Mathematics - Binomial Theorem Question with Solution | TestHub

$$\begin{array}{rl}& {\mathrm{T}}_{r+1}={}^{15}{\mathrm{C}}_r{\left(x^2\right)}^{15-r}\cdot {\left(2x^{-1}\right)}^r\\ & ={}^{15}{\mathrm{C}}_r\times (2{)}^r\times x^{30-3r}\end{array}$$ For independent term, $30-3r=0\Rightarrow r=10$ Hence the term independent of $x$, $${\mathrm{T}}_{11}={}^{15}{\mathrm{C}}_{10}\times (2{)}^{10}$$ For term involve $x^{15},30-3r=15\Rightarrow r=5$ Hence coefficient of $x^{15}={}^{15}{\mathrm{C}}_5\times (2{)}^5$ Required ratio $$\begin{array}{rl}& =\frac{{}^{15}{\mathrm{C}}_5\times (2{)}^5}{{}^{15}{\mathrm{C}}_{10}\times (2{)}^{10}}=\frac{\frac{15!}{10!5!}}{\frac{15!}{5!10!}\times (2{)}^5}\\ & =1:32\\ & \end{array}$$