The coefficient of $x^{2}$ in the expansion of $(1 + 4 x + x^{2})^{1 / 2}$ is =

- 3

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- 2

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2

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- \frac{3}{2}

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Solution

The correct option is C $- \frac{3}{2}$
Given, $(1 + 4 x + x^{2})^{1 / 2}$ $= (1 + (4 x + x^{2}))^{1 / 2}$
$= 1 + \frac{1}{2} (4 x + x^{2}) - \frac{1}{4.2!} (4 x + x^{2})^{2} . . . \infty$
$= 1 + \frac{1}{2} (4 x + x^{2}) - \frac{1}{8} (x^{4} + 16 x^{2} + 8 x^{3}) . . . \infty$ ..(i)
Hence we get coefficient of $x^{2}$ from Eq (i) as.
$\frac{1}{2} - \frac{16}{8}$
$= \frac{1}{2} - 2$
$= \frac{- 3}{2}$

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