Let the function $g : ( - \infty , \infty ) \rightarrow \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$ be given by $g ( u ) = 2 \tan ^ { - 1 } \left( e ^ { u } \right) - \frac { \pi } { 2 }$. Then, $g$ is
(A) even and is strictly increasing in $(0 , \infty)$
(B) odd and is strictly decreasing in $( - \infty , \infty )$
(C) odd and is strictly increasing in $( - \infty , \infty )$
(D) neither even nor odd, but is strictly increasing in $( - \infty , \infty )$

The total number of local maxima and local minima of the function $$f ( x ) = \begin{cases} ( 2 + x ) ^ { 3 } , & - 3 < x \leq - 1 \\ x ^ { 2 / 3 } , & - 1 < x < 2 \end{cases}$$ is
(A) 0
(B) 1
(C) 2
(D) 3

Consider the function $f : ( - \infty , \infty ) \rightarrow ( - \infty , \infty )$ defined by
$$f ( x ) = \frac { x ^ { 2 } - a x + 1 } { x ^ { 2 } + a x + 1 } , 0 < a < 2 .$$
Which of the following is true?
(A) $f ( x )$ is decreasing on $( - 1,1 )$ and has a local minimum at $x = 1$
(B) $f ( x )$ is increasing on $( - 1,1 )$ and has a local maximum at $x = 1$
(C) $f ( x )$ is increasing on $( - 1,1 )$ but has neither a local maximum nor a local minimum at $x = 1$
(D) $f ( x )$ is decreasing on $( - 1,1 )$ but has neither a local maximum nor a local minimum at $x = 1$

The maximum value of the function $f(x)=2x^{3}-15x^{2}+36x-48$ on the set $A=\left\{x\mid x^{2}+20\leq9x\right\}$ is

Let $p(x)$ be a polynomial of degree 4 having extremum at $x=1,2$ and $$\lim_{x\rightarrow0}\left(1+\frac{p(x)}{x^{2}}\right)=2.$$ Then the value of $p(2)$ is

Let f be a function defined on $\mathbf { R }$ (the set of all real numbers) such that $\mathrm { f } ^ { \prime } ( \mathrm { x } ) = 2010 ( \mathrm { x } - 2009 ) ( \mathrm { x } - 2010 ) ^ { 2 } ( \mathrm { x } - 2011 ) ^ { 3 } ( \mathrm { x } - 2012 ) ^ { 4 }$, for all $\mathrm { x } \in \mathbf { R }$.
If $g$ is a function defined on $\mathbf { R }$ with values in the interval $( 0 , \infty )$ such that
$$\mathrm { f } ( \mathrm { x } ) = \ell n ( \mathrm {~g} ( \mathrm { x } ) ) \text {, for all } \mathrm { x } \in \mathbf { R } \text {, }$$
then the number of points in $\mathbf { R }$ at which $g$ has a local maximum is

Consider the polynomial
$$f ( x ) = 1 + 2 x + 3 x ^ { 2 } + 4 x ^ { 3 }$$
Let s be the sum of all distinct real roots of $\mathrm { f } ( \mathrm { x } )$ and let $\mathrm { t } = | \mathrm { s } |$.
The function $f ^ { \prime } ( x )$ is
A) increasing in $\left( - t , - \frac { 1 } { 4 } \right)$ and decreasing in $\left( - \frac { 1 } { 4 } , t \right)$
B) decreasing in $\left( - t , - \frac { 1 } { 4 } \right)$ and increasing in $\left( - \frac { 1 } { 4 } , t \right)$
C) increasing in (-t, t)
D) decreasing in (-t, t)

Let $f , g$ and $h$ be real-valued functions defined on the interval $[ 0,1 ]$ by $f ( x ) = e ^ { x ^ { 2 } } + e ^ { - x ^ { 2 } } , g ( x ) = x e ^ { x ^ { 2 } } + e ^ { - x ^ { 2 } }$ and $h ( x ) = x ^ { 2 } e ^ { x ^ { 2 } } + e ^ { - x ^ { 2 } }$. If $a , b$ and $c$ denote, respectively, the absolute maximum of $f , g$ and $h$ on $[ 0,1 ]$, then
A) $\mathrm { a } = \mathrm { b }$ and $\mathrm { c } \neq \mathrm { b }$
B) a $=$ c and a $\neq$ b
C) $a \neq b$ and $c \neq b$
D) $a = b = c$

A rectangular sheet of fixed perimeter with sides having their lengths in the ratio $8 : 15$ is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is 100, the resulting box has maximum volume. Then the lengths of the sides of the rectangular sheet are
(A) 24
(B) 32
(C) 45
(D) 60

