For any complex number, if |Z| = 1, then the value of \(2(Z\ +\ \ba

1.	For any complex number, if \|Z\| = 1, then the value of \(2(Z\ +\ \bar{Z})\ -\ 2(\frac{1}{Z}\ +\ \frac{1}{\bar{Z}})\) will be1. 12. 03. -14. 2
Answer» 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.

For any complex number, if |Z| = 1, then the value of \(2(Z\ +\ \bar{Z})\ -\ 2(\frac{1}{Z}\ +\ \frac{1}{\bar{Z}})\) will be1. 12. 03. -14. 2

Answer» 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.

Discussion

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