Question:
If $\alpha$ and $\beta$ are the roots of the equation $x^{2} - x + 1 = 0$, then $\alpha^{2009} + \beta^{2009} =$
If $\alpha$ and $\beta$ are the roots of the equation $x^{2} - x + 1 = 0$, then $\alpha^{2009} + \beta^{2009} =$
Updated On: Jul 28, 2022
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The Correct Option is B
Solution and Explanation
$x^{2} - x + 1 = 0 \quad\Rightarrow x = \frac{1\pm\sqrt{1-4}}{2}$
$x = \frac{1\pm \sqrt{3} i}{2}$
$\alpha = \frac{1}{2} + i \frac{\sqrt{3}}{2},\quad \beta = \frac{1}{2} - \frac{i\sqrt{3}}{2}$
$\alpha =cos\frac{\pi }{3} + i\,sin \frac{\pi }{3},\quad\beta = cos\frac{\pi }{3} - i\,sin \frac{\pi }{3}$
$\alpha^{2009} + \beta^{2009} = 2cos\, 2009 \left(\frac{\pi }{3}\right)$
$= 2cos\left[668\pi+\pi +\frac{2\pi }{3}\right] = 2cos \left(\pi +\frac{2\pi }{3}\right)$
$= - 2cos \frac{2\pi }{3} = -2\left(-\frac{1}{2}\right) = 1$
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Concepts Used:
Complex Numbers and Quadratic Equations
Complex Number: Any number that is formed as a+ib is called a complex number. For example: 9+3i,7+8i are complex numbers. Here i = -1. With this we can say that i² = 1. So, for every equation which does not have a real solution we can use i = -1.
Quadratic equation: A polynomial that has two roots or is of the degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b and c are the real numbers.
