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Q)

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

$\begin{array}{1 1} 1:31 \\ 1:30 \\ 1:32 \\ 1:33 \end{array}$

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A)
General term in $(-1)^r.(1-x^2)^{10}$ is $^{10}C_rx^{2r}$
For coefficient of $x^{10}$, $2r=10$ $\Rightarrow\:\:r=5$
$\therefore$ Coefficient of $x^{10}$ in $(1-x^2)^{10}$ is $-^{10}C_5$
Similarly the general term in $(x-\large\frac{2}{x}$$)^{10}$ is $(-1)^r.^{10}C_r.x^{10-r}.2^r.x^{-r}$
$=(-1)^r.^{10}C_r.2^r.x^{10-2r}$
For independent term in this expansion, $10-2r=0$ or $r=5$
$\therefore\:$ The independent term in $(x-\large\frac{2}{x}$$)^{10}$ is $-^{10}C_5.2^5$
$\Rightarrow\:$ The ratio of coeff. of $x^{10}$ in $(1-x^2)^{10}$ and the independent term in $(x-\large\frac{2}{x}$$)^{10}$ is
$-^{10}C_5\: :\:-^{10}C_5.2^5=1:32$
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