Practice Test

1) A vertical stick 20 m long casts a shadow 10 m long on the ground. At the same time, a tower casts the shadow 50 m long on the ground. Find the height of the tower.

2) Two poles of height 6 m and 11 m stand vertically upright on a plane ground. If the distance between their foot is 12 m, find the distance between their tops.

3) The radius of a circle is 9 cm and length of one of its chords is 14 cm. Find the distance of the chord from the center.

4) Find the length of a chord that is at a distance of 12 cm from the center of a circle of radius 13 cm.

5) In a right angled triangle, find the hypotenuse if base and perpendicular are respectively 36015 cm and 48020 cm.

6) The inner circumfernece of a circle track is 440 cm. The track is 14 cm wide. Find the diamter of the outer circly of the trak.

7) A race track is in the form of a ring whose inner and outer circumference are 352 meter and 396 meter respectively. Find the width of the track.

8) The outer circumference of a circular track is 220 meter. The track is 7 meter wide everywhere. Calculate the cost of levelling the track at the rate of 50 paise per square meter.

9) Find the area of a quadrant of a circle whose circumference is 44 cm

10) A pit 7.5 meter long, 6 meter wide and 1.5 meter deep is dug in a field. Find the volume of soil removed in cubic meters.

11) Find the length of the longest pole that can be placed in an indoor stadium 24 meter long, 18 meter wide and 16 meter high.

12) The whole surface of a rectangular block is 8788 square cm. If length, breadth and height are in the ratio of 4 : 3 : 2, find length.

13) Three metal cubes with edges 6 cm, 8 cm and 10 cm respectively are melted together and formed into a single cube. Find the side of the resulting cube.

14) Find curved and total surface area of a conical flask of radius 6 cm and height 8 cm.

15) The diameters of two cones are equal. If their slant height be in the ratio 5 : 7, find the ratio of their curved surface areas.

16) The curved surface area of a cone is 2376 square cm and its slant height is 18 cm. Find the diameter.

17) The ratio of radii of a cylinder to a that of a cone is 1 : 2. If their heights are equal, find the ratio of their volumes?

18) The circumference of a circle exceeds its diameter by 16.8 cm. Find the circumference of the circle.

19) A bicycle wheel makes 5000 revolutions in moving 11 km. What is the radius of the wheel?

20) The surface areas of two spheres are in the ratio of 1 : 4. Find the ratio of their volumes.

21) The radii of two spheres are in the ratio of 1 : 2. Find the ratio of their surface areas.

22) A sphere of radius r has the same volume as that of a cone with a circular base of radius r. Find the height of cone.

23) A road that is 7 m wide surrounds a circular path whose circumference is 352 m. What will be the area of the road?

24) In a shower, 10 cm of rain falls. What will be the volume of water that falls on 1 hectare area of ground?

25) Seven equal cubes each of side 5 cm are joined end to end. Find the surface area of the resulting cuboid.

26) In a swimming pool measuring 90 m by 40 m, 150 men take a dip. If the average displacement of water by a man is 8 cubic meters, what will be rise in water level?

27) How many meters of cloth 5 m wide will be required to make a conical tent, the radius of whose base is 7 m and height is 24 m?

28) Two cones have their heights in the ratio 1 : 2 and the diameters of their bases are in the ratio 2 : 1. What will be the ratio of their volumes?

29) A closed wooden box measure externally 10 cm long, 8 cm broad and 6 cm high. Thickness of wood is 0.5 cm. Find the volume of wood used.

30) The largest cone is formed at the base of a cube of side measuring 7 cm. Find the ratio of volume of cone to cube.

31) A spherical cannon ball, 28 cm in diameter, is melted and cast into a right circular conical mould the base of which is 35 cm in diameter. Find the height of the cone correct up to two places of decimals.

32) Find the area of the circle circumscribed about a square each side of which is 10 cm.

33) Find the radius of the circle inscribed in a triangle whose sides are 8 cm, 15 cm and 17 cm.

34) A cube whose edge is 20 cm long has circle on each its faces painted black. What is the total area of unpainted surface of the cube if the circles are of the largest area possible?

35) A wire is looped in the form of a circle of radius 28 cm. It is bent again into a square form. What will be the length of the diagonal of the largest square possible thus?

36) The perimeter of a sector of a circle of radius 5.7 m is 27.2 m. Find the area of the sector.

37) The dimensions of a field are 20 m by 9 m. A pit 10 m long, 4.5 m wide and 3 m deep is dug in one corner of the field and the earth removed has been evenly spread over the remaining area of the field. What will be the rise in the height of field as a result of this operation?

38) The sides of a triangle are 21, 20 and 13 cm. Find the area of the larger triangle into which the given triangle is divided by the perpendicular upon the longest side from the opposite vertex.

39) A circular tent is cylindrical to a height of 3 meters and conical above it. If its diameter is 105 m and the slant height of he conical portion is 53 m, calculate the length of the canvas 5 m wide to make the required tent.

