Lines and angles, Circles, Right Triangles, Area of Plane figures, Volume of solids and Coordinate Geometry
MULTIPLE CHOICE QUESTIONS (Part a)
1. The measure of an angle is 35o . What is the measure of the complement of this angle? a. 40o b. 60o c. 55o d. 145o Sol: Correct option is (c)
Let the angle given be x
x 35o
Let the complement of x be y Then x y 90o y 90o x y 55o 2. The measure of an angle is 75o . What is the measure of the supplement of this angle? a. 15o b. 105o c. 110o d. 50o
Sol:
Correct option is (b)
Let the angle given be x
x 75o
Let the complement of x be y Then x y 180o y 180o x y 105o 3. The measure of an angle is five times of its complement. What is the measure of the angle? a. 50o b. 70o c. 30o d. 75o Sol: Correct option is (d)
Let the angle be x and its complement be y It is given that x 5 y Also x y 90o
5 y y 90o 6 y 90o y 15o x 75o
4. An angle is half of its supplement. What is the measure of this angle? a. 20 o b. 60o c. 90o d. 110o Sol: Correct option is (b)
Let the angle be x and its supplement be y Then x y 180o and x y 2x x 2 x 180o 3 x 180o x 60o 5. If the two complementary angles are congruent, then what is the measure of each angle? a. 60o b. 30o c. 45o d. 90o Sol: Correct option is (c)
y 2
Let one angle be x and its complement be y As both are congruent, then x y Also x y 90o
x x 90o 2 x 90o
x 45o y 45o 6. If the two supplementary angles are congruent, then what is the measure of each angle? a. 60o b. 45o c. 30o d. 90o Sol: Correct option is (d)
Let one angle be x and its supplement be y As both are congruent x y Also x y 180o
x x 180o 2 x 180o x 90o y 90o
7. Find the value of x in the above figure a. 8 b. 3 c. 5 d. 9 Sol: Correct option is (a)
As vertically opposite angles are equal, therefore
x 3 2 x 11 x 2x 8 8 2x x x 8
8. What is the value of x in the above figure? a. 40 b. 50 c. 60 d. 10 Sol: Correct option is (b)
As both marked angles are supplementary angles, therefore
(3 x 15) 45 180
3 x 30 180 3 x 150 x 50o
9. 9.What is the value of y in the above diagram? a. 60o b. 30o c. 90o d. 65o Sol: Correct option is (d)
As 2 x 50 and x 40 are both supplementary angels, therefore
(2 x 50) ( x 40) 180 3 x 90 180 x 90o
Also
2 y 2 x 50 2 y 180 50 2 y 130 y 65o
10. The supplement of an angle is four times its complement. What is the measure of the angle?
a. 60o b. 70o c. 90o d. 105o Sol: Correct option is (a)
Let the angle be x and its supplement be y and its complement be z Then x y 180 and x z 90
y z 90 Also y 4 z 4 z z 90 z 30 y 120
Also x y 180 x 180 120 60o
11. In above figure aIIb and c is a transversal. What are the angles whose measure is same as that of 5 a. 1, 3, 6 b. 7, 3, 2 c. 6, 8 d. 1, 2
Sol:
Correct option is (b)
7 5 As both are vertically opposite angles
Also 7 3 5 3 Also 3 2 5 2
12. In above diagram lIIm and n is the transversal. What is the value of x in the above diagram? a. 40 b. 60 c. 50 d. 30 Sol: Correct option is (c)
Here y 2 x 1 As y and 2 x 1 are vertically opposite angles Also (2 x 10 (2 x 19) 180
4 x 20 180 4 x 200 x 50
13. If in above diagram lIIm and nIIo , then which of the following statements are true? a. 1 4 3 b. 1 3 6 c. 1 4 10 d. 3 6 11 Sol: Correct option is (d)
3 6 As both are alternate interior angles.
