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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
