Hard Mensuration (2D/3D) Quiz - Concept of Maths (Set 3)

Question 1

A sphere $S_1$ is inscribed in a cube $C$, and another sphere $S_2$ circumscribes the same cube $C$. What is the ratio of the volume of $S_1$ to the volume of $S_2$?

Accepted Answer

C) $1 : 3\sqrt{3}$

Answer

A) $1 : 3$

Answer

B) $1 : \sqrt{3}$

Answer

D) $1 : 9$

Question 2

A right circular cone is cut into two parts by a plane parallel to its base, passing through the midpoint of its axis. What is the ratio of the volume of the smaller cone to the volume of the frustum formed?

Accepted Answer

B) $1 : 7$

Answer

A) $1 : 4$

Answer

C) $1 : 8$

Answer

D) $1 : 9$

Question 3

A wire of length $L$ is bent into the shape of an equilateral triangle. If the same wire is instead bent into the shape of a circle, find the ratio of the area of the triangle to the area of the circle.

Accepted Answer

A) $\pi\sqrt{3} : 9$

Answer

B) $\pi : 3\sqrt{3}$

Answer

C) $3\sqrt{3} : \pi$

Answer

D) $\sqrt{3} : 4\pi$

Question 4

The radius of a cylinder is increased by 25% and its height is decreased by 40%. What is the percentage change in its volume?

Accepted Answer

D) 6.25% decrease

Answer

A) 15% increase

Answer

B) 10% decrease

Answer

C) 5.25% increase

Question 5

A hollow spherical shell has an inner radius of 3 cm and an outer radius of 6 cm. It is melted and recast into a solid cylinder of height 28 cm. What is the radius of the cylinder?

Accepted Answer

C) 3 cm

Answer

A) 1.5 cm

Answer

B) 2 cm

Answer

D) 4.5 cm

Question 6

A sector of a circle with radius 12 cm and central angle $120^\circ$ is rolled up so that the two bounding radii are joined together to form a cone. Find the slant height of the cone.

Accepted Answer

B) 12 cm

Answer

A) 4 cm

Answer

C) 8 cm

Answer

D) $4\sqrt{3}$ cm

Question 7

Three circles of radius 7 cm each are placed such that each circle touches the other two externally. What is the area of the region enclosed between the three circles?

Accepted Answer

A) $49\sqrt{3} - 77$

Answer

B) $49\sqrt{3} - 154$

Answer

C) $98\sqrt{3} - 77$

Answer

D) $147 - 49\pi$

Question 8

A rectangular sheet of dimensions 22 cm $ imes$ 10 cm is rolled along its length to form a cylinder. If the same sheet is rolled along its breadth, what is the ratio of the volume of the first cylinder to the second?

Accepted Answer

D) $11 : 5$

Answer

A) $1 : 1$

Answer

B) $5 : 11$

Answer

C) $1 : 2.2$

Question 9

If the surface area of a cube and a sphere are equal, what is the ratio of the volume of the cube to the volume of the sphere?

Accepted Answer

A) $\sqrt{\pi} : \sqrt{6}$

Answer

B) $\sqrt{6} : \sqrt{\pi}$

Answer

C) $\pi : 6$

Answer

D) $1 : 1$

Question 10

A regular square pyramid has a base edge of 10 cm and a vertical height of 12 cm. Find its lateral surface area.

Accepted Answer

C) 260 cm$^2$

Answer

A) 240 cm$^2$

Answer

B) 360 cm$^2$

Answer

D) 300 cm$^2$