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Question

Mathematics Question on Complex Numbers and Quadratic Equations

If z(2−i)=(3+i)z(2-i)=(3+i), then z20z^{20} is equal to

A

2102^{10}

B

−210-2^{10}

C

2202^{20}

D

−220-2^{20}

Answer

−210-2^{10}

Explanation

Solution

We have, z(2−i)=(3+i)z\left(2-i\right)=\left(3+i\right) ⇒ z=(3+i2−i)×(2+i2+i)\Rightarrow\, z=\left(\frac{3+i}{2-i}\right)\times\left(\frac{2+i}{2+i}\right) =5+5i5=\frac{5+5i}{5} ⇒ z=1+i\Rightarrow\, z=1+i ⇒ z2=2i\Rightarrow\, z^{2}=2i ⇒ z20\Rightarrow\, z^{20} =−210=-2^{10}