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Mensuration Test - 12

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Mensuration Test - 12

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Question 1
1 / -0

What is the volume of a right circular cone with base radius p and height h?

A
3p²h

B
p²h

C
3ph²

D
ph²

Solution

Given:
Radius of the cone (r) = p
Height of the cone = h
Volume of the cone = r²h
=
=
Question 2
1 / -0

If the volume of a hemisphere is cm³, its radius is

A
4 cm

B
2 cm

C
cm

D
cm

Solution

Volume of the hemisphere = cm³
We know that volume of the hemisphere = r³, where r is the radius of the hemisphere.

r³ = 8
r = 2 cm
Question 3
1 / -0

The heights of two cones are equal. If the ratio of their radii is 3 : 2, the ratio of their volumes is

A
9 : 4

B
27 : 8

C
3 : 2

D
4 : 9

Solution

Volume of cone =
Volume of the 1^st cone =
Volume of the 2^ndcone =
Also, h₁ = h₂(given)
And
Ratio,
= [ h₁ = h₂]
=
Question 4
1 / -0

The surface area of a sphere with radius 5 cm is

A
50 cm²

B
cm²

C
100 cm²

D
25 cm²

Solution

Given: Radius = 5 cm
Surface area of the sphere = 4r²
= 4(5)²cm²
= 4 25 cm²
= 100 cm²
Question 5
1 / -0

The curved surface area of a right circular cone, whose base radius is 3 cm and slant height is 4 cm, is

A
24 cm²

B
12 cm²

C
48 cm²

D
21 cm²

Solution

Given: Base radius (r) of the cone = 3 cm
Slant height () of the cone = 4 cm
Curved surface area of the cone = r
= 3 4
= 12 cm²
Question 6
1 / -0

The total surface area of a right circular cone is 330 cm². If the radius of its base is 7 cm, then find its slant height.

A
10 cm

B
15 cm

C
9 cm

D
8 cm

Solution

Given:
Radius of the base of the cone = 7 cm
Total surface area = 330 cm²
Total surface area = r(r + )
330 cm² = cm
= 15 cm – 7 cm
= 8 cm
Question 7
1 / -0

The radius of a sphere with volume cm³ is ______.

A
cm

B
4 cm

C
4 cm

D
2 cm

Solution

Volume of sphere =

8 = r³
r = 2 cm
Question 8
1 / -0

The ratio of the radii of two spheres is 2 : 1. What is the ratio of the surface areas of the two spheres?

A
2 : 1

B
4 : 1

C
1 : 4

D
8 : 1

E
1 : 6

Solution

Given: Radii of the two spheres are in the ratio 2 : 1.
If r₁ is the radius of the 1^st sphere and r₂ is the radius of the 2^ndsphere, then .
Now,
=
=
= ∴ Ratio = 4 : 1
Question 9
1 / -0

The radii of two hemispheres are 2 cm and 4 cm. The ratio of their volumes is ____.

A
1 : 2

B
8 : 1

C
4 : 16

D
1 : 8

Solution

We know that volume of hemisphere =
Given, Radius (r₁) of the I^st hemisphere = 2 cm
Radius (r₂) of the II^nd hemisphere = 4 cm
Ratio of volume =
=
=
=
=
Question 10
1 / -0

Two right circular cones have equal slant height. If the ratio of their base radii is 5 : 3, then the ratio of their curved surface areas is

A
3 : 5

B
25 : 9

C
5 : 3

D
9 : 25

Solution

Given: ( are the slant heights of the first and second cones, respectively.)
And (r₁ and r₂ are the base radii of the first and second cones, respectively.)
Also, curved surface area of a cone =
Ratio of curved surface areas of the two cones = Curved surface area of the first cone/Curved surface area of the second cone
=
=
= [ ]
Ratio = 5 : 3
Question 11
1 / -0

If the base radius of a cone is 5 cm and its slant height is 8 cm, its total surface area is ______.

A
65 cm²

B
75 cm²

C
40 cm²

D
130 cm²

Solution

Total surface area of cone =
r = radius of base of cone
= slant height
Given r = 5 cm
= 8 cm
Total surface area = r (r + )
= 5 (5 + 8)
= 5 (13)
= 65 cm²
Question 12
1 / -0

The height of a right circular cone, whose base radius is 5 cm and volume is 25 cm³, is

A
3 cm

B
3.75 cm

C
4 cm

D
2.25 cm

Solution

Volume of a cone =
where r = radius of the cone
h = height of the cone
^{According to the question:}25 =
25 =

h = 3 cm
Question 13
1 / -0

The curved surface area of a hemisphere with radius 2 cm is

A
6 cm²

B
8 cm²

C
12 cm²

D
16 cm²

Solution

Curved surface area of a hemisphere = 2r², where 'r' is the radius of the hemisphere.
= 2 (2)²
= 2 4 cm²
= 8 cm²
Question 14
1 / -0

