when } x=0\end{array}\right.$ is continuous, then $k$ is equal to

[NA Jan. 7, 2020 (II)]

Sol.

Q. Let $f: \mathrm{R} \rightarrow \mathrm{R}$ be a function defined by $f(x)=\max \left\{x, x^{2}\right\}$. Let $\mathrm{S}$ denote the set of all points in $\mathrm{R}$, where $f$ is not differentiable. Then:

(a) $\{0,1\}$

(b) $\{0\}$

(c) $\phi$ (an empty set)

(d) $\{1\}$

[Sep. 06, 2020 (II)]

Sol.

Q. If the function $f(x)\left\{\begin{array}{ll}k_{1}(x-\pi)^{2}-1, & x \leq \pi \\ k_{2} \cos x, & x>\pi\end{array}\right.$ is twice differentiable, then the ordered pair $\left(k_{1}, k_{2}\right)$ is equal to:

(a) $\left(\frac{1}{2}, 1\right)$

(b) $(1,0)$

(c) $\left(\frac{1}{2},-1\right)$

(d) $(1,1)$

[Sep. 05, 2020 (I)]

Sol.

Q. Let $f$ be a twice differentiable function on $(1,6)$. If $f(2)=8$, $f^{\prime}(2)=5, f^{\prime}(x) \geq 1$ and $f^{\prime \prime}(x) \geq 4$, for all $x \in(1,6)$, then :

(a) $f(5)+f^{\prime}(5) \leq 26$

(b) $f(5)+f^{\prime}(5) \geq 28$

(c) $f^{\prime}(5)+f^{\prime \prime}(5) \leq 20$

(d) $f(5) \leq 10$

[Sep. 04, 2020 (I)]

Sol.

Q. Suppose a differentiable function $f(x)$ satisfies the identity $f(x+y)=f(x)+f(y)+x y^{2}+x^{2} y$, for all real $x$ and $y$. If $\lim _{x \rightarrow 0} \frac{f(x)}{x}=1$, then $f^{\prime}(3)$ is equal to

[NA Sep. 04, 2020 (I)]

Sol.

Q. The function $f(x)=\left\{\begin{array}{l}\frac{\pi}{4}+\tan ^{-1} x,|x| \leq 1 \\ \frac{1}{2}(|x|-1),|x|>1\end{array}\right.$ is:

(a) continuous on $\mathbf{R}-\{1\}$ and differentiable on $\mathbf{R}-\{-1,1\}$

(b) both continuous and differentiable on $\mathbf{R}-\{1\}$.

(c) continuous on $\mathbf{R}-\{-1\}$ and differentiable on $\mathbf{R}-\{-1,1\}$

(d) both continuous and differentiable on $\mathbf{R}-\{-1\}$.

[Sep. 04, 2020 (II)]

Sol.

Q. If $f(x)=\left\{\begin{array}{lll}\frac{\sin (a+2) x+\sin x}{x} & ; & x<0 \\ b & ; & x=0 \\ \frac{\left(x+3 x^{2}\right)^{1 / 3}-x^{1 / 3}}{x^{4 / 3}} & ; & x>0\end{array}\right.$

is continuous at $x=0$, then $a+2 b$ is equal to:

(a) 1

(b) $-1$

(c) 0

(d) $-2$

[Jan. 9, 2020 (I)]

Sol.

Q. Let $f$ and $g$ be differentiable functions on $\mathbf{R}$ such that fog is the identity function. If for some $a, b \in \mathbf{R}, g^{\prime}(a)=5$ and $g(a)=b$, then $f^{\prime}(b)$ is equal to:

(a) $\frac{1}{5}$

(b) 1

(c) 5

(d) $\frac{2}{5}$

[Jan.9,2020 (II)]

Sol.

On this page you will get Continuity and differentiability JEE Main Previous year Questions with complete and detailed solution. These solutions are arranged in chronological order.

Continuity and differentiability JEE Main Previous year Questions with solution

Q. Let $f(x)=x .\left[\frac{x}{2}\right]$, for $-10<x<10$, where $[t]$ denotes the greatest integer function. Then the number of points of discontinuity of $f$ is equal to

[NA Sep. 05, 2020 (I)]

Sol.

Q. If a function $f(x)$ defined by

$f(x)=\left\{\begin{array}{ll}a e^{x}+b e^{-x}, & -1 \leq x<1 \\ c x^{2} & , 1 \leq x \leq 3 \\ a x^{2}+2 c x, & 3<x \leq 4\end{array}\right.$ be continuous for some $a, b, c \in \mathbf{R}$ and $f^{\prime}(0)+f^{\prime}(2)=e$, then the value of $a$ is :

(a) $\frac{1}{e^{2}-3 e+13}$

(b) $\frac{e}{e^{2}-3 e-13}$

(c) $\frac{e}{e^{2}+3 e+13}$

(d) $\frac{e}{e^{2}-3 e+13}$

[Sep. 02, 2020 (I)]

Sol.

Q. Let $[t]$ denote the greatest integer $\leq t$ and $\lim _{x \rightarrow 0} x\left[\frac{4}{x}\right]=\mathrm{A}$. Then the function, $f(x)=\left[x^{2}\right] \sin (\pi x)$ is discontinuous, when $x$ is equal to :

(a) $\sqrt{\mathrm{A}+1}$

(b) $\sqrt{\mathrm{A}+5}$

(c) $\sqrt{\mathrm{A}+21}$

(d) $\sqrt{\mathrm{A}}$

[Jan. 9, 2020 (II)]

Sol.

Q. If the function $f$ defined on $\left(-\frac{1}{3}, \frac{1}{3}\right)$ by

$f(x)=\left\{\begin{array}{l}\frac{1}{x} \log _{e}\left(\frac{1+3 x}{1-2 x}\right), \text { when } x \neq 0 \\ k, \text { when } x=0\end{array}\right.$ is continuous, then $k$ is equal to

[NA Jan. 7, 2020 (II)]

Sol.

Q. Let $f: \mathrm{R} \rightarrow \mathrm{R}$ be a function defined by $f(x)=\max \left\{x, x^{2}\right\}$. Let $\mathrm{S}$ denote the set of all points in $\mathrm{R}$, where $f$ is not differentiable. Then:

(a) $\{0,1\}$

(b) $\{0\}$

(c) $\phi$ (an empty set)

(d) $\{1\}$

[Sep. 06, 2020 (II)]

Sol.

Q. If the function $f(x)\left\{\begin{array}{ll}k_{1}(x-\pi)^{2}-1, & x \leq \pi \\ k_{2} \cos x, & x>\pi\end{array}\right.$ is twice differentiable, then the ordered pair $\left(k_{1}, k_{2}\right)$ is equal to:

(a) $\left(\frac{1}{2}, 1\right)$

(b) $(1,0)$

(c) $\left(\frac{1}{2},-1\right)$

(d) $(1,1)$

[Sep. 05, 2020 (I)]

Sol.

