If |x| is so small that all terms containing $x^{2}$ and higher powers of x can be neglected, then the approximate value of $\frac{{(3 - 5 x)}^{1 / 2}}{{(5 - 3 x)}^{2}}$ , where $x = \frac{1}{\sqrt{363}},$ is

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$\frac{\sqrt{3}}{25}$

$\frac{1 + 30 \sqrt{3}}{750}$

$\frac{1 + 30 \sqrt{3}}{75}$

$\frac{1 - 30 \sqrt{3}}{75}$

answer is D.

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Detailed Solution

Given $= {(3 - 5 x)}^{1 / 2} {(5 - 3 x)}^{- 2}$

$= \sqrt{3} {(1 - \frac{5}{3} x)}^{1 / 2} \frac{1}{25} {(1 - \frac{3}{5} x)}^{- 2} = \frac{\sqrt{3}}{25} (1 - \frac{5}{6} x) (1 + \frac{6}{5} x) = \frac{\sqrt{3}}{25} (1 + \frac{6}{5} x - \frac{5}{6} x) = \frac{\sqrt{3}}{25} (1 + \frac{11}{30} x) = \frac{\sqrt{3}}{25} (1 + \frac{11}{30} \frac{1}{11 \sqrt{3}}) = \frac{1 + 30 \sqrt{3}}{750}$