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7.Binomial Theorem
hard
If the sum of the coefficients of all the positive even powers of $x$ in the binomial expansion of $\left(2 x^{3}+\frac{3}{x}\right)^{10}$ is $5^{10}-\beta \cdot 3^{9}$, then $\beta$ is equal to
A
$36$
B
$75$
C
$89$
D
$83$
(JEE MAIN-2022)
Solution
$T_{r+1}={ }^{10} C_{r}\left(2 x^{3}\right)^{10-r}\left(\frac{3}{x}\right)^{r}$
$={ }^{10} C_{r} 2^{10-r} 3^{r} x^{30-4 r}$
Put $r=0,1,2, \ldots 7$ and we get $\beta=83$
Standard 11
Mathematics