A circle $C$ of radius 1 is inscribed in an equilateral triangle $P Q R$. The points of contact of $C$ with the sides $P Q , Q R , R P$ are $D , E , F$, respectively. The line $P Q$ is given by the equation $\sqrt { 3 } x + y - 6 = 0$ and the point $D$ is $\left( \frac { 3 \sqrt { 3 } } { 2 } , \frac { 3 } { 2 } \right)$. Further, it is given that the origin and the centre of $C$ are on the same side of the line $P Q$.
Points $E$ and $F$ are given by
(A) $\left( \frac { \sqrt { 3 } } { 2 } , \frac { 3 } { 2 } \right) , ( \sqrt { 3 } , 0 )$
(B) $\left( \frac { \sqrt { 3 } } { 2 } , \frac { 1 } { 2 } \right) , ( \sqrt { 3 } , 0 )$
(C) $\left( \frac { \sqrt { 3 } } { 2 } , \frac { 3 } { 2 } \right) , \left( \frac { \sqrt { 3 } } { 2 } , \frac { 1 } { 2 } \right)$
(D) $\left( \frac { 3 } { 2 } , \frac { \sqrt { 3 } } { 2 } \right) , \left( \frac { \sqrt { 3 } } { 2 } , \frac { 1 } { 2 } \right)$

A circle $C$ of radius 1 is inscribed in an equilateral triangle $P Q R$. The points of contact of $C$ with the sides $P Q , Q R , R P$ are $D , E , F$, respectively. The line $P Q$ is given by the equation $\sqrt { 3 } x + y - 6 = 0$ and the point $D$ is $\left( \frac { 3 \sqrt { 3 } } { 2 } , \frac { 3 } { 2 } \right)$. Further, it is given that the origin and the centre of $C$ are on the same side of the line $P Q$.
Equations of the sides $Q R , R P$ are
(A) $y = \frac { 2 } { \sqrt { 3 } } x + 1 , y = - \frac { 2 } { \sqrt { 3 } } x - 1$
(B) $y = \frac { 1 } { \sqrt { 3 } } x , y = 0$
(C) $y = \frac { \sqrt { 3 } } { 2 } x + 1 , y = - \frac { \sqrt { 3 } } { 2 } x - 1$
(D) $y = \sqrt { 3 } x , y = 0$

Consider the lines given by
$$\begin{aligned} & L _ { 1 } : x + 3 y - 5 = 0 \\ & L _ { 2 } : 3 x - k y - 1 = 0 \\ & L _ { 3 } : 5 x + 2 y - 12 = 0 \end{aligned}$$
Match the Statements / Expressions in Column I with the Statements / Expressions 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) $L _ { 1 } , L _ { 2 } , L _ { 3 }$ are concurrent, if
(B) One of $L _ { 1 } , L _ { 2 } , L _ { 3 }$ is parallel to at least one of the other two, if
(C) $L _ { 1 } , L _ { 2 } , L _ { 3 }$ form a triangle, if
(D) $L _ { 1 } , L _ { 2 } , L _ { 3 }$ do not form a triangle, if
Column II
(p) $k = - 9$
(q) $k = - \frac { 6 } { 5 }$
(r) $k = \frac { 5 } { 6 }$
(s) $k = 5$

The locus of the orthocentre of the triangle formed by the lines $$\begin{aligned} &(1+p)x-py+p(1+p)=0\\ &(1+q)x-qy+q(1+q)=0 \end{aligned}$$ and $y=0$, where $p\neq q$, is
(A) a hyperbola
(B) a parabola
(C) an ellipse
(D) a straight line

Tangents are drawn from the point $P ( 3,4 )$ to the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$ touching the ellipse at points A and B.
The orthocenter of the triangle $P A B$ is
A) $\left( 5 , \frac { 8 } { 7 } \right)$
B) $\left( \frac { 7 } { 5 } , \frac { 25 } { 8 } \right)$
C) $\left( \frac { 11 } { 5 } , \frac { 8 } { 5 } \right)$
D) $\left( \frac { 8 } { 25 } , \frac { 7 } { 5 } \right)$

