Question

Find the term independent of $x$ in the expansion of ${\left(2x-\frac{1}{x}\right)}^{10}$.

• A. $-8064$
• B. $-8004$
• C. $8065$
• D. $-8060$

Answer 1

Answer: (A)

Let $(r+1{)}^{th}$ term be independent of $x$ in the given expansion
$\therefore T_{r+1}={}^{10}{C}_r{\left(2x\right)}^{10-r}{\left(-\frac{1}{x}\right)}^r$
$={}^{10}{C}_r{2}^{10-r}{\left(-1\right)}^r{x}^{10-2r}$
For this term to be independent of $x$, put $10-2r=0$
$\Rightarrow 2r=10$
$\Rightarrow r=5$
$\therefore T_6={\left(-1\right)}^5{}^{10}{C}_5{2}^{10-5}$
$=\frac{10\times 9\times 8\times 7\times 6}{5\times 4\times 3\times 2}\cdot 32=-8064$