Let $f:(0,\infty) \rightarrow \mathbb{R}$ be a differentiable function such that $f'(x) = 2 - \frac{f(x)}{x}$ for all $x \in (0,\infty)$ and $f(1) \neq 1$. Then
(A) $\lim_{x \rightarrow 0+} f'\left(\frac{1}{x}\right) = 1$
(B) $\lim_{x \rightarrow 0+} xf\left(\frac{1}{x}\right) = 2$
(C) $\lim_{x \rightarrow 0+} x^2 f'(x) = 0$
(D) $|f(x)| \leq 2$ for all $x \in (0,2)$

A solution curve of the differential equation $\left(x^2 + xy + 4x + 2y + 4\right)\frac{dy}{dx} - y^2 = 0, x > 0$, passes through the point $(1,3)$. Then the solution curve
(A) intersects $y = x + 2$ exactly at one point
(B) intersects $y = x + 2$ exactly at two points
(C) intersects $y = (x+2)^2$
(D) does NOT intersect $y = (x+3)^2$

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ and $g : \mathbb { R } \rightarrow \mathbb { R }$ be two non-constant differentiable functions. If $$f ^ { \prime } ( x ) = \left( e ^ { ( f ( x ) - g ( x ) ) } \right) g ^ { \prime } ( x ) \text { for all } x \in \mathbb { R }$$ and $f ( 1 ) = g ( 2 ) = 1$, then which of the following statement(s) is (are) TRUE?
(A) $f ( 2 ) < 1 - \log _ { \mathrm { e } } 2$
(B) $f ( 2 ) > 1 - \log _ { \mathrm { e } } 2$
(C) $g ( 1 ) > 1 - \log _ { \mathrm { e } } 2$
(D) $g ( 1 ) < 1 - \log _ { e } 2$

Let $f : [ 0 , \infty ) \rightarrow \mathbb { R }$ be a continuous function such that $$f ( x ) = 1 - 2 x + \int _ { 0 } ^ { x } e ^ { x - t } f ( t ) d t$$ for all $x \in [ 0 , \infty )$. Then, which of the following statement(s) is (are) TRUE?
(A) The curve $y = f ( x )$ passes through the point $( 1,2 )$
(B) The curve $y = f ( x )$ passes through the point $( 2 , - 1 )$
(C) The area of the region $\left\{ ( x , y ) \in [ 0,1 ] \times \mathbb { R } : f ( x ) \leq y \leq \sqrt { 1 - x ^ { 2 } } \right\}$ is $\frac { \pi - 2 } { 4 }$
(D) The area of the region $\left\{ ( x , y ) \in [ 0,1 ] \times \mathbb { R } : f ( x ) \leq y \leq \sqrt { 1 - x ^ { 2 } } \right\}$ is $\frac { \pi - 1 } { 4 }$

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a differentiable function with $f ( 0 ) = 0$. If $y = f ( x )$ satisfies the differential equation
$$\frac { d y } { d x } = ( 2 + 5 y ) ( 5 y - 2 )$$
then the value of $\lim _ { x \rightarrow - \infty } f ( x )$ is $\_\_\_\_$ .

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a differentiable function with $f ( 0 ) = 1$ and satisfying the equation
$$f ( x + y ) = f ( x ) f ^ { \prime } ( y ) + f ^ { \prime } ( x ) f ( y ) \text { for all } x , y \in \mathbb { R }$$
Then, the value of $\log _ { e } ( f ( 4 ) )$ is $\_\_\_\_$ .

For any real numbers $\alpha$ and $\beta$, let $y _ { \alpha , \beta } ( x ) , x \in \mathbb { R }$, be the solution of the differential equation $$\frac { d y } { d x } + \alpha y = x e ^ { \beta x } , \quad y ( 1 ) = 1$$ Let $S = \left\{ y _ { \alpha , \beta } ( x ) : \alpha , \beta \in \mathbb { R } \right\}$. Then which of the following functions belong(s) to the set $S$ ?
(A) $f ( x ) = \frac { x ^ { 2 } } { 2 } e ^ { - x } + \left( e - \frac { 1 } { 2 } \right) e ^ { - x }$
(B) $f ( x ) = - \frac { x ^ { 2 } } { 2 } e ^ { - x } + \left( e + \frac { 1 } { 2 } \right) e ^ { - x }$
(C) $f ( x ) = \frac { e ^ { x } } { 2 } \left( x - \frac { 1 } { 2 } \right) + \left( e - \frac { e ^ { 2 } } { 4 } \right) e ^ { - x }$
(D) $f ( x ) = \frac { e ^ { x } } { 2 } \left( \frac { 1 } { 2 } - x \right) + \left( e + \frac { e ^ { 2 } } { 4 } \right) e ^ { - x }$

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 $y ( x )$ be the solution of the differential equation
$$x ^ { 2 } \frac { d y } { d x } + x y = x ^ { 2 } + y ^ { 2 } , \quad x > \frac { 1 } { e }$$
satisfying $y ( 1 ) = 0$. Then the value of $2 \frac { ( y ( e ) ) ^ { 2 } } { y \left( e ^ { 2 } \right) }$ is $\_\_\_\_$.

