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 $f : ( 0,1 ) \rightarrow \mathbb { R }$ be a function defined by $f ( x ) = \frac { 1 } { 1 - e ^ { - x } }$, and $g ( x ) = ( f ( - x ) - f ( x ) )$. Consider two statements (I) $g$ is an increasing function in $( 0,1 )$ (II) $g$ is one-one in $( 0,1 )$ Then,
(1) Only (I) is true
(2) Only (II) is true
(3) Neither (I) nor (II) is true
(4) Both (I) and (II) are true

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) $\left( - \frac { 9 } { 2 } , \frac { 9 } { 2 } \right)$

If the functions $f(x) = \frac{x^{3}}{3} + 2bx + \frac{ax^{2}}{2}$ and $g(x) = \frac{x^{3}}{3} + ax + bx^{2}$, $a \neq 2b$ have a common extreme point, then $a + 2b + 7$ is equal to
(1) 4
(2) $\frac{3}{2}$
(3) 3
(4) 6

Let $x = 2$ be a local minima of the function $f ( x ) = 2 x ^ { 4 } - 18 x ^ { 2 } + 8 x + 12 , x \in ( - 4,4 )$. If $M$ is local maximum value of the function $f$ in $( - 4,4 )$, then $M =$
(1) $12 \sqrt { 6 } - \frac { 33 } { 2 }$
(2) $12 \sqrt { 6 } - \frac { 31 } { 2 }$
(3) $18 \sqrt { 6 } - \frac { 33 } { 2 }$
(4) $18 \sqrt { 6 } - \frac { 31 } { 2 }$

The maximum area of a triangle whose one vertex is at $( 0,0 )$ and the other two vertices lie on the curve $y = - 2 x ^ { 2 } + 54$ at points $( x , y )$ and $( - x , y )$ where $\mathrm { y } > 0$ is :
(1) 88
(2) 122
(3) 92
(4) 108

For the function $f ( x ) = ( \cos x ) - x + 1 , x \in \mathbb { R }$, between the following two statements (S1) $f ( x ) = 0$ for only one value of $x$ in $[ 0 , \pi ]$. (S2) $f ( x )$ is decreasing in $\left[ 0 , \frac { \pi } { 2 } \right]$ and increasing in $\left[ \frac { \pi } { 2 } , \pi \right]$.
(1) Both (S1) and (S2) are correct.
(2) Both (S1) and (S2) are incorrect.
(3) Only (S2) is correct.
(4) Only (S1) is correct.

Let $f ( x ) = 4 \cos ^ { 3 } x + 3 \sqrt { 3 } \cos ^ { 2 } x - 10$. The number of points of local maxima of $f$ in interval $( 0,2 \pi )$ is
(1) 3
(2) 4
(3) 1
(4) 2

The number of critical points of the function $f ( x ) = ( x - 2 ) ^ { 2 / 3 } ( 2 x + 1 )$ is
(1) 1
(2) 2
(3) 0
(4) 3

The function $f ( x ) = 2 x + 3 x ^ { \frac { 2 } { 3 } } , x \in R$, has
(1) exactly one point of local minima and no point of local maxima
(2) exactly one point of local maxima and no point of local minima
(3) exactly one point of local maxima and exactly one point of local minima
(4) exactly two points of local maxima and exactly one point of local minima

If the function $f : (-\infty, -1] \rightarrow [a, b]$ defined by $f(x) = e^{x^3 - 3x + 1}$ is one-one and onto, then the distance of the point $P(2b+4, a+2)$ from the line $x + e^{-3}y = 4$ is:

Let $\mathrm { g } ( \mathrm { x } ) = 3 \mathrm { f } ^ { \mathrm { x } } + \mathrm { f } ( 3 - \mathrm { x } )$ and $\mathrm { f } ^ { \prime \prime } ( \mathrm { x } ) > 0$ for all $\mathrm { x } \in ( 0,3 )$. If g is decreasing in ( $0 , \alpha$ ) and increasing in $( \alpha , 3 )$, then $8 \alpha$ is
(1) 24
(2) 0
(3) 18
(4) 20

If the function $f ( x ) = \left( \frac { 1 } { x } \right) ^ { 2 x } ; x > 0$ attains the maximum value at $x = \frac { 1 } { \mathrm { e } }$ then:
(1) $\mathrm { e } ^ { \pi } < \pi ^ { \mathrm { e } }$
(2) $\mathrm { e } ^ { \pi } > \pi ^ { \mathrm { e } }$
(3) $( 2 e ) ^ { \pi } > \pi ^ { ( 2 e ) }$
(4) $\mathrm { e } ^ { 2 \pi } < ( 2 \pi ) ^ { \mathrm { e } }$

