Q75 / 150 · MHT CET 2023 (13 May Shift 1) · Mathematics

The value of $\frac{\mathrm{i}^{248}+\mathrm{i}^{246}+\mathrm{i}^{244}+\mathrm{i}^{242}+\mathrm{i}^{240}}{\mathrm{i}^{249}+\mathrm{i}^{247}+\mathrm{i}^{245}+\mathrm{i}^{243}+\mathrm{i}^{241}}$, $(\mathrm{i}=\sqrt{-1})$ is

  • A
    $\mathrm{i}$
  • B
    1
  • C
    -1
  • Correct answer
    D
    $-\mathrm{i}$

Solution

$\begin{aligned} & \frac{i^{245}+i^{246}+i^{244}+i^{242}+i^{240}}{i^{243}+i^{247}+i^{245}+i^{243}+i^{241}} \\ & =\frac{i^{240}\left(i^8+i^6+i^4+i^2+1\right)}{i^{241}\left(i^8+i^6+i^4+i^2+1\right)} \\ & =\frac{i^{200}}{i^{241}} \\ & =\frac{1}{i} \\ & =\frac{i}{i^2}=-i\end{aligned}$

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

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