Consider the function $f : ( - \infty , \infty ) \rightarrow ( - \infty , \infty )$ defined by
$$f ( x ) = \frac { x ^ { 2 } - a x + 1 } { x ^ { 2 } + a x + 1 } , 0 < a < 2 .$$
Let
$$g ( x ) = \int _ { 0 } ^ { e ^ { x } } \frac { f ^ { \prime } ( t ) } { 1 + t ^ { 2 } } d t$$
Which of the following is true?
(A) $g ^ { \prime } ( x )$ is positive on $( - \infty , 0 )$ and negative on $( 0 , \infty )$
(B) $g ^ { \prime } ( x )$ is negative on $( - \infty , 0 )$ and positive on $( 0 , \infty )$
(C) $g ^ { \prime } ( x )$ changes sign on both $( - \infty , 0 )$ and $( 0 , \infty )$
(D) $g ^ { \prime } ( x )$ does not change sign on $( - \infty , \infty )$

Match the Statements / Expressions in Column I with the Statements / Expressions in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS.
Column I
(A) The minimum value of $\frac { x ^ { 2 } + 2 x + 4 } { x + 2 }$ is
(B) Let $A$ and $B$ be $3 \times 3$ matrices of real numbers, where $A$ is symmetric, $B$ is skew-symmetric, and $( A + B ) ( A - B ) = ( A - B ) ( A + B )$. If $( A B ) ^ { t } = ( - 1 ) ^ { k } A B$, where $( A B ) ^ { t }$ is the transpose of the matrix $A B$, then the possible values of $k$ are
(C) Let $a = \log _ { 3 } \log _ { 3 } 2$. An integer $k$ satisfying $1 < 2 ^ { \left( - k + 3 ^ { - a } \right) } < 2$, must be less than
(D) If $\sin \theta = \cos \varphi$, then the possible values of $\frac { 1 } { \pi } \left( \theta \pm \varphi - \frac { \pi } { 2 } \right)$ are
Column II
(p) 0
(q) 1
(r) 2
(s) 3

For the function $$f(x)=x\cos\frac{1}{x},\quad x\geq1,$$ (A) for at least one $x$ in the interval $[1,\infty),f(x+2)-f(x)<2$
(B) $\lim_{x\rightarrow\infty}f^{\prime}(x)=1$
(C) for all $x$ in the interval $[1,\infty),f(x+2)-f(x)>2$
(D) $f^{\prime}(x)$ is strictly decreasing in the interval $[1,\infty)$

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

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 $-4 < a < 4$

A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of $V \mathrm {~mm} ^ { 3 }$, has a 2 mm thick solid wall and is open at the top. The bottom of the container is a solid circular disc of thickness 2 mm and is of radius equal to the outer radius of the container. If the volume of the material used to make the container is minimum when the inner radius of the container is 10 mm, then the value of $\frac { V } { 250 \pi }$ is

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 $f : \mathbb { R } \rightarrow ( 0 , \infty )$ 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 $f : \mathbb { R } \rightarrow \mathbb { R }$ is a differentiable function such that $f ^ { \prime } ( x ) > 2 f ( x )$ for all $x \in \mathbb { R }$, and $f ( 0 ) = 1$, then
[A] $f ( x )$ is increasing in $( 0 , \infty )$
[B] $f ( x )$ is decreasing in $( 0 , \infty )$
[C] $f ( x ) > e ^ { 2 x }$ in $( 0 , \infty )$
[D] $f ^ { \prime } ( x ) < e ^ { 2 x }$ in $( 0 , \infty )$

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 $x = 0$

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 $x \in (0, 5)$

Consider all rectangles lying in the region
$$\left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : 0 \leq x \leq \frac { \pi } { 2 } \text { and } 0 \leq y \leq 2 \sin ( 2 x ) \right\}$$
and having one side on the $x$-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is
(A) $\frac { 3 \pi } { 2 }$
(B) $\pi$
(C) $\frac { \pi } { 2 \sqrt { 3 } }$
(D) $\frac { \pi \sqrt { 3 } } { 2 }$

