Mensuration — Assertion-Reason Type Questions

Assertion (A): The surface area of largest sphere that can be inscribed in a hollow cube of side 'a' cm is πa² cm².

Reason (R): The surface area of sphere of radius 'r' is $\dfrac{4}{3}πr^3$.

1. Assertion (A) is true, but Reason (R) is false.

2. Assertion (A) is false, but Reason (R) is true.

3. Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

Given, a sphere that can be inscribed in a hollow cube.

The maximum diameter of the sphere that can be inscribed is equal to the side of the cube (a).

The radius of the sphere is half of the diameter: r = $\dfrac{a}{2}$

The surface area of a sphere is given by 4πr².

So, reason(R) is false.

The surface area of a sphere = 4 x π x $\Big(\dfrac{a}{2}\Big)^2$

= 4 x π x $\dfrac{a^2}{4}$

= πa² cm².

So, assertion (A) is true.

Thus, Assertion (A) is true, but Reason (R) is false.

Hence, option 1 is the correct option.

Assertion (A): From a solid wooden cylinder of height 15 cm and diameter 14 cm, of conical cavity of same height and same base diameter is hollowed out. The volume of the cone is 770 cm³.

Reason (R): The volume of a cylinder of height h and radius r is πr²h.

Answer

Given,

Height of cylinder,(H) = 15 cm

Diameter of cylinder, (D) = 14 cm

Radius of cylinder, r = $\dfrac{D}{2} = \dfrac{14}{2}$ = 7 cm

Height of cone,(h) = 15 cm

Radius of cone, (r) = 7 cm

The volume of the cone is given by the formula = $\dfrac{1}{3}$ πr²h

$= \dfrac{1}{3} \times \dfrac{22}{7} \times 7^2 \times 15 \\[1em] = \dfrac{1}{3} \times 22 \times 7 \times 15 \\[1em] = 22 \times 7 \times 3\\[1em] = 462$

So, assertion (A) is false.

The volume of a cylinder of height h and radius r is given by the formula; πr²h.

So, reason (R) is true.

Thus, Assertion (A) is false, but Reason (R) is true.

Hence, option 2 is the correct option.

Assertion (A): From a solid wooden cylinder of height 15 cm and diameter 14 cm, a hemispherical depression of same base diameter is carved out. The volume of remaining wood is $1591\dfrac{1}{3}$ cm³.

Reason (R): Volume of a cylinder of radius r and height h is πr²h and the volume of a hemisphere of radius r is $\dfrac{2}{3}$ πr³.

Answer

Given,

Height of cylinder,(H) = 15 cm

Diamter of cylinder, (D) = 14 cm

Radius of cylinder, r = $\dfrac{D}{2} = \dfrac{14}{2}$ = 7 cm

Radius of hemisphere, (r) = 7 cm

The volume of a cylinder is πr²h and the volume of a hemisphere is $\dfrac{2}{3}$ πr³.

So, reason (R) is true.

Volume of remaining wood = Volume of cylinder - Volume of hemisphere

$= πR^2H - \dfrac{2}{3}πr^3\\[1em] = \dfrac{22}{7} \times 7^2 \times 15 - \dfrac{2}{3}\times \dfrac{22}{7} \times 7^3\\[1em] = 22 \times 7 \times 15 - \dfrac{2}{3}\times 22 \times 7^2\\[1em] = 2310 - \dfrac{2156}{3}\\[1em] = \dfrac{6930 - 2156}{3}\\[1em] = \dfrac{4774}{3}\\[1em] = 1591\dfrac{1}{3}\\[1em]$

So, assertion (A) is true.

Thus, both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

Hence, option 3 is the correct option.