4.02r - OCR Spec

AQA FP2 2008 January Q1

8 marks Standard +0.3

1

Express $4 + 4 \mathrm { i }$ in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$.
Solve the equation $$z ^ { 5 } = 4 + 4 i$$ giving your answers in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$.

AQA FP2 2009 January Q8

11 marks Standard +0.8

8

Show that $$\left( z ^ { 4 } - \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( z ^ { 4 } - \mathrm { e } ^ { - \mathrm { i } \theta } \right) = z ^ { 8 } - 2 z ^ { 4 } \cos \theta + 1$$ (2 marks)
Hence solve the equation $$z ^ { 8 } - z ^ { 4 } + 1 = 0$$ giving your answers in the form $\mathrm { e } ^ { \mathrm { i } \phi }$, where $- \pi < \phi \leqslant \pi$.
Indicate the roots on an Argand diagram.

AQA FP2 2006 June Q7

17 marks Challenging +1.2

7

Find the six roots of the equation $z ^ { 6 } = 1$, giving your answers in the form $\mathrm { e } ^ { \mathrm { i } \phi }$, where $- \pi < \phi \leqslant \pi$.
It is given that $w = \mathrm { e } ^ { \mathrm { i } \theta }$, where $\theta \neq n \pi$.
1. Show that $\frac { w ^ { 2 } - 1 } { w } = 2 \mathrm { i } \sin \theta$.
2. Show that $\frac { w } { w ^ { 2 } - 1 } = - \frac { \mathrm { i } } { 2 \sin \theta }$.
3. Show that $\frac { 2 \mathrm { i } } { w ^ { 2 } - 1 } = \cot \theta - \mathrm { i }$.
4. Given that $z = \cot \theta - \mathrm { i }$, show that $z + 2 \mathrm { i } = z w ^ { 2 }$.
1. Explain why the equation $$( z + 2 \mathrm { i } ) ^ { 6 } = z ^ { 6 }$$ has five roots.
2. Find the five roots of the equation $$( z + 2 \mathrm { i } ) ^ { 6 } = z ^ { 6 }$$ giving your answers in the form $a + \mathrm { i } b$.

AQA FP2 2007 June Q8

13 marks Challenging +1.2

8

1. Given that $z ^ { 6 } - 4 z ^ { 3 } + 8 = 0$, show that $z ^ { 3 } = 2 \pm 2 \mathrm { i }$.
2. Hence solve the equation $$z ^ { 6 } - 4 z ^ { 3 } + 8 = 0$$ giving your answers in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$.
Show that, for any real values of $k$ and $\theta$, $$\left( z - k \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( z - k \mathrm { e } ^ { - \mathrm { i } \theta } \right) = z ^ { 2 } - 2 k z \cos \theta + k ^ { 2 }$$
Express $z ^ { 6 } - 4 z ^ { 3 } + 8$ as the product of three quadratic factors with real coefficients.

AQA FP2 2009 June Q1

8 marks Standard +0.8

1 Given that $z = 2 \mathrm { e } ^ { \frac { \pi \mathrm { i } } { 12 } }$ satisfies the equation $$z ^ { 4 } = a ( 1 + \sqrt { 3 } i )$$ where $a$ is real:

find the value of $a$;
find the other three roots of this equation, giving your answers in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$.

AQA Further Paper 1 2020 June Q6

9 marks Standard +0.8

6 Let $w$ be the root of the equation $z ^ { 7 } = 1$ that has the smallest argument $\alpha$ in the interval $0 < \alpha < \pi$ 6

Prove that $w ^ { n }$ is also a root of the equation $z ^ { 7 } = 1$ for any integer $n$. 6
Prove that $1 + w + w ^ { 2 } + w ^ { 3 } + w ^ { 4 } + w ^ { 5 } + w ^ { 6 } = 0$ 6
Show the positions of $w , w ^ { 2 } , w ^ { 3 } , w ^ { 4 } , w ^ { 5 }$, and $w ^ { 6 }$ on the Argand diagram below.
[0pt] [2 marks] \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-08_835_898_1802_571} 6
Prove that $$\cos \frac { 2 \pi } { 7 } + \cos \frac { 4 \pi } { 7 } + \cos \frac { 6 \pi } { 7 } = - \frac { 1 } { 2 }$$

OCR Further Pure Core 2 2021 June Q4

9 marks Standard +0.8

4 In this question you must show detailed reasoning.
The complex number $- 4 + i \sqrt { 48 }$ is denoted by $z$.

Determine the cube roots of $z$, giving the roots in exponential form. The points which represent the cube roots of $z$ are denoted by $A , B$ and $C$ and these form a triangle in an Argand diagram.
Write down the angles that any lines of symmetry of triangle $A B C$ make with the positive real axis, justifying your answer.