The volume (in unit³) of a hemisphere with radius 2a units is

A
a³

B
a³

C
a³

D
a³

Solution

Given: radius (r) of the hemisphere = 2a units
Volume (in unit³) of the hemisphere = r³
=
= 8a³
=
Question 15
1 / -0

The surface area of a sphere of radius 3 cm is

A
24 cm²

B
36 cm²

C
18 cm²

D
9 cm²

Solution

Given:
Radius of the sphere (r) = 3 cm
Surface area of the sphere = 4r²
= 4(3)²cm²
= 36 cm²
Question 16
1 / -0

The volume of a sphere of radius 3 cm is _______.

A
36 cm³

B
^₂₄ cm³

C
18 cm³

D
12 cm³

Solution

Given, radius (r) of sphere = 3 cm
Volume of sphere = r³
=
= (3 3 3)
= x 27 = 36 cm³
Question 17
1 / -0

The slant height and the base radius of a cone are 'a' and 'b', respectively. Its curved surface area is

A
ab

B
a²b

C
ab²

D
b(a + b)

Solution

Given:
Slant height () of the cone = a
Base radius of the cone = b
Curved surface area of the cone = r = ab
Question 18
1 / -0

If the total surface area of a right circular cone with base radius 6 cm is 96 cm², then its altitude is

A
4 cm

B
16 cm

C
8 cm

D
10 cm

Solution

Total surface area of the cone =r(r + )
So, 96 = 6(6 + )

16 = 6 +
= 16 - 6
= 10 cm
Now,
h =
=
=
=
=
= 8 cm
Question 19
1 / -0

The volume of a right circular cone, whose base radius is 3 cm and height is 5 cm, is

A
60 cm³

B
45 cm³

C
25 cm³

D
15 cm³

Solution

Volume of cone = r²h (r is the base radius of the cone and h is the height of the cone)
= (3)² 5
= (3 3) 5
= 15 cm³
Question 20
1 / -0

If the radius of a hemisphere is 3 cm, then its total surface area is

A
27 cm²

B
9 cm²

C
18 cm²

D
36 cm²

Solution

Given:
Radius of the hemisphere (r) = 3 cm
Total surface area of the hemisphere = 3r²
= 3(3)²cm²
= 3 3 3 cm²
= 27 cm²
Question 21
1 / -0

The radius of a hemisphere with volume 18 cm³ is

A
cm

B
3 cm

C
9 cm

D
cm

Solution

Given, volume of hemisphere = 18 cm³
Volume of hemisphere = r³
18 = r³
18 = r³
r³ = 27
r = 3 cm
Question 22
1 / -0

If the surface area of a sphere is 616 cm², then its radius is

A
22 cm

B
cm

C
7 cm

D
cm

Solution

Surface area of the sphere = 4r²
Or 616 cm² = 4r²
616 cm² = 4 r² ( = )
cm²= r²
49 cm² = r²
r = 7 cm
Question 23
1 / -0

The ratio of the volumes of two spheres of radii 2a and 3a is _____.

A
8 : 27

B
27 : 8

C
4 : 9

D
2 : 3

Solution

Radius (r₁) of the I^st sphere = 2a
Radius (r₂) of the II^nd sphere = 3a
Then, ratio of volumes =
=
=
= =
Ratio = 8 : 27
Question 24
1 / -0

The curved surface area of a right circular cone is 22 cm². If the slant height of the cone is 7 cm, then find its base radius.

A
5 cm

B
1 cm

C
2 cm

D
3 cm

Solution

Given:
Slant height () of the cone = 7 cm
And curved surface area of the cone = 22 cm²
Curved surface area of the cone = r
Or 22 cm² = cm

r = 1 cm
Question 25
1 / -0

If the curved surface area of a hemisphere is cm², its radius is _____.

A
cm

B
cm

C
cm

D
cm

Solution

Given, curved surface area of hemisphere =
We know that curved surface area of hemisphere = 2r²
Therefore, = 2r²
r² =
r = cm
Question 26
1 / -0

If the total surface area of a hemisphere is 48 cm², its radius is _____.

A
cm

B
cm

C
4 cm

D
8 cm

Solution

Given, total surface area of hemisphere = 48 cm²
We know that total surface area of hemisphere = 3r²
Thus, 48 = 3r²

16 = r²
r = 4 cm [ Length is always positive.]
Question 27
1 / -0

The radius and slant height of a right circular cone are 7 cm and 10 cm respectively. Its total surface area is _____.

