2025 SAJC P1 Q10

Do not use a calculator in answering this question.

1. Find the complex numbers \(z\) and \(w\) which satisfy the following simultaneous equations.\[|z| + 5w = 0\]\[iz - 4w = -4 + 7i\]Give your answers in the form of \(a + ib\), where \(a\) and \(b\) are real constants.[5]
2. The point \(A\) on the Argand diagram represents the complex number \(u\).
   
   1. On the copy of the Argand diagram in the Printed Answer booklet, plot the point \(B\) to represent the complex number \(-u\).[1]

The points \(C\) and \(D\) represent the complex numbers \(v - u\) and \((v - u)^*\) respectively, where \(v\) is an unknown complex number. It is also given that \(\angle CDA = 90{}^\circ \).

2. By using the Argand diagram or otherwise, state the value of \(\mathrm{Im}(v)\) and justify your answer.[2]