Question

If the fourth term in the expansion of ${(x+x^{{\log }_{2}x})}^7$ is $4480,$ then the value of $x$ where $x\in N$ is equal to:

• A. $4$
• B. $3$
• C. $2$
• D. $1$

Answer 1

The correct option is C $2$

Given : ${(x+x^{{\log }_{2}x})}^7$
Now, we know
${}^7{C}_3{x}^4{\left(x^{{\log }_{2}x}\right)}^3=4480\Rightarrow 35x^4{\left(x^{{\log }_{2}x}\right)}^3=4480\Rightarrow x^4{\left(x^{{\log }_{2}x}\right)}^3=128\Rightarrow x^{4+3{\log }_{2}x}=128$
Taking ${\log }_2$, we get
$\Rightarrow (4+3{\log }_{2}x){\log }_{2}x=7$
Let ${\log }_{2}x=y$
$\Rightarrow 4y+3y^2=7\Rightarrow 3y^2+4y-7=0\Rightarrow (y-1)(3y+7)=0\Rightarrow y=1,-\frac{7}{3}\Rightarrow {\log }_{2}x=1,-\frac{7}{3}\Rightarrow x=2,x=2^{-7/3}$
Hence, from the given options $x=2.$