Also 3 11 As both are corresponding angles. Hence 3 6 11
14. In above diagram if AB=AC, then which of the following is ture? a. no bisects lAB b. BAC 90o c. ABC BAC d. none of the above Sol: Correct option is (a)
As AB=AC ABC ACB Also ABC ACB lAB lAB 2ABC Also ABC BAn lAB 2BAn BAn
1 lAB 2
15. In above diagram SR II PQ and SM PQ. What is the value of x? a. 40 o b. 30o c. 70o d. 60o
Sol:
Correct option is (c)
SMP 90o
We know that SPM SMP 20 180
SPM 180 20 90 SPM 70
Also x SPM as both are corresponding angles Hence x 70o
16. In above diagram PQ II RS and RQ II TS. What is the value of x? a. 78o b. 85o c. 66o d. 36o Sol: Correct option is (a)
RSQ 36o As both are corresponding angles
Also RST QRS 66o Also x RST RSQ 180o
x 180 66 36 78o
17. In above figure PQ II RS II UT. What are the values of x and y respectively. a. 20, 30 b. 10, 40 c. 10, 30 d. 30, 40 Sol: Correct option is (c)
Here PQR QRS 180
130 2 x y 180 2 x y 50
Also RST STU 180
120 3 x y 180 30 x y 60
Solving above equations of x and y
x 10 and y 30
18. In above figure PQ II SR and PS II QR. What are the values of x and y respectively a. 50, 100 b. 50, 70 c. 60, 100 d. 60, 70 Sol: Correct option is (a)
QRU RQP 2 x 40 x 10 x 50
Also SPQ RQP 180
y 20 x 10 180 y 180 120 10 x y 150 x y 150
19. In above figure RT II SU. What is the value of y in the above figure? a. 130 b. 160
c. 75 d. 120 Sol: Correct option is (d)
Here 80 40 x 2 x 180
3 x 60 x 20
We know that
PQT 40 x 180 PQT 120
Also PQT y
y 120
20. In above figure SQ PQ . What are the values of x and y respectively. a. 20, 30 b. 60, 40 c. 30, 60 d. 40, 60 Sol: Correct option is (c)
Here (2 x y ) 60 180
2 x y 120
Also y x 60 90 180
y x 30
Solving above equations
x 30 and y 60
21. What is the area of the above figure? a. 42 b. 35 c. 38 d. 36 Sol: Correct option is (b)
Area of above figure can be split into 2 parts Area = 9 (4 1) 8 1 27 8 35
22. What is the area of the above figure? a. 18 b. 72 c. 36 d. 24 Sol: Correct option is (c)
Area = 3 2 6 2 9 2
6 12 18 36
23. What is the area of the above figure? Given PQRS is a rhombus and POQ 90o
a. 100 b. 180 c. 150 d. 120 Sol: Correct option is (d)
1 d1 d 2 2
Area of rhombus =
Here d1 12 12 24 Also PO PQ 2 OQ 2 132 122 25 5
d 2 5 5 10
1 Area 24 10 120 2
24. What is the area of equilateral triangle whose perimeter is 24. a. 27.68 b. 30.21 c. 4864 d. 29.60 Sol: Correct option is (a)
Here PQ=QR=RP=8 Also PQS PRS QS 4
PS PQ 2 QS 2 82 42 48 6.92
Area =
1 8 6.92 27.68 2
25. What is the area of isosceles triangle with sides 13, 13 and 10 a. 80 b. 120 c. 60 d. 40 Sol: Correct option is (c)
Here PQS PRS
QS 5
Also PS 132 52
PS 144 12 1 Area 10 12 60 2
26. What is the area of a isosceles right triangle whose hypotenuse is 50. a. 694.61 b. 587.38 c. 798.6 d. 624.81 Sol: Correct option is (d)
Here PQ=QR=a Also a 2 a 2 50 2 2a 2 2500
a 35.35
Area =
1 35.35 35.35 624.81 2
27. What is the area of an equilateral triangle whose height is 100? a. 4832.81 b. 5773. 67 c. 8192.74 d. 384.96
Sol:
Correct option is (b)
PS=100 Let PS=2a QS=a
PS (20) 2 a 2 3a 2
3a 100
a
100 3 1 200 100 2 3
Area =
Area = 5773.67 28. A square PQRS is inscribed in a circle of radius 4. What is the area of square? a. 64 b. 16 c. 4 d. 32 Sol: Correct option is (d)
Each diagonal of square = 8 Hence Area =
1 8 8 32 2
29. One diagonal of a rhombus is 4 times the other diagonal and the area of diagonal is 72. What is the measure of diagonals? a. 24, 12 b. 4, 16 c. 8, 32 d. 6, 24 Sol: Correct option is (d)
Let one diagonal = d Other diagonal = 4d Area =
1 d 4d 72 2
d 2 36 d 6
Hence measure of first diagonal = 6 and measure of second diagonal = 24 30. The height of a triangle is 3 times its base. What is the length of base if area is 54?
