N2023 Specimen Paper P1 Q7

Do not use a calculator in answering this question.

1. Showing your working, find the complex numbers \(z\) and \(w\) which satisfy the simultaneous equations
   \(2iz+\left( 1-2i \right)w=4\)
   \(\left( 1+i \right)z+\left( 2+i \right)w=3\)[6]
2. 
   The points \(A\) and \(B\) on the Argand diagram represent the complex numbers \({{w}_{1}}\) and \({{w}_{2}}\) respectively.
   1. On the copy of the Argand diagram in the Printed Answer Booklet, plot the point \(C\) to represent the complex number \({{w}_{3}}\), where \({{w}_{3}}={{w}_{1}}+{{w}_{2}}\). Show clearly the geometrical relationship between the points \(A\), \(B\) and \(C\).[2]
      The points \({A}'\), \({B}'\) and \({C}'\) represent the complex numbers \(\mathbf{i}\,{{w}_{1}}\), \(\mathbf{i}\,{{w}_{2}}\) and \(\mathbf{i}\,{{w}_{3}}\) respectively.
   2. On the same Argand diagram in the Printed Answer Booklet, plot the points \({A}'\), \({B}'\) and \({C}'\). State the transformation that maps the points \(A\), \(B\) and \(C\) onto the points \({A}'\), \({B}'\) and \({C}'\).[2]
   3. The transformation in part (ii) maps point \({A}'\) onto the point \({A}''\). Determine, with justification, whether \({A}''\) represents the complex conjugate of \({{w}_{1}}\).[1]