Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be given by $$f ( x ) = \left\{ \begin{aligned} x ^ { 5 } + 5 x ^ { 4 } + 10 x ^ { 3 } + 10 x ^ { 2 } + 3 x + 1 , & x < 0 \\ x ^ { 2 } - x + 1 , & 0 \leq x < 1 \\ \frac { 2 } { 3 } x ^ { 3 } - 4 x ^ { 2 } + 7 x - \frac { 8 } { 3 } , & 1 \leq x < 3 \\ ( x - 2 ) \log _ { e } ( x - 2 ) - x + \frac { 10 } { 3 } , & x \geq 3 \end{aligned} \right.$$ Then which of the following options is/are correct?
(A) $f$ is increasing on $( - \infty , 0 )$
(B) $f ^ { \prime }$ has a local maximum at $x = 1$
(C) $f$ is onto
(D) $f ^ { \prime }$ is NOT differentiable at $x = 1$

Let the function $f: (0, \pi) \rightarrow \mathbb{R}$ be defined by $$f(\theta) = (\sin\theta + \cos\theta)^{2} + (\sin\theta - \cos\theta)^{4}$$ Suppose the function $f$ has a local minimum at $\theta$ precisely when $\theta \in \{\lambda_{1}\pi, \ldots, \lambda_{r}\pi\}$, where $0 < \lambda_{1} < \cdots < \lambda_{r} < 1$. Then the value of $\lambda_{1} + \cdots + \lambda_{r}$ is $\_\_\_\_$

Let $f_1 : (0, \infty) \to \mathbb{R}$ and $f_2 : (0, \infty) \to \mathbb{R}$ be defined by $$f_1(x) = \int_0^x \prod_{j=1}^{21} (t - j)^j \, dt, \quad x > 0$$ and $$f_2(x) = 98(x-1)^{50} - 600(x-1)^{49} + 2450, \quad x > 0,$$ where, for any positive integer $n$ and real numbers $a_1, a_2, \ldots, a_n$, $\prod_{i=1}^n a_i$ denotes the product of $a_1, a_2, \ldots, a_n$. Let $m_i$ and $n_i$, respectively, denote the number of points of local minima and the number of points of local maxima of function $f_i$, $i = 1, 2$, in the interval $(0, \infty)$.
The value of $2m_1 + 3n_1 + m_1 n_1$ is ____.

Let $f_1 : (0, \infty) \to \mathbb{R}$ and $f_2 : (0, \infty) \to \mathbb{R}$ be defined by $$f_1(x) = \int_0^x \prod_{j=1}^{21} (t - j)^j \, dt, \quad x > 0$$ and $$f_2(x) = 98(x-1)^{50} - 600(x-1)^{49} + 2450, \quad x > 0,$$ where, for any positive integer $n$ and real numbers $a_1, a_2, \ldots, a_n$, $\prod_{i=1}^n a_i$ denotes the product of $a_1, a_2, \ldots, a_n$. Let $m_i$ and $n_i$, respectively, denote the number of points of local minima and the number of points of local maxima of function $f_i$, $i = 1, 2$, in the interval $(0, \infty)$.
The value of $6m_2 + 4n_2 + 8m_2 n_2$ is ____.

Let $f _ { 1 } : ( 0 , \infty ) \rightarrow \mathbb { R }$ and $f _ { 2 } : ( 0 , \infty ) \rightarrow \mathbb { R }$ be defined by $$f _ { 1 } ( x ) = \int _ { 0 } ^ { x } \prod _ { j = 1 } ^ { 21 } ( t - j ) ^ { j } d t , \quad x > 0$$ and $$f _ { 2 } ( x ) = 98 ( x - 1 ) ^ { 50 } - 600 ( x - 1 ) ^ { 49 } + 2450 , \quad x > 0$$ where, for any positive integer $n$ and real numbers $a _ { 1 } , a _ { 2 } , \ldots , a _ { n } , \prod _ { i = 1 } ^ { n } a _ { i }$ denotes the product of $a _ { 1 } , a _ { 2 } , \ldots , a _ { n }$. Let $m _ { i }$ and $n _ { i }$, respectively, denote the number of points of local minima and the number of points of local maxima of function $f _ { i } , i = 1,2$, in the interval $( 0 , \infty )$. The value of $2 m _ { 1 } + 3 n _ { 1 } + m _ { 1 } n _ { 1 }$ is $\_\_\_\_$.

