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) $\sqrt { 2 }$

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 $x = 0$

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 $x = - 1$ 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

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 (in $\mathrm { cm } ^ { 2 }$) of this cone is :
(1) $8 \sqrt { 2 } \pi$
(2) $6 \sqrt { 2 } \pi$
(3) $8 \sqrt { 3 } \pi$
(4) $6 \sqrt { 3 } \pi$

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) $8 \sqrt { 2 } \pi$

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 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) $( - \sqrt { 2 } , \sqrt { 2 } )$

Moment of inertia of a cylinder of mass m, length L and radius R about an axis passing through its centre and perpendicular to the axis of the cylinder is $I = M \left( \frac { R ^ { 2 } } { 4 } + \frac { L ^ { 2 } } { 12 } \right)$. If such a cylinder is to be made for a given mass of a material, the ratio $\frac { L } { R }$ for it to have minimum possible $I$ is:
(1) $\frac { 2 } { 3 }$
(2) $\frac { 3 } { 2 }$
(3) $\sqrt { \frac { 3 } { 2 } }$
(4) $\sqrt { \frac { 2 } { 3 } }$

The value of $c$, in the Lagrange's mean value theorem for the function $f ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 8 x + 11$, when $x \in [ 0,1 ]$, is
(1) $\frac { 4 - \sqrt { 5 } } { 3 }$
(2) $\frac { 4 - \sqrt { 7 } } { 3 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { \sqrt { 7 } - 2 } { 3 }$

Let $f$ be any function continuous on $[ a , b ]$ and twice differentiable on $( a , b )$. If all $x \in ( a , b ) , f ^ { \prime } ( x ) > 0$ and $f ^ { \prime \prime } ( x ) < 0$, then for any $c \in ( a , b ) , \frac { f ( c ) - f ( a ) } { f ( b ) - f ( c ) }$
(1) $\frac { b + a } { b - a }$
(2) 1
(3) $\frac { b - c } { c - a }$
(4) $\frac { c - a } { b - c }$