For all $x > 0$, let $y _ { 1 } ( x ) , y _ { 2 } ( x )$, and $y _ { 3 } ( x )$ be the functions satisfying
$$\begin{aligned} & \frac { d y _ { 1 } } { d x } - ( \sin x ) ^ { 2 } y _ { 1 } = 0 , \quad y _ { 1 } ( 1 ) = 5 \\ & \frac { d y _ { 2 } } { d x } - ( \cos x ) ^ { 2 } y _ { 2 } = 0 , \quad y _ { 2 } ( 1 ) = \frac { 1 } { 3 } \\ & \frac { d y _ { 3 } } { d x } - \left( \frac { 2 - x ^ { 3 } } { x ^ { 3 } } \right) y _ { 3 } = 0 , \quad y _ { 3 } ( 1 ) = \frac { 3 } { 5 e } \end{aligned}$$
respectively. Then
$$\lim _ { x \rightarrow 0 ^ { + } } \frac { y _ { 1 } ( x ) y _ { 2 } ( x ) y _ { 3 } ( x ) + 2 x } { e ^ { 3 x } \sin x }$$
is equal to $\_\_\_\_$ .

The population $p(t)$ at time $t$ of a certain mouse species satisfies the differential equation $\frac{dp(t)}{dt} = 0.5\,p(t) - 450$. If $p(0) = 850$, then the time at which the population becomes zero is
(1) $\ln 18$
(2) $\ln 9$
(3) $\frac{1}{2}\ln 18$
(4) $2\ln 18$

If $\frac { d y } { d x } + y \tan x = \sin 2 x$ and $y ( 0 ) = 1$, then $y ( \pi )$ is equal to
(1) - 1
(2) 5
(3) 1
(4) - 5

Let $y(x)$ be the solution of the differential equation $(x \log x) \frac{dy}{dx} + y = 2x \log x$, $(x \geq 1)$. Then $y(e)$ is equal to:
(1) $e$
(2) $0$
(3) $2$
(4) $2e$

Let $y ( x )$ be the solution of the differential equation $( x \log x ) \frac { d y } { d x } + y = 2 x \log x , ( x \geq 1 )$. Then $y ( e )$ is equal to
(1) $2 e$
(2) $e$
(3) 0
(4) 2

The solution of the differential equation $\frac { d y } { d x } + \frac { y } { 2 } \sec x = \frac { \tan x } { 2 y }$, where $0 \leq x < \frac { \pi } { 2 }$ and $y ( 0 ) = 1$, is given by
(1) $y ^ { 2 } = 1 + \frac { x } { \sec x + \tan x }$
(2) $y = 1 + \frac { x } { \sec x + \tan x }$
(3) $y = 1 - \frac { x } { \sec x + \tan x }$
(4) $y ^ { 2 } = 1 - \frac { x } { \sec x + \tan x }$

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

Let $y = y ( x )$ be the solution of the differential equation $\sin x \frac { d y } { d x } + y \cos x = 4 x , x \in ( 0 , \pi )$. If $y \left( \frac { \pi } { 2 } \right) = 0$, then $y \left( \frac { \pi } { 6 } \right)$ is equal to
(1) $- \frac { 4 } { 9 } \pi ^ { 2 }$
(2) $\frac { 4 } { 9 \sqrt { 3 } } \pi ^ { 2 }$
(3) $\frac { - 8 } { 9 \sqrt { 3 } } \pi ^ { 2 }$
(4) $- \frac { 8 } { 9 } \pi ^ { 2 }$

