41. The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball is
(A) 75
(B) 150
(C) 210
(D) 243
ANSWER : B
- Let $f ( x ) = \left\{ \begin{array} { r l } x ^ { 2 } \left| \cos \frac { \pi } { x } \right| , & x \neq 0 \\ 0 , & x = 0 \end{array} , x \in \mathbb { R } \right.$, then $f$ is
(A) differentiable both at $x = 0$ and at $x = 2$
(B) differentiable at $x = 0$ but not differentiable at $x = 2$
(C) not differentiable at $x = 0$ but differentiable at $x = 2$
(D) differentiable neither at $x = 0$ nor at $x = 2$
ANSWER : B
- The function $f : [ 0,3 ] \rightarrow [ 1,29 ]$, defined by $f ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 36 x + 1$, is
(A) one-one and onto.
(B) onto but not one-one.
(C) one-one but not onto.
(D) neither one-one nor onto.
ANSWER: B
- If $\lim _ { x \rightarrow \infty } \left( \frac { x ^ { 2 } + x + 1 } { x + 1 } - a x - b \right) = 4$, then
(A) $a = 1 , b = 4$
(B) $a = 1 , b = - 4$
(C) $a = 2 , b = - 3$
(D) $a = 2 , b = 3$ - Let $z$ be a complex number such that the imaginary part of $z$ is nonzero and $a = z ^ { 2 } + z + 1$ is real. Then $a$ cannot take the value
(A) - 1
(B) $\frac { 1 } { 3 }$
(C) $\frac { 1 } { 2 }$
(D) $\frac { 3 } { 4 }$
ANSWER : D
- The ellipse $E _ { 1 } : \frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$ is inscribed in a rectangle $R$ whose sides are parallel to the coordinate axes. Another ellipse $E _ { 2 }$ passing through the point $( 0,4 )$ circumscribes the rectangle $R$. The eccentricity of the ellipse $E _ { 2 }$ is
(A) $\frac { \sqrt { 2 } } { 2 }$
(B) $\frac { \sqrt { 3 } } { 2 }$
(C) $\frac { 1 } { 2 }$
(D) $\frac { 3 } { 4 }$
ANSWER : C
- Let $P = \left[ a _ { i j } \right]$ be a $3 \times 3$ matrix and let $Q = \left[ b _ { i j } \right]$, where $b _ { i j } = 2 ^ { i + j } a _ { i j }$ for $1 \leq i , j \leq 3$. If the determinant of $P$ is 2 , then the determinant of the matrix $Q$ is
(A) $2 ^ { 10 }$
(B) $2 ^ { 11 }$
(C) $2 ^ { 12 }$
(D) $2 ^ { 13 }$
ANSWER : D
- The integral $\int \frac { \sec ^ { 2 } x } { ( \sec x + \tan x ) ^ { 9 / 2 } } d x$ equals (for some arbitrary constant $K$ )
(A) $- \frac { 1 } { ( \sec x + \tan x ) ^ { 11 / 2 } } \left\{ \frac { 1 } { 11 } - \frac { 1 } { 7 } ( \sec x + \tan x ) ^ { 2 } \right\} + K$
(B) $\frac { 1 } { ( \sec x + \tan x ) ^ { 11 / 2 } } \left\{ \frac { 1 } { 11 } - \frac { 1 } { 7 } ( \sec x + \tan x ) ^ { 2 } \right\} + K$
(C) $- \frac { 1 } { ( \sec x + \tan x ) ^ { 11 / 2 } } \left\{ \frac { 1 } { 11 } + \frac { 1 } { 7 } ( \sec x + \tan x ) ^ { 2 } \right\} + K$
(D) $\frac { 1 } { ( \sec x + \tan x ) ^ { 11 / 2 } } \left\{ \frac { 1 } { 11 } + \frac { 1 } { 7 } ( \sec x + \tan x ) ^ { 2 } \right\} + K$
ANSWER : C
- The point $P$ is the intersection of the straight line joining the points $Q ( 2,3,5 )$ and $R ( 1 , - 1,4 )$ with the plane $5 x - 4 y - z = 1$. If $S$ is the foot of the perpendicular drawn from the point $T ( 2,1,4 )$ to $Q R$, then the length of the line segment $P S$ is
(A) $\frac { 1 } { \sqrt { 2 } }$
(B) $\sqrt { 2 }$
(C) 2
(D) $2 \sqrt { 2 }$
ANSWER : A
- The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line $4 x - 5 y = 20$ to the circle $x ^ { 2 } + y ^ { 2 } = 9$ is
(A) $20 \left( x ^ { 2 } + y ^ { 2 } \right) - 36 x + 45 y = 0$
(B) $20 \left( x ^ { 2 } + y ^ { 2 } \right) + 36 x - 45 y = 0$
(C) $36 \left( x ^ { 2 } + y ^ { 2 } \right) - 20 x + 45 y = 0$
(D) $36 \left( x ^ { 2 } + y ^ { 2 } \right) + 20 x - 45 y = 0$
ANSWER : A
SECTION II: Multiple Correct Answer(s) Type
This section contains 5 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct. 51. Let $\theta , \varphi \in [ 0,2 \pi ]$ be such that $2 \cos \theta ( 1 - \sin \varphi ) = \sin ^ { 2 } \theta \left( \tan \frac { \theta } { 2 } + \cot \frac { \theta } { 2 } \right) \cos \varphi - 1$, $\tan ( 2 \pi - \theta ) > 0$ and $- 1 < \sin \theta < - \frac { \sqrt { 3 } } { 2 }$. Then $\varphi$ cannot satisfy
