Question

The term independent of $x$ in the binomial expansion of $\left(1-\frac{1}{x}+3x^5\right){\left(2x^2-\frac{1}{x}\right)}^8$ is

• A. 496
• B. -496
• C. 400
• D. -400

Answer 1

Answer: (C)

Given expression $=\left(1-\frac{1}{x}+3x^5\right){\left(2x^2-\frac{1}{x}\right)}^8$
General term in ${\left(2x^2-\frac{1}{x}\right)}^8={}^8{C}_r{\left(2x^2\right)}^{8-r}{\left(\frac{-1}{x}\right)}^r$
$\therefore 16-3r=0\Rightarrow r=\frac{16}{3}$ which is not possible
and $16-3r=1\Rightarrow r=5$
and $16-3r=-5\Rightarrow r=7$
$\therefore$ Term independent of $x=-{}^8{C}_5{2}^3(-1{)}^5+3\cdot {}^8{C}_7\cdot 2(-1{)}^7$
$=56\times 8-48$
$=400$