2021 SJI P1 Q8

1. Consider the quadratic equation \(2z^2-(2-2\mathrm{i})z-5\mathrm{i}=0\).
   1. Show that \((2-2\mathrm{i})^2=-8\mathrm{i}\).[1]
   2. Write down \((2+2\mathrm{i})^2\) in Cartesian form.[1]
   3. Using the quadratic formula, show that the roots of the quadratic equation are given by \(\dfrac12(1-\mathrm{i})\pm\sqrt{2\mathrm{i}}\).[3]
   4. Using the result in (a)(ii), express each of the roots in the form \(a+\mathrm{i}b\), where \(a\) and \(b\) are real numbers.[4]

Let \(z=\cos\theta+\mathrm{i}\sin\theta\).

2. 1. Find \(|z|\).[2]
   2. Deduce that \(\dfrac1z=z^*\), where \(z^*\) is the conjugate of \(z\).[1]
   3. Find \(z+\dfrac1z\).[2]
   4. Show that \(z^2+\dfrac1{z^2}=2\cos2\theta\).[3]
3. It is given that each of the four roots of the equation \(5z^4-11z^3+16z^2-11z+5=0\) has modulus equal to \(1\).
   1. Using the results in (b), show that \[10\cos^2\theta-11\cos\theta+3=0.\][3]
   2. Hence find these roots.[7]