45. Let $O ( 0,0 ) , P ( 3,4 ) , Q ( 6,0 )$ be the vertices of the triangle $O P Q$. The point $R$ inside the triangle $O P Q$ is such that the triangles $O P R , P Q R , O Q R$ are of equal area. The coordinates of $R$ are
(A) $\left( \frac { 4 } { 3 } , 3 \right)$
(B) $\left( 3 , \frac { 2 } { 3 } \right)$
(C) $\left( 3 , \frac { 4 } { 3 } \right)$
(D) $\left( \frac { 4 } { 3 } , \frac { 2 } { 3 } \right)$
Answer ◯ [Figure] [Figure] [Figure]
(A)
(B)
(C)
(D)

46. If $| z | = 1$ and $z \neq \pm 1$, then all the values of $\frac { z } { 1 - z ^ { 2 } }$ lie on
(A) a line not passing through the origin
(B) $| z | = \sqrt { 2 }$
(C) the $x$-axis
(D) the $y$-axis

Answer

[Figure]
(A)
[Figure]
(B)
[Figure]
(C)
[Figure]
(D)

\setcounter{enumi}{46}
1. Let $E ^ { c }$ denote the complement of an event $E$. Let $E , F , G$ be pairwise independent events with $P ( G ) > 0$ and $P ( E \cap F \cap G ) = 0$. Then $P \left( E ^ { c } \cap F ^ { c } \mid G \right)$ equals
   (A) $P \left( E ^ { c } \right) + P \left( F ^ { c } \right)$
   (B) $P \left( E ^ { c } \right) - P \left( F ^ { c } \right)$
   (C) $P \left( E ^ { c } \right) - P ( F )$
   (D) $P ( E ) - P \left( F ^ { c } \right)$

Answer ◯
(A)
[Figure]
(B)
[Figure]
(C)
◯
(D)

48. $\frac { d ^ { 2 } x } { d y ^ { 2 } }$ equals
(A) $\left( \frac { d ^ { 2 } y } { d x ^ { 2 } } \right) ^ { - 1 }$
(B) $- \left( \frac { d ^ { 2 } y } { d x ^ { 2 } } \right) ^ { - 1 } \left( \frac { d y } { d x } \right) ^ { - 3 }$
(C) $\left( \frac { d ^ { 2 } y } { d x ^ { 2 } } \right) \left( \frac { d y } { d x } \right) ^ { - 2 }$
(D) $- \left( \frac { d ^ { 2 } y } { d x ^ { 2 } } \right) \left( \frac { d y } { d x } \right) ^ { - 3 }$

Answer

◯ ◯
(A)
(B)
(C)
(D)

49. The differential equation $\frac { d y } { d x } = \frac { \sqrt { 1 - y ^ { 2 } } } { y }$ determines a family of circles with
(A) variable radii and a fixed centre at $( 0,1 )$
(B) variable radii and a fixed centre at $( 0 , - 1 )$
(C) fixed radius 1 and variable centres along the $x$-axis
(D) fixed radius 1 and variable centres along the $y$-axis Answer O O O O
(A)
(B)
(C)
(D)

