If the point on the curve $y ^ { 2 } = 6 x$, nearest to the point $\left( 3 , \frac { 3 } { 2 } \right)$ is $( \alpha , \beta )$, then $2 ( \alpha + \beta )$ is equal to $\underline{\hspace{1cm}}$.

Let $f : [ - 1,1 ] \rightarrow R$ be defined as $f ( x ) = a x ^ { 2 } + b x + c$ for all $x \in [ - 1,1 ]$, where $a , b , c \in R$ such that $f ( - 1 ) = 2 , f ^ { \prime } ( - 1 ) = 1$ and for $x \in ( - 1,1 )$ the maximum value of $f ^ { \prime \prime } ( x )$ is $\frac { 1 } { 2 }$. If $f ( x ) \leq \alpha , x \in [ - 1,1 ]$, then the least value of $\alpha$ is equal to

If $R$ is the least value of $a$ such that the function $f ( x ) = x ^ { 2 } + \mathrm { a } x + 1$ is increasing on $[ 1,2 ]$ and $S$ is the greatest value of $a$ such that the function $f ( x ) = x ^ { 2 } + a x + 1$ is decreasing on $[ 1,2 ]$, then the value of $| R - S |$ is

Let $f : [ - 1,1 ] \rightarrow R$ be defined as $f ( x ) = a x ^ { 2 } + b x + c$ for all $x \in [ - 1,1 ]$, where $a , b , c \in R$ such that $f ( - 1 ) = 2 , f ^ { \prime } ( - 1 ) = 1$ and for $x \in ( - 1,1 )$ the maximum value of $f ^ { \prime \prime } ( x )$ is $\frac { 1 } { 2 }$. If $f ( x ) \leq \alpha , x \in [ - 1,1 ]$, then the least value of $\alpha$ is equal to

Let $f ( x )$ be a polynomial of degree 6 in $x$, in which the coefficient of $x ^ { 6 }$ is unity and it has extrema at $x = - 1$ and $x = 1$. If $\lim _ { x \rightarrow 0 } \frac { f ( x ) } { x ^ { 3 } } = 1$, then $5 \cdot f ( 2 )$ is equal to

Let $x , y > 0$. If $x ^ { 3 } y ^ { 2 } = 2 ^ { 15 }$, then the least value of $3 x + 2 y$ is
(1) 30
(2) 32
(3) 36
(4) 40

If the minimum value of $f(x) = \frac{5x^2}{2} + \frac{\alpha}{x^5}$, $x > 0$, is 14, then the value of $\alpha$ is equal to
(1) 32
(2) 64
(3) 128
(4) 256

If the absolute maximum value of the function $f(x) = (x ^ { 2 } - 2x + 7) e ^ { (4x ^ { 3 } - 12x ^ { 2 } - 180x + 31)}$ in the interval $[-3,0]$ is $f(\alpha)$, then
(1) $\alpha = 0$
(2) $\alpha = - 3$
(3) $\alpha \in (-1,0)$
(4) $\alpha \in (-3,-1)$

The function $f ( x ) = x e ^ { x ( 1 - x ) } , x \in R$, is
(1) increasing in $\left( - \frac { 1 } { 2 } , 1 \right)$
(2) decreasing in $\left( \frac { 1 } { 2 } , 2 \right)$
(3) increasing in $\left( - 1 , - \frac { 1 } { 2 } \right)$
(4) decreasing in $\left( - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$

Let $f ( x ) = 3 ^ { \left( x ^ { 2 } - 2 \right) ^ { 3 } + 4 } , \mathrm { x } \in R$. Then which of the following statements are true? $P : x = 0$ is a point of local minima of $f$ $Q : x = \sqrt { 2 }$ is a point of inflection of $f$ $R : f ^ { \prime }$ is increasing for $x > \sqrt { 2 }$
(1) Only $P$ and $Q$
(2) Only $P$ and $R$
(3) Only $Q$ and $R$
(4) All $P , Q$ and $R$

The curve $y(x) = ax ^ { 3 } + bx ^ { 2 } + cx + 5$ touches the $x$-axis at the point $P(-2,0)$ and cuts the $y$-axis at the point $Q$, where $y'$ is equal to 3. Then the local maximum value of $y(x)$ is
(1) $\frac { 27 } { 4 }$
(2) $\frac { 29 } { 4 }$
(3) $\frac { 37 } { 4 }$
(4) $\frac { 9 } { 2 }$

The sum of the absolute maximum and absolute minimum values of the function $f ( x ) = \tan ^ { - 1 } ( \sin x - \cos x )$ in the interval $[ 0 , \pi ]$ is
(1) $0$
(2) $\tan ^ { - 1 } \left( \frac { 1 } { \sqrt { 2 } } \right) - \frac { \pi } { 4 }$
(3) $\cos ^ { - 1 } \left( \frac { 1 } { \sqrt { 3 } } \right) - \frac { \pi } { 4 }$
(4) $\frac { - \pi } { 12 }$

Let $\lambda ^ { * }$ be the largest value of $\lambda$ for which the function $f _ { \lambda } ( x ) = 4 \lambda x ^ { 3 } - 36 \lambda x ^ { 2 } + 36 x + 48$ is increasing for all $x \in \mathbb { R }$. Then $f _ { \lambda ^ { * } } ( 1 ) + f _ { \lambda ^ { * } } ( - 1 )$ is equal to:
(1) 36
(2) 48
(3) 64
(4) 72