a. 8 b. 4 c. 6 d. 3 Sol: Correct option is (c)
Let height = h and base = b h=3b
1 3 Area b 3b b 2 2 2 3 54 2 b 2 54 b 2 b 2 36 b 6 2 3
31. What is the area of a triangle whose sides are 4, 6, 8? a. 12.68 b. 18.92 c. 11.61 d. 7.99 Sol: Correct option is (c)
A S ( S a)( S b)( S c )
Here S
a b c 4 68 9 2 2
A 9(9 4)(9 6)(9 8) 9(5)(3)(1) 11.61 32. What is the area of a trapezoid with height 11 and base 14 and 6
a. 55 b. 150 c. 19 d. 110 Sol: Correct option is (d)
1 Area h (b1 b2 ) 2 1 1 11 (14 6) 11 20 110 2 2
33. The median of a trapezoid divides into two regions. What is the ration of the areas of these two regions? a. 2 b. 3 c. 1 d. 4 Sol: Correct option is (c)
Here ST is median Also SM PQ and let SM =h
1 Area of SPT PQ h 2
Let PQ=2a Also area of STQR
1 (a a ) h ah 2
1 Also area of SPT 2a h ah 2
Hence ratio =
ah 1 ah
34. What is the area of above drawn trapezoid? a. 49 b. 38 c. 32 d. 78 Sol: Correct option is (b)
PQ=TR=8, SR=11 ST=3 PT 52 32 16 4 Here QR=4
1 Area (8 11) 4 38 2
35. What is the area of a trapezoid with legs 10, 10 and bases 17 and 5? a. 88 b. 102 c. 72 d. 34 Sol: Correct option is (a)
Here TU=5, also RT=US and RT+US=12 RT US 6 Here PT = 102 62 64 8
1 Area 8 (17 5) 88 2
36. What is the area of a trapezoid whose bases are 12 and 26. The base angles of the trapezoid are 45o ? a. 109 b. 84 c. 79 d. 133
Sol:
Correct option is (d)
Here TU=12, also RT=TS=7
PT tan 45o 1 PT RT 7 RT 1 Hence Area 7 (12 26) 133 2
37. An isosceles trapezoid has bases 8 and 14 and its area is 44. What is the height of the trapezoid? a. 2 b. 4 c. 17 d. 9 Sol: Correct option is (b)
Here Area = 44
1 Area h (8 14) 11h 2
11h 44 h 4
38. An isosceles trapezoid has bases 8 and 14. Its area is 44. What is the perimeter of trapezoid? a. 16 b. 82 c. 32 d. 28 Sol: Correct option is (c)
Here Area = 44
1 Area h (14 8) 11h 2
11h 44 h 4
Now PT =4, also RT=US=3
PR 42 32 25 5
Hence perimeter = 8+14+5+5=32 39. The complement of an angle is 50o more than the angle itself. What is, the measure of the angle?
