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<ol class="abc-list"><li><ol class="i-list"><li><div class="list-item-paragraph">Showing your working, find the complex numbers \(v\) and \(w\) which satisfy the following simultaneous equations.</div><div class="list-item-paragraph" style="text-align: center">\(v + 2w = 5\)<br>\(3v - w^* = 1 + 5\mathbf{i}\)<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Points \(O\), \(V\) and \(W\) on an Argand diagram represent the complex numbers \(0\), \(v\) and \(w\) respectively. Plot these points on an Argand diagram and state the transformation that maps the point \(W\) onto \(V\).</div></li></ol></li><li><div class="list-item-paragraph">The roots of the equation \(z^4 - 2z^3 - 3z^2 + az + b = 0\), where&nbsp;\(a\) and \(b\) are real numbers, are \(z_1\), \(z_2\), \(z_3\) and \(z_4\). It is given that&nbsp;\(z_1^2 + z_2^2 + z_3^2 + z_4^2 &lt; 0\). Explain why at most two of \(z_1\), \(z_2\), \(z_3\) and \(z_4\) are real.<span class="inline-marks">[2]</span></div></li></ol>

The roots of the equation \(z^4 - 2z^3 - 3z^2 + az + b = 0\), where \(a\) and \(b\) are real numbers, are \(z_1\), \(z_2\), \(z_3\) and \(z_4\). It is given that \(z_1^2 + z_2^2 + z_3^2 + z_4^2 < 0\). Explain why at most two of \(z_1\), \(z_2\), \(z_3\) and \(z_4\) are real.

Complex Numbers - General

Complex Numbers

H2 Math

International Baccalaureate Higher Level Mathematics

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