Let $f , g$ be continuous functions from $[ 0 , \infty )$ to itself, $$h ( x ) = \int _ { 2 ^ { x } } ^ { 3 ^ { x } } f ( t ) d t , x > 0$$ and $$F ( x ) = \int _ { 0 } ^ { h ( x ) } g ( t ) d t , x > 0$$ If $F ^ { \prime }$ is the derivative of $F$, then for $x > 0$,
(A) $F ^ { \prime } ( x ) = g ( h ( x ) )$.
(B) $F ^ { \prime } ( x ) = g ( h ( x ) ) \left[ f \left( 3 ^ { x } \right) - f \left( 2 ^ { x } \right) \right]$.
(C) $F ^ { \prime } ( x ) = g ( h ( x ) ) \left[ x 3 ^ { x - 1 } f \left( 3 ^ { x } \right) - x 2 ^ { x - 1 } f \left( 2 ^ { x } \right) \right]$.
(D) $F ^ { \prime } ( x ) = g ( h ( x ) ) \left[ 3 ^ { x } f \left( 3 ^ { x } \right) \ln 3 - 2 ^ { x } f \left( 2 ^ { x } \right) \ln 2 \right]$.

48. $\frac { d ^ { 2 } x } { d y ^ { 2 } }$ equals
(A) $\left( \frac { d ^ { 2 } y } { d x ^ { 2 } } \right) ^ { - 1 }$
(B) $- \left( \frac { d ^ { 2 } y } { d x ^ { 2 } } \right) ^ { - 1 } \left( \frac { d y } { d x } \right) ^ { - 3 }$
(C) $\left( \frac { d ^ { 2 } y } { d x ^ { 2 } } \right) \left( \frac { d y } { d x } \right) ^ { - 2 }$
(D) $- \left( \frac { d ^ { 2 } y } { d x ^ { 2 } } \right) \left( \frac { d y } { d x } \right) ^ { - 3 }$

Answer

◯ ◯
(A)
(B)
(C)
(D)

Let $m$ and $n$ be two positive integers greater than 1 . If
$$\lim _ { \alpha \rightarrow 0 } \left( \frac { e ^ { \cos \left( \alpha ^ { n } \right) } - e } { \alpha ^ { m } } \right) = - \left( \frac { e } { 2 } \right)$$
then the value of $\frac { m } { n }$ is

Let $\alpha, \beta \in \mathbb{R}$ be such that $\lim_{x \rightarrow 0} \frac{x^2 \sin(\beta x)}{\alpha x - \sin x} = 1$. Then $6(\alpha + \beta)$ equals

Let the function $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined by $f ( x ) = x ^ { 3 } - x ^ { 2 } + ( x - 1 ) \sin x$ and let $g : \mathbb { R } \rightarrow \mathbb { R }$ be an arbitrary function. Let $f g : \mathbb { R } \rightarrow \mathbb { R }$ be the product function defined by $( f g ) ( x ) = f ( x ) g ( x )$. Then which of the following statements is/are TRUE?
(A) If $g$ is continuous at $x = 1$, then $f g$ is differentiable at $x = 1$
(B) If $f g$ is differentiable at $x = 1$, then $g$ is continuous at $x = 1$
(C) If $g$ is differentiable at $x = 1$, then $f g$ is differentiable at $x = 1$
(D) If $f g$ is differentiable at $x = 1$, then $g$ is differentiable at $x = 1$

Let $k \in \mathbb { R }$. If $\lim _ { x \rightarrow 0 + } ( \sin ( \sin k x ) + \cos x + x ) ^ { \frac { 2 } { x } } = e ^ { 6 }$, then the value of $k$ is
(A) 1
(B) 2
(C) 3
(D) 4

If $f^{\prime}(x) = \sin(\log x)$ and $y = f\left(\frac{2x+3}{3-2x}\right)$, then $\frac{dy}{dx}$ at $x = 1$ is equal to (the question continues with answer options as given in the paper).

If $y = \sec\left(\tan^{-1}x\right)$, then $\frac{dy}{dx}$ at $x = 1$ is equal to
(1) 1
(2) $\sqrt{2}$
(3) $\frac{1}{\sqrt{2}}$
(4) $\frac{1}{2}$

$\lim _ { x \rightarrow 0 } \frac { \sin \left( \pi \cos ^ { 2 } x \right) } { x ^ { 2 } }$ is equal to
(1) $- \pi$
(2) $\pi$
(3) $\frac { \pi } { 2 }$
(4) 1

If $y = e ^ { n x }$, then $\frac { d ^ { 2 } y } { d x ^ { 2 } } \cdot \frac { d ^ { 2 } x } { d y ^ { 2 } }$ is equal to:
(1) $n e ^ { - n x }$
(2) $- n e ^ { - n x }$
(3) $n e ^ { n x }$
(4) 1

If the function $g(x) = \begin{cases} k\sqrt{x+1}, & 0 \leq x \leq 3 \\ mx + 2, & 3 < x \leq 5 \end{cases}$ is differentiable, then the value of $k + m$ is:
(1) $2$
(2) $\frac{16}{5}$
(3) $\frac{10}{3}$
(4) $4$

$\lim _ { x \rightarrow 0 } \frac { ( 1 - \cos 2 x ) ^ { 2 } } { 2 x \tan x - x \tan 2 x }$ is
(1) 2
(2) $- \frac { 1 } { 2 }$
(3) $- 2$
(4) $\frac { 1 } { 2 }$

