Let $f_1 : \mathbb{R} \rightarrow \mathbb{R}$, $f_2 : [0,\infty) \rightarrow \mathbb{R}$, $f_3 : \mathbb{R} \rightarrow \mathbb{R}$ and $f_4 : \mathbb{R} \rightarrow [0,\infty)$ be defined by
$$f_1(x) = \begin{cases} |x| & \text{if } x < 0 \\ e^x & \text{if } x \geq 0 \end{cases}$$
$$f_2(x) = x^2;$$
$$f_3(x) = \begin{cases} \sin x & \text{if } x < 0 \\ x & \text{if } x \geq 0 \end{cases}$$
and
$$f_4(x) = \begin{cases} f_2(f_1(x)) & \text{if } x < 0 \\ f_2(f_1(x)) - 1 & \text{if } x \geq 0 \end{cases}$$
List I (functions) P. $f_4$ is Q. $f_3$ is R. $f_2 \circ f_1$ is S. $f_2$ is
List II (properties)
1. onto but not one-one
2. neither continuous nor one-one
3. differentiable but not one-one
4. continuous and one-one
P Q R S
(A) 3142
(B) 1342
(C) 3124
(D) 1324

Let $f : \left[ - \frac { 1 } { 2 } , 2 \right] \rightarrow \mathbb { R }$ and $g : \left[ - \frac { 1 } { 2 } , 2 \right] \rightarrow \mathbb { R }$ be functions defined by $f ( x ) = \left[ x ^ { 2 } - 3 \right]$ and $g ( x ) = | x | f ( x ) + | 4 x - 7 | f ( x )$, where $[ y ]$ denotes the greatest integer less than or equal to $y$ for $y \in \mathbb { R }$. Then
(A) $f$ is discontinuous exactly at three points in $\left[ - \frac { 1 } { 2 } , 2 \right]$
(B) $f$ is discontinuous exactly at four points in $\left[ - \frac { 1 } { 2 } , 2 \right]$
(C) $g$ is NOT differentiable exactly at four points in $\left( - \frac { 1 } { 2 } , 2 \right)$
(D) $g$ is NOT differentiable exactly at five points in $\left( - \frac { 1 } { 2 } , 2 \right)$

Let $[x]$ be the greatest integer less than or equals to $x$. Then, at which of the following point(s) the function $f(x) = x\cos(\pi(x + [x]))$ is discontinuous?
[A] $x = -1$
[B] $x = 0$
[C] $x = 1$
[D] $x = 2$

Let $f ( x ) = \frac { 1 - x ( 1 + | 1 - x | ) } { | 1 - x | } \cos \left( \frac { 1 } { 1 - x } \right)$ for $x \neq 1$. Then
[A] $\lim _ { x \rightarrow 1 ^ { - } } f ( x ) = 0$
[B] $\lim _ { x \rightarrow 1 ^ { - } } f ( x )$ does not exist
[C] $\lim _ { x \rightarrow 1 ^ { + } } f ( x ) = 0$
[D] $\lim _ { x \rightarrow 1 ^ { + } } f ( x )$ does not exist

Let $f(x) = x + \log_e x - x\log_e x$, $x \in (0, \infty)$.
- Column 1 contains information about zeros of $f(x)$, $f'(x)$ and $f''(x)$. - Column 2 contains information about the limiting behavior of $f(x)$, $f'(x)$ and $f''(x)$ at infinity. - Column 3 contains information about increasing/decreasing nature of $f(x)$ and $f'(x)$.

Column 1	Column 2	Column 3
(I) $f(x) = 0$ for some $x \in (1, e^2)$	(i) $\lim_{x\to\infty} f(x) = 0$	(P) $f$ is increasing in $(0,1)$
(II) $f'(x) = 0$ for some $x \in (1, e)$	(ii) $\lim_{x\to\infty} f(x) = -\infty$	(Q) $f$ is decreasing in $(e, e^2)$
(III) $f'(x) = 0$ for some $x \in (0,1)$	(iii) $\lim_{x\to\infty} f'(x) = -\infty$	(R) $f'$ is increasing in $(0,1)$
(IV) $f''(x) = 0$ for some $x \in (1, e)$	(iv) $\lim_{x\to\infty} f''(x) = 0$	(S) $f'$ is decreasing in $(e, e^2)$

Which of the following options is the only INCORRECT combination?
[A] (I) (iii) (P)
[B] (II) (iv) (Q)
[C] (III) (i) (R)
[D] (II) (iii) (P)

