Consider the parabola $C: y^2 = 4x$ and the straight line $L: y = x + 2$. Let $P$ be a variable point on $L$. Draw the two tangents from $P$ to $C$ and let $Q_1$ and $Q_2$ denote the two points of contact on $C$. Let $Q$ be the mid-point of the line segment joining $Q_1$ and $Q_2$. Find the locus of $Q$ as $P$ moves along $L$.

Let $y = x + c _ { 1 } , y = x + c _ { 2 }$ be the two tangents to the ellipse $x ^ { 2 } + 4 y ^ { 2 } = 1$. What is the value of $\left| c _ { 1 } - c _ { 2 } \right|$?
(A) $\sqrt { 2 }$
(B) $\sqrt { 5 }$
(C) $\frac { \sqrt { 5 } } { 2 }$
(D) 1

Consider a triangle with vertices $( 0,0 ) , ( 1,2 )$ and $( - 4,2 )$. Let $A$ be the area of the triangle and $B$ be the area of the circumcircle of the triangle. Then $\frac { B } { A }$ equals
(A) $\frac { \pi } { 2 }$.
(B) $\frac { 5 \pi } { 4 }$.
(C) $\frac { 3 } { \sqrt { 2 } } \pi$.
(D) $2 \pi$.

For real numbers $a , b , c , d , a ^ { \prime } , b ^ { \prime } , c ^ { \prime } , d ^ { \prime }$, consider the system of equations $$\begin{aligned} a x ^ { 2 } + a y ^ { 2 } + b x + c y + d & = 0 \\ a ^ { \prime } x ^ { 2 } + a ^ { \prime } y ^ { 2 } + b ^ { \prime } x + c ^ { \prime } y + d ^ { \prime } & = 0 \end{aligned}$$ If $S$ denotes the set of all real solutions $( x , y )$ of the above system of equations, then the number of elements in $S$ can never be
(A) 0.
(B) 1.
(C) 2.
(D) 3.

The angle subtended at the origin by the common chord of the circles $x^2 + y^2 - 6x - 6y = 0$ and $x^2 + y^2 = 36$ is
(A) $\pi/2$
(B) $\pi/4$
(C) $\pi/3$
(D) $2\pi/3$

Consider points of the form $\left(n, n^k\right)$, where $n$ and $k$ are integers with $n \geq 0$, $k \geq 1$. How many such points are strictly inside the circle of radius 10 with centre at the origin?
(A) 11
(B) 12
(C) 15
(D) 17

In the adjoining figure, $C$ is the centre of the circle drawn, $A, F, E$ lie on the circle and $BCDF$ is a rectangle. If $\frac{DE}{AB} = 2$, then $\frac{FE}{FA}$ equals
(A) $\sqrt{\frac{3}{2}}$
(B) $\sqrt{2}$
(C) $\sqrt{\frac{5}{2}}$
(D) $\sqrt{3}$

Consider a circle with centre $O$. Two chords $A B$ and $C D$ extended intersect at a point $P$ outside the circle. If $\angle A O C = 43 ^ { \circ }$ and $\angle B P D = 18 ^ { \circ }$, then the value of $\angle B O D$ is
(a) $36 ^ { \circ }$.
(B) $29 ^ { \circ }$.
(C) $7 ^ { \circ }$.
(D) $25 ^ { \circ }$.

A triangle $A B C$ has a fixed base $B C$. If $A B : A C = 1 : 2$, then the locus of the vertex $A$ is
(a) a circle whose centre is the midpoint of $B C$.
(b) a circle whose centre is on the line $B C$ but not the midpoint of $B C$.
(c) a straight line.
(d) none of the above.

6. Determine the equation of the spherical surface $S$, with centre on the line $r: \left\{ \begin{array}{l} x = t \\ y = t \\ z = t \end{array} \right. t \in \mathbb{R}$ tangent to the plane $\pi: 3x - y - 2z + 14 = 0$ at the point $T(-4, 0, 1)$.

2. Consider the spherical surface with equation $( x - 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } + z ^ { 2 } = 1$ and the plane $\pi$ with equation $x - 2 y - 2 z + d = 0$. Discuss, as the real parameter $d$ varies, whether the plane $\pi$ is secant, tangent or external to the spherical surface. Determine the value of the parameter $d$ so that $\pi$ divides the sphere into two equal parts.

