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  4. If the constant term, in binomial expansion of (2 xr+(1/x2))10 is 180 , then r is equal to

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Q. If the constant term, in binomial expansion of $\left(2 x^{r}+\frac{1}{x^{2}}\right)^{10}$ is $180$ , then $r$ is equal to ______

JEE MainJEE Main 2021Binomial Theorem

Solution:

$\left(2 x^{r}+\frac{1}{x^{2}}\right)^{10} $
General term $={ }^{10} C_{R}\left(2 x^{2}\right)^{10-R} x^{-2 R} $
$\Rightarrow 2^{10-R^{10}} C_{R}=180 \ldots \ldots .(1) $
$\&(10-R) r-2 R=0$
$r=\frac{2 R}{10-R} $
$r=\frac{2(R-10)}{10-R}+\frac{20}{10-R} $
$\Rightarrow r=-2+\frac{20}{10-R} \ldots \ldots . .(2)$
$R=8$ or $5$ reject equation (1) not satisfied
At $R=8$
$2^{10-R }{}^{10} C_{R}=180 \Rightarrow r=8$