Find the complex number \(z\) in the form \(x+iy\), where \(x,y\in\mathbb{R}\), such that \(\displaystyle \frac{10iz}{z^*-1+2i}=1+7i\).[3]
The cubic equation \(az^3+bz^2-2=0\) has real coefficients \(a\) and \(b\), \(a\ne0\). The roots of the equation are \(z_1\), \(z_2\) and \(z_3\), with \(z_1\) being a real root, \(z_2\) and \(z_3\) are non-real roots. If \(|z_2-z_3|=\sqrt3\) and \(z^2+cz+1\), where \(c>0\), is the quadratic factor of the cubic function in \(z\), find the roots of the cubic equation.[5]
The point \(A\) represents a fixed complex number \(a\) such that \(-\frac\pi2<\arg a<-\frac\pi4\). The complex numbers \(ia\) and \(a+ia\) are represented by the points \(B\) and \(C\) respectively.
On a clearly labelled Argand diagram, show the points \(A\), \(B\) and \(C\). Deduce the shape of the quadrilateral \(OACB\).[4]
The set of points on the quadrilateral \(OACB\) is represented by \(z\). If \(a=1-\sqrt3i\), find the exact maximum \(|z|\).[2]
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Answer:(a) \(z=-2+i\) (b) \(1\), \(\frac{-1+i\sqrt3}{2}\), \(\frac{-1-i\sqrt3}{2}\) (c) \(OACB\) is a square; \(\max|z|=2\sqrt2\)