50. Let $\vec { a } , \vec { b } , \vec { c }$ be unit vectors such that $\vec { a } + \vec { b } + \vec { c } = \overrightarrow { 0 }$. Which one of the following is correct?
(A) $\vec { a } \times \vec { b } = \vec { b } \times \vec { c } = \vec { c } \times \vec { a } = \overrightarrow { 0 }$
(B) $\vec { a } \times \vec { b } = \vec { b } \times \vec { c } = \vec { c } \times \vec { a } \neq \overrightarrow { 0 }$
(C) $\vec { a } \times \vec { b } = \vec { b } \times \vec { c } = \vec { a } \times \vec { c } \neq \overrightarrow { 0 }$
(D) $\vec { a } \times \vec { b } , \vec { b } \times \vec { c } , \vec { c } \times \vec { a }$ are mutually perpendicular
Answer [Figure] [Figure] ◯ ◯
(A)
(B)
(C)
(D) 51. Let $A B C D$ be a quadrilateral with area 18, with side $A B$ parallel to the side $C D$ and $A B = 2 C D$. Let $A D$ be perpendicular to $A B$ and $C D$. If a circle is drawn inside the quadrilateral $A B C D$ touching all the sides, then its radius is
(A) 3
(B) 2
(C) $\frac { 3 } { 2 }$
(D) 1
Answer [Figure] [Figure] [Figure] [Figure]
(A)
(B)
(C)
(D) 52. Let $f ( x ) = \frac { x } { \left( 1 + x ^ { n } \right) ^ { 1 / n } }$ for $n \geq 2$ and $g ( x ) = \underbrace { f \circ f \circ \cdots \circ f ) } _ { f \text { occurs } n \text { times } } ( x )$. Then $\int x ^ { n - 2 } g ( x ) d x$ equals
(A) $\frac { 1 } { n ( n - 1 ) } \left( 1 + n x ^ { n } \right) ^ { 1 - \frac { 1 } { n } } + K$
(B) $\frac { 1 } { n - 1 } \left( 1 + n x ^ { n } \right) ^ { 1 - \frac { 1 } { n } } + K$
(C) $\frac { 1 } { n ( n + 1 ) } \left( 1 + n x ^ { n } \right) ^ { 1 + \frac { 1 } { n } } + K$
(D) $\frac { 1 } { n + 1 } \left( 1 + n x ^ { n } \right) ^ { 1 + \frac { 1 } { n } } + K$ Answer [Figure] ◯ ◯
(A)
(B)
(C)
(D) 53. The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is
(A) 360
(B) 192
(C) 96
(D) 48
Answer [Figure]
(A) [Figure]
(B) [Figure]
(C) [Figure]
(D) 54. Consider the planes $3 x - 6 y - 2 z = 15$ and $2 x + y - 2 z = 5$.
STATEMENT-1 : The parametric equations of the line of intersection of the given planes are $x = 3 + 14 t , y = 1 + 2 t , z = 15 t$.

because

STATEMENT-2 : The vector $14 \hat { i } + 2 \hat { j } + 15 \hat { k }$ is parallel to the line of intersection of given planes.
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True

Answer

[Figure]
(A)
(B)
(C)
(D) 55. STATEMENT-1 : The curve $y = \frac { - x ^ { 2 } } { 2 } + x + 1$ is symmetric with respect to the line $x = 1$. because STATEMENT-2 : A parabola is symmetric about its axis.
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True Answer
(A) [Figure] [Figure] [Figure]
(A)
(B)
(C)
(D) 56. Let $f ( x ) = 2 + \cos x$ for all real $x$.
STATEMENT-1 : For each real $t$, there exists a point $c$ in $[ t , t + \pi ]$ such that $f ^ { \prime } ( c ) = 0$. because STATEMENT-2 : $f ( t ) = f ( t + 2 \pi )$ for each real $t$.
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True
Answer
[Figure]
(A)
[Figure]
(B)
[Figure]
(C)
◯
(D) 57. Lines $L _ { 1 } : y - x = 0$ and $L _ { 2 } : 2 x + y = 0$ intersect the line $L _ { 3 } : y + 2 = 0$ at $P$ and $Q$, respectively. The bisector of the acute angle between $L _ { 1 }$ and $L _ { 2 }$ intersects $L _ { 3 }$ at $R$.
STATEMENT-1 : The ratio $P R : R Q$ equals $2 \sqrt { 2 } : \sqrt { 5 }$.

because

STATEMENT-2 : In any triangle, bisector of an angle divides the triangle into two similar triangles.
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True Answer
[Figure]
(A)
[Figure]
(B)
(B)
[Figure]
(D)

\setcounter{enumi}{57}
1. Which one of the following statements is correct?
   (A) $G _ { 1 } > G _ { 2 } > G _ { 3 } > \cdots$
   (B) $G _ { 1 } < G _ { 2 } < G _ { 3 } < \cdots$
   (C) $G _ { 1 } = G _ { 2 } = G _ { 3 } = \cdots$
   (D) $G _ { 1 } < G _ { 3 } < G _ { 5 } < \cdots$ and $G _ { 2 } > G _ { 4 } > G _ { 6 } > \cdots$ Answer ◯
   (A) [Figure]
   (B) [Figure]
   (C) ◯
   (D)
2. Which one of the following statements is correct?
   (A) $A _ { 1 } > A _ { 2 } > A _ { 3 } > \cdots$
   (B) $A _ { 1 } < A _ { 2 } < A _ { 3 } < \cdots$
   (C) $A _ { 1 } > A _ { 3 } > A _ { 5 } > \cdots$ and $A _ { 2 } < A _ { 4 } < A _ { 6 } < \cdots$
   (D) $A _ { 1 } < A _ { 3 } < A _ { 5 } < \cdots$ and $A _ { 2 } > A _ { 4 } > A _ { 6 } > \cdots$

