QUESTION

CSAT

Easy

Maths

Prelims 2023

A cuboid of dimensions $7\text{cm} \times 5\text{cm} \times 3\text{cm}$ is painted red, green and blue colour on each pair of opposite faces of dimensions $7\text{cm} \times 5\text{cm}$ , $5\text{cm} \times 3\text{cm}$ , $7\text{cm} \times 3\text{cm}$ respectively. Then the cuboid is cut and separated into various cubes each of side length $1\text{cm}$ . Which of the following statements is/are correct?

There are exactly 15 small cubes with no paint on any face.
There are exactly 6 small cubes with exactly two faces, one painted with blue and the other with green.

Select the correct answer using the code given below:

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Explanation

A cuboid of dimensions $7 \text{ cm} \times 5 \text{ cm} \times 3 \text{ cm}$ will have:

Two faces of dimensions $7 \text{ cm} \times 5 \text{ cm}$ . These faces are painted red.
Two faces of dimensions $5 \text{ cm} \times 3 \text{ cm}$ . These faces are painted green.
Two faces of dimensions $7 \text{ cm} \times 3 \text{ cm}$ . These faces are painted blue.

Statement 1: Only the internal cubes will have no paint on any face. So, we basically need to find out the number of internal cubes in the cuboid. Number of internal cubes in the cuboid = $(7 - 2) \times (5 - 2) \times (3 - 2) = 5 \times 3 \times 1 = 15$

So, Statement 1 is correct.

Statement 2: We need to find the number of cubes having only two painted faces, one with blue and the other with green colour. These cubes can be found in the middle of the four $3 \text{ cm}$ edges. There is only one such cube at the middle of each of these four $3 \text{ cm}$ edges. So, there are only 4 such cubes.