For a positive real number $\alpha$, let $S_\alpha$ denote the set of points $(x, y)$ satisfying $$|x|^\alpha + |y|^\alpha = 1$$ A positive number $\alpha$ is said to be good if the points in $S_\alpha$ that are closest to the origin lie only on the coordinate axes. Then
(A) all $\alpha$ in $(0,1)$ are good and others are not good.
(B) all $\alpha$ in $(1,2)$ are good and others are not good.
(C) all $\alpha > 2$ are good and others are not good.
(D) all $\alpha > 1$ are good and others are not good.

Let $S = \left\{ x - y \mid x , y \text{ are real numbers with } x ^ { 2 } + y ^ { 2 } = 1 \right\}$. Then the maximum number in the set $S$ is
(A) 1
(B) $\sqrt { 2 }$
(C) $2 \sqrt { 2 }$
(D) $1 + \sqrt { 2 }$.

If $x , y$ are positive real numbers such that $3 x + 4 y < 72$, then the maximum possible value of $12 x y ( 72 - 3 x - 4 y )$ is:
(A) 12240
(B) 13824
(C) 10656
(D) 8640

35. The maximum value of $( \cos a 1 ) \cdot ( \cos a 2 ) \cdot \ldots \cdot \cdot ( \cos a n )$, under the restrictions $0 \leq \mathrm { a } 1 , \mathrm { a } 2 , \ldots \ldots , \mathrm { an } \leq \pi / 2$ and $( \cot \mathrm { a } 1 )$. ( $\cot \mathrm { a } 2$ ). ..... ( $\cot \mathrm { an } ) = 1$ is:
(A) $1 / 2 n / 2$
(B) $1 / 2 n$
(C) $1 / 2 n$
(D) 1

The maximum value of the function $f(x)=2x^{3}-15x^{2}+36x-48$ on the set $A=\left\{x\mid x^{2}+20\leq9x\right\}$ is

Let $f , g$ and $h$ be real-valued functions defined on the interval $[ 0,1 ]$ by $f ( x ) = e ^ { x ^ { 2 } } + e ^ { - x ^ { 2 } } , g ( x ) = x e ^ { x ^ { 2 } } + e ^ { - x ^ { 2 } }$ and $h ( x ) = x ^ { 2 } e ^ { x ^ { 2 } } + e ^ { - x ^ { 2 } }$. If $a , b$ and $c$ denote, respectively, the absolute maximum of $f , g$ and $h$ on $[ 0,1 ]$, then
A) $\mathrm { a } = \mathrm { b }$ and $\mathrm { c } \neq \mathrm { b }$
B) a $=$ c and a $\neq$ b
C) $a \neq b$ and $c \neq b$
D) $a = b = c$

If $f ( x ) = \left| \begin{array} { c c c } \cos ( 2 x ) & \cos ( 2 x ) & \sin ( 2 x ) \\ - \cos x & \cos x & - \sin x \\ \sin x & \sin x & \cos x \end{array} \right|$, then
[A] $f ^ { \prime } ( x ) = 0$ at exactly three points in $( - \pi , \pi )$
[B] $f ^ { \prime } ( x ) = 0$ at more than three points in $( - \pi , \pi )$
[C] $f ( x )$ attains its maximum at $x = 0$
[D] $f ( x )$ attains its minimum at $x = 0$

If $p$ and $q$ are positive real numbers such that $p ^ { 2 } + q ^ { 2 } = 1$, then the maximum value of ( $p + q$ ) is
(1) 2
(2) $1 / 2$
(3) $\frac { 1 } { \sqrt { 2 } }$
(4) $\sqrt { 2 }$

Let $I = \int _ { a } ^ { b } \left( x ^ { 4 } - 2 x ^ { 2 } \right) d x$. If $I$ is minimum then the ordered pair $( a , b )$ is
(1) $( 0 , \sqrt { 2 } )$
(2) $( \sqrt { 2 } , - \sqrt { 2 } )$
(3) $( - \sqrt { 2 } , 0 )$
(4) $( - \sqrt { 2 } , \sqrt { 2 } )$

The minimum value of $2 ^ { \sin x } + 2 ^ { \cos x }$ is:
(1) $2 ^ { - 1 + \frac { 1 } { \sqrt { 2 } } }$
(2) $2 ^ { - 1 + \sqrt { 2 } }$
(3) $2 ^ { 1 - \sqrt { 2 } }$
(4) $2 ^ { 1 - \frac { 1 } { \sqrt { 2 } } }$

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 $f : [ - 1,1 ] \rightarrow R$ 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 $M$ and $m$ respectively be the maximum and minimum values of the function $f ( x ) = \tan ^ { - 1 } ( \sin x + \cos x )$ in $\left[ 0 , \frac { \pi } { 2 } \right]$. Then the value of $\tan ( M - m )$ is equal to: (1) $2 - \sqrt { 3 }$ (2) $3 - 2 \sqrt { 2 }$ (3) $3 + 2 \sqrt { 2 }$ (4) $2 + \sqrt { 3 }$

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

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) $\frac { 9 - \sqrt { 17 } } { 2 }$

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) $15 - \pi$

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 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) $\frac { - \pi } { 12 }$

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 $f(x) = (x+3)^2(x-2)^3$, $x \in [-4, 4]$. 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 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 $\_\_\_\_$

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) $\mathrm { e } ^ { 2 \pi } < ( 2 \pi ) ^ { \mathrm { e } }$

Q72. Let the sum of the maximum and the minimum values of the function $f ( x ) = \frac { 2 x ^ { 2 } - 3 x + 8 } { 2 x ^ { 2 } + 3 x + 8 }$ be $\frac { \mathrm { m } } { \mathrm { n } }$, where $\operatorname { gcd } ( \mathrm { m } , \mathrm { n } ) = 1$. Then $\mathrm { m } + \mathrm { n }$ is equal to :
(1) 195
(2) 201
(3) 217
(4) 182
Q73. Let $f : \mathbf { R } \rightarrow \mathbf { R }$ be a function given by $f ( x ) = \left\{ \begin{array} { l l } \frac { 1 - \cos 2 x } { x ^ { 2 } } , & x < 0 \\ \alpha , & x = 0 \\ \frac { \beta \sqrt { 1 - \cos x } } { x } , & x > 0 \end{array} \right.$, where $\alpha , \beta \in \mathbf { R }$. If $f$ is continuous at $x = 0$, then $\alpha ^ { 2 } + \beta ^ { 2 }$ is equal to :
(1) 3
(2) 12
(3) 48
(4) 6

Q74. 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