A
374 cm²

B
364 cm²

C
220 cm²

D
748 cm²

Solution

Given, radius (r) of cone = 7 cm
Slant height (h) of cone = 10 cm
Total surface area of cone = r(r + )
= 7 (7 + 10)
= cm²
= 374 cm²
Question 28
1 / -0

The volume of a cone is 32 cm³. If its height is 6 cm, its base radius is _____.

A
cm

B
4 cm

C
cm

D
8 cm

Solution

Volume of cone = r²h
32 = r²x 6
= r²
r² = 16
r = 4 cm
Question 29
1 / -0

If the radius of a sphere is 3k, its surface area is _____.

A
9 k²

B
36 k²

C
18 k²

D
27 k²

Solution

Given, radius (r) of sphere = 3k
Surface area of sphere = 4r²
= 4 (3k)²
= 4 (9k²)
= 36k²
Question 30
1 / -0

A sphere with radius r is cut into two halves by a plane passing through the centre of the sphere. Two hemispheres of same radii are formed. Which of the following relations is true in this case?

A
Surface area of the sphere = Sum of the surface areas of the two hemispheres

B
Surface area of the sphere = Sum of the curved surface areas of the two hemispheres

C
Surface area of each hemisphere = Surface area of the sphere

D
Surface area of the sphere = Sum of the total surface areas of the two hemispheres

Solution

Surface area of sphere = 4r²
And curved surface area of hemisphere = 2r²
A sphere is cut into two hemispheres.
So, surface area of sphere = C.S.A of the I^st hemisphere + C.S.A of the II^nd hemisphere
4r² = 2r² + 2r²
4r² = 4r²
L.H.S = R.H.S
Question 31
1 / -0

If the radius of a hemisphere is cm, its total surface area is ______.

A
cm²

B
cm²

C
cm²

D
cm²

Solution

Given, radius (r) of hemisphere = cm
Then, total surface area of hemisphere = 3r²
= 3
= cm²
Question 32
1 / -0

The volume (in terms of ) of a sphere with radius 2 cm is

A
cm³

B
cm³

C
cm³

D
cm³

Solution

Given, radius (r) of sphere = 2 cm
Volume of sphere = r³
= 8 cm³
= cm³
Question 33
1 / -0

The curved surface area of a right circular cone with base radius 2 cm and slant height 4 cm is

A
12 cm²

B
cm²

C
8 cm²

D
cm²

Solution

Base radius (r) of the cone = 2 cm
Slant height () of the cone = 4 cm
Curved surface area of the cone = r
= 2 cm 4 cm
= 8 cm²
Question 34
1 / -0

The base radius of a cone is 8 cm and the slant height is 12 cm. Its total surface area is _____.

A
160 cm²

B
96 cm²

C
140 cm²

D
192 cm²

Solution

Given, base radius (r) of cone = 8 cm
Slant height of cone = 12 cm
Then, total surface area = r (r + )
= 8 (8 + 12)
= 8 (20)
= 160 cm²
Question 35
1 / -0

The radii of two hemispheres are 3 cm and 4 cm. What will be the ratio of their curved surface areas?

A
16 : 9

B
3 : 4

C
9 : 16

D
27 : 64

Solution

Given radius (r₁) of the I^st hemisphere = 3 cm
Radius (r₂) of the II^nd hemisphere = 4 cm
Then, ratio =
=
=
=
Question 36
1 / -0

If the radius of a hemisphere is cm, its total surface area is _____.

A
cm²

B
cm²

C
cm²

D
cm²

Solution

Given, radius (r) of hemisphere = cm
Then, total surface area = 3r²
= 3
= 3
= cm²
Question 37
1 / -0

The radius of a sphere with volume 36 cm³ is _____.

A
3 cm

B
cm³

C
6 cm

D
2 cm

Solution

Volume of sphere = 36
r³ = 36
r³ = 36
r³ = 3 3 3
Thus, r = 3 cm
Question 38
1 / -0

If the radius of a hemisphere is 6 cm, its volume is _____.

A
108 cm³

B
72 cm³

C
144 cm³

D
288 cm³

Solution

Given, radius (r) of hemisphere = 6 cm
Volume of hemisphere = r³
= (6)³
= 2 3 6 6
= 144 cm³
Question 39
1 / -0

If the radius of a hemisphere is 5 cm, its curved surface area is _____.

A
75 cm²

B
100 cm²

C
25 cm²

D
50 cm²

Solution

Given, radius of hemisphere = 5 cm
Curved surface area of hemisphere = 2r²
= 2 (5)²
= 2 25
= 50 cm²
Question 40
1 / -0

The ratio of the radii of two hemispheres is 5 : 3. The ratio of their total surface areas is _____.

A
9 : 25

B
25 : 9

C
5 : 3

D
125 : 27

Solution

Given ratio,
Now ratio,
=
=
=
=
Required ratio = 25 : 9

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