7 cm × 5 cm × 3 cm विमाओं वाले एक घनाभ के क्रमशः 7 cm × 5 cm, 5 cm × 3 cm, 7 cm × 3 cm विमाओं वाले सम्मुख फलकों के प्रत्येक युग्म को लाल, हरे और नीले रंग से रँगा गया है । तब इस घनाभ को काटकर प्रत्येक 1 cm भुजा के विभिन्न घन अलग कर दिए जाते हैं । निम्नलिखित कथनों में से कौन-सा/से सही है/हैं ?
- 1.ऐसे ठीक-ठीक 15 छोटे घन हैं जिनके किसी भी फलक पर कोई रंग नहीं है ।
- 2.ऐसे ठीक-ठीक 6 छोटे घन हैं जिनके ठीक-ठीक दो फलक, एक नीले और दूसरा हरे रंग से, रँगे हुए हैं ।
नीचे दिए गए कूट का प्रयोग कर सही उत्तर चुनिए :
Read this question in English
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 · (b) 2 only · (c) Both 1 and 2 · (d) Neither 1 nor 2
उत्तर और व्याख्या · explanation in English
Answer: (a) केवल 1
The unpainted cubes form an inner block of 5 × 3 × 1 = 15. Green (5 × 3) and blue (7 × 3) faces meet only along the four edges of length 3 cm, and each such edge gives 3 − 2 = 1 cube with exactly those two faces painted — 4 cubes, not 6.
- No paint: remove one layer from each side: (7 − 2) × (5 − 2) × (3 − 2) = 5 × 3 × 1 = 15. Statement 1 is correct.
- Green faces are the 5 × 3 ends (across the 7 cm length); blue faces are the 7 × 3 sides (across the 5 cm width).
- A green face meets a blue face along an edge 3 cm long; a cuboid has 4 such edges.
- Each 3 cm edge has 3 cubes; the 2 end ones are corners with three painted faces, leaving 1 cube per edge.
- Blue-and-green-only cubes = 4 × 1 = 4, not 6. Statement 2 is incorrect.
- Check: two-face cubes in all = 4(7 − 2) + 4(5 − 2) + 4(3 − 2) = 20 + 12 + 4 = 36, and the green–blue share is the 4 on the 3 cm edges.
- ✓ 1. Inner block (7 − 2)(5 − 2)(3 − 2) = 15 cubes.
- ✗ 2. Green–blue edges are the four 3 cm edges, each giving one such cube: 4 in all.
Remember · Painted cuboid a × b × c: unpainted = (a − 2)(b − 2)(c − 2); cubes on an edge with two colours = edge length − 2.
Question and answer: UPSC's official GS Paper II (2023, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·