4.02l Geometrical effects: conjugate, addition, subtraction

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1. Solve the equation \(\omega + 2 + 7 \mathrm { i } = 3 \omega ^ { * } - \mathrm { i }\).
2. Prove algebraically that, for non-zero \(z , z = - z ^ { * }\) if and only if \(z\) is purely imaginary.
3. The complex numbers \(z\) and \(z ^ { * }\) are represented on an Argand diagram by the points \(A\) and \(B\) respectively.
   1. State, for any \(z\), the single transformation which transforms \(A\) to \(B\).
   2. Use a geometric argument to prove that \(z = z ^ { * }\) if and only if \(z\) is purely real.

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1. The Argand diagram below shows the two points which represent two complex numbers, \(z _ { 1 }\) and \(z _ { 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{20816f61-154d-4491-9d2d-4c62687bf81e-02_321_592_276_347} On the copy of the diagram in the Resource Materials.
   • draw an appropriate shape to illustrate the geometrical effect of adding \(z _ { 1 }\) and \(z _ { 2 }\),
   • indicate with a cross \(( \times )\) the location of the point representing the complex number \(z _ { 1 } + z _ { 2 }\).
   • You are given that \(\arg z _ { 3 } = \frac { 1 } { 4 } \pi\) and \(\arg z _ { 4 } = \frac { 3 } { 8 } \pi\).
   In each part, sketch and label the points representing the numbers \(z _ { 3 } , z _ { 4 }\) and \(z _ { 3 } z _ { 4 }\) on the diagram in the Resource Materials. You should join each point to the origin with a straight line.
   1. \(\left| z _ { 3 } \right| = 1.5\) and \(\left| z _ { 4 } \right| = 1.2\)
   2. \(\left| z _ { 3 } \right| = 0.7\) and \(\left| z _ { 4 } \right| = 0.5\)

The complex number \(2 + \mathrm{i}\) is denoted by \(u\). Its complex conjugate is denoted by \(u^*\).

1. Show, on a sketch of an Argand diagram with origin \(O\), the points \(A\), \(B\) and \(C\) representing the complex numbers \(u\), \(u^*\) and \(u + u^*\) respectively. Describe in geometrical terms the relationship between the four points \(O\), \(A\), \(B\) and \(C\). [4]
2. Express \(\frac{u}{u^*}\) in the form \(x + \mathrm{i}y\), where \(x\) and \(y\) are real. [3]
3. By considering the argument of \(\frac{u}{u^*}\), or otherwise, prove that $$\tan^{-1}\left(\frac{4}{3}\right) = 2\tan^{-1}\left(\frac{1}{2}\right).$$ [2]

The transformation \(T\) from the complex \(z\)-plane to the complex \(w\)-plane is given by $$w = \frac{z + 1}{z + i}, \quad z \neq -i.$$

1. Show that \(T\) maps points on the half-line \(\arg(z) = \frac{\pi}{4}\) in the \(z\)-plane into points on the circle \(|w| = 1\) in the \(w\)-plane. [4]
2. Find the image under \(T\) in the \(w\)-plane of the circle \(|z| = 1\) in the \(z\)-plane. [6]
3. Sketch on separate diagrams the circle \(|z| = 1\) in the \(z\)-plane and its image under \(T\) in the \(w\)-plane. [2]
4. Mark on your sketches the point \(P\), where \(z = i\), and its image \(Q\) under \(T\) in the \(w\)-plane. [2]