For the function $f ( x ) = 4 \log _ { e } ( x - 1 ) - 2 x ^ { 2 } + 4 x + 5 , x > 1$, which one of the following is NOT correct?
(1) $f ( x )$ is increasing in $( 1,2 )$ and decreasing in $( 2 , \infty )$
(2) $f ( x ) = - 1$ has exactly two solutions
(3) $f ^ { \prime } ( \mathrm { e } ) - f ^ { \prime \prime } ( 2 ) < 0$
(4) $f ( x ) = 0$ has a root in the interval $( e , e + 1 )$

Consider a cuboid of sides $2 x , 4 x$ and $5 x$ and a closed hemisphere of radius $r$. If the sum of their surface areas is constant $k$, then the ratio $x : r$, for which the sum of their volumes is maximum, is
(1) $2 : 5$
(2) $19 : 45$
(3) $3 : 8$
(4) $19 : 15$

The sum of the absolute minimum and the absolute maximum values of the function $f ( x ) = \left| 3 x - x ^ { 2 } + 2 \right| - x$ in the interval $[ - 1 , 2 ]$ is
(1) $\frac { \sqrt { 17 } + 3 } { 2 }$
(2) $\frac { \sqrt { 17 } + 5 } { 2 }$
(3) 5
(4) $\frac { 9 - \sqrt { 17 } } { 2 }$

If $m$ and $n$ respectively are the number of local maximum and local minimum points of the function $f ( x ) = \int _ { 0 } ^ { x ^ { 2 } } \frac { t ^ { 2 } - 5 t + 4 } { 2 + e ^ { t } } d t$, then the ordered pair $( m , n )$ is equal to
(1) $( 2,3 )$
(2) $( 3,2 )$
(3) $( 2,2 )$
(4) $( 3,4 )$

Let $f ( x ) = \left\{ \begin{array} { c c } x ^ { 3 } - x ^ { 2 } + 10 x - 7 , & x \leq 1 \\ - 2 x + \log _ { 2 } \left( b ^ { 2 } - 4 \right) , & x > 1 \end{array} \right.$ Then the set of all values of $b$, for which $f ( x )$ has maximum value at $x = 1$, is:
(1) $( - 6 , - 2 )$
(2) $( 2,6 )$
(3) $[ - 6 , - 2 ) \cup ( 2,6 ]$
(4) $[ - \sqrt { 6 } , - 2 ) \cup ( 2 , \sqrt { 6 } ]$

Let a function $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined as: $f ( x ) = \begin{cases} \int _ { 0 } ^ { x } ( 5 - | t - 3 | ) d t , & x > 4 \\ x ^ { 2 } + b x , & x \leq 4 \end{cases}$ where $b \in \mathbb { R }$. If $f$ is continuous at $x = 4$, then which of the following statements is NOT true?
(1) $f$ is not differentiable at $x = 4$
(2) $f ^ { \prime } ( 3 ) + f ^ { \prime } ( 5 ) = \frac { 35 } { 4 }$
(3) $f$ is increasing in $\left( - \infty , \frac { 1 } { 8 } \right) \cup ( 8 , \infty )$
(4) $f$ has a local minima at $x = \frac { 1 } { 8 }$

If the maximum value of $a$, for which the function $f _ { a } ( x ) = \tan ^ { - 1 } 2 x - 3 a x + 7$ is non-decreasing in $\left[ - \frac { \pi } { 6 } , \frac { \pi } { 6 } \right]$, is $\bar { a }$, then $f _ { \bar { a } } \left( \frac { \pi } { 8 } \right)$ is equal to
(1) $8 - \frac { 9 \pi } { 49 + \pi ^ { 2 } }$
(2) $8 - \frac { 4 \pi } { 94 + \pi ^ { 2 } }$
(3) $8 \frac { 1 + \pi ^ { 2 } } { 9 + \pi ^ { 2 } }$
(4) $8 - \frac { \pi } { 4 }$

A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is equal to (if the full question text was truncated, this is the standard formulation of this problem).

Let $f ( x ) = 2 \cos ^ { - 1 } x + 4 \cot ^ { - 1 } x - 3 x ^ { 2 } - 2 x + 10 , x \in [ - 1 , 1 ]$. If $[ a , b ]$ is the range of the function, then $4a - b$ is equal to
(1) 11
(2) $11 - \pi$
(3) $11 + \pi$
(4) $15 - \pi$

Let $S$ be the set of all $a \in N$ such that the area of the triangle formed by the tangent at the point $P(b, c)$, $b, c \in N$, on the parabola $y^2 = 2ax$ and the lines $x = b$, $y = 0$ is 16 unit$^2$, then $\sum_{a \in S} a$ is equal to $\_\_\_\_$.

A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in $\mathrm{cm}^2$) is equal to
(1) 800
(2) 675
(3) 1025
(4) 900

Let $f(x) = \int_0^x t(t-1)(t-2)\,dt$, $x > 0$. Then the number of points in the interval $(0, 3)$ at which $f(x)$ has a local maximum is $\_\_\_\_$.