4.02b Express complex numbers: cartesian and modulus-argument forms

Given that \(\frac{z + 2i}{z - \lambda i} = i\), where \(\lambda\) is a positive, real constant,

1. show that \(z = \left( \frac{\lambda}{2} + 1 \right) + i \left( \frac{\lambda}{2} - 1 \right)\). [5]

Given also that \(\arg z = \arctan \frac{1}{3}\), calculate

1. the value of \(\lambda\), [3]
2. the value of \(|z|^2\). [2]

The complex number \(z\) is defined by $$z = \frac{a + 2i}{a - 1}, \quad a \in \mathbb{R}, a > 0 .$$ Given that the real part of \(z\) is \(\frac{1}{2}\) , find

1. the value of \(a\), [4]
2. the argument of \(z\), giving your answer in radians to 2 decimal places. [3]

Solve the equation $$z^2 = 4\sqrt{2} - 4\sqrt{2}i,$$ giving your answers in the form \(r(\cos \theta + i \sin \theta)\), where \(-\pi < \theta \leq \pi\). [6]

\(z = 5\sqrt{3} - 5i\) Find

1. \(|z|\), [1]
2. \(\arg(z)\), in terms of \(\pi\). [2]

$$w = 2\left[\cos \frac{\pi}{4} + i\sin \frac{\pi}{4}\right]$$ Find

1. \(\left|\frac{w}{z}\right|\), [1]
2. \(\arg\left(\frac{w}{z}\right)\), in terms of \(\pi\). [2]

A complex number \(z\) is represented by the point \(P\) in the Argand diagram. Given that $$|z - 3i| = 3,$$

1. sketch the locus of \(P\). [2]
2. Find the complex number \(z\) which satisfies both \(|z - 3i| = 3\) and \(\arg (z - 3i) = \frac{3}{4}\pi\). [4] The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac{2i}{w}.$$
3. Show that \(T\) maps \(|z - 3i| = 3\) to a line in the \(w\)-plane, and give the cartesian equation of this line. [5]

The roots of the equation \(z^3 - 1 = 0\) are denoted by \(1, \omega\) and \(\omega^2\).

1. Sketch an Argand diagram to show these roots. [1]
2. Show that \(1 + \omega + \omega^2 = 0\). [2]
3. Hence evaluate
   1. \((2 + \omega)(2 + \omega^2)\), [2]
   2. \(\frac{1}{2 + \omega} + \frac{1}{2 + \omega^2}\). [2]
4. Hence find a cubic equation, with integer coefficients, which has roots \(2, \frac{1}{2 + \omega}\) and \(\frac{1}{2 + \omega^2}\). [4]

Find the cube roots of \(\frac{1}{2}\sqrt{3} + \frac{1}{2}i\), giving your answers in the form \(\cos \theta + i \sin \theta\), where \(0 \leqslant \theta < 2\pi\). [4]

The cube roots of 1 are denoted by \(1\), \(\omega\) and \(\omega^2\), where the imaginary part of \(\omega\) is positive.

1. Show that \(1 + \omega + \omega^2 = 0\). [2]

\includegraphics{figure_1} In the diagram, \(ABC\) is an equilateral triangle, labelled anticlockwise. The points \(A\), \(B\) and \(C\) represent the complex numbers \(z_1\), \(z_2\) and \(z_3\) respectively.

1. State the geometrical effect of multiplication by \(\omega\) and hence explain why \(z_1 - z_3 = \omega(z_3 - z_2)\). [4]
2. Hence show that \(z_1 + \omega z_2 + \omega^2 z_3 = 0\). [2]

Given that \(z_1 = 2\left(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6}\right)\) and \(z_2 = 2\left(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4}\right)\)

1. Find the value of \(|z_1z_2|\) [1 mark]
2. Find the value of \(\arg\left(\frac{z_1}{z_2}\right)\) [1 mark]
3. Sketch \(z_1\) and \(z_2\) on the Argand diagram below, labelling the points as \(P\) and \(Q\) respectively. [2 marks]
4. A third complex number \(w\) satisfies both \(|w| = 2\) and \(-\pi < \arg w < 0\) Given that \(w\) is represented on the Argand diagram as the point \(R\), find the angle \(PRQ\). Fully justify your answer. [3 marks]

Express the complex number \(1 - i\sqrt{3}\) in modulus-argument form. Tick \((\checkmark)\) one box. [1 mark] \(2\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)\) \(2\left(\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3}\right)\) \(2\left(\cos\left(-\frac{\pi}{3}\right) + i\sin\left(-\frac{\pi}{3}\right)\right)\) \(2\left(\cos\left(-\frac{2\pi}{3}\right) + i\sin\left(-\frac{2\pi}{3}\right)\right)\)

In this question you must show detailed reasoning. The complex number \(7 - 4\text{i}\) is denoted by \(z\).

