Let $f ( x ) = x ^ { 2 } + \frac { 1 } { x ^ { 2 } }$ and $g ( x ) = x - \frac { 1 } { x } , x \in R - \{ - 1,0,1 \}$. If $h ( x ) = \frac { f ( x ) } { g ( x ) }$, then the local minimum value of $h ( x )$ is:
(1) $2 \sqrt { 2 }$
(2) 3
(3) - 3
(4) $- 2 \sqrt { 2 }$

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

The shortest distance between the line $y = x$ and the curve $y^2 = x - 2$ is
(1) $\frac{7}{4\sqrt{2}}$
(2) $\frac{7}{8}$
(3) $\frac{11}{4\sqrt{2}}$
(4) 2

If $f ( x )$ is a non-zero polynomial of degree four, having local extreme points at $x = - 1,0,1$; then the set $S = \{ x \in R : f ( x ) = f ( 0 ) \}$ contains exactly
(1) Two irrational and two rational numbers
(2) Four rational numbers
(3) Two irrational and one rational number
(4) Four irrational numbers

If the function $f$ given by $f ( x ) = x ^ { 3 } - 3 ( a - 2 ) x ^ { 2 } + 3 a x + 7$, for some $a \in R$ is increasing in $( 0,1 ]$ and decreasing in $[ 1,5 )$, then a root of the equation, $\frac { f ( x ) - 14 } { ( x - 1 ) ^ { 2 } } = 0 , ( x \neq 1 )$ is :
(1) 7
(2) - 7
(3) 6
(4) 5

If $S_1$ and $S_2$ are respectively the sets of local minimum and local maximum points of the function, $f(x) = 9x^4 + 12x^3 - 36x^2 + 25$, $x \in R$, then
(1) $S_1 = \{-2\}$; $S_2 = \{0, 1\}$
(2) $S_1 = \{-1\}$; $S_2 = \{0, 2\}$
(3) $S_1 = \{-2, 0\}$; $S_2 = \{1\}$
(4) $S_1 = \{-2, 1\}$; $S_2 = \{0\}$

The shortest distance between the point $\left( \frac { 3 } { 2 } , 0 \right)$ and the curve $y = \sqrt { x } , ( x > 0 )$, is
(1) $\frac { \sqrt { 3 } } { 2 }$
(2) $\frac { 5 } { 4 }$
(3) $\frac { 3 } { 2 }$
(4) $\frac { \sqrt { 5 } } { 2 }$

The height of a right circular cylinder of maximum volume inscribed in a sphere of radius 3 is:
(1) $\sqrt { 3 }$
(2) $\frac { 2 } { 3 } \sqrt { 3 }$
(3) $\sqrt { 6 }$
(4) $2 \sqrt { 3 }$

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 $f ( x )$ be a polynomial of degree 5 such that $x = \pm 1$ are its critical points. If $\lim _ { x \rightarrow 0 } \left( 2 + \frac { f ( x ) } { x ^ { 3 } } \right) = 4$, then which one of the following is not true?
(1) $f$ is an odd function
(2) $f ( 1 ) - 4 f ( - 1 ) = 4$
(3) $x = 1$ is a point of local minimum and $x = - 1$ is a point of local maximum
(4) $x = 1$ is a point of local maxima of $f$

The function, $f ( x ) = ( 3 x - 7 ) x ^ { \frac { 2 } { 3 } } , x \in \mathrm { R }$, is increasing for all $x$ lying in
(1) $( - \infty , 0 ) \cup \left( \frac { 14 } { 15 } , \infty \right)$
(2) $( - \infty , 0 ) \cup \left( \frac { 3 } { 7 } , \infty \right)$
(3) $\left( - \infty , \frac { 14 } { 15 } \right)$
(4) $\left( - \infty , - \frac { 14 } { 15 } \right) \cup ( 0 , \infty )$

Let a function $f : [ 0,5 ] \rightarrow R$ be continuous, $f ( 1 ) = 3$ and $F$ be defined as: $F ( x ) = \int _ { 1 } ^ { x } t ^ { 2 } g ( t ) d t$, where $g ( t ) = \int _ { 1 } ^ { t } f ( u ) d u$. Then for the function $F ( x )$, the point $x = 1$ is:
(1) a point of local minima
(2) not a critical point
(3) a point of local maxima
(4) a point of inflection

If $p(x)$ be a polynomial of degree three that has a local maximum value 8 at $x = 1$ and a local minimum value 4 at $x = 2$ then $p(0)$ is equal to
(1) 6
(2) $-12$
(3) 24
(4) 12

If $x = 1$ is a critical point of the function $f(x) = (3x^2 + ax - 2 - a)e^x$, then
(1) $x = 1$ and $x = -\frac{2}{3}$ are local minima of $f$
(2) $x = 1$ and $x = -\frac{2}{3}$ is a local maxima of $f$
(3) $x = 1$ is a local maxima and $x = -\frac{2}{3}$ is a local minima of $f$
(4) $x = 1$ is a local minima and $x = -\frac{2}{3}$ are local maxima of $f$