Let the functions $f: (-1,1) \rightarrow \mathbb{R}$ and $g: (-1,1) \rightarrow (-1,1)$ be defined by $$f(x) = |2x - 1| + |2x + 1| \quad \text{and} \quad g(x) = x - [x],$$ where $[x]$ denotes the greatest integer less than or equal to $x$. Let $f \circ g: (-1,1) \rightarrow \mathbb{R}$ be the composite function defined by $(f \circ g)(x) = f(g(x))$. Suppose $c$ is the number of points in the interval $(-1,1)$ at which $f \circ g$ is NOT continuous, and suppose $d$ is the number of points in the interval $(-1,1)$ at which $f \circ g$ is NOT differentiable. Then the value of $c + d$ is $\_\_\_\_$

For any real number $t$, let $\lfloor t \rfloor$ be the largest integer less than or equal to $t$. Then the number of points of discontinuity of the function $x \mapsto \lfloor x^2 - 3 \rfloor$ for $x \in (-\infty, 0)$ is ____.
Let $f: [-1, 3] \to \mathbb{R}$ be defined as $$f(x) = \begin{cases} |x| + [x], & -1 \leq x < 1 \\ x + |x|, & 1 \leq x < 2 \\ x + [x], & 2 \leq x \leq 3 \end{cases}$$ where $[t]$ denotes the greatest integer less than or equal to $t$. The number of points of discontinuity of $f$ in the interval $(-1, 3)$ is ____.

Let $| M |$ denote the determinant of a square matrix $M$. Let $g : \left[ 0 , \frac { \pi } { 2 } \right] \rightarrow \mathbb { R }$ be the function defined by
$$g ( \theta ) = \sqrt { f ( \theta ) - 1 } + \sqrt { f \left( \frac { \pi } { 2 } - \theta \right) - 1 }$$
where $$f ( \theta ) = \frac { 1 } { 2 } \left| \begin{array} { c c c } 1 & \sin \theta & 1 \\ - \sin \theta & 1 & \sin \theta \\ - 1 & - \sin \theta & 1 \end{array} \right| + \left| \begin{array} { c c c } \sin \pi & \cos \left( \theta + \frac { \pi } { 4 } \right) & \tan \left( \theta - \frac { \pi } { 4 } \right) \\ \sin \left( \theta - \frac { \pi } { 4 } \right) & - \cos \frac { \pi } { 2 } & \log _ { e } \left( \frac { 4 } { \pi } \right) \\ \cot \left( \theta + \frac { \pi } { 4 } \right) & \log _ { e } \left( \frac { \pi } { 4 } \right) & \tan \pi \end{array} \right|$$
Let $p ( x )$ be a quadratic polynomial whose roots are the maximum and minimum values of the function $g ( \theta )$, and $p ( 2 ) = 2 - \sqrt { 2 }$. Then, which of the following is/are TRUE ?
(A) $p \left( \frac { 3 + \sqrt { 2 } } { 4 } \right) < 0$
(B) $p \left( \frac { 1 + 3 \sqrt { 2 } } { 4 } \right) > 0$
(C) $p \left( \frac { 5 \sqrt { 2 } - 1 } { 4 } \right) > 0$
(D) $\quad p \left( \frac { 5 - \sqrt { 2 } } { 4 } \right) < 0$

Let $f : ( 0,1 ) \rightarrow \mathbb { R }$ be the function defined as $f ( x ) = [ 4 x ] \left( x - \frac { 1 } { 4 } \right) ^ { 2 } \left( x - \frac { 1 } { 2 } \right)$, where $[ x ]$ denotes the greatest integer less than or equal to $x$. Then which of the following statements is(are) true?
(A) The function $f$ is discontinuous exactly at one point in $( 0,1 )$
(B) There is exactly one point in $( 0,1 )$ at which the function $f$ is continuous but NOT differentiable
(C) The function $f$ is NOT differentiable at more than three points in $( 0,1 )$
(D) The minimum value of the function $f$ is $- \frac { 1 } { 512 }$

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function defined by
$$f ( x ) = \left\{ \begin{array} { c l } x ^ { 2 } \sin \left( \frac { \pi } { x ^ { 2 } } \right) , & \text { if } x \neq 0 \\ 0 , & \text { if } x = 0 \end{array} \right.$$
Then which of the following statements is TRUE?
(A) $f ( x ) = 0$ has infinitely many solutions in the interval $\left[ \frac { 1 } { 10 ^ { 10 } } , \infty \right)$.
(B) $f ( x ) = 0$ has no solutions in the interval $\left[ \frac { 1 } { \pi } , \infty \right)$.
(C) The set of solutions of $f ( x ) = 0$ in the interval $\left( 0 , \frac { 1 } { 10 ^ { 10 } } \right)$ is finite.
(D) $f ( x ) = 0$ has more than 25 solutions in the interval $\left( \frac { 1 } { \pi ^ { 2 } } , \frac { 1 } { \pi } \right)$.