a. 30o b. 20o c. 40o d. 15o Sol: Correct option is (b)
Let angle be x and its complement be y
x y 90
Also y x 50
x x 50 90 x 20o
40. The supplement of an angle is 40o more than the angle itself, what is the measure of the angle? a. 90o b. 130o c. 70o d. 60o Sol: Correct option is (c)
Let the measure of angle = x Let the measure of its supplement = y
x y 180
Also y x 40
x x 40 180 x 70o
41. What is the area of a circle whose diameter is 20? a. 215 b. 38 c. 97 d. 100 Sol: Correct option is (d)
Here diameter = 20
radius 20 10 2
Area r 2 (10) 2 100
42. What is the circumference of the circle whose radius is 8? a. 16 b. 8 c. 9 d. 3 Sol: Correct option is (a)
Circumference = 2 r 2 (8) 16 43. What is the area of a circle whose circumference is 14 ? a. 37 b. 36 c. 16 d. 49
Sol:
Correct option is (d)
Area = r 2 Circumference = 2 r 14
r 7
Area r 2 (7)2 49
44. PQRS is a square with side 8. QTRQ is a semicircle with diameter QR. What is the area of the whole region? a. 92.82 b. 67.74 c. 89.12 d. 64 Sol: Correct option is (c)
Area of square = a 2 8 8 64 Area of semicircle =
QR 8 4 2 2 1 (4) 2 8 25.12 2 1 2 r 2
Here r =
Area of semicircle =
Area of whole region = 64+25.12=89.12
45. In above figure PQ=6 and QR=8. Semicircles are drawn on each side of right angled triangle. What is the area of the whole region? ( 3.14 ) a. 82.7 b. 39.6 c. 39.1 d. 78.5 Sol: Correct option is (d)
1 6 6 24 2 1 (3) 2 4.5 14.13 2 1 (4)2 8 25.12 2
Area of triangle =
Area of 1st semicircle =
Area of 2nd semicircle =
Also PR 82 62 100 10 Area of 3rd semicircle =
1 (5) 2 12.5 39.25 2
Area of whose region = 14.13+25.12+39.25=78.5 46. A wheel has radius 4. How far will the car travel in 5 minutes, if the wheel rotates once every 30 seconds? ( 3.14 )
a. 251.2 b. 217.84 c. 79.38 d. 118.60 Sol: Correct option is (a)
Circumference = 2 r 8 25.12 No of rotations in 5 minutes = 2 rotations/minute 5 minutes = 10 rotations Hence distance traveled = 25.12 x 10 = 251.2 a
47. What is the area of shaded portion in above figure? a. 20.82 b. 25.64 c. 50.24 d. 96
Sol:
Correct option is (b)
Area of rectangle = 12 8 96 Area of semicircle =
1 2 1 r 3.14 16 25.12 2 2
Area of circle = r 2 (16) 50.24 Area of shaded region = 96-25.12-50.24=20.64 48. What is the area of shaded region?
a. 96 b. 314 c. 200 d. 114 Sol: Correct option is (d)
90 (20) 2 314 360
Area of sector PQR=
1 Area of PQR 20 20 200 2 Area of shaded region = 314=200=114
49. What is the length of arc ? ADB a. 36.64 b. 42.81 c. 39.82 d. 7.54
Sol:
Correct option is (a)
Circumference = 2 r 2 3.14 10 62.8
150 Length of arc AB 62.8 26.16 360
Hence length of arc 62.80 26.16 36.64 ADB 50. A circle has area 100 m 2 . If the sector of the circle has area 20 m 2 . What is the measure of the angle of the sector? a. 36o b. 72o c. 150o d. 60o Sol: Correct option is (b)
Area of sector = 20 m 2 Let measure of angle = x Then area of sector =
x 100 m 2 360
x 100 20 360
x 72o
51. What is the lateral area of a cylinder having radius 6 cm and height 8 cm? a. 48 cm 2 b. 96 cm2 c. 112 cm2 d. 39 cm 2 Sol: Correct option is (b)
Lateral area = 2 rh 2 6 8 96 cm 2 52. What is the total area of the cylinder having radius 7 cm and height 3 cm? a. 45 cm 2 b. 80 cm 2 c. 160 cm2 d. 140 cm2 Sol: Correct option is (d)
Total area = Lateral area + 2 x Area of base Lateral area = 2 rh 2 7 3 42 cm2 Area of base = r 2 (7)2 49 cm 2 Total area = (42 2 49 ) 140cm2 53. What is the volume of cylinder whose radius is 5 cm and height is 6 cm? a. 50 cm3 b. 36 cm3 c. 150 cm3 d. 25 cm3
Sol:
Correct option is (c)
Volume = r 2 h 6 5 5 150 cm3 54. What is the lateral area of a cone whose radius is 4 cm and height is 3 cm? a. 10 cm2 b. 5 cm 2 c. 17 cm2 d. 20 cm 2 Sol: Correct option is (d)
Slant height l 32 42 25 5 Lateral area = rl 4 5 20 cm 2 55. What is the total area of a cone whose radius is 3 cm and height is 4 cm? a. 24 cm2 b. 15 cm2 c. 6 cm 2 d. 18 cm2 Sol: Correct option is (a)
Slant height l 32 42 25 5 Total area = Lateral area + 2 x Area of base Lateral area = rl 3 5 15 Area of base = r 2 (3)2 9 Total area = 15 9 24 cm 2 56. What is the volume of a cone whose radius is 3 cm and height is 7 cm?
