Question:medium

A right circular cone (PQR) is cut into two parts, cone (C) and frustum (F) by a plane parallel to base.
What is the ratio of volume of C to the volume of F?
Statement 1: PQR is twice the radius of C
Statement 2: PQR is been cut off at the middle of its height
Directions: This question has a problem and two statements numbered (1) and (2) giving certain information. You have to decide if the information given in the statements is sufficient for answering the problem. Indicate your answer

Updated On: Jun 30, 2026
  • statement (1) alone is sufficient to answer the question
  • statement (2) alone is sufficient to answer the question
  • both the statements together are needed to answer the question
  • either statement (1) alone or statement (2) alone is sufficient to answer the question
  • neither statement (1) nor statement (2) suffices to answer the question
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The Correct Option is D

Solution and Explanation

The correct answer is option (D):
either statement (1) alone or statement (2) alone is sufficient to answer the question

Let's analyze the problem. We have a right circular cone PQR cut into a smaller cone C and a frustum F. We want to find the ratio of the volume of cone C to the volume of frustum F.

Statement 1: PQR is twice the radius of C.

Let the radius of cone C be r and its height be h. Then, the radius of the original cone PQR is 2r. By similar triangles (formed by the height and radius), the height of the original cone PQR is 2h.

The volume of cone C is (1/3) * pi * r^2 * h.
The volume of cone PQR is (1/3) * pi * (2r)^2 * (2h) = (1/3) * pi * 4r^2 * 2h = 8 * (1/3) * pi * r^2 * h.
The volume of frustum F is the volume of PQR minus the volume of C, which is 8 * (1/3) * pi * r^2 * h - (1/3) * pi * r^2 * h = 7 * (1/3) * pi * r^2 * h.
The ratio of the volume of C to the volume of F is [(1/3) * pi * r^2 * h] / [7 * (1/3) * pi * r^2 * h] = 1/7.
Therefore, statement 1 alone is sufficient.

Statement 2: PQR is been cut off at the middle of its height.

Let the radius of cone C be r and its height be h. Let the radius of the original cone PQR be R and its height be H. Since the cut is made at the middle of the height, h = H/2. Because the cutting plane is parallel to the base, the smaller cone C and the larger cone PQR are similar. Therefore, r/R = h/H = (H/2) / H = 1/2. So, R = 2r.

The volume of cone C is (1/3) * pi * r^2 * h.
The volume of cone PQR is (1/3) * pi * R^2 * H = (1/3) * pi * (2r)^2 * (2h) = (1/3) * pi * 4r^2 * 2h = 8 * (1/3) * pi * r^2 * h.
The volume of frustum F is the volume of PQR minus the volume of C, which is 8 * (1/3) * pi * r^2 * h - (1/3) * pi * r^2 * h = 7 * (1/3) * pi * r^2 * h.
The ratio of the volume of C to the volume of F is [(1/3) * pi * r^2 * h] / [7 * (1/3) * pi * r^2 * h] = 1/7.
Therefore, statement 2 alone is sufficient.

Since either statement 1 or statement 2 alone provides enough information to calculate the ratio, the correct answer is: "either statement (1) alone or statement (2) alone is sufficient to answer the question".
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