Let $( x , y )$ be any point on the parabola $y ^ { 2 } = 4 x$. Let $P$ be the point that divides the line segment from $( 0,0 )$ to $( x , y )$ in the ratio $1 : 3$. Then the locus of $P$ is
(A) $x ^ { 2 } = y$
(B) $y ^ { 2 } = 2 x$
(C) $y ^ { 2 } = x$
(D) $x ^ { 2 } = 2 y$

41. The equation of a plane passing through the line of intersection of the planes $x + 2 y + 3 z = 2$ and $x - y + z = 3$ and at a distance $\frac { 2 } { \sqrt { 3 } }$ from the point $( 3,1 , - 1 )$ is
(A) $5 x - 11 y + z = 17$
(B) $\sqrt { 2 } x + y = 3 \sqrt { 2 } - 1$
(C) $x + y + z = \sqrt { 3 }$
(D) $x - \sqrt { 2 } y = 1 - \sqrt { 2 }$

ANSWER : A

1. Let $P Q R$ be a triangle of area $\triangle$ with $a = 2 , b = \frac { 7 } { 2 }$ and $c = \frac { 5 } { 2 }$, where $a , b$ and $c$ are the lengths of the sides of the triangle opposite to the angles at $P , Q$ and $R$ respectively. Then $\frac { 2 \sin P - \sin 2 P } { 2 \sin P + \sin 2 P }$ equals
   (A) $\frac { 3 } { 4 \Delta }$
   (B) $\frac { 45 } { 4 \Delta }$
   (C) $\left( \frac { 3 } { 4 \Delta } \right) ^ { 2 }$
   (D) $\left( \frac { 45 } { 4 \Delta } \right) ^ { 2 }$

ANSWER : C

1. If $\vec { a }$ and $\vec { b }$ are vectors such that $| \vec { a } + \vec { b } | = \sqrt { 29 }$ and $\vec { a } \times ( 2 \hat { i } + 3 \hat { j } + 4 \hat { k } ) = ( 2 \hat { i } + 3 \hat { j } + 4 \hat { k } ) \times \vec { b }$, then a possible value of $( \vec { a } + \vec { b } ) \cdot ( - 7 \hat { i } + 2 \hat { j } + 3 \hat { k } )$ is
   (A) 0
   (B) 3
   (C) 4
   (D) 8
2. If $P$ is a $3 \times 3$ matrix such that $P ^ { T } = 2 P + I$, where $P ^ { T }$ is the transpose of $P$ and $I$ is the $3 \times 3$ identity matrix, then there exists a column matrix $X = \left[ \begin{array} { c } x \\ y \\ z \end{array} \right] \neq \left[ \begin{array} { l } 0 \\ 0 \\ 0 \end{array} \right]$ such that
   (A) $P X = \left[ \begin{array} { l } 0 \\ 0 \\ 0 \end{array} \right]$
   (B) $P X = X$
   (C) $P X = 2 X$
   (D) $P X = - X$

ANSWER : D

1. Let $\alpha$ (a) and $\beta$ (a) be the roots of the equation $( \sqrt [ 3 ] { 1 + a } - 1 ) x ^ { 2 } + ( \sqrt { 1 + a } - 1 ) x + ( \sqrt [ 6 ] { 1 + a } - 1 ) = 0$ where $a > - 1$. Then $\lim _ { a \rightarrow 0 ^ { + } } \alpha ( a )$ and $\lim _ { a \rightarrow 0 ^ { + } } \beta ( a )$ are
   (A) $- \frac { 5 } { 2 }$ and 1
   (B) $- \frac { 1 } { 2 }$ and - 1
   (C) $- \frac { 7 } { 2 }$ and 2
   (D) $- \frac { 9 } { 2 }$ and 3

