1
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In a circle with center O and radius 1 cm

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

- (π/3√3)
^{1/2} - (π/4)
^{1/2} - (π/6)
^{1/2} - (π/4√3)
^{1/2}

2
###
In a parallelogram ABCD of area 72 sq cm, the sides CD and AD have lengths 9 cm and 16 cm

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

- 18√3
- 24√3
- 32√3
- 12√3

3
###
Let ABCD be a rectangle inscribed in a circle of radius 13 cm

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?

- 24, 10
- 25, 9
- 24, 12
- 25, 10

4
###
Points E, F, G, H lie on the sides AB, BC, CD, and DA

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

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

5
###
In a circle, two parallel chords on the same side of a diameter

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

- √12
- √14
- √13
- √11

6
###
A circle is inscribed in a given square and another circle is circumscribed about the square

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?

- 2 : 3
- 3 : 4
- 1 : 2
- 1 : 4

7
###
A semicircle is drawn with AB as its diameter. From C, a point on AB, a line perpendicular

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:

- 32 π
- 40.5 π
- 50 π
- 81 π

8
###
Consider obtuse angled triangles with sides 8 cm, 15 cm and x cm

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

- 5
- 10
- 15
- 21

9
###
If in the figure below, angle XYZ = 90° and the length of the arc XZ = 10π

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

- 10π
- 25π
- 100π
- None of the above

10
###
AB is a chord of a circle. The length of AB is 24 cm

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?

- 17 cm
- 18 cm
- 19 cm
- 20 cm
- 21 cm

11
###
AB, CD and EF are three parallel lines, in that order

AB, CD and EF are three parallel lines, in that order. Let d_{1} and d_{2} be the distances from CD to AB and EF respectively. d_{1} and d_{2} are integers, where d_{1} : d_{2} = 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?

- slightly less than 10 units
- 10 units
- slightly greater than 10 units
- slightly less than 20 units
- slightly greater than 20 units

12
###
The points (2, 5) and (6, 3) are two end points of a diagonal of a rectangle

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

- -5
- -6
- -7
- -8