Let $f _ { 1 } : ( 0 , \infty ) \rightarrow \mathbb { R }$ and $f _ { 2 } : ( 0 , \infty ) \rightarrow \mathbb { R }$ be defined by $$f _ { 1 } ( x ) = \int _ { 0 } ^ { x } \prod _ { j = 1 } ^ { 21 } ( t - j ) ^ { j } d t , \quad x > 0$$ and $$f _ { 2 } ( x ) = 98 ( x - 1 ) ^ { 50 } - 600 ( x - 1 ) ^ { 49 } + 2450 , \quad x > 0$$ where, for any positive integer $n$ and real numbers $a _ { 1 } , a _ { 2 } , \ldots , a _ { n } , \prod _ { i = 1 } ^ { n } a _ { i }$ denotes the product of $a _ { 1 } , a _ { 2 } , \ldots , a _ { n }$. Let $m _ { i }$ and $n _ { i }$, respectively, denote the number of points of local minima and the number of points of local maxima of function $f _ { i } , i = 1,2$, in the interval $( 0 , \infty )$. The value of $6 m _ { 2 } + 4 n _ { 2 } + 8 m _ { 2 } n _ { 2 }$ is $\_\_\_\_$.

Let $\psi _ { 1 } : [ 0 , \infty ) \rightarrow \mathbb { R } , \quad \psi _ { 2 } : [ 0 , \infty ) \rightarrow \mathbb { R } , \quad f : [ 0 , \infty ) \rightarrow \mathbb { R }$ and $g : [ 0 , \infty ) \rightarrow \mathbb { R }$ be functions such that $f ( 0 ) = g ( 0 ) = 0$, $$\begin{gathered} \psi _ { 1 } ( x ) = e ^ { - x } + x , \quad x \geq 0 \\ \psi _ { 2 } ( x ) = x ^ { 2 } - 2 x - 2 e ^ { - x } + 2 , \quad x \geq 0 \\ f ( x ) = \int _ { - x } ^ { x } \left( | t | - t ^ { 2 } \right) e ^ { - t ^ { 2 } } d t , \quad x > 0 \end{gathered}$$ and $$g ( x ) = \int _ { 0 } ^ { x ^ { 2 } } \sqrt { t } e ^ { - t } d t , \quad x > 0$$ Which of the following statements is TRUE ?
(A) $f ( \sqrt { \ln 3 } ) + g ( \sqrt { \ln 3 } ) = \frac { 1 } { 3 }$
(B) For every $x > 1$, there exists an $\alpha \in ( 1 , x )$ such that $\psi _ { 1 } ( x ) = 1 + \alpha x$
(C) For every $x > 0$, there exists a $\beta \in ( 0 , x )$ such that $\psi _ { 2 } ( x ) = 2 x \left( \psi _ { 1 } ( \beta ) - 1 \right)$
(D) $f$ is an increasing function on the interval $\left[ 0 , \frac { 3 } { 2 } \right]$

Let $\psi _ { 1 } : [ 0 , \infty ) \rightarrow \mathbb { R } , \quad \psi _ { 2 } : [ 0 , \infty ) \rightarrow \mathbb { R } , \quad f : [ 0 , \infty ) \rightarrow \mathbb { R }$ and $g : [ 0 , \infty ) \rightarrow \mathbb { R }$ be functions such that $f ( 0 ) = g ( 0 ) = 0$, $$\begin{gathered} \psi _ { 1 } ( x ) = e ^ { - x } + x , \quad x \geq 0 \\ \psi _ { 2 } ( x ) = x ^ { 2 } - 2 x - 2 e ^ { - x } + 2 , \quad x \geq 0 \\ f ( x ) = \int _ { - x } ^ { x } \left( | t | - t ^ { 2 } \right) e ^ { - t ^ { 2 } } d t , \quad x > 0 \end{gathered}$$ and $$g ( x ) = \int _ { 0 } ^ { x ^ { 2 } } \sqrt { t } e ^ { - t } d t , \quad x > 0$$ Which of the following statements is TRUE ?
(A) $\psi _ { 1 } ( x ) \leq 1$, for all $x > 0$
(B) $\psi _ { 2 } ( x ) \leq 0$, for all $x > 0$
(C) $f ( x ) \geq 1 - e ^ { - x ^ { 2 } } - \frac { 2 } { 3 } x ^ { 3 } + \frac { 2 } { 5 } x ^ { 5 }$, for all $x \in \left( 0 , \frac { 1 } { 2 } \right)$
(D) $g ( x ) \leq \frac { 2 } { 3 } x ^ { 3 } - \frac { 2 } { 5 } x ^ { 5 } + \frac { 1 } { 7 } x ^ { 7 }$, for all $x \in \left( 0 , \frac { 1 } { 2 } \right)$

