A cuboid of dimensions 7 cm × 5 cm × 3 cm is painted red, green and blue colour on each pair of opposite faces of dimensions 7 cm × 5 cm, 5 cm × 3 cm, 7 cm × 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 ✓ UPSC's answer
- (b) 2 only
- (c) Both 1 and 2
- (d) Neither 1 nor 2
Why the answer is (a)
• The cuboid dimensions are 7 cm × 5 cm × 3 cm, so the number of unpainted cubes (interior) is (7-2) × (5-2) × (3-2) = 5 × 3 × 1 = 15, making statement 1 correct.
• The faces are painted as follows: 7×5 faces are Red, 5×3 faces are Green, and 7×3 faces are Blue.
• Cubes with exactly two painted faces lie on the edges where two colored faces meet.
• The edge between a Blue face (7×3) and a Green face (5×3) has a length of 3 cm (the common dimension).
• The number of such cubes is (3 - 2) = 1 per edge. There are 4 such edges, so there are 4 cubes with one Blue and one Green face, not 6, making statement 2 incorrect.
• Therefore, only statement 1 is correct, which corresponds to option (a).
Why the other options are wrong
- (b) 2 only
- Statement 2 is incorrect because there are only 4 cubes with one blue and one green face, not 6.
- (c) Both 1 and 2
- Statement 2 is incorrect, so both statements cannot be correct.
- (d) Neither 1 nor 2
- Statement 1 is correct, so it is not the case that neither statement is correct.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2023, held on 28 May 2023. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.