Answer

[Figure] ◯ ◯
(A)
(B)
(C)
(D) 60. Which one of the following statements is correct?
(A) $H _ { 1 } > H _ { 2 } > H _ { 3 } > \cdots$
(B) $H _ { 1 } < H _ { 2 } < H _ { 3 } < \cdots$
(C) $H _ { 1 } > H _ { 3 } > H _ { 5 } > \cdots$ and $H _ { 2 } < H _ { 4 } < H _ { 6 } < \cdots$
(D) $H _ { 1 } < H _ { 3 } < H _ { 5 } < \cdots$ and $H _ { 2 } > H _ { 4 } > H _ { 6 } > \cdots$

M61-63: Paragraph for Question Nos. 61 to 63

If a continuous function $f$ defined on the real line $\mathbf { R }$, assumes positive and negative values in $\mathbf { R }$ then the equation $f ( x ) = 0$ has a root in $\mathbf { R }$. For example, if it is known that a continuous function $f$ on $\mathbf { R }$ is positive at some point and its minimum value is negative then the equation $f ( x ) = 0$ has a root in $\mathbf { R }$. Consider $f ( x ) = k e ^ { x } - x$ for all real $x$ where $k$ is a real constant. Answer ◯
(A) [Figure]
(B) O
(C) [Figure]
(D) 61. The line $y = x$ meets $y = k e ^ { x }$ for $k \leq 0$ at
(A) no point
(B) one point
(C) two points
(D) more than two points
Answer
[Figure]
(A)
[Figure]
(B)
[Figure]
(C)
[Figure]
(D)

\setcounter{enumi}{61}
1. The positive value of $k$ for which $k e ^ { x } - x = 0$ has only one root is
   (A) $\frac { 1 } { e }$
   (B) 1
   (C) $e$
   (D) $\log _ { e } 2$

Answer [Figure] [Figure] [Figure]
(A)
(B)
(C)
(D) 63. For $k > 0$, the set of all values of $k$ for which $k e ^ { x } - x = 0$ has two distinct roots is
(A) $\left( 0 , \frac { 1 } { e } \right)$
(B) $\left( \frac { 1 } { e } , 1 \right)$
(C) $\left( \frac { 1 } { e } , \infty \right)$
(D) $( 0,1 )$
Answer ◯ ◯ ◯
(A)
(B)
(C)
(D) 64. Let $f ( x ) = \frac { x ^ { 2 } - 6 x + 5 } { x ^ { 2 } - 5 x + 6 }$.
Match the expressions/statements in Column I with expressions/statements in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS.

Column I

(A) If $- 1 < x < 1$, then $f ( x )$ satisfies
(B) If $1 < x < 2$, then $f ( x )$ satisfies
(C) If $3 < x < 5$, then $f ( x )$ satisfies
(D) If $x > 5$, then $f ( x )$ satisfies

Column II

(p) $0 < f ( x ) < 1$
(q) $f ( x ) < 0$
(r) $f ( x ) > 0$
(s) $f ( x ) < 1$
Answer [Figure] 65. Let $( x , y )$ be such that
$$\sin ^ { - 1 } ( a x ) + \cos ^ { - 1 } ( y ) + \cos ^ { - 1 } ( b x y ) = \frac { \pi } { 2 }$$
Match the statements in Column I with statements in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS.

Column I

(A) If $a = 1$ and $b = 0$, then $( x , y )$
(B) If $a = 1$ and $b = 1$, then $( x , y )$
(C) If $a = 1$ and $b = 2$, then $( x , y )$
(D) If $a = 2$ and $b = 2$, then $( x , y )$
(p) lies on the circle $x ^ { 2 } + y ^ { 2 } = 1$
(q) lies on $\left( x ^ { 2 } - 1 \right) \left( y ^ { 2 } - 1 \right) = 0$
(r) lies on $y = x$
(s) lies on $\left( 4 x ^ { 2 } - 1 \right) \left( y ^ { 2 } - 1 \right) = 0$

Column II

\setcounter{enumi}{65}
1. Match the statements in Column I with the properties in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS.

Column I

(A) Two intersecting circles
(B) Two mutually external circles
(C) Two circles, one strictly inside the other
(D) Two branches of a hyperbola

Column II

(p) have a common tangent
(q) have a common normal
(r) do not have a common tangent
(s) do not have a common normal
Answer [Figure]