Let $S$ be the set of all twice differentiable functions $f$ from $\mathbb { R }$ to $\mathbb { R }$ such that $\frac { d ^ { 2 } f } { d x ^ { 2 } } ( x ) > 0$ for all $x \in ( - 1,1 )$. For $f \in S$, let $X _ { f }$ be the number of points $x \in ( - 1,1 )$ for which $f ( x ) = x$. Then which of the following statements is(are) true?
(A) There exists a function $f \in S$ such that $X _ { f } = 0$
(B) For every function $f \in S$, we have $X _ { f } \leq 2$
(C) There exists a function $f \in S$ such that $X _ { f } = 2$
(D) There does NOT exist any function $f$ in $S$ such that $X _ { f } = 1$

Let $\mathbb { R }$ denote the set of all real numbers. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined by
$$f ( x ) = \begin{cases} \frac { 6 x + \sin x } { 2 x + \sin x } & \text { if } x \neq 0 \\ \frac { 7 } { 3 } & \text { if } x = 0 \end{cases}$$
Then which of the following statements is (are) TRUE?

(A)	The point $x = 0$ is a point of local maxima of $f$
(B)	The point $x = 0$ is a point of local minima of $f$
(C)	Number of points of local maxima of $f$ in the interval $[ \pi , 6 \pi ]$ is 3
(D)	Number of points of local minima of $f$ in the interval $[ 2 \pi , 4 \pi ]$ is 1

Let $\mathbb { R }$ denote the set of all real numbers. For a real number $x$, let $[ x ]$ denote the greatest integer less than or equal to $x$. Let $n$ denote a natural number.
Match each entry in List-I to the correct entry in List-II and choose the correct option.

List-I

(P) The minimum value of $n$ for which the function
$$f ( x ) = \left[ \frac { 10 x ^ { 3 } - 45 x ^ { 2 } + 60 x + 35 } { n } \right]$$
is continuous on the interval $[ 1,2 ]$, is (Q) The minimum value of $n$ for which
$$g ( x ) = \left( 2 n ^ { 2 } - 13 n - 15 \right) \left( x ^ { 3 } + 3 x \right)$$
$x \in \mathbb { R }$, is an increasing function on $\mathbb { R }$, is (R) The smallest natural number $n$ which is greater than 5, such that $x = 3$ is a point of local minima of
$$h ( x ) = \left( x ^ { 2 } - 9 \right) ^ { n } \left( x ^ { 2 } + 2 x + 3 \right) ,$$
is (S) Number of $x _ { 0 } \in \mathbb { R }$ such that
$$l ( x ) = \sum _ { k = 0 } ^ { 4 } \left( \sin | x - k | + \cos \left| x - k + \frac { 1 } { 2 } \right| \right) ,$$
$x \in \mathbb { R }$, is NOT differentiable at $x _ { 0 }$, is

List-II

(1) 8
(2) 9
(3) 5
(4) 6
(5) 10

(A)	$( \mathrm { P } ) \rightarrow ( 1 )$	$( \mathrm { Q } ) \rightarrow ( 3 )$	$( \mathrm { R } ) \rightarrow ( 2 )$	$( \mathrm { S } ) \rightarrow ( 5 )$
(B)	$( \mathrm { P } ) \rightarrow ( 2 )$	$( \mathrm { Q } ) \rightarrow ( 1 )$	$( \mathrm { R } ) \rightarrow ( 4 )$	$( \mathrm { S } ) \rightarrow ( 3 )$
(C)	$( \mathrm { P } ) \rightarrow ( 5 )$	$( \mathrm { Q } ) \rightarrow ( 1 )$	$( \mathrm { R } ) \rightarrow ( 4 )$	$( \mathrm { S } ) \rightarrow ( 3 )$
(D)	$( \mathrm { P } ) \rightarrow ( 2 )$	$( \mathrm { Q } ) \rightarrow ( 3 )$	$( \mathrm { R } ) \rightarrow ( 1 )$	$( \mathrm { S } ) \rightarrow ( 5 )$

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) $\sqrt { 2 }$

The function $f ( x ) = \tan ^ { - 1 } ( \sin x + \cos x )$ is an increasing function in
(1) $\left( \frac { \pi } { 4 } , \frac { \pi } { 2 } \right)$
(2) $\left( - \frac { \pi } { 2 } , \frac { \pi } { 4 } \right)$
(3) $\left( 0 , \frac { \pi } { 2 } \right)$
(4) $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$

