Question

The term in dependent of $x$ in
$(1+x+2x^3){(\frac{3x^2}{2}-\frac{1}{3x})}^9$ is

• A. $\frac{25}{54}$
• B. $\frac{17}{54}$
• C. $\frac{1}{6}$
• D. $-\frac{17}{54}$

Answer 1

The correct option is B $\frac{17}{54}$

To find the term independent of $x$ in

$(1+x+2x^3){\left(\frac{3x^2}{2}-\frac{1}{3x}\right)}^9$,

we have the term free from $x$ in the expansion of ${\left(\frac{3x^2}{2}-\frac{1}{3x}\right)}^9$ is the $3^{th}$ term whose co-efficient is ${}^9{C}_6{\left(\frac{3}{2}\right)}^3.{\left(\frac{1}{3}\right)}^6=\frac{7}{18}$ and no term containing $\frac{1}{x}$ and the term containing $\frac{1}{x^3}$ is the $2^{th}$ term, whose co-efficient is ${}^9{C}_7{\left(\frac{3}{2}\right)}^2.-{\left(\frac{1}{3}\right)}^7=-\frac{1}{27}$.

The required term is $\frac{7}{18}+2(-\frac{1}{27})=\frac{17}{54}$.