Let $f : \left[ \frac { 1 } { 2 } , 1 \right] \rightarrow \mathbb { R }$ (the set of all real numbers) be a positive, non-constant and differentiable function such that $f ^ { \prime } ( x ) < 2 f ( x )$ and $f \left( \frac { 1 } { 2 } \right) = 1$. Then the value of $\int _ { 1/2 } ^ { 1 } f ( x ) d x$ lies in the interval
(A) $( 2 e - 1,2 e )$
(B) $( e - 1,2 e - 1 )$
(C) $\left( \frac { e - 1 } { 2 } , e - 1 \right)$
(D) $\left( 0 , \frac { e - 1 } { 2 } \right)$

Let $f : [a,b] \rightarrow [1, \infty)$ be a continuous function and let $g : \mathbb{R} \rightarrow \mathbb{R}$ be defined as $$g(x) = \begin{cases} 0 & \text{if } x < a \\ \int_{a}^{x} f(t)\, dt & \text{if } a \leq x \leq b \\ \int_{a}^{b} f(t)\, dt & \text{if } x > b \end{cases}$$ Then
(A) $g(x)$ is continuous but not differentiable at $a$
(B) $g(x)$ is differentiable on $\mathbb{R}$
(C) $g(x)$ is continuous but not differentiable at $b$
(D) $g(x)$ is continuous and differentiable at either $a$ or $b$ but not both

Given that for each $a \in (0,1)$,
$$\lim_{h \rightarrow 0^+} \int_{h}^{1-h} t^{-a}(1-t)^{a-1}\, dt$$
exists. Let this limit be $g(a)$. In addition, it is given that the function $g(a)$ is differentiable on $(0,1)$.
The value of $g\left(\frac{1}{2}\right)$ is
(A) $\pi$
(B) $2\pi$
(C) $\frac{\pi}{2}$
(D) $\frac{\pi}{4}$

The value of $$\int_{0}^{1} 4x^3 \left\{\frac{d^2}{dx^2}\left(1 - x^2\right)^5\right\} dx$$ is

List I	List II
P. The number of polynomials $f(x)$ with non-negative integer coefficients of degree $\leq 2$, satisfying $f(0) = 0$ and $\int_{0}^{1} f(x)\,dx = 1$, is	1. 8
Q. The number of points in the interval $[-\sqrt{13}, \sqrt{13}]$ at which $f(x) = \sin(x^2) + \cos(x^2)$ attains its maximum value, is	2. 2
R. $\int_{-2}^{2} \frac{3x^2}{1+e^x}\,dx$ equals	3. 4
S. $\dfrac{\displaystyle\int_{-\frac{1}{2}}^{\frac{1}{2}} \cos 2x \log\left(\frac{1+x}{1-x}\right)dx}{\displaystyle\int_{0}^{\frac{1}{2}} \cos 2x \log\left(\frac{1+x}{1-x}\right)dx}$ equals	4. 0

P Q R S
(A) 3241
(B) 2341
(C) 3214
(D) 2314

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function defined by $f ( x ) = \left\{ \begin{array} { l l } { [ x ] , } & x \leq 2 \\ 0 , & x > 2 \end{array} \right.$, where $[ x ]$ is the greatest integer less than or equal to $x$. If $I = \int _ { - 1 } ^ { 2 } \frac { x f \left( x ^ { 2 } \right) } { 2 + f ( x + 1 ) } d x$, then the value of $( 4 I - 1 )$ is

The option(s) with the values of $a$ and $L$ that satisfy the following equation is(are)
$$\frac { \int _ { 0 } ^ { 4 \pi } e ^ { t } \left( \sin ^ { 6 } a t + \cos ^ { 4 } a t \right) d t } { \int _ { 0 } ^ { \pi } e ^ { t } \left( \sin ^ { 6 } a t + \cos ^ { 4 } a t \right) d t } = L ?$$
(A) $\quad a = 2 , L = \frac { e ^ { 4 \pi } - 1 } { e ^ { \pi } - 1 }$
(B) $\quad a = 2 , L = \frac { e ^ { 4 \pi } + 1 } { e ^ { \pi } + 1 }$
(C) $\quad a = 4 , L = \frac { e ^ { 4 \pi } - 1 } { e ^ { \pi } - 1 }$
(D) $\quad a = 4 , L = \frac { e ^ { 4 \pi } + 1 } { e ^ { \pi } + 1 }$

