UPSC NDA II 2015 Mathematics Paper

Concept:Use the relationships between coefficients and roots of a quadratic equation: sum $S = \alpha + \beta = -\frac{b}{a}$ and product $P = \alpha\beta = \frac{c}{a}$. Then express the given condition in terms of $S$ and $P$.Explanation:Let the roots be $\alpha$ and $\beta$ of $ax^{2} + bx + c = 0$ with $a \neq 0$.Sum of roots: $\alpha + \beta = -\frac{b}{a}$.Product of roots: $\alpha\beta = \frac{c}{a}$.Given: sum of roots equals sum of their squares:$\alpha + \beta = \alpha^{2} + \beta^{2}$.Rewrite $\alpha^{2} + \beta^{2} = (\alpha+\beta)^{2} - 2\alpha\beta$.Substitute: $-\frac{b}{a} = \left(-\frac{b}{a}\right)^{2} - 2\cdot\frac{c}{a}$.Simplify: $-\frac{b}{a} = \frac{b^{2}}{a^{2}} - \frac{2c}{a}$.Multiply by $a^{2}$: $-ab = b^{2} - 2ac$.Rearrange: $b^{2} + ab = 2ac$.Answer:$ab + b^{2} = 2ac$, which matches option C.