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  4. The sum of the roots (real or complex) of the equation x2001+((1/2)-x)2001=0 is

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Q. The sum of the roots (real or complex) of the equation $x^{2001}+\left(\frac{1}{2}-x\right)^{2001}=0$ is

Binomial Theorem

Solution:

$x^{2001}-\left[{ }^{2001} C_{0} x^{2001}-{ }^{2001} C_{1} x^{2000} \cdot \frac{1}{2}+{ }^{2001} C_{2} x^{1999} \cdot \frac{1}{4} \ldots\right]=0$
${ }^{2001} C_{1} x^{2000} \cdot \frac{1}{2}-{ }^{2001} C_{2} x^{1999} \cdot \frac{1}{4} \ldots=0$
sum of the roots $={ }^{2001} C_{2} \cdot \frac{1}{4} \cdot \frac{2}{{ }^{2001} C_{1}}$
$=\frac{2001 \cdot 2000}{4 \cdot 2001}=500$