Let $f ^ { \prime } ( x ) = \frac { 192 x ^ { 3 } } { 2 + \sin ^ { 4 } \pi x }$ for all $x \in \mathbb { R }$ with $f \left( \frac { 1 } { 2 } \right) = 0$. If $m \leq \int _ { 1 / 2 } ^ { 1 } f ( x ) d x \leq M$, then the possible values of $m$ and $M$ are
(A) $m = 13 , M = 24$
(B) $\quad m = \frac { 1 } { 4 } , M = \frac { 1 } { 2 }$
(C) $m = - 11 , M = 0$
(D) $m = 1 , M = 12$

The value of $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \frac { x ^ { 2 } \cos x } { 1 + e ^ { x } } d x$ is equal to
(A) $\frac { \pi ^ { 2 } } { 4 } - 2$
(B) $\frac { \pi ^ { 2 } } { 4 } + 2$
(C) $\pi ^ { 2 } - e ^ { \frac { \pi } { 2 } }$
(D) $\pi ^ { 2 } + e ^ { \frac { \pi } { 2 } }$

The total number of distinct $x \in [0,1]$ for which $\int_0^x \frac{t^2}{1+t^4}\,dt = 2x - 1$ is

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f(0) = 0$, $f\left(\frac{\pi}{2}\right) = 3$ and $f'(0) = 1$. If $$g(x) = \int_x^{\frac{\pi}{2}} \left[f'(t)\operatorname{cosec} t - \cot t\operatorname{cosec} t\, f(t)\right] dt$$ for $x \in \left(0, \frac{\pi}{2}\right]$, then $\lim_{x \rightarrow 0} g(x) =$

If $$I = \frac { 2 } { \pi } \int _ { - \pi / 4 } ^ { \pi / 4 } \frac { d x } { \left( 1 + e ^ { \sin x } \right) ( 2 - \cos 2 x ) }$$ then $27 I ^ { 2 }$ equals

Which of the following inequalities is/are TRUE?
(A) $\int _ { 0 } ^ { 1 } x \cos x \, d x \geq \frac { 3 } { 8 }$
(B) $\int _ { 0 } ^ { 1 } x \sin x \, d x \geq \frac { 3 } { 10 }$
(C) $\int _ { 0 } ^ { 1 } x ^ { 2 } \cos x \, d x \geq \frac { 1 } { 2 }$
(D) $\int _ { 0 } ^ { 1 } x ^ { 2 } \sin x \, d x \geq \frac { 2 } { 9 }$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that its derivative $f'$ is continuous and $f(\pi) = -6$.
If $F: [0, \pi] \rightarrow \mathbb{R}$ is defined by $F(x) = \int_{0}^{x} f(t)\, dt$, and if $$\int_{0}^{\pi} \left(f'(x) + F(x)\right) \cos x\, dx = 2$$ then the value of $f(0)$ is $\_\_\_\_$

Let $g_i : [\pi/8, 3\pi/8] \to \mathbb{R}$, $i = 1, 2$, and $f : [\pi/8, 3\pi/8] \to \mathbb{R}$ be functions such that $$g_1(x) = 1, \quad g_2(x) = |4x - \pi|, \quad f(x) = \sin^2 x,$$ for all $x \in [\pi/8, 3\pi/8]$. Define $$S_i = \int_{\pi/8}^{3\pi/8} f(x) \cdot g_i(x) \, dx, \quad i = 1, 2.$$
The value of $\frac{16 S_1}{\pi}$ is ____.

