Consider a quadratic equation $a x ^ { 2 } + b x + c = 0$, where $2 a + 3 b + 6 c = 0$ and let $g ( x ) = a \frac { x ^ { 3 } } { 3 } + b \frac { x ^ { 2 } } { 2 } + c x$. Statement 1: The quadratic equation has at least one root in the interval $( 0,1 )$. Statement 2: The Rolle's theorem is applicable to function $g ( x )$ on the interval $[ 0,1 ]$.
(1) Statement 1 is false, Statement 2 is true.
(2) Statement 1 is true, Statement 2 is false.
(3) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.
(4) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.

At present, a firm is manufacturing 2000 items. It is estimated that the rate of change of production $P$ w.r.t. additional number of workers $x$ is given by $\frac{dP}{dx} = 100 - 12\sqrt{x}$. If the firm employs 25 more workers, then the new level of production of items is
(1) 3500
(2) 4500
(3) 2500
(4) 3000

If the volume of a spherical ball is increasing at the rate of $4 \pi \mathrm { cc } / \mathrm { sec }$ then the rate of increase of its radius (in $\mathrm { cm } / \mathrm { sec }$), when the volume is $288 \pi \mathrm { cc }$ is
(1) $\frac { 1 } { 9 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 1 } { 24 }$
(4) $\frac { 1 } { 36 }$

If $y = \left[ x + \sqrt { x ^ { 2 } - 1 } \right] ^ { 15 } + \left[ x - \sqrt { x ^ { 2 } - 1 } \right] ^ { 15 }$, then $\left( x ^ { 2 } - 1 \right) \frac { d ^ { 2 } y } { d x ^ { 2 } } + x \frac { d y } { d x }$ is equal to
(1) $224 y ^ { 2 }$
(2) $125 y$
(3) $225 y$
(4) $225 y ^ { 2 }$

If a right circular cone, having maximum volume, is inscribed in a sphere of radius $3$ cm, then the curved surface area (in $\mathrm { cm } ^ { 2 }$) of this cone is :
(1) $8 \sqrt { 2 } \pi$
(2) $6 \sqrt { 2 } \pi$
(3) $8 \sqrt { 3 } \pi$
(4) $6 \sqrt { 3 } \pi$

Let $S$ be the set of all values of $x$ for which the tangent to the curve $y = f ( x ) = x ^ { 3 } - x ^ { 2 } - 2 x$ at ( $x , y$ ) is parallel to the line segment joining the points $( 1 , f ( 1 ) )$ and $( - 1 , f ( - 1 ) )$, then $S$ is equal to
(1) $\left\{ - \frac { 1 } { 3 } , - 1 \right\}$
(2) $\left\{ - \frac { 1 } { 3 } , 1 \right\}$
(3) $\left\{ \frac { 1 } { 3 } , 1 \right\}$
(4) $\left\{ \frac { 1 } { 3 } , - 1 \right\}$

The value of $c$, in the Lagrange's mean value theorem for the function $f ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 8 x + 11$, when $x \in [ 0,1 ]$, is
(1) $\frac { 4 - \sqrt { 5 } } { 3 }$
(2) $\frac { 4 - \sqrt { 7 } } { 3 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { \sqrt { 7 } - 2 } { 3 }$

Let $f$ be any function continuous on $[ a , b ]$ and twice differentiable on $( a , b )$. If all $x \in ( a , b ) , f ^ { \prime } ( x ) > 0$ and $f ^ { \prime \prime } ( x ) < 0$, then for any $c \in ( a , b ) , \frac { f ( c ) - f ( a ) } { f ( b ) - f ( c ) }$
(1) $\frac { b + a } { b - a }$
(2) 1
(3) $\frac { b - c } { c - a }$
(4) $\frac { c - a } { b - c }$

Let the function, $f : [-7, 0] \rightarrow R$ be continuous on $[-7, 0]$ and differentiable on $(-7, 0)$. If $f(-7) = -3$ and $f ^ { \prime } (x) \leq 2$ for all $x \in (-7, 0)$, then for all such functions $f$, $f(-1) + f(0)$ lies in the interval
(1) $(-\infty, 20]$
(2) $[-3, 11]$
(3) $(-\infty, 11]$
(4) $[-6, 20]$

