MULTIPLE CHOICE QUESTIONS (Part b)
1. The content of 3 cylinders each of radius 2 cm and height 1 cm is to be poured into a cone of height 9 cm. What is the radius of the cone? a. 2 cm b. 4 cm c. 6 cm d. 1 cm Sol: Correct option is (a)
Volume of 3 cylinders = 3 (2) 2 1 12 cm 2 Volume of cone =
1 r 2 9 3 r 2 3
3 r 2 12 r 2 4 r 2 cm
2. A 90o sector is cut out of a circular metal sheet of radius 4 cm and bent to form the lateral surface of a cone. What is the volume of the cone? ( 3.14 )? a. 3.85 cm3 b. 9.34 cm3 c. 6.74 cm3 d. 4.05 cm3 Sol: Correct option is (d) 90 1 Area of sector = (4) 2 16 4 360 4 Lateral area of cone = rl
rl 4 rl 4
Also l 4 r 1 Also h l 2 r 2 16 1 15 Hence volume =
1 (1) 2 15 3
1 3.14 3.87 4.05cm3 3
3. In PQR , PQ=3, QR=4 and PR=5. The triangle is rotated about QR in space. What is the volume of solid formed? a. 8 b. 9 c. 16 d. 25 Sol: Correct option is (c)
The shape of a solid formed will be cone. Here r =4 and h =3
1 Volume (4) 2 3 16 3
4. What is the volume of a sphere whose radius is 3 cm?
a. 12 cm3 b. 36 cm3 c. 24 cm 3 d. 8 cm3 Sol: Correct option s (b)
4 3 r 3
Volume =
4 (3)3 36 cm3 3
5. What is the area of the sphere whose radius is 4 cm? a. 32 cm 2 b. 16 cm 2 c. 4 cm 2 d. 64 cm 2 Sol: Correct option is (d)
Area = 4 r 2 4 (4) 2 64 cm2 6. What is the volume of the sphere whose area is 144cm 2 ? a. 144 cm3 b. 72 cm 3 c. 288 cm 3 d. 36 cm3
Sol: Correct option is (c) Area = 4 r 2
4 r 2 144 r 2 36 r 6
Volume =
4 3 4 r (6) 3 288 cm3 3 3
7. What is the area of the sphere whose volume is 36 cm3 ? a. 36 cm 2 b. 72 cm 2 c. 144 cm 2 d. 18 cm 2 Sol: Correct option is (a)
4 3 r 3
Volume =
4 r 3 36 3 r3 36 3 27 4
r 3
Area = 4 r 2 4 r 2 4 (3)2 36 cm 2 8. A circle with radius 5 cm is rotated about its diameter. What is the volume of the solid formed?
a. 134.62 b. 181.9 c. 166.66 d. 125 Sol: Correct option is (c)
The shape of solid formed is that of a sphere Volume =
4 3 4 r (5)3 166.66 3 3
9. A cylinder of height 6 cm is inscribed in a sphere of radius 5 cm. What is the volume of the cylinder? a. 96 cm3 b. 16 cm3 c. 24 cm 3 d.
8 3 cm
Sol:
Correct option is (a)
Radius of cylinder =
52 32 4cm
Hence volume = (4) 2 6 96 cm3 10. What is the area of the shaded portion?
a. 10 b. 15 c. 8 d. 20
Sol:
Correct option is (d)
Area of shaded portion = (6) 2 (4) 2 (36 16) 20 11. A circle of radius 4 cm is inscribed inside a square PQRS. What is the area of shaded portion?
a. 12.82cm 2
b. 13.76cm 2 c. 18cm 2 d. 19.8cm 2
Sol:
Correct option is (b)
Area of square = (8)2 64cm2 Area of circle = (4) 2 16 cm2 50.24cm 2 Area of shaded portion = 64 50.24 13.76cm 2
12.
In above figure PQ=5 and PS=3. What is the value of RQ? a. 4 b. 6 c. 8 d. 10 Sol: Correct option is (c)
QS= PQ 2 PS 2 52 32 16 4 Also QS=QR SR 4 QR 8
13.
In above figure if m PSR 130o . What is the value of x? a. 50o b. 65o c. 75o d. 90o Sol:
x
Correct option is (b)
1 1 1 PQR mPSR 130 65o 2 2 2
14.