Q. Let $f$ be a twice differentiable function on $(1,6)$. If $f(2)=8$, $f^{\prime}(2)=5, f^{\prime}(x) \geq 1$ and $f^{\prime \prime}(x) \geq 4$, for all $x \in(1,6)$, then :

(a) $f(5)+f^{\prime}(5) \leq 26$

(b) $f(5)+f^{\prime}(5) \geq 28$

(c) $f^{\prime}(5)+f^{\prime \prime}(5) \leq 20$

(d) $f(5) \leq 10$

[Sep. 04, 2020 (I)]

Sol.

Q. Suppose a differentiable function $f(x)$ satisfies the identity $f(x+y)=f(x)+f(y)+x y^{2}+x^{2} y$, for all real $x$ and $y$. If $\lim _{x \rightarrow 0} \frac{f(x)}{x}=1$, then $f^{\prime}(3)$ is equal to

[NA Sep. 04, 2020 (I)]

Q. The function $f(x)=\left\{\begin{array}{l}\frac{\pi}{4}+\tan ^{-1} x,|x| \leq 1 \\ \frac{1}{2}(|x|-1),|x|>1\end{array}\right.$ is:

(a) continuous on $\mathbf{R}-\{1\}$ and differentiable on $\mathbf{R}-\{-1,1\}$

(b) both continuous and differentiable on $\mathbf{R}-\{1\}$.

(c) continuous on $\mathbf{R}-\{-1\}$ and differentiable on $\mathbf{R}-\{-1,1\}$

(d) both continuous and differentiable on $\mathbf{R}-\{-1\}$.

[Sep. 04, 2020 (II)]

Sol.

Q. If $f(x)=\left\{\begin{array}{lll}\frac{\sin (a+2) x+\sin x}{x} & ; & x<0 \\ b & ; & x=0 \\ \frac{\left(x+3 x^{2}\right)^{1 / 3}-x^{1 / 3}}{x^{4 / 3}} & ; & x>0\end{array}\right.$

is continuous at $x=0$, then $a+2 b$ is equal to:

(a) 1

(b) $-1$

(c) 0

(d) $-2$

[Jan. 9, 2020 (I)]

Sol.

Q. Let $f$ and $g$ be differentiable functions on $\mathbf{R}$ such that fog is the identity function. If for some $a, b \in \mathbf{R}, g^{\prime}(a)=5$ and $g(a)=b$, then $f^{\prime}(b)$ is equal to:

(a) $\frac{1}{5}$

(b) 1

(c) 5

(d) $\frac{2}{5}$

[Jan.9,2020 (II)]

Sol.

Q. Let $\mathrm{S}$ be the set of all functions $f:[0,1] \rightarrow R$, which are continuous on $[0,1]$ and differentiable on $(0,1)$. Then for every $f$ in $S$, there exists a $c \in(0,1)$, depending on $f$, such that:

(a) $|f(c)-f(1)|<(1-c)\left|f^{\prime}(c)\right|$

(b) $\frac{f(1)-f(c)}{1-c}=f^{\prime}(c)$

(c) $|f(c)+f(1)|<(1+c)\left|f^{\prime}(c)\right|$

(d) $|f(c)-f(1)|<\left|f^{\prime}(c)\right|$

[Jan. 8,2020 (II)]

Sol.

Q. Let the function, $f:[-7,0] \rightarrow R$ be continuous on $[-7,0]$ and differentiable on $(-7,0)$. If $f(-7)=-3$ and $f^{\prime}(x) \mathrm{d} ” 2$, for all $x \in(-7,0)$, then for all such functions $f, f^{\prime}(-1)+f(0)$ lies in the interval:

(a) $(-\infty, 20]$

(b) $[-3,11]$

(c) $(-\infty, 11]$

(d) $[-6,20]$

[Jan. 7, 2020 (I)]

Sol.

Q. Let $S$ be the set of points where the function,

$f(x)=|2-| x-3||, x \in \boldsymbol{R}$, is not differentiable.

Then $\sum_{x \in S} f(f(x))$ is equal to

[NAJan. 7, 2020(I)]

Sol.

Q. The derivative of $\tan ^{-1}\left(\frac{\sqrt{1+x^{2}}-1}{x}\right)$ with respect to $\tan ^{-1}\left(\frac{2 x \sqrt{1-x^{2}}}{1-2 x^{2}}\right)$ at $x=\frac{1}{2}$ is :

(a) $\frac{2 \sqrt{3}}{5}$

(b) $\frac{\sqrt{3}}{12}$

(c) $\frac{2 \sqrt{3}}{3}$

(d) $\frac{\sqrt{3}}{10}$

[Sep. 05,2020 (II)]

Sol.

Q. If $(a+\sqrt{2} b \cos x)(a-\sqrt{2} b \cos y)=a^{2}-b^{2}$, where $a>b>0$, then $\frac{d x}{d y}$ at $\left(\frac{\pi}{4}, \frac{\pi}{4}\right)$ is:

(a) $\frac{a-2 b}{a+2 b}$

(b) $\frac{a-b}{a+b}$

(c) $\frac{a+b}{a-b}$

(d) $\frac{2 a+b}{2 a-b}$

[Sep. 04, 2020 (I)]

Sol.

Q. If $y=\sum_{k=1}^{6} k \cos ^{-1}\left\{\frac{3}{5} \cos k x-\frac{4}{5} \sin k x\right\}$, then $\frac{d y}{d x}$ at $x=0$ is

[NA Sep. 02, 2020 (II)]

Sol.

Q. If $x=2 \sin \theta-\sin 2 \theta$ and $y=2 \cos \theta-\cos 2 \theta, \theta \in[0,2 \pi]$, then $\frac{d^{2} y}{d x^{2}}$ at $\theta=\pi$ is :

(a) $\frac{3}{4}$

(b) $-\frac{3}{8}$

(c) $\frac{3}{2}$

(d) $-\frac{3}{4}$

[Jan. 9, 2020 (II)]

Sol.

Q. If $y(\alpha)=\sqrt{2\left(\frac{\tan \alpha+\cot \alpha}{1+\tan ^{2} \alpha}\right)+\frac{1}{\sin ^{2} \alpha}}, \alpha \in\left(\frac{3 \pi}{4}, \pi\right)$, then $\frac{d y}{d \alpha}$ at $\alpha=\frac{5 \pi}{6}$ is:

(a) 4

(b) $\frac{4}{3}$

(c) $-4$

(d) $-\frac{1}{4}$

[Jan. 7, 2020 (I)]

Sol.

Q. Let $y=y(x)$ be $a$ function of $x$ satisfying

$y \sqrt{1-x^{2}}=k-x \sqrt{1-y^{2}}$ where $k$ is $a$ constant and $y\left(\frac{1}{2}\right)=-\frac{1}{4}$. Then $\frac{d y}{d x}$ at $x=\frac{1}{2}$, is equal to:

(a) $-\frac{\sqrt{5}}{4}$

(b) $-\frac{\sqrt{5}}{2}$

(c) $\frac{2}{\sqrt{5}}$

(d) $\frac{\sqrt{5}}{2}$

[Jan. 7, 2020 (II)]

Sol.

