Question

The Value of x, for which the 6-th term in the expansion of

${\left\{2^{lo{g}_2\sqrt{(9^{y-1}+7)}+\frac{1}{2^{\left[\right(1/5\left)lo{g}_2\right(3^{x-1}+1\left)\right]}}}\right\}}^7$is 84, is equal to:

• A. 4
• B. 3
• C. 2 or 1
• D. 2 or 3

Answer 1

The correct option is C

2 or 1

${\left\{2^{lo{g}_2\sqrt{(9^{y-1}+7)}+\frac{1}{2^{\left[\right(1/5\left)lo{g}_2\right(3^{x-1}+1\left)\right]}}}\right\}}^7$

= ${\left\{\right(9^{X-1}+7{)}^{1/2}+\frac{1}{(3^{x-1}+1{)}^{1/5}}\}}^7$

$\therefore$ $T_6=T_{5+1}={}^7{C}_5.[9^{X-1}+7]\frac{1}{(36X-1+1)}=84$

$\therefore$ $9^{x-1}+7=4(3^{X-1}+1)$

$\therefore$ Putting $3^x=y$ , we ger

$\frac{y^2}{9}+7=4(\frac{y}{3}=1)$

$\Rightarrow$ $y^2+63=12y+36\Rightarrow y^2-12y+27=0$

$\Rightarrow$ y = 3,9.

y = 3 $\Rightarrow$ $3^x=3\Rightarrow x=1$

y = 9 $\Rightarrow$ $3^x=3^2\Rightarrow x=2$

$\therefore$ x = 1 or 2.