Square roots with follow-up application

10

1. Use an algebraic method to find the square roots of the complex number \(16 + 30 \mathrm { i }\).
2. Use your answers to part (i) to solve the equation \(z ^ { 2 } - 2 z - ( 15 + 30 \mathrm { i } ) = 0\), giving your answers in the form \(x + \mathrm { i } y\).

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 \(x ^ { 4 } - 10 x ^ { 2 } + 169 = 0\).

8 The complex number \(a\) is such that \(a ^ { 2 } = 5 - 12 \mathrm { i }\).

1. Use an algebraic method to find the two possible values of \(a\).
2. Sketch on a single Argand diagram the two possible loci given by \(| z - a | = | a |\).

7

1. Use an algebraic method to find the square roots of the complex number \(5 + 12 \mathrm { i }\). You must show sufficient working to justify your answers.
2. Hence solve the quadratic equation \(x ^ { 2 } - 4 x - 1 - 12 \mathrm { i } = 0\).

5 In this question you must show detailed reasoning.

1. Use an algebraic method to find the square roots of \(- 16 + 30 \mathrm { i }\).
2. By finding the cube of one of your answers to part (a) determine a cube root of \(\frac { - 99 + 5 i } { 4 }\). Give your answer in the form \(a + b \mathrm { i }\).

10 In this question you must show detailed reasoning.

1. You are given that \(- 1 + \mathrm { i }\) is a root of the equation \(z ^ { 3 } = a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers. Find \(a\) and \(b\).
2. Find all the roots of the equation in part (a), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r\) and \(\theta\) are exact.
3. Chris says "the complex roots of a polynomial equation come in complex conjugate pairs". Explain why this does not apply to the polynomial equation in part (a).

7 In this question you must show detailed reasoning.

1. Find the square roots of the number \(528 + 46 \mathrm { i }\) giving your answers in the form \(a + b \mathrm { i }\).
2. \(\quad 3 + 2 \mathrm { i }\) is a root of the equation \(x ^ { 3 } - a x + 78 = 0\), where \(a\) is a real number. Find the value of \(a\).

The complex number \(z\), where \(0 < \arg z < \frac{1}{2}\pi\), is such that \(z^2 = 3 + 4\text{i}\).

1. Use an algebraic method to find \(z\). [5]
2. Show that \(z^3 = 2 + 11\text{i}\). [1]

The complex number \(w\) is the root of the equation $$w^6 - 4w^3 + 125 = 0$$ for which \(-\frac{1}{2}\pi < \arg w < 0\).

1. Find \(w\). [5]