In above figure PQR 90o and PR=8 cm. What is the value of QP? a. 4 2cm b. 2 2cm c. 4 3cm d. 3cm Sol: Correct option is (a)
Here PS=SR=4 cm Also QPR QRP 45o
tan 45o QS QS PS 4cm PS
Hence QP 42 42 4 2cm 15. In below figure SO= 10 2 cm. What is the value of SR?
a. 10
b. 20 c. 25 d. 43
Sol:
Correct option is (b)
Here OSR 45o ST 1 OS 10 2 cos 45o ST OS 2 2 2 ST 10 SR 10 10 20 16. What are the values of x and y in below figure?
a. 60, 75
b. 20, 60 c. 60,100 d. 100, 20 Sol:
x 60 o
Correct option is (c)
Also SPR 20 o
x y SPR 180o
Also
y 180 60 20 100o
17. What are the values of y and z in above below figure respectively?
a. 40, 60 b. 60, 40 c. 80, 60 d. 80, 40 Sol: Correct option is (d)
y = 2 40 800
Also z =
1 y 40o 2
18. What is the value of a and b in below figure respectively?
a. 65o , 40o b. 55o , 40 o c. 50o , 65o d. 65o , 75o Sol: a Correct option is (c) 1 (2 50o ) 50o 2
a 50o
1 PRQ 130 65o 2
b 180 50 65 b 65o 19. What are the values of a and b in below figure respectively? PQ=QR
a. 75o , 65o b. 65o , 65o c. 45o , 75o d. 45o , 45o Sol: Correct option is (b)
1 PQR 100 50o 2
As PQ=QR QPR QRP
QPR a 65o b 65o 1 (180 50) 65 2
20. What is the value of x and y in the below figure respectively?
a. 140o , 75o b. 50o , 75o c. 75o ,140o d. 50o ,140 o Sol: Correct option is (d)
1 x 100 50o 2 PRQ 180 60 50 70o y 2PRQ 2 70 140o
21. What is the value of x in the below figure?
a. 60 o b. 35o c. 30o d. 45 o Sol:
x
Correct option is (c)
1 1 (100 40) (60) 30o 2 2
22. What is the value of y in the below figure?
a. 60 o b. 35o c. 30o d. 45 o Sol: y Correct option is (a) 1 (130 (360 130 120)) 2 1 1 (130 110) (20) 10o 2 2
y
23. What is the value of x in below figure?
a. 4 b. 8 c. 6 d. 2 Sol: Correct option is (c)
x 6 18 2 x 18 2 6 6
24. What is the value of x in the below figure?
a. 2
b. 4 c. 9 d. 1 Sol: Correct option is (d)
Here (6+2) 2 (15 x ) x
16 15x x 2
x 2 15 x 16 0 ( x 1)( x 16) 0 x 1 x 16
25. What is the value of x in the below figure?
a. 4 b. 6 c. 3 d. 2 Sol: Correct option is (b)
Here (5 4) 4 x 2
x2 9 4 x 2 36 x 6 26. What is the value of x in the below figure?
8 3 4 b. 3
a.
c.
8 5 3 5
d.
Sol:
Correct option is (a)
Here (8 x x) x 82
9 x x 82 82 8 x 9 3
x2
27. What is the value of x in the below figure if PQ QR ?
a. 4 b. 2 c. 6 d. 8
Sol:
Correct option is (c)
102 x 2 ( x 2)2 x 2 x 2 4 4 x 100 2 x 2 4 x 96 0 x 2 2 x 48 0 ( x 6)( x 8) 0 x6 x 8 28. What is the value of x in the below figure?
a. 1 b. 3 c. 5 d. 2 Sol: Correct option is (d)
x 8 42 x 8 16 x 16 2 8
29. What is the value of x in the below figure?
a. 8 b. 6 c. 5 d. 3 Sol: Correct option is (b)
Here PR 2 RS RQ x 2 4 (5 4) x 2 36 x6 30. What area the values of a, b and c respectively in the below figure? a. 4 2, 2, 4 b. 4, 4 2, 4 2 c. 4, 2, 4 2 d. 2,3, 4 2
Sol: Here
Correct option is (b)
b2 4 4 b 4
Also a 2 8 4 32 a 32 4 2 c 2 8 4 32 c 32 4 2
31. What is the value of x in the below figure?
a. 4 2 b. 8 c. 8 2 d. 8 Sol: Correct option is (c)
2 leg 8 2 8 2
Hypotenuse =
32. What is the value of x in the below figure?
a. 8 b. 2 c. 6 d. 4 Sol: Correct option is (a)
IN figure 30 o 60 o 90 o is triangle Hypotenuse = 2 shorterleg 2 4 8 33. What is the value of x in the below figure?
