UPSC CDS 2 2023 Math Paper

Concept:Given expressions for $p$, $q$, $r$ in terms of $x$, $y$, $z$, we evaluate $p x^{2} + q y^{2} + r z^{2}$ by substituting simple numeric values to match options.Explanation:Choose $x = 1$, $y = 1$, $z = 2$.Compute $p = x^{4} - y^{2}z^{2} = 1^{4} - (1^{2} \times 2^{2}) = 1 - 4 = -3$.Compute $q = y^{4} - z^{2}x^{2} = 1^{4} - (2^{2} \times 1^{2}) = 1 - 4 = -3$.Compute $r = z^{4} - x^{2}y^{2} = 2^{4} - (1^{2} \times 1^{2}) = 16 - 1 = 15$.Now evaluate the target expression: $p x^{2} + q y^{2} + r z^{2} = (-3)(1^{2}) + (-3)(1^{2}) + 15(2^{2}) = (-3) + (-3) + 60 = 54$.Check each option with the same values:Option A: $(x^{2} + y^{2} + z^{2})(p + q + r) = (1 + 1 + 4)(-3 - 3 + 15) = 6 \times 9 = 54$. Matches.Option B: $-(x^{2}+y^{2}+z^{2})(p+q+r) = -6 \times 9 = -54$. Does not match.Option C: $(y^{2}+z^{2}-x^{2})(r-q-p) = (1+4-1)(15 - (-3) - (-3)) = 4 \times 15 = 60$. Does not match.Option D: $(x^{2}+y^{2}-z^{2})(p-q-r) = (1+1-4)(-3 - (-3) - 15) = (-2) \times (-15) = 30$. Does not match.Only option A gives 54, the same as the target expression.Answer:$p x^{2} + q y^{2} + r z^{2} = (x^{2} + y^{2} + z^{2})(p + q + r)$ (Option A).