UPSC CDS 1 2023 Math Paper

Concept:The parity (odd/even nature) of $p^2 + q^2 + q$ depends only on $p$, because $q^2 + q$ is always even.Explanation:We need to check whether $p^2 + q^2 + q$ is odd.First, simplify: $q^2 + q = q(q+1)$. This product of two consecutive integers is always even.Thus, $p^2 + q^2 + q$ has the same parity as $p^2$ (since adding an even number does not change parity).So the expression is odd if and only if $p$ is odd (because $p^2$ is odd when $p$ is odd, even when $p$ is even).Now evaluate the statements:Statement I: $2p + q$ is odd. Since $2p$ is always even, $q$ must be odd for the sum to be odd.Statement II: $q - 2p$ is odd. Again, $2p$ is even, so $q$ must be odd to make the difference odd.Both statements independently tell us that $q$ is odd.However, they give no information about the parity of $p$.Without knowing whether $p$ is odd or even, we cannot decide if $p^2 + q^2 + q$ is odd.Using both statements together still only tells us $q$ is odd, not the parity of $p$.Therefore, the question cannot be answered even by using both statements together.Answer:Option D.