Q. For all twice differentiable functios $f: \mathrm{R} \rightarrow \mathrm{R}$, with $f(0)=f(1)=f^{\prime}(0)=0$

(a) $f^{\prime \prime}(x) \neq 0$ at every point $x \in(0,1)$

(b) $f^{\prime \prime}(x)=0$, for some $x \in(0,1)$

(c) $f^{\prime \prime}(0)=0$

(d) $f^{\prime \prime}(x)=0$, at every point $x \in(0,1)$

[Sep. 06, 2020 (II)]

Sol.

Q. If $y^{2}+\log _{e}\left(\cos ^{2} x\right)=y, x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, then :

(a) $y^{\prime \prime}(0)=0$

(b) $\left|y^{\prime}(0)\right|+\left|y^{\prime \prime}(0)\right|=1$

(c) $\left|y^{\prime \prime}(0)\right|=2$

(d) $\left|y^{\prime}(0)\right|+\left|y^{\prime \prime}(0)\right|=3$

[Sep. 03, 2020 (I)]

Sol.

Q. If $c$ is a point at which Rolle’s theorem holds for the function, $f(x)=\log _{e}\left(\frac{x^{2}+a}{7 x}\right)$ in the interval $[3,4]$, where $\alpha \in R$, then $f^{\prime \prime}(c)$ is equal to:

(a) $-\frac{1}{12}$

(b) $\frac{1}{12}$

(c) $-\frac{1}{24}$

(d) $\frac{\sqrt{3}}{7}$

[Jan. 8, 2020 (I)]

Sol.

Q. Let $x^{k}+y^{k}=a^{k},(a, k>0)$ and $\frac{d y}{d x}+\left(\frac{y}{x}\right)^{\frac{1}{3}}=0$, then $k$ is:

(a) $\frac{3}{2}$

(b) $\frac{4}{3}$

(c) $\frac{2}{3}$

(d) $\frac{1}{3}$

[Jan. 7, 2020 (I)]

Sol.

Q. The value of $c$ in the Lagrange’s mean value theorem for the function $f(x)=x^{3}-4 x^{2}+8 x+11$, when $x \in[0,1]$ is:

(a) $\frac{4-\sqrt{5}}{3}$

(b) $\frac{4-\sqrt{7}}{3}$

(c) $\frac{2}{3}$

(d) $\frac{\sqrt{7}-2}{3}$

[Jan. 7, 2020 (II)]

Sol.

2019

Q. If the function $\mathrm{f}$ defined on $\left(\frac{\pi}{6}, \frac{\pi}{3}\right)$ by $f(x)=\left\{\begin{array}{cc}\frac{\sqrt{2} \cos x-1}{\cot x-1}, & x \neq \frac{\pi}{4} \\ k, & x=\frac{\pi}{4}\end{array}\right.$ is continuous, then $\mathrm{k}$ is equal to:

(a) 2

(b) $\frac{1}{2}$

(c) 1

(d) $\frac{1}{\sqrt{2}}$

[April 09, $2019($ I)]

Sol.

Q. If $f(x)=[x]-\left[\frac{x}{4}\right], x \in \mathrm{R}$, where $[x]$ denotes the greatest integer function, then:

(a) $f$ is continuous at $x=4$.

(b) $\lim _{x \rightarrow 4+} f(x)$ exists but $\lim _{x \rightarrow 4-} f(x)$ does not exist.

(c) Both $\lim _{x \rightarrow 4-} f(x)$ and $\lim _{x \rightarrow 4+} f(x)$ exist but are not equal.

(d) $\lim _{x \rightarrow 4-} f(x)$ exists but $\lim _{x \rightarrow 4+} f(x)$ does not exist.

[April 09, 2019 (II)]

Sol.

Q. If the function

$$f(x)=\left\{\begin{array}{l} a|\pi-x|+1, x \leq 5 \\ b|x-\pi|+3, x>5 \end{array}\right.$$

is continuous at $x=5$, then the value of $a-b$ is:

(a) $\frac{2}{\pi+5}$

(b) $\frac{-2}{\pi+5}$

(c) $\frac{2}{\pi-5}$

(d) $\frac{2}{5-\pi}$

[April 09, 2019 (II)]

Sol.

Q. Let $f:[-1,3] \rightarrow \mathrm{R}$ be defined as

$$f(x)=\left\{\begin{array}{lc} |x|+[x], & -1 \leq \mathrm{x}<1 \\ x+|x|, & 1 \leq x<2 \\ x+[x], & 2 \leq x \leq 3 \end{array}\right.$$

where $[t]$ denotes the greatest integer less than or equal to $t$. Then, $f$ is discontinuous at :

(a) only one point

(b) only two points

(c) only three points

(d) four or more points

[April 08, 2019 (II)]

Sol.

Q. Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a function defined as

$$f(x)=\left\{\begin{array}{ccc} 5, & \text { if } & x \leq 1 \\ \mathrm{a}+\mathrm{b} x, & \text { if } & 1<x<3 \\ \mathrm{~b}+5 x, & \text { if } & 3 \leq x<5 \\ 30, & \text { if } & x \geq 5 \end{array}\right.$$

Then, $f$ is :

(a) continuous if $\mathrm{a}=5$ and $\mathrm{b}=5$

(b) continuous if $a=-5$ and $b=10$

(c) continous if $a=0$ and $b=5$

(d) not continuous for any values of $a$ and $b$

$[$ Jan 09, 2019 (I)]

Sol.

Q. If $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{cc}\frac{\sin (\mathrm{p}+1) x+\sin x}{x}, & x<0 \\ \mathrm{q} & , x=0 \\ \frac{\sqrt{x+x^{2}}-\sqrt{x}}{x^{3 / 2}}, & x>0\end{array}\right.$

is continuous at $\mathrm{x}=0$, then the ordered pair $(\mathrm{p}, \mathrm{q})$ is equal to:

(a) $\left(-\frac{3}{2},-\frac{1}{2}\right)$

(b) $\left(-\frac{1}{2}, \frac{3}{2}\right)$

(c) $\left(-\frac{3}{2}, \frac{1}{2}\right)$

(d) $\left(\frac{5}{2}, \frac{1}{2}\right)$

[April 10, 2019 (I)]

Sol.

Q. Let $f(x)=\log _{\mathrm{e}}(\sin x),(0<x<\pi)$ and $g(x)=\sin ^{-1}\left(e^{-x}\right),(x \geq 0)$. If $\alpha$ is a positive real number such that $a=(f \circ g)^{\prime}(\alpha)$ and $b=(f \circ g)(\alpha)$, then:

(a) $a \alpha^{2}+b \alpha+a=0$

(b) $a \alpha^{2}-b \alpha-a=1$

(c) $a \alpha^{2}-b \alpha-\mathrm{a}=0$

(d) $a \alpha^{2}+\mathrm{b} \alpha-a=-2 a^{2}$

[April 10, 2019 (II)]

Sol.

