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Q.

If the sum of the coefficients of all the positive even powers of x in the binomial expansion of ${(2 x^{3} + \frac{3}{x})}^{10}$ is $5^{10} - β \cdot 3^{9}$ , then $β$ is equal to ________

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answer is 83.

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Detailed Solution

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$G e n e r a l t e r m = T_{r + 1} = {}^{10}C_{r} {(2 x^{3})}^{10 - r} {(\frac{3}{x})}^{r} = {}^{10}C_{r} 2^{10 - r} 3^{r} x^{30 - 4 r}$

Put r = 0,1,2, …… 7 and we get $β$ = 83

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If the sum of the coefficients of all the positive even powers of x in the binomial expansion of 2x3+3x10 is 510-β·39, then β is equal to ________