Question

The term independent of x in the expression of ${\left(\sqrt{x}-\frac{2}{\sqrt{x}}\right)}^{18}$

• A. $\left(\frac{18}{9}\right)2^{12}$
• B. $\left(\frac{18}{6}\right)2^6$
• C. $\left(\frac{18}{6}\right)2^8$
• D. $\left(\frac{-18}{9}\right)2^9$

Answer 1

The correct option is D $\left(\frac{-18}{9}\right)2^9$

$(\sqrt{x}-\frac{2}{\sqrt{x}}{)}^{18}$

The $(r+1)th$ term of the above expansion is

$t_{r+1}{=}^{18}{C}_r\left(\sqrt{x}{)}^{18-r}\right(\frac{-2}{\sqrt{x}}{)}^r$

${=}^{18}{C}_r{x}^{\frac{18-r}{2}}(-2{)}^r{x}^{\frac{-r}{2}}$

${=}^{18}{C}_r{x}^{\frac{18-2r}{2}}(-2{)}^r$

For a term independent of x, we need

$\frac{18-2r}{2}=0$

$\therefore r=9$

$\therefore$It is the $(r+1)th$ term $=10th$ term which is x independent and is equal to ${}^{18}{C}_9(-2{)}^9-(-1{)}^{18}{C}_9{2}^9$.