1. All the vertices of a rectangle lie on a circle of radius \( R \). If one side of the rectangle is “a”, then its area is
a) \( a\sqrt{4R^2 – a^2} \)
b) \( \frac{a}{2}\sqrt{4R^2 – a^2} \)
c) \( \frac{a}{2}\sqrt{R^2 – a^2} \)
d) \( a\sqrt{R^2 – a^2} \)
2. The medians of triangle ABC intersect at O. If the area of triangle ABC is 72 sq cm, then the area (in sq cm) of one of the six small triangles formed is
a) 18
b) 24
c) 12
d) 9
3. In triangle ABC, AB=AC=6 cm and a circle with BC as diameter passes through A. Another circle with Centre A and radius 6 cm is drawn. The area (in sq cm) of their overlapping region is
a) \( 9\pi \)
b) \( 18\pi \)
c) \( 12\pi \)
d) \( 6\pi \)
4. A rhombus has perimeter 52 m and area 120 sq m. The cost of laying wire along both diagonals at ₹100 per m is
a) 5200
b) 4800
c) 5000
d) 5400
5. If the area of a rhombus is 24 sq cm and one diagonal is 6 cm, then the other diagonal (in cm) is
a) 6
b) 10
c) 8
d) 12
6. An equilateral triangle of side 0.02 m inscribes a square of maximum possible area. That square again inscribes an equilateral triangle of same side as the square, and so on. The area (in sq m) of the innermost square after one such repetition is
a) 0.0001
b) 0.000075
c) 0.00005
d) 0.0002
7. three equal circles of radius 6 cm touch each other externally. A fourth circle is drawn touching all three externally. What is the radius (in cm, nearest whole number) of the fourth circle?
(a) 1
(b) 2
(c) 3
(d) 4
8. A square of maximum area is inscribed in a semicircle of radius \( r \). The maximum area of the square is
a) \( r^2 \)
b) \( 2r^2 \)
c) \( \frac{3}{2} r^2 \)
d) \( \frac{4}{5} r^2 \)
9. The wheels of bicycles A and B have radii 25 cm and 35 cm respectively. A makes 4000 more revolutions than B while covering a distance D. The value of D (in meters) is
a) \( 8800\pi \)
b) \( 14000\pi \)
c) \( 7000\pi \)
d) \( 5600\pi \)
10. In triangle ABC, AB = AC and area = 36 sq cm. If centroid, circumcenter and orthocenter are collinear, then triangle ABC is
a) scalene
b) obtuse
c) equilateral
d) right angled
11. A right prism has a regular hexagonal base of side a and height h. If doubling the height increases total surface area by 60%, then the ratio a : h is
a) 1:1
b) 2:1
c) 3:2
d) 4:1
12. A regular polygon has n sides. If its circumradius equals twice its apothem, then n equals
a) 4
b) 6
c) 8
d) 12
13. The circumradius of a regular hexagon of side a is R and its inradius is r. Then R : r equals
a) 1 : 1
b) \( 2 : \sqrt{3} \)
c) \( \sqrt{3} : 2 \)
d) 3 : 2
14. A regular hexagon is inscribed in a circle. The ratio of the area of the hexagon to the area of the circle is
a) \( \frac{3\sqrt{3}}{2\pi} \)
b) \( \frac{\sqrt{3}}{\pi} \)
c) \( \frac{2}{\pi} \)
d) \( \frac{3}{\pi} \)
15. In square ABCD of side a, points E, F, G, H are midpoints. The ratio of area of square EFGH to area of ABCD is
a) 1:2
b) 1:3
c) 1:4
d) 2:3
16. Squares are formed successively by joining midpoints. If S₁ has side a, the sum of the areas of all such infinite squares equals
a) \( 2a^2 \)
b) \( a^2 \)
c) \( 3a^2 \)
d) \( 4a^2 \)
17. A square of side b is divided into mmm smaller squares. From each smaller square, the largest possible circle is cut. In another process, one largest possible circle is cut from the original square. What is the ratio of total scrap area in the first process to that in the second?
