Question

For any complex number, if |Z| = 1, then the value of  \(2(Z\ +\ \bar{Z})\ -\ 2(\frac{1}{Z}\ +\ \frac{1}{\bar{Z}})\) will be
1. 1
2. 0
3. -1
4. 2

Answer 1

Correct Answer - Option 2 : 0

Concept:

Properties of |Z|: If Z = x + iy is a complex number then the following properties are applicable for |Z|.

1. \(|Z|\ =\ |\bar{Z}|\)

2. \(|z|^2 \ =\ Z̅{Z}\)

3. \(|\overline{z_1\ +\ z_2}|\ =\ |\bar{Z_1}\ +\ \bar{Z_2}|\)

Calculation:

Given that,

|Z| = 1 

⇒ |Z|² = 1

⇒ Z Z̅  = 1                  (∵ \(|z|^2 \ =\ Z̅{Z}\))

\(⇒ Z = \frac{1}{\bar{Z}}\)        ----(1)

Therefore, the value of \(2(Z\ +\ \bar{Z})\ -\ 2(\frac{1}{Z}\ +\ \frac{1}{\bar{Z}})\)

=  \(2(Z\ -\ \frac{1}{\bar{Z}})\ -\ 2(\ \bar{Z}\ -\ \frac{1}{Z})\)

But, from equation (1) \( Z = \frac{1}{\bar{Z}}\)

⇒ \(2(Z\ -\ \frac{1}{\bar{Z}})\ -\ 2(\ \bar{Z}\ -\ \frac{1}{Z})\ =\ 0\)

Hence, option 2 is correct.