Let $b$ be a nonzero real number. Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function such that $f(0) = 1$. If the derivative $f'$ of $f$ satisfies the equation $$f'(x) = \frac{f(x)}{b^{2} + x^{2}}$$ for all $x \in \mathbb{R}$, then which of the following statements is/are TRUE?
(A) If $b > 0$, then $f$ is an increasing function
(B) If $b < 0$, then $f$ is a decreasing function
(C) $f(x)f(-x) = 1$ for all $x \in \mathbb{R}$
(D) $f(x) - f(-x) = 0$ for all $x \in \mathbb{R}$

If $y ( x )$ is the solution of the differential equation
$$x d y - \left( y ^ { 2 } - 4 y \right) d x = 0 \text { for } x > 0 , \quad y ( 1 ) = 2$$
and the slope of the curve $y = y ( x )$ is never zero, then the value of $10 y ( \sqrt { 2 } )$ is $\_\_\_\_$ .

For $x \in \mathbb { R }$, let $y ( x )$ be a solution of the differential equation $$\left( x ^ { 2 } - 5 \right) \frac { d y } { d x } - 2 x y = - 2 x \left( x ^ { 2 } - 5 \right) ^ { 2 }$$ such that $y ( 2 ) = 7$. Then the maximum value of the function $y ( x )$ is

Let $f ( x )$ be a continuously differentiable function on the interval $( 0 , \infty )$ such that $f ( 1 ) = 2$ and
$$\lim _ { t \rightarrow x } \frac { t ^ { 10 } f ( x ) - x ^ { 10 } f ( t ) } { t ^ { 9 } - x ^ { 9 } } = 1$$
for each $x > 0$. Then, for all $x > 0 , f ( x )$ is equal to
(A) $\frac { 31 } { 11 x } - \frac { 9 } { 11 } x ^ { 10 }$
(B) $\frac { 9 } { 11 x } + \frac { 13 } { 11 } x ^ { 10 }$
(C) $\frac { - 9 } { 11 x } + \frac { 31 } { 11 } x ^ { 10 }$
(D) $\frac { 13 } { 11 x } + \frac { 9 } { 11 } x ^ { 10 }$

The normal to a curve at $P ( x , y )$ meets the $x$-axis at $G$. If the distance of $G$ from the origin is twice the abscissa of $P$, then the curve is a
(1) ellipse
(2) parabola
(3) circle
(4) pair of straight lines

The differential equation of all circles passing through the origin and having their centres on the $x$-axis is
(1) $x ^ { 2 } = y ^ { 2 } + x y \frac { d y } { d x }$
(2) $x ^ { 2 } = y ^ { 2 } + 3 x y \frac { d y } { d x }$
(3) $y ^ { 2 } = x ^ { 2 } + 2 x y \frac { d y } { d x }$
(4) $y ^ { 2 } = x ^ { 2 } - 2 x y \frac { d y } { d x }$

Statement-1: The slope of the tangent at any point P on a parabola, whose axis is the axis of $x$ and vertex is at the origin, is inversely proportional to the ordinate of the point P. Statement-2: The system of parabolas $y ^ { 2 } = 4 a x$ satisfies a differential equation of degree 1 and order 1.
(1) Statement-1 is true; Statement-2 is true; Statement-2 is a correct explanation for statement-1.
(2) Statement-1 is true; Statement-2 is true; Statement-2 is not a correct explanation for statement-1.
(3) Statement-1 is true; Statement-2 is false.
(4) Statement-1 is false; Statement-2 is true.

If the function $f ( x ) = \left\{ \begin{array} { c l } \frac { \sqrt { 2 + \cos x } - 1 } { ( \pi - x ) ^ { 2 } } , & x \neq \pi \\ k , & x = \pi \end{array} \right.$ is continuous at $x = \pi$, then $k$ equals
(1) $\frac { 1 } { 4 }$
(2) 0
(3) 2
(4) $\frac { 1 } { 2 }$

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 the population of rabbits surviving at a time $t$ be governed by the differential equation $\frac { d p ( t ) } { d t } = \frac { 1 } { 2 } \{ p ( t ) - 400 \}$. If $p ( 0 ) = 100$, then $p ( t )$ equals
(1) $600 - 500 e ^ { \frac { t } { 2 } }$
(2) $400 - 300 e ^ { \frac { - t } { 2 } }$
(3) $400 - 300 e ^ { t / 2 }$
(4) $300 - 200 e ^ { \frac { - t } { 2 } }$

If the differential equation representing the family of all circles touching $x$-axis at the origin is $\left( x ^ { 2 } - y ^ { 2 } \right) \frac { d y } { d x } = g ( x ) y$, then $g ( x )$ equals
(1) $\frac { 1 } { 2 } x ^ { 2 }$
(2) $2 x$
(3) $\frac { 1 } { 2 } x$
(4) $2 x ^ { 2 }$

If a curve $y = f(x)$ passes through the point $(1,-1)$ and satisfies the differential equation, $y(1+xy)dx = x\,dy$, then $f\left(-\frac{1}{2}\right)$ is equal to: (1) $-\frac{2}{5}$ (2) $-\frac{4}{5}$ (3) $\frac{2}{5}$ (4) $\frac{4}{5}$

