UPSC CDS 2 2023 Math Paper

Concept:Each statement provides a relation (product or sum) that uniquely determines the other root of a quadratic equation given one root.Explanation:Let the given root be $r_1 = \frac{-4-\sqrt{10}}{2}$.Using Statement-I: Product of roots is $-\frac{3}{2}(3+\sqrt{10})$.Let the other root be $r_2$. Then $r_1 \times r_2 = -\frac{3}{2}(3+\sqrt{10})$.Solve: $r_2 = \frac{-\frac{3}{2}(3+\sqrt{10})}{r_1} = \frac{-\frac{3}{2}(3+\sqrt{10})}{\frac{-4-\sqrt{10}}{2}} = \frac{-3(3+\sqrt{10})}{-4-\sqrt{10}}$.This gives a unique value for $r_2$. Thus Statement-I alone is sufficient.Using Statement-II: Sum of roots is $-1$.Then $r_1 + r_2 = -1$.Solve: $r_2 = -1 - r_1 = -1 - \frac{-4-\sqrt{10}}{2} = -1 + \frac{4+\sqrt{10}}{2} = \frac{-2 + 4 + \sqrt{10}}{2} = \frac{2+\sqrt{10}}{2}$.This gives a unique value for $r_2$. Thus Statement-II alone is also sufficient.Therefore, the question can be answered using either statement alone.Answer:Option B (the question can be answered by using either statement alone).