Question

The term independent of $x$ in the product $(4+x+7x^2){(x-\frac{3}{x})}^{11}$

• A. $7^{11}{C}_6$
• B. $3^6{.}^{11}{C}_6$
• C. $3^5{.}^{11}{C}_5$
• D. $-12\cdot 2^{11}$

Answer 1

The correct option is B $3^6{.}^{11}{C}_6$

$(4+x+7x^2){(x-\frac{3}{x})}^{11}$
$4{(x-\frac{3}{x})}^{11}+x.{(x-\frac{3}{x})}^{11}+7x^2{(x-\frac{3}{x})}^{11}$
term independent of x in above will be 4 x coefficient of $x^{\circ }$ in expansion of ${(x-\frac{3}{x})}^{11}$
+ 1 x coefficient of $x^{-1}$ in expansion of ${(x-\frac{3}{x})}^{11}$
+7 x coefficient of $x^{-2}$ in expansion of ${(x-\frac{3}{x})}^{11}$
$\therefore x^r$ in expansion of ${(x-\frac{3}{x})}^{11}$
$\widehat{11}{C}_r{x}^{11-r}{\left(\frac{-3}{x}\right)}^r\Rightarrow \widehat{11}{C}_r.x^{11-2r}.(-3{)}^r$
$x^{\circ }$ will not exist in expansion of ${(x-\frac{3}{x})}^{11}$ for integral r $x^{-1}$ will occur at r = 6 $\therefore$ coefficient of $x^{-1}{=}^{1}1C_6(-3{)}^6=3^6{.}^{1}1C_6$
Also $x^2$ will not exist in expansion of ${(x-\frac{3}{x})}^{11}$ for integral r
$\therefore$ term independent of x in expansion will be = $3^6.11C_6$