Question

If $z=(1+i\sqrt{3}{)}^{n/2},$ where $n$ is a positive integer, then

• A. if $z$ is real, then least value of $|z|$ is $16$
• B. the smallest value of $n,$ for which $z$ is real, is $6$
• C. if $z$ is real, then least value of $|z|$ is $8$
• D. least real value of $z$ is $-8$

Answer 1

Answer: (B), (C)

The correct options are
B the smallest value of $n,$ for which $z$ is real, is $6$
C if $z$ is real, then least value of $|z|$ is $8$

We know that,
${\omega }^2=\frac{-1-i\sqrt{3}}{2}\Rightarrow 1+i\sqrt{3}=-2{\omega }^2$

$\therefore z=(1+i\sqrt{3}{)}^{n/2}=(-2{\omega }^2{)}^{n/2}$
$\Rightarrow z=(-2{)}^{n/2}\cdot {\omega }^n$

${\omega }^n$ will be real, if $n$ will be a multiple of $3.$
$(-2{)}^{n/2}$ will be real if $n$ will be a multiple of $2.$

Hence for $z$ to be real, the least value of $n$ is $6.$

Least real value of $|z|$ will be
$=|(-2{)}^3{\omega }^6|=|-8|=8$