Q177 / 270 · NDA 2020 · Mathematics

If $p^2, q^2$ and $r^2$ (where $\left.p, q, r \gt 0\right)$ are in GP, then which of the following is/are correct? 1. $\quad p, q$ and $r$ are in GP. 2. $\ln p, \ln q$ and $\ln r$ are in AP . Select the correct answer using the code given below :

  • A
    1 only
  • B
    2 only
  • Correct answer
    C
    Both 1 and 2
  • D
    Neither 1 nor 2

Solution

(1) $p^2, q^2$ and $r^2$ in G.P. Then, $\left(q^2\right)^2=p^2 \cdot r^2 \Rightarrow q^2=\left(p^2 \cdot r^2\right)^{\frac{1}{2}}$ $\Rightarrow q^2=p \cdot r$ Hence, $p, q$ and $r$ in G.P. (2) As, $q^2=p \cdot r$ Taking $\log$ on both sides, we have $\begin{aligned} & \ln \left(q^2\right)=\ln (p \cdot r) \\ & 2 \ln (q)=\ln p+\ln r \end{aligned}$ Hence, $\ln p, \ln q$ and $\ln r$ are in A.P.

Hence, the correct answer is option: 'C'.

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