4.02k Argand diagrams: geometric interpretation

11

1. Use de Moivre's theorem to prove that \(\sin 5 \theta \equiv s \left( 16 s ^ { 4 } - 20 s ^ { 2 } + 5 \right)\), where \(s = \sin \theta\), and deduce that $$\sin \frac { 2 \pi } { 5 } = \sqrt { \frac { 5 + \sqrt { 5 } } { 8 } } .$$ The complex number \(\omega = 16 ( - 1 + \mathrm { i } \sqrt { 3 } )\).
2. State the value of \(| \omega |\) and find \(\arg \omega\) as a rational multiple of \(\pi\).
3. 1. Determine the five roots of the equation \(z ^ { 5 } = \omega\), giving your answers in the form ( \(\mathrm { r } , \theta\) ), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
   2. These five roots are represented in the complex plane by the points \(A , B , C , D\) and \(E\). Show these points on an Argand diagram, and find the area of the pentagon \(A B C D E\) in an exact surd form.[3]

9 The complex number 3-4i is denoted by \(z\). Giving your answers in the form \(x + \mathrm { i } y\), and showing clearly how you obtain them, find

1. \(2 z + z ^ { * }\),
2. \(\frac { 5 } { z }\).
3. Show \(z\) and \(z ^ { * }\) on an Argand diagram.

11

1. Use de Moivre's theorem to prove that \(\sin 5 \theta \equiv s \left( 16 s ^ { 4 } - 20 s ^ { 2 } + 5 \right)\), where \(s = \sin \theta\), and deduce that \(\sin \frac { 2 \pi } { 5 } = \sqrt { \frac { 5 + \sqrt { 5 } } { 8 } }\). The complex number \(\omega = 16 ( - 1 + \mathrm { i } \sqrt { 3 } )\).
2. State the value of \(| \omega |\) and find \(\arg \omega\) as a rational multiple of \(\pi\).
3. 1. Determine the five roots of the equation \(z ^ { 5 } = \omega\), giving your answers in the form \(( \mathrm { r } , \theta )\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
   2. These five roots are represented in the complex plane by the points \(A , B , C , D\) and \(E\). Show these points on an Argand diagram, and find the area of the pentagon \(A B C D E\) in an exact surd form.

9 The complex number \(3 - 4 \mathrm { i }\) is denoted by \(z\). Giving your answers in the form \(x + \mathrm { i } y\), and showing clearly how you obtain them, find

1. \(2 z + z ^ { * }\),
2. \(\frac { 5 } { z }\).
3. Show \(z\) and \(z ^ { * }\) on an Argand diagram.

10

1. Use de Moivre's theorem to prove that \(\sin 5 \theta \equiv s \left( 16 s ^ { 4 } - 20 s ^ { 2 } + 5 \right)\), where \(s = \sin \theta\), and deduce that \(\sin \frac { 2 \pi } { 5 } = \sqrt { \frac { 5 + \sqrt { 5 } } { 8 } }\). The complex number \(\omega = 16 ( - 1 + \mathrm { i } \sqrt { 3 } )\).
2. State the value of \(| \omega |\) and find \(\arg \omega\) as a rational multiple of \(\pi\).
3. (a) Determine the five roots of the equation \(z ^ { 5 } = \omega\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leq \pi\).
   (b) These five roots are represented in the complex plane by the points \(A , B , C , D\) and \(E\). Show these points on an Argand diagram, and find the area of the pentagon \(A B C D E\) in an exact surd form.

2

1. On a single Argand diagram, sketch and clearly label each of the following loci:
   1. \(| z | = 4\),
   2. \(\quad \arg ( z + 4 \mathrm { i } ) = \frac { 1 } { 4 } \pi\).
   3. On the same Argand diagram, shade the region \(R\) defined by the inequalities $$| z | \leqslant 4 \quad \text { and } \quad 0 \leqslant \arg ( z + 4 i ) \leqslant \frac { 1 } { 4 } \pi$$

The complex numbers \(u\) and \(v\) are defined by \(u = -4 + 2\text{i}\) and \(v = 3 + \text{i}\).

1. Find \(\frac{u}{v}\) in the form \(x + \text{i}y\), where \(x\) and \(y\) are real. [3]
2. Hence express \(\frac{u}{v}\) in the form \(re^{\text{i}\theta}\), where \(r\) and \(\theta\) are exact. [2]

In an Argand diagram, with origin \(O\), the points \(A\), \(B\) and \(C\) represent the complex numbers \(u\), \(v\) and \(2u + v\) respectively.

1. State fully the geometrical relationship between \(OA\) and \(BC\). [2]
2. Prove that angle \(AOB = \frac{3}{4}\pi\). [2]

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]

Throughout this question the use of a calculator is not permitted. The complex number \(2 - \mathrm{i}\) is denoted by \(u\).