1. Giving your answers in the form \(a + b\text{i}\), where \(a\) and \(b\) are rational numbers, find the following.
   1. \(3z - 4z^*\) [2]
   2. \((z + 1 - 3\text{i})^2\) [2]
   3. \(\frac{z + 1}{z - 1}\) [2]
2. Express \(z\) in modulus-argument form giving the modulus exactly and the argument correct to 3 significant figures. [3]
3. The complex number \(\omega\) is such that \(z\omega = \sqrt{585}(\cos(0.5) + \text{i}\sin(0.5))\). Find the following.
   [3]

In this question you must show detailed reasoning.

1. Solve the equation \(x^2 - 6x + 58 = 0\). Give your solutions in the form \(a + bi\) where \(a\) and \(b\) are real numbers. [3]
2. Determine, in exact form, \(\arg(-10 + (5\sqrt{12})i)^3\). [3]

Fig. 9 shows a sketch of the region OPQ of the Argand diagram defined by $$\left\{z : |z| \leq 4\sqrt{2}\right\} \cap \left\{z : -\frac{1}{4}\pi \leq \arg z \leq \frac{1}{4}\pi\right\}.$$ \includegraphics{figure_9}

1. Find, in modulus-argument form, the complex number represented by the point P. [2]
2. Find, in the form \(a + ib\), where \(a\) and \(b\) are exact real numbers, the complex number represented by the point Q. [3]
3. In this question you must show detailed reasoning. Determine whether the points representing the complex numbers
   lie within this region. [4]

The complex number \(z_1\) is \(1+ i\) and the complex number \(z_2\) has modulus 4 and argument \(\frac{\pi}{3}\).

1. Express \(z_2\) in the form \(a + bi\), giving \(a\) and \(b\) in exact form. [2]
2. Express \(\frac{z_2}{z_1}\) in the form \(c + di\), giving \(c\) and \(d\) in exact form. [2]

A complex number is defined by \(z = -3 + 4\mathrm{i}\).

1. 1. Express \(z\) in the form \(r(\cos\theta + \mathrm{i}\sin\theta)\), where \(-\pi \leqslant \theta \leqslant \pi\).
   2. Express \(\bar{z}\), the complex conjugate of \(z\), in the form \(r(\cos\theta + \mathrm{i}\sin\theta)\). [4]

Another complex number is defined as \(w = \sqrt{5}(\cos 2.68 + \mathrm{i}\sin 2.68)\).

1. Express \(zw\) in the form \(r(\cos\theta + \mathrm{i}\sin\theta)\). [3]

Solve the equation \(2z + iz = \frac{-1 + 7i}{2 + i}\).

1. Give your answer in Cartesian form [7]
2. Give your answer in modulus-argument form. [4]

A complex number is defined by \(z = 3 + 4\mathrm{i}\).

1. Express \(z\) in the form \(z = re^{i\theta}\), where \(-\pi \leqslant \theta \leqslant \pi\). [3]
2. 1. Find the Cartesian coordinates of the vertices of the triangle formed by the cube roots of \(z\) when plotted in an Argand diagram. Give your answers correct to two decimal places.
   2. Write down the geometrical name of the triangle. [5]

Fig. 9 shows a sketch of the region OPQ of the Argand diagram defined by $$\{z : |z| \leq 4\sqrt{2}\} \cap \left\{z : \frac{1}{4}\pi \leq \arg z \leq \frac{3}{4}\pi\right\}.$$ \includegraphics{figure_9}

1. Find, in modulus-argument form, the complex number represented by the point P. [2]
2. Find, in the form \(a + ib\), where \(a\) and \(b\) are exact real numbers, the complex number represented by the point Q. [3]
3. In this question you must show detailed reasoning. Determine whether the points representing the complex numbers
   lie within this region. [4]