Let $G$ be a circle of radius $R > 0$. Let $G _ { 1 } , G _ { 2 } , \ldots , G _ { n }$ be $n$ circles of equal radius $r > 0$. Suppose each of the $n$ circles $G _ { 1 } , G _ { 2 } , \ldots , G _ { n }$ touches the circle $G$ externally. Also, for $i = 1,2 , \ldots , n - 1$, the circle $G _ { i }$ touches $G _ { i + 1 }$ externally, and $G _ { n }$ touches $G _ { 1 }$ externally. Then, which of the following statements is/are TRUE ?
(A) If $n = 4$, then $( \sqrt { 2 } - 1 ) r < R$
(B) If $n = 5$, then $r < R$
(C) If $n = 8$, then $( \sqrt { 2 } - 1 ) r < R$
(D) If $n = 12$, then $\sqrt { 2 } ( \sqrt { 3 } + 1 ) r > R$

Consider the parabola $y ^ { 2 } = 4 x$. Let $S$ be the focus of the parabola. A pair of tangents drawn to the parabola from the point $P = ( - 2,1 )$ meet the parabola at $P _ { 1 }$ and $P _ { 2 }$. Let $Q _ { 1 }$ and $Q _ { 2 }$ be points on the lines $S P _ { 1 }$ and $S P _ { 2 }$ respectively such that $P Q _ { 1 }$ is perpendicular to $S P _ { 1 }$ and $P Q _ { 2 }$ is perpendicular to $S P _ { 2 }$. Then, which of the following is/are TRUE?
(A) $\quad S Q _ { 1 } = 2$
(B) $\quad Q _ { 1 } Q _ { 2 } = \frac { 3 \sqrt { 10 } } { 5 }$
(C) $\quad P Q _ { 1 } = 3$
(D) $\quad S Q _ { 2 } = 1$

Consider the ellipse
$$\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 3 } = 1$$
Let $H ( \alpha , 0 ) , 0 < \alpha < 2$, be a point. A straight line drawn through $H$ parallel to the $y$-axis crosses the ellipse and its auxiliary circle at points $E$ and $F$ respectively, in the first quadrant. The tangent to the ellipse at the point $E$ intersects the positive $x$-axis at a point $G$. Suppose the straight line joining $F$ and the origin makes an angle $\phi$ with the positive $x$-axis.
List-I (I) If $\phi = \frac { \pi } { 4 }$, then the area of the triangle $F G H$ is (II) If $\phi = \frac { \pi } { 3 }$, then the area of the triangle $F G H$ is (III) If $\phi = \frac { \pi } { 6 }$, then the area of the triangle $F G H$ is (IV) If $\phi = \frac { \pi } { 12 }$, then the area of the triangle $F G H$ is
List-II (P) $\frac { ( \sqrt { 3 } - 1 ) ^ { 4 } } { 8 }$ (Q) 1 (R) $\frac { 3 } { 4 }$ (S) $\frac { 1 } { 2 \sqrt { 3 } }$ (T) $\frac { 3 \sqrt { 3 } } { 2 }$
The correct option is:
(A) (I) → (R); (II) → (S); (III) → (Q); (IV) → (P)
(B) (I) → (R); (II) → (T); (III) → (S); (IV) → (P)
(C) (I) → (Q); (II) → (T); (III) → (S); (IV) → (P)
(D) (I) → (Q); (II) → (S); (III) → (Q); (IV) → (P)

Let $T _ { 1 }$ and $T _ { 2 }$ be two distinct common tangents to the ellipse $E : \frac { x ^ { 2 } } { 6 } + \frac { y ^ { 2 } } { 3 } = 1$ and the parabola $P : y ^ { 2 } = 12 x$. Suppose that the tangent $T _ { 1 }$ touches $P$ and $E$ at the points $A _ { 1 }$ and $A _ { 2 }$, respectively and the tangent $T _ { 2 }$ touches $P$ and $E$ at the points $A _ { 4 }$ and $A _ { 3 }$, respectively. Then which of the following statements is(are) true?
(A) The area of the quadrilateral $A _ { 1 } A _ { 2 } A _ { 3 } A _ { 4 }$ is 35 square units
(B) The area of the quadrilateral $A _ { 1 } A _ { 2 } A _ { 3 } A _ { 4 }$ is 36 square units
(C) The tangents $T _ { 1 }$ and $T _ { 2 }$ meet the $x$-axis at the point $( - 3,0 )$
(D) The tangents $T _ { 1 }$ and $T _ { 2 }$ meet the $x$-axis at the point $( - 6,0 )$

