If sum of coefficients of all positive even powers of x in binomial expansion of left(2 x^{3}+frac{3

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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

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