9709 - 8 marks | Maths4U

A parameter controls the number of distinct angles

8 written marks

For real \(k\), consider \(\cos^2x-(k+1)\cos x+k=0\), with \(0\le x\lt 2\pi\).

Factor the quadratic in \(u=\cos x\).

List every solution when (i) \(k=-1\), (ii) \(k=0\), and (iii) \(k=2\).

Find all \(k\) for which the original equation has exactly three distinct solutions. Explain both boundary values.

Complete-answer checking, worked solutions and indicative written criteria.