Do not use a calculator in answering this question.
One of the roots of the equation \(z^3+az^2+bz-1=0\), where \(a\) and \(b\) are real numbers, is \(-\frac12+\frac{i\sqrt3}{2}\).
Express \(z^3+az^2+bz-1\) as the product of a linear factor and a quadratic factor with real coefficients and find the values of \(a\) and \(b\).[4]
Hence state the other roots of the equation \(z^3+az^2+bz-1=0\).[1]
Consider \(z=-\frac12+\frac{i\sqrt3}{2}\), which is a root of the equation \(z^3+az^2+bz-1=0\), and using the results in part (i), show that \(1+z^5+z^7=0\).[2]
Do not use a calculator in answering this question.
It is given that \(z_1=1+i\sqrt3\) and \(z_2=\sqrt3-i\).
Sketch on an Argand diagram with origin \(O\), showing the points \(A\), \(B\) and \(C\) representing \(z_1\), \(z_2\) and \(z_1+z_2\) respectively.[2]
Show that \(OACB\) is a square.[2]
Hence by considering \(\arg(z_1)\) and \(\arg(z_1+z_2)\), show that \(\tan\frac{\pi}{12}=2-\sqrt3\).[3]
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