A line $L : y = m x + 3$ meets $y$-axis at $E ( 0,3 )$ and the arc of the parabola $y ^ { 2 } = 16 x$, $0 \leq y \leq 6$ at the point $F \left( x _ { 0 } , y _ { 0 } \right)$. The tangent to the parabola at $F \left( x _ { 0 } , y _ { 0 } \right)$ intersects the $y$-axis at $G \left( 0 , y _ { 1 } \right)$. The slope $m$ of the line $L$ is chosen such that the area of the triangle $E F G$ has a local maximum.
Match List I with List II and select the correct answer using the code given below the lists:
List I

• [P.] $m =$
• [Q.] Maximum area of $\triangle EFG$ is
• [R.] $y_0 =$
• [S.] $y_1 =$

List II

1. $\frac{1}{2}$
2. $4$
3. $2$
4. $1$

Codes:

	P	Q	R	S
(A)	4	1	2	3
(B)	3	4	1	2
(C)	1	3	2	4
(D)	1	3	4	2

Let $a \in \mathbb{R}$ and let $f : \mathbb{R} \rightarrow \mathbb{R}$ be given by $$f(x) = x^5 - 5x + a$$ Then
(A) $f(x)$ has three real roots if $a > 4$
(B) $f(x)$ has only one real root if $a > 4$
(C) $f(x)$ has three real roots if $a < -4$
(D) $f(x)$ has three real roots if $-4 < a < 4$

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ and $g : \mathbb{R} \rightarrow \mathbb{R}$ be respectively given by $f(x) = |x| + 1$ and $g(x) = x^2 + 1$. Define $h : \mathbb{R} \rightarrow \mathbb{R}$ by $$h(x) = \begin{cases} \max\{f(x), g(x)\} & \text{if } x \leq 0 \\ \min\{f(x), g(x)\} & \text{if } x > 0 \end{cases}$$ The number of points at which $h(x)$ is not differentiable is

Consider the hyperbola $H : x ^ { 2 } - y ^ { 2 } = 1$ and a circle $S$ with center $N \left( x _ { 2 } , 0 \right)$. Suppose that $H$ and $S$ touch each other at a point $P \left( x _ { 1 } , y _ { 1 } \right)$ with $x _ { 1 } > 1$ and $y _ { 1 } > 0$. The common tangent to $H$ and $S$ at $P$ intersects the $x$-axis at point $M$. If ( $l , m$ ) is the centroid of the triangle $\triangle P M N$, then the correct expression(s) is(are)
(A) $\frac { d l } { d x _ { 1 } } = 1 - \frac { 1 } { 3 x _ { 1 } ^ { 2 } }$ for $x _ { 1 } > 1$
(B) $\frac { d m } { d x _ { 1 } } = \frac { x _ { 1 } } { 3 \left( \sqrt { x _ { 1 } ^ { 2 } - 1 } \right) }$ for $x _ { 1 } > 1$
(C) $\frac { d l } { d x _ { 1 } } = 1 + \frac { 1 } { 3 x _ { 1 } ^ { 2 } }$ for $x _ { 1 } > 1$
(D) $\frac { d m } { d y _ { 1 } } = \frac { 1 } { 3 }$ for $y _ { 1 } > 0$

Let $f , g : [ - 1,2 ] \rightarrow \mathbb { R }$ be continuous functions which are twice differentiable on the interval $( - 1,2 )$. Let the values of $f$ and $g$ at the points $- 1,0$ and 2 be as given in the following table:

	$x = - 1$	$x = 0$	$x = 2$
$f ( x )$	3	6	0
$g ( x )$	0	1	- 1

In each of the intervals $( - 1,0 )$ and $( 0,2 )$ the function $( f - 3 g ) ^ { \prime \prime }$ never vanishes. Then the correct statement(s) is(are)
(A) $\quad f ^ { \prime } ( x ) - 3 g ^ { \prime } ( x ) = 0$ has exactly three solutions in $( - 1,0 ) \cup ( 0,2 )$
(B) $f ^ { \prime } ( x ) - 3 g ^ { \prime } ( x ) = 0$ has exactly one solution in $( - 1,0 )$
(C) $f ^ { \prime } ( x ) - 3 g ^ { \prime } ( x ) = 0$ has exactly one solution in $( 0,2 )$
(D) $f ^ { \prime } ( x ) - 3 g ^ { \prime } ( x ) = 0$ has exactly two solutions in ( $- 1,0$ ) and exactly two solutions in ( 0,2 )