$\lim_{x \to \frac{\pi}{2}} \dfrac{\cot x - \cos x}{(\pi - 2x)^3}$ equals
(1) $\dfrac{1}{24}$
(2) $\dfrac{1}{16}$
(3) $\dfrac{1}{8}$
(4) $\dfrac{1}{4}$

$\lim _ { x \rightarrow 0 } \frac { ( 27 + x ) ^ { \frac { 1 } { 3 } } - 3 } { 9 - ( 27 + x ) ^ { \frac { 2 } { 3 } } }$ equals
(1) $- \frac { 1 } { 6 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 1 } { 3 }$
(4) $- \frac { 1 } { 3 }$

If $f ( x ) = \left| \begin{array} { c c c } \cos x & x & 1 \\ 2 \sin x & x ^ { 2 } & 2 x \\ \tan x & x & 1 \end{array} \right|$, then $\lim _ { x \rightarrow 0 } \frac { f ^ { \prime } ( x ) } { x }$
(1) does not exist
(2) exists and is equal to $-2$
(3) exists and is equal to 0
(4) exists and is equal to 2

$f ( x ) = \left| \begin{array} { c c c } \cos x & x & 1 \\ 2 \sin x & x ^ { 2 } & 2 x \\ \tan x & x & 1 \end{array} \right|$, then $\lim _ { x \rightarrow 0 } \frac { f ^ { \prime } ( x ) } { x }$
(1) Exists and is equal to - 2
(2) Does not exist
(3) Exist and is equal to 0
(4) Exists and is equal to 2

$\lim _ { x \rightarrow 1 ^ { - } } \frac { \sqrt { \pi } - \sqrt { 2 \sin ^ { - 1 } x } } { \sqrt { 1 - x } }$ is equal to
(1) $\sqrt { \pi }$
(2) $\sqrt { \frac { 2 } { \pi } }$
(3) $\frac { 1 } { \sqrt { 2 \pi } }$
(4) $\sqrt { \frac { \pi } { 2 } }$

Let $f : R \rightarrow R$ be a differentiable function satisfying $f ^ { \prime } ( 3 ) + f ^ { \prime } ( 2 ) = 0$. Then $\lim _ { x \rightarrow 0 } \left( \frac { 1 + f ( 3 + x ) - f ( 3 ) } { 1 + f ( 2 - x ) - f ( 2 ) } \right)$ is equal to
(1) 1
(2) e
(3) $e ^ { 2 }$
(4) $e ^ { - 1 }$

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 :
(1) $4 e ^ { 2 }$
(2) $2 e$
(3) $4 e$
(4) $2 e ^ { 2 }$

Let, $f : R \rightarrow R$ be a function such that $f ( x ) = x ^ { 3 } + x ^ { 2 } f \prime ( 1 ) + x f \prime \prime ( 2 ) + f \prime \prime \prime ( 3 ) , \forall x \in R$. Then $f ( 2 )$ equals
(1) 30
(2) 8
(3) $- 4$
(4) $- 2$

If $f ( 1 ) = 1 , f ^ { \prime } ( 1 ) = 3$, then the derivative of $f ( f ( f ( x ) ) ) + ( f ( x ) ) ^ { 2 }$ at $x = 1$ is:
(1) 9
(2) 12
(3) 15
(4) 33

$\lim _ { x \rightarrow 0 } \frac { \sin ^ { 2 } \left( \pi \cos ^ { 4 } x \right) } { x ^ { 4 } }$ is equal to :
(1) $2 \pi ^ { 2 }$
(2) $\pi ^ { 2 }$
(3) $4 \pi ^ { 2 }$
(4) $4 \pi$

The function $f ( x ) = \left| x ^ { 2 } - 2 x - 3 \right| \cdot \mathrm { e } ^ { 9 x ^ { 2 } - 12 x + 4 }$ is not differentiable at exactly :
(1) Four points
(2) Two points
(3) three points
(4) one point

If the function $f ( x ) = \begin{cases} \frac { 1 } { x } \log _ { \mathrm { e } } \left( \frac { 1 + \frac { x } { a } } { 1 - \frac { x } { b } } \right) & , x < 0 \\ k & , x = 0 \\ \frac { \cos ^ { 2 } x - \sin ^ { 2 } x - 1 } { \sqrt { x ^ { 2 } + 1 } - 1 } & , x > 0 \end{cases}$ is continuous at $x = 0$, then $\frac { 1 } { a } + \frac { 1 } { b } + \frac { 4 } { k }$ is equal to:
(1) 4
(2) 5
(3) $- 4$
(4) $- 5$

Let $f$ be a non-negative function in $[ 0,1 ]$ and twice differentiable in $( 0,1 )$. If $\int _ { 0 } ^ { x } \sqrt { 1 - \left( f ^ { \prime } ( t ) \right) ^ { 2 } } \mathrm { dt } = \int _ { 0 } ^ { x } f ( \mathrm { t } ) \mathrm { dt } , 0 \leq x \leq 1$ and $f ( 0 ) = 0$, then $\lim _ { x \rightarrow 0 } \frac { 1 } { x ^ { 2 } } \int _ { 0 } ^ { x } f ( \mathrm { t } ) \mathrm { dt } :$
(1) does not exist
(2) equals 0
(3) equals 1
(4) equals $\frac { 1 } { 2 }$