Question

In the expansion of ${\left(3^{-x/4}+3^{5x/4}\right)}^n$ the sum of binomial coefficient is $64$ and term with the greatest binomial coefficient exceeds the third by $(n-1)$, the value of $x$ must be

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

Answer: (A)

To get sum of coefficients put $x=0$. Given that sum of coefficients is
$2^n=64$
or $n=6$
The greatest binomial coefficient is ${}^6{C}_3$.
Now given that
$T_4-T_3=6-1=5$
$\Rightarrow {}^6{C}_3{\left(3^{-x/4}\right)}^3{\left(3^{5x/4}\right)}^3-{}^6{C}_2{\left(3^{-x/4}\right)}^2{\left(3^{5x/4}\right)}^4=5$
which is satisfied by $x=0$.