40) A steel sphere of radius 4 cm is drawn into a wire of diameter 4 mm. Find the length of wire.

41) A cylinder and a cone having equal diameter of their bases are placed in the Qutab Minar one on the other, with the cylinder placed in the bottom. If their curved surface area are in the ratio of 8 : 5, find the ratio of their heights. Assume the height of the cylinder to be equal to the radius of Qutab Minar. (Assume Qutab Minar to be having same radius throughout).

42) If the curved surface area of a cone is thrice that of another cone and slant height of the second cone is thrice that of the first, find the ratio of the area of their base.

43) A solid sphere of radius 6 cm is melted into a hollow cylinder of uniform thickness. If the external radius of the base of the cylinder is 5 cm and its height is 32 cm, find the uniform thickness of the cylinder.

44) A hollow sphere of external and internal diameters 6 cm and 4 cm respectively is melted into a cone of base diameter 8 cm. Find the height of the cone

45) Three equal cubes are placed adjacently in a row. Find the ratio of total surface area of the new cuboid to that of the sum of the surface areas of the three cubes.

46) The minute hand of a clock is 10 cm long. Find the area of the face of the clock described by the minute hand between 9 a. m. and 9 : 35 a. m.

47) A toy is in the shape of a right circular cylinder with a hemisphere on one end and a cone on the other. The height and radius of the cylindrical part are 13 cm and 5 cm respectively. The radii of the hemispherical and conical parts are the same as that of the cylindrical part. Calculate the surface area of the toy if the height of conical part is 12 cm.

48) A solid wooden toy is in the form of a cone mounted on a hemisphere. If the radii of the hemisphere is 4.2 cm and the total height of the toy is 10.2 cm, find the volume of wood used in the toy.

49) A cylinder container whose diameter is 12 cm and height is 15 cm, is filled with ice cream. The whole ice-cream is distributed to 10 children in equal cones having hemispherical tops. If the height of the conical portion is twice the diameter of its base, find the diameter of the ice-cream cone.

50) A cone, a hemisphere and a cylinder stand on equal bases and have the same height. What is the ratio of their volumes?

51) A right elliptical cylinder full of petrol has its widest elliptical side 2.4 m and the shortest 1.6 m. Its height is 7 m. Find the time required to empty half the tank through a hose of diameter 4 cm if the rate of flow of petrol is 120 m/min.

52) The radius of a right circular cylinder is increased by 50%. Find the percentage increase in volume.

53) Water flows out at the rate of 10 m/min from a cylindrical pipe of diameter 5 min. Find the time taken to fill a conical tank whose diameter at the surface is 40 cm and depth 24 cm.

54) The section of a solid right circular cone by a plane containing vertex and perpendicular to base is an equilateral triangle of side 12 cm. Find the volume of the cone.

55) Iron weights 8 times the weight of oak. Find the diameter of an iron ball whose weight is equal to that of a ball of oak 18 cm in diameter.

56) Find the area of the triangle inscribed in a circle circumscribed by a square made by joining the mid-points of the adjacent sides of a square of side a.

57) Find the ratio of the diameter of the circles inscribed in and circumscribing an equilateral triangles to its height.

58) A piece of paper is in the shape of a right angled triangle. This paper is cut along a line that is parallel to the hypotenuse, leaving a smaller triangle. The cut was done in such a manner that there was a 35% reduction in the length of the hypotenuse of the triangle. If the area of the original triangle was 54 square inches before the cut, what is the area (in square inches) of the smaller triangle?

59) Consider two different cloth-cutting processes. In the first one, n square cloth pieces of the same size are cut from a square cloth piece of the size a; then a circle of maximum possible area is cut from each of the smaller squares. In the second process, only one circle of maximum possible area is cut from a square of the same size and the process ends there. The cloth pieces remaining after cutting the circles are scrapped in both the processes. The ratio of the total area of scrap cloth generated in the former to that in the latter is:

60) The length of the circumference of a circle equals the perimeter of a triangle of equal sides. This is also equal of the perimeter of a square and also the perimeter of a regular pentagon. The areas covered by the circle, triangle, pentagon and square are c, t, p, and s respectively. Then :

61) A certain city has a circular wall around it with four gates pointing north, south, east and west. A house stands outside the city, three km north of the north gate, and it can just be seen from a point nine km east of the south gate. What is the diameter of the wall that surrounds the city?

62) What is the number of distinct triangles with integral valued sides and perimeter as 12?

63) What is the area that can be grazed by the cow if the length of the rope is 8 m?

64) What is the area that can be grazed by the cow if the length of the rope is 12 m?

65) The value of each of a set of coins varies as the square of its diameter, if its thickness remains constant, varies as the thickness, if the diameter remains constant. If the diameter of two coins are in the ratio 3 : 4, what should be the ratio of their thicknesses be if the value of the second is four times that of the first?