Let $f _ { 1 } : ( 0 , \infty ) \rightarrow \mathbb { R }$ and $f _ { 2 } : ( 0 , \infty ) \rightarrow \mathbb { R }$ be defined by $$f _ { 1 } ( x ) = \int _ { 0 } ^ { x } \prod _ { j = 1 } ^ { 21 } ( t - j ) ^ { j } d t , \quad x > 0$$ 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 $2 m _ { 1 } + 3 n _ { 1 } + m _ { 1 } n _ { 1 }$ is $\_\_\_\_$.

Let $g_i : [\pi/8, 3\pi/8] \to \mathbb{R}$, $i = 1, 2$, and $f : [\pi/8, 3\pi/8] \to \mathbb{R}$ be functions such that $$g_1(x) = 1, \quad g_2(x) = |4x - \pi|, \quad f(x) = \sin^2 x,$$ for all $x \in [\pi/8, 3\pi/8]$. Define $$S_i = \int_{\pi/8}^{3\pi/8} f(x) \cdot g_i(x) \, dx, \quad i = 1, 2.$$
The value of $\frac{48 S_2}{\pi^2}$ is ____.

Let $f _ { 1 } : ( 0 , \infty ) \rightarrow \mathbb { R }$ and $f _ { 2 } : ( 0 , \infty ) \rightarrow \mathbb { R }$ be defined by $$f _ { 1 } ( x ) = \int _ { 0 } ^ { x } \prod _ { j = 1 } ^ { 21 } ( t - j ) ^ { j } d t , \quad x > 0$$ 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 $6 m _ { 2 } + 4 n _ { 2 } + 8 m _ { 2 } n _ { 2 }$ is $\_\_\_\_$.

For any real number $x$, let $\lfloor x \rfloor$ denote the largest integer less than or equal to $x$. If $$I = \int_0^{10} \left\lfloor \frac{10x}{x+1} \right\rfloor dx,$$ then the value of $9I$ is ____.

Let $g _ { i } : \left[ \frac { \pi } { 8 } , \frac { 3 \pi } { 8 } \right] \rightarrow \mathbb { R } , i = 1,2$, and $f : \left[ \frac { \pi } { 8 } , \frac { 3 \pi } { 8 } \right] \rightarrow \mathbb { R }$ be functions such that $$g _ { 1 } ( x ) = 1 , g _ { 2 } ( x ) = | 4 x - \pi | \text { and } f ( x ) = \sin ^ { 2 } x , \text { for all } x \in \left[ \frac { \pi } { 8 } , \frac { 3 \pi } { 8 } \right]$$ Define $$S _ { i } = \int _ { \frac { \pi } { 8 } } ^ { \frac { 3 \pi } { 8 } } f ( x ) \cdot g _ { i } ( x ) d x , \quad i = 1,2$$ The value of $\frac { 16 S _ { 1 } } { \pi }$ is $\_\_\_\_$.

Let $g _ { i } : \left[ \frac { \pi } { 8 } , \frac { 3 \pi } { 8 } \right] \rightarrow \mathbb { R } , i = 1,2$, and $f : \left[ \frac { \pi } { 8 } , \frac { 3 \pi } { 8 } \right] \rightarrow \mathbb { R }$ be functions such that $$g _ { 1 } ( x ) = 1 , g _ { 2 } ( x ) = | 4 x - \pi | \text { and } f ( x ) = \sin ^ { 2 } x , \text { for all } x \in \left[ \frac { \pi } { 8 } , \frac { 3 \pi } { 8 } \right]$$ Define $$S _ { i } = \int _ { \frac { \pi } { 8 } } ^ { \frac { 3 \pi } { 8 } } f ( x ) \cdot g _ { i } ( x ) d x , \quad i = 1,2$$ The value of $\frac { 48 S _ { 2 } } { \pi ^ { 2 } }$ is $\_\_\_\_$.

