Let a solution $y = y ( x )$ of the differential equation
$$x \sqrt { x ^ { 2 } - 1 } d y - y \sqrt { y ^ { 2 } - 1 } d x = 0$$
satisfy $y ( 2 ) = \frac { 2 } { \sqrt { 3 } }$. STATEMENT-1 : $y ( x ) = \sec \left( \sec ^ { - 1 } x - \frac { \pi } { 6 } \right)$ and STATEMENT-2 : $y ( x )$ is given by
$$\frac { 1 } { y } = \frac { 2 \sqrt { 3 } } { x } - \sqrt { 1 - \frac { 1 } { x ^ { 2 } } }$$
(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
(B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
(C) STATEMENT-1 is True, STATEMENT-2 is False
(D) STATEMENT-1 is False, STATEMENT-2 is True

Let $f$ be a non-negative function defined on the interval $[ 0,1 ]$. If
$$\int _ { 0 } ^ { x } \sqrt { 1 - \left( f ^ { \prime } ( t ) \right) ^ { 2 } } d t = \int _ { 0 } ^ { x } f ( t ) d t , \quad 0 \leq x \leq 1 ,$$
and $f ( 0 ) = 0$, then
(A) $f \left( \frac { 1 } { 2 } \right) < \frac { 1 } { 2 }$ and $f \left( \frac { 1 } { 3 } \right) > \frac { 1 } { 3 }$
(B) $f \left( \frac { 1 } { 2 } \right) > \frac { 1 } { 2 }$ and $f \left( \frac { 1 } { 3 } \right) > \frac { 1 } { 3 }$
(C) $f \left( \frac { 1 } { 2 } \right) < \frac { 1 } { 2 }$ and $f \left( \frac { 1 } { 3 } \right) < \frac { 1 } { 3 }$
(D) $f \left( \frac { 1 } { 2 } \right) > \frac { 1 } { 2 }$ and $f \left( \frac { 1 } { 3 } \right) < \frac { 1 } { 3 }$

Match the statements/expressions given in Column I with the values given in Column II.
Column I
(A) The number of solutions of the equation $$xe^{\sin x}-\cos x=0$$ in the interval $\left(0,\frac{\pi}{2}\right)$
(B) Value(s) of $k$ for which the planes $kx+4y+z=0,4x+ky+2z=0$ and $2x+2y+z=0$ intersect in a straight line
(C) Value(s) of $k$ for which $$|x-1|+|x-2|+|x+1|+|x+2|=4k$$ has integer solution(s)
(D) If $$y^{\prime}=y+1\text{ and }y(0)=1$$ then value(s) of $y(\ln2)$
Column II
(p) 1
(q) 2
(r) 3
(s) 4
(t) 5

Let $f:\mathbf{R}\rightarrow\mathbf{R}$ be a continuous function which satisfies $$f(x)=\int_{0}^{x}f(t)\,dt.$$ Then the value of $f(\ln5)$ is

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 be a real-valued differentiable function on $\mathbf { R }$ (the set of all real numbers) such that $f ( 1 ) = 1$. If the $y$-intercept of the tangent at any point $P ( x , y )$ on the curve $y = f ( x )$ is equal to the cube of the abscissa of $P$, then the value of $f ( - 3 )$ is equal to

A curve passes through the point $\left( 1 , \frac { \pi } { 6 } \right)$. Let the slope of the curve at each point $( x , y )$ be $\frac { y } { x } + \sec \left( \frac { y } { x } \right) , x > 0$. Then the equation of the curve is
(A) $\quad \sin \left( \frac { y } { x } \right) = \log x + \frac { 1 } { 2 }$
(B) $\quad \operatorname { cosec } \left( \frac { y } { x } \right) = \log x + 2$
(C) $\quad \sec \left( \frac { 2 y } { x } \right) = \log x + 2$
(D) $\quad \cos \left( \frac { 2 y } { x } \right) = \log x + \frac { 1 } { 2 }$