(A) $0 < \varphi < \frac { \pi } { 2 }$
(B) $\frac { \pi } { 2 } < \varphi < \frac { 4 \pi } { 3 }$
(C) $\frac { 4 \pi } { 3 } < \varphi < \frac { 3 \pi } { 2 }$
(D) $\frac { 3 \pi } { 2 } < \varphi < 2 \pi$
ANSWER : ACD
- Let $S$ be the area of the region enclosed by $y = e ^ { - x ^ { 2 } } , y = 0 , x = 0$, and $x = 1$. Then
(A) $S \geq \frac { 1 } { e }$
(B) $S \geq 1 - \frac { 1 } { e }$
(C) $S \leq \frac { 1 } { 4 } \left( 1 + \frac { 1 } { \sqrt { e } } \right)$
(D) $S \leq \frac { 1 } { \sqrt { 2 } } + \frac { 1 } { \sqrt { e } } \left( 1 - \frac { 1 } { \sqrt { 2 } } \right)$
ANSWER : ABD
- A ship is fitted with three engines $E _ { 1 } , E _ { 2 }$ and $E _ { 3 }$. The engines function independently of each other with respective probabilities $\frac { 1 } { 2 } , \frac { 1 } { 4 }$ and $\frac { 1 } { 4 }$. For the ship to be operational at least two of its engines must function. Let $X$ denote the event that the ship is operational and let $X _ { 1 } , X _ { 2 }$ and $X _ { 3 }$ denote respectively the events that the engines $E _ { 1 } , E _ { 2 }$ and $E _ { 3 }$ are functioning. Which of the following is (are) true ?
(A) $P \left[ X _ { 1 } { } ^ { c } \mid X \right] = \frac { 3 } { 16 }$
(B) $P$ [Exactly two engines of the ship are functioning $\mid X ] = \frac { 7 } { 8 }$
(C) $P \left[ X \mid X _ { 2 } \right] = \frac { 5 } { 16 }$
(D) $P \left[ X \mid X _ { 1 } \right] = \frac { 7 } { 16 }$
ANSWER : BD
- Tangents are drawn to the hyperbola $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 4 } = 1$, parallel to the straight line $2 x - y = 1$. The points of contact of the tangents on the hyperbola are
(A) $\left( \frac { 9 } { 2 \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } \right)$
(B) $\left( - \frac { 9 } { 2 \sqrt { 2 } } , - \frac { 1 } { \sqrt { 2 } } \right)$
(C) $( 3 \sqrt { 3 } , - 2 \sqrt { 2 } )$
(D) $( - 3 \sqrt { 3 } , 2 \sqrt { 2 } )$
ANSWER : AB
- If $y ( x )$ satisfies the differential equation $y ^ { \prime } - y \tan x = 2 x \sec x$ and $y ( 0 ) = 0$, then
(A) $y \left( \frac { \pi } { 4 } \right) = \frac { \pi ^ { 2 } } { 8 \sqrt { 2 } }$
(B) $y ^ { \prime } \left( \frac { \pi } { 4 } \right) = \frac { \pi ^ { 2 } } { 18 }$
(C) $y \left( \frac { \pi } { 3 } \right) = \frac { \pi ^ { 2 } } { 9 }$
(D) $y ^ { \prime } \left( \frac { \pi } { 3 } \right) = \frac { 4 \pi } { 3 } + \frac { 2 \pi ^ { 2 } } { 3 \sqrt { 3 } }$
ANSWER : AD
SECTION III : Integer Answer Type
This section contains $\mathbf { 5 }$ questions. The answer to each question is a single digit integer, ranging from 0 to 9 (both inclusive). 56. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined as $f ( x ) = | x | + \left| x ^ { 2 } - 1 \right|$. The total number of points at which $f$ attains either a local maximum or a local minimum is
ANSWER : 5
- The value of $6 + \log _ { \frac { 3 } { 2 } } \left( \frac { 1 } { 3 \sqrt { 2 } } \sqrt { 4 - \frac { 1 } { 3 \sqrt { 2 } } \sqrt { 4 - \frac { 1 } { 3 \sqrt { 2 } } \sqrt { 4 - \frac { 1 } { 3 \sqrt { 2 } } \ldots } } } \right)$ is
ANSWER : 4
- Let $p ( x )$ be a real polynomial of least degree which has a local maximum at $x = 1$ and a local minimum at $x = 3$. If $p ( 1 ) = 6$ and $p ( 3 ) = 2$, then $p ^ { \prime } ( 0 )$ is
ANSWER : 9
- If $\vec { a } , \vec { b }$ and $\vec { c }$ are unit vectors satisfying $| \vec { a } - \vec { b } | ^ { 2 } + | \vec { b } - \vec { c } | ^ { 2 } + | \vec { c } - \vec { a } | ^ { 2 } = 9$, then $| 2 \vec { a } + 5 \vec { b } + 5 \vec { c } |$ is
ANSWER : 3
- Let $S$ be the focus of the parabola $y ^ { 2 } = 8 x$ and let $P Q$ be the common chord of the circle $x ^ { 2 } + y ^ { 2 } - 2 x - 4 y = 0$ and the given parabola. The area of the triangle $P Q S$ is
ANSWER : 4