The ratio of the coefficient of $x^{10}$ in $(1 - x^{2})^{10}$ and the term independent of $x$ in ${(x - \frac{2}{x})}^{10}$ is

32 : 1

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- 32 : 1

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- 1 : 32

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1 : 32

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Solution

The correct option is A $1 : 32$
Given $(1 - x^{2})^{10}$
$T_{r + 1} = (- 1)^{r}^{10} C_{r} x^{2 r}$ ...(i)
For the coefficient of $x^{10}$
$2 r = 10$
$r = 5$
$T_{6} = -^{10} C_{5} x^{10}$ ...(a)
$T_{r + 1} = (- 1)^{r}^{10} C_{r} x^{10 - 2 r} 2^{r}$ ...(ii)
For the term independent of $x$
$2 r = 10$
$r = 5$
Substituting in (ii), we get
$T_{6} = -^{10} C_{5} 2^{5}$ ...(b)
Hence $\frac{a}{b} = 1 : 32$

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