Let $\psi _ { 1 } : [ 0 , \infty ) \rightarrow \mathbb { R } , \quad \psi _ { 2 } : [ 0 , \infty ) \rightarrow \mathbb { R } , \quad f : [ 0 , \infty ) \rightarrow \mathbb { R }$ and $g : [ 0 , \infty ) \rightarrow \mathbb { R }$ be functions such that $f ( 0 ) = g ( 0 ) = 0$, $$\begin{gathered} \psi _ { 1 } ( x ) = e ^ { - x } + x , \quad x \geq 0 \\ \psi _ { 2 } ( x ) = x ^ { 2 } - 2 x - 2 e ^ { - x } + 2 , \quad x \geq 0 \\ f ( x ) = \int _ { - x } ^ { x } \left( | t | - t ^ { 2 } \right) e ^ { - t ^ { 2 } } d t , \quad x > 0 \end{gathered}$$ and $$g ( x ) = \int _ { 0 } ^ { x ^ { 2 } } \sqrt { t } e ^ { - t } d t , \quad x > 0$$ Which of the following statements is TRUE ?
(A) $f ( \sqrt { \ln 3 } ) + g ( \sqrt { \ln 3 } ) = \frac { 1 } { 3 }$
(B) For every $x > 1$, there exists an $\alpha \in ( 1 , x )$ such that $\psi _ { 1 } ( x ) = 1 + \alpha x$
(C) For every $x > 0$, there exists a $\beta \in ( 0 , x )$ such that $\psi _ { 2 } ( x ) = 2 x \left( \psi _ { 1 } ( \beta ) - 1 \right)$
(D) $f$ is an increasing function on the interval $\left[ 0 , \frac { 3 } { 2 } \right]$

Let $\psi _ { 1 } : [ 0 , \infty ) \rightarrow \mathbb { R } , \quad \psi _ { 2 } : [ 0 , \infty ) \rightarrow \mathbb { R } , \quad f : [ 0 , \infty ) \rightarrow \mathbb { R }$ and $g : [ 0 , \infty ) \rightarrow \mathbb { R }$ be functions such that $f ( 0 ) = g ( 0 ) = 0$, $$\begin{gathered} \psi _ { 1 } ( x ) = e ^ { - x } + x , \quad x \geq 0 \\ \psi _ { 2 } ( x ) = x ^ { 2 } - 2 x - 2 e ^ { - x } + 2 , \quad x \geq 0 \\ f ( x ) = \int _ { - x } ^ { x } \left( | t | - t ^ { 2 } \right) e ^ { - t ^ { 2 } } d t , \quad x > 0 \end{gathered}$$ and $$g ( x ) = \int _ { 0 } ^ { x ^ { 2 } } \sqrt { t } e ^ { - t } d t , \quad x > 0$$ Which of the following statements is TRUE ?
(A) $\psi _ { 1 } ( x ) \leq 1$, for all $x > 0$
(B) $\psi _ { 2 } ( x ) \leq 0$, for all $x > 0$
(C) $f ( x ) \geq 1 - e ^ { - x ^ { 2 } } - \frac { 2 } { 3 } x ^ { 3 } + \frac { 2 } { 5 } x ^ { 5 }$, for all $x \in \left( 0 , \frac { 1 } { 2 } \right)$
(D) $g ( x ) \leq \frac { 2 } { 3 } x ^ { 3 } - \frac { 2 } { 5 } x ^ { 5 } + \frac { 1 } { 7 } x ^ { 7 }$, for all $x \in \left( 0 , \frac { 1 } { 2 } \right)$

For any real number $x$, let $[ x ]$ denote the largest integer less than or equal to $x$. If $$I = \int _ { 0 } ^ { 10 } \left[ \sqrt { \frac { 10 x } { x + 1 } } \right] d x$$ then the value of $9 I$ is $\_\_\_\_$.

The greatest integer less than or equal to
$$\int _ { 1 } ^ { 2 } \log _ { 2 } \left( x ^ { 3 } + 1 \right) d x + \int _ { 1 } ^ { \log _ { 2 } 9 } \left( 2 ^ { x } - 1 \right) ^ { \frac { 1 } { 3 } } d x$$
is $\_\_\_\_$ .