NDA 2 2024 Maths, question 8

Topic: Algebra

Q8

Let $p=\ln (x)$ , $q=\ln (x^{3})$ and $r=\ln (x^{5})$ , where $x>1$ . Which of the following statements is/are correct?

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

II. $p,q$ and $r$ can never be in GP.

Select the answer using the code given below.

(a) I only
(b) II only
(c) Both I and IICorrect answer
(d) Neither I nor II

Solution.

Answer: (c) Both I and II

Let L=ln x>0.
Then p=L, q=3L and r=5L, with equal differences 2L, so they are in AP.
A GP would require q²=pr, or 9L²=5L², impossible for L>0.
Both statements are true.