ANSWER : B

1. Four fair dice $D _ { 1 } , D _ { 2 } , D _ { 3 }$ and $D _ { 4 }$, each having six faces numbered $1,2,3,4,5$ and 6 , are rolled simultaneously. The probability that $D _ { 4 }$ shows a number appearing on one of $D _ { 1 } , D _ { 2 }$ and $D _ { 3 }$ is
   (A) $\frac { 91 } { 216 }$
   (B) $\frac { 108 } { 216 }$
   (C) $\frac { 125 } { 216 }$
   (D) $\frac { 127 } { 216 }$

ANSWER : A

1. The value of the integral $\int _ { - \pi / 2 } ^ { \pi / 2 } \left( x ^ { 2 } + \ln \frac { \pi + x } { \pi - x } \right) \cos x \mathrm {~d} x$ is
   (A) 0
   (B) $\frac { \pi ^ { 2 } } { 2 } - 4$
   (C) $\frac { \pi ^ { 2 } } { 2 } + 4$
   (D) $\frac { \pi ^ { 2 } } { 2 }$

ANSWER : B

1. Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ be in harmonic progression with $a _ { 1 } = 5$ and $a _ { 20 } = 25$. The least positive integer $n$ for which $a _ { n } < 0$ is
   (A) 22
   (B) 23
   (C) 24
   (D) 25

SECTION II : Paragraph Type

This section contains $\mathbf { 6 }$ multiple choice questions relating to three paragraphs with two questions on each paragraph. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Paragraph for Questions 49 and 50

Let $a _ { n }$ denote the number of all $n$-digit positive integers formed by the digits 0,1 or both such that no consecutive digits in them are 0 . Let $b _ { n } =$ the number of such $n$-digit integers ending with digit 1 and $c _ { n } =$ the number of such $n$-digit integers ending with digit 0 .

For $a > b > c > 0$, the distance between $( 1,1 )$ and the point of intersection of the lines $a x + b y + c = 0$ and $b x + a y + c = 0$ is less than $2 \sqrt { 2 }$. Then
(A) $a + b - c > 0$
(B) $a - b + c < 0$
(C) $a - b + c > 0$
(D) $a + b - c < 0$

Let $\mathbb { R } ^ { 3 }$ denote the three-dimensional space. Take two points $P = ( 1,2,3 )$ and $Q = ( 4,2,7 )$. Let $\operatorname { dist } ( X , Y )$ denote the distance between two points $X$ and $Y$ in $\mathbb { R } ^ { 3 }$. Let
$$\begin{gathered} S = \left\{ X \in \mathbb { R } ^ { 3 } : ( \operatorname { dist } ( X , P ) ) ^ { 2 } - ( \operatorname { dist } ( X , Q ) ) ^ { 2 } = 50 \right\} \text { and } \\ T = \left\{ Y \in \mathbb { R } ^ { 3 } : ( \operatorname { dist } ( Y , Q ) ) ^ { 2 } - ( \operatorname { dist } ( Y , P ) ) ^ { 2 } = 50 \right\} \end{gathered}$$
Then which of the following statements is (are) TRUE?
(A) There is a triangle whose area is 1 and all of whose vertices are from $S$.
(B) There are two distinct points $L$ and $M$ in $T$ such that each point on the line segment $L M$ is also in $T$.
(C) There are infinitely many rectangles of perimeter 48, two of whose vertices are from $S$ and the other two vertices are from $T$.
(D) There is a square of perimeter 48, two of whose vertices are from $S$ and the other two vertices are from $T$.

Let $P \left( x _ { 1 } , y _ { 1 } \right)$ and $Q \left( x _ { 2 } , y _ { 2 } \right)$ be two distinct points on the ellipse
$$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$$
such that $y _ { 1 } > 0$, and $y _ { 2 } > 0$. Let $C$ denote the circle $x ^ { 2 } + y ^ { 2 } = 9$, and $M$ be the point $( 3,0 )$.
Suppose the line $x = x _ { 1 }$ intersects $C$ at $R$, and the line $x = x _ { 2 }$ intersects C at $S$, such that the $y$-coordinates of $R$ and $S$ are positive. Let $\angle R O M = \frac { \pi } { 6 }$ and $\angle S O M = \frac { \pi } { 3 }$, where $O$ denotes the origin $( 0,0 )$. Let $| X Y |$ denote the length of the line segment $X Y$.
Then which of the following statements is (are) TRUE?

