Let a line with slope of $60 ^ { \circ }$ be drawn through the focus $F$ of the parabola $y ^ { 2 } = 8 ( x + 2 )$. If the two points of intersection of the line with the parabola are $A$ and $B$ and the perpendicular bisector of the chord $A B$ intersects the $x$-axis at the point $P$, then the length of the segment PF is (a) $16 / 3$ (b) $8 / 3$ (c) $16 \sqrt{3} / 3$ (d) $8 \sqrt{3}$
Let $C _ { 1 } , C _ { 2 }$ and $C _ { 3 }$ be three circles lying in the same quadrant, each touching both the axes. Suppose also that $C _ { 1 }$ touches $C _ { 2 }$ and $C _ { 2 }$ touches $C _ { 3 }$. If the area of the smallest circle is 1 unit, then area of the largest circle is (a) $\{ ( \sqrt{2} + 1 ) / ( \sqrt{2} - 1 ) \} ^ { 4 }$ (b) $( 1 + \sqrt{2} ) ^ { 2 }$ (c) $( 2 + \sqrt{2} ) ^ { 2 }$ (d) $2 ^ { 4 }$
Let $A$ be the fixed point $(0,4)$ and $B$ be a moving point $(2t, 0)$. Let $M$ be the mid-point of $AB$ and let the perpendicular bisector of $AB$ meet the $y$-axis at $R$. The locus of the mid-point $P$ of $MR$ is (A) $y + x ^ { 2 } = 2$ (B) $x ^ { 2 } + ( y - 2 ) ^ { 2 } = 1 / 4$ (C) $( y - 2 ) ^ { 2 } - x ^ { 2 } = 1 / 4$ (D) none of the above
Let $n$ be a positive integer. Consider a square $S$ of side $2n$ units. 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) $8n - 2$
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) $4/\sqrt{3}$
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) $25^\circ$
Let $P$ be a point on the ellipse $x^2 + 4y^2 = 4$ which does not lie on the axes. If the normal at the point $P$ intersects the major and minor axes at $C$ and $D$ respectively, then the ratio $PC : PD$ equals (A) 2 (B) $1/2$ (C) 4 (D) $1/4$
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
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) $r \left( 1 + \operatorname{cosec} \frac{\pi}{n} \right)$
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.
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.
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 $n \geq 3$ 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) $r \left( 1 + \operatorname { cosec } \frac { \pi } { n } \right)$.
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) $r \left( 1 + \operatorname { cosec } \frac { \pi } { n } \right)$.
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.
The minimum area of the triangle formed by any tangent to the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ and the coordinate axes is (A) $a b$ (B) $\frac { a ^ { 2 } + b ^ { 2 } } { 2 }$ (C) $\frac { ( a + b ) ^ { 2 } } { 2 }$ (D) $\frac { a ^ { 2 } + a b + b ^ { 2 } } { 3 }$