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Question:

Direction: Each of these items has a question followed by two statements, I and II. Mark the answer

1. if the question can be answered with the help of one statement alone.
2. if the question can be answered with the help of any one statement independently.
3. if the question can be answered with the help of both statements together.
4. if the question cannot be answered even with the help of both statements together.

What is the ratio of the volume of the given right circular cone to the one obtained from it?

I. The smaller cone is obtained by passing a plane parallel to the base and dividing the original height in the ratio 1 : 2.
II. The height and the base of the new cone are one-third those of the original cone.

Explanation:

Let V₁ be the original volume and r₁ and h₁be the radius and height of the cone respectively.

V₁ = $(\frac{1}{3})$ × π × (r₁)² × h₁. Consider statement I. If the cone is cut parallel to base and dividing the height in
the ratio 1 : 2, then r₂ = $(\frac{1}{2}) \times r_{1}$ and h₂ = $(\frac{1}{2})$ × h₁, where r₂ and h₂ are the radius and height of the new
cone respectively. If V₂ is the volume of new cone, then V2 = $(\frac{1}{3})$ × π(r₂)²× h₂

$= (\frac{1}{3}) \times π \times {(\frac{1}{2} \times r_{1})}^{2} \times ((\frac{1}{2}) h_{1}) = (\frac{1}{8}) \times V_{1}$

Hence, statement I alone is sufficient to answer the question (as we get the ratio as 1 : 8).

Similarly, based on statement II alone, we can find the ratio (which will be 1 : 27).

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