1. It is given that \(u\) is a root of the equation \(x^3 + ax^2 - 3x + b = 0\), where the constants \(a\) and \(b\) are real. Find the values of \(a\) and \(b\). [4]
2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying both the inequalities \(|z - u| < 1\) and \(|z| < |z + \mathrm{i}|\). [4]

Throughout this question the use of a calculator is not permitted.

1. The complex numbers \(u\) and \(v\) satisfy the equations $$u + 2v = 2i \quad \text{and} \quad iu + v = 3.$$ Solve the equations for \(u\) and \(v\), giving both answers in the form \(x + iy\), where \(x\) and \(y\) are real. [5]
2. On an Argand diagram, sketch the locus representing complex numbers \(z\) satisfying \(|z + i| = 1\) and the locus representing complex numbers \(w\) satisfying \(\arg(w - 2) = \frac{\pi}{4}\). Find the least value of \(|z - w|\) for points on these loci. [5]

The complex numbers \(z_1\) and \(z_2\) are given by $$z_1 = 3 + 5\text{i} \quad \text{and} \quad z_2 = -2 + 6\text{i}$$

1. Show \(z_1\) and \(z_2\) on a single Argand diagram. [2]
2. Without using your calculator and showing all stages of your working,
   1. determine the value of \(|z_1|\) [1]
   2. express \(\frac{z_1}{z_2}\) in the form \(a + b\text{i}\), where \(a\) and \(b\) are fully simplified fractions. [3]
3. Hence determine the value of \(\arg \frac{z_1}{z_2}\) Give your answer in radians to 2 decimal places. [2]

Given that \(z_1 = 3 + 2i\) and \(z_2 = \frac{12 - 5i}{z_1}\).

1. Find \(z_2\) in the form \(a + ib\), where \(a\) and \(b\) are real. [2]
2. Show, on an Argand diagram, the point \(P\) representing \(z_1\) and the point \(Q\) representing \(z_2\). [2]
3. Given that \(O\) is the origin, show that \(\angle POQ = \frac{\pi}{2}\). [2]

The circle passing through the points \(O\), \(P\) and \(Q\) has centre \(C\). Find

1. the complex number represented by \(C\), [2]
2. the exact value of the radius of the circle. [2]

$$f(x) = (4x^2 + 9)(x^2 - 2x + 5)$$

1. Find the four roots of \(f(x) = 0\) [4]
2. Show the four roots of \(f(x) = 0\) on a single Argand diagram. [2]

Given that \(z = 3 + 4i\) and \(w = -1 + 7i\).

1. find \(|w|\). [1]

The complex numbers \(z\) and \(w\) are represented by the points \(A\) and \(B\) on an Argand diagram.

1. Show points \(A\) and \(B\) on an Argand diagram. [1]
2. Prove that \(\triangle OAB\) is an isosceles right-angled triangle. [5]
3. Find the exact value of \(\arg \left( \frac{z}{w} \right)\). [3]

Given that \(z = 3 - 3i\) express, in the form \(a + ib\), where \(a\) and \(b\) are real numbers,

1. \(z^2\), [2]
2. \(\frac{1}{z}\), [2]
3. Find the exact value of each of \(|z|\), \(|z^2|\) and \(\left|\frac{1}{z}\right|\). [2]

The complex numbers \(z\), \(z^2\) and \(\frac{1}{z}\) are represented by the points \(A\), \(B\) and \(C\) respectively on an Argand diagram. The real number 1 is represented by the point \(D\), and \(O\) is the origin.

1. Show the points \(A\), \(B\), \(C\) and \(D\) on an Argand diagram. [2]
2. Prove that \(\triangle OAB\) is similar to \(\triangle OCD\). [3]

Given that \(z = 2 - 2i\) and \(w = -\sqrt{3} + i\),

1. find the modulus and argument of \(wz^2\). [6]
2. Show on an Argand diagram the points \(A\), \(B\) and \(C\) which represent \(z\), \(w\) and \(wz^2\) respectively, and determine the size of angle \(BOC\). [4]

Given that \(z = 1 + \sqrt{3}i\) and that \(\frac{w}{z} = 2 + 2i\), find

1. \(w\) in the form \(a + ib\), where \(a, b \in \mathbb{R}\), [3]
2. the argument of \(w\), [2]
3. the exact value for the modulus of \(w\). [2]

On an Argand diagram, the point \(A\) represents \(z\) and the point \(B\) represents \(w\).

1. Draw the Argand diagram, showing the points \(A\) and \(B\). [2]
2. Find the distance \(AB\), giving your answer as a simplified surd. [2]