Which of the following statements are true?(A) If the diagonal of a c…

The correct answer is Option (1) → (A) only

For a cube of edge \(a\): diagonal \(=a\sqrt{3}\), volume \(=a^3\) and total surface area \(=6a^2\). A block can be cut into equal cubes only if the cube’s edge divides all three dimensions, so the fewest cubes come from the largest such edge, the HCF of the dimensions.

(A) If the diagonal of a cube is \(6\sqrt{3}\) cm, then its volume and surface area are equal. Correct. The volume and the surface area come to the same number, 216 (in cm³ and cm²):

• \(a\sqrt{3}=6\sqrt{3}\), so \(a=6\,\mathrm{cm}\)

• Volume \(=6^3=216\,\mathrm{cm^3}\)

• Surface area \(=6\times6^2=216\,\mathrm{cm^2}\)

(B) If a rectangular block having dimensions 6 cm × 12 cm × 15 cm is cut into an exact number of equal cubes, then the least possible number of such cubes is 27. Incorrect. The least number is 40, not 27:

• HCF of 6, 12 and 15 \(=3\), so the largest cube has an edge of 3 cm

• Number of cubes \(\displaystyle =\frac{6}{3}\times\frac{12}{3}\times\frac{15}{3}=2\times4\times5=40\)