Find the complex numbers $w$ and $z$ which satisfy the simultaneous equations $wz=\mathbf{i}$

Question

Find the complex numbers $w$ and $z$ which satisfy the simultaneous equations $wz=\mathbf{i}$ and $w-2\mathbf{i}\,z=2$.[4] The complex number $v$ satisfies $vv^*=3$ and $\arg(v/v^*)=-\pi/2$, where $v^*$ is the complex conjugate of $v$. Arguments are principal values in $(-\pi,\pi]$. Find the modulus of $v$ and all possible values of its argument.[3] Find the smallest positive integer $n$ such that $v^n$ is real for either possible value of $v$.[2]

Accepted Answer

(a) $(z,w)=((1+i)/2,\,1+i)$ or $((-1+i)/2,\,1-i)$. (b)(i) $|v|=\sqrt3,\ \arg v=-\pi/4$ or $3\pi/4$. (ii) $n=4$.

Simultaneous Equations Involving Complex Numbers and Conjugates

Simultaneous Equations Involving Complex Numbers and Conjugates

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