Let $a$ be a real number. The number of distinct solutions $(x, y)$ of the system of equations $(x - a)^2 + y^2 = 1$ and $x^2 = y^2$, can only be
(A) $0, 1, 2, 3, 4$ or 5
(B) 0, 1 or 3
(C) $0, 1, 2$ or 4
(D) $0, 2, 3$, or 4

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

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

Let $P$ be a variable point on a circle $C$ and $Q$ be a fixed point outside $C$. If $R$ is the mid-point of the line segment $PQ$, then the locus of $R$ is
(A) a circle
(B) an ellipse
(C) a line segment
(D) segment of a parabola

A circle of radius $r$ is inscribed in a circular sector. The chord of the sector has length $a$. If the circle touches the chord and the two radii of the sector, find the relation between $a$ and $r$.
(A) $a = 8r/5$ (B) $a = 5r/8$ (C) $a = 4r/3$ (D) $a = 3r/4$

An isosceles triangle with base 6 cms. and base angles $30 ^ { \circ }$ each is inscribed in a circle. A second circle, which is situated outside the triangle, touches the first circle and also touches the base of the triangle at its midpoint. Find its radius.

If a circle intersects the hyperbola $y = 1 / x$ at four distinct points $\left( x _ { i } , y _ { i } \right) , i = 1,2,3,4$, then prove that $x _ { 1 } x _ { 2 } = y _ { 3 } y _ { 4 }$.

An isosceles triangle with base 6 cms. and base angles $30 ^ { \circ }$ each is inscribed in a circle. A second circle, which is situated outside the triangle, touches the first circle and also touches the base of the triangle at its midpoint. Find its radius.

If a circle intersects the hyperbola $y = 1 / x$ at four distinct points $\left( x _ { i } , y _ { i } \right) , i = 1,2,3,4$, then prove that $x _ { 1 } x _ { 2 } = y _ { 3 } y _ { 4 }$.

Let $a$ be a real number. The number of distinct solutions $( x , y )$ of the system of equations $( x - a ) ^ { 2 } + y ^ { 2 } = 1$ and $x ^ { 2 } = y ^ { 2 }$, can only be
(a) $0,1,2,3,4$ or 5
(b) 0, 1 or 3
(c) 0, 1, 2 or 4
(d) $0,2,3$, or 4

Let $a$ be a real number. The number of distinct solutions $( x , y )$ of the system of equations $( x - a ) ^ { 2 } + y ^ { 2 } = 1$ and $x ^ { 2 } = y ^ { 2 }$, can only be
(a) $0,1,2,3,4$ or 5
(b) 0, 1 or 3
(c) 0, 1, 2 or 4
(d) $0,2,3$, or 4

Let $P$ be a variable point on a circle $C$ and $Q$ be a fixed point outside $C$. If $R$ is the mid-point of the line segment $P Q$, then the locus of $R$ is
(a) a circle
(b) an ellipse
(c) a line segment
(d) segment of a parabola.

Let $P$ be a variable point on a circle $C$ and $Q$ be a fixed point outside $C$. If $R$ is the mid-point of the line segment $P Q$, then the locus of $R$ is
(a) a circle
(b) an ellipse
(c) a line segment
(d) segment of a parabola.

Let $n$ be a positive integer. Consider a square $S$ of side $2n$ units with sides parallel to the coordinate axes. Divide $S$ into $4n^2$ unit squares by drawing $2n-1$ horizontal and $2n-1$ vertical lines one unit apart. A circle of diameter $2n-1$ is drawn with its centre at the intersection of the two diagonals of the square $S$. How many of these unit squares contain a portion of the circumference of the circle?
(A) $4n - 2$
(B) $4n$
(C) $8n - 4$
(D) $8n - 2$

Let $n$ be a positive integer. Consider a square $S$ of side $2n$ units with sides parallel to the coordinate axes. Divide $S$ into $4 n ^ { 2 }$ unit squares by drawing $2n - 1$ horizontal and $2n - 1$ vertical lines one unit apart. A circle of diameter $2n - 1$ is drawn with its centre at the intersection of the two diagonals of the square $S$. How many of these unit squares contain a portion of the circumference of the circle?
(A) $4 n - 2$
(B) $4 n$
(C) $8 n - 4$
(D) $8 n - 2$

An isosceles triangle with base 6 cms. and base angles $30^\circ$ each is inscribed in a circle. A second circle touches the first circle and also touches the base of the triangle at its midpoint. If the second circle is situated outside the triangle, then its radius (in cms.) is
(A) $3\sqrt{3}/2$
(B) $\sqrt{3}/2$
(C) $\sqrt{3}$
(D) $4/\sqrt{3}$

An isosceles triangle with base 6 cms. and base angles $30 ^ { \circ }$ each is inscribed in a circle. A second circle touches the first circle and also touches the base of the triangle at its midpoint. If the second circle is situated outside the triangle, then its radius (in cms.) is
(A) $3 \sqrt { 3 } / 2$
(B) $\sqrt { 3 } / 2$
(C) $\sqrt { 3 }$
(D) $4 / \sqrt { 3 }$

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

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 $ABC$ has a fixed base $BC$. If $AB : AC = 1 : 2$, then the locus of the vertex $A$ is
(A) a circle whose centre is the midpoint of $BC$
(B) a circle whose centre is on the line $BC$ but not the midpoint of $BC$
(C) a straight line
(D) none of the above

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

Let $P$ be a variable point on a circle $C$ and $Q$ be a fixed point outside $C$. If $R$ is the mid-point of the line segment $PQ$, then the locus of $R$ is
(A) a circle
(B) an ellipse
(C) a line segment
(D) segment of a parabola

Let $P$ be a variable point on a circle $C$ and $Q$ be a fixed point outside $C$. If $R$ is the mid-point of the line segment $P Q$, then the locus of $R$ is
(A) a circle
(B) an ellipse
(C) a line segment
(D) segment of a parabola

Let $n \geq 3$ be an integer. Assume that inside a big circle, exactly $n$ small circles of radius $r$ can be drawn so that each small circle touches the big circle and also touches both its adjacent small circles. Then, the radius of the big circle is
(A) $r \operatorname{cosec} \frac{\pi}{n}$
(B) $r \left( 1 + \operatorname{cosec} \frac{2\pi}{n} \right)$
(C) $r \left( 1 + \operatorname{cosec} \frac{\pi}{2n} \right)$
(D) $r \left( 1 + \operatorname{cosec} \frac{\pi}{n} \right)$

Let $n \geq 3$ be an integer. Assume that inside a big circle, exactly $n$ small circles of radius $r$ can be drawn so that each small circle touches the big circle and also touches both its adjacent small circles. Then, the radius of the big circle is
(A) $r \operatorname { cosec } \frac { \pi } { n }$
(B) $r \left( 1 + \operatorname { cosec } \frac { 2 \pi } { n } \right)$
(C) $r \left( 1 + \operatorname { cosec } \frac { \pi } { 2 n } \right)$
(D) $r \left( 1 + \operatorname { cosec } \frac { \pi } { n } \right)$