Q. Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be differentiable at $\mathrm{c} \in \mathbf{R}$ and $\mathrm{f}(\mathrm{c})=0$. If $g(x)=|f(x)|$, then at $x=c, g$ is :

(a) not differentiable if $\mathrm{f}^{\prime}(\mathrm{c})=0$

(b) differentiable if $f^{\prime \prime}(c) \neq 0$

(c) differentiable if $f^{\prime}(\mathrm{c})=0$

(d) not differentiable

[April 10, 2019 (I)]

Sol.

Q. Let $f(x)=15-|x-10| ; x \in R$. Then the set of all values of $x$, at which the function, $g(x)=f(f(x))$ is not differentiable, is:

(a) $\{5,10,15\}$

(b) $\{10,15\}$

(c) $\{5,10,15,20\}$

(d) $\{10\}$

[April 09, 2019 (I)]

Sol.

Q. If $f(1)=1, f^{\prime}(1)=3$, then the derivative of $f(f(f(x)))+(f(x))^{2}$ at $x=1$ is:

(a) 33

(b) 12

(c) 15

(d) 9

[April 08,2019(II)]

Sol.

Q. Let $f$ be a differentiable function such that $f(1)=2$ and $f^{\prime}(x)=f(x)$ for all $x \in R$. If $h(x)=f(f(x))$, then $h^{\prime}(1)$ is equal to :

(a) $2 \mathrm{e}^{2}$

(b) $4 \mathrm{e}$

(c) $2 \mathrm{e}$

(d) $4 \mathrm{e}^{2}$

[Jan. 12, 2019 (II)]

Sol.

Q. Let $f(x)=\left\{\begin{array}{cc}-1, & -2 \leq x<0 \\ x^{2}-1, & 0 \leq x \leq 2\end{array}\right.$ and $g(x)=|f(x)|+f(|x|)$. Then, in the interval $(-2,2), g$ is :

(a) differentiable at all points

(b) not continuous

(c) not differentiable at two points

(d) not differentiable at one point

Sol.

Q. If $x \log _{e}\left(\log _{e} x\right)-x^{2}+y^{2}=4(y>0)$, then $\frac{d y}{d x}$ at $\mathrm{x}=\mathrm{e}$ is equal to :

(a) $\frac{(1+2 e)}{2 \sqrt{4+e^{2}}}$

(b) $\frac{(2 e-1)}{2 \sqrt{4+e^{2}}}$

c) $\frac{(1+2 e)}{\sqrt{4+e^{2}}}$

(d) $\frac{e}{\sqrt{4+e^{2}}}$

[Jan. 11, 2019 (I)]

Sol.

Q. Let $\mathrm{K}$ be the set of all real values of $x$ where the function $f(x)=\sin |x|-|x|+2(x-\pi) \cos |x|$ is not differentiable. Then the set $\mathrm{K}$ is equal to :

(a) $\phi$ (an empty set)

(b) $\{\pi\}$

(c) $\{0\}$

(d) $\{0, \pi\}$

[Jan. 11, 2019 (II)]

Sol.

Q. Let $f(x)=\left\{\begin{array}{cc}\max \left\{|x|, x^{2}\right\} & |x| \leq 2 \\ 8-2|x|, & 2<|x| \leq 4\end{array}\right.$

Let $\mathrm{S}$ be the set of points in the interval $(-4,4)$ at which $f$ is not differentiable. Then $\mathrm{S}$ :

(a) is an empty set

(b) equals $\{-2,-1,0,1,2\}$

(c) equals $\{-2,-1,1,2\}$

(d) equals $\{-2,2\}$

[Jan 10,2019(I)]

Sol.

Q. Let $f:(-1,1) \rightarrow \mathbf{R}$ be a function defined by $f(x)=\max$ $\left\{-|x|,-\sqrt{1-x^{2}}\right\}$. If $\mathrm{K}$ be the set of all points at which $f$ is not differentiable, then $\mathrm{K}$ has exactly:

(a) five elements

(b) one element

(c) three elements

(d) two elements

[Jan. 10, 2019 (II)]

Sol.

Q. If $\mathrm{e}^{y}+x y=e$, the ordered pair $\left(\frac{d y}{d x}, \frac{d^{2} y}{d x^{2}}\right)$ at $x=0$ is equal to :

(a) $\left(\frac{1}{e},-\frac{1}{e^{2}}\right)$

(b) $\left(-\frac{1}{e}, \frac{1}{e^{2}}\right)$

(c) $\left(\frac{1}{e}, \frac{1}{e^{2}}\right)$

(d) $\left(-\frac{1}{e},-\frac{1}{e^{2}}\right)$

[April 12, 2019 (I)]

Sol.

Q. The derivative of $\tan ^{-1}\left(\frac{\sin x-\cos x}{\sin x+\cos x}\right)$, with respect to $\frac{x}{2}$, where $\left(x \in\left(0, \frac{\pi}{2}\right)\right)$ is :

(a) 1

(b) $\frac{2}{3}$

(c) $\frac{1}{2}$

(d) 2

[April 12, 2019 (II)]

Sol.

Q. If $2 y=\left(\cot ^{-1}\left(\frac{\sqrt{3} \cos x+\sin x}{\cos x-\sqrt{3} \sin x}\right)\right)^{2}, x \in\left(0, \frac{\pi}{2}\right)$ then $\frac{d y}{d x}$ is equal to :

(a) $\frac{\pi}{6}-x$

(b) $x-\frac{\pi}{6}$

(c) $\frac{\pi}{3}-x$

(d) $2 x-\frac{\pi}{3}$

[April 08, 2019 (I)]

Sol.

Q. Let $S$ be the set of all points in $(-\pi, \pi)$ at which the function $\mathrm{f}(\mathrm{x})=\min \{\sin \mathrm{x}, \cos \mathrm{x}\}$ is not differentiable. Then $S$ is a subset of which of the following?

(a) $\left\{-\frac{\pi}{4}, 0, \frac{\pi}{4}\right\}$

(b) $\left\{-\frac{3 \pi}{4},-\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{\pi}{4}\right\}$

(c) $\left\{-\frac{\pi}{2},-\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{2}\right\}$

(d) $\left\{-\frac{3 \pi}{4},-\frac{\pi}{2}, \frac{\pi}{2}, \frac{3 \pi}{4}\right\}$

[Jan. 12, 2019 (I)]

Sol.