Let $F : \mathbb { R } \rightarrow \mathbb { R }$ be a thrice differentiable function. Suppose that $F ( 1 ) = 0 , F ( 3 ) = - 4$ and $F ^ { \prime } ( x ) < 0$ for all $x \in ( 1 / 2,3 )$. Let $f ( x ) = x F ( x )$ for all $x \in \mathbb { R }$. The correct statement(s) is(are)
(A) $f ^ { \prime } ( 1 ) < 0$
(B) $f ( 2 ) < 0$
(C) $f ^ { \prime } ( x ) \neq 0$ for any $x \in ( 1,3 )$
(D) $f ^ { \prime } ( x ) = 0$ for some $x \in ( 1,3 )$

The least value of $\alpha \in \mathbb{R}$ for which $4\alpha x^2 + \frac{1}{x} \geq 1$, for all $x > 0$, is
(A) $\frac{1}{64}$
(B) $\frac{1}{32}$
(C) $\frac{1}{27}$
(D) $\frac{1}{25}$

Let $f ( x ) = \lim _ { n \rightarrow \infty } \left( \frac { n ^ { n } ( x + n ) \left( x + \frac { n } { 2 } \right) \cdots \left( x + \frac { n } { n } \right) } { n ! \left( x ^ { 2 } + n ^ { 2 } \right) \left( x ^ { 2 } + \frac { n ^ { 2 } } { 4 } \right) \cdots \left( x ^ { 2 } + \frac { n ^ { 2 } } { n ^ { 2 } } \right) } \right) ^ { \frac { x } { n } }$, for all $x > 0$. Then
(A) $f \left( \frac { 1 } { 2 } \right) \geq f ( 1 )$
(B) $f \left( \frac { 1 } { 3 } \right) \leq f \left( \frac { 2 } { 3 } \right)$
(C) $f ^ { \prime } ( 2 ) \leq 0$
(D) $\frac { f ^ { \prime } ( 3 ) } { f ( 3 ) } \geq \frac { f ^ { \prime } ( 2 ) } { f ( 2 ) }$

Let $f : \mathbb { R } \rightarrow ( 0 , \infty )$ and $g : \mathbb { R } \rightarrow \mathbb { R }$ be twice differentiable functions such that $f ^ { \prime \prime }$ and $g ^ { \prime \prime }$ are continuous functions on $\mathbb { R }$. Suppose $f ^ { \prime } ( 2 ) = g ( 2 ) = 0 , \quad f ^ { \prime \prime } ( 2 ) \neq 0$ and $g ^ { \prime } ( 2 ) \neq 0$. If $\lim _ { x \rightarrow 2 } \frac { f ( x ) g ( x ) } { f ^ { \prime } ( x ) g ^ { \prime } ( x ) } = 1$, then
(A) $f$ has a local minimum at $x = 2$
(B) $f$ has a local maximum at $x = 2$
(C) $f ^ { \prime \prime } ( 2 ) > f ( 2 )$
(D) $f ( x ) - f ^ { \prime \prime } ( x ) = 0$ for at least one $x \in \mathbb { R }$

If $f : \mathbb { R } \rightarrow \mathbb { R }$ is a twice differentiable function such that $f ^ { \prime \prime } ( x ) > 0$ for all $x \in \mathbb { R }$, and $f \left( \frac { 1 } { 2 } \right) = \frac { 1 } { 2 } , f ( 1 ) = 1$, then
[A] $f ^ { \prime } ( 1 ) \leq 0$
[B] $0 < f ^ { \prime } ( 1 ) \leq \frac { 1 } { 2 }$
[C] $\frac { 1 } { 2 } < f ^ { \prime } ( 1 ) \leq 1$
[D] $f ^ { \prime } ( 1 ) > 1$

If $f ( x ) = \left| \begin{array} { c c c } \cos ( 2 x ) & \cos ( 2 x ) & \sin ( 2 x ) \\ - \cos x & \cos x & - \sin x \\ \sin x & \sin x & \cos x \end{array} \right|$, then
[A] $f ^ { \prime } ( x ) = 0$ at exactly three points in $( - \pi , \pi )$
[B] $f ^ { \prime } ( x ) = 0$ at more than three points in $( - \pi , \pi )$
[C] $f ( x )$ attains its maximum at $x = 0$
[D] $f ( x )$ attains its minimum at $x = 0$

Let $f(x) = x + \log_e x - x\log_e x$, $x \in (0, \infty)$.
- Column 1 contains information about zeros of $f(x)$, $f'(x)$ and $f''(x)$. - Column 2 contains information about the limiting behavior of $f(x)$, $f'(x)$ and $f''(x)$ at infinity. - Column 3 contains information about increasing/decreasing nature of $f(x)$ and $f'(x)$.

