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 $