Let $A _ { 1 } , A _ { 2 } , A _ { 3 } , \ldots , A _ { 8 }$ be the vertices of a regular octagon that lie on a circle of radius 2. Let $P$ be a point on the circle and let $P A _ { i }$ denote the distance between the points $P$ and $A _ { i }$ for $i = 1,2 , \ldots , 8$. If $P$ varies over the circle, then the maximum value of the product $P A _ { 1 } \cdot P A _ { 2 } \cdots P A _ { 8 }$, is

Let $C _ { 1 }$ be the circle of radius 1 with center at the origin. Let $C _ { 2 }$ be the circle of radius $r$ with center at the point $A = ( 4,1 )$, where $1 < r < 3$. Two distinct common tangents $P Q$ and $S T$ of $C _ { 1 }$ and $C _ { 2 }$ are drawn. The tangent $P Q$ touches $C _ { 1 }$ at $P$ and $C _ { 2 }$ at $Q$. The tangent $S T$ touches $C _ { 1 }$ at $S$ and $C _ { 2 }$ at $T$. Mid points of the line segments $P Q$ and $S T$ are joined to form a line which meets the $x$-axis at a point $B$. If $A B = \sqrt { 5 }$, then the value of $r ^ { 2 }$ is

Consider the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$. Let $S ( p , q )$ be a point in the first quadrant such that $\frac { p ^ { 2 } } { 9 } + \frac { q ^ { 2 } } { 4 } > 1$. Two tangents are drawn from $S$ to the ellipse, of which one meets the ellipse at one end point of the minor axis and the other meets the ellipse at a point $T$ in the fourth quadrant. Let $R$ be the vertex of the ellipse with positive $x$-coordinate and $O$ be the center of the ellipse. If the area of the triangle $\triangle O R T$ is $\frac { 3 } { 2 }$, then which of the following options is correct?
(A) $q = 2 , p = 3 \sqrt { 3 }$
(B) $q = 2 , p = 4 \sqrt { 3 }$
(C) $q = 1 , p = 5 \sqrt { 3 }$
(D) $q = 1 , p = 6 \sqrt { 3 }$

Let $A _ { 1 } , B _ { 1 } , C _ { 1 }$ be three points in the $xy$-plane. Suppose that the lines $A _ { 1 } C _ { 1 }$ and $B _ { 1 } C _ { 1 }$ are tangents to the curve $y ^ { 2 } = 8 x$ at $A _ { 1 }$ and $B _ { 1 }$, respectively. If $O = ( 0,0 )$ and $C _ { 1 } = ( - 4,0 )$, then which of the following statements is (are) TRUE?
(A) The length of the line segment $OA _ { 1 }$ is $4 \sqrt { 3 }$
(B) The length of the line segment $A _ { 1 } B _ { 1 }$ is 16
(C) The orthocenter of the triangle $A _ { 1 } B _ { 1 } C _ { 1 }$ is $( 0,0 )$
(D) The orthocenter of the triangle $A _ { 1 } B _ { 1 } C _ { 1 }$ is $( 1,0 )$

A normal with slope $\frac { 1 } { \sqrt { 6 } }$ is drawn from the point $( 0 , - \alpha )$ to the parabola $x ^ { 2 } = - 4 a y$, where $a > 0$. Let $L$ be the line passing through $( 0 , - \alpha )$ and parallel to the directrix of the parabola. Suppose that $L$ intersects the parabola at two points $A$ and $B$. Let $r$ denote the length of the latus rectum and $s$ denote the square of the length of the line segment $AB$. If $r : s = 1 : 16$, then the value of $24 a$ is $\_\_\_\_$ .

