Question

In the expansion of ${\left(\frac{2}{x^3}+\left(a^2+4a+6\right)x\right)}^{20},a\in R$ identify which of the following statement is(are) correct?

• A. The least value of the term independent of $x$ is ${}^{20}{C}_5{2}^{20}$.
• B. The least value of the term independent of $x$ is ${}^{20}{C}_{15}{2}^{15}$.
• C. $\underset{a\to -2}{\mathrm{Lim}}\frac{\text{ coefficient of }{x}^{20}-\text{ coefficient of }{x}^{-60}}{(a+2{)}^2}=20\cdot 2^{19}$
• D. $\underset{a\to -2}{\mathrm{Lim}}\frac{\text{ coefficient of }{x}^{20}-\text{ coefficient of }{x}^{-60}}{(a+2{)}^2}=10\cdot 2^{19}$

Answer 1

Answer: (A), (C)

${\left(\frac{2}{x^3}+\left(a^2+4a+6\right)x\right)}^{20}$
$T_{r+1}={}^{20}{C}_r\cdot {\left(\frac{2}{x^3}\right)}^{20-r}\cdot {\left(a^2+4a+6\right)}^r\cdot x^r$
For term to be independent of $x$
$-60+3r+r=0\Rightarrow r=15$
$\therefore T_{16}={}^{20}{C}_{15}\cdot 2^5\cdot {\left(a^2+4a+6\right)}^{15}$
${T_{16}|}_{\text{least }}={}^{20}{C}_5\cdot 2^{20}$
$\underset{a\to -2}{\mathrm{Lim}}\frac{\text{ coefficient of }{x}^{20}-\text{ coefficient of }{x}^{-60}}{(a+2{)}^2}$
$\underset{a\to -2}{\mathrm{Lim}}\frac{{\left(a^2+4a+6\right)}^{20}-2^{20}}{(a+2{)}^2}$
$\underset{a\to -2}{\mathrm{Lim}}\frac{20{\left(a^2+4a+6\right)}^{19}\cdot (2a+4)}{2(a+2)}=20\cdot 2^{19}$