2022 A-Level H2 Mathematics
Papers 1 & 221 questions11 video solutions
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Question 1Complex Numbers4 marks
Do not use a calculator in answering this question.
The complex numbers \(z\) and \(w\) satisfy the following equations.
\[\mathrm{i}z+2w=-1\]
\[(2-\mathrm{i})z+\mathrm{i}w=6\]
Find \(z\) and \(w\), giving your answers in the form \(a+\mathrm{i}b\) where \(a\) and \(b\) are real numbers.[4]
\(2[(2-\mathrm{i})z+\mathrm{i}w]-\mathrm{i}[\mathrm{i}z+2w]=12+\mathrm{i}\)
\((5-2\mathrm{i})z=12+\mathrm{i}\)
\(z=\frac{(12+\mathrm{i})(5+2\mathrm{i})}{29}=2+\mathrm{i}\)
\(2w=-1-\mathrm{i}(2+\mathrm{i})=-2\mathrm{i}\)
\(w=-\mathrm{i}\)
Final Answer:
\(z=2+\mathrm{i},\quad w=-\mathrm{i}\)