Q. For $\mathrm{x}>1$, if $(2 x)^{2 y}=4 e^{2 x-2 y}$, then $\left(1+\log _{e} 2 x\right)^{2} \frac{d y}{d x}$ is equal to:

(a) $\frac{x \log _{e} 2 x-\log _{e} 2}{x}$

(b) $\log _{\mathrm{e}} 2 \mathrm{x}$

(c) $\frac{x \log _{e} 2 x+\log _{e} 2}{x}$

(d) $x \log _{\mathrm{e}} 2 \mathrm{x}$

$[$ Jan. 12, 2019 (I)]

Sol.

Q. Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a function such that $f(x)=x^{3}+x^{2} f^{\prime}(1)+x f^{\prime \prime}(2)+f^{\prime \prime \prime}(3), x \in \mathbf{R}$. Then $f(2)$ equals:

(a) $-4$

(b) 30

(c) $-2$

(d) 8

[Jan 10, 2019 (I)]

Sol.

Q. If $x=3 \tan t$ and $y=3 \sec t$, then the value of $\frac{d^{2} y}{d x^{2}}$ at $\mathrm{t}=\frac{\pi}{4}$, is:

(a) $\frac{1}{3 \sqrt{2}}$

(b) $\frac{1}{6 \sqrt{2}}$

(c) $\frac{3}{2 \sqrt{2}}$

(d) $\frac{1}{6}$

$[$ Jan. 09, 2019(II)]

Sol.

2018

Q. If the function $f$ defined as

$$f(x)=\frac{1}{x}-\frac{k-1}{e^{2 x}-1}$$

$x \neq 0$, is continuous at $x=0$,

then the ordered pair $(k, f(0))$ is equal to?

(a) $(3,1)$

(b) $(3,2)$

(c) $\left(\frac{1}{3}, 2\right)$

(d) $(2,1)$

[Online April 16, 2018]

Sol.

Q. Let $f(x)=\left\{\begin{array}{cc}(x-1)^{\frac{1}{2-x}}, & x>1, x \neq 2 \\ k, & x=2\end{array}\right.$

The value of $k$ for which $f$ is continuous at $x=2$ is

(a) $e^{-2}$

(b) $e$

(c) $e^{-1}$

(d) 1

[Online April 15, 2018]

Sol.

Q. Let $\mathrm{S}=\left\{\mathrm{t} \in \mathrm{R}: \mathrm{f}(\mathrm{x})=|\mathrm{x}-\pi|\left(\mathrm{e}^{|\mathrm{x}|}-1\right) \sin |\mathrm{x}|\right.$ is not differentiable at $\mathrm{t}\}$. Then the set $\mathrm{S}$ is equal to :

(a) $\{0\}$

(b) $\{\pi\}$

(c) $\{0, \pi\}$

(d) $\phi($ an empty set)

[2018]

Sol.

Q. Let $S=\left\{(\lambda, \mu) \in R \times R: f(t)=\left(|\lambda| \mathrm{e}^{|t|}-\mu\right)\right.$. $\sin (2|t|), t \in R$, is a differentiable function $\}$. Then $S$ is a subest of?

(a) $R \times[0, \infty)$

(b) $(-\infty, 0) \times R$

(c) $[0, \infty) \times R$

(d) $R \times(-\infty, 0)$

[Online April 15, 2018]

Sol.

Q. If $x=\sqrt{2^{\operatorname{cosec}^{-1} t}}$ and $y=\sqrt{2^{\sec ^{-1} t}}(|\mathrm{t}| \geq 1)$, then $\frac{d y}{d x}$ is equal to.

(a) $\frac{y}{x}$

(b) $-\frac{y}{x}$

(c) $-\frac{x}{y}$

(d) $\frac{x}{y}$

[Online April 16, 2018]

Sol.

Q. If $f(x)=\left|\begin{array}{ccc}\cos x & x & 1 \\ 2 \sin x & x^{2} & 2 x \\ \tan x & x & 1\end{array}\right|$, then $\lim _{x \rightarrow 0} \frac{f^{\prime}(\mathrm{x})}{x}$

(a) Exists and is equal to $-2$

(b) Does not exist

(c) Exist and is equal to 0

(d) Exists and is equal to 2

[Online April 15, 2018]

Sol.

Q. If $f(x)=\sin ^{-1}\left(\frac{2 \times 3^{x}}{1+9^{x}}\right)$, then $f^{\prime}\left(-\frac{1}{2}\right)$ equals.

(a) $\sqrt{3} \log _{e} \sqrt{3}$

(b) $-\sqrt{3} \log _{e} \sqrt{3}$

(c) $-\sqrt{3} \log _{e} 3$

(d) $\sqrt{3} \log _{e} 3$

[Online April 15, 2018]

Sol.

Q. If $x^{2}+y^{2}+\sin y=4$, then the value of $\frac{d^{2} y}{d x^{2}}$ at the point $(-2,0)$ is

(a) $-34$

(b) $-32$

(c) $-2$

(d) 4

[Online April 15, 2018]

Sol.

Q. Let $\mathrm{S}$ be the set of all functions $f:[0,1] \rightarrow R$, which are continuous on $[0,1]$ and differentiable on $(0,1)$. Then for every $f$ in $S$, there exists a $c \in(0,1)$, depending on $f$, such that:

(a) $|f(c)-f(1)|<(1-c)\left|f^{\prime}(c)\right|$

(b) $\frac{f(1)-f(c)}{1-c}=f^{\prime}(c)$

(c) $|f(c)+f(1)|<(1+c)\left|f^{\prime}(c)\right|$

(d) $|f(c)-f(1)|<\left|f^{\prime}(c)\right|$

[Jan. 8,2020 (II)]

Sol.

Q. Let the function, $f:[-7,0] \rightarrow R$ be continuous on $[-7,0]$ and differentiable on $(-7,0)$. If $f(-7)=-3$ and $f^{\prime}(x) \mathrm{d} ” 2$, for all $x \in(-7,0)$, then for all such functions $f, f^{\prime}(-1)+f(0)$ lies in the interval:

(a) $(-\infty, 20]$

(b) $[-3,11]$

(c) $(-\infty, 11]$

(d) $[-6,20]$

[Jan. 7, 2020 (I)]

Sol.

Q. Let $S$ be the set of points where the function,

$f(x)=|2-| x-3||, x \in \boldsymbol{R}$, is not differentiable.

Then $\sum_{x \in S} f(f(x))$ is equal to

[NAJan. 7, 2020(I)]

Sol.

Q. The derivative of $\tan ^{-1}\left(\frac{\sqrt{1+x^{2}}-1}{x}\right)$ with respect to $\tan ^{-1}\left(\frac{2 x \sqrt{1-x^{2}}}{1-2 x^{2}}\right)$ at $x=\frac{1}{2}$ is :

(a) $\frac{2 \sqrt{3}}{5}$

(b) $\frac{\sqrt{3}}{12}$

(c) $\frac{2 \sqrt{3}}{3}$

(d) $\frac{\sqrt{3}}{10}$

[Sep. 05,2020 (II)]

Sol.

