Complex Numbers Arithmetic

5 Show that \(( 2 + \mathrm { i } ) ^ { 3 }\) is \(2 + 11 \mathrm { i }\) [0pt] [3 marks]

3 The complex numbers \(z\) and \(w\) are given by \(z = 5 - 2 \mathrm { i }\) and \(w = 3 + 7 \mathrm { i }\). Giving your answers in the form \(x + \mathrm { i } y\) and showing clearly how you obtain them, find

1. \(4 z - 3 w\),
2. \(z ^ { * } w\).

8 The complex number \(5 + 4 \mathrm { j }\) is denoted by \(\alpha\).

1. Find \(\alpha ^ { 2 }\) and \(\alpha ^ { 3 }\), showing your working.
2. The real numbers \(q\) and \(r\) are such that \(\alpha ^ { 3 } + \mathrm { q } \alpha ^ { 2 } + 11 \alpha + \mathrm { r } = 0\). Find \(q\) and \(r\). Let \(\mathrm { f } ( \mathrm { z } ) = \mathrm { z } ^ { 3 } + \mathrm { qz } ^ { 2 } + 11 \mathrm { z } + \mathrm { r }\), where \(q\) and \(r\) are as in part (ii).
3. Solve the equation \(\mathrm { f } ( z ) = 0\).
4. Solve the equation \(z ^ { 4 } + q z ^ { 3 } + 11 z ^ { 2 } + r z = z ^ { 3 } + q z ^ { 2 } + 11 z + r\).

Solve the equation \(2z - 5iz^* = 12\). [4]

3 It is given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers.

1. Find, in terms of \(x\) and \(y\), the real and imaginary parts of $$\mathrm { i } ( z + 7 ) + 3 \left( z ^ { * } - \mathrm { i } \right)$$
2. Hence find the complex number \(z\) such that $$\mathrm { i } ( z + 7 ) + 3 \left( z ^ { * } - \mathrm { i } \right) = 0$$

Given that \(z\) is the complex number \(x + iy\) and satisfies $$|z| + z = 6 - 2i$$ find the value of \(x\) and the value of \(y\). [4]

1

1. Express \(( 1 + 8 i ) ( 2 - i )\) in the form \(x + i y\), showing clearly how you obtain your answer.
2. Hence express \(\frac { 1 + 8 i } { 2 + i }\) in the form \(x + i y\).

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.

2 In this question you must show detailed reasoning.

1. Express \(\frac { 8 + \mathrm { i } } { 2 - \mathrm { i } }\) in the form \(\mathrm { a } + \mathrm { bi }\) where \(a\) and \(b\) are real.
2. Solve the equation \(4 x ^ { 2 } - 8 x + 5 = 0\). Give your answer(s) in the form \(\mathrm { c } + \mathrm { di }\) where \(c\) and \(d\) are real.

Use an algebraic method to find the square roots of the complex number \(21 - 20i\). [6]

1 Without using a calculator, determine the possible values of \(a\) and \(b\) for which \(( a + \mathrm { i } b ) ^ { 2 } = 21 - 20 \mathrm { i }\).

In this question you must show detailed reasoning. Use an algebraic method to find the square roots of \(-77 - 36\text{i}\). [6]

2 In this question you must show detailed reasoning. You are given that \(x = 2 + 5 \mathrm { i }\) is a root of the equation \(x ^ { 3 } - 2 x ^ { 2 } + 21 x + 58 = 0\).
Solve the equation.

Given that \(1 - i\) is a root of the equation \(z^3 - 3z^2 + 4z - 2 = 0\), find the other two roots. Tick \((\checkmark)\) one box. [1 mark] \(-1 + i\) and \(-1\) \(1 + i\) and \(1\) \(-1 + i\) and \(1\) \(1 + i\) and \(-1\)

4 In this question you must show detailed reasoning. You are given that \(\mathrm { f } ( \mathrm { z } ) = 4 \mathrm { z } ^ { 4 } - 12 \mathrm { z } ^ { 3 } + 41 \mathrm { z } ^ { 2 } - 128 \mathrm { z } + 185\) and that \(2 + \mathrm { i }\) is a root of the equation \(f ( z ) = 0\).

1. Express \(\mathrm { f } ( \mathrm { z } )\) as the product of two quadratic factors with integer coefficients.
2. Solve \(f ( z ) = 0\). Two loci on an Argand diagram are defined by \(C _ { 1 } = \left\{ z : | z | = r _ { 1 } \right\}\) and \(C _ { 2 } = \left\{ z : | z | = r _ { 2 } \right\}\) where \(r _ { 1 } > r _ { 2 }\). You are given that two of the points representing the roots of \(\mathrm { f } ( \mathrm { z } ) = 0\) are on \(C _ { 1 }\) and two are on \(C _ { 2 } . R\) is the region on the Argand diagram between \(C _ { 1 }\) and \(C _ { 2 }\).
3. Find the exact area of \(R\).
4. \(\omega\) is the sum of all the roots of \(\mathrm { f } ( \mathrm { z } ) = 0\). Determine whether or not the point on the Argand diagram which represents \(\omega\) lies in \(R\).

