Let $f$ be a differentiable function such that $x^2 f(x) - x = 4\int_0^x t\, f(t)\, dt$, $f(1) = \frac{2}{3}$. Then $18\, f(3)$ is equal to
(1) 210
(2) 160
(3) 150
(4) 180

The slope of tangent at any point $(x, y)$ on a curve $y = y(x)$ is $\frac{x^2 + y^2}{2xy}$, $x > 0$. If $y(2) = 0$, then a value of $y(8)$ is
(1) $-4\sqrt{2}$
(2) $2\sqrt{3}$
(3) $-2\sqrt{3}$
(4) $4\sqrt{3}$

The area enclosed by the closed curve $C$ given by the differential equation $\frac{dy}{dx} + \frac{x + a}{y - 2} = 0$, $y(1) = 0$ is $4\pi$. Let $P$ and $Q$ be the points of intersection of the curve $C$ and the $y$-axis. If normals at $P$ and $Q$ on the curve $C$ intersect $x$-axis at points $R$ and $S$ respectively, then the length of the line segment $RS$ is
(1) $2\sqrt{3}$
(2) $\frac{2\sqrt{3}}{3}$
(3) 2
(4) $\frac{4\sqrt{3}}{3}$

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a differentiable function that satisfies the relation $f ( x + y ) = f ( x ) + f ( y ) - 1 , \forall x , y \in \mathbb { R }$. If $f ^ { \prime } ( 0 ) = 2$, then $| f ( - 2 ) |$ is equal to

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

Let $\int _ { 0 } ^ { x } \sqrt { 1 - \left( y ^ { \prime } ( t ) \right) ^ { 2 } } d t = \int _ { 0 } ^ { x } y ( t ) d t , 0 \leq x \leq 3 , y \geq 0 , y ( 0 ) = 0$. Then at $x = 2 , y ^ { \prime \prime } + y + 1$ is equal to
(1) 1
(2) 2
(3) $\sqrt { 2 }$
(4) $1 / 2$

The differential equation of the family of circles passing through the origin and having centre at the line $y = x$ is :
(1) $\left( x ^ { 2 } - y ^ { 2 } + 2 x y \right) \mathrm { d } x = \left( x ^ { 2 } - y ^ { 2 } - 2 x y \right) \mathrm { d } y$
(2) $\left( x ^ { 2 } + y ^ { 2 } + 2 x y \right) \mathrm { d } x = \left( x ^ { 2 } + y ^ { 2 } - 2 x y \right) \mathrm { d } y$
(3) $\left( x ^ { 2 } + y ^ { 2 } - 2 x y \right) \mathrm { d } x = \left( x ^ { 2 } + y ^ { 2 } + 2 x y \right) \mathrm { d } y$
(4) $\left( x ^ { 2 } - y ^ { 2 } + 2 x y \right) \mathrm { d } x = \left( x ^ { 2 } - y ^ { 2 } + 2 x y \right) \mathrm { d } y$

Suppose the solution of the differential equation $\frac { d y } { d x } = \frac { ( 2 + \alpha ) x - \beta y + 2 } { \beta x - 2 \alpha y - ( \beta \gamma - 4 \alpha ) }$ represents a circle passing through origin. Then the radius of this circle is:
(1) 2
(2) $\sqrt { 17 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { \sqrt { 17 } } { 2 }$

If the solution of the differential equation $( 2 x + 3 y - 2 ) d x + ( 4 x + 6 y - 7 ) d y = 0 , y ( 0 ) = 3$, is $\alpha x + \beta y + 3 \log _ { e } | 2 x + 3 y - \gamma | = 6$, then $\alpha + 2 \beta + 3 \gamma$ is equal to $\_\_\_\_$.

Let $y = y ( x )$ be the solution of the differential equation $( x + y + 2 ) ^ { 2 } d x = d y , y ( 0 ) = - 2$. Let the maximum and minimum values of the function $y = y ( x )$ in $\left[ 0 , \frac { \pi } { 3 } \right]$ be $\alpha$ and $\beta$, respectively. If $( 3 \alpha + \pi ) ^ { 2 } + \beta ^ { 2 } = \gamma + \delta \sqrt { 3 } , \gamma , \delta \in \mathbb { Z }$, then $\gamma + \delta$ equals $\_\_\_\_$

Let $Y = Y(X)$ be a curve lying in the first quadrant such that the area enclosed by the line $Y - y = Y'(x)(X - x)$ and the coordinate axes, where $(x, y)$ is any point on the curve, is always $\frac{-y^2}{2Y'(x)} + 1$, $Y'(x) \neq 0$. If $Y(1) = 1$, then $12\,Y(2)$ equals $\underline{\hspace{1cm}}$.

Let $\alpha | x | = | y | \mathrm { e } ^ { x y - \beta } , \alpha , \beta \in \mathbf { N }$ be the solution of the differential equation $x \mathrm {~d} y - y \mathrm {~d} x + x y ( x \mathrm {~d} y + y \mathrm {~d} x ) = 0$, $y ( 1 ) = 2$. Then $\alpha + \beta$ is equal to $\_\_\_\_$

Let a curve $y = f ( x )$ pass through the points $( 0,5 )$ and $\left( \log _ { e } 2 , k \right)$. If the curve satisfies the differential equation $2 ( 3 + y ) e ^ { 2 x } d x - \left( 7 + e ^ { 2 x } \right) d y = 0$, then $k$ is equal to
(1) 4
(2) 32
(3) 8
(4) 16

Let $f(x) = \frac{2^{x+2} + 16}{2^{2x+1} + 2^{x+4} + 32}$. Then the value of $8\left(f\left(\frac{1}{15}\right) + f\left(\frac{2}{15}\right) + \ldots + f\left(\frac{59}{15}\right)\right)$ is equal to
(1) 92
(2) 118
(3) 102
(4) 108

Let for some function $\mathrm { y } = f ( x ) , \int _ { 0 } ^ { x } t f ( t ) d t = x ^ { 2 } f ( x ) , x > 0$ and $f ( 2 ) = 3$. Then $f ( 6 )$ is equal to
(1) 1
(2) 3
(3) 6
(4) 2

Let $f:(0,\infty) \rightarrow \mathbf{R}$ be a twice differentiable function. If for some $\mathrm{a} \neq 0$, $\int_0^1 f(\lambda x)\,\mathrm{d}\lambda = \mathrm{a}f(x)$, $f(1) = 1$ and $f(16) = \frac{1}{8}$, then $16 - f'\left(\frac{1}{16}\right)$ is equal to \_\_\_\_ .

