Complex Numbers for you

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

(a) 4

(b) 3

(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

This entry was written by Nalin Pithwa, posted on July 31, 2015 at 1:59 pm, filed under Cnennai Math Institute Entrance Exam, Complex Numbers, IITJEE Mains, ISI Kolkatta Entrance Exam, RMO and tagged complex numbers. Bookmark the permalink. Follow any comments here with the RSS feed for this post. Post a comment or leave a trackback: Trackback URL.

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