UPSC CDS 1 2023 Math Paper

Concept:If $2$ is a root of $ax^2 + bx + c = 0$, the other root depends on the sum and product of roots, but the coefficients are not uniquely determined without additional information.Explanation:Let the roots be $\alpha = 2$ and $\beta$.Sum of roots: $\alpha + \beta = -\frac{b}{a}$.Product of roots: $\alpha\beta = \frac{c}{a}$.Statement I says $\frac{c}{a} = 1$. So $2\beta = 1$, giving $\beta = \frac{1}{2}$.The quadratic is $k\left[x^2 - \left(2+\frac{1}{2}\right)x + 2\cdot\frac{1}{2}\right] = k\left[x^2 - \frac{5}{2}x + 1\right] = 0$.This can be written as $k[2x^2 - 5x + 2] = 0$.Comparing with $ax^2 + bx + c = 0$, we get $a = 2k$, $b = -5k$, $c = 2k$, where $k$ can be any non-zero constant.Thus $a, b, c$ are not unique.Statement II says $\frac{b}{a} = -\frac{5}{2}$. Then $-\frac{b}{a} = \frac{5}{2}$, so sum of roots $2 + \beta = \frac{5}{2}$, giving $\beta = \frac{1}{2}$ again.This leads to the same family of quadratics $a = 2k$, $b = -5k$, $c = 2k$, with no unique values.Even using both statements together, we only know the ratio of coefficients, not their actual values.Therefore, the question cannot be answered.Answer:D (The question cannot be answered even by using both statements together).