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If $z$ is a complex number of unit modulus and argument $\theta ,$ then $ \, arg \left(\frac{1 + z}{1 + \bar{z}}\right)$ is equal to
If $z$ is a complex number of unit modulus and argument $\theta ,$ then $ \, arg \left(\frac{1 + z}{1 + \bar{z}}\right)$ is equal to
Updated On: Sep 30, 2024
- $-\theta$
- $\frac{\pi}{2}-\theta$
- $\theta$
- $\pi - \theta $
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The Correct Option is C
Solution and Explanation
The correct option is(C): θ.
Given, |z | = 1, arg 2 =\(\theta \therefore \, z=e^{i\theta}\)
But \(\bar{z}=\frac{1}{z}\)
\(\therefore arg \bigg( \frac{1+z}{1+\frac{1}{2}}\bigg)=arg (z) =\theta\)
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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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