RD Sharma 2020 solution class 9 chapter 21 Surface Area and Volume of a Sphere MCQS

MCQS

Page-21.24

Question 1:

Mark the correct alternative in each of the following:

In a sphere the number of faces is

(a) 1

(b) 2

(c) 3

(d) 4

Answer 1:

A sphere has only a single face. Since there are no sides of a sphere, it has a single continuous face.

Therefore, the correct option is (a)

Question 2:

The total surface area of a hemisphere of radius r is

(a) ${\mathrm{\pi r}}^2$

(b) 2${\mathrm{\pi r}}^2$

(c) 3${\mathrm{\pi r}}^2$

(d) 4${\mathrm{\pi r}}^2$

Answer 2:

The curved surface area of a hemisphere of radius r is. So, the total surface area of a hemisphere will be the sum of the curved surface area and the area of the base.

Total surface area of a hemisphere of radius r =

Therefore, the correct option is (c)

Question 3:

The ratio of the total surface area of a sphere and a hemisphere of same radius is

(a) 2 : 1

(b) 3 : 2

(c) 4 : 1

(d) 4 : 3

Answer 3:

In the given question,

The total surface area of a sphere (S₁) =

The total surface area of a hemisphere (S₂) =

So the ratio of the total surface area of a sphere and a hemisphere will be,

Therefore, the ratio of the surface areas is. So, the correct option is (d)

Page-21.25

Question 4:

A sphere and a cube are of the same height. The ratio of their volumes is

(a) 3 : 4

(b) 21 : 11

(c) 4 : 3

(d) 11 : 21

Answer 4:

In the given problem, we have a sphere and a cube of equal heights. So, let the diameter of the sphere and side of the cube be x units.

So, volume of the sphere (V₁) =

Volume of the cube (V₂) =

So, to find the ratio of the volumes,

Therefore, the ratio of the volumes of sphere and cube of equal heights is . So, the correct option is (d).

Question 5:

The largest sphere is cut off from a cube of side 6 cm. The volume of the sphere will be

(a) 27$\mathrm{\pi }$ cm³

(b) 36$\mathrm{\pi }$ cm³

(c) 108$\mathrm{\pi }$ cm³

(d) 12$\mathrm{\pi }$ cm³

Answer 5:

In the given problem, the largest sphere is carved out of a cube and we have to find the volume of the sphere.

Side of a cube = 6 cm

So, for the largest sphere in a cube, the diameter of the sphere will be equal to side of the cube.

Therefore, diameter of the sphere = 6 cm

Radius of the sphere = 3 cm

Now, the volume of the sphere =

Therefore, the volume of the largest sphere inside the given cube is . So, the correct option is (b).

Question 6:

A cylindrical rod whose height is 8 times of its radius is melted and recast into spherical balls of same radius. The number of balls will be

(a) 4

(b) 3

(c) 6

(d) 8

Answer 6:

In the given problem, we have a cylindrical rod of the given dimensions:

Radius of the base (r_c) = x units

Height of the cylinder (h) = 8x units

So, the volume of the cylinder (V_c) =

Now, this cylinder is remolded into spherical balls of same radius. So let us take the number of balls be y.

Total volume of y spheres (V_s) =

So, the volume of the cylinder will be equal to the total volume of y number of balls.

We get,

Therefore, the number of balls that will be made is. So, the correct option is (c)

Question 7:

If the ratio of volumes of two spheres is 1 : 8, then the ratio of their surface areas is

(a) 1 : 2

(b) 1 : 4

(c) 1 : 8

(d) 1 : 16

Answer 7:

Here, we are given that the ratio of the two spheres of ratio 1:8

Let us take,

The radius of 1^st sphere = r₁

The radius of 1^st sphere = r₂

So,

Volume of 1^st sphere (V₁) =

Volume of 2^nd sphere (V₂) =

Now,

Now, let us find the surface areas of the two spheres

Surface area of 1^st sphere (S₁) =

Surface area of 2^nd sphere (S₂) =

So, Ratio of the surface areas,

Using (1), we get,

Therefore, the ratio of the spheres is. So, the correct option is (b)

Question 8:

If the surface area of a sphere is 144π m², then its volume (in m³) is
 
(a) 288 π

(b) 316 π

(c) 300 π

(d) 188 π

Answer 8:

In the given problem,

Surface area of a sphere = m²

So,

Now, using the formula volume of the sphere, we get

Therefore, volume of the sphere is. Therefore the correct option is (a)

Question 9:

If a solid sphere of radius 10 cm is moulded into 8 spherical solid balls of equal radius, then the surface area of each ball (in sq.cm) is

(a) 100 π

(b) 75 π

(c) 60 π

(d) 50 π

Answer 9:

In the given problem, Let the radius of smaller spherical balls which can be made from a bigger ball be x units.

