UPSC NDA II 2025 Mathematics Paper

Concept:The equation $x^{2}+3y=0$ represents a parabola. Its orientation is determined by the coefficient sign, axis by squared variable, and latus rectum by standard form.Explanation:Rewrite as $y = -\frac{x^{2}}{3}$, which is of the form $y = ax^{2}$ with $a = -\frac{1}{3}$.Statement I: Since $a < 0$, the parabola opens downwards, not upwards. Hence, Statement I is incorrect.Statement II: The term $x^{2}$ indicates axis along the y‑axis, so axis is $x=0$. Statement II is correct.Statement III: For $y=ax^{2}$, latus rectum length is $\frac{4}{|a|} = \frac{4}{|-1/3|} = 12$, and its equation is $y = \frac{1}{4a}$? Actually, for $y=ax^{2}$, latus rectum is horizontal line $y = \frac{1}{4a}$? Wait, standard parabola $x^{2}=4py$ has latus rectum $y=p$. Here $x^{2} = -3y$ so $4p = -3$, $p = -\frac{3}{4}$. Latus rectum equation is $y = p = -\frac{3}{4}$, which simplifies to $4y+3=0$, not $4y-3=0$. Therefore, Statement III is incorrect.Only Statement II is correct.Answer:One (Option B).