Question

In the expansion of ${(x^3+{3.2}^{-{\log }_2{x}^3})}^{11}$

• A. there appears a term with the power $x^2$
• B. there does not appear a term with the power $x^2$
• C. there appear a term with the power $x^{-3}$
• D. the ratio of the co-efficient of $x^3$ to that of $x^{-3}$ is $\frac{1}{3}$

Answer 1

Answer: (B), (C), (D)

The correct options are
B there does not appear a term with the power $x^2$
C there appear a term with the power $x^{-3}$
D the ratio of the co-efficient of $x^3$ to that of $x^{-3}$ is $\frac{1}{3}$

${(x^3+{3.2}^{-{\log }_2{x}^3})}^{11}={(x^3+\frac{3}{x^3})}^{11}$
$T_{r+1}{=}^{11}{C}_r(x^3{)}^{11-r}{\left(\frac{3}{x^3}\right)}^r={}^{11}{C}_r(x{)}^{33-6r}(3{)}^r$
Now $33-6r=2$ $\Rightarrow$ $6r=31$ (not possible)
$33-6r=-3$ $\Rightarrow$ $r=6$
$33-6r=3$ $\Rightarrow$ $r=5$
$\therefore \frac{coeff.of{x}^3}{coeff.of{x}^{-3}}=\frac{{}^{11}{C}_5{3}^5}{{}^{11}{C}_6{3}^6}=\frac{1}{3}$