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  4. If Re(1+iy)3=-26, where y is a real number ,then the value of |y| is

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Q. If $Re\left(1+iy\right)^{3}=-26,$ where $y$ is a real number ,then the value of $\left|y\right|$ is

KEAMKEAM 2015Complex Numbers and Quadratic Equations

Solution:

Let $ z=(1+i y)^{3} $
$=1+3\, i \,y+3\, i^{2} y^{2}+i^{3} y^{3} $
$=1+3\,i \,y-3 y^{2}-i \,y^{3}$
$\left(1-3 y^{2}\right)+\left(3 y-y^{3}\right) i $
Now, $ \operatorname{Re}(z)=-26 $
$\Rightarrow 1-3 y^{2} =-26 $
$\Rightarrow 27=3 y^{2} \Rightarrow y^{2}=9 $
$ \Rightarrow y=\pm 3 $ or $|y|=3$