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  4. The term independent of x in the expansion of (. 1 - x .)2(. x + (1/x) .)10 is

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Q. The term independent of $x$ in the expansion of $\left(\right. 1 - x \left.\right)^{2}\left(\right. x + \frac{1}{x} \left.\right)^{10}$ is

NTA AbhyasNTA Abhyas 2022

Solution:

We have, $(1-x)^{2}\left(x+\frac{1}{x}\right)^{10}$ $ \begin{array}{l} =\left(1-2 x+x^{2}\right)\left(x^{10}+{ }^{10} C_{1} x^{9} \cdot \frac{1}{x}+{ }^{10} C_{2} x^{8} \times \frac{1}{x^{2}}+\ldots+\frac{1}{x^{10}}\right) \\ =\left(1-2 x+x^{2}\right)\left(x^{10}+{ }^{10} C_{1} x^{8}+{ }^{10} C_{2} x^{6}+\ldots+{ }^{10} C_{9} \times \frac{1}{x^{8}}+\frac{1}{x^{10}}\right) \end{array} $
So, the term independent of $x$ is
$ \begin{array}{l} { }^{10} C_{5}+(-2) \times 0+{ }^{10} C_{6}={ }^{10} C_{5}+{ }^{10} C_{6} \\ ={ }^{11} C_{5} \end{array} $