The question is followed by three statements I, II and III. You have to…

2024

The question is followed by three statements I, II and III. You have to determine which statement or statements (s) is/are sufficient/necessary to answer the question and mark answer accordingly. The slant height of a cone is x cm. If side of a square is (x+5) cm, then find the area of the square.
I. The ratio of diameter of cone to its height is 3 : 2.
II. The base and height of a cylinder is same as base and height of the cone respectively. Volume of the cylinder is 4500𝜋 cm3.
III. Total surface area of the cone is 600𝜋 cm2 and curved surface area of the cone is 375𝜋 cm2.

  1. A.

    Statement II and III together sufficient to answer the question

  2. B.

    Statement III alone is sufficient to answer the question

  3. C.

    All the three statements taken together are necessary to answer the question

  4. D.

    Statement I and II together sufficient but III alone not sufficient

  5. E.

    Either statement I and II together or statement III alone is sufficient to answer the question.

Show answer & explanation

Correct answer: E

Concept

This is a data-sufficiency item. The square's area is (x+5)2, so the whole question reduces to one thing: can we pin down the cone's slant height x from the information given? A set of statements is sufficient exactly when it forces a single numerical value of x. For a cone, the slant height obeys l2 = r2 + h2, curved surface area = πrl, total surface area = πr2 + πrl, and an equal-base-equal-height cylinder has volume πr2h. Any route that fixes r (and h where needed) fixes x.

Evaluating each statement

The statement giving TSA = 600π and CSA = 375π:

  1. Base area = TSA − CSA = 600π − 375π = 225π, so πr2 = 225π ⇒ r2 = 225 ⇒ r = 15.

  2. CSA = πrl = 375π ⇒ 15l = 375 ⇒ l = 25, hence x = 25.

This single statement alone fixes x = 25.

The statement giving diameter : height = 3 : 2, taken with the statement giving cylinder volume = 4500π:

  1. Diameter : height = 3 : 2 ⇒ 2r : h = 3 : 2 ⇒ h = 4r/3 (a relationship only, not a value).

  2. Equal-base-equal-height cylinder volume πr2h = 4500π ⇒ r2h = 4500.

  3. Substitute h = 4r/3: r2·(4r/3) = 4500 ⇒ 4r3/3 = 4500 ⇒ r3 = 3375 ⇒ r = 15, then h = 20.

  4. Slant l = √(r2 + h2) = √(225 + 400) = √625 = 25, hence x = 25.

Sufficiency map (ruling out every other pattern)

  • Ratio statement alone: gives only h = 4r/3, no value for r — insufficient.

  • Cylinder-volume statement alone: gives r2h = 4500, two unknowns — insufficient.

  • Surface-area statement alone: fixes r = 15 and l = 25 — sufficient by itself.

  • Ratio + cylinder-volume together: fixes r = 15, h = 20, l = 25 — sufficient.

  • So the surface-area statement on its own and the ratio-plus-volume pair are each sufficient, independently of one another.

Cross-check and result

Both independent routes give x = 25, so the square's side = x + 5 = 30 and its area = 302 = 900 cm2. The data is therefore sufficient through two separate routes — the single surface-area statement on its own, or the ratio statement paired with the cylinder-volume statement.

Explore the full course: Ctet Paper 2