Question

The ratio of coefficient of $x^{15}$ to the term independent of $x$ in the expansion of ${(x^2+\frac{1}{2x})}^{15}$ is

• A. $16:1$
• B. $1:32$
• C. $1:16$
• D. $32:1$

Answer 1

The correct option is D $32:1$

General term $T_{r+1}={}^{15}{C}_r(x^2{)}^{15-r}{\left(\frac{1}{2x}\right)}^r$
$=\frac{{}^{15}{C}_r}{2^r}\times x^{30-2r}\times x^{-r}$

$={}^{15}{C}_r\times 2^{-r}\times x^{30-3r}$
Coefficient of $x^{15}\to 30-3r=15$
$\Rightarrow r=5$
Therefore, coefficient of $x^{15}$ is
${}^{15}{C}_5\times 2^{-5}$

Coefficient of $x^0\to 30-3r=0$
$\Rightarrow r=10$
Therefore, coefficient of the term independent of $x$ is ${}^{15}{C}_{10}\times 2^{-10}$
$\text{Ratio}=\frac{{}^{15}{C}_5\times 2^{-5}}{{}^{15}{C}_{10}\times 2^{-10}}=\frac{32}{1}=32:1$