4.02k Argand diagrams: geometric interpretation

In an Argand diagram, the points \(A\) and \(B\) are represented by the complex numbers \(-3 + 2i\) and \(5 - 4i\) respectively. The points \(A\) and \(B\) are the end points of a diameter of a circle \(C\).

1. Find the equation of \(C\), giving your answer in the form $$|z - a| = b \quad a \in \mathbb{C}, \, b \in \mathbb{R}$$ [3]

The circle \(D\), with equation \(|z - 2 - 3i| = 2\), intersects \(C\) at the points representing the complex numbers \(z_1\) and \(z_2\)

1. Find the complex numbers \(z_1\) and \(z_2\) [6]

$$f(z) = 3z^3 + pz^2 + 57z + q$$ where \(p\) and \(q\) are real constants. Given that \(3 - 2\sqrt{2}i\) is a root of the equation \(f(z) = 0\)

1. show all the roots of \(f(z) = 0\) on a single Argand diagram, [7]
2. find the value of \(p\) and the value of \(q\). [3]

In this question you must show detailed reasoning. The complex numbers \(z_1\) and \(z_2\) are given by \(z_1 = 2 - 3i\) and \(z_2 = a + 4i\) where \(a\) is a real number.

1. Express \(z_1\) in modulus-argument form, giving the modulus in exact form and the argument correct to 3 significant figures. [3]
2. Find \(z_1z_2\) in terms of \(a\), writing your answer in the form \(c + id\). [2]
3. The real and imaginary parts of a complex number on an Argand diagram are \(x\) and \(y\) respectively. Given that the point representing \(z_1z_2\) lies on the line \(y = x\), find the value of \(a\). [2]
4. Given instead that \(z_1z_2 = (z_1z_2)^*\) find the value of \(a\). [2]

\(C\) is the locus of numbers, \(z\), for which \(\ln\left(\frac{z + 7i}{z - 24}\right) = \frac{1}{4}\). By writing \(z = x + iy\) give a complete description of the shape of \(C\) on an Argand diagram. [7]

In an Argand diagram the points representing the numbers \(2 + 3i\) and \(1 - i\) are two adjacent vertices of a square \(S\).

1. Find the area of \(S\). [3]
2. Find all the possible pairs of numbers represented by the other two vertices of \(S\). [4]

In this question you must show detailed reasoning. Solve the equation \(4z^2 - 20z + 169 = 0\). Give your answers in modulus-argument form. [5]

The loci \(C_1\) and \(C_2\) are given by \(|z - (3 + 2i)| = 2\) and \(\arg(z - (3 + 2i)) = \frac{5\pi}{6}\) respectively.

1. Sketch \(C_1\) and \(C_2\) on a single Argand diagram. [4]
2. Find, in surd form, the number represented by the point of intersection of \(C_1\) and \(C_2\). [3]
3. Indicate, by shading, the region of the Argand diagram for which $$|z - (3 + 2i)| \leq 2 \text{ and } \frac{5\pi}{6} \leq \arg(z - (3 + 2i)) \leq \pi.$$ [2]

In this question you must show detailed reasoning. The distinct numbers \(\omega_1\) and \(\omega_2\) both satisfy the quadratic equation \(4x^2 + 4x + 17 = 0\).

1. Write down the value of \(\omega_1 \omega_2\). [1]
2. \(A\), \(B\) and \(C\) are the points on an Argand diagram which represent \(\omega_1\), \(\omega_2\) and \(\omega_1 \omega_2\). Find the area of triangle \(ABC\). [6]

The complex number \(2 + i\) is denoted by \(z\).

1. Show that \(z^2 = 3 + 4i\). [2]
2. Plot the following on the Argand diagram in the Printed Answer Booklet.
   [1]
3. State the relationship between \(|z^2|\) and \(|z|\). [1]
4. State the relationship between \(\arg(z^2)\) and \(\arg(z)\). [1]

C is the locus of numbers, \(z\), for which \(\text{Im}\left(\frac{z + 7i}{z - 24}\right) = \frac{1}{4}\). By writing \(z = x + iy\) give a complete description of the shape of C on an Argand diagram. [7]

Draw the region of the Argand diagram for which \(|z - 3 - 4i| \leq 5\) and \(|z| \leq |z - 2|\). [4]

The complex number \(z_1\) is such that \(z_1 = a + ib\), where \(a\) and \(b\) are positive real numbers.

1. Given that \(z_1^2 = 2 + 2i\), show that \(a = \sqrt{\sqrt{2} + 1}\) and find the exact value of \(b\) in a similar form. [5]

The complex number \(z_2\) is such that \(z_2 = -a + ib\).

1. 1. Determine \(\arg z_2\) as a rational multiple of \(\pi\). [You may use the result \(\tan(\frac{1}{8}\pi) = \sqrt{2} - 1\).] [2]
   2. The point \(P_n\) in an Argand diagram represents the complex number \(z_2^n\), for positive integers \(n\). Find the least value of \(n\) for which \(P_n\) lies on the half-line with equation $$\arg(z) = \frac{1}{4}\pi.$$ [3]

The complex numbers \(z_1\) and \(z_2\) are given by \(z_1 = 2 + i\) and \(z_2 = 3 + 4i\).

1. Verify that \(|z_1| + |z_2| > |z_1 + z_2|\). [4]
2. Sketch on an Argand diagram the locus \(|z - z_1| = 2\). [2]

The region \(R\) of an Argand diagram is defined by the inequalities $$0 \leqslant \arg(z + 4\mathrm{i}) \leqslant \frac{1}{4}\pi \quad \text{and} \quad |z| \leqslant 4.$$ Draw a clearly labelled diagram to illustrate \(R\). [4]

11 Answer only one of the following two alternatives. EITHER State the fifth roots of unity in the form \(\cos \theta + \mathrm { i } \sin \theta\), where \(- \pi < \theta \leqslant \pi\). Simplify $$\left( x - \left[ \cos \frac { 2 } { 5 } \pi + i \sin \frac { 2 } { 5 } \pi \right] \right) \left( x - \left[ \cos \frac { 2 } { 5 } \pi - i \sin \frac { 2 } { 5 } \pi \right] \right) .$$ Hence find the real factors of $$x ^ { 5 } - 1$$ Express the six roots of the equation $$x ^ { 6 } - x ^ { 3 } + 1 = 0$$ as three conjugate pairs, in the form \(\cos \theta \pm \mathrm { i } \sin \theta\). Hence find the real factors of $$x ^ { 6 } - x ^ { 3 } + 1$$ OR Given that $$y ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 3 y ^ { 3 } = 25 \mathrm { e } ^ { - 2 x }$$ and that \(v = y ^ { 3 }\), show that $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} v } { \mathrm {~d} x } + 9 v = 75 \mathrm { e } ^ { - 2 x }$$ Find the particular solution for \(y\) in terms of \(x\), given that when \(x = 0 , y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\). \end{document}