Geometry

In a circle with center O and radius 1 cm, an arc AB makes an angle 60 degrees at O. Let R be the region bounded by the radii OA, OB and the arc AB. If C and D are two points
on OA and OB, respectively, such that OC = OD and the area of triangle OCD is half that of R, then the length of OC, in cm, is

1. (π/3√3)^1/2
2. (π/4)^1/2
3. (π/6)^1/2
4. (π/4√3)^1/2

In a parallelogram ABCD of area 72 sq cm, the sides CD and AD have lengths 9 cm and 16 cm, respectively. Let P be a point on CD such that AP is perpendicular to CD. Then the area, in sq cm, of triangle APD is

1. 18√3
2. 24√3
3. 32√3
4. 12√3

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

1. 24, 10
2. 25, 9
3. 24, 12
4. 25, 10

Points E, F, G, H lie on the sides AB, BC, CD, and DA, respectively, of a square ABCD. If EFGH is also a square whose area is 62.5% of that of ABCD and CG is longer than EB, then the ratio of length of EB to that of CG is

1. 2 : 5
2. 4 : 9
3. 3 : 8
4. 1 : 3

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

1. √12
2. √14
3. √13
4. √11

A circle is inscribed in a given square and another circle is circumscribed about the square. What is the ratio of the area of the inscribed circle to that of the circumscribed circle?

1. 2 : 3
2. 3 : 4
3. 1 : 2
4. 1 : 4

A semicircle is drawn with AB as its diameter. From C, a point on AB, a line perpendicular to AB is drawn meeting the circumference of the semi-circle at D. Given that AC = 2 cm and CD = 6 cm, the area of the semicircle (in sq. cm) will be:

1. 32 π
2. 40.5 π
3. 50 π
4. 81 π

Consider obtuse angled triangles with sides 8 cm, 15 cm and x cm. If x is an integer then how many such triangles exist?

1. 5
2. 10
3. 15
4. 21

If in the figure below, angle XYZ = 90° and the length of the arc XZ = 10π, then the area of the sector XYZ is

1. 10π
2. 25π
3. 100π
4. None of the above

AB is a chord of a circle. The length of AB is 24 cm. P is the midpoint of AB. Perpendiculars from P on either side of the chord meets the circle at M and N respectively. If PM < PN and PM = 8 cm. then what will be the length of PN?

1. 17 cm
2. 18 cm
3. 19 cm
4. 20 cm
5. 21 cm

AB, CD and EF are three parallel lines, in that order. Let d₁ and d₂ be the distances from CD to AB and EF respectively. d₁ and d₂ are integers, where d₁ : d₂ = 2 : 1. P is a point on AB, Q and S are points on CD and R is a point on EF. If the area of the quadrilateral PQRS is 30 square units, what is the value of QR when value of SR is the least?

1. slightly less than 10 units
2. 10 units
3. slightly greater than 10 units
4. slightly less than 20 units
5. slightly greater than 20 units

The points (2, 5) and (6, 3) are two end points of a diagonal of a rectangle. If the other diagonal has the equation y = 3x + c, then c is

1. -5
2. -6
3. -7
4. -8