If one of the diameters of the circle $x ^ { 2 } + y ^ { 2 } - 10 x + 4 y + 13 = 0$ is a chord of another circle $C$, whose center is the point of intersection of the lines $2 x + 3 y = 12$ and $3 x - 2 y = 5$, then the radius of the circle $C$ is
(1) $\sqrt { 20 }$
(2) 4
(3) 6
(4) $3 \sqrt { 2 }$

Let $\left( 5 , \frac { a } { 4 } \right)$, be the circumcenter of a triangle with vertices $A ( a , - 2 ) , B ( a , 6 )$ and $C \left( \frac { a } { 4 } , - 2 \right)$. Let $\alpha$ denote the circumradius, $\beta$ denote the area and $\gamma$ denote the perimeter of the triangle. Then $\alpha + \beta + \gamma$ is
(1) 60
(2) 53
(3) 62
(4) 30

Let $A$ and $B$ be two finite sets with $m$ and $n$ elements respectively. The total number of subsets of the set $A$ is 56 more than the total number of subsets of $B$. Then the distance of the point $\mathrm { P } ( \mathrm { m } , \mathrm { n } )$ from the point $\mathrm { Q } ( - 2 , - 3 )$ is
(1) 10
(2) 6
(3) 4
(4) 8

If $x^2 - y^2 + 2hxy + 2gx + 2fy + c = 0$ is the locus of a point, which moves such that it is always equidistant from the lines $x + 2y + 7 = 0$ and $2x - y + 8 = 0$, then the value of $g + c + h - f$ equals
(1) 14
(2) 6
(3) 8
(4) 29

Let $C$ be a circle with radius $\sqrt { 10 }$ units and centre at the origin. Let the line $x + y = 2$ intersects the circle C at the points P and Q . Let MN be a chord of C of length 2 unit and slope - 1 . Then, a distance (in units) between the chord PQ and the chord MN is
(1) $3 - \sqrt { 2 }$
(2) $\sqrt { 2 } + 1$
(3) $\sqrt { 2 } - 1$
(4) $2 - \sqrt { 3 }$

Let the locus of the mid points of the chords of circle $x^2 + (y-1)^2 = 1$ drawn from the origin intersect the line $x + y = 1$ at $P$ and $Q$. Then, the length of $PQ$ is:
(1) $\frac{1}{\sqrt{2}}$
(2) $\sqrt{2}$
(3) $\frac{1}{2}$
(4) 1

A circle is inscribed in an equilateral triangle of side of length 12 . If the area and perimeter of any square inscribed in this circle are $m$ and $n$, respectively, then $m + n ^ { 2 }$ is equal to
(1) 408
(2) 414
(3) 396
(4) 312

Four distinct points $( 2 \mathrm { k } , 3 \mathrm { k } ) , ( 1,0 ) , ( 0,1 )$ and $( 0,0 )$ lie on a circle for $k$ equal to:
(1) $\frac { 2 } { 13 }$
(2) $\frac { 3 } { 13 }$
(3) $\frac { 5 } { 13 }$
(4) $\frac { 1 } { 13 }$

Let $A(\alpha, 0)$ and $B(0, \beta)$ be the points on the line $5x + 7y = 50$. Let the point $P$ divide the line segment $AB$ internally in the ratio $7:3$. Let $3x - 25 = 0$ be a directrix of the ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and the corresponding focus be $S$. If from $S$, the perpendicular on the $x$-axis passes through $P$, then the length of the latus rectum of $E$ is equal to
(1) $\frac{25}{3}$
(2) $\frac{32}{9}$
(3) $\frac{25}{9}$
(4) $\frac{32}{5}$

Let $A B C D$ and $A E F G$ be squares of side 4 and 2 units, respectively. The point $E$ is on the line segment AB and the point F is on the diagonal AC . Then the radius r of the circle passing through the point F and touching the line segments BC and CD satisfies:
(1) $r = 0$
(2) $2 r ^ { 2 } - 4 r + 1 = 0$
(3) $2 r ^ { 2 } - 8 r + 7 = 0$
(4) $r ^ { 2 } - 8 r + 8 = 0$

Let the circles $C _ { 1 } : ( x - \alpha ) ^ { 2 } + ( y - \beta ) ^ { 2 } = r _ { 1 } ^ { 2 }$ and $C _ { 2 } : ( x - 8 ) ^ { 2 } + \left( y - \frac { 15 } { 2 } \right) ^ { 2 } = r _ { 2 } ^ { 2 }$ touch each other externally at the point $( 6,6 )$. If the point $( 6,6 )$ divides the line segment joining the centres of the circles $C _ { 1 }$ and $C _ { 2 }$ internally in the ratio $2 : 1$, then $( \alpha + \beta ) + 4 \left( r _ { 1 } ^ { 2 } + r _ { 2 } ^ { 2 } \right)$ equals
(1) 125
(2) 130
(3) 110
(4) 145