Let the straight line $y = 2 x$ touch a circle with center $( 0 , \alpha ) , \alpha > 0$, and radius $r$ at a point $A _ { 1 }$. Let $B _ { 1 }$ be the point on the circle such that the line segment $A _ { 1 } B _ { 1 }$ is a diameter of the circle. Let $\alpha + r = 5 + \sqrt { 5 }$.
Match each entry in List-I to the correct entry in List-II.
List-I
(P) $\alpha$ equals
(Q) $r$ equals
(R) $A _ { 1 }$ equals
(S) $B _ { 1 }$ equals
List-II
(1) $( - 2,4 )$
(2) $\sqrt { 5 }$
(3) $( - 2,6 )$
(4) 5
(5) $( 2,4 )$
The correct option is
(A) $( \mathrm { P } ) \rightarrow ( 4 )$, $( \mathrm { Q } ) \rightarrow ( 2 )$, $( \mathrm { R } ) \rightarrow ( 1 )$, $( \mathrm { S } ) \rightarrow ( 3 )$
(B) $( \mathrm { P } ) \rightarrow ( 2 )$, $( \mathrm { Q } ) \rightarrow ( 4 )$, $( \mathrm { R } ) \rightarrow ( 1 )$, $( \mathrm { S } ) \rightarrow ( 3 )$
(C) $( \mathrm { P } ) \rightarrow ( 4 )$, $( \mathrm { Q } ) \rightarrow ( 2 )$, $( \mathrm { R } ) \rightarrow ( 5 )$, $( \mathrm { S } ) \rightarrow ( 3 )$
(D) $( \mathrm { P } ) \rightarrow ( 2 )$, $( \mathrm { Q } ) \rightarrow ( 4 )$, $( \mathrm { R } ) \rightarrow ( 3 )$, $( \mathrm { S } ) \rightarrow ( 5 )$

Let $S$ denote the locus of the point of intersection of the pair of lines
$$\begin{gathered} 4 x - 3 y = 12 \alpha \\ 4 \alpha x + 3 \alpha y = 12 \end{gathered}$$
where $\alpha$ varies over the set of non-zero real numbers. Let $T$ be the tangent to $S$ passing through the points $( p , 0 )$ and $( 0 , q ) , q > 0$, and parallel to the line $4 x - \frac { 3 } { \sqrt { 2 } } y = 0$.
Then the value of $p q$ is

(A)

$- 6 \sqrt { 2 }$

(B)

$- 3 \sqrt { 2 }$

(C)

$- 9 \sqrt { 2 }$

(D)

$- 12 \sqrt { 2 }$

Let $S$ denote the locus of the mid-points of those chords of the parabola $y ^ { 2 } = x$, such that the area of the region enclosed between the parabola and the chord is $\frac { 4 } { 3 }$. Let $\mathcal { R }$ denote the region lying in the first quadrant, enclosed by the parabola $y ^ { 2 } = x$, the curve $S$, and the lines $x = 1$ and $x = 4$.
Then which of the following statements is (are) TRUE?

(A)	$( 4 , \sqrt { 3 } ) \in S$
(B)	$( 5 , \sqrt { 2 } ) \in S$
(C)	Area of $\mathcal { R }$ is $\frac { 14 } { 3 } - 2 \sqrt { 3 }$
(D)	Area of $\mathcal { R }$ is $\frac { 14 } { 3 } - \sqrt { 3 }$

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 $9 \left| N _ { 1 } P \right| = 4 \left| N _ { 1 } R \right|$

Consider a family of circles which are passing through the point $( - 1,1 )$ and are tangent to $x -$ axis. If $( h , k )$ are the co-ordinates of the centre of the circles, then the set of values of $k$ is given by the interval
(1) $0 < \mathrm { k } < 1 / 2$
(2) $k \geq 1 / 2$
(3) $- 1 / 2 \leq k \leq 1 / 2$
(4) $k \leq 1 / 2$

