A metal block of density ‘D’ and mass ‘M’, in the form of a cuboid, is beaten…

Concept: This is a data-sufficiency question built on volume conservation. When a solid block is reshaped — beaten into a sheet, then rolled into a cylinder — without losing any material, its volume stays exactly the same at every stage. A statement (or combination of statements) is “sufficient” only when it lets every unknown needed for the final quantity be pinned down to a number; if a required link is missing (such as an unstated density), a related quantity cannot substitute for it, however precisely it is given.

Application — using Statement I and Statement II:

1. Volume of the cuboid from Statement I: 10 cm × 5 cm × 12 cm = 600 cm3.

2. Beating it into a square sheet of side S and thickness t conserves this volume: S2 × t = 600 cm3.

3. From Statement II, t = 1.5 cm, so S2 = 600 / 1.5 = 400, giving S = 20 cm.

4. Rolling this square sheet into a cylinder of the same thickness keeps one side of the square as the cylinder’s height (h = 20 cm) and turns the other side into the mean circumference of the tube: 2π × rmean = 20 cm, so rmean = 10/π cm.

5. The tube’s wall thickness is t = 1.5 cm, so the inner radius = rmean − t/2 = 10/π − 0.75 ≈ 2.43 cm.

6. Every quantity above came only from Statement I (the dimensions) and Statement II (the thickness) — Statement III (the mass) was never needed to reach this number.

Cross-check — does Statement III help on its own? Statement III alone only gives mass M = 216 kg; converting mass to volume needs volume = mass ÷ density, and no numeric density is stated anywhere in the question. So mass alone can never be turned into a volume, and it cannot replace or add to what Statements I and II already establish.

Result: Statements I and II together are enough to compute the inner radius, and Statement III adds nothing beyond what they already fix — so Statement I and Statement II together are sufficient.