1. Sketch this diagram and state fully the geometrical relationship between \(O B\) and \(A C\).
2. Find, in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real, the complex number \(\frac { u } { v }\).
3. Prove that angle \(A O B = \frac { 3 } { 4 } \pi\).

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]

7. $$f ( z ) = z ^ { 3 } + z ^ { 2 } + p z + q$$ where \(p\) and \(q\) are real constants.
The equation \(f ( z ) = 0\) has roots \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) When plotted on an Argand diagram, the points representing \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) form the vertices of a triangle of area 35 Given that \(z _ { 1 } = 3\), find the values of \(p\) and \(q\).

2 The complex number \(7 + 3 \mathrm { i }\) is denoted by \(z\). Find

1. \(| z |\) and \(\arg z\),
2. \(\frac { z } { 4 - \mathrm { i } }\), showing clearly how you obtain your answer.

2 The complex number \(7 + 3 \mathrm { i }\) is denoted by \(z\). Find

1. \(| z |\) and \(\arg z\),
2. \(\frac { z } { 4 - \mathrm { i } }\), showing clearly how you obtain your answer.

1. The complex number \(z\) is given by

$$z = - 7 + 3 i$$ Find

1. \(| z |\)
2. \(\arg z\), giving your answer in radians to 2 decimal places. Given that \(\frac { z } { 1 + \mathrm { i } } + w = 3 - 6 \mathrm { i }\)
3. find the complex number \(w\), giving your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers. You must show all your working.
4. Show the points representing \(z\) and \(w\) on a single Argand diagram.

2 You are given that \(\alpha = - 3 + 4 \mathrm { j }\).

1. Calculate \(\alpha ^ { 2 }\).
2. Express \(\alpha\) in modulus-argument form.

2 You are given that \(\alpha = - 3 + 4 \mathrm { j }\).

1. Calculate \(\alpha ^ { 2 }\).
2. Express \(\alpha\) in modulus-argument form.

8 Let \(z = \cos \theta + \mathrm { i } \sin \theta\). Show that $$1 + z = 2 \cos \frac { 1 } { 2 } \theta \left( \cos \frac { 1 } { 2 } \theta + \mathrm { i } \sin \frac { 1 } { 2 } \theta \right)$$ By considering \(( 1 + z ) ^ { n }\), where \(n\) is a positive integer, deduce the sum of the series $$\binom { n } { 1 } \sin \theta + \binom { n } { 2 } \sin 2 \theta + \ldots + \binom { n } { n } \sin n \theta$$

2 Find the roots of the quadratic equation \(z ^ { 2 } - 4 z + 13 = 0\).
Find the modulus and argument of each root.

2

1. Find the roots of the quadratic equation \(z ^ { 2 } - 4 z + 7 = 0\), simplifying your answers as far as possible.
2. Represent these roots on an Argand diagram.

8

1. Find the roots of the equation \(z ^ { 2 } - z + 1 = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. Obtain the modulus and argument of each root.
3. Show that each root also satisfies the equation \(z ^ { 3 } = - 1\).

1 The complex number \(x + \mathrm { i } y\) is denoted by \(z\). Express \(3 z z ^ { * } - | z | ^ { 2 }\) in terms of \(x\) and \(y\).

Find real numbers \(a\) and \(b\) such that \((a - 3i)(5 - i) = b - 17i\). [5]

14 Show that the two values of \(b\) given on line 36 are equivalent.

\(p(z) = z^4 + 3z^2 + az + b\), \(a \in \mathbb{R}\), \(b \in \mathbb{R}\) \(2 - 3i\) is a root of the equation \(p(z) = 0\)

1. Express \(p(z)\) as a product of quadratic factors with real coefficients. [5 marks]
2. Solve the equation \(p(z) = 0\). [1 mark]

9

1. Use an algebraic method to find the square roots of the complex number \(5 + 12 \mathrm { i }\).
2. Find \(( 3 - 2 \mathrm { i } ) ^ { 2 }\).
3. Hence solve the quartic equation \(x ^ { 4 } - 10 x ^ { 2 } + 169 = 0\).