a. 60 o b. 90o c. 30o d. 45 o Sol: sin x Correct option is (c) PQ 5 1 PR 10 2
x 30o 34. PQRS is a parallelogram. What are the values of x and y respectively in the below figure?
a. 30o ,8 b. 45o ,5 c. 45o ,8 d. 60o ,3 Sol: Correct option is (c)
Here SPR 90o
x 180 90 45 45o
Also y= 5+3 =8 35. What is the value of x and y in the below figure if PQRS is a parallelogram?
a. 4, 2 b. 3, 4 c. 2, 3 d. 1, 4 Sol: Correct option is (b)
Here 2x+y=10 and 2+2y =11
2 x 4 y 22 3 y 12 y 4 x 3
36. In the figure given below what is the value of x if PQRS is a trapezoid and PT=TQ and SU=UR
a. 6 b. 3 c. 2 d. 9 Sol:
x
Correct option is (d)
1 1 (8 10) (18) 9 2 2
37. PQRS is a square of side 8 cm. What is the area of the shaded portion?
a. 32cm 2 b. 64cm 2 c. 16cm 2 d. 2cm 2 Sol: Correct option is (a)
Area of square = 8 8 64cm 2 Area of triangle =
1 8 8 32cm 2 2
Area of shaded portion = 32cm 2 101. What is the distance between the points (2, 5) and (5, 9)? a. 2 b. 4 c. 5 d. 11 Sol: Correct option is (c)
d (5 2) 2 (9 5) 2 32 42 d 25 5
38. What is the center of the circle whose equation is given as ( x 2) 2 ( y 5)2 16 ? a. 1, 5 b. 4, 2 c. 2, 5 d. 5, 4 Sol: Correct option is (c)
In equation ( x a ) 2 ( y b)2 r 2 (a, b) is the center Hence (2, 5) is the center. 39. What is the radius of the circle whose equation is ( x 1) 2 ( y 3) 2 25 ? a. 3 b. 2 c. 1 d. 6 Sol: Correct option is (d)
In equation ( x a ) 2 ( y b)2 r 2 , r is the radius of the circle Here
r 2 25 r 5
40. What is the equation of the circle having origin as center and 4 as radius?
a. x 2 ( y 1) 2 16 b. x 2 y 2 16 c. x 2 ( y 2) 2 4 d. x 2 y 2 4 Sol: Correct option is (b)
Equation can be written as
( x 0)2 ( y 0) 2 42 x 2 y 2 16
41. What is the distance between the centers of the circles whose equations are ( x 1) 2 ( y 3) 2 25 and ( x 9) 2 ( y 9) 2 144 ? a. 10 b. 25 c. 15 d. 7 Sol: Correct option is (a)
Center of 1st circle is (1, 3) Center of 2nd circle is (9,9) Distance between them =
(9 1) 2 (9 3) 2 64 36 100 10
42. Find the area of the square whose 2 corresponding vertices are (3, 7) and (7, 10)?
a. 16 b. 25 c. 64 d. 100 Sol: Correct option is (b)
(7 3) 2 (10 7) 2 42 32 5
Length of side = Area = (5) 2 25
43. What is the slope of the line which passes through (1, 3) and (5, 5)? a. 4 b.
1 3
c. 1 d.
1 2
Sol:
Correct option is (d)
y2 y1 5 3 2 1 x2 x1 5 1 4 2
Slope =
44. What is the value of tan x in the figure given below?
a.
4 3 3 2
b.
c. 3 d.
1 3
Sol:
Correct option is (a)
tan x o = Slope of line
Slope =
y2 y1 3 ( 1) 4 3 0 3 x2 x1
45. What is the slope of the line which is parallel to the line having slope 2? a. 1 b. 3 c. 4 d. 2 Sol: Correct option is (d)
As lines are parallel hence they have equal slope, hence slope of both lines is 2.
1 46. What is the slope of line l which is perpendicular to the line n having slope ? 2
a. -1 b. -2 c. 2 d. Sol:
1 3
Correct option is (b)
Let slope of line l m1 Let slope of line n m2 Here m1m2 1 Also m2
1 2
m1
1 1 2
m1 2
47. What is the slope of altitude AT of ABC , where A (2, 4), B (0, 0) and C (1, 8)? a.
1 4 1 8
b.
c. 4 d. 2 Sol: Correct option is (b)
Here Slope of BC = Let slope of AT = m
m 8 1 m 1 8
80 8 1 0
48. What is the mid point of the segment joining points (2, 3) and (8, 11)? a. (5, 7) b. (7, 8) c. (4, 7) d. (3, 11) Sol: Correct option is (a)
28 x' 5 2 11 3 y' 7 2
Hence mid point is (5, 7) 49. If the mid point of the segment joining A (4, 7) and B is (5, 8) then what are the coordinates of b?
