If $f: R \rightarrow R$ is a function defined by $f(x) = e^{|x|} - e^{-x}$ / $e^{x} + e^{-x}$, then $f$ is:
(1) bijective
(2) $f$ is monotonically increasing on $(0, \infty)$
(3) $f$ is monotonically decreasing on $(0, \infty)$
(4) not differentiable at $x = 0$

Let $A ( 1,4 )$ and $B ( 1 , - 5 )$ be two points. Let $P$ be a point on the circle $( ( x - 1 ) ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 1$, such that $( P A ) ^ { 2 } + ( P B ) ^ { 2 }$ have maximum value, then the points, $P , A$ and $B$ lie on
(1) a hyperbola
(2) a straight line
(3) an ellipse
(4) a parabola

If $\lim _ { x \rightarrow \infty } \left( \sqrt { x ^ { 2 } - x + 1 } - a x \right) = b$, then the ordered pair $( a , b )$ is: (1) $\left( 1 , - \frac { 1 } { 2 } \right)$ (2) $\left( - 1 , \frac { 1 } { 2 } \right)$ (3) $\left( - 1 , - \frac { 1 } { 2 } \right)$ (4) $\left( 1 , \frac { 1 } { 2 } \right)$

Let $f : R \rightarrow R$ be defined as $f ( x ) = \begin{cases} 2 \sin \left( - \frac { \pi x } { 2 } \right) , & \text { if } x < - 1 \\ \left| a x ^ { 2 } + x + b \right| , & \text { if } - 1 \leq x \leq 1 \\ \sin ( \pi x ) , & \text { if } x > 1 \end{cases}$ If $f ( x )$ is continuous on $R$, then $a + b$ equals :
(1) 1
(2) 3
(3) - 3
(4) - 1

Consider the function $f : R \rightarrow R$ defined by $f ( x ) = \left\{ \begin{array} { c c } \left( 2 - \sin \left( \frac { 1 } { x } \right) \right) | x | , & x \neq 0 \\ 0 , & x = 0 \end{array} \right.$. Then $f$ is:
(1) monotonic on $( - \infty , 0 ) \cup ( 0 , \infty )$
(2) not monotonic on $( - \infty , 0 )$ and $( 0 , \infty )$
(3) monotonic on $( 0 , \infty )$ only
(4) monotonic on $( - \infty , 0 )$ only

A function $f$ is defined on $[ - 3,3 ]$ as $$f ( x ) = \left\{ \begin{array} { c } \min \left\{ | x | , 2 - x ^ { 2 } \right\} , - 2 \leq x \leq 2 \\ { [ | x | ] , 2 < | x | \leq 3 } \end{array} \right.$$ where $[ x ]$ denotes the greatest integer $\leq x$. The number of points, where $f$ is not differentiable in $( - 3,3 )$ is $\underline{\hspace{1cm}}$.

The equation of a common tangent to the parabolas $y = x ^ { 2 }$ and $y = -(x - 2) ^ { 2 }$ is
(1) $y = 4 x - 2$
(2) $y = 4 x - 1$
(3) $y = 4 x + 1$
(4) $y = 4 x + 2$

Let $f ( x ) = \begin{cases} \frac { \sin ( x - [ x ] ) } { x - [ x ] } , & x \in ( - 2 , - 1 ) \\ \max ( 2 x , 3 [ | x | ] ) , & | x | < 1 \\ 1 , & \text { otherwise } \end{cases}$ where $[ t ]$ denotes greatest integer $\leq t$. If $m$ is the number of points where $f$ is not continuous and $n$ is the number of points where $f$ is not differentiable, the ordered pair $( m , n )$ is:
(1) $( 3,3 )$
(2) $( 2,4 )$
(3) $( 2,3 )$
(4) $( 3,4 )$

Let $f ( x ) = \min \{ 1,1 + x \sin x \} , 0 \leq x \leq 2 \pi$. If $m$ is the number of points, where $f$ is not differentiable and $n$ is the number of points, where $f$ is not continuous, then the ordered pair $( m , n )$ is equal to
(1) $( 2,0 )$
(2) $( 1,0 )$
(3) $( 1,1 )$
(4) $( 2,1 )$

Let $f(x) = \begin{cases} \left\lfloor 4x ^ { 2 } - 8x + 5 \right\rfloor, & \text{if } 8x ^ { 2 } - 6x + 1 \geq 0 \\ \left\lfloor 4x ^ { 2 } - 8x + 5 \right\rfloor, & \text{if } 8x ^ { 2 } - 6x + 1 < 0 \end{cases}$, where $\lfloor \alpha \rfloor$ denotes the greatest integer less than or equal to $\alpha$. Then the number of points in $R$ where $f$ is not differentiable is $\_\_\_\_$.

Let $R$ be the focus of the parabola $y ^ { 2 } = 20 x$ and the line $y = m x + c$ intersect the parabola at two points $P$ and $Q$. Let the point $G ( 10 , 10 )$ be the centroid of the triangle $P Q R$. If $c - m = 6$, then $P Q ^ { 2 }$ is
(1) 296
(2) 325
(3) 317
(4) 346

Let $[ x ]$ denote the greatest integer function and $f ( x ) = \max \{ 1 + x + [ x ] , 2 + x , x + 2 [ x ] \} , 0 \leq x \leq 2$, where $m$ is the number of points in $( 0,2 )$ where $f$ is not continuous and $n$ be the number of points in $( 0,2 )$, where $f$ is not differentiable. Then $( m + n ) ^ { 2 } + 2$ is equal to
(1) 2
(2) 11
(3) 6
(4) 3

The range of the function $f(x) = \sqrt{3 - x} + \sqrt{2 + x}$ is
(1) $[\sqrt{5}, \sqrt{10}]$
(2) $[2\sqrt{2}, \sqrt{11}]$
(3) $[\sqrt{5}, \sqrt{13}]$
(4) $[\sqrt{2}, \sqrt{7}]$

Let $[ x ]$ be the greatest integer $\leq x$. Then the number of points in the interval $( - 2,1 )$ where the function $f ( x ) = | [ x ] | + \sqrt { x - [ x ] }$ is discontinuous, is $\_\_\_\_$ .

