Do not use a calculator in answering this question. Showing your working, find the complex numbers \

Question

Do not use a calculator in answering this question. 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] The points $A$ and $B$ on the Argand diagram represent the complex numbers ${{w}_{1}}$ and ${{w}_{2}}$ respectively. 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. 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] 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]

Accepted Answer

(a) z = (1-3i)/2, w = (3+i)/5; (b)(i) OBCA is parallelogram; (b)(ii) Diagram; (b)(iii) A" is rotation not reflection

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