Q261 / 270 · NDA 2015 (Phase 2) · Mathematics

The geometric mean of the observations $\mathrm{x}_1, \mathrm{x}_2, \mathrm{x}_3, \ldots . . \mathrm{x}_{\mathrm{n}}$ is $\mathrm{G}_1$, The geometric mean of the observations $\mathrm{y}_1, \mathrm{y}_2, \mathrm{y}_3, \ldots \mathrm{y}_{\mathrm{n}}$ is $\mathrm{G}_2$. The geometric mean of observations $\frac{x_1}{y_1}, \frac{x_2}{y_2}, \frac{x_3}{y_3}, \ldots . \frac{x_n}{y_n}$ is

  • A
    $\mathrm{G}_1 \mathrm{G}_2$
  • B
    $\ln \left(\mathrm{G}_1 \mathrm{G}_2\right)$
  • Correct answer
    C
    $\frac{\mathrm{G}_1}{\mathrm{G}_2}$
  • D
    $\ln \left(\frac{G_1}{G_2}\right)$

Solution

$\begin{aligned} & \mathrm{G}_1=\left[\mathrm{x}_1 \times \mathrm{x}_2 \times \mathrm{x}_3 \times \ldots \ldots \times \mathrm{x}_{\mathrm{n}}\right]^{1 / n} \\ & \mathrm{G}_2=\left[\mathrm{y}_1 \times \mathrm{y}_2 \times \mathrm{y}_3 \times \ldots \ldots \times \mathrm{y}_{\mathrm{n}}\right]^{1 / n} \\ & \Rightarrow \frac{\mathrm{G}_1}{\mathrm{G}_2}=\left[\frac{\mathrm{x}_1}{\mathrm{y}_1} \times \frac{\mathrm{x}_2}{\mathrm{y}_2} \times \frac{\mathrm{x}_3}{\mathrm{y}_3} \times \ldots \ldots \ldots \times \frac{\mathrm{x}_{\mathrm{n}}}{\mathrm{y}_{\mathrm{n}}}\right]^{1 / n} \\ & \therefore \frac{\mathrm{G}_1}{\mathrm{G}_2} \text { is the G.M of } \frac{\mathrm{x}_1}{\mathrm{y}_1}, \frac{\mathrm{x}_2}{\mathrm{y}_2}, \frac{\mathrm{x}_3}{\mathrm{y}_3}, \ldots \ldots \ldots \ldots . \frac{\mathrm{x}_{\mathrm{n}}}{\mathrm{y}_{\mathrm{n}}}\end{aligned}$

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

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