Mathematics - Binomial Theorem Question with Solution | TestHub

$T_{r+1}={}^{10}C_r{\left(t{x}^{\left(1/5\right)}\right)}^{10-r}{\left[\frac{{\left(1-x\right)}^{\frac{1}{10}}}{t}\right]}^r$

$={}^{10}C_r{\left(t\right)}^{10-2r}{\left(x\right)}^{\frac{\left(10-r\right)}{5}}{\left(1-x\right)}^{\frac{r}{10}}$

For the term independent of $t$, 

$\Rightarrow 10-2r=0$

$\Rightarrow r=5$

$T_6=f\left(x\right)={}^{10}C_5\left(x\sqrt{1-x}\right);$ for maximum

$f^{\text{'}}\left(x\right)=0\Rightarrow x=\frac{2}{3}\&f^{\text{'}\text{'}}\left(\frac{2}{3}\right)<0$

so $f{\left(x\right)}_{max.}={}^{10}C_5\left(\frac{2}{3}\right)·\frac{1}{\sqrt{3}}$

$=\frac{2.10!}{3\sqrt{3}{\left(5!\right)}^2}$