2025 HCI P1 Q12

Do not use a graphing calculator for this question.
It is given that \(\mathrm{f}(z) = z^4 - 6z^2 + k\), where \(k\) is a non-zero constant.

1. If \(k\) is a purely imaginary number, determine, with justification, whether \(\mathrm{f}(z) = 0\) can have real roots.[1]
2. Show that \(\mathrm{f}(-z) = \mathrm{f}(z)\).[1]
3. Given that \(2 + i\) is a root of the equation \(\mathrm{f}(z) = 0\), determine the value of \(k\). Hence, or otherwise, find the remaining roots, showing your workings clearly.[6]

Use the value of \(k\) found in part (c) for the rest of this question.

4. Given that the product of all the roots found in part (c) is \(D\), find the value of \(D\), showing your workings clearly.[2]
5. A complex number \(w_1\) satisfies the equation \(kw^4 - 6w^2 + 1 = 0\). Given that \(w_1\) can be obtained from \(2 + i\), find \(w_1\).[2]