Question: What is the ratio of the lengths of diagonals of a rhombus?<br>Statement-I: One diagonal of the rhombus is equal to its side.<br>Statement-II: The longer diagonal of the rhombus is equal to \sqrt{3} times its side.

Consider the following for the next ten (10) items that follow : Each item contains a Question followed by two Statements. Answer each item using the following instructions : Choose option (a) If the Question can be answered by one of the Statements alone, but not by the other. (b) If the Question can be answered by either Statement alone. (c) If the Question can be answered by using both the Statements together, but cannot be answered by using either Statement alone. (d) If the Question cannot be answered even by using both Statements together.

  1. A. If the Question can be answered by one of the Statements alone, but not by the other.
  2. B. If the Question can be answered by either Statement alone. ✓
  3. C. If the Question can be answered by using both the Statements together, but cannot be answered by using either Statement alone.
  4. D. If the Question cannot be answered even by using both Statements together.

Correct Answer: B. If the Question can be answered by either Statement alone.

Explanation

In a rhombus with side a and diagonals d_1, d_2, the relation d_1^2 + d_2^2 = 4a^2 holds. Statement-I states d_1 = a, so a^2 + d_2^2 = 4a^2, giving d_2 = a\sqrt{3}. The ratio is 1:\sqrt{3}. Statement-II states d_2 = a\sqrt{3}, so d_1^2 + 3a^2 = 4a^2, giving d_1 = a. The ratio is again 1:\sqrt{3}. Since both statements individually allow us to find the ratio, the question can be answered by either statement alone.

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