UPSC NDA I 2025 Mathematics Paper

Concept:The problem uses trigonometric identities to simplify a sum of products of sines and cosines. Key ideas: product-to-sum formulas and sum-to-product formulas.Explanation:First, rewrite each term: $p = \sin 35^\circ$, $q = \sin 25^\circ$, $r = \sin(-95^\circ) = -\cos 5^\circ$.Then $pq + qr + rp = (\sin35^\circ \sin25^\circ) + (\sin25^\circ)(-\cos5^\circ) + (-\cos5^\circ)(\sin35^\circ)$.Use product-to-sum identities:$\sin A \sin B = \frac{1}{2}[\cos(A-B)-\cos(A+B)]$,$\sin A \cos B = \frac{1}{2}[\sin(A+B)+\sin(A-B)]$.Compute each term:$\sin35^\circ \sin25^\circ = \frac{1}{2}(\cos10^\circ - \cos60^\circ) = \frac{1}{2}(\cos10^\circ - \frac12)$.$-\sin25^\circ \cos5^\circ = -\frac12(\sin30^\circ + \sin20^\circ) = -\frac12(\frac12 + \sin20^\circ)$.$-\cos5^\circ \sin35^\circ = -\frac12(\sin40^\circ + \sin30^\circ) = -\frac12(\sin40^\circ + \frac12)$.Sum them: $\frac12\cos10^\circ - \frac14 - \frac14 - \frac12\sin20^\circ - \frac12\sin40^\circ - \frac14 = \frac12\cos10^\circ - \frac12(\sin20^\circ + \sin40^\circ) - \frac34$.Apply sum-to-product: $\sin20^\circ + \sin40^\circ = 2\sin30^\circ\cos10^\circ = 2\cdot\frac12\cdot\cos10^\circ = \cos10^\circ$.Thus expression becomes $\frac12\cos10^\circ - \frac12\cos10^\circ - \frac34 = -\frac34$.Answer:$pq+qr+rp = -\dfrac{3}{4}$, which corresponds to option A.