${ } ^ { n - 1 } C _ { r } = \left( k ^ { 2 } - 8 \right) ^ { n } C _ { r + 1 }$ if and only if:
(1) $2 \sqrt { 2 } < k \leq 3$
(2) $2 \sqrt { 3 } < \mathrm { k } \leq 3 \sqrt { 2 }$
(3) $2 \sqrt { 3 } < \mathrm { k } < 3 \sqrt { 3 }$
(4) $2 \sqrt { 2 } < \mathrm { k } < 2 \sqrt { 3 }$

Let $f : [ - 1,2 ] \rightarrow \mathbf { R }$ be given by $f ( x ) = 2 x ^ { 2 } + x + \left[ x ^ { 2 } \right] - [ x ]$, where $[ t ]$ denotes the greatest integer less than or equal to $t$. The number of points, where $f$ is not continuous, is :
(1) 5
(2) 6
(3) 3
(4) 4

Let $f(x) = |2x^2 + 5x - 3|$, $x \in \mathbb{R}$. If $m$ and $n$ denote the number of points where $f$ is not continuous and not differentiable respectively, then $m + n$ is equal to:
(1) 5
(2) 2
(3) 0
(4) 3

Let $[ \mathrm { t } ]$ denote the greatest integer less than or equal to t. Let $f : [ 0 , \infty ) \rightarrow \mathbf { R }$ be a function defined by $f ( x ) = \left[ \frac { x } { 2 } + 3 \right] - [ \sqrt { x } ]$. Let $S$ be the set of all points in the interval $[ 0,8 ]$ at which $f$ is not continuous. Then $\sum _ { \mathrm { a } \in \mathrm { S } } \mathrm { a }$ is equal to $\_\_\_\_$

Let $\mathrm { A } = \{ ( x , y ) \in \mathbf { R } \times \mathbf { R } : | x + y | \geqslant 3 \}$ and $\mathrm { B } = \{ ( x , y ) \in \mathbf { R } \times \mathbf { R } : | x | + | y | \leq 3 \}$. If $\mathrm { C } = \{ ( x , y ) \in \mathrm { A } \cap \mathrm { B } : x = 0$ or $y = 0 \}$, then $\sum _ { ( x , y ) \in \mathrm { C } } | x + y |$ is :
(1) 15
(2) 24
(3) 18
(4) 12

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function defined by $f ( x ) = ( 2 + 3 a ) x ^ { 2 } + \left( \frac { a + 2 } { a - 1 } \right) x + b , a \neq 1$. If $f ( x + \mathrm { y } ) = f ( x ) + f ( \mathrm { y } ) + 1 - \frac { 2 } { 7 } x \mathrm { y }$, then the value of $28 \sum _ { i = 1 } ^ { 5 } | f ( i ) |$ is
(1) 545
(2) 715
(3) 735
(4) 675

$\lim _ { x \rightarrow \infty } \frac { \left( 2 x ^ { 2 } - 3 x + 5 \right) ( 3 x - 1 ) ^ { \frac { x } { 2 } } } { \left( 3 x ^ { 2 } + 5 x + 4 \right) \sqrt { ( 3 x + 2 ) ^ { x } } }$ is equal to :
(1) $\frac { 2 } { \sqrt { 3 \mathrm { e } } }$
(2) $\frac { 2 \mathrm { e } } { \sqrt { 3 } }$
(3) $\frac { 2 } { 3 \sqrt { e } }$
(4) $\frac { 2 e } { 3 }$

If $7 = 5 + \frac{1}{7}(5 + \alpha) + \frac{1}{7^{2}}(5 + 2\alpha) + \frac{1}{7^{3}}(5 + 3\alpha) + \cdots \infty$, then the value of $\alpha$ is:
(1) $\frac{6}{7}$
(2) 6
(3) $\frac{1}{7}$
(4) 1

Let the function $f ( x ) = \left( x ^ { 2 } + 1 \right) \left| x ^ { 2 } - a x + 2 \right| + \cos | x |$ be not differentiable at the two points $x = \alpha = 2$ and $x = \beta$. Then the distance of the point $( \alpha , \beta )$ from the line $12 x + 5 y + 10 = 0$ is equal to :
(1) 5
(2) 4
(3) 3
(4) 2

The number of real solution(s) of the equation $x^{2} + 3x + 2 = \min\{|x - 3|, |x + 2|\}$ is:
(1) 1
(2) 0
(3) 2
(4) 3

Q72. Let $f : [ - 1,2 ] \rightarrow \mathbf { R }$ be given by $f ( x ) = 2 x ^ { 2 } + x + \left[ x ^ { 2 } \right] - [ x ]$, where $[ t ]$ denotes the greatest integer less than or equal to $t$. The number of points, where $f$ is not continuous, is :
(1) 5
(2) 6
(3) 3
(4) 4