Question

If z = -1 + i√2, then which one of the following is not true ?
Option:
A) Re(z⁴)+Re(z³)=-2
B)Im(z⁴)+Im(z³)=5√2
C)Im(z⁴)-Re(z³)=4√2-5
D)Re(z⁴)-Im(z³)=√2-7​

Answer 1

Answer:

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Answer 2

Answer:

D) Re($z^{4}$) - Im($z^{3}$) = $\sqrt{2}$ - 7

Step-by-step explanation:

The given complex number is z = -1 + i$\sqrt{2}$

To find $z^{2}$, consider

$z^{2} = (-1+i\sqrt{2} ) (-1+i\sqrt{2} )$

$z^{2}=1-i\sqrt{2}-i\sqrt{2}-2$

z² = -1 - i2√2

To find z³, consider

z³ = z² × z

$z^{3}=(-1-i2\sqrt{2} ) (-1+i\sqrt{2} )$

$z^{3}=1+i2\sqrt{2}-i\sqrt{2}+4$

$z^{3}=5+i\sqrt{2}$

To find $z^{4}$, consider

$z^{4}$ =  z³ × z

$z^{4}=(5 +i\sqrt{2} )(-1+i\sqrt{2} )$

$z^{4}=-7+i4\sqrt{2}$

Now check the each of the given options

A) Re($z^{4}$) + Re($z^{3}$) = - 7 + 5 = -2

Therefore (A) is true

B) Im($z^{4}$)+Im($z^{3}$) = 4$\sqrt{2}$ +$\sqrt{2}$ = 5$\sqrt{2}$

Therefore (B) is true

C) Im($z^{4}$) - Re($z^{3}$) = 4$\sqrt{2}$ - 5

Therefore (C) is true

D) Re($z^{4}$) - Im($z^{3}$) = - 7 - $\sqrt{2}$ ≠ $\sqrt{2}$ - 7

Therefore, (D) is not true