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Two tangents drawn from a point $P$ touch a circle with center $O$ at points $Q$ and $R$. Points $A$ and $B$ lie on $PQ$ and $PR$, respectively, such that $AB$ is also a tangent to the same circle. If $\angle AOB = 50^\circ$, then $\angle APB$, in degrees, equals

In a circle with center $C$ and radius $6\sqrt{2}$ cm, $PQ$ and $SR$ are two parallel chords separated by one of the diameters. If $\angle PQC = 45^\circ$, and the ratio of the perpendicular distance of $PQ$ and $SR$ from $C$ is $3:2$, then the area, in sq. cm, of the quadrilateral $PQRS$ is

A circular plot of land is divided into two regions by a chord of length $10\sqrt{3}$ meters such that the chord subtends an angle of $120^\circ$ at the center. Then, the area, in square meters, of the smaller region is

Three circles of equal radii touch (but not cross) each other externally. Two other circles, X and Y, are drawn such that both touch (but not cross) each of the three previous circles. If the radius of X is more than that of Y, the ratio of the radii of X and Y is

In the XY-plane, the area, in sq. units, of the region defined by the inequalities $y \geq x+4$ and $-4 \leq x^{2}+y^{2}+4(x-y) \leq 0$ is

A rectangle with the largest possible area is drawn inside a semicircle of radius $2$ cm. Then, the ratio of the lengths of the largest to the smallest side of this rectangle is

Let $C$ be the circle $x^2 + y^2 + 4x - 6y - 3 = 0$ and $L$ be the locus of the point of intersection of a pair of tangents to $C$ with the angle between the two tangents equal to $60°$. Then, the point at which $L$ touches the line $x = 6$ is

In a triangle $ABC, AB = AC = 8$cm. $A$ circle drawn with $BC$ as diameter passes through $A$. Another circle drawn with center at $A$ passes through $B$ and $C$. Then the area, in sq. cm, of the overlapping region between the two circles is

All the vertices of a rectangle lie on a circle of radius $R$. If the perimeter of the rectangle is $P$, then the area of the rectangle is

The vertices of a triangle are $(0,0)$, and $(4,0)$ and $(3,9)$. The area of the circle passing through these three points is

Let $C1$ and $C2$ be concentric circles such that the diameter of $C1$ is $2 \mathrm{~cm}$ longer than that of $C2$. If a chord of $C1$ has length $6 \mathrm{~cm}$ and is a tangent of $C2$, then the diameter, in cm, of $C1$ is

Let $C$ be a circle of radius $5$ meters having center at $O$. Let $P Q$ be a chord of $C$ that passes through points $A$ and $B$ where $A$ is located $4$ meters north of $O$ and $B$ is located $3$ meters east of $O$. Then, the length of $P Q$, in meters, is nearest to

A circle is inscribed in a rhombus with diagonals $12 \mathrm{~cm}$ and $16 \mathrm{~cm}$. The ratio of the area of circle to the area of rhombus is

On a rectangular metal sheet of area $135 \mathrm{sq}$ in, a circle is painted such that the circle touches two opposite sides. If the are the sheet left unpainted is two-thirds of the painted area then the perimeter of the rectangle in inches is

Two circles, each of radius $4$ cm, touch externally. Each of these two circles is touched externally by a third circle. If these three circles have a common tangent, then the radius of the third circle, in cm, is

In a circle of radius $11$ cm, $CD$ is a diameter and $AB$ is a chord of length $20.5$ cm. If $AB$ and $CD$ intersect at a point $E$ inside the circle and $CE$ has length $7$ cm, then the difference of the lengths of $BE$ and $AE$, in cm, is

$AB$ is a diameter of a circle of radius $5 \mathrm{~cm}$. Let $P$ and $Q$ be two points on the circle so that the length of $PB$ is $6 \mathrm{~cm}$, and the length of $AP$ is twice that of $AQ$ . Then the length, in cm , of $QB$ is nearest to

A chord of length $5 \mathrm{~cm}$ subtends an angle of $60^{\circ}$ at the centre of a circle. The length, in cm , of a chord that subtends an angle of $120^{\circ}$ at the centre of the same circle is

From a rectangle $ABCD$ of area $768 \text{ sq cm}$, a semicircular part with diameter $AB$ and area $72\pi \text{ sq cm}$ is removed. The perimeter of the leftover portion, in cm, is

On a triangle $A B C$, a circle with diameter $B C$ is drawn, intersecting $A B$ and $A C$ at points $P$ and $Q$, respectively. If the lengths of $A B, A C$, and $C P$ are $30 \mathrm{~cm}, 25 \mathrm{~cm}$, and $20 \mathrm{~cm}$ respectively, then the length of $BQ$ , in cm , is

Let $ABCD$ be a rectangle inscribed in a circle of radius $13 \mathrm{~cm}$. Which one of the following pairs can represent, in cm , the possible length and breadth of $ABCD$ ?

In a circle, two parallel chords on the same side of a diameter have lengths $4$ cm and $6$ cm. If the distance between these chords is $1$ cm, then the radius of the circle, in cm, is

$ABCD$ is a quadrilateral inscribed in a circle with centre O. If $\angle COD = 120$ degrees and $\angle BAC = 30$ degrees, then the value of $\angle BCD$ (in degrees) is