5. The number of common tangents to the circles $x 2 + y 2 = 4$ and $x 2 + y 2 - 6 x - y = 24$ is :
(A) 0
(B) 1
(C) 3
(D) 4

5. Let $\mathrm { T } 1 , \mathrm {~T} 2$ be two tangents drawn from $( - 2,0 )$ onto the circle $\mathrm { C } : \mathrm { x } 2 + \mathrm { y } 2 = 1$. Determine the circles touching C and having T1, T2 as their pair of tangents. Further, find the equations of all possible common tangents to these circles, when taken two at a time.

6. Consider the family of circles $x 2 + y 2 = r 2,2 < r < 5$. If in the first quadrant, the common tangent to a circle of this family and the ellipse $4 \times 2 + 25 y 2 = 100$ meets the coordinate axes at A and B , then find the equation of the locus of the mid point of AB .

16. If two distinct chords, drawn from the point ( $p , q$ ) on the circle $x 2 + y 2 = p x + q y$ (where $p q { } ^ { 1 } 0$ ) are bisected by the $x$-axis, then :
(A) $\mathrm { p } 2 = \mathrm { q } 2$
(B) $p 2 = 8 q 2$
(C) $p 2 < 8 q 2$
(D) $p 2 > 8 q 2$

26. Let L1 be a straight line passing through the origin and L2 be the straight line $x + y =$ 1. If the intercepts made by the circle $x 2 + y 2 - x + 3 y = 0$ on L1 and L2 are equal, then which of the following equations can represent L1?
(A) $x + y = 0$
(B) $x - y = 0$
(B) $x + 7 y = 0$
(D) $x - 7 y = 0$

30. On the ellipse $4 x 2 + 9 y 2 = 1$, the points at which the tangents are parallel to the line $8 \mathrm { x } = 9 \mathrm { y }$ are :
(A) $( 2 / 5,1 / 5 )$
B) $( - 2 / 5,1 / 5 )$
(C) $( - 2 / 5 , - 1 / 5 )$
(D) $( 2 / 5 , - 1 / 5 )$

9. If the system of equations $x - k y - z = 0 , k x - y - z = 0 , x + y - z = 0$ has a non-zero solution, then possible values of $k$ are :
(A) $- 1,2$
(B) 1,2
(C) 0,1
(D) $- 1,1$

20. If $f ( x ) = \left\{ \begin{array} { c c } e ^ { \cos x } \sin x & \text { for } | x | \leq 2 , \\ 2 & \text { otherwise, } \end{array} \right.$ then $\int _ { - 2 } ^ { 3 } f ( x ) d x =$
(A) 0
(B) 1
(C) 2
(D) 3

23. Let $f ( x ) = \int \operatorname { ex } ( x - 1 ) ( x - 2 ) d x$. Then $f$ decreases in the interval :
(A) $( \infty , - 2 )$
(B) $( - 2 , - 1 )$
(C) $( 1,2 )$
(D) $( 2 , + \infty )$

4. Let $2 \times 2 + \underset { 2 } { 2 } - 3 \times y = 0$ be the equation of a pair of tangents drawn from the origin O to a circle of radius 3 with centre in the first quadrant. If A is one of the points of contact, find the length of OA .

9. Let Cl and C 2 be two circles with C 2 lying inside Cl . A circle C lying inside C1touches C1 internally and C2 externally. Identify the locus of the centre of C.

10. The equation of the common tangent touching the circle $( x - 3 ) 2 + y 2 = 9$ and the parabola $y 2 = 4 x$ above the $x$-axis is :
(A) $\sqrt { } 3 y = 3 x + 1$
(B) $\sqrt { } 3 y = - ( x + 3 )$
(C) $\sqrt { } 3 y = x + 3$
(D) $\sqrt { } 3 y = - ( 3 x + 1 )$

12. Let $A B$ be a chord of the circle $x 2 + y 2 = r 2$ subtending a right angle at the centre. Then the locus of the centroid of the triangle PAB as P moves on the circle is:
(A) A parabola
(B) A circle
(C) An ellipse
(D) A pair of straight lines

14. The equation of the directrix of the parabola $y 2 + 4 y + 4 x + 2 = 0$ is:
(A) $x = - 1$
(B) $x = 1$
(C) $x = - 3 / 2$
(D) $x = 3 / 2$