a. 7 cm3 b. 63 cm3 c. 9 cm3 d. 21 cm3 Sol: Correct option is (d)
1 2 1 r h (3) 2 7 21 cm3 3 3
Volume of cone =
57. The radius of cylinder is twice the height of cylinder. What is the height of cylinder if its volume is 32 cm3 ? a. 6 cm b. 3 cm c. 7 cm d. 2 cm Sol: Correct option is (d)
Volume = r 2 h Also r = 2h h
r 2
3
r r Volume r 2 2 2
r3 32 r 3 64 r 4 2
r 4 h 2cm 2 2
Also h
58. The total area of a cylinder is 8 cm2 . The height of a cylinder is 3 times the radius of the cylinder. What is the height of the cylinder?
a. 4 cm b. 3 cm c. 8 cm d. 1 cm Sol: Correct option is (b)
Total area = 2 rh 2 r 2 Also h=3r
Total area = 2 r (3r ) 2 r 2 6 r 2 2 r 2 8 r 2
8 r 2 8 r2 1 r 1 Also h=3r h 3 1 3cm 59. What is the ratio of the volume of the cylinder to that of volume of a cone having same radius and height as that of a cylinder? a. 4:1 b. 1:2 c. 3:1 d. 1:1 Sol: Correct option is (c)
Volume of cylinder = r 2 h Volume of cone =
1 2 r h 3
Ratio =
r 2h 3 3 :1 1 2 r h 1 3
60. The radius of the cylinder is increased to 2 times but its height is decreased to half. What is the ratio of the new volume of the cylinder to the old one? a. 2:1 b. 3:1 c. 4:1 d. 1:4 Sol: Correct option is (a)
V1 r 2 h V2 (r ' ) 2 h '
r ' 2r and h '
V2 (2r ) 2
V2 2 2 :1 V1 1
h 2 h 2 r 2 h 2V1 2
61. Oil is being poured into a cylindrical vessel at the rate of 3 m3 / min ute . In how much minutes the cylinder of radius 2 m and height 2.2 m will get failed? ( 3.14 ) a. 86 min b. 74 min c. 93 min d. 113 min Sol: Correct option is (c)
Volume = r 2 h 3.14 2 2 22 276.32m3 No of minute = 276.32m3 92.1min utes 93min utes 3m3 / min ute
62. Two cylinders of same height and radius 3 cm and 4 cm are containing water inside them. This water is to be poured into a new cylinder of same height but different radius. What should be the radius of new cylinder? a. 5 cm b. 3 cm c. 2 cm d. 7 cm Sol: Correct option is (a)
Let height of all cylinders be (h) Volume of two cylinders combined = (3)2 h (4) 2 h 25h 25 h Volume of new cylinder = r 2 h
r 2 h 25 h r 2 25 r 5 cm
63. Water contained in the cone of radius 4 cm and height 3 cm is to be poured into a cylinder of height 1 cm. What should be the radius of the cylinder? a. 2 cm b. 4 cm c. 8 cm d. 1 cm Sol: Correct option is (b)
1 2 r h 3
Volume of cone =
1 = (4) 2 h 16 3
Volume of cylinder = r 2 h
r 2 h 16 r 2 h 16 r 2 (1) 16 r 4cm