The area (in sq. units) of the largest rectangle $ABCD$ whose vertices $A$ and $B$ lie on the $x$-axis and vertices $C$ and $D$ lie on the parabola, $y = x ^ { 2 } - 1$ below the $x$-axis, is:
(1) $\frac { 2 } { 3 \sqrt { 3 } }$
(2) $\frac { 1 } { 3 \sqrt { 3 } }$
(3) $\frac { 4 } { 3 }$
(4) $\frac { 4 } { 3 \sqrt { 3 } }$

If $P$ is a point on the parabola $y = x ^ { 2 } + 4$ which is closest to the straight line $y = 4 x - 1$, then the coordinates of $P$ are:
(1) $( - 2,8 )$
(2) $( 1,5 )$
(3) $( 2,8 )$
(4) $( 3,13 )$

The triangle of maximum area that can be inscribed in a given circle of radius ' $r$ ' is :
(1) An equilateral triangle having each of its side of length $\sqrt { 3 } r$.
(2) An isosceles triangle with base equal to $2 r$.
(3) An equilateral triangle of height $\frac { 2 r } { 3 }$.
(4) A right angle triangle having two of its sides of length $2 r$ and $r$.

Let $f$ be a real valued function, defined on $R - \{ - 1,1 \}$ and given by $f ( x ) = 3 \log _ { \mathrm { e } } \left| \frac { x - 1 } { x + 1 } \right| - \frac { 2 } { x - 1 }$. Then in which of the following intervals, function $f ( x )$ is increasing?
(1) $( - \infty , - 1 ) \cup \left( \left[ \frac { 1 } { 2 } , \infty \right) - \{ 1 \} \right)$
(2) $( - \infty , \infty ) - \{ - 1,1 \}$
(3) $\left( - 1 , \frac { 1 } { 2 } \right]$
(4) $\left( - \infty , \frac { 1 } { 2 } \right] - \{ - 1 \}$

The sum of all the local minimum values of the twice differentiable function $f : R \rightarrow R$ defined by $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - \frac { 3 f ^ { \prime \prime } ( 2 ) } { 2 } x + f ^ { \prime \prime } ( 1 )$ is:
(1) - 22
(2) 5
(3) - 27
(4) 0

A box open from top is made from a rectangular sheet of dimension $a \times b$ by cutting squares each of side $x$ from each of the four corners and folding up the flaps. If the volume of the box is maximum, then $x$ is equal to: (1) $\frac { a + b + \sqrt { a ^ { 2 } + b ^ { 2 } - a b } } { 6 }$ (2) $\frac { a + b - \sqrt { a ^ { 2 } + b ^ { 2 } - a b } } { 12 }$ (3) $\frac { a + b - \sqrt { a ^ { 2 } + b ^ { 2 } - a b } } { 6 }$ (4) $\frac { a + b - \sqrt { a ^ { 2 } + b ^ { 2 } + a b } } { 6 }$

If Rolle's theorem holds for the function $f ( x ) = x ^ { 3 } - a x ^ { 2 } + b x - 4 , x \in [ 1,2 ]$ with $f ^ { \prime } \left( \frac { 4 } { 3 } \right) = 0$, then ordered pair $( a , b )$ is equal to :
(1) $( - 5 , - 8 )$
(2) $( - 5,8 )$
(3) $( 5,8 )$
(4) $( 5 , - 8 )$

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) $2 + \sqrt { 3 }$

Let $a$ be a real number such that the function $f ( x ) = a x ^ { 2 } + 6 x - 15 , x \in R$ is increasing in $( - \infty , \frac { 3 } { 4 } )$ and decreasing in $\left( \frac { 3 } { 4 } , \infty \right)$. Then the function $g ( x ) = a x ^ { 2 } - 6 x + 15 , x \in R$ has a
(1) local maximum at $x = - \frac { 3 } { 4 }$
(2) local minimum at $x = - \frac { 3 } { 4 }$
(3) local maximum at $x = \frac { 3 } { 4 }$
(4) local minimum at $x = \frac { 3 } { 4 }$

The range of $a \in R$ for which the function $f ( x ) = ( 4 a - 3 ) \left( x + \log _ { e } 5 \right) + 2 ( a - 7 ) \cot \left( \frac { x } { 2 } \right) \sin ^ { 2 } \left( \frac { x } { 2 } \right) , x \neq 2 n \pi , n \in N$, has critical points, is :
(1) $( - 3,1 )$
(2) $\left[ - \frac { 4 } { 3 } , 2 \right]$
(3) $[ 1 , \infty )$
(4) $( - \infty , - 1 ]$