If the function $f ( x ) = 2 x ^ { 3 } - 9 x ^ { 2 } + 12 \mathrm { a } ^ { 2 } x + 1 , \mathrm { a } > 0$ has a local maximum at $x = \alpha$ and a local minimum at $x = \alpha ^ { 2 }$, then $\alpha$ and $\alpha ^ { 2 }$ are the roots of the equation : (1) $x ^ { 2 } - 6 x + 8 = 0$ (2) $x ^ { 2 } + 6 x + 8 = 0$ (3) $8 x ^ { 2 } + 6 x - 1 = 0$ (4) $8 x ^ { 2 } - 6 x + 1 = 0$

The function $f ( x ) = \frac { x } { x ^ { 2 } - 6 x - 16 } , x \in \mathbb { R } - \{ - 2,8 \}$
(1) decreases in $( - 2,8 )$ and increases in $( - \infty , - 2 ) \cup ( 8 , \infty )$
(2) decreases in $( - \infty , - 2 ) \cup ( - 2,8 ) \cup ( 8 , \infty )$
(3) decreases in $( - \infty , - 2 )$ and increases in $( 8 , \infty )$
(4) increases in $( - \infty , - 2 ) \cup ( - 2,8 ) \cup ( 8 , \infty )$

The interval in which the function $f ( x ) = x ^ { x } , x > 0$, is strictly increasing is
(1) $\left( 0 , \frac { 1 } { e } \right]$
(2) $( 0 , \infty )$
(3) $\left[ \frac { 1 } { e } , \infty \right)$
(4) $\left[ \frac { 1 } { e ^ { 2 } } , 1 \right)$

Let $f(x) = (x+3)^2(x-2)^3$, $x \in [-4, 4]$. If $M$ and $m$ are the maximum and minimum values of $f$, respectively in $[-4, 4]$, then the value of $M - m$ is:
(1) 600
(2) 392
(3) 608
(4) 108

Let $f ( x ) = 3 \sqrt { x - 2 } + \sqrt { 4 - x }$ be a real valued function. If $\alpha$ and $\beta$ are respectively the minimum and the maximum values of $f$, then $\alpha ^ { 2 } + 2 \beta ^ { 2 }$ is equal to
(1) 42
(2) 38
(3) 24
(4) 44

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a thrice differentiable function such that $f ( 0 ) = 0 , f ( 1 ) = 1 , f ( 2 ) = - 1 , f ( 3 ) = 2$ and $f ( 4 ) = - 2$. Then, the minimum number of zeros of $\left( 3 f ^ { \prime } f ^ { \prime \prime } + f f ^ { \prime \prime \prime } \right) ( x )$ is $\_\_\_\_$

Let the maximum and minimum values of $\left( \sqrt { 8 x - x ^ { 2 } - 12 } - 4 \right) ^ { 2 } + ( x - 7 ) ^ { 2 } , x \in \mathbf { R }$ be M and m , respectively. Then $\mathrm { M } ^ { 2 } - \mathrm { m } ^ { 2 }$ is equal to $\_\_\_\_$

Let A be the region enclosed by the parabola $y ^ { 2 } = 2 x$ and the line $x = 24$. Then the maximum area of the rectangle inscribed in the region A is $\_\_\_\_$

Let the set of all values of $p$, for which $f ( x ) = \left( p ^ { 2 } - 6 p + 8 \right) \left( \sin ^ { 2 } 2 x - \cos ^ { 2 } 2 x \right) + 2 ( 2 - p ) x + 7$ does not have any critical point, be the interval $( a , b )$. Then $16 a b$ is equal to $\_\_\_\_$

The sum of all local minimum values of the function
$$f ( x ) = \left\{ \begin{array} { l r } 1 - 2 x , & x < - 1 \\ \frac { 1 } { 3 } ( 7 + 2 | x | ) , & - 1 \leq x \leq 2 \\ \frac { 11 } { 18 } ( x - 4 ) ( x - 5 ) , & x > 2 \end{array} \right.$$
is
(1) $\frac { 157 } { 72 }$
(2) $\frac { 131 } { 72 }$
(3) $\frac { 171 } { 72 }$
(4) $\frac { 167 } { 72 }$

Let $f ( x ) = \int _ { 0 } ^ { x ^ { 2 } } \frac { \mathrm { t } ^ { 2 } - 8 \mathrm { t } + 15 } { \mathrm { e } ^ { t } } \mathrm { dt } , x \in \mathbf { R }$. Then the numbers of local maximum and local minimum points of $f$, respectively, are :
(1) 2 and 3
(2) 2 and 2
(3) 3 and 2
(4) 1 and 3

Let $(2, 3)$ be the largest open interval in which the function $f(x) = 2\log_{\mathrm{e}}(x - 2) - x^{2} + ax + 1$ is strictly increasing and $(\mathrm{b}, \mathrm{c})$ be the largest open interval, in which the function $\mathrm{g}(x) = (x - 1)^{3}(x + 2 - \mathrm{a})^{2}$ is strictly decreasing. Then $100(a + b - c)$ is equal to:
(1) 420
(2) 360
(3) 160
(4) 280