Question

The term independent of $x$ in the expression of $\left(1-x^2+3x^3\right){\left(\frac{5}{2}{x}^3-\frac{1}{5x^2}\right)}^{11},x\ne 0$ is

• A. $\frac{7}{40}$
• B. $\frac{33}{200}$
• C. $\frac{39}{200}$
• D. $\frac{11}{50}$

Answer 1

Answer: (B)

$\left(1-x^2+3x^3\right){\left(\frac{5}{2}{x}^3-\frac{1}{5x^2}\right)}^{11}$
General term of ${\left(\frac{5}{2}{x}^3-\frac{1}{5x^2}\right)}^{11}$ is
${}^{11}{C}_r{\left(\frac{5}{2}{x}^3\right)}^{11-r}{\left(-\frac{1}{5x^2}\right)}^r$
General term is ${}^{11}{C}_r{\left(\frac{5}{2}\right)}^{11-r}{\left(-\frac{1}{5}\right)}^r{x}^{33-5r}$
Now, term independent of $x$
$1\times$ coefficient of $x^0$ in ${\left(\frac{5}{2}{x}^3-\frac{1}{5x^2}\right)}^{11}$
$-1\times$ coefficient of $x^{-2}$ in ${\left(\frac{5}{2}{x}^3-\frac{1}{5x^2}\right)}^{11}+$
$3\times$ coefficient of $x^{-3}$ in ${\left(\frac{5}{2}{x}^3-\frac{1}{5x^2}\right)}^{11}$
for coefficient of $x^{0}33-5r=0$ not possible
for coefficient of $x^{-2}33-5r=-2$
$35=5r\Rightarrow r=7$
for coefficient of $x^{-3}33-5r=-3$
$36=5r$ not possible
So term independent of $x$ is
$(-1{)}^{11}{C}_7{\left(\frac{5}{2}\right)}^4{\left(-\frac{1}{5}\right)}^7=\frac{33}{200}$