Question

Find the term independent of x in the expansion of
a) ${(\sqrt{\frac{x}{3}}+\frac{\sqrt{3}}{2x^2})}^{10}$
b) ${(2x^2-\frac{3}{x^3})}^{25}$

Answer 1

${(\sqrt{\frac{x}{3}}+\frac{\sqrt{3}}{2x^2})}^{10}$
General term = $t_{r+1}{=}^{10}{C}_r{\left(\frac{x}{3}\right)}^{\frac{10-r}{2}}{\left(\frac{\sqrt{3}}{2x^2}\right)}^r$
${=}^{10}{C}_r{\left(\frac{1}{3}\right)}^{\frac{10-r}{2}}(3{)}^{\frac{r}{2}}{\left(\frac{1}{2}\right)}^r.x^{\frac{10-r}{2}-2r}$
for term independent of x
$\frac{10-r}{2}-2r=0$
$\frac{10=5r}{2}=0\Rightarrow 5r=10$
$r=2$