(a) m : 1
(b) 1 : m
(c) \( \pi : 4 \)
(d) 1 : 1
18. A right-angled triangle is cut by a line parallel to its hypotenuse, forming a smaller similar triangle. The hypotenuse is reduced by 20%. If the original area was 50 cm², what is the area of the smaller triangle?
(a) 40
(b) 22
(c) 32
(d) 30
19. In a circle of radius 7 cm, a tangent at point P makes an angle of 60° with chord PQ. If PQ = 7 cm, what is the approximate perimeter of triangle OPQ (O is centre)?
(a) 21 cm
(b) 24 cm
(c) 28 cm
(d) 30 cm
20. A rectangle is drawn such that both sides are independently and uniformly chosen from (0, b). What is the probability that its diagonal exceeds b?
(a) 25%
(b) 40%
(c) 21%
(d) 50%
21. In parallelogram OABC, O is (0,0) and A is (6,0). The area is 18 sq units and point C lies on the line x = 3. Find coordinates of B.
(a) (6,3)
(b) (3,6)
(c) (9,6)
(d) (9,3)
22. An equilateral triangle has area \( 48\sqrt{3} \) cm². A point P inside the triangle is such that the areas of triangles APD and BPD are each \( 12\sqrt{3} \) cm². What is the area (in cm²) of triangle CPD?
(a) 12
(b) 24
(c) 36
(d) 48
23. A circle of radius 3 cm is drawn on 1 cm × 1 cm grid paper. Only complete squares inside the circle are counted. If the true area exceeds the counted area by d, what is the maximum possible value of d (approx.)?
(a) 10
(b) 8
(c) 12
(d) 7
24. Under the same setup as above, what is the minimum possible value of d (approx)?
(a) 5
(b) 7
(c) 9
(d) 4
25. The circumference of a cylinder is 4 m and height is 15 m. An insect climbs in a spiral such that one complete round increases height by 3 m. What is the total distance travelled?
(a) 20 m
(b) 25 m
(c) 30 m
(d) 35 m
26. In triangle XYZ, XY = 13 cm, XZ = 15 cm. The altitude from X to YZ is 12 cm. What is the radius of the circumcircle (in cm)?
(a) 8
(b) 6.5
(c) 7.5
(d) 9
27. A square tin sheet of side 16 inches is converted into an open box by cutting equal squares of side x inches from each corner and folding up the sides. If x is an integer, what value of x maximizes the volume?
(a) 2
(b) 3
(c) 4
(d) 5
28. The circumference of a circle equals the perimeter of an equilateral triangle and also that of a square. If their areas are c, t and s respectively, which of the following represents the increasing order of their areas (1 for smallest, 3 for largest)?
(a) c < s < t
(b) t < s < c
(c) s < t < c
(d) c < t < s
29. A 30 m by 30 m square lawn is mowed in 1 m wide strips starting from a corner and going around inward. After how many complete rounds will half the lawn be mowed?
(a) 5
(b) 4
(c) 6
(d) 3
30. Four horses are tied at the four corners of a square field of side 20 m. Each rope is 10 m long so adjacent horses can just meet. There is a circular pond of area 25 m² at the centre. What is the ungrazed area (approx, in m²)?
(a) 50
(b) 75
(c) 86
(d) 100
31. A boy saves a distance equal to half the longer side by walking along the diagonal instead of along two adjacent sides of a rectangle. If the longer side is 8 units, what is the shorter side (in units)?
(a) 3
(b) 4
(c) 5
(d) 6
32. A string 50 cm long is cut into three pieces. The longest piece is twice the middle piece and the shortest piece is 10 cm shorter than the longest piece. What is the length (in cm) of the shortest piece?
