A cuboid of dimensions 7cm × 5cm × 3cm is painted red, green and blue colour on each pair of opposite faces of dimensions 7cm × 5cm × 5cm, 5cm × 3cm, 7cm × 3cm respectively. Then the cuboid is cut and separated into various cubes each of side length 1cm. 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:

1. A. 1 Only
2. B. 2 Only
3. C. Both 1 and 2
4. D. Neither 1 nor 2

Answer: A

Explanation

The cuboid has dimensions L=7cm, W=5cm, H=3cm. It is cut into 1cm cubes.
Statement 1: ‘There are exactly 15 small cubes with no paint on any face.’ Unpainted cubes are the inner cubes. Their dimensions are (L-2) × (W-2) × (H-2). Number of unpainted cubes = (7-2) × (5-2) × (3-2) = 5 × 3 × 1 = 15. So, Statement 1 is correct.
Statement 2: ‘There are exactly 6 small cubes with exactly two faces, one painted with blue and the other with green.’ The faces are painted as follows: 7×5 faces (Red), 5×3 faces (Green), 7×3 faces (Blue). Cubes with exactly two faces painted blue and green must be along the edges where a blue face and a green face meet. These are the edges of length 3cm (where the 7×3 face meets the 5×3 face). A cuboid has 4 such edges. For an edge of length ‘n’ cm, the number of cubes with exactly two painted faces (excluding corner cubes) is (n-2). So, for each 3cm edge, the number of cubes with two painted faces (blue and green) = (3-2) = 1. Since there are 4 such edges, the total number of such cubes = 4 × 1 = 4. The statement claims there are 6 such cubes, which is incorrect. Therefore, only Statement 1 is correct. This question tests spatial reasoning and visualization of 3D objects, a common type in CSAT.