$f ( x ) = \left\{ \begin{array} { c l } \frac { \sin ( a + 2 ) x + \sin x } { x } & ; x < 0 \\ b & ; x = 0 \\ \frac { \left( x + 3 x ^ { 2 } \right) ^ { 1 / 3 } - x ^ { 1 / 3 } } { x ^ { 1 / 3 } } & ; x > 0 \end{array} \right.$ is continuous at $x = 0$, then $a + 2 b$ is equal to:
(1) 1
(2) - 1
(3) 0
(4) - 2

Let $P$ be a variable point on the parabola $y = 4 x ^ { 2 } + 1$. Then, the locus of the mid-point of the point $P$ and the foot of the perpendicular drawn from the point $P$ to the line $y = x$ is:
(1) $( 3 x - y ) ^ { 2 } + ( x - 3 y ) + 2 = 0$
(2) $2 ( 3 x - y ) ^ { 2 } + ( x - 3 y ) + 2 = 0$
(3) $( 3 x - y ) ^ { 2 } + 2 ( x - 3 y ) + 2 = 0$
(4) $2 ( x - 3 y ) ^ { 2 } + ( 3 x - y ) + 2 = 0$

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) $\left( 1 , \frac { 1 } { 2 } \right)$

$\lim _ { n \rightarrow \infty } \left( 1 + \frac { 1 + \frac { 1 } { 2 } + \ldots\ldots + \frac { 1 } { n } } { n ^ { 2 } } \right) ^ { n }$ is equal to
(1) $\frac { 1 } { e }$
(2) 0
(3) $\frac { 1 } { 2 }$
(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 $\underline{\hspace{1cm}}$.

Let a function $g : [ 0,4 ] \rightarrow R$ be defined as $g ( x ) = \left\{ \begin{array} { c c } \max \left\{ t ^ { 3 } - 6 t ^ { 2 } + 9 t - 3 \right\} , & 0 \leq x \leq 3 \\ 0 \leq t \leq x & \\ 4 - x, & 3 < x \leq 4 \end{array} \right.$ then the number of points in the interval $( 0,4 )$ where $g ( x )$ is NOT differentiable, is $\underline{\hspace{1cm}}$.

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 $f : \mathbb{R} \rightarrow \mathbb{R}$ be defined as $$f(x) = \begin{cases} \left[e^x\right], & x < 0 \\ ae^x + [x-1], & 0 \leq x < 1 \\ b + [\sin(\pi x)], & 1 \leq x < 2 \\ \left[e^{-x}\right] - c, & x \geq 2 \end{cases}$$ where $a, b, c \in \mathbb{R}$ and $[t]$ denotes greatest integer less than or equal to $t$. Then, which of the following statements is true?
(1) There exists $a, b, c \in \mathbb{R}$ such that $f$ is continuous
(2) If $f$ is discontinuous at exactly one point, then $a + b + c = 1$
(3) If $f$ is discontinuous at exactly one point, then $a + b + c \neq 1$
(4) $f$ is discontinuous at at least two points, for any values of $a, b, c \in \mathbb{R}$

The number of distinct real roots of the equation $x ^ { 7 } - 7 x - 2 = 0$ is
(1) 5
(2) 7
(3) 1
(4) 3

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 $\_\_\_\_$.

The combined equation of the two lines $ax + by + c = 0$ and $a'x + b'y + c' = 0$ can be written as $(ax + by + c)(a'x + b'y + c') = 0$. The equation of the angle bisectors of the lines represented by the equation $2x^2 + xy - 3y^2 = 0$ is
(1) $3x^2 + 5xy + 2y^2 = 0$
(2) $x^2 - y^2 + 10xy = 0$
(3) $3x^2 + xy - 2y^2 = 0$
(4) $x^2 - y^2 - 10xy = 0$

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) $(-\infty, -2) \cup (2, \infty)$

If $f(x) = \frac{\tan^{-1}x + \log_e 123}{x \log_e 1234 - \tan^{-1}x}$, $x > 0$, then the least value of $f(f(x)) + f\!\left(f\!\left(\frac{4}{x}\right)\right)$ is
(1) 0
(2) 8
(3) 2
(4) 4

The set of values of $a$ for which $\lim _ { x \rightarrow a } ( [ x - 5 ] - [ 2 x + 2 ] ) = 0$, where $[ \zeta ]$ denotes the greatest integer less than or equal to $\zeta$ is equal to
(1) $( - 7.5 , - 6.5 )$
(2) $( - 7.5 , - 6.5 ]$
(3) $[ - 7.5 , - 6.5 ]$
(4) $[ - 7.5 , - 6.5 )$

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 $\_\_\_\_$ .