(A)	The equation of the line joining $P$ and $Q$ is $2 x + 3 y = 3 ( 1 + \sqrt { 3 } )$
(B)	The equation of the line joining $P$ and $Q$ is $2 x + y = 3 ( 1 + \sqrt { 3 } )$
(C)	If $N _ { 2 } = \left( x _ { 2 } , 0 \right)$, then $3 \left| N _ { 2 } Q \right| = 2 \left| N _ { 2 } S \right|$
(D)	If $N _ { 1 } = \left( x _ { 1 } , 0 \right)$, then $9 \left| N _ { 1 } P \right| = 4 \left| N _ { 1 } R \right|$

Let $A ( h , k ) , B ( 1,1 )$ and $C ( 2,1 )$ be the vertices of a right angled triangle with $A C$ as its hypotenuse. If the area of the triangle is 1 , then the set of values which ' k ' can take is given by
(1) $\{ 1,3 \}$
(2) $\{ 0,2 \}$
(3) $\{ - 1,3 \}$
(4) $\{ - 3 , - 2 \}$

Let $P = ( - 1,0 ) , Q = ( 0,0 )$ and $R = ( 3,3 \sqrt { 3 } )$ be three points. The equation of the bisector of the angle PQR
(1) $\sqrt { 3 } x + y = 0$
(2) $x + \frac { \sqrt { 3 } } { 2 } y = 0$
(3) $\frac { \sqrt { 3 } } { 2 } x + y = 0$
(4) $x + \sqrt { 3 } y = 0$

If one of the lines of $m y ^ { 2 } + \left( 1 - m ^ { 2 } \right) x y - m x ^ { 2 } = 0$ is a bisector of the angle between the lines $x y = 0$, then $m$ is
(1) $- 1 / 2$
(2) - 2
(3) 1
(4) 2

The lines $L_{1}: y-x=0$ and $L_{2}: 2x+y=0$ 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 $R$. This question has Statement-1 and Statement-2. Of the four choices given after the statements, choose the one that best describes the two statements. Statement-1: The ratio $PR:RQ$ equals $2\sqrt{2}:\sqrt{5}$. Statement-2: In any triangle, bisector of an angle divides the triangle into two similar triangles.
(1) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(2) Statement-1 is true, Statement-2 is false.
(3) Statement-1 is false, Statement-2 is true.
(4) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

If the point $(1, a)$ lies between the straight lines $x + y = 1$ and $2(x+y) = 3$ then $a$ lies in interval
(1) $\left(\frac{3}{2}, \infty\right)$
(2) $\left(1, \frac{3}{2}\right)$
(3) $(-\infty, 0)$
(4) $\left(0, \frac{1}{2}\right)$

If the straight lines $x + 3 y = 4,3 x + y = 4$ and $x + y = 0$ form a triangle, then the triangle is
(1) scalene
(2) equilateral triangle
(3) isosceles
(4) right angled isosceles

If two vertices of a triangle are $(5, -1)$ and $(-2, 3)$ and its orthocentre is at $(0, 0)$, then the third vertex is
(1) $(4, -7)$
(2) $(-4, -7)$
(3) $(-4, 7)$
(4) $(4, 7)$

The point of intersection of the lines $\left( a ^ { 3 } + 3 \right) x + a y + a - 3 = 0$ and $\left( a ^ { 5 } + 2 \right) x + ( a + 2 ) y + 2 a + 3 = 0$ (a real) lies on the $y$-axis for
(1) no value of $a$
(2) more than two values of $a$
(3) exactly one value of $a$
(4) exactly two values of $a$