$$z = -2 + i.$$

1. Express in the form \(a + ib\)
   1. \(\frac{1}{z}\)
   2. \(z^2\). [4]
2. Show that \(|z^2 - z| = 5\sqrt{2}\). [2]
3. Find \(\arg (z^2 - z)\). [2]
4. Display \(z\) and \(z^2 - z\) on a single Argand diagram. [2]

$$z = \frac{a + 3i}{2 + ai}, \quad a \in \mathbb{R}.$$

1. Given that \(a = 4\), find \(|z|\). [3]
2. Show that there is only one value of \(a\) for which \(\arg z = \frac{\pi}{4}\), and find this value. [6]

7 Given that \(z\) is a complex number, prove that \(z z ^ { * } = | z | ^ { 2 }\).

3. $$\mathrm { f } ( z ) = z ^ { 4 } + a z ^ { 3 } + 6 z ^ { 2 } + b z + 65$$ where \(a\) and \(b\) are real constants.
Given that \(z = 3 + 2 \mathbf { i }\) is a root of the equation \(\mathrm { f } ( z ) = 0\), show the roots of \(\mathrm { f } ( z ) = 0\) on a single Argand diagram.

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

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]

3 One root of the quadratic equation \(x ^ { 2 } + a x + b = 0\), where \(a\) and \(b\) are real, is the complex number \(4 - 3 \mathrm { i }\). Find the values of \(a\) and \(b\).

Find the complex number \(z\) such that $$5iz + 3z^* + 16 = 8i$$ Give your answer in the form \(a + bi\), where \(a\) and \(b\) are real. [6 marks]

5 In this question you must show detailed reasoning.

1. Express \(( 2 + 3 \mathrm { i } ) ^ { 3 }\) in the form \(a + \mathrm { i } b\).
2. Hence verify that \(2 + 3\) i is a root of the equation \(3 z ^ { 3 } - 8 z ^ { 2 } + 23 z + 52 = 0\).
3. Express \(3 z ^ { 3 } - 8 z ^ { 2 } + 23 z + 52\) as the product of a linear factor and a quadratic factor with real coefficients.

3 Given that \(z = 1\) is the real root of the equation \(z ^ { 3 } - 1 = 0\), find the two complex roots.

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]

Solve the quadratic equation \(x^2 - 4x - 1 - 12i = 0\) writing your solutions in the form \(a + bi\). [8]

3. Given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers, solve the equation $$( z - 2 i ) \left( z ^ { * } - 2 i \right) = 21 - 12 i$$ where \(z ^ { * }\) is the complex conjugate of \(z\).

1. The complex number \(z\) is given by \(z = 3 + \lambda \mathrm { i }\), where \(\lambda\) is a positive constant. The complex conjugate of \(z\) is denoted by \(\bar { z }\).

Given that \(z ^ { 2 } + \bar { z } ^ { 2 } = 2\), find the value of \(\lambda\).

1 The complex number \(3 + a \mathrm { i }\), where \(a\) is real, is denoted by \(z\). Given that \(\arg z = \frac { 1 } { 6 } \pi\), find the value of \(a\) and hence find \(| z |\) and \(z ^ { * } - 3\).

8 Two complex numbers are given by \(\alpha = 2 - \mathrm { j }\) and \(\beta = - 1 + 2 \mathrm { j }\).

1. Find \(\alpha + \beta , \alpha \beta\) and \(\frac { \alpha } { \beta }\) in the form \(a + b \mathrm { j }\), showing your working.
2. Find the modulus of \(\alpha\), leaving your answer in surd form. Find also the argument of \(\alpha\).
3. Sketch the locus \(| z - \alpha | = 2\) on an Argand diagram.
4. On a separate Argand diagram, sketch the locus \(\arg ( z - \beta ) = \frac { 1 } { 4 } \pi\).

1. Given that \(z = a + b \mathrm { i }\) is a complex number where \(a\) and \(b\) are real constants,
   1. show that \(z z ^ { * }\) is a real number.
   Given that
   • \(z z ^ { * } = 18\)
   • \(\frac { z } { z ^ { * } } = \frac { 7 } { 9 } + \frac { 4 \sqrt { 2 } } { 9 } \mathrm { i }\)
   • determine the possible complex numbers \(z\)

1 Show that \(\frac { 5 } { 2 - 4 \mathrm { i } } = \frac { 1 } { 2 } + \mathrm { i }\).

9 Solve the equation \(z ^ { 3 } + 6 z - 20 = 0\). Find the modulus and argument of each root and illustrate the roots on an Argand diagram.

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 \(2 - i\) is denoted by \(z\).

1. Find \(|z|\) and \(\arg z\). [2]
2. Given that \(az + bz^* = 4 - 8i\), find the values of the real constants \(a\) and \(b\). [5]