Question

The coefficient of the term independent of $x$ in the expansion of ${\left\{\left[\frac{\left(x+1\right)}{\left(x^{{\displaystyle \frac{2}{3}}}-x^{{\displaystyle \frac{1}{3}}}+1\right)}\right]-\left[\frac{\left(x-1\right)}{\left(x-x^{{\displaystyle \frac{1}{2}}}\right)}\right]\right\}}^{10}$

• A. $210$
• B. $105$
• C. $70$
• D. $112$

Answer 1

The correct option is A

$210$

Explanation for the correct option:

We have to find coefficient of term independent of $x$ of given expression.

Given expression is

${\left\{\left[\frac{\left(x+1\right)}{\left(x^{{\displaystyle \frac{2}{3}}}-x^{{\displaystyle \frac{1}{3}}}+1\right)}\right]-\left[\frac{\left(x-1\right)}{\left(x-x^{{\displaystyle \frac{1}{2}}}\right)}\right]\right\}}^{10}$

We can write it as,

$$\begin{gathered} {\left\{\left[\frac{{\left(x^{{\displaystyle \frac{1}{3}}}\right)}^3+1^3}{\left(x^{{\displaystyle \frac{2}{3}}}-x^{{\displaystyle \frac{1}{3}}}+1\right)}\right]-\left[\frac{\left\{{\left(\sqrt{x}\right)}^2-1\right\}}{\sqrt{x}\left(\sqrt{x}-1\right)}\right]\right\}}^{10} \\ ={\left\{\left[\frac{\left(\left(x^{\frac{1}{3}}\right)+1\right)\left(x^{\frac{2}{3}}-x^{\frac{1}{3}}+1\right)}{\left(x^{{\displaystyle \frac{2}{3}}}-x^{{\displaystyle \frac{1}{3}}}+1\right)}\right]-\left[\frac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)}\right]\right\}}^{10} \\ ={\left[\left(\left(x^{\frac{1}{3}}\right)+1\right)-\frac{\left(\sqrt{x}+1\right)}{\sqrt{x}}\right]}^{10} \\ ={\left[x^{\frac{1}{3}}-x^{\frac{-1}{2}}\right]}^{10} \end{gathered}$$

The general term is;

$$\begin{gathered} T_{r+1}={}^{10}C_r{\left(x^{\frac{1}{3}}\right)}^{10-r}{\left(-x^{\frac{-1}{2}}\right)}^r \\ {T}_{r+1}={}^{10}C_r{\left(-1\right)}^r{x}^{\frac{10-r}{3}-\frac{r}{2}} \end{gathered}$$

For independent of x we will put,

$$\begin{gathered} \frac{10-r}{3}-\frac{r}{2}=0 \\ \Rightarrow 20-2r-3r=0 \\ \Rightarrow 20-5r=0 \\ \Rightarrow r=4 \end{gathered}$$

The coefficient is

$T_5={}^{10}C_4{\left(-1\right)}^4=210$

Hence option(A) is correct answer