UPSC CDS 2 2022 Math Paper

Concept:Using the quadratic formula to find the roots and checking each given expression.Explanation:The given equation is $qx^{2} - 2px + q = 0$.Using the quadratic formula $x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$ with $a = q$, $b = -2p$, $c = q$, we get:$x = \frac{2p \pm \sqrt{4p^{2} - 4q^{2}}}{2q} = \frac{p \pm \sqrt{p^{2} - q^{2}}}{q}$. This is the actual root expression.Check statement 1: $x = \frac{\sqrt{p+q} + \sqrt{p-q}}{\sqrt{p+q} - \sqrt{p-q}}$.Rationalise the denominator by multiplying numerator and denominator by $\sqrt{p+q} + \sqrt{p-q}$:$x = \frac{(\sqrt{p+q} + \sqrt{p-q})^{2}}{(p+q) - (p-q)} = \frac{p+q + p-q + 2\sqrt{(p+q)(p-q)}}{2q} = \frac{2p + 2\sqrt{p^{2} - q^{2}}}{2q} = \frac{p + \sqrt{p^{2} - q^{2}}}{q}$.This matches one of the two possible roots from the quadratic formula. So statement 1 is correct.Check statement 2: $x = \frac{p + \sqrt{p^{2} - q^{2}}}{q}$.This directly matches the positive root from the quadratic formula. So statement 2 is correct.Check statement 3: $x = \frac{\sqrt{p+q}}{\sqrt{p-q}}$.Substituting into the equation or comparing with the root form shows that this expression cannot equal either $\frac{p \pm \sqrt{p^{2} - q^{2}}}{q}$ for general $p>q$. So statement 3 is incorrect.Hence, only two of the given values satisfy the equation.Answer:Only two values (Option B).