# Consider the following in respect of a complex number z: 1. $\rm {\overline{\left(z^{-1}\right)}}=(\bar{z})^{-1}$ 2. zz-1 = |z|2 Which of the above

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Consider the following in respect of a complex number z:

1. $\rm {\overline{\left(z^{-1}\right)}}=(\bar{z})^{-1}$

2. zz-1 = |z|2

Which of the above is/are correct?

1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2

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Correct Answer - Option 1 : 1 only

Concept:

Consider z = a + ib

Multiplicative identity z.z-1 = 1

zz̅ = a2 + b2 = (a + ib) (a - ib)

Explanation

Given:

1. $\rm {\overline{\left(z^{-1}\right)}}=(\bar{z})^{-1}$

consider z = a - ib

⇒ z-1 = $\rm \frac 1 z = \frac 1 {a - ib}$ =$\rm \frac{a + ib}{a^{2} + b^{2}}$

⇒ $\rm \overline{z^{-1}}$ = $\rm \frac{a - ib}{a^{2} + b^{2}}$      ....(1)

⇒ $\rm \overline{z}$ = a + ib

⇒ ($\rm \overline{z}$)-1 = $\rm \frac 1 {\bar{z}} = \frac 1 {a +ib}$$\rm \frac{a - ib}{a^{2} + b^{2}}$      ....(2)

from eq(1) and (2)

statement 1 is correct.

2. zz-1 = |z|2

consider z = a - ib

⇒ z-1 = $\rm \frac{a + ib}{a^{2} + b^{2}}$

⇒ z.z-1 = (a - ib)$\rm \frac{a + ib}{a^{2} + b^{2}}$ = $\rm \frac{a^{2} + b^{2}}{a^{2} + b^{2}} =1$

Statement 2 is not correct