Question: ABCD is a parallelogram with \angle ABC = 60^\circ. If the area of the parallelogram is 7\sqrt{3} square units, then what is the perimeter of the parallelogram? Statement I: The lengths of the sides AB and DA are prime numbers. Statement II: The lengths of the sides are natural numbers each greater than 1 unit.

A Question is given followed by two Statements I and II. Consider the Question and the Statements. Which one of the following is correct in respect of the above Question and the Statements?

  1. A. The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
  2. B. The Question can be answered by using either Statement alone ✓
  3. C. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
  4. D. The Question can be answered even without using any of the Statements

Correct Answer: B. The Question can be answered by using either Statement alone

Explanation

Area = AB \times BC \times \sin(60^\circ) = 7\sqrt{3}, resulting in AB \times BC = 14. Statement I identifies the sides as prime numbers, uniquely making them 2 and 7 (Perimeter = 18). Statement II asserts they are natural numbers > 1, identically leaving only 2 and 7 as factors. Either statement works alone.

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