Q. If $(a+\sqrt{2} b \cos x)(a-\sqrt{2} b \cos y)=a^{2}-b^{2}$, where $a>b>0$, then $\frac{d x}{d y}$ at $\left(\frac{\pi}{4}, \frac{\pi}{4}\right)$ is:

(a) $\frac{a-2 b}{a+2 b}$

(b) $\frac{a-b}{a+b}$

(c) $\frac{a+b}{a-b}$

(d) $\frac{2 a+b}{2 a-b}$

[Sep. 04, 2020 (I)]

Sol.

Q. If $y=\sum_{k=1}^{6} k \cos ^{-1}\left\{\frac{3}{5} \cos k x-\frac{4}{5} \sin k x\right\}$, then $\frac{d y}{d x}$ at $x=0$ is

[NA Sep. 02, 2020 (II)]

Sol.

Q. If $x=2 \sin \theta-\sin 2 \theta$ and $y=2 \cos \theta-\cos 2 \theta, \theta \in[0,2 \pi]$, then $\frac{d^{2} y}{d x^{2}}$ at $\theta=\pi$ is :

(a) $\frac{3}{4}$

(b) $-\frac{3}{8}$

(c) $\frac{3}{2}$

(d) $-\frac{3}{4}$

[Jan. 9, 2020 (II)]

Sol.

Q. If $y(\alpha)=\sqrt{2\left(\frac{\tan \alpha+\cot \alpha}{1+\tan ^{2} \alpha}\right)+\frac{1}{\sin ^{2} \alpha}}, \alpha \in\left(\frac{3 \pi}{4}, \pi\right)$, then $\frac{d y}{d \alpha}$ at $\alpha=\frac{5 \pi}{6}$ is:

(a) 4

(b) $\frac{4}{3}$

(c) $-4$

(d) $-\frac{1}{4}$

[Jan. 7, 2020 (I)]

Sol.

Q. Let $y=y(x)$ be $a$ function of $x$ satisfying

$y \sqrt{1-x^{2}}=k-x \sqrt{1-y^{2}}$ where $k$ is $a$ constant and $y\left(\frac{1}{2}\right)=-\frac{1}{4}$. Then $\frac{d y}{d x}$ at $x=\frac{1}{2}$, is equal to:

(a) $-\frac{\sqrt{5}}{4}$

(b) $-\frac{\sqrt{5}}{2}$

(c) $\frac{2}{\sqrt{5}}$

(d) $\frac{\sqrt{5}}{2}$

[Jan. 7, 2020 (II)]

Sol.

Q. For all twice differentiable functios $f: \mathrm{R} \rightarrow \mathrm{R}$, with $f(0)=f(1)=f^{\prime}(0)=0$

(a) $f^{\prime \prime}(x) \neq 0$ at every point $x \in(0,1)$

(b) $f^{\prime \prime}(x)=0$, for some $x \in(0,1)$

(c) $f^{\prime \prime}(0)=0$

(d) $f^{\prime \prime}(x)=0$, at every point $x \in(0,1)$

[Sep. 06, 2020 (II)]

Sol.

Q. If $y^{2}+\log _{e}\left(\cos ^{2} x\right)=y, x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, then :

(a) $y^{\prime \prime}(0)=0$

(b) $\left|y^{\prime}(0)\right|+\left|y^{\prime \prime}(0)\right|=1$

(c) $\left|y^{\prime \prime}(0)\right|=2$

(d) $\left|y^{\prime}(0)\right|+\left|y^{\prime \prime}(0)\right|=3$

[Sep. 03, 2020 (I)]

Sol.

Q. If $c$ is a point at which Rolle’s theorem holds for the function, $f(x)=\log _{e}\left(\frac{x^{2}+a}{7 x}\right)$ in the interval $[3,4]$, where $\alpha \in R$, then $f^{\prime \prime}(c)$ is equal to:

(a) $-\frac{1}{12}$

(b) $\frac{1}{12}$

(c) $-\frac{1}{24}$

(d) $\frac{\sqrt{3}}{7}$

[Jan. 8, 2020 (I)]

Sol.

Q. Let $x^{k}+y^{k}=a^{k},(a, k>0)$ and $\frac{d y}{d x}+\left(\frac{y}{x}\right)^{\frac{1}{3}}=0$, then $k$ is:

(a) $\frac{3}{2}$

(b) $\frac{4}{3}$

(c) $\frac{2}{3}$

(d) $\frac{1}{3}$

[Jan. 7, 2020 (I)]

Sol.

Q. The value of $c$ in the Lagrange’s mean value theorem for the function $f(x)=x^{3}-4 x^{2}+8 x+11$, when $x \in[0,1]$ is:

(a) $\frac{4-\sqrt{5}}{3}$

(b) $\frac{4-\sqrt{7}}{3}$

(c) $\frac{2}{3}$

(d) $\frac{\sqrt{7}-2}{3}$

[Jan. 7, 2020 (II)]

Sol.

### 2019

Q. If the function $\mathrm{f}$ defined on $\left(\frac{\pi}{6}, \frac{\pi}{3}\right)$ by $f(x)=\left\{\begin{array}{cc}\frac{\sqrt{2} \cos x-1}{\cot x-1}, & x \neq \frac{\pi}{4} \\ k, & x=\frac{\pi}{4}\end{array}\right.$ is continuous, then $\mathrm{k}$ is equal to:

(a) 2

(b) $\frac{1}{2}$

(c) 1

(d) $\frac{1}{\sqrt{2}}$

[April 09, $2019($ I)]

Sol.

Q. If $f(x)=[x]-\left[\frac{x}{4}\right], x \in \mathrm{R}$, where $[x]$ denotes the greatest integer function, then:

(a) $f$ is continuous at $x=4$.

(b) $\lim _{x \rightarrow 4+} f(x)$ exists but $\lim _{x \rightarrow 4-} f(x)$ does not exist.

(c) Both $\lim _{x \rightarrow 4-} f(x)$ and $\lim _{x \rightarrow 4+} f(x)$ exist but are not equal.

(d) $\lim _{x \rightarrow 4-} f(x)$ exists but $\lim _{x \rightarrow 4+} f(x)$ does not exist.

[April 09, 2019 (II)]

Sol.

Q. If the function

$$f(x)=\left\{\begin{array}{l} a|\pi-x|+1, x \leq 5 \\ b|x-\pi|+3, x>5 \end{array}\right.$$

is continuous at $x=5$, then the value of $a-b$ is:

(a) $\frac{2}{\pi+5}$

(b) $\frac{-2}{\pi+5}$

(c) $\frac{2}{\pi-5}$

(d) $\frac{2}{5-\pi}$

[April 09, 2019 (II)]

Sol.

Q. Let $f:[-1,3] \rightarrow \mathrm{R}$ be defined as

$$f(x)=\left\{\begin{array}{lc} |x|+[x], & -1 \leq \mathrm{x}<1 \\ x+|x|, & 1 \leq x<2 \\ x+[x], & 2 \leq x \leq 3 \end{array}\right.$$

where $[t]$ denotes the greatest integer less than or equal to $t$. Then, $f$ is discontinuous at :

(a) only one point

(b) only two points

(c) only three points

(d) four or more points

[April 08, 2019 (II)]

Sol.

