If $\mathrm { P } ( 1 ) = 0$ and $( \mathrm { dP } ( \mathrm { x } ) ) / \mathrm { dx } > \mathrm { P } ( \mathrm { x } )$ for all $\mathrm { x } > 1$ then prove that $\mathrm { P } ( \mathrm { x } ) > 0$ for all $\mathrm { x } > 1$.

Match the statements/expressions in Column I with the open intervals in Column II.
Column I
(A) Interval contained in the domain of definition of non-zero solutions of the differential equation $( x - 3 ) ^ { 2 } y ^ { \prime } + y = 0$
(B) Interval containing the value of the integral $$\int _ { 1 } ^ { 5 } ( x - 1 ) ( x - 2 ) ( x - 3 ) ( x - 4 ) ( x - 5 ) d x$$ (C) Interval in which at least one of the points of local maximum of $\cos ^ { 2 } x + \sin x$ lies
(D) Interval in which $\tan ^ { - 1 } ( \sin x + \cos x )$ is increasing
Column II
(p) $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$
(q) $\left( 0 , \frac { \pi } { 2 } \right)$
(r) $\left( \frac { \pi } { 8 } , \frac { 5 \pi } { 4 } \right)$
(s) $\left( 0 , \frac { \pi } { 8 } \right)$
(t) $( - \pi , \pi )$

Let $f : [ 0,1 ] \rightarrow \mathbb { R }$ (the set of all real numbers) be a function. Suppose the function $f$ is twice differentiable, $f ( 0 ) = f ( 1 ) = 0$ and satisfies $f ^ { \prime \prime } ( x ) - 2 f ^ { \prime } ( x ) + f ( x ) \geq e ^ { x } , x \in [ 0,1 ]$.
Which of the following is true for $0 < x < 1$?
(A) $0 < f ( x ) < \infty$
(B) $- \frac { 1 } { 2 } < f ( x ) < \frac { 1 } { 2 }$
(C) $- \frac { 1 } { 4 } < f ( x ) < 1$
(D) $- \infty < f ( x ) < 0$

For $x \in \mathbb { R }$, let the function $y ( x )$ be the solution of the differential equation
$$\frac { d y } { d x } + 12 y = \cos \left( \frac { \pi } { 12 } x \right) , \quad y ( 0 ) = 0$$
Then, which of the following statements is/are TRUE ?
(A) $y ( x )$ is an increasing function
(B) $y ( x )$ is a decreasing function
(C) There exists a real number $\beta$ such that the line $y = \beta$ intersects the curve $y = y ( x )$ at infinitely many points
(D) $y ( x )$ is a periodic function

Let $f : R \rightarrow R$ be a function such that $| f ( x ) | \leq x ^ { 2 }$, for all $x \in R$. Then, at $x = 0 , f$ is
(1) differentiable but not continuous
(2) neither continuous nor differentiable
(3) continuous as well as differentiable
(4) continuous but not differentiable

Let $S = \left\{ ( \lambda , \mu ) \in R \times R : f ( t ) = \left( | \lambda | e ^ { | t | } - \mu \right) \sin ( 2 | t | ) , t \in R \right.$ is a differentiable function $\}$. Then, $S$ is a subset of :
(1) $( - \infty , 0 ) \times R$
(2) $R \times [ 0 , \infty )$
(3) $[ 0 , \infty ) \times R$
(4) $R \times ( - \infty , 0 )$

Let $S = \left\{ ( \lambda , \mu ) \in R \times R : f ( t ) = \left( | \lambda | e ^ { t } - \mu \right) \cdot \sin ( 2 | t | ) , t \in R \right.$, is a differentiable function $\}$. Then $S$ is a subest of?
(1) $R \times [ 0 , \infty )$
(2) $( - \infty , 0 ) \times R$
(3) $[ 0 , \infty ) \times R$
(4) $R \times ( - \infty , 0 )$

Let $y = y _ { 1 } ( x )$ and $y = y _ { 2 } ( x )$ be two distinct solutions of the differential equation $\frac { d y } { d x } = x + y$, with $y _ { 1 } ( 0 ) = 0$ and $y _ { 2 } ( 0 ) = 1$ respectively. Then, the number of points of intersection of $y = y _ { 1 } ( x )$ and $y = y _ { 2 } ( x )$ is
(1) 0
(2) 1
(3) 2
(4) 3

The number of solutions, of the equation $e^{\sin x} - 2e^{-\sin x} = 2$ is
(1) 2
(2) more than 2
(3) 1
(4) 0

$\frac{1}{x + 1} + x - 1 = \frac{1}{x^{2}}$
Given that, which of the following is the expression $x^{3} - 1$ equal to?
A) $\frac{2}{x - 1}$ B) $\frac{1}{x}$ C) $\frac{x - 1}{x}$ D) $-x$ E) $\frac{1}{x + 1}$

A function f defined on the set of natural numbers is defined for every n as
$$f ( n ) = \begin{cases} 5 n + 40 , & 0 \leq n < 10 \\ f ( n - 10 ) , & n \geq 10 \end{cases}$$
Example: $f ( 23 ) = f ( 13 ) = f ( 3 ) = 5 \cdot 3 + 40 = 55$
Accordingly, what is the sum of the two-digit numbers AB that satisfy the equation $f ( A B ) = A B$?
A) 75 B) 80 C) 90 D) 100 E) 105

$$f ( x ) = \left\{ \begin{array} { l l l } 10 - x ^ { 2 } & , & x < 0 \\ a x + b & , & 0 \leq x \leq 3 \\ ( 1 - x ) ^ { 2 } & , & x > 3 \end{array} \right.$$
The function is continuous on the set of real numbers.
Accordingly, what is the sum $\mathbf { a } + \mathbf { b }$?
A) 16 B) 15 C) 12 D) 9 E) 8

Let a be a real number. A function f is defined on the set of real numbers as
$$f ( x ) = \left\{ \begin{array} { c c c } a - x & , & x < 1 \\ 5 x - 4 & , & 1 \leq x \leq 5 \\ ( x - a ) ^ { 2 } + 12 & , & x > 5 \end{array} \right.$$
If there is only one point where the function f is not continuous, what is the value of
$$f ( 7 ) - f ( 0 )$$?
A) 3
B) 4
C) 5
D) 6
E) 7