Geometry

Solids

Medium

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:

(A) only

(B) only

Both (A) and (B)

Neither (A) nor (B)

✅ Correct Option: 1

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.

2026: 29 May Shift 2

Find the total surface area of a solid hemisphere of diameter 14 cm.

2023: 19 June Shift 1

A cone of height 24 cm and base radius 6 cm is made of modelling clay. A child reshapes it in the form of a sphere. The radius of the sphere is :

2025: 20 May Shift 1

The base radius of a cone is 21 cm and the height is 28cm, then which of the following is/are correct?

(A) Slant height = 35 cm

(B) Slant height = 33 cm

(D) Curved Surface Area = 2310 cm²

Choose the correct answer from the options given below:

2025: 3 June Shift 2

Consider the surface area of the following:

(A) A cube having each side as 6 cm.

(B) A cylinder with a diameter of base 14 cm and length 80 cm.

(D) A sphere of radius 10.5 cm.

The surface area of these in decreasing order are:

Choose the correct answer from the options given below:

2026: 6 June Shift 1

Three solid cubes of sides 1 cm, 6 cm, and 8 cm are melted to form a new cube. The surface area of the cube so formed is:

2025: 3 June Shift 1

If a cone and sphere have equal radii and volumes, then determine the ratio of the diameter of the sphere to the height of the cone?

Solution

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