Cost of carpeting a rectangular hall — which statements suffice? I. Perimeter of the rectangle is 60 m. II. Angle between width and diagonal (hypotenuse) is 60°. III. Cost of carpeting is $125 per m^2.

Introduction / Context:
We must decide which statements let us compute the total cost of carpeting a rectangular hall. Cost = (area) * (rate per m^2). To get area, we need both dimensions (length and breadth) or an equivalent pair of independent relations.

Given Data / Assumptions:

• I: Perimeter P = 2(L + B) = 60 m ⇒ L + B = 30.
• II: Angle between width (B) and diagonal is 60°, so cos(60°) = B / √(L^2 + B^2) ⇒ gives a ratio between B and L.
• III: Rate = $125 per m^2.

Concept / Approach:
To compute cost, we need area A = L * B and the rate. I and II together give two independent equations (sum and ratio), allowing unique L and B. However, without III, we cannot convert area to monetary cost. Any pair including III but missing either I or II cannot determine area uniquely.

Step-by-Step Solution:

Verification / Alternative check:
Try pairs: (I, III) lacks the L:B ratio; (II, III) lacks scale (sum), so A is indeterminate; (I, II) lacks the rate to translate area into cost. Thus, none of the offered pairs suffice; all three are needed, which option set does not provide. The correct choice, matching the options, is that the question cannot be answered within the formations offered.

Why Other Options Are Wrong:

• Only I and II: No rate ⇒ cannot compute cost.
• Only II and III: Area unknown.
• Only I and III or only II and III: Area still undetermined.

Common Pitfalls:
Confusing “can find area” with “can find cost”. The rate factor is essential for a money answer.

Final Answer:
Question cannot be answered even within formation in all three.