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) $- \infty < f ( x ) < 0$

Consider the family of all circles whose centers lie on the straight line $y = x$. If this family of circles is represented by the differential equation $P y ^ { \prime \prime } + Q y ^ { \prime } + 1 = 0$, where $P , Q$ are functions of $x , y$ and $y ^ { \prime }$ (here $y ^ { \prime } = \frac { d y } { d x } , y ^ { \prime \prime } = \frac { d ^ { 2 } y } { d x ^ { 2 } }$), then which of the following statements is (are) true?
(A) $P = y + x$
(B) $P = y - x$
(C) $P + Q = 1 - x + y + y ^ { \prime } + \left( y ^ { \prime } \right) ^ { 2 }$
(D) $P - Q = x + y - y ^ { \prime } - \left( y ^ { \prime } \right) ^ { 2 }$

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 $y = (x+3)^2$

If $y = y ( x )$ satisfies the differential equation
$$8 \sqrt { x } ( \sqrt { 9 + \sqrt { x } } ) d y = ( \sqrt { 4 + \sqrt { 9 + \sqrt { x } } } ) ^ { - 1 } d x , \quad x > 0$$
and $y ( 0 ) = \sqrt { 7 }$, then $y ( 256 ) =$
[A] 3
[B] 9
[C] 16
[D] 80

Let $f : ( 0 , \pi ) \rightarrow \mathbb { R }$ be a twice differentiable function such that
$$\lim _ { t \rightarrow x } \frac { f ( x ) \sin t - f ( t ) \sin x } { t - x } = \sin ^ { 2 } x \text { for all } x \in ( 0 , \pi )$$
If $f \left( \frac { \pi } { 6 } \right) = - \frac { \pi } { 12 }$, then which of the following statement(s) is (are) TRUE?
(A) $f \left( \frac { \pi } { 4 } \right) = \frac { \pi } { 4 \sqrt { 2 } }$
(B) $f ( x ) < \frac { x ^ { 4 } } { 6 } - x ^ { 2 }$ for all $x \in ( 0 , \pi )$
(C) There exists $\alpha \in ( 0 , \pi )$ such that $f ^ { \prime } ( \alpha ) = 0$
(D) $f ^ { \prime \prime } \left( \frac { \pi } { 2 } \right) + f \left( \frac { \pi } { 2 } \right) = 0$

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 $\Gamma$ denote a curve $y = y ( x )$ which is in the first quadrant and let the point $( 1,0 )$ lie on it. Let the tangent to $\Gamma$ at a point $P$ intersect the $y$-axis at $Y _ { P }$. If $P Y _ { P }$ has length 1 for each point $P$ on $\Gamma$, then which of the following options is/are correct?
(A) $y = \log _ { e } \left( \frac { 1 + \sqrt { 1 - x ^ { 2 } } } { x } \right) - \sqrt { 1 - x ^ { 2 } }$
(B) $\quad x y ^ { \prime } + \sqrt { 1 - x ^ { 2 } } = 0$
(C) $y = - \log _ { e } \left( \frac { 1 + \sqrt { 1 - x ^ { 2 } } } { x } \right) + \sqrt { 1 - x ^ { 2 } }$
(D) $x y ^ { \prime } - \sqrt { 1 - x ^ { 2 } } = 0$

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 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 $f : [ 1 , \infty ) \rightarrow \mathbb { R }$ be a differentiable function such that $f ( 1 ) = \frac { 1 } { 3 }$ and $3 \int _ { 1 } ^ { x } f ( t ) d t = x f ( x ) - \frac { x ^ { 3 } } { 3 } , x \in [ 1 , \infty )$. Let $e$ denote the base of the natural logarithm. Then the value of $f ( e )$ is
(A) $\frac { e ^ { 2 } + 4 } { 3 }$
(B) $\frac { \log _ { e } 4 + e } { 3 }$
(C) $\frac { 4 e ^ { 2 } } { 3 }$
(D) $\frac { e ^ { 2 } - 4 } { 3 }$

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) $\frac { 13 } { 11 x } + \frac { 9 } { 11 } x ^ { 10 }$

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 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) $y ^ { 2 } = x ^ { 2 } - 2 x y \frac { d y } { d x }$

A particle is moving with velocity $\vec { v } = K ( y \hat { i } + x \hat { j } )$, where $K$ is a constant. The general equation for its path is
(1) $y = x ^ { 2 } +$ constant
(2) $y ^ { 2 } = x +$ constant
(3) $x y =$ constant
(4) $y ^ { 2 } = x ^ { 2 } +$ constant

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) $2\ln 18$

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.