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ and $g : \mathbb { R } \rightarrow \mathbb { R }$ be functions defined by
$$f ( x ) = \left\{ \begin{array} { l l } x | x | \sin \left( \frac { 1 } { x } \right) , & x \neq 0 , \\ 0 , & x = 0 , \end{array} \quad \text { and } \quad g ( x ) = \begin{cases} 1 - 2 x , & 0 \leq x \leq \frac { 1 } { 2 } \\ 0 , & \text { otherwise } \end{cases} \right.$$
Let $a , b , c , d \in \mathbb { R }$. Define the function $h : \mathbb { R } \rightarrow \mathbb { R }$ by
$$h ( x ) = a f ( x ) + b \left( g ( x ) + g \left( \frac { 1 } { 2 } - x \right) \right) + c ( x - g ( x ) ) + d g ( x ) , x \in \mathbb { R }$$
Match each entry in List-I to the correct entry in List-II.
List-I
(P) If $a = 0 , b = 1 , c = 0$, and $d = 0$, then
(Q) If $a = 1 , b = 0 , c = 0$, and $d = 0$, then
(R) If $a = 0 , b = 0 , c = 1$, and $d = 0$, then
(S) If $a = 0 , b = 0 , c = 0$, and $d = 1$, then
List-II
(1) $h$ is one-one.
(2) $h$ is onto.
(3) $h$ is differentiable on $\mathbb { R }$.
(4) the range of $h$ is $[ 0,1 ]$.
(5) the range of $h$ is $\{ 0,1 \}$.
The correct option is
(A) $(\mathrm{P}) \rightarrow (4)$, $(\mathrm{Q}) \rightarrow (3)$, $(\mathrm{R}) \rightarrow (1)$, $(\mathrm{S}) \rightarrow (2)$
(B) $(\mathrm{P}) \rightarrow (5)$, $(\mathrm{Q}) \rightarrow (2)$, $(\mathrm{R}) \rightarrow (4)$, $(\mathrm{S}) \rightarrow (3)$
(C) $(\mathrm{P}) \rightarrow (5)$, $(\mathrm{Q}) \rightarrow (3)$, $(\mathrm{R}) \rightarrow (2)$, $(\mathrm{S}) \rightarrow (4)$
(D) $(\mathrm{P}) \rightarrow (4)$, $(\mathrm{Q}) \rightarrow (2)$, $(\mathrm{R}) \rightarrow (1)$, $(\mathrm{S}) \rightarrow (3)$

Let $f : R \rightarrow R$ be a function defined by $f ( x ) = \operatorname { Min } \{ x + 1 , | x | + 1 \}$. Then which of the following is true?
(1) $f ( x ) \geq 1$ for all $x \in R$
(2) $f ( x )$ is not differentiable at $x = 1$
(3) $f ( x )$ is differentiable everywhere
(4) $f ( x )$ is not differentiable at $x = 0$

Statement 1: A function $f: R \rightarrow R$ is continuous at $x_{0}$ if and only if $\lim_{x \rightarrow x_{0}} f(x)$ exists and $\lim_{x \rightarrow x_{0}} f(x) = f(x_{0})$. Statement 2: A function $f: R \rightarrow R$ is discontinuous at $x_{0}$ if and only if, $\lim_{x \rightarrow x_{0}} f(x)$ exists and $\lim_{x \rightarrow x_{0}} f(x) \neq f(x_{0})$.
(1) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1.
(2) Statement 1 is false, Statement 2 is true.
(3) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation of Statement 1.
(4) Statement 1 is true, Statement 2 is false.

The range of the function $f ( x ) = \frac { x } { 1 + | x | } , x \in R$, is
(1) $R$
(2) $( - 1,1 )$
(3) $R - \{ 0 \}$
(4) $[ - 1,1 ]$

The function $f ( x ) = | \sin 4 x | + | \cos 2 x |$, is a periodic function with a fundamental period
(1) $\pi$
(2) $2 \pi$
(3) $\frac { \pi } { 4 }$
(4) $\frac { \pi } { 2 }$