Here,

The radius of the bigger ball (r₁) = 10 cm

The radius of the smaller ball (r₂) = x cm

The number of smaller balls = 8

So, volume of the big ball is equal to the volume of 8 small balls.

Volume of the big balls = volume of the 8 small balls

Further, solving for x, we get,

Now, surface area of a small ball of radius 5 cm =

Therefore, the surface area of the small spherical ball is. So, the correct option is (a).

Question 10:

If a sphere is inscribed in a cube, then the ratio of the volume of the sphere to the volume of the cube is

(a) π : 2

(b) π : 3

(c) π : 4

(d) π : 6

Answer 10:

In the given problem, we are given a sphere inscribed in a cube. So, here we need to find the ratio between the volume of a sphere and volume of a cube. This means that the diameter of the sphere will be equal to the side of the cube. Let us take the diameter as d.

Here,

Volume of a sphere (V₁) =

Volume of a cube (V₂) =

Now, the ratio of the volume of sphere to the volume of the cube =

So, the ratio of the volume of sphere to the volume of the cube is . Therefore, the correct option is (d)

Question 11:

If a solid sphere of radius r is melted and cast into the shape of a solid cone of height r, then the radius of the base of the cone is

(a) 2r

(b) 3r

(c) r

(d) 4r

Answer 11:

In the given problem, we have a solid sphere which is remolded into a solid cone such that the radius of the sphere is equal to the height of the cone. We need to find the radius of the base of the cone.

Here, radius of the solid sphere (r_s) = r cm

Height of the solid cone (h) = r cm

Let the radius of the base of cone (r_c) = x cm

So, the volume of cone will be equal to the volume of the solid sphere.

Therefore, we get,

Therefore, radius of the base of the cone is. So, the correct option is (a).

Question 12:

A sphere is placed inside a right circular cylinder so as to touch the top, base and lateral surface of the cylinder. If the radius of the sphere is r, then the volume of the cylinder is

(a) 4πr³

(b) $\frac{8}{3}$πr³

(c) 2πr³

(d) 8πr³

Answer 12:

In the given problem, we have a sphere inscribed in a cylinder such that it touches the top, base and the lateral surface of the cylinder. This means that the height and the diameter of the cylinder are equal to the diameter of the sphere.

So, if the radius of the sphere = r

The radius of the cylinder (r_c)= r

The height of the cylinder (h) = 2r

Therefore, Volume of the cylinder =

So, the volume of the cylinder is. Therefore, the correct option is (c).

Question 13:

The ratio between the volume of a sphere and volume of a circumscribing right circular cylinder is

(a) 2 : 1

(b) 1 : 1

(c) 2 : 3

(d) 1 : 2

Answer 13:

In the given problem, we need to find the ratio between the volume of a sphere and volume of a circumscribing right circular cylinder. This means that the diameter of the sphere and the cylinder are the same. Let us take the diameter as d.

Here,

Volume of a sphere (V₁) =

As the cylinder is circumscribing the height of the cylinder will also be equal to the height of the sphere. So,

Volume of a cylinder (V₂) =

Now, the ratio of the volume of sphere to the volume of the cylinder =

So, the ratio of the volume of sphere to the volume of the cylinder is . Therefore, the correct option is (c)

Question 14:

A cone and a hemisphere have equal bases and equal volumes the ratio of their heights is

(a) 1 : 2

(b) 2 : 1

(c) 4 : 1

(d) $\sqrt{2}$ : 1

Answer 14:

In the given problem, we are given a cone and a hemisphere which have equal bases and have equal volumes. We need to find the ratio of their heights.

So,

Let the radius of the cone and hemisphere be x cm.

Also, height of the hemisphere is equal to the radius of the hemisphere.

Now, let the height of the cone = h cm

So, the ratio of the height of cone to the height of the hemisphere =

Here, Volume of the hemisphere = volume of the cone

Therefore, the ratio of the heights of the cone and the hemisphere is. So, the correct option is (b)

Question 15:

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

(a) 1 : 2 : 3

(b) 2 : 1 : 3

(c) 2 : 3 : 1

(d) 3 : 2 : 1

Answer 15:

In the given problem, we are given a cone, a hemisphere and a cylinder which stand on equal bases and have equal heights. We need to find the ratio of their volumes.

So,

Let the radius of the cone, cylinder and hemisphere be x cm.

Now, the height of the hemisphere is equal to the radius of the hemisphere. So, the height of the cone and the cylinder will also be equal to the radius.

Therefore, the height of the cone, hemisphere and cylinder = x cm

Now, the next step is to find the volumes of each of these.

Volume of a cone (V₁) =

Volume of a hemisphere (V₂) =

Volume of a cylinder (V₃) =

So, now the ratio of their volumes = (V₁) : (V₂) : (V₃)

Therefore, the ratio of the volumes of the given cone, hemisphere and the cylinder is . So, the correct option is (a).