CDS 2023 PYQ with Answer — Question: What is the other root of the quadratic equation with real…

2023 · Q92

Question: What is the other root of the quadratic equation with real coefficients if one of the roots is (−4 − √10)/2?

Statement IThe product of the roots is (−3 + √10)/2.

Statement IIThe sum of roots of quadratic equation is −1. Mark option (a) if the question can be answered by using one of the statements alone, but cannot be answered using the other statement alone. Mark option (b) if the question can be answered by using either statement alone. Mark option (c) if the question can be answered by using both the statements together, but cannot be answered using either statement alone. Mark option (d) if the question cannot be answered even by using both the statements together.

Elementary MathematicsAlgebra

AThe question can be answered by using one of the statements alone, but cannot be answered using the other statement alone
BThe question can be answered by using either statement alone
CThe question can be answered by using both the statements together, but cannot be answered using either statement alone
DThe question cannot be answered even by using both the statements together

Explanation

This is a data-sufficiency question about a quadratic with real coefficients. Note the given root (−4 − √10)/2 is a real number, so the conjugate-root shortcut does not apply. Statement I gives the product of the roots, and the other root is (product) ÷ (given root) — one linear division, so Statement I alone suffices. Statement II gives the sum of the roots, and the other root is (sum) − (given root), so Statement II alone also suffices. Since either statement alone answers the question, the correct choice is option (b). Options (a), (c) and (d) misjudge sufficiency: each statement independently determines the unknown root.