Let $f$ be a differentiable function such that $2(x+2)^2 f(x) - 3(x+2)^2 = 10\int_0^x (t+2)f(t)\,dt$, $x \geq 0$. Then $f(2)$ is equal to $\underline{\hspace{2cm}}$.

Q75. The solution curve, of the differential equation $2 y \frac { \mathrm {~d} y } { \mathrm {~d} x } + 3 = 5 \frac { \mathrm {~d} y } { \mathrm {~d} x }$, passing through the point $( 0,1 )$ is a conic, whose vertex lies on the line:
(1) $2 x + 3 y = 9$
(2) $2 x + 3 y = - 9$
(3) $2 x + 3 y = - 6$
(4) $2 x + 3 y = 6$

Q76. The differential equation of the family of circles passing through the origin and having centre at the line $y = x$ is :
(1) $\left( x ^ { 2 } - y ^ { 2 } + 2 x y \right) \mathrm { d } x = \left( x ^ { 2 } - y ^ { 2 } - 2 x y \right) \mathrm { d } y$
(2) $\left( x ^ { 2 } + y ^ { 2 } + 2 x y \right) \mathrm { d } x = \left( x ^ { 2 } + y ^ { 2 } - 2 x y \right) \mathrm { d } y$
(3) $\left( x ^ { 2 } + y ^ { 2 } - 2 x y \right) \mathrm { d } x = \left( x ^ { 2 } + y ^ { 2 } + 2 x y \right) \mathrm { d } y$
(4) $\left( x ^ { 2 } - y ^ { 2 } + 2 x y \right) \mathrm { d } x = \left( x ^ { 2 } - y ^ { 2 } + 2 x y \right) \mathrm { d } y$

Q76. Suppose the solution of the differential equation $\frac { d y } { d x } = \frac { ( 2 + \alpha ) x - \beta y + 2 } { \beta x - 2 \alpha y - ( \beta \gamma - 4 \alpha ) }$ represents a circle passing through origin. Then the radius of this circle is :
(1) 2
(2) $\sqrt { 17 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { \sqrt { 17 } } { 2 }$

Q76. The solution of the differential equation $\left( x ^ { 2 } + y ^ { 2 } \right) \mathrm { d } x - 5 x y \mathrm {~d} y = 0 , y ( 1 ) = 0$, is :
(1) $\left| x ^ { 2 } - 2 y ^ { 2 } \right| ^ { 6 } = x$
(2) $\left| x ^ { 2 } - 4 y ^ { 2 } \right| ^ { 6 } = x$
(3) $\left| x ^ { 2 } - 4 y ^ { 2 } \right| ^ { 5 } = x ^ { 2 }$
(4) $\left| x ^ { 2 } - 2 y ^ { 2 } \right| ^ { 5 } = x ^ { 2 }$

Q77. If the solution $y = y ( x )$ of the differential equation $\left( x ^ { 4 } + 2 x ^ { 3 } + 3 x ^ { 2 } + 2 x + 2 \right) \mathrm { d } y - \left( 2 x ^ { 2 } + 2 x + 3 \right) \mathrm { d } x = 0$ satisfies $y ( - 1 ) = - \frac { \pi } { 4 }$, then $y ( 0 )$ is equal to :
(1) $\frac { \pi } { 2 }$
(2) $- \frac { \pi } { 2 }$
(3) 0
(4) $\frac { \pi } { 4 }$

Q88. Let $y = y ( x )$ be the solution of the differential equation $( x + y + 2 ) ^ { 2 } d x = d y , y ( 0 ) = - 2$. Let the maximum and minimum values of the function $y = y ( x )$ in $\left[ 0 , \frac { \pi } { 3 } \right]$ be $\alpha$ and $\beta$, respectively. If $( 3 \alpha + \pi ) ^ { 2 } + \beta ^ { 2 } = \gamma + \delta \sqrt { 3 } , \gamma , \delta \in \mathbb { Z }$, then $\gamma + \delta$ equals $\_\_\_\_$

Q89. Let $\alpha | x | = | y | \mathrm { e } ^ { x y - \beta } , \alpha , \beta \in \mathbf { N }$ be the solution of the differential equation $x \mathrm {~d} y - y \mathrm {~d} x + x y ( x \mathrm {~d} y + y \mathrm {~d} x ) = 0$, $y ( 1 ) = 2$. Then $\alpha + \beta$ is equal to $\_\_\_\_$

If $y = y(x)$ and $\left(1 + x^{2}\right)dy + \left(1 - \tan^{-1}x\right)dx = 0$ and $y(0) = 1$ then $\mathbf{y}(1)$ is
(A) $\frac{\pi^{2}}{32} + \frac{\pi}{4} + 1$ (B) $\frac{\pi^{2}}{32} - \frac{\pi}{2} + 1$ (C) $\frac{\pi^{2}}{32} + \frac{\pi}{2} - 1$ (D) $\frac{\pi^{2}}{32} - \frac{\pi}{4} + 1$