A possible value of ^{prime}x^{prime} , for which ninth term in expansion of left{3^{log _{3} √{25

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7.Binomial Theorem

hard

A possible value of $^{\prime}x^{\prime}$, for which the ninth term in the expansion of $\left\{3^{\log _{3} \sqrt{25^{x-1}+7}}+3^{\left(-\frac{1}{8}\right) \log _{3}\left(5^{x-1}+1\right)}\right\}^{10}$ in the increasing powers of $3^{\left(-\frac{1}{8}\right) \log _{3}\left(5^{x-1}+1\right)}$ is equal to $180$ , is:

A

$2$

B

$1$

C

$0$

D

$-1$

(JEE MAIN-2021)

Solution

${ }^{10} \mathrm{C}_{8}\left(25^{(x-1)}+7\right) \times\left(5^{(x-1)}+1\right)^{-1}=180$

$\Rightarrow \frac{25^{x-1}+7}{5^{(x-1)}+1}=4$

$\Rightarrow \frac{t^{2}+7}{t+1}=4$

$\Rightarrow t=1,3=5^{x-1}$

$\Rightarrow x-1=0 \text { (one of the possible value) }$

$\Rightarrow x=1$

Standard 11

Mathematics

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