The imaginary part of $\left( \frac{1}{2} + \frac{1}{2}i\right)^{10} $ is
- 0
- $\frac{1}{30}$
- $\frac{1}{31}$
- $\frac{1}{32}$
The Correct Option is D
Solution and Explanation
We have,
\(= \left(\frac{1}{2} +\frac{1}{2}i\right)^{10} = \frac{1}{2^{10}} \left(1+i\right)^{10}\)
\(= \frac{1}{2^{10}} \left[\left(1+i\right)^{2}\right]^{5} = \frac{1}{2^{10}} \left(2i\right)^{5}\)
\(= \frac{1}{2^{10}} \times2^{5}i = \frac{1}{32} i\)
Imaginary part \(= \frac{1}{32}\)
Therefore, the Correct Option is (D): \(= \frac{1}{32}\)
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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.
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