Let $f : \mathbb{R} \rightarrow (0, \infty)$ be strictly increasing function such that $\lim_{x \rightarrow \infty} \dfrac{f(7x)}{f(x)} = 1$. Then, the value of $\lim_{x \rightarrow \infty} \left(\dfrac{f(5x)}{f(x)} - 1\right)$ is equal to
(1) 4
(2) 0
(3) $\dfrac{7}{5}$
(4) 1

Consider the function $f : \left[ \frac { 1 } { 2 } , 1 \right] \rightarrow \mathrm { R }$ defined by $f ( x ) = 4 \sqrt { 2 } x ^ { 3 } - 3 \sqrt { 2 } x - 1$. Consider the statements (I) The curve $y = f ( x )$ intersects the $x$-axis exactly at one point (II) The curve $y = f ( x )$ intersects the $x$-axis at $x = \cos \frac { \pi } { 12 }$ Then
(1) Only (II) is correct
(2) Both (I) and (II) are incorrect
(3) Only (I) is correct
(4) Both (I) and (II) are correct

Consider the function $\mathrm { f } : ( 0,2 ) \rightarrow \mathrm { R }$ defined by $\mathrm { f } ( \mathrm { x } ) = \frac { \mathrm { x } } { 2 } + \frac { 2 } { \mathrm { x } }$ and the function $\mathrm { g } ( \mathrm { x } )$ defined by $\mathrm { gx } = \begin{array} { c c } \min \{ \mathrm { f } ( \mathrm { t } ) \} , & 0 < \mathrm { t } \leq \mathrm { x } \text { and } 0 < \mathrm { x } \leq 1 \\ \frac { 3 } { 2 } + \mathrm { x } , & 1 < \mathrm { x } < 2 \end{array}$. Then
(1) g is continuous but not differentiable at $\mathrm { x } = 1$
(2) g is not continuous for all $\mathrm { x } \in ( 0,2 )$
(3) g is neither continuous nor differentiable at $\mathrm { x } = 1$
(4) $g$ is continuous and differentiable for all $x \in ( 0,2 )$

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

For $\mathrm { a } , \mathrm { b } > 0$, let $f ( x ) = \left\{ \begin{array} { c l } \frac { \tan ( ( \mathrm { a } + 1 ) x ) + \mathrm { b } \tan x } { x } , & x < 0 \\ 3 , & x = 0 \\ \frac { \sqrt { \mathrm { a } x + \mathrm { b } ^ { 2 } x ^ { 2 } } - \sqrt { \mathrm { a } x } } { \mathrm {~b} \sqrt { \mathrm { a } } \sqrt { x } } , & x > 0 \end{array} \right.$ be a continous function at $x = 0$. Then $\frac { \mathrm { b } } { \mathrm { a } }$ is equal to : (1) 6 (2) 4 (3) 5 (4) 8

Let the range of the function $f ( x ) = \frac { 1 } { 2 + \sin 3 x + \cos 3 x } , x \in \mathbb { R }$ be $[ a , b ]$. If $\alpha$ and $\beta$ are respectively the A.M. and the G.M. of $a$ and $b$, then $\frac { \alpha } { \beta }$ is equal to
(1) $\pi$
(2) $\sqrt { \pi }$
(3) 2
(4) $\sqrt { 2 }$

Consider the function $\mathrm { f } ( \mathrm { x } ) = \left\{ \begin{array} { c l } \frac { \mathrm { a } \left( 7 \mathrm { x } - 12 - \mathrm { x } ^ { 2 } \right) } { \mathrm { b } \left| \mathrm { x } ^ { 2 } - 7 \mathrm { x } + 12 \right| } & , \mathrm { x } < 3 \\ 2 ^ { \frac { \sin ( \mathrm { x } - 3 ) } { \mathrm { x } - [ \mathrm { x } ] } } & , \mathrm { x } > 3 \\ \mathrm {~b} & , \mathrm { x } = 3 \end{array} \right.$, where $[ \mathrm { x } ]$ denotes the greatest integer less than or equal to x. If S denotes the set of all ordered pairs $( \mathrm { a } , \mathrm { b } )$ such that $\mathrm { f } ( \mathrm { x } )$ is continuous at $x = 3$, then the number of elements in S is:
(1) 2
(2) Infinitely many
(3) 4
(4) 1

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 $[x]$ denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function $f(x) = [x] + |x - 2|$, $-2 < x < 3$, is not continuous and not differentiable. Then $\mathrm{m} + \mathrm{n}$ is equal to:
(1) 6
(2) 8
(3) 9
(4) 7