66) Then M is

67) The total distance walked by Safdar is:

68) The sides of a triangle are 5, 12 and 13 units. A rectangle is constructed, which has an area twice the area of the triangle, and has a width of 10 units. Then, the perimeter of the rectangle is:

69) The diameter of a hollow cone is equal to the diameter of a spherical ball. If the ball is placed at the base of the cone, what portion of the ball will be inside the cone?

70) The length of a ladder is exactly equal to the height of the wall it is leaning against. If lower end of the ladder is kept on a platform of height 3 feet and the platform is kept 9 feet away from the wall, the upper end of the ladder coincides with the top of the wall. Then, the height of the wall is:

71) A cube of side 12 cm is painted red on all the faces and then cut into smaller cubes, each of side 3 cm. What is the total number of smaller cubes having none of their faces painted?

72) From a circular sheet of paper with a radius 20 cm, four circles of radius 5 cm each are cut out. What is the ratio of the cut to the uncut portion?

73) Four friends Mani, Sunny, Honey and Funny start from four towns Mindain, Sindian, Hindian and Findian respectively. The four towns are at the four corners of an imaginary rectangle. They meet at a point which falls inside this imaginary rectangle. At that point three of them (Mani, Sunny and Honey has traveled distance of 40 m, 50 m and 60 m respectively). The maximum distance that Funny could have traveled is near about:

74) The length of the common chord of two circles of radii 15 cm and 20 cm whose centers are 25 cm apart, is (in cm):

75) A square tin sheet of side 24 inches is converted into a box with open top in the following steps: The sheet is placed horizontally. Then, equal sized squares, each of side x inches, are cut from the four corners of the sheet. Finally, the four resulting sides are bent vertically upwards in the shape of box. If x is an integer, then what value of x maximize the volume of the box?

76) A square, whose side is 2 meters, has its corners cut away so as to form an octagon with all sides equal. Then length of each side of the octagon, in meters is :

77) A rectangular pool 30 meters wide and 50 meters long is surrounded by a walkway of uniform width. If the area of the walkway is 516 square meters, how wide, in meters, is the walkway?

78) A farmer has decided to build a wire fence along one straight side of his property. For this purpose he has planned to place several fence-posts at 12 m intervals, with posts fixed at both ends of the side. After he bought the posts and wire, he found that the number of posts he had bought was 5 less than required. However, he discovered that the number of posts he had bought would be just sufficient if he spaced them 16 m apart. What is the length of the side of his property and how many posts did he buy?

79) In a triangle XYZ, X Y = 6, YZ = 8 and XZ = 10. A perpendicular dropped from Y, meet the side XZ at M. A circle of radius YM (with center Y) is drawn. If the circle cuts XY and YZ at P and Q respectively, then QZ : XP is equal to :

80) A ladder leans against a vertical wall. The top of the ladder is 12 m above the ground. When the bottom of the ladder is moved 4 m farther away from the wall, the top of the ladder rests against the foot of the wall. What is the length of the ladder?

81) The perimeter of right triangle is 36 and the sum of the square of its sides is 450. The area of the right triangle is

82) A ladder reaches a window that is 8 cm above the ground on one side of the street. Keeping its foot on the same point, the ladder is twined to the other side of the street to reach a window 12 cm high. Find the width of the street if the ladder is 13 m.

83) All the following is true except:

84) The area of the circle circumscribing three circles of unit radius touching each other is

85) There are two spheres and one cube. The cube is inside the bigger sphere and the smaller sphere is inside the cube. Find the ratio of surface areas of the bigger sphere to the smaller sphere?

86) The area of the largest triangle that can be inscribed in a semi circle whose radius is

87) A cube of side 16 cm is painted red on all the faces and then cut into smaller cubes, each of side 4 cm. What is the total number of smaller cubes having none of their faces painted?

88) There is an equilateral triangle of side 32 cm. The mid-points of the sides are joined to form another triangle, whose mid-points are again joined to form still another triangle. This process is continued for 'n' number of times. The sum of the perimeters of all the triangles is 180 cm. Find the value of n.

89) Ram Singh has a rectangular plot of land of dimensions 30 m * 40 m. He wants to construct a unique swimming pool which is in the shape of an equilateral triangle. Find the area of the largest swimming pool which he can have?

90) The perimeter of a triangle is 105 cm. The ratio of its altitudes is 3 : 5 : 6. Find the sides of the triangles.

91) Rizwan gave his younger sister a rectangular sheet of paper. He halved it by folding it at the mid point of its longer side. The piece of paper again became a rectangle whose longer and shorter sides were in the same proportion as the longer and shorter sides of the original rectangle. If the shorter side of the original rectangle was 4 cm, find the diagonal of the smaller rectangle?

92) A rectangular hall, 50 m in length and 75 m in width has to be paved with square tiles of equal size. What is the minimum number of tiles required?

93) In triangle ABC we have angle A = 100 degree and B = C = 40. The side AB is produced to a point D so that B lies between A and D and AD = BC. Then angle BCD = ?

94) The sides of a cyclic quadrilateral are 9, 10, 12 and 16. If one of its diagonals is 14, then find the other diagonal?