For $x \in R , x \neq 0$, if $y ( x )$ is a differentiable function such that $x \int _ { 1 } ^ { x } y ( t ) d t = ( x + 1 ) \int _ { 1 } ^ { x } t y ( t ) d t$, then $y ( x )$ equals (where $C$ is a constant)
(1) $C x ^ { 3 } e ^ { \frac { 1 } { x } }$
(2) $\frac { C } { x ^ { 2 } } e ^ { - \frac { 1 } { x } }$
(3) $\frac { C } { x } e ^ { - \frac { 1 } { x } }$
(4) $\frac { C } { x ^ { 3 } } e ^ { - \frac { 1 } { x } }$

If a curve $y = f(x)$ passes through the point $(1, -1)$ and satisfies the differential equation, $y(1 + xy) dx = x\, dy$, then $f\left(-\frac{1}{2}\right)$ is equal to:
(1) $-\frac{2}{5}$
(2) $-\frac{4}{5}$
(3) $\frac{2}{5}$
(4) $\frac{4}{5}$

The solution of the differential equation $\frac{dy}{dx} + \frac{y}{2}\sec^2 x = \frac{\tan x \sec^2 x}{2y}$, where $y(0) = 1$, is given by:
(1) $y^2 = 1 + \frac{\tan x}{x}$
(2) $y^2 = 1 + \tan x$
(3) $y = 1 - \tan x$
(4) $y^2 = 1 - \tan x$

If $( 2 + \sin x ) \frac { d y } { d x } + ( y + 1 ) \cos x = 0$ and $y ( 0 ) = 1$, then $y \left( \frac { \pi } { 2 } \right)$ is equal to:
(1) $\frac { 1 } { 3 }$
(2) $- \frac { 2 } { 3 }$
(3) $- \frac { 1 } { 3 }$
(4) $\frac { 4 } { 3 }$

The curve satisfying the differential equation, $y d x - \left( x + 3 y ^ { 2 } \right) d y = 0$ and passing through the point $( 1,1 )$ also passes through the point
(1) $\left( \frac { 1 } { 4 } , - \frac { 1 } { 2 } \right)$
(2) $\left( - \frac { 1 } { 3 } , \frac { 1 } { 3 } \right)$
(3) $\left( \frac { 1 } { 4 } , \frac { 1 } { 2 } \right)$
(4) $\left( \frac { 1 } { 3 } , - \frac { 1 } { 3 } \right)$

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 $f:[0,1] \rightarrow R$ be such that $f(xy) = f(x) \cdot f(y)$, for all $x,y \in [0,1]$, and $f(0) \neq 0$. If $y = y(x)$ satisfies the differential equation, $\frac{dy}{dx} = f(x)$ with $y(0) = 1$ then $y\left(\frac{1}{4}\right) + y\left(\frac{3}{4}\right)$ is equal to:
(1) 5
(2) 2
(3) 3
(4) 4

Given that the slope of the tangent to a curve $y = y ( x )$ at any point $( x , y )$ is $\frac { 2 y } { x ^ { 2 } }$. If the curve passes through the centre of the circle $x ^ { 2 } + y ^ { 2 } - 2 x - 2 y = 0$, then its equation is
(1) $x ^ { 2 } \log _ { e } | y | = - 2 ( x - 1 )$
(2) $x \log _ { e } | y | = 2 ( x - 1 )$
(3) $x \log _ { e } | y | = - 2 ( x - 1 )$
(4) $x \log _ { e } | y | = x - 1$

If a curve passes through the point $( 1 , - 2 )$ and has slope of the tangent at any point $( x , y )$ on it as $\frac { x ^ { 2 } - 2 y } { x }$, then the curve also passes through the point
(1) $( \sqrt { 3 } , 0 )$
(2) $( - 1,2 )$
(3) $( - \sqrt { 2 } , 1 )$
(4) $( 3,0 )$

Let $x ^ { k } + y ^ { k } = a ^ { k }$, $(a, k > 0)$ and $\frac { d y } { d x } + \left( \frac { y } { x } \right) ^ { \frac { 1 } { 3 } } = 0$, then $k$ is
(1) $\frac { 3 } { 2 }$
(2) $\frac { 4 } { 3 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 1 } { 3 }$

Let $f : ( 0 , \infty ) \rightarrow ( 0 , \infty )$ be a differentiable function such that $f ( 1 ) = e$ and $\lim _ { t \rightarrow x } \frac { t ^ { 2 } f ^ { 2 } ( x ) - x ^ { 2 } f ^ { 2 } ( t ) } { t - x } = 0$. If $f ( x ) = 1$, then $x$ is equal to:
(1) $\frac { 1 } { e }$
(2) $2 e$
(3) $\frac { 1 } { 2 e }$
(4) $e$

If $y = y(x)$ is the solution of the differential equation, $e ^ { y } \left( \frac { d y } { d x } - 1 \right) = e ^ { x }$ such that $y(0) = 0$, then $y(1)$ is equal to
(1) $1 + \log _ { e } 2$
(2) $2 + \log _ { e } 2$
(3) $2e$
(4) $\log _ { e } 2$