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  4. The term independent of x in the expansion of (1+x+2 x2)(3 x2-(1/3 x2))4 is

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Q. The term independent of $x$ in the expansion of $\left(1+x+2 x^2\right)\left(3 x^2-\frac{1}{3 x^2}\right)^4$ is

Binomial Theorem

Solution:

general term $-\left(1+x+2 x^2\right)^4 C_r\left(3 x^2\right)^{4-r}\left(\frac{-1}{3 x^2}\right)^r$
$= \left(1+x+2 x^2\right)^4 C_r 3^{4-r}\left(\frac{-1}{3}\right)^r x^{8-4 r}$
$={ }^4 C_r 3^{4-r}\left(\frac{-1}{3}\right)^r x^{0-4 r}+{ }^4 C_r 3^{4-r}\left(\frac{-1}{3}\right)^r x^{9-4 r}+{ }^4 C_r 3^{4-r} 2\left(\frac{-1}{3}\right)^r x^{10-4 r}$
For independent term of $x$
$ 8-4 r=0 $
$ \Rightarrow r=2 $
and $ 9-4 r=0$
$r=\frac{9}{4}$ Not possible
and $10-4 r=0$
$r=\frac{5}{2}$ Not possible
term$={ }^4 C_2 \cdot 3^2 \times \frac{1}{3^2}=6$