Question

The ratio of the coefficient of $x^2$ to the coefficient of $x^{10}$ in the expansion of ${(x^5+4\cdot 3^{-{\log }_{\sqrt{3}}\sqrt{x^3}})}^{10}$ is

• A. $4:7$
• B. $10:3$
• C. $3:10$
• D. $7:4$

Answer 1

The correct option is B $10:3$

${(x^5+4\cdot 3^{-{\log }_{\sqrt{3}}\sqrt{x^3}})}^{10}={(x^5+4x^{-3})}^{10}$

$T_{r+1}={}^{10}{C}_r\left(x^5{)}^{10-r}\right(4x^{-3}{)}^r={}^{10}{C}_r(4{)}^r\cdot x^{50-8r}\text{Now,}50-8r=2\Rightarrow r=6\text{\&}50-8r=10\Rightarrow r=5$

$\therefore \text{Required ratio}=\frac{{}^{10}{C}_6\cdot 4^6}{{}^{10}{C}_5\cdot 4^5}=\frac{10}{3}$