Let $L$ be the line $y = 2 x$, in the two dimensional plane. Statement 1: The image of the point $( 0,1 )$ in $L$ is the point $\left( \frac { 4 } { 5 } , \frac { 3 } { 5 } \right)$. Statement 2: The points $( 0,1 )$ and $\left( \frac { 4 } { 5 } , \frac { 3 } { 5 } \right)$ lie on opposite sides of the line $L$ and are at equal distance from it.
(1) Statement 1 is true, Statement 2 is false.
(2) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.
(3) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.
(4) Statement 1 is false, Statement 2 is true.

The line parallel to $x$-axis and passing through the point of intersection of lines $a x + 2 b y + 3 b = 0$ and $b x - 2 a y - 3 a = 0$, where $( a , b ) \neq ( 0,0 )$ is
(1) above $x$-axis at a distance $2/3$ from it
(2) above $x$-axis at a distance $3/2$ from it
(3) below $x$-axis at a distance $3/2$ from it
(4) below $x$-axis at a distance $2/3$ from it

Consider the straight lines $$\begin{aligned} & L _ { 1 } : x - y = 1 \\ & L _ { 2 } : x + y = 1 \\ & L _ { 3 } : 2 x + 2 y = 5 \\ & L _ { 4 } : 2 x - 2 y = 7 \end{aligned}$$ The correct statement is
(1) $L _ { 1 } \left\| L _ { 4 } , L _ { 2 } \right\| L _ { 3 } , L _ { 1 }$ intersect $L _ { 4 }$.
(2) $L _ { 1 } \perp L _ { 2 } , L _ { 1 } \| L _ { 3 } , L _ { 1 }$ intersect $L _ { 2 }$.
(3) $L _ { 1 } \perp L _ { 2 } , L _ { 2 } \| L _ { 3 } , L _ { 1 }$ intersect $L _ { 4 }$.
(4) $L _ { 1 } \perp L _ { 2 } , L _ { 1 } \perp L _ { 3 } , L _ { 2 }$ intersect $L _ { 4 }$.

A line is drawn through the point $(1,2)$ to meet the coordinate axes at $P$ and $Q$ such that it forms a triangle of area $\frac{9}{2}$ sq. units with the coordinate axes. The equation of the line $PQ$ is
(1) $x+2y=5$
(2) $3x+y=5$
(3) $x+2y=4$
(4) $2x+y=4$

If two vertical poles 20 m and 80 m high stand apart on a horizontal plane, then the height (in m) of the point of intersection of the lines joining the top of each pole to the foot of other is
(1) 16
(2) 18
(3) 50
(4) 15

A light ray emerging from the point source placed at $\mathrm { P } ( 1,3 )$ is reflected at a point Q in the axis of $x$. If the reflected ray passes through the point $R ( 6,7 )$, then the abscissa of $Q$ is:
(1) 1
(2) 3
(3) $\frac { 7 } { 2 }$
(4) $\frac { 5 } { 2 }$

Let $\theta _ { 1 }$ be the angle between two lines $2 x + 3 y + c _ { 1 } = 0$ and $- x + 5 y + c _ { 2 } = 0$ and $\theta _ { 2 }$ be the angle between two lines $2 x + 3 y + c _ { 1 } = 0$ and $- x + 5 y + c _ { 3 } = 0$, where $c _ { 1 } , c _ { 2 } , c _ { 3 }$ are any real numbers: Statement-1: If $c _ { 2 }$ and $c _ { 3 }$ are proportional, then $\theta _ { 1 } = \theta _ { 2 }$. Statement-2: $\theta _ { 1 } = \theta _ { 2 }$ for all $c _ { 2 }$ and $c _ { 3 }$.
(1) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation of Statement-1.
(2) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation of Statement-1.
(3) Statement-1 is false; Statement-2 is true.
(4) Statement-1 is true; Statement-2 is false.