Let $a, b \in \mathbb{R}$ be such that the function $f$ given by $f(x) = \ln|x| + bx^{2} + ax$, $x \neq 0$ has extreme values at $x = -1$ and $x = 2$. Statement 1: $f$ has local maximum at $x = -1$ and at $x = 2$. Statement 2: $a = \frac{1}{2}$ and $b = \frac{-1}{4}$.
(1) Statement 1 is false, Statement 2 is true
(2) Statement 1 is true, Statement 2 is false
(3) Statement 1 is true, Statement 2 is the correct explanation for Statement 1
(4) Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation for Statement 1

If $f$ \& $g$ are differentiable functions in $[ 0,1 ]$ satisfying $f ( 0 ) = 2 = g ( 1 ) , g ( 0 ) = 0$ \& $f ( 1 ) = 6$, then for some $c \in ] 0,1 [$
(1) $f ^ { \prime } ( c ) = g ^ { \prime } ( c )$
(2) $f ^ { \prime } ( c ) = 2 g ^ { \prime } ( c )$
(3) $2 f ^ { \prime } ( c ) = g ^ { \prime } ( c )$
(4) $2 f ^ { \prime } ( c ) = 3 g ^ { \prime } ( c )$

If $x = - 1$ and $x = 2$ are extreme points of $f ( x ) = \alpha \log | x | + \beta x ^ { 2 } + x$, then
(1) $\alpha = 2 , \beta = - \frac { 1 } { 2 }$
(2) $\alpha = 2 , \beta = \frac { 1 } { 2 }$
(3) $\alpha = - 6 , \beta = \frac { 1 } { 2 }$
(4) $\alpha = - 6 , \beta = - \frac { 1 } { 2 }$

If the Rolle's theorem holds for the function $f ( x ) = 2 x ^ { 3 } + a x ^ { 2 } + b x$ in the interval $[ - 1,1 ]$ for the point $c = \frac { 1 } { 2 }$, then the value of $2 a + b$ is:
(1) $-1$
(2) 2
(3) 1
(4) $-2$

Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = \begin{cases} k - 2x, & \text{if } x \leq -1 \\ 2x + 3, & \text{if } x > -1 \end{cases}$. If $f$ has a local minimum at $x = -1$, then a possible value of $k$ is:
(1) $0$
(2) $-\frac{1}{2}$
(3) $-1$
(4) $1$

Let $f ( x )$ be a polynomial of degree four and having its extreme values at $x = 1$ and $x = 2$. If $\lim _ { x \rightarrow 0 } \left[ 1 + \frac { f ( x ) } { x ^ { 2 } } \right] = 3$, then $f ( 2 )$ is equal to
(1) 4
(2) - 8
(3) - 4
(4) 0

A wire of length 2 units is cut into two parts which are bent respectively to form a square of side $= x$ units and a circle of radius $= r$ units. If the sum of the areas of the square and the circle so formed is minimum, then: (1) $2x = (\pi + 4)r$ (2) $(4-\pi)x = \pi r$ (3) $x = 2r$ (4) $2x = r$

Let $f ( x ) = \sin ^ { 4 } x + \cos ^ { 4 } x$. Then, $f$ is an increasing function in the interval:
(1) $]\frac { 5 \pi } { 8 } , \frac { 3 \pi } { 4 } [$
(2) $]\frac { \pi } { 2 } , \frac { 5 \pi } { 8 } [$
(3) $]\frac { \pi } { 4 } , \frac { \pi } { 2 } [$
(4) $]0 , \frac { \pi } { 4 } [$

A wire of length 2 units is cut into two parts which are bent respectively to form a square of side $= x$ units and a circle of radius $= r$ units. If the sum of the areas of the square and the circle so formed is minimum, then:
(1) $2x = (\pi + 4)r$
(2) $(4 - \pi)x = \pi r$
(3) $x = 2r$
(4) $2x = r$

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is:
(1) 12.5
(2) 10
(3) 25
(4) 30

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is:
(1) 12.5
(2) 10
(3) 25
(4) 30

A man in a car at location Q on a straight highway is moving with speed v . He decides to reach a point $P$ in a field at a distance $d$ from highway (point $M$) as shown in the figure. Speed of the car in the field is half to that on the highway. What should be the distance RM, so that the time taken to reach $P$ is minimum? [Figure]
(1) $\frac { \mathrm { d } } { \sqrt { 3 } }$
(2) $\frac { d } { 2 }$
(3) $\frac { d } { \sqrt { 2 } }$
(4) d