Which of the following statements are true? (A) If the… - CUET UG 2025 General Test PYQ

CUET UG 2025 — General Test Geometry

2025mcqmedium

Which of the following statements are true?

(A) If the diagonal of a cube is $6\sqrt{3}$ cm, then its volume and surface area are equal.

(B) If a rectangular block having dimensions 6 cm x 12 cm x 15 cm is cut into an exact number of equal cubes, then the least possible number of such cubes is 27.

Choose the correct answer from the options given below:

Official previous-year question

Held on 13 May 2025 · Verified 13 Jul 2026.

Options

A
(A) only
B
(B) only
C
Both (A) and (B)
D
Neither (A) nor (B)

Did you get this right?

Solution

For a cube with side length $a$ , the body diagonal is $a\sqrt{3}$ .

Given diagonal $= 6\sqrt{3}$ cm:

$a\sqrt{3} = 6\sqrt{3}$

$a = 6$ cm

Volume of cube $= a^3$

Volume $= 6^3$

Volume $= 216$ cm³

Surface area of cube $= 6a^2$

Surface area $= 6 \times 6^2$

Surface area $= 6 \times 36$

Surface area $= 216$ cm²

Volume and surface area are numerically equal.

Statement (A) is true.

To cut the block into equal cubes, the side of each cube must divide all three dimensions.

The largest such cube has side $= \text{GCD}(6, 12, 15)$

$6 = 2 \times 3$

$12 = 2^2 \times 3$

$15 = 3 \times 5$

$\text{GCD} = 3$ cm

Number of cubes $= \dfrac{6}{3} \times \dfrac{12}{3} \times \dfrac{15}{3}$

Number of cubes $= 2 \times 4 \times 5$

Number of cubes $= 40$

The minimum number of cubes is 40, not 27.

Statement (B) is false.

Only statement (A) is true.

Your note

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