Let $y = y ( x )$ be the solution of the differential equation $\frac { d y } { d x } + 2 y = f ( x )$, where $f ( x ) = \left\{ \begin{array} { l l } 1 , & x \in [ 0,1 ] \\ 0 , & \text { otherwise } \end{array} \right.$. If $y ( 0 ) = 0$, then $y \left( \frac { 3 } { 2 } \right)$ is
(1) $\frac { e ^ { 2 } - 1 } { e ^ { 3 } }$
(2) $\frac { 1 } { 2 e }$
(3) $\frac { e ^ { 2 } + 1 } { 2 e ^ { 4 } }$
(4) $\frac { e ^ { 2 } - 1 } { 2 e ^ { 3 } }$

Let $y = y ( x )$ be the solution of the differential equation $\frac { d y } { d x } + 2 y = f ( x )$, where
$$f ( x ) = \left\{ \begin{array} { l c } 1 , & x \in [ 0,1 ] \\ 0 , & \text { otherwise } \end{array} \right.$$
If $y ( 0 ) = 0$, then $y \left( \frac { 3 } { 2 } \right)$ is
(1) $\frac { e ^ { 2 } - 1 } { 2 e ^ { 3 } }$
(2) $\frac { e ^ { 2 } - 1 } { e ^ { 3 } }$
(3) $\frac { 1 } { 2 e }$
(4) $\frac { e ^ { 2 } + 1 } { 2 e ^ { 4 } }$

The area (in sq. units) of the region $A = \{(x,y) \in R \times R \mid 0 \leq x \leq 3, 0 \leq y \leq 4, y \leq x^2 + 3x\}$ is
(1) $\frac{26}{3}$
(2) $8$
(3) $\frac{53}{6}$
(4) $\frac{59}{6}$

If $\frac { d y } { d x } + \frac { 3 } { \cos ^ { 2 } x } y = \frac { 1 } { \cos ^ { 2 } x } , x \in \left( - \frac { \pi } { 3 } , \frac { \pi } { 3 } \right)$, and $y \left( \frac { \pi } { 4 } \right) = \frac { 4 } { 3 }$, then $y \left( - \frac { \pi } { 4 } \right)$ equals
(1) $\frac { 1 } { 3 }$
(2) $\frac { 1 } { 3 } + e ^ { 3 }$
(3) $\frac { 1 } { 3 } + e ^ { 6 }$
(4) $- \frac { 4 } { 3 }$

The solution of the differential equation $x \frac { d y } { d x } + 2 y = x ^ { 2 } , ( x \neq 0 )$ with $y ( 1 ) = 1$, is
(1) $y = \frac { x ^ { 3 } } { 5 } + \frac { 1 } { 5 x ^ { 2 } }$
(2) $y = \frac { 3 } { 4 } x ^ { 2 } + \frac { 1 } { 4 x ^ { 2 } }$
(3) $y = \frac { x ^ { 2 } } { 4 } + \frac { 3 } { 4 x ^ { 2 } }$
(4) $y = \frac { 4 } { 5 } x ^ { 3 } + \frac { 1 } { 5 x ^ { 2 } }$

If $\cos x \frac { d y } { d x } - y \sin x = 6 x , \left( 0 < x < \frac { \pi } { 2 } \right)$ and $y \left( \frac { \pi } { 3 } \right) = 0$, then $y \left( \frac { \pi } { 6 } \right)$ is equal to
(1) $- \frac { \pi ^ { 2 } } { 4 \sqrt { 3 } }$
(2) $\frac { \pi ^ { 2 } } { 2 \sqrt { 3 } }$
(3) $- \frac { \pi ^ { 2 } } { 2 }$
(4) $- \frac { \pi ^ { 2 } } { 2 \sqrt { 3 } }$

Let $y = y(x)$ be the solution of the differential equation, $\frac{2 + \sin x}{y + 1} \cdot \frac{dy}{dx} = -\cos x, y > 0, y(0) = 1$. If $y(\pi) = a$ and $\frac{dy}{dx}$ at $x = \pi$ is $b$, then the ordered pair $(a, b)$ is equal to
(1) $\left(2, \frac{3}{2}\right)$
(2) $(1, -1)$
(3) $(1, 1)$
(4) $(2, 1)$

The solution of the differential equation $\frac { d y } { d x } - \frac { y + 3 x } { \log _ { e } ( y + 3 x ) } + 3 = 0$ is (where $C$ is a constant of integration)
(1) $x - \frac { 1 } { 2 } \left( \log _ { e } ( y + 3 x ) \right) ^ { 2 } = C$
(2) $x - \log _ { e } ( y + 3 x ) = C$
(3) $y + 3 x - \frac { 1 } { 2 } \left( \log _ { e } x \right) ^ { 2 } = C$
(4) $x - 2 \log _ { e } ( y + 3 x ) = C$