Question

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

• A. ${}^{11}{C}_6{3}^5$
• B. ${}^{11}{C}_5{3}^5$
• C. ${}^{11}{C}_6{3}^6$
• D. None of these

Answer 1

The correct option is C

${}^{11}{C}_6{3}^6$

The general term of $(x-\frac{3}{x}{)}^{11}$ is

$T_{r+1}={}^{11}{C}_r{x}^r{\left(\frac{-3}{x}\right)}^{11-r}$

$={}^{11}{C}_r{x}^r(-3{)}^{11-r}{x}^{r-11}$

$={}^{11}{C}_r{x}^{2r-11}(-3{)}^{11-r}$

To get term independent of $x$, its exponent should be equal to zero.

When $4$ is multiplied to $T_{r+1}$, the exponent of $x$ is $2r-11$ which can never be zero for any non-zero integral value of $r$.

Similarly, when $x^2$ is multiplied to $T_{r+1}$, the exponent of $x$ is $2r-9$ which can never be zero for any non-zero integral value of $r$.

But, when $x$ is multiplied to $T_{r+1}$, the exponent of $x$ is $2r-10$ which can be zero for $r=5$.

$2r-10=0\Rightarrow r=5$

So, the term independent of $x$ is $T_{5+1}={}^{11}{C}_5(-3{)}^{11-5}={}^{11}{C}_5{3}^6={}^{11}{C}_6{3}^6$