If $\mathrm { P } ( 6,1 )$ be the orthocentre of the triangle whose vertices are $\mathrm { A } ( 5 , - 2 ) , \mathrm { B } ( 8,3 )$ and $\mathrm { C } ( \mathrm { h } , \mathrm { k } )$, then the point $C$ lies on the circle:
(1) $x ^ { 2 } + y ^ { 2 } - 61 = 0$
(2) $x ^ { 2 } + y ^ { 2 } - 52 = 0$
(3) $x ^ { 2 } + y ^ { 2 } - 65 = 0$
(4) $x ^ { 2 } + y ^ { 2 } - 74 = 0$

If the image of the point $( - 4,5 )$ in the line $x + 2 y = 2$ lies on the circle $( x + 4 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = r ^ { 2 }$, then r is equal to: (1) 2 (2) 3 (3) 1 (4) 4

Let $P$ be a point on the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$. Let the line passing through $P$ and parallel to $y$-axis meet the circle $x^2 + y^2 = 9$ at point $Q$ such that $P$ and $Q$ are on the same side of the $x$-axis. Then, the eccentricity of the locus of the point $R$ on $PQ$ such that $PR:RQ = 4:3$ as $P$ moves on the ellipse, is:
(1) $\frac{11}{19}$
(2) $\frac{13}{21}$
(3) $\frac{\sqrt{139}}{23}$
(4) $\frac{\sqrt{13}}{7}$

Let a variable line passing through the centre of the circle $x^2 + y^2 - 16x - 4y = 0$, meet the positive coordinate axes at the point $A$ and $B$. Then the minimum value of $OA + OB$, where $O$ is the origin, is equal to
(1) 12
(2) 18
(3) 20
(4) 24

Let $C$ be the circle of minimum area touching the parabola $y = 6 - x ^ { 2 }$ and the lines $y = \sqrt { 3 } | x |$. Then, which one of the following points lies on the circle $C$ ?
(1) $( 1,2 )$
(2) $( 1,1 )$
(3) $( 2,2 )$
(4) $( 2,4 )$

If the shortest distance of the parabola $y ^ { 2 } = 4 x$ from the centre of the circle $x ^ { 2 } + y ^ { 2 } - 4 x - 16 y + 64 = 0$ is $d$ , then $\mathrm { d } ^ { 2 }$ is equal to:
(1) 16
(2) 24
(3) 20
(4) 36

Let the circle $C _ { 1 } : x ^ { 2 } + y ^ { 2 } - 2 ( x + y ) + 1 = 0$ and $C _ { 2 }$ be a circle having centre at $( - 1,0 )$ and radius 2 . If the line of the common chord of $\mathrm { C } _ { 1 }$ and $\mathrm { C } _ { 2 }$ intersects the $y$-axis at the point P , then the square of the distance of P from the centre of $\mathrm { C } _ { 1 }$ is :
(1) 2
(2) 1
(3) 4
(4) 6

If the locus of the point, whose distances from the point $( 2,1 )$ and $( 1,3 )$ are in the ratio $5 : 4$, is $a x ^ { 2 } + b y ^ { 2 } + c x y + d x + e y + 170 = 0$, then the value of $a ^ { 2 } + 2 b + 3 c + 4 d + e$ is equal to:
(1) 37
(2) 437
(3) $- 27$
(4) 5

The length of the chord of the ellipse $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } = 1$, whose mid point is $\left( 1 , \frac { 2 } { 5 } \right)$, is equal to:
(1) $\frac { \sqrt { 1691 } } { 5 }$
(2) $\frac { \sqrt { 2009 } } { 5 }$
(3) $\frac { \sqrt { 1741 } } { 5 }$
(4) $\frac { \sqrt { 1541 } } { 5 }$

If the sum of squares of all real values of $\alpha$, for which the lines $2 x - y + 3 = 0,6 x + 3 y + 1 = 0$ and $\alpha x + 2 y - 2 = 0$ do not form a triangle is $p$, then the greatest integer less than or equal to $p$ is $\_\_\_\_$ .

Let $A , B$ and $C$ be three points on the parabola $y ^ { 2 } = 6 x$ and let the line segment $A B$ meet the line $L$ through $C$ parallel to the $x$-axis at the point $D$. Let $M$ and $N$ respectively be the feet of the perpendiculars from $A$ and $B$ on $L$. Then $\left( \frac { A M \cdot B N } { C D } \right) ^ { 2 }$ is equal to $\_\_\_\_$

Equations of two diameters of a circle are $2 x - 3 y = 5$ and $3 x - 4 y = 7$. The line joining the points $\left( - \frac { 22 } { 7 } , - 4 \right)$ and $\left( - \frac { 1 } { 7 } , 3 \right)$ intersects the circle at only one point $P ( \alpha , \beta )$. Then $17 \beta - \alpha$ is equal to

Let $\alpha = \sum_{k=0}^{n} \frac{\binom{n}{k}^2}{k+1}$ and $\beta = \sum_{k=0}^{n-1} \frac{\binom{n}{k}\binom{n}{k+1}}{k+2}$. If $5\alpha = 6\beta$, then $n$ equals $\underline{\hspace{1cm}}$.

Let a line perpendicular to the line $2 x - y = 10$ touch the parabola $y ^ { 2 } = 4 ( x - 9 )$ at the point $P$. The distance of the point $P$ from the centre of the circle $x ^ { 2 } + y ^ { 2 } - 14 x - 8 y + 56 = 0$ is $\_\_\_\_$