Q. Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a function defined as

$$f(x)=\left\{\begin{array}{ccc} 5, & \text { if } & x \leq 1 \\ \mathrm{a}+\mathrm{b} x, & \text { if } & 1<x<3 \\ \mathrm{~b}+5 x, & \text { if } & 3 \leq x<5 \\ 30, & \text { if } & x \geq 5 \end{array}\right.$$

Then, $f$ is :

(a) continuous if $\mathrm{a}=5$ and $\mathrm{b}=5$

(b) continuous if $a=-5$ and $b=10$

(c) continous if $a=0$ and $b=5$

(d) not continuous for any values of $a$ and $b$

$[$ Jan 09, 2019 (I)]

Sol.

Q. If $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{cc}\frac{\sin (\mathrm{p}+1) x+\sin x}{x}, & x<0 \\ \mathrm{q} & , x=0 \\ \frac{\sqrt{x+x^{2}}-\sqrt{x}}{x^{3 / 2}}, & x>0\end{array}\right.$

is continuous at $\mathrm{x}=0$, then the ordered pair $(\mathrm{p}, \mathrm{q})$ is equal to:

(a) $\left(-\frac{3}{2},-\frac{1}{2}\right)$

(b) $\left(-\frac{1}{2}, \frac{3}{2}\right)$

(c) $\left(-\frac{3}{2}, \frac{1}{2}\right)$

(d) $\left(\frac{5}{2}, \frac{1}{2}\right)$

[April 10, 2019 (I)]

Sol.

Q. Let $f(x)=\log _{\mathrm{e}}(\sin x),(0<x<\pi)$ and $g(x)=\sin ^{-1}\left(e^{-x}\right),(x \geq 0)$. If $\alpha$ is a positive real number such that $a=(f \circ g)^{\prime}(\alpha)$ and $b=(f \circ g)(\alpha)$, then:

(a) $a \alpha^{2}+b \alpha+a=0$

(b) $a \alpha^{2}-b \alpha-a=1$

(c) $a \alpha^{2}-b \alpha-\mathrm{a}=0$

(d) $a \alpha^{2}+\mathrm{b} \alpha-a=-2 a^{2}$

[April 10, 2019 (II)]

Sol.

Q. Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be differentiable at $\mathrm{c} \in \mathbf{R}$ and $\mathrm{f}(\mathrm{c})=0$. If $g(x)=|f(x)|$, then at $x=c, g$ is :

(a) not differentiable if $\mathrm{f}^{\prime}(\mathrm{c})=0$

(b) differentiable if $f^{\prime \prime}(c) \neq 0$

(c) differentiable if $f^{\prime}(\mathrm{c})=0$

(d) not differentiable

[April 10, 2019 (I)]

Sol.

Q. Let $f(x)=15-|x-10| ; x \in R$. Then the set of all values of $x$, at which the function, $g(x)=f(f(x))$ is not differentiable, is:

(a) $\{5,10,15\}$

(b) $\{10,15\}$

(c) $\{5,10,15,20\}$

(d) $\{10\}$

[April 09, 2019 (I)]

Sol.

Q. If $f(1)=1, f^{\prime}(1)=3$, then the derivative of $f(f(f(x)))+(f(x))^{2}$ at $x=1$ is:

(a) 33

(b) 12

(c) 15

(d) 9

[April 08,2019(II)]

Sol.

Q. Let $f$ be a differentiable function such that $f(1)=2$ and $f^{\prime}(x)=f(x)$ for all $x \in R$. If $h(x)=f(f(x))$, then $h^{\prime}(1)$ is equal to :

(a) $2 \mathrm{e}^{2}$

(b) $4 \mathrm{e}$

(c) $2 \mathrm{e}$

(d) $4 \mathrm{e}^{2}$

[Jan. 12, 2019 (II)]

Sol.

Q. Let $f(x)=\left\{\begin{array}{cc}-1, & -2 \leq x<0 \\ x^{2}-1, & 0 \leq x \leq 2\end{array}\right.$ and $g(x)=|f(x)|+f(|x|)$. Then, in the interval $(-2,2), g$ is :

(a) differentiable at all points

(b) not continuous

(c) not differentiable at two points

(d) not differentiable at one point

Sol.

Q. If $x \log _{e}\left(\log _{e} x\right)-x^{2}+y^{2}=4(y>0)$, then $\frac{d y}{d x}$ at $\mathrm{x}=\mathrm{e}$ is equal to :

(a) $\frac{(1+2 e)}{2 \sqrt{4+e^{2}}}$

(b) $\frac{(2 e-1)}{2 \sqrt{4+e^{2}}}$

c) $\frac{(1+2 e)}{\sqrt{4+e^{2}}}$

(d) $\frac{e}{\sqrt{4+e^{2}}}$

[Jan. 11, 2019 (I)]

Sol.

Q. Let $\mathrm{K}$ be the set of all real values of $x$ where the function $f(x)=\sin |x|-|x|+2(x-\pi) \cos |x|$ is not differentiable. Then the set $\mathrm{K}$ is equal to :

(a) $\phi$ (an empty set)

(b) $\{\pi\}$

(c) $\{0\}$

(d) $\{0, \pi\}$

[Jan. 11, 2019 (II)]

Sol.

Q. Let $f(x)=\left\{\begin{array}{cc}\max \left\{|x|, x^{2}\right\} & |x| \leq 2 \\ 8-2|x|, & 2<|x| \leq 4\end{array}\right.$

Let $\mathrm{S}$ be the set of points in the interval $(-4,4)$ at which $f$ is not differentiable. Then $\mathrm{S}$ :

(a) is an empty set

(b) equals $\{-2,-1,0,1,2\}$

(c) equals $\{-2,-1,1,2\}$

(d) equals $\{-2,2\}$

[Jan 10,2019(I)]

Sol.

Q. Let $f:(-1,1) \rightarrow \mathbf{R}$ be a function defined by $f(x)=\max$ $\left\{-|x|,-\sqrt{1-x^{2}}\right\}$. If $\mathrm{K}$ be the set of all points at which $f$ is not differentiable, then $\mathrm{K}$ has exactly:

(a) five elements

(b) one element

(c) three elements

(d) two elements

[Jan. 10, 2019 (II)]

Sol.

Q. If $\mathrm{e}^{y}+x y=e$, the ordered pair $\left(\frac{d y}{d x}, \frac{d^{2} y}{d x^{2}}\right)$ at $x=0$ is equal to :

(a) $\left(\frac{1}{e},-\frac{1}{e^{2}}\right)$

(b) $\left(-\frac{1}{e}, \frac{1}{e^{2}}\right)$

(c) $\left(\frac{1}{e}, \frac{1}{e^{2}}\right)$

(d) $\left(-\frac{1}{e},-\frac{1}{e^{2}}\right)$

[April 12, 2019 (I)]

Sol.

