The modulus of $\frac{1-i}{3+i}+\frac{4i}{5}$ is
- $\sqrt{5}$ unit
- $\frac{\sqrt{11}}{5}$ unit
- $\frac{\sqrt{5}}{5}$ unit
- $\frac{\sqrt{12}}{5}$ unit
The Correct Option is C
Solution and Explanation
Let assume :
\(z=\frac{1-i}{3+i}+\frac{4i}{5}\)
\(⇒\frac{5-5i+12i-4}{5(3+i)}=\frac{i+7i}{5(3+i)}\)
\(=\frac{(1+7i)(3-i)}{5(9+1)}\)
\(=\frac{10+20i}{50}=\frac{i+2i}{5}\)
Therefore, \(|z|=\sqrt{(\frac{1}{5})^2+(\frac{2}{5})^2}\)
\(=\frac{1}{5}\sqrt{1+4}=\frac{\sqrt5}{5}\)
So, the correct option is (C) : $\frac{\sqrt{5}}{5}$ unit
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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.
