UPSC NDA I 2024 Mathematics Paper

Concept:The sign of trigonometric products determines the type of triangle based on angle ranges.Explanation:For triangle $ABC$, $A + B + C = \pi$ and each angle is between $0$ and $\pi$.Statement 1: If $\cot A \cdot \cot B \cdot \cot C > 0$, all three cotangents must be positive because more than one negative is impossible (would require two obtuse angles, sum > $\pi$).$\cot x > 0$ only for $0 < x < \frac{\pi}{2}$, so all angles are acute.Hence, the triangle is acute‑angled. Statement 1 is correct.Statement 2: If $\tan A \cdot \tan B \cdot \tan C > 0$, either all three tangents are positive or two are negative and one positive.In a triangle, at most one angle can be obtuse ($> \frac{\pi}{2}$).$\tan x > 0$ for $0 < x < \frac{\pi}{2}$ (acute) and $\tan x < 0$ for $\frac{\pi}{2} < x < \pi$ (obtuse).If one angle is obtuse, its tangent is negative, the other two acute have positive tangents → product negative.If all angles are acute, product is positive, but that indicates an acute triangle, not obtuse.Two obtuse angles cannot exist (sum > $\pi$). So product positive never indicates an obtuse triangle. Statement 2 is incorrect.Answer:Only statement 1 is correct. The correct option is A.