Column 1	Column 2	Column 3
(I) $f(x) = 0$ for some $x \in (1, e^2)$	(i) $\lim_{x\to\infty} f(x) = 0$	(P) $f$ is increasing in $(0,1)$
(II) $f'(x) = 0$ for some $x \in (1, e)$	(ii) $\lim_{x\to\infty} f(x) = -\infty$	(Q) $f$ is decreasing in $(e, e^2)$
(III) $f'(x) = 0$ for some $x \in (0,1)$	(iii) $\lim_{x\to\infty} f'(x) = -\infty$	(R) $f'$ is increasing in $(0,1)$
(IV) $f''(x) = 0$ for some $x \in (1, e)$	(iv) $\lim_{x\to\infty} f''(x) = 0$	(S) $f'$ is decreasing in $(e, e^2)$

Which of the following options is the only CORRECT combination?
[A] (I) (i) (P)
[B] (II) (ii) (Q)
[C] (III) (iii) (R)
[D] (IV) (iv) (S)

Let $f(x) = x + \log_e x - x\log_e x$, $x \in (0, \infty)$.
- Column 1 contains information about zeros of $f(x)$, $f'(x)$ and $f''(x)$. - Column 2 contains information about the limiting behavior of $f(x)$, $f'(x)$ and $f''(x)$ at infinity. - Column 3 contains information about increasing/decreasing nature of $f(x)$ and $f'(x)$.

Column 1	Column 2	Column 3
(I) $f(x) = 0$ for some $x \in (1, e^2)$	(i) $\lim_{x\to\infty} f(x) = 0$	(P) $f$ is increasing in $(0,1)$
(II) $f'(x) = 0$ for some $x \in (1, e)$	(ii) $\lim_{x\to\infty} f(x) = -\infty$	(Q) $f$ is decreasing in $(e, e^2)$
(III) $f'(x) = 0$ for some $x \in (0,1)$	(iii) $\lim_{x\to\infty} f'(x) = -\infty$	(R) $f'$ is increasing in $(0,1)$
(IV) $f''(x) = 0$ for some $x \in (1, e)$	(iv) $\lim_{x\to\infty} f''(x) = 0$	(S) $f'$ is decreasing in $(e, e^2)$

Which of the following options is the only CORRECT combination?
[A] (I) (ii) (R)
[B] (II) (iii) (S)
[C] (III) (iv) (P)
[D] (IV) (i) (S)

For every twice differentiable function $f : \mathbb { R } \rightarrow [ - 2,2 ]$ with $( f ( 0 ) ) ^ { 2 } + \left( f ^ { \prime } ( 0 ) \right) ^ { 2 } = 85$, which of the following statement(s) is (are) TRUE?
(A) There exist $r , s \in \mathbb { R }$, where $r < s$, such that $f$ is one-one on the open interval ( $r , s$ )
(B) There exists $x _ { 0 } \in ( - 4,0 )$ such that $\left| f ^ { \prime } \left( x _ { 0 } \right) \right| \leq 1$
(C) $\lim _ { x \rightarrow \infty } f ( x ) = 1$
(D) There exists $\alpha \in ( - 4,4 )$ such that $f ( \alpha ) + f ^ { \prime \prime } ( \alpha ) = 0$ and $f ^ { \prime } ( \alpha ) \neq 0$

Let $$f(x) = \frac{\sin\pi x}{x^2}, \quad x > 0$$
Let $x_1 < x_2 < x_3 < \cdots < x_n < \cdots$ be all the points of local maximum of $f$ and $y_1 < y_2 < y_3 < \cdots < y_n < \cdots$ be all the points of local minimum of $f$. Then which of the following options is/are correct?
(A) $x_1 < y_1$
(B) $x_{n+1} - x_n > 2$ for every $n$
(C) $x_n \in \left(2n, 2n + \frac{1}{2}\right)$ for every $n$
(D) $|x_n - y_n| > 1$ for every $n$

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be given by $f(x) = (x-1)(x-2)(x-5)$. Define $$F(x) = \int_0^x f(t)\,dt, \quad x > 0$$
Then which of the following options is/are correct?
(A) $F$ has a local minimum at $x = 1$
(B) $F$ has a local maximum at $x = 2$
(C) $F$ has two local maxima and one local minimum in $(0, \infty)$
(D) $F(x) \neq 0$ for all $x \in (0, 5)$