2025 RI P1 Q8

Do not use a calculator in answering this question.

1. The point \(P\) on the Argand diagram represents the complex number \(w\) given by \(w = u + \mathrm{i}v\), where \(u\) and \(v\) are real and positive.
   1. Explain algebraically why \(\arg(kw) = \arg(w)\) for any real constant \(k > 1\).[1]

   Points \(Q\) and \(R\) represent the complex numbers \(kw\) and \(\mathrm{i}kw\), where \(k\) is a real constant and \(k > 1\).

   2. On the same Argand diagram in the Printed Answer Booklet, plot the points \(Q\) and \(R\). Show clearly the geometrical relationship between the points \(P\), \(Q\) and \(R\).

      [2]

2. In another Argand diagram, the points \(A\), \(B\) and \(C\) represent the complex numbers \(z\), \(\mathrm{f}(z)\) and \(\mathrm{f}(\mathrm{f}(z))\) respectively, where \(\mathrm{f}(z) = z^2 - 2z\).
   1. Show that \(\mathrm{f}(\mathrm{f}(z)) - \mathrm{f}(z) = z(z - 2)(z - 3)(z + 1)\).[2]
   2. It is given that \(ABC\) is a right-angled triangle, described in an anticlockwise sense, with a right angle at \(B\), and \(BC = mBA\), where \(m\) is a positive real constant.

      By considering \(\frac{\mathrm{f}(\mathrm{f}(z)) - \mathrm{f}(z)}{\mathrm{f}(z) - z}\), or otherwise, show that \((z - 2)(z + 1) = mi\).

      [2]
   3. In the case where \(z = x + 2i\), where \(x\) is a positive real number, find \(x\) and \(m\). Hence obtain the complex number represented by the point \(B\).[5]