Let the function $f: (0, \pi) \rightarrow \mathbb{R}$ be defined by $$f(\theta) = (\sin\theta + \cos\theta)^{2} + (\sin\theta - \cos\theta)^{4}$$ Suppose the function $f$ has a local minimum at $\theta$ precisely when $\theta \in \{\lambda_{1}\pi, \ldots, \lambda_{r}\pi\}$, where $0 < \lambda_{1} < \cdots < \lambda_{r} < 1$. Then the value of $\lambda_{1} + \cdots + \lambda_{r}$ is $\_\_\_\_$

Let $f_1 : (0, \infty) \to \mathbb{R}$ and $f_2 : (0, \infty) \to \mathbb{R}$ be defined by $$f_1(x) = \int_0^x \prod_{j=1}^{21} (t - j)^j \, dt, \quad x > 0$$ and $$f_2(x) = 98(x-1)^{50} - 600(x-1)^{49} + 2450, \quad x > 0,$$ where, for any positive integer $n$ and real numbers $a_1, a_2, \ldots, a_n$, $\prod_{i=1}^n a_i$ denotes the product of $a_1, a_2, \ldots, a_n$. Let $m_i$ and $n_i$, respectively, denote the number of points of local minima and the number of points of local maxima of function $f_i$, $i = 1, 2$, in the interval $(0, \infty)$.
The value of $2m_1 + 3n_1 + m_1 n_1$ is ____.

Let $f_1 : (0, \infty) \to \mathbb{R}$ and $f_2 : (0, \infty) \to \mathbb{R}$ be defined by $$f_1(x) = \int_0^x \prod_{j=1}^{21} (t - j)^j \, dt, \quad x > 0$$ and $$f_2(x) = 98(x-1)^{50} - 600(x-1)^{49} + 2450, \quad x > 0,$$ where, for any positive integer $n$ and real numbers $a_1, a_2, \ldots, a_n$, $\prod_{i=1}^n a_i$ denotes the product of $a_1, a_2, \ldots, a_n$. Let $m_i$ and $n_i$, respectively, denote the number of points of local minima and the number of points of local maxima of function $f_i$, $i = 1, 2$, in the interval $(0, \infty)$.
The value of $6m_2 + 4n_2 + 8m_2 n_2$ is ____.

Let $S$ be the set of all twice differentiable functions $f$ from $\mathbb { R }$ to $\mathbb { R }$ such that $\frac { d ^ { 2 } f } { d x ^ { 2 } } ( x ) > 0$ for all $x \in ( - 1,1 )$. For $f \in S$, let $X _ { f }$ be the number of points $x \in ( - 1,1 )$ for which $f ( x ) = x$. Then which of the following statements is(are) true?
(A) There exists a function $f \in S$ such that $X _ { f } = 0$
(B) For every function $f \in S$, we have $X _ { f } \leq 2$
(C) There exists a function $f \in S$ such that $X _ { f } = 2$
(D) There does NOT exist any function $f$ in $S$ such that $X _ { f } = 1$

Let $\mathbb { R }$ denote the set of all real numbers. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined by
$$f ( x ) = \begin{cases} \frac { 6 x + \sin x } { 2 x + \sin x } & \text { if } x \neq 0 \\ \frac { 7 } { 3 } & \text { if } x = 0 \end{cases}$$
Then which of the following statements is (are) TRUE?

(A)	The point $x = 0$ is a point of local maxima of $f$
(B)	The point $x = 0$ is a point of local minima of $f$
(C)	Number of points of local maxima of $f$ in the interval $[ \pi , 6 \pi ]$ is 3
(D)	Number of points of local minima of $f$ in the interval $[ 2 \pi , 4 \pi ]$ is 1