Let $a , b \in R , ( a \neq 0 )$. If the function $f$, defined as $$f ( x ) = \left\{ \begin{array} { c } \frac { 2 x ^ { 2 } } { a } , 0 \leq x < 1 \\ a , 1 \leq x < \sqrt { 2 } \\ \frac { 2 b ^ { 2 } - 4 b } { x ^ { 3 } } , \sqrt { 2 } \leq x < 8 \end{array} \right.$$ is continuous in the interval $[ 0 , \infty )$, then an ordered pair $( a , b )$ can be
(1) $( - \sqrt { 2 } , 1 - \sqrt { 3 } )$
(2) $( \sqrt { 2 } , - 1 + \sqrt { 3 } )$
(3) $( \sqrt { 2 } , 1 - \sqrt { 3 } )$
(4) $( - \sqrt { 2 } , 1 + \sqrt { 3 } )$

The number of distinct real roots of the equation $x^4 - 4x^3 + 12x^2 + x - 1 = 0$ is: (1) 2 (2) 3 (3) 0 (4) 4

$\lim _ { x \rightarrow 3 } \frac { \sqrt { 3 x } - 3 } { \sqrt { 2 x - 4 } - \sqrt { 2 } }$ is equal to
(1) $\frac { 1 } { \sqrt { 2 } }$
(2) $\frac { 1 } { 2 \sqrt { 2 } }$
(3) $\frac { \sqrt { 3 } } { 2 }$
(4) $\sqrt { 3 }$

For each $t \in R$, let $[ t ]$ be the greatest integer less than or equal to $t$. Then $\lim _ { x \rightarrow 0 ^ { + } } x \left( \left[ \frac { 1 } { x } \right] + \left[ \frac { 2 } { x } \right] + \ldots + \left[ \frac { 15 } { x } \right] \right)$
(1) does not exist (in $R$ )
(2) is equal to 0
(3) is equal to 15
(4) is equal to 120

If $f ( x ) = [ x ] - \left[ \frac { x } { 4 } \right] , x \in R$, where $[ x ]$ denotes the greatest integer function, then:
(1) $\lim _ { x \rightarrow 4 + } f ( x )$ exists but $\lim _ { x \rightarrow 4 - } f ( x )$ does not exist
(2) $f$ is continuous at $x = 4$
(3) $\lim _ { x \rightarrow 4 - } f ( x )$ exists but $\lim _ { x \rightarrow 4 + } f ( x )$ does not exist
(4) Both $\lim _ { x \rightarrow 4 - } f ( x )$ and $\lim _ { x \rightarrow 4 + } f ( x )$ exist but are not equal

Let $f ( x ) = \left\{ \begin{array} { c c } \max \left( | x | , x ^ { 2 } \right) , & | x | \leq 2 \\ 8 - 2 | x | , & 2 < | x | \leq 4 \end{array} \right.$. Let $S$ be the set of points in the interval $( - 4,4 )$ at which $f$ is not differentiable. Then $S$
(1) equals $\{ - 2 , - 1,0,1,2 \}$
(2) equals $\{ - 2,2 \}$
(3) is an empty set
(4) equal $\{ - 2 , - 1,1,2 \}$

Let $f : [ - 1,3 ] \rightarrow \mathrm { R }$ be defined as $$f ( x ) = \begin{cases} |x| + [x], & -1 \leq x < 1 \\ x + |x|, & 1 \leq x < 2 \\ x + [x], & 2 \leq x \leq 3 \end{cases}$$ where $[t]$ denotes the greatest integer less than or equal to $t$. Then, $f$ is discontinuous at:
(1) Only one point
(2) Only two points
(3) Four or more points
(4) Only three points

If the function $f ( x ) = \left\{ \begin{array} { l } a | \pi - x | + 1 , x \leq 5 \\ b | x - \pi | + 3 , x > 5 \end{array} \right.$ is continuous at $x = 5$, then the value of $a - b$ is:
(1) $\frac { 2 } { 5 - \pi }$
(2) $\frac { - 2 } { \pi + 5 }$
(3) $\frac { 2 } { \pi + 5 }$
(4) $\frac { 2 } { \pi - 5 }$

Let $[ t ]$ denote the greatest integer $\leq t$. If $\lambda \varepsilon R - \{ 0,1 \} , \quad \lim _ { x \rightarrow 0 } \left| \frac { 1 - x + | x | } { \lambda - x + [ x ] } \right| = L$, then $L$ is equal to
(1) 1
(2) 2
(3) $\frac { 1 } { 2 }$
(4) 0

Let $[ t ]$ denote the greatest integer $\leq t$ and $\lim _ { x \rightarrow 0 } x \left[ \frac { 4 } { x } \right] = A$. Then the function, $f ( x ) = \left[ x ^ { 2 } \right] \sin ( \pi x )$ is discontinuous, when $x$ is equal to:
(1) $\sqrt { A + 1 }$
(2) $\sqrt { A + 5 }$
(3) $\sqrt { A + 21 }$
(4) $\sqrt { A }$