Q. The derivative of $\tan ^{-1}\left(\frac{\sin x-\cos x}{\sin x+\cos x}\right)$, with respect to $\frac{x}{2}$, where $\left(x \in\left(0, \frac{\pi}{2}\right)\right)$ is :

(a) 1

(b) $\frac{2}{3}$

(c) $\frac{1}{2}$

(d) 2

[April 12, 2019 (II)]

Sol.

Q. If $2 y=\left(\cot ^{-1}\left(\frac{\sqrt{3} \cos x+\sin x}{\cos x-\sqrt{3} \sin x}\right)\right)^{2}, x \in\left(0, \frac{\pi}{2}\right)$ then $\frac{d y}{d x}$ is equal to :

(a) $\frac{\pi}{6}-x$

(b) $x-\frac{\pi}{6}$

(c) $\frac{\pi}{3}-x$

(d) $2 x-\frac{\pi}{3}$

[April 08, 2019 (I)]

Sol.

Q. Let $S$ be the set of all points in $(-\pi, \pi)$ at which the function $\mathrm{f}(\mathrm{x})=\min \{\sin \mathrm{x}, \cos \mathrm{x}\}$ is not differentiable. Then $S$ is a subset of which of the following?

(a) $\left\{-\frac{\pi}{4}, 0, \frac{\pi}{4}\right\}$

(b) $\left\{-\frac{3 \pi}{4},-\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{\pi}{4}\right\}$

(c) $\left\{-\frac{\pi}{2},-\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{2}\right\}$

(d) $\left\{-\frac{3 \pi}{4},-\frac{\pi}{2}, \frac{\pi}{2}, \frac{3 \pi}{4}\right\}$

[Jan. 12, 2019 (I)]

Sol.

Q. For $\mathrm{x}>1$, if $(2 x)^{2 y}=4 e^{2 x-2 y}$, then $\left(1+\log _{e} 2 x\right)^{2} \frac{d y}{d x}$ is equal to:

(a) $\frac{x \log _{e} 2 x-\log _{e} 2}{x}$

(b) $\log _{\mathrm{e}} 2 \mathrm{x}$

(c) $\frac{x \log _{e} 2 x+\log _{e} 2}{x}$

(d) $x \log _{\mathrm{e}} 2 \mathrm{x}$

$[$ Jan. 12, 2019 (I)]

Sol.

Q. Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a function such that $f(x)=x^{3}+x^{2} f^{\prime}(1)+x f^{\prime \prime}(2)+f^{\prime \prime \prime}(3), x \in \mathbf{R}$. Then $f(2)$ equals:

(a) $-4$

(b) 30

(c) $-2$

(d) 8

[Jan 10, 2019 (I)]

Sol.

Q. If $x=3 \tan t$ and $y=3 \sec t$, then the value of $\frac{d^{2} y}{d x^{2}}$ at $\mathrm{t}=\frac{\pi}{4}$, is:

(a) $\frac{1}{3 \sqrt{2}}$

(b) $\frac{1}{6 \sqrt{2}}$

(c) $\frac{3}{2 \sqrt{2}}$

(d) $\frac{1}{6}$

$[$ Jan. 09, 2019(II)]

Sol.

2018

Q. If the function $f$ defined as

$$f(x)=\frac{1}{x}-\frac{k-1}{e^{2 x}-1}$$

$x \neq 0$, is continuous at $x=0$,

then the ordered pair $(k, f(0))$ is equal to?

(a) $(3,1)$

(b) $(3,2)$

(c) $\left(\frac{1}{3}, 2\right)$

(d) $(2,1)$

[Online April 16, 2018]

Sol.

Q. Let $f(x)=\left\{\begin{array}{cc}(x-1)^{\frac{1}{2-x}}, & x>1, x \neq 2 \\ k, & x=2\end{array}\right.$

The value of $k$ for which $f$ is continuous at $x=2$ is

(a) $e^{-2}$

(b) $e$

(c) $e^{-1}$

(d) 1

[Online April 15, 2018]

Sol.

Q. Let $\mathrm{S}=\left\{\mathrm{t} \in \mathrm{R}: \mathrm{f}(\mathrm{x})=|\mathrm{x}-\pi|\left(\mathrm{e}^{|\mathrm{x}|}-1\right) \sin |\mathrm{x}|\right.$ is not differentiable at $\mathrm{t}\}$. Then the set $\mathrm{S}$ is equal to :

(a) $\{0\}$

(b) $\{\pi\}$

(c) $\{0, \pi\}$

(d) $\phi($ an empty set)

[2018]

Sol.

Q. Let $S=\left\{(\lambda, \mu) \in R \times R: f(t)=\left(|\lambda| \mathrm{e}^{|t|}-\mu\right)\right.$. $\sin (2|t|), t \in R$, is a differentiable function $\}$. Then $S$ is a subest of?

(a) $R \times[0, \infty)$

(b) $(-\infty, 0) \times R$

(c) $[0, \infty) \times R$

(d) $R \times(-\infty, 0)$

[Online April 15, 2018]

Sol.

Q. If $x=\sqrt{2^{\operatorname{cosec}^{-1} t}}$ and $y=\sqrt{2^{\sec ^{-1} t}}(|\mathrm{t}| \geq 1)$, then $\frac{d y}{d x}$ is equal to.

(a) $\frac{y}{x}$

(b) $-\frac{y}{x}$

(c) $-\frac{x}{y}$

(d) $\frac{x}{y}$

[Online April 16, 2018]

Sol.

Q. If $f(x)=\left|\begin{array}{ccc}\cos x & x & 1 \\ 2 \sin x & x^{2} & 2 x \\ \tan x & x & 1\end{array}\right|$, then $\lim _{x \rightarrow 0} \frac{f^{\prime}(\mathrm{x})}{x}$

(a) Exists and is equal to $-2$

(b) Does not exist

(c) Exist and is equal to 0

(d) Exists and is equal to 2

[Online April 15, 2018]

Sol.

Q. If $f(x)=\sin ^{-1}\left(\frac{2 \times 3^{x}}{1+9^{x}}\right)$, then $f^{\prime}\left(-\frac{1}{2}\right)$ equals.

(a) $\sqrt{3} \log _{e} \sqrt{3}$

(b) $-\sqrt{3} \log _{e} \sqrt{3}$

(c) $-\sqrt{3} \log _{e} 3$

(d) $\sqrt{3} \log _{e} 3$

[Online April 15, 2018]

Sol.

Q. If $x^{2}+y^{2}+\sin y=4$, then the value of $\frac{d^{2} y}{d x^{2}}$ at the point $(-2,0)$ is

(a) $-34$

(b) $-32$

(c) $-2$

(d) 4

[Online April 15, 2018]

Sol.

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