Mathematics - Binomial Theorem Question with Solution | TestHub

General term of ${\left(\frac{5}{2}{x}^3-\frac{1}{5x^2}\right)}^{11}$ is 

$T_{r+1}={}^{11}C_r{\left(\frac{5}{2}{x}^3\right)}^{11-r}{\left(-\frac{1}{5x^2}\right)}^r$ $={}^{11}C_r{\left(-1\right)}^r·\frac{5^{11-2r}}{2^{11-r}}·x^{33-5r}$

The term independent of $x$ in the expansion of $\left(1-x^2+3x^3\right){\left(\frac{5}{2}{x}^3-\frac{1}{5x^2}\right)}^{11}$ will be the coefficient of $x^0$ in ${\left(\frac{5}{2}{x}^3-\frac{1}{5x^2}\right)}^{11}$$-$ coefficient of $x^{-2}$ in ${\left(\frac{5}{2}{x}^3-\frac{1}{5x^2}\right)}^{11}+3\times$ coefficient of $x^{-3}$ in ${\left(\frac{5}{2}{x}^3-\frac{1}{5x^2}\right)}^{11}$

$=-{}^{11}C_7{\left(-1\right)}^7·\frac{5^{-3}}{2^4}=\frac{330}{5^3·2^4}=\frac{33}{200}$