Volume & Surface Area of Solids — Assertion-Reason Type Questions

Assertion (A) : Slant height of a cone of height 4 cm and radius 3 cm is (4 + 3) cm = 7 cm.

Reason (R) : Curved surface area of a cone of radius r and slant height l is πrl.

1. Both A and R are true, and R is the correct explanation of A.

2. Both A and R are true, but R is not the correct explanation of A.

3. A is true, but R is false.

4. A is false, but R is true.

Answer

Given, radius r = 3 cm and height h = 4 cm.

By formula, l² = h² + r² = 4² + 3² = 16 + 9 = 25

⇒ l = $\sqrt{25}$ = 5 cm.

So, the slant height is 5 cm, not (4 + 3) = 7 cm.

∴ Assertion (A) is false.

The curved surface area of a cone of radius r and slant height l is πrl.

∴ Reason (R) is true.

Hence, option 4 is the correct option.

Assertion (A) : The maximum volume of a cone that can be carved out of a solid hemisphere of radius r is $\dfrac{1}{3}$ πr³.

Reason (R) : For a cone of radius r and height h, slant height is given by $\sqrt{h^2 + r^2}$.

1. Both A and R are true, and R is the correct explanation of A.

2. Both A and R are true, but R is not the correct explanation of A.

3. A is true, but R is false.

4. A is false, but R is true.

Answer

The cone of maximum volume that can be carved out of a solid hemisphere of radius r has the same base radius r and the same height r.

Volume of cone = $\dfrac{1}{3}$ πr²h = $\dfrac{1}{3}$ πr² × r = $\dfrac{1}{3}$ πr³.

∴ Assertion (A) is true.

For a cone of radius r and height h, slant height l = $\sqrt{h^2 + r^2}$.

∴ Reason (R) is true.

Reason (R) is a correct statement, but it states the slant-height formula, which is not used to establish the maximum volume in the Assertion. So, R is not the correct explanation of A.

Hence, option 2 is the correct option.

Assertion (A) : The total surface area of a right circular cone of slant height 13 cm and radius 5 cm is 90 π cm².

Reason (R) : Curved surface area of a right circular cone is given by πr(l + r).

1. Both A and R are true, and R is the correct explanation of A.

2. Both A and R are true, but R is not the correct explanation of A.

3. A is true, but R is false.

4. A is false, but R is true.

Answer

Given, radius r = 5 cm and slant height l = 13 cm.

Total surface area of cone = πr(l + r) = π × 5 × (13 + 5) = π × 5 × 18 = 90π cm².

∴ Assertion (A) is true.

The expression πr(l + r) is the formula for the total surface area of a cone, whereas the curved surface area of a cone is πrl. So, Reason (R) wrongly calls πr(l + r) the curved surface area.

∴ Reason (R) is false.

Hence, option 3 is the correct option.

Assertion (A) : Two solid spheres of radii 2 cm and 4 cm are melted and recast into a cone of radius 6 cm. The height of the cone so obtained will be 8 cm.

Reason (R) : When we convert one solid into another, the volume of the two solids remains the same.

1. Both A and R are true, and R is the correct explanation of A.

2. Both A and R are true, but R is not the correct explanation of A.

3. A is true, but R is false.

4. A is false, but R is true.

Answer

Total volume of the two spheres = $\dfrac{4}{3}$ π(2)³ + $\dfrac{4}{3}$ π(4)³ = $\dfrac{4}{3}$ π(8 + 64) = $\dfrac{4}{3}$ π × 72 = 96π cm³.

Let the height of the cone be h cm. Radius of cone = 6 cm.

Volume of cone = $\dfrac{1}{3}$ π(6)²h = 12πh.

On recasting, the volume remains the same.

⇒ 12πh = 96π ⇒ h = 8 cm.

∴ Assertion (A) is true.

When one solid is converted (melted and recast) into another, the volume remains unchanged, and this is exactly the principle used to find the height.

∴ Reason (R) is true and is the correct explanation of A.

Hence, option 1 is the correct option.

A solid sphere is cut into two identical hemispheres.

Assertion (A): The total volume of two hemispheres is equal to the volume of the original sphere.

Reason (R): The total surface area of two hemispheres together is equal to the surface area of the original sphere.

1. Both A and R are true, and R is the correct explanation of A.

2. Both A and R are true, but R is not the correct explanation of A.

3. A is true, but R is false.

4. A is false, but R is true.

Answer

By formula,

Volume of sphere = $\dfrac{4}{3} \pi r^3$

Given,

A solid sphere is cut into two identical hemispheres.

Volume of hemisphere = $\dfrac{2}{3} \pi r^3$

Volume of two identical hemispheres = $2 \times \dfrac{2}{3} \pi r^3$

= $\dfrac{4}{3} \pi r^3$

Thus, volume of a sphere = volume of two identical hemispheres.

So assertion (A) is true.

We know that,

Surface area of sphere = 4πr²

When a sphere is cut into two hemispheres, two new flat circular surfaces are created,

Surface area of a single hemisphere = Curved surface area + Area of its flat circular face

= 2πr² + πr²

= 3πr²

Total surface area of two hemispheres = 2 × 3πr² = 6πr².

Thus, the surface area of the original sphere ≠ the total surface area of the two hemispheres.

So reason (R) is false.

(A) is true, (R) is false.

Hence, option 3 is the correct option.