a. (2, 3)
b. (4, 8) c. (6, 9) d. (4, 9) Sol: Correct option is (c)
Let the coordinates of B be (x, y) Then
5 x4 x 10 4 6 2
Also
8 y7 x 16 7 9 2
Coordinates of B are (6, 9) 50. What are the coordinates of the point at which two lines having equations x-y =3 and 2x+y=24 intersect each other a. (6, 4) b. (9, 6) c. (8, 4) d. (9, 12) Sol: Correct option is (b)
Coordinates of the point can be derived by solving both equations x-y =3 and 2x = y =24 Solving these equations x = 9 and y = 6 Hence coordinates are (9, 6) 51. What are the x and y intercepts of the line 3x + 4y = 12?
a. 3, 2 b. 4, 2 c. 4, 3 d. 2, 6 Sol: Correct option is (c)
For x- intercept y =0
3x 12 x 4
For y intercept x = 0
4 y 12 y 3
52. What is the slope of the line 2x + 3y = 8?
a.
2 3 1 2 3 2 2 5
b.
c.
d.
Sol:
Correct option is 9a)
Here equation of line is 2x + 3y = 8
3 y 8 2x y 2 8 x 3 3 2 3
m
53. What is the equation of the line passing through (2, 4) and having slope 3? a. y = 4x +1 b. y = 3x-2 c. y = 3x-5 d. y = x-9 Sol: Correct option is (b)
Equation of line is
y y1 m( x x1 ) y 4 3( x 2) y 3x 6 4 y 3x 2
54. What is the equation of line which passes through (2, 5) and is parallel to y = 5x-8? a. y = 2(x-3) b. y = x +4 c. y = 2x -1 d. y = 5 (x-1) Sol: Correct option is (d)
Equation of line is y y1 m( x x1 ) m=5
y 5 5( x 2) y 5 x 10 5 y 5x 5 y 5( x 1)
55. What is the equation of line passing through (2, 5) and perpendicular to 1 line y x 7 ? 2 a. x + 2y + 3 b. 2x – y = 10 c. 2x + y = 9 d. 2x + 3y = 14 Sol: Correct option is (c)
Equation of line is y y1 m( x x1 ) y1 5, x1 2 Also m = -2
y 5 2( x 2) y 2 x 4 5 2 x y 9
56. What is the equation of line passing through points (2, 7) and (6, 3)?
a. x + y =9 b. x + 2y = 3 c. x + y = 11 d. 2x – y = 13 Sol: Correct option is (a)
73 4 1 2 6 4
Here m =
y 7 1( x 2) y x 2 7 x y 9
57. What is the equation of horizontal line through (5, 4)? a. x = 8 b. y =4 c. y = 3x d. y +x =9 Sol: Correct option is (b)
As the line is horizontal, hence m = 0
y 4 0( x 5) y4
58. What is the equation of a vertical line passing through (7, 11)?
a. y =11 b. x =11 c. y = 2x d. x = 7 Sol: Correct option is (d)
As the line is vertical, hence the value of x remains constant x 7 59. What is the equation of the line passing through (9, 12) and which makes a 45 o angle measured counter clockwise from the positive x-axis? a. y = x + 3 b. 2x + y = 11 c. y = 2x – 3 d. x = y -4 Sol: Correct option is (a)
Here m = tan 45o 1 Hence equation of line is y -12 = 1 (x-9)
y x3
60. What is the equation of the perpendicular bisector of the segment joining (2, 4) and (4, 8)?
a. 2x + 3 Y = 16 b. x + 2y = 15 c. x – y = 4 d. x = 3y Sol: Correct option is (b)
84 2 42 1 2
Here slope of segment =
Slope of perpendicular bisector =
Perpendicular bisector passes through (3, 6) Hence equation of perpendicular bisector is
y6 1 ( x 3) 2
2 y 12 x 3 x 2 y 15
61. What is the equation of the perpendicular bisector of the segment joining (3, 7) and (5, 13)? a. x + 3y = 34 b. 2x + y = 19 c. x = 3y d. y = 24 Sol: Correct option is (a)
13 7 6 3 53 2
Slope of segment =
Slope of perpendicular bisector =
1 3
Also perpendicular bisector passes through (4, 10) Hence equation of perpendicular bisector is
y 10 1 ( x 4) 3
3 y 30 x 4 x 3 y 34