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 $R$

$f , g : R \rightarrow R$ be two real valued function defined as $f ( x ) = \left\{ \begin{array} { c l } - | x + 3 | & , x < 0 \\ e ^ { x } & , x \geq 0 \end{array} \right.$ and $g ( x ) = \left\{ \begin{array} { l l } x ^ { 2 } + k _ { 1 } x , & x < 0 \\ 4 x + k _ { 2 } , & x \geq 0 \end{array} \right.$, where $k _ { 1 }$ and $k _ { 2 }$ are real constants. If $gof$ is differentiable at $x = 0$, then $gof ( - 4 ) + gof ( 4 )$ is equal to
(1) $4 \left( e ^ { 4 } + 1 \right)$
(2) $2 \left( 2 e ^ { 4 } + 1 \right)$
(3) $4 e ^ { 4 }$
(4) $2 \left( 2 e ^ { 4 } - 1 \right)$

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) $19 : 15$

Let $P$ and $Q$ be any points on the curves $( x - 1 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 1$ and $y = x ^ { 2 }$, respectively. The distance between $P$ and $Q$ is minimum for some value of the abscissa of $P$ in the interval
(1) $\left( 0 , \frac { 1 } { 4 } \right)$
(2) $\left( \frac { 1 } { 2 } , \frac { 3 } { 4 } \right)$
(3) $\left( \frac { 1 } { 4 } , \frac { 1 } { 2 } \right)$
(4) $\left( \frac { 3 } { 4 } , 1 \right)$

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 $x = \frac { 1 } { 8 }$

Let $f : R \rightarrow R$ be a function defined by : $f ( x ) = \left\{ \begin{array} { c c } \max _ { t \leq x } \left\{ t ^ { 3 } - 3 t \right\} ; & x \leq 2 \\ x ^ { 2 } + 2 x - 6 ; & 2 < x < 3 \\ { [ x - 3 ] + 9 ; } & 3 \leq x \leq 5 \\ 2 x + 1 ; & x > 5 \end{array} \right.$ Where $[ t ]$ is the greatest integer less than or equal to $t$. Let $m$ be the number of points where $f$ is not differentiable and $I = \int _ { - 2 } ^ { 2 } f ( x ) d x$. Then the ordered pair ( $m , I$ ) is equal to
(1) $\left( 3 , \frac { 27 } { 4 } \right)$
(2) $\left( 3 , \frac { 23 } { 4 } \right)$
(3) $\left( 4 , \frac { 27 } { 4 } \right)$
(4) $\left( 4 , \frac { 23 } { 4 } \right)$

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) $8 - \frac { \pi } { 4 }$

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 the function $f ( x ) = 2 x ^ { 2 } - \log _ { e } x , x > 0$, be decreasing in $( 0 , a )$ and increasing in $( a , 4 )$. A tangent to the parabola $y ^ { 2 } = 4 a x$ at a point $P$ on it passes through the point $( 8 a , 8 a - 1 )$ but does not pass through the point $\left( - \frac { 1 } { a } , 0 \right)$. If the equation of the normal at $P$ is $\frac { x } { \alpha } + \frac { y } { \beta } = 1$, then $\alpha + \beta$ is equal to $\_\_\_\_$.

Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = \frac{x^2 + 2}{x^2 + 1}$. Then which of the following is NOT true?
(1) $f(x)$ has a minimum at $x = 0$
(2) $f(x)$ is an even function
(3) $f(x)$ is strictly increasing for $x > 0$
(4) $f(x)$ is onto

Let $f(x) = \begin{cases} x^3 - x^2 + 10x - 7, & x \leq 1 \\ -2x + \log_2(b^2 - 4), & x > 1 \end{cases}$. Then the set of all values of $b$, for which $f(x)$ has maximum value at $x = 1$, is
(1) $(-2, -1]$
(2) $[-2, -1) \cup (1, 2]$
(3) $(-2, 2)$
(4) $(-\infty, -2) \cup (2, \infty)$

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 $\_\_\_\_$.

A wire of length 20 m is to be cut into two pieces. A piece of length $\ell_1$ is bent to make a square of area $A_1$ and the other piece of length $\ell_2$ is made into a circle of area $A_2$. If $2A_1 + 3A_2$ is minimum then $\pi\ell_1 : \ell_2$ is equal to:
(1) 6:1
(2) $3:1$
(3) $1:6$
(4) $4:1$

Let the function $f ( x ) = 2 x ^ { 3 } + ( 2 p - 7 ) x ^ { 2 } + 3 ( 2 p - 9 ) x - 6$ have a maxima for some value of $x < 0$ and a minima for some value of $x > 0$. Then, the set of all values of $p$ is
(1) $\left( \frac { 9 } { 2 } , \infty \right)$
(2) $\left( 0 , \frac { 9 } { 2 } \right)$
(3) $\left( - \infty , \frac { 9 } { 2 } \right)$
(4) $\left( - \frac { 9 } { 2 } , \frac { 9 } { 2 } \right)$

Let $f$ and $g$ be twice differentiable functions on $R$ such that $f ^ { \prime \prime } ( x ) = g ^ { \prime \prime } ( x ) + 6 x$ $f ^ { \prime } ( 1 ) = 4 g ^ { \prime } ( 1 ) - 3 = 9$ $f ( 2 ) = 3 g ( 2 ) = 12$ Then which of the following is NOT true ? (1) $g ( - 2 ) - f ( - 2 ) = 20$ (2) If $- 1 < x < 2$, then $| f ( x ) - g ( x ) | < 8$ (3) $\left| f ^ { \prime } ( x ) - g ^ { \prime } ( x ) \right| < 6 \Rightarrow - 1 < x < 1$ (4) There exists $x _ { 0 } \in \left( 1 , \frac { 3 } { 2 } \right)$ such that $f \left( x _ { 0 } \right) = g \left( x _ { 0 } \right)$