Correct Option (a)
When a solid cube of side N units is painted on all its faces and then cut into smaller cubes of 1 unit side, the number of small cubes with exactly two painted faces is determined by the formula 12 × (N-2).
In this specific problem, the side of the large cube is 3 cm, which means N = 3. Applying the formula:
Number of cubes with exactly two painted faces = 12 × (3 - 2) = 12 × 1 = 12.
These cubes are situated along the edges of the original large cube, excluding the corner cubes.
Incorrect Options:
- Option (b) 8: This value corresponds to the number of small cubes with exactly three painted faces. These are the corner cubes of the original large cube, and there are always 8 such cubes for any N ≥ 2.
- Option (c) 6: This value represents the number of small cubes with exactly one painted face. These cubes are located at the center of each face of the original large cube. For N=3, the relevant formula is 6 × (N-2)², which yields 6 × (3-2)² = 6 × 1² = 6.
- Option (d) 4: This number does not correspond to a standard count of small cubes with 0, 1, 2, or 3 painted faces when a 3x3x3 cube is formed from a larger painted cube.