the sum of the roots (real or complex) of the equation 
X²⁰⁰¹+[1/2-X]²⁰⁰¹=0 is ???
1: 2000
2: 2001
3: 1000
4: 500

Dear Student,
Please find below the solution to the asked query:

$$\begin{gathered} We have \\ {x}^{2001}+{\left(\frac{1}{2}-x\right)}^{2001}=0 \\ \Rightarrow x^{2001}-{\left(x-\frac{1}{2}\right)}^{2001}=0 \end{gathered}$$

$$\begin{gathered} =\frac{{ }^{2001}{\mathrm{C}}_2 \left({\displaystyle \frac{1}{2}}\right)}{{\left({\displaystyle \frac{1}{2}}\right)}^{2001}} \\ \mathrm{which} \mathrm{is} \mathrm{not} \mathrm{equal} \mathrm{to} \mathrm{any} \mathrm{of} \mathrm{the} \mathrm{options}. \mathrm{Kindly} \mathrm{recheck} \mathrm{your} \mathrm{question} \mathrm{and} \\ \mathrm{get} \mathrm{back} \mathrm{to} \mathrm{us}. \end{gathered}$$

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