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One of the roots of the equation \(2x^3-5x^2+22x-10=0\) is \(\frac{1}{2}\). Find the other two roots of the equation.[3]
The complex number \(z\) is given by \(z=a+bi\), where \(a\) and \(b\) are real numbers such that \(a<0\) and \(b>0\). It is also given that \(\arg(z)=\theta\). In an Argand diagram, points \(P\) and \(Q\) represent \(z\) and \(iz\) respectively.
On an Argand diagram, mark point \(P\), indicating \(\arg(z)\). On the same diagram, mark point \(Q\), showing the relationship between the two points.[2]
Hence, find \(\arg(iz)\) in terms of \(\theta\).[1]
Showing your working, find the complex numbers \(v\) and \(w\) which satisfy the simultaneous equations. \(\begin{aligned}2v-iw&=3\\(1+2i)v+2w&=5+i\end{aligned}\) Give your answers in the form \(x+iy\) where \(x\) and \(y\) are real numbers.[4]
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