CSAT 2023

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

1. There are exactly 15 small cubes with no paint on any face.
2. 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:
(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2
Correct Answer: (a) 1 only

Question from UPSC Prelims 2023 CSAT

Explanation : 

Cuboid Painting and Cubes Separation

The given cuboid is of dimensions 7 cm x 5 cm x 3 cm.

1. The small cubes with no paint on any face will be the ones that are not on the surface of the cuboid. These will be the cubes that are in the interior of the cuboid. The interior of the cuboid will have dimensions (7-2) cm x (5-2) cm x (3-2) cm = 5 cm x 3 cm x 1 cm = 15 cubes. So, the first statement is correct.

2. The small cubes with exactly two faces painted will be the ones that are on the edges of the cuboid but not at the corners. These will be painted with the colors of the two faces that meet at that edge. The edges with blue and green paint are of length 5 cm and 3 cm. But the corners of these edges will not have exactly two faces painted. So, we subtract 2 from each edge length to get the number of cubes with exactly two faces painted. This gives us (5-2) + (3-2) = 3 + 1 = 4 cubes. So, the second statement is incorrect.

Therefore, the correct answer is (a) 1 only.

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