NDA 2 2025 Maths, question 1

Topic: Algebra

If $p^{x}=q^{y}=r^{z}$ , where $x,y$ and $z$ are in GP, then consider the following statements :

I. $p,q$ and $r$ are in AP.

II. $\ln p,\ln q$ and $\ln r$ are in GP.

Which of the statements given above is/are correct?

Solution.

Answer: (b) II only

Let $p^{x} = q^{y} = r^{z} = k$ . Then $x\ln p = y\ln q = z\ln r = \ln k$ .
Since $y^{2} = xz$ , we get $(\ln q)^{2} = (\ln p)(\ln r)$ , so $\ln p, \ln q, \ln r$ are in GP: II is correct.
I fails: with $x, y, z = 1, 2, 4$ and $p, q, r = 16, 4, 2$ , the condition holds but 16, 4, 2 are not in AP.

Trap: This assumes positive bases and nonzero GP terms, which the question takes for granted.