The function $f ( x ) = \left\{ \begin{array} { l l } \frac { \pi } { 4 } + \tan ^ { - 1 } x , & | x | \leq 1 \\ \frac { 1 } { 2 } ( | x | - 1 ) , & | x | > 1 \end{array} \right.$ is:
(1) continuous on $R - \{ 1 \}$ and differentiable on $R - \{ - 1,1 \}$.
(2) both continuous and differentiable on $R - \{ 1 \}$
(3) continuous on $R - \{ - 1 \}$ and differentiable on $R - \{ - 1,1 \}$
(4) both continuous and differentiable on $R - \{ - 1 \}$

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) $e$

Let $f : ( - 1 , \infty ) \rightarrow R$ be defined by $f ( 0 ) = 1$ and $f ( x ) = \frac { 1 } { x } \log _ { e } ( 1 + x ) , x \neq 0$. Then the function $f$
(1) Decreases in $( - 1,0 )$ and increases in $( 0 , \infty )$
(2) Increases in $( - 1 , \infty )$
(3) Increases in $( - 1,0 )$ and decreases in $( 0 , \infty )$
(4) Decreases in $( - 1 , \infty )$

The value of $\lim _ { x \rightarrow 0 ^ { + } } \frac { \cos ^ { - 1 } \left( x - [ x ] ^ { 2 } \right) \cdot \sin ^ { - 1 } \left( x - [ x ] ^ { 2 } \right) } { x - x ^ { 3 } }$, where $[ x ]$ denotes the greatest integer $\leq x$ is:
(1) $\pi$
(2) 0
(3) $\frac { \pi } { 4 }$
(4) $\frac { \pi } { 2 }$

Let $f : S \rightarrow S$ where $S = ( 0 , \infty )$ be a twice differentiable function such that $f ( x + 1 ) = x f ( x )$. If $g : S \rightarrow R$ be defined as $g ( x ) = \log _ { \mathrm { e } } f ( x )$, then the value of $\left| g ^ { \prime \prime } ( 5 ) - g ^ { \prime \prime } ( 1 ) \right|$ is equal to :
(1) $\frac { 205 } { 144 }$
(2) $\frac { 197 } { 144 }$
(3) $\frac { 187 } { 144 }$
(4) 1

The shortest distance between the line $x - y = 1$ and the curve $x ^ { 2 } = 2 y$ is:
(1) $\frac { 1 } { 2 }$
(2) $\frac { 1 } { \sqrt { 2 } }$
(3) $\frac { 1 } { 2 \sqrt { 2 } }$
(4) 0

If the normal to the curve $y ( x ) = \int _ { 0 } ^ { x } \left( 2 t ^ { 2 } - 15 t + 10 \right) d t$ at a point $( a , b )$ is parallel to the line $x + 3 y = - 5 , a > 1$, then the value of $| a + 6 b |$ is equal to $\_\_\_\_$.

The number of points, where the function $f : R \rightarrow R , f ( x ) = | x - 1 | \cos | x - 2 | \sin | x - 1 | + ( x - 3 ) \left| x ^ { 2 } - 5 x + 4 \right|$, is NOT differentiable, is
(1) 1
(2) 2
(3) 3
(4) 4

Let $f ( x ) = \min \{ 1,1 + x \sin x \} , 0 \leq x \leq 2 \pi$. If $m$ is the number of points, where $f$ is not differentiable and $n$ is the number of points, where $f$ is not continuous, then the ordered pair $( m , n )$ is equal to
(1) $( 2,0 )$
(2) $( 1,0 )$
(3) $( 1,1 )$
(4) $( 2,1 )$

A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semivertical angle is $\tan ^ { - 1 } \frac { 3 } { 4 }$. Water is poured in it at a constant rate of 6 cubic meter per hour. The rate (in square meter per hour), at which the wet curved surface area of the tank is increasing, when the depth of water in the tank is 4 meters, is $\_\_\_\_$ .

