Question

The value of coefficient which is independent from $x$ in ${\left(2x-\frac{3}{x^2}\right)}^6,$ is :

• A. -2916
• B. 4860
• C. 2160
• D. none of these

Answer 1

Answer: (C)

The general term in the expansion of
${\left(2x-\frac{3}{x^2}\right)}^6$ is
$T_{r+1}={}^6{C}_r{(2x)}^{6-r}{\left(-\frac{3}{x^2}\right)}^r$
$={(-1)}^r{}^6{C}_r{2}^{6-r}{.3}^r{x}^{6-3r}$
$\therefore$ The term independent of $x,$ if
$6-3r=0$ $\Rightarrow$ $r=2$
$\therefore$ $T_3={}^6{C}_2{.2}^4{.3}^2=2160$