Q8 / 120 · NDA 2024 (Phase 2) · Mathematics

Let $p=\operatorname{In}(x), q=\ln \left(x^3\right)$ and $\mathrm{r}=\ln \left(x^5\right)$, where $x>1$. Which of the following statements is/are correct? A. $p, q$ and $r$ are in AP. B. $p, q$ and $r$ can never be in GP. Select the answer using the code given below:

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

Solution

We have given, $\mathrm{p}=\ell \mathrm{n} \mathrm{x}, \mathrm{q}=\ell \mathrm{n} \mathrm{x}^3=3 \ell \mathrm{n} \mathrm{x}$ $\mathrm{r}=\ell \ln \mathrm{x}^5=5 \ln \mathrm{x}$ Clearly, $\mathrm{q}-\mathrm{p}=\mathrm{r}-\mathrm{q}=2$ lnx $\Rightarrow \mathrm{p}, \mathrm{q}$ and r are in A.P. Also, since $\frac{\mathrm{q}}{\mathrm{p}} \neq \frac{\mathrm{r}}{\mathrm{q}} \Rightarrow \mathrm{p}, \mathrm{q}$ and r can never be in GP.

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

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