UPSC NDA I 2025 Mathematics Paper

Concept:A triangle is obtuse if one angle exceeds $90^{\circ}$. The cosine rule helps find the angle opposite the longest side. The sum of the other two angles is $180^{\circ}$ minus the obtuse angle.Explanation:Given sides: $5$ cm, $7$ cm, $3$ cm. The longest side is $7$ cm.Use cosine rule for angle $B$ opposite side $7$:$\cos B = \frac{5^{2}+3^{2}-7^{2}}{2 \times 5 \times 3} = \frac{25+9-49}{30} = \frac{-15}{30} = -\frac{1}{2}$.Thus $B = \cos^{-1}(-\frac12) = 120^{\circ}$ — obtuse angle. Statement I is correct.Sum of the other two acute angles: $A + C = 180^{\circ} - 120^{\circ} = 60^{\circ}$, which is less than $90^{\circ}$, so it is acute. Statement II is also correct.Answer:C. Both I and II