UPSC NDA 2 2019 Mathematics Solved Paper

Concept:For any two positive numbers, the arithmetic mean is always greater than or equal to the geometric mean (AM ≥ GM).Explanation:Consider statement 1: $\cos \theta$ and $\sec \theta$ are reciprocals, so both are positive or both negative. Using AM ≥ GM:$\frac{\cos \theta + \sec \theta}{2} \ge \sqrt{\cos \theta \cdot \sec \theta}$Since $\cos \theta \cdot \sec \theta = 1$, we get $\frac{\cos \theta + \sec \theta}{2} \ge 1$, so $\cos \theta + \sec \theta \ge 2$.Hence it can never equal 1.5 (because 1.5 < 2). Statement 1 is correct.Consider statement 2: $\tan \theta$ and $\cot \theta$ are reciprocals, both positive when $\theta$ is in first quadrant. Similarly,$\frac{\tan \theta + \cot \theta}{2} \ge \sqrt{\tan \theta \cdot \cot \theta} = 1$, so $\tan \theta + \cot \theta \ge 2$.Thus it can never be less than 2. Statement 2 is also correct.Answer:Both statements are correct, so option C (Both 1 and 2) is the answer.