If $\alpha > \beta > 0$ are the roots of the equation $a x ^ { 2 } + b x + 1 = 0$, and $\lim _ { x \rightarrow \frac { 1 } { \alpha } } \left( \frac { 1 - \cos \left( x ^ { 2 } + b x + a \right) } { 2 ( 1 - \alpha x ) ^ { 2 } } \right) ^ { \frac { 1 } { 2 } } = \frac { 1 } { k } \left( \frac { 1 } { \beta } - \frac { 1 } { \alpha } \right)$, then $k$ is equal to
(1) $2 \beta$
(2) $\alpha$
(3) $2 \alpha$
(4) $\beta$

Let $g(x) = f(x) + f(1 - x)$ and $f ^ { \prime \prime } (x) > 0 , x \in (0,1)$. If $g$ is decreasing in the interval $(0 , \alpha)$ and increasing in the interval $(\alpha , 1)$, then $\tan ^ { - 1 } (2\alpha) + \tan ^ { - 1 } \left( \frac { 1 } { \alpha } \right) + \tan ^ { - 1 } \left( \frac { \alpha + 1 } { \alpha } \right)$ is equal to
(1) $\pi$
(2) $\frac { 5\pi } { 4 }$
(3) $\frac { 3\pi } { 4 }$
(4) $\frac { 3\pi } { 2 }$

The co-ordinates of a particle moving in $x - y$ plane are given by : $x = 2 + 4 \mathrm { t } , y = 3 \mathrm { t } + 8 \mathrm { t } ^ { 2 }$. The motion of the particle is :
(1) uniformly accelerated having motion along a parabolic path.
(2) uniform motion along a straight line.
(3) uniformly accelerated having motion along a straight line.
(4) non-uniformly accelerated.

A particle is moving in one dimension (along $x$ axis) under the action of a variable force. It's initial position was 16 m right of origin. The variation of its position $x$ with time $t$ is given as $x = - 3 t ^ { 3 } + 18 t ^ { 2 } + 16t$, where $x$ is in m and $t$ is in s. The velocity of the particle when its acceleration becomes zero is $\_\_\_\_$ $\mathrm { m } \mathrm { s } ^ { - 1 }$.

Let $g : R \rightarrow R$ be a non constant twice differentiable such that $g ^ { \prime } \left( \frac { 1 } { 2 } \right) = g ^ { \prime } \left( \frac { 3 } { 2 } \right)$. If a real valued function $f$ is defined as $f ( x ) = \frac { 1 } { 2 } [ g ( x ) + g ( 2 - x ) ]$, then
(1) $f ^ { \prime \prime } ( x ) = 0$ for atleast two $x$ in $( 0,2 )$
(2) $f ^ { \prime \prime } ( x ) = 0$ for exactly one $x$ in $( 0,1 )$
(3) $f ^ { \prime \prime } ( x ) = 0$ for no $x$ in $( 0,1 )$
(4) $f ^ { \prime } \left( \frac { 3 } { 2 } \right) + f ^ { \prime } \left( \frac { 1 } { 2 } \right) = 1$

Q1 Let $a > 0$. Consider two curves
$$C_1: y = e^{6x}$$ $$C_2: y = ax^2.$$
We are to find the condition on $a$ such that there exist two straight lines, each of which is tangent to both $C_1$ and $C_2$.
The equation of the tangent to $C_1$ at a point $(t, e^{6t})$ is
$$y = \mathbf{A}e^{6t}x - e^{6t}(\mathbf{B}t - \mathbf{C})$$
This is tangent also to $C_2$ under the condition that the quadratic equation
$$ax^2 = \mathbf{A}e^{6t}x - e^{6t}(\mathbf{B}t - \mathbf{C})$$
has just one solution. Hence, the equation
$$\mathbf{D}e^{12t} - ae^{6t}(\mathbf{E}t - \mathbf{F}) = 0$$
must hold for $a$ and $t$. From this equation we obtain
$$a = \frac{\mathbf{D}}{\mathbf{E}t - \mathbf{F}}e^{6t}$$
Let $f(t)$ denote the right side of this equation. The condition under which there exist two straight lines each of which is tangent to both $C_1$ and $C_2$, is that the straight line $s = a$ intersects the graph of $s = f(t)$ at two points.
Now, the derivative of $f(t)$ is
$$f'(t) = \frac{108e^{6t}(\mathbf{G}t - \mathbf{H})}{(\mathbf{E}t - \mathbf{F})^2}.$$
Hence the condition on $a$ that we are seeking is
$$a > \square e^{\square}.$$
Note that $\lim_{t \to \infty} \frac{e^t}{t} = \infty$.