(a) 5
(b) 10
(c) 15
(d) 20
33. A rectangular pool 30 m long and 10 m wide is surrounded by a uniform walkway. If the area of the walkway is 400 m², what is its width (in m)?
(a) 4
(b) 5
(c) 6
(d) 3
34. A triangle has sides 13 and 14 units, and its area is 84 square units. What is the length (in units) of the third side?
(a) 12
(b) 15
(c) 16
(d) 17
35. A quadrilateral has two adjacent sides of 20 m and 15 m at right angle. The other two sides are 17 m each. What is the area (in m²)?
(a) 250
(b) 280
(c) 300
(d) 320
36. A circular city wall has four gates at the cardinal directions. A house is 4 km north of the north gate and is just visible from a point 8 km east of the south gate. What is the diameter (in km) of the wall?
(a) 8
(b) 10
(c) 12
(d) 14
37. A square of side 4 m has equal corners cut off to form a regular octagon. What is the length (in m) of each side of the octagon?
(a) 1
(b) 2
(c) 3
(d) 4
38. How many distinct triangles with integral sides have perimeter 12?
(a) 2
(b) 3
(c) 4
(d) 5
39. A circle has unit radius. Five adjacent sectors have total area equal to one-fourth of the area of the circle. Each sector has twice the area of the previous one. What is the angle (in degrees) subtended by the smallest sector at the centre?
(a) 6
(b) 3
(c) 9
(d) 12
40. Two concentric circles have radii 10 cm and 5 cm. A square is inscribed in the outer circle and circumscribes the inner circle. What is the ratio of the perimeter of the outer circle to that of the square (nearest whole number)?
(a) 2
(b) 3
(c) 4
(d) 5
41. A cylinder has height 8 cm and radius 2 cm. A string is wound tightly around it without gaps making 4 complete turns. What is the length (in cm) of the string (nearest whole number)?
(a) 40
(b) 45
(c) 50
(d) 55
42. A rectangular sheet is used to make a closed cylindrical vessel with minimum wastage. What is the ratio of wastage area to utilised area (in simplest whole number form, a:b)?
(a) 1:2
(b) 1:3
(c) 2:3
(d) 1:4
43. The diagonals of the three rectangular faces of a brick are in the ratio \( 2 : \sqrt{5} : 3 \). What is the ratio of the shortest edge to the longest edge?
(a) 1:2
(b) 1:3
(c) 2:3
(d) 3:4
44. A right circular cone has height 15 ft and base diameter 10 ft. The top is cut parallel to the base at a height 10 ft from the base. What is the volume (in cubic ft, nearest whole number) of the remaining frustum?
(a) 400
(b) 550
(c) 600
(d) 650
45. The radius of a cone is twice its height. A cube of maximum volume is cut from it. What is the ratio of the volume of the cone to the cube (nearest whole number)?
(a) 2
(b) 3
(c) 4
(d) 5
46. The radius of a cone is 6 cm and height is 12 cm. If reducing radius by x gives same volume change as reducing height by x, what is x (in cm)?
(a) 1
(b) 2
(c) 3
(d) 4
47. 125 small cubes of 1 cm³ each are arranged to form a cuboid with minimum surface area. What is the diagonal (in cm) of the cuboid?
(a) 7
(b) 8
(c) 9
(d) 10
48. A solid sphere of radius 6 cm is melted and recast into cones of the same radius and height 8 cm. How many complete cones can be formed?
(a) 4
(b) 5
(c) 6
(d) 7
49. A cone has base radius 6 cm and height 12 cm. A cylinder is placed inside with its base on the cone’s base. What is the maximum height (in cm) of the cylinder?
(a) 4
(b) 6
(c) 8
(d) 10
50. In a cube of side 4 cm, the three face diagonals meeting at a vertex are taken as the sides of a triangle. What is the radius (in cm) of the circle circumscribing this triangle?
(a) 3
(b) 4
(c) 5
(d) 6