UPSC NDA I 2022 Mathematics Solved Paper

Concept:Using the trigonometric ratio $\tan \theta = \frac{\text{perpendicular}}{\text{base}}$ and analyzing sign of $\sin \theta$ based on the quadrant of $\theta$.Explanation:Given $\tan \theta = -\frac{5}{12}$.In a right triangle, perpendicular $= 5$, base $= 12$.By Pythagoras theorem: hypotenuse $= \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$.Thus, $\sin \theta = \frac{\text{perpendicular}}{\text{hypotenuse}} = \pm \frac{5}{13}$ (the sign depends on the quadrant).Since $\tan \theta$ is negative, $\theta$ lies in the $2^{\text{nd}}$ or $4^{\text{th}}$ quadrant.In the $2^{\text{nd}}$ quadrant, $\sin \theta > 0$, so $\sin \theta = \frac{5}{13}$.In the $4^{\text{th}}$ quadrant, $\sin \theta < 0$, so $\sin \theta = -\frac{5}{13}$.Hence, both $\frac{5}{13}$ and $-\frac{5}{13}$ are possible values of $\sin \theta$.Answer:Option C: $\frac{5}{13}$ or $-\frac{5}{13}$