The equation of a tangent to the parabola $y ^ { 2 } = 8 x$ is $y = x + 2$. The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is
(1) $( - 1,1 )$
(2) $( 0,2 )$
(3) $( 2,4 )$
(4) $( - 2,0 )$

The two circles $x^{2}+y^{2}=ax$ and $x^{2}+y^{2}=c^{2}$ $(c>0)$ touch each other if
(1) $|a|=c$
(2) $a=2c$
(3) $|a|=2c$
(4) $2|a|=c$

The area of triangle formed by the lines joining the vertex of the parabola, $x^{2} = 8y$, to the extremities of its latus rectum is
(1) 2
(2) 8
(3) 1
(4) 4

The equation of the circle passing through the point $( 1,2 )$ and through the points of intersection of $x ^ { 2 } + y ^ { 2 } - 4 x - 6 y - 21 = 0$ and $3 x + 4 y + 5 = 0$ is given by
(1) $x ^ { 2 } + y ^ { 2 } + 2 x + 2 y + 11 = 0$
(2) $x ^ { 2 } + y ^ { 2 } - 2 x + 2 y - 7 = 0$
(3) $x ^ { 2 } + y ^ { 2 } + 2 x - 2 y - 3 = 0$
(4) $x ^ { 2 } + y ^ { 2 } + 2 x + 2 y - 11 = 0$

If the line $y = m x + 1$ meets the circle $x ^ { 2 } + y ^ { 2 } + 3 x = 0$ in two points equidistant from and on opposite sides of $x$-axis, then
(1) $3 m + 2 = 0$
(2) $3 m - 2 = 0$
(3) $2 m + 3 = 0$
(4) $2 m - 3 = 0$

If $P_{1}$ and $P_{2}$ are two points on the ellipse $\frac{x^{2}}{4} + y^{2} = 1$ at which the tangents are parallel to the chord joining the points $(0, 1)$ and $(2, 0)$, then the distance between $P_{1}$ and $P_{2}$ is
(1) $2\sqrt{2}$
(2) $\sqrt{5}$
(3) $2\sqrt{3}$
(4) $\sqrt{10}$

Statement 1: $y = m x - \frac { 1 } { m }$ is always a tangent to the parabola, $y ^ { 2 } = - 4 x$ for all non-zero values of $m$. Statement 2: Every tangent to the parabola, $y ^ { 2 } = - 4 x$ will meet its axis at a point whose abscissa is nonnegative.
(1) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation of Statement 1.
(2) Statement 1 is false, Statement 2 is true.
(3) Statement 1 is true, Statement 2 is false.
(4) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1.

The number of common tangents of the circles given by $x ^ { 2 } + y ^ { 2 } - 8 x - 2 y + 1 = 0$ and $x ^ { 2 } + y ^ { 2 } + 6 x + 8 y = 0$ is
(1) one
(2) four
(3) two
(4) three

The equation of the circle passing through the point $(1,0)$ and $(0,1)$ and having the smallest radius is
(1) $x^{2}+y^{2}-2x-2y+1=0$
(2) $x^{2}+y^{2}+2x+2y-7=0$
(3) $x^{2}+y^{2}-x-y=0$
(4) $x^{2}+y^{2}+x+y-2=0$

The normal at $\left( 2 , \frac { 3 } { 2 } \right)$ to the ellipse, $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 3 } = 1$ touches a parabola, whose equation is
(1) $y ^ { 2 } = - 104 x$
(2) $y ^ { 2 } = 14 x$
(3) $y ^ { 2 } = 26 x$
(4) $y ^ { 2 } = - 14 x$

The length of the diameter of the circle which touches the $x$-axis at the point $(1,0)$ and passes through the point $(2,3)$ is
(1) $\frac{10}{3}$
(2) $\frac{3}{5}$
(3) $\frac{6}{5}$
(4) $\frac{5}{3}$