Complex Numbers for you

If iz^{3}+z^{2}-z+i=0, then |z| equals

(a) 4

(b) 3

(c) 2

(d) 1.

Solution.

We can write the given equation as

z^{3}+\frac{1}{i}z^{2}-\frac{1}{i}z+1=0, or

z^{3}-iz^{2}+iz-i^{2}=0

\Longrightarrow z^{2}(z-i)+i(z-i)=0

\Longrightarrow (z^{2}+i)(z-i)=0 \Longrightarrow z^{2}=-i, z=i

\Longrightarrow |z|^{2}=|-i| and |z|=|i|

\Longrightarrow |z|^{2}=1 and |z|=1

\Longrightarrow |z|=1

Answer. Option d.

More later,

Nalin Pithwa

 

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