4.02k Argand diagrams: geometric interpretation

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]

In this question, \(w\) denotes the complex number \(\cos \frac{2\pi}{5} + i \sin \frac{2\pi}{5}\).

1. Express \(w^2\), \(w^3\) and \(w^4\) in polar form, with arguments in the interval \(0 \leq \theta < 2\pi\). [4]
2. The points in an Argand diagram which represent the numbers $$1, \quad 1 + w, \quad 1 + w + w^2, \quad 1 + w + w^2 + w^3, \quad 1 + w + w^2 + w^3 + w^4$$ are denoted by \(A\), \(B\), \(C\), \(D\), \(E\) respectively. Sketch the Argand diagram to show these points and join them in the order stated. (Your diagram need not be exactly to scale, but it should show the important features.) [4]
3. Write down a polynomial equation of degree 5 which is satisfied by \(w\). [1]

It is given that \(z = e^{i\theta}\), where \(0 < \theta < 2\pi\), and \(w = \frac{1+z}{1-z}\).

1. Prove that \(w = i \cot \frac{1}{2}\theta\). [3]
2. Sketch separate Argand diagrams to show the locus of \(z\) and the locus of \(w\). You should show the direction in which each locus is described when \(\theta\) increases in the interval \(0 < \theta < 2\pi\). [3]

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]

The locus of points \(L_1\) satisfies the equation \(|z| = 2\) The locus of points \(L_2\) satisfies the equation \(\arg(z + 4) = \frac{\pi}{4}\)

1. Sketch \(L_1\) on the Argand diagram below. \includegraphics{figure_18} [1 mark]
2. Sketch \(L_2\) on the Argand diagram above. [1 mark]
3. The complex number \(a + ib\), where \(a\) and \(b\) are real, lies on \(L_1\) The complex number \(c + id\), where \(c\) and \(d\) are real, lies on \(L_2\) Calculate the least possible value of the expression $$(c - a)^2 + (d - b)^2$$ [3 marks]

The Argand diagram shows the solutions to the equation \(z^5 = 1\) \includegraphics{figure_3}

1. Solve the equation $$z^5 = 1$$ giving your answers in the form \(z = \cos\theta + i\sin\theta\), where \(0 \leq \theta < 2\pi\) [2 marks]
2. Explain why the points on an Argand diagram which represent the solutions found in part (a) are the vertices of a regular pentagon. [2 marks]
3. Show that if \(c = \cos\theta\), where \(z = \cos\theta + i\sin\theta\) is a solution to the equation \(z^5 = 1\), then \(c\) satisfies the equation $$16c^5 - 20c^3 + 5c - 1 = 0$$ [5 marks]
4. The Argand diagram on page 22 is repeated below. \includegraphics{figure_4} Explain, with reference to the Argand diagram, why the expression $$16c^5 - 20c^3 + 5c - 1$$ has a repeated quadratic factor. [3 marks]
5. \(O\) is the centre of a regular pentagon \(ABCDE\) such that \(OA = OB = OC = OD = OE = 1\) unit. The distance from \(O\) to \(AB\) is \(h\) By solving the equation \(16c^5 - 20c^3 + 5c - 1 = 0\), show that $$h = \frac{\sqrt{5} + 1}{4}$$ [5 marks]

The diagram below shows a locus on an Argand diagram. \includegraphics{figure_2} Which of the equations below represents the locus shown above? Circle your answer. [1 mark] \(|z - 2 + 3\mathrm{i}| = 2 \quad |z + 2 - 3\mathrm{i}| = 2 \quad |z - 2 + 3\mathrm{i}| = 4 \quad |z + 2 - 3\mathrm{i}| = 4\)

A circle \(C\) in the complex plane has equation \(|z - 2 - 5\mathrm{i}| = a\) The point \(z_1\) on \(C\) has the least argument of any point on \(C\), and \(\arg(z_1) = \frac{\pi}{4}\) Prove that \(a = \frac{3\sqrt{2}}{2}\) [6 marks]

The region \(R\) on an Argand diagram satisfies both \(|z + 2\text{i}| \leq 3\) and \(-\frac{\pi}{6} \leq \arg(z) \leq \frac{\pi}{2}\)

1. Sketch \(R\) on the Argand diagram below. [3 marks] \includegraphics{figure_10a}
2. Find the maximum value of \(|z|\) in the region \(R\), giving your answer in exact form. [5 marks]

The Argand diagram below shows a circle \(C\) \includegraphics{figure_17}

1. Write down the equation of the locus of \(C\) in the form $$|z - w| = a$$ where \(w\) is a complex number whose real and imaginary parts are integers, and \(a\) is an integer. [2 marks]
2. It is given that \(z_1\) is a complex number representing a point on \(C\). Of all the complex numbers which represent points on \(C\), \(z_1\) has the least argument.
   1. Find \(|z_1|\) Give your answer in an exact form. [3 marks]
   2. Show that \(\arg z_1 = \arcsin\left(\frac{6\sqrt{3} - 2}{13}\right)\) [4 marks]

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]

In this question you must show detailed reasoning.

1. Explain why all cubic equations with real coefficients have at least one real root. [2]
2. Points representing the three roots of the equation \(z^3 + 9z^2 + 27z + 35 = 0\) are plotted on an Argand diagram. Find the exact area of the triangle which has these three points as its vertices. [7]

The complex number \(z\) is represented by the point \(P(x, y)\) in the Argand diagram and $$|z - 4 - \mathrm{i}| = |z + 2|.$$

1. Find the equation of the locus of \(P\). [4]
2. Give a geometric interpretation of the locus of \(P\). [1]

The complex number \(z\) is represented by the point \(P(x, y)\) in an Argand diagram and $$|z - 3| = 2|z + i|.$$ Show that the locus of \(P\) is a circle and determine its radius and the coordinates of its centre. [9]

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]

You are given that \(z = 3 - 4\mathrm{i}\).

1. Find
   [3] On an Argand diagram the complex number \(w\) is represented by the point \(A\) and \(w^*\) is represented by the point \(B\).
2. Describe the geometrical relationship between the points \(A\) and \(B\). [2]