UPSC NDA I 2025 Mathematics Paper

Concept:An arithmetic progression (AP) has a constant difference, so if three terms are in AP, the middle term equals the average of the other two.A geometric progression (GP) has a constant ratio, so the square of the middle term equals the product of the first and third.A harmonic progression (HP) means the reciprocals of the terms are in AP.Explanation:Given p, 1, q are in AP.$2(1) = p + q$ ⇒ $p + q = 2$.Given p, 2, q are in GP.$2^2 = p \times q$ ⇒ $pq = 4$.Check statement I: p, 4, q are in HP. This means $1/p$, $1/4$, $1/q$ are in AP.For AP condition: $2 \times (1/4) = 1/p + 1/q$.Left side: $1/2$. Right side: $\frac{p+q}{pq} = \frac{2}{4} = \frac{1}{2}$. True.So statement I is correct.Statement II directly says $1/p$, $1/4$, $1/q$ are in AP, which is exactly the same condition. Hence it is also correct.Both statements are true.Answer:Both I and II are correct. Option C.