Pre-U Pre-U 9795/1 2010 June Q10

11 marks Challenging +1.2

10 One root of the equation $z ^ { 5 } - 1 = 0$ is the complex number $\omega = \mathrm { e } ^ { \frac { 2 } { 5 } \pi \mathrm { i } }$.

Show that
1. $\quad \omega ^ { 5 } = 1$,
2. $\quad \omega + \omega ^ { 2 } + \omega ^ { 3 } + \omega ^ { 4 } = - 1$,
3. $\quad \omega + \omega ^ { 4 } = 2 \cos \frac { 2 } { 5 } \pi$, and write down a similar expression for $\omega ^ { 2 } + \omega ^ { 3 }$.
4. Using these results, find the values of $\cos \frac { 2 } { 5 } \pi + \cos \frac { 4 } { 5 } \pi$ and $\cos \frac { 2 } { 5 } \pi \times \cos \frac { 4 } { 5 } \pi$, and deduce a quadratic equation, with integer coefficients, which has roots $$\cos \frac { 2 } { 5 } \pi \quad \text { and } \quad \cos \frac { 4 } { 5 } \pi$$

Pre-U Pre-U 9795/1 2012 June Q11

11 marks Standard +0.8

11 The complex number $w = ( \sqrt { 3 } - 1 ) + \mathrm { i } ( \sqrt { 3 } + 1 )$.

Determine, showing full working, the exact values of $| w |$ and $\arg w$.
[0pt] [You may use the result that $\tan \left( \frac { 5 } { 12 } \pi \right) = 2 + \sqrt { 3 }$.]
(a) Find, in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, the three roots, $z _ { 1 } , z _ { 2 }$ and $z _ { 3 }$, of the equation $z ^ { 3 } = w$.
(b) Determine $z _ { 1 } z _ { 2 } z _ { 3 }$ in the form $a + \mathrm { i } b$.
(c) Mark the points representing $z _ { 1 } , z _ { 2 }$ and $z _ { 3 }$ on a sketch of the Argand diagram. Show that they form an equilateral triangle, $\Delta _ { 1 }$, and determine the side-length of $\Delta _ { 1 }$.
(d) The points representing $k z _ { 1 } , k z _ { 2 }$ and $k z _ { 3 }$ form $\Delta _ { 2 }$, an equilateral triangle which is congruent to $\Delta _ { 1 }$, and one of whose vertices lies on the positive real axis. Write down a suitable value for the complex constant $k$.

Pre-U Pre-U 9795/1 2016 June Q7

9 marks Standard +0.3

7

Find all values of $z$ for which $z ^ { 3 } = 2 + 2 \mathrm { i }$. Give your answers in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $\theta$ is an exact multiple of $\pi$ in the interval $0 < \theta < 2 \pi$.
The vertices of a triangle in the Argand diagram correspond to the three roots of the equation $z ^ { 3 } = 2 + 2 \mathrm { i }$. Sketch the triangle and determine its area.

Pre-U Pre-U 9795/1 2016 Specimen Q11

11 marks Challenging +1.8

11

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 $\omega = 16 ( - 1 + \mathrm { i } \sqrt { 3 } )$.
State the value of $| \omega |$ and find $\arg \omega$ as a rational multiple of $\pi$.
(a) Determine the five roots of the equation $z ^ { 5 } = \omega$, giving your answers in the form ( $\mathrm { r } , \theta$ ), where $r > 0$ and $- \pi < \theta \leqslant \pi$.
(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.

Pre-U Pre-U 9794/1 2018 June Q3

4 marks Easy -1.2

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

Pre-U Pre-U 9795/1 2019 Specimen Q11

8 marks Challenging +1.8

11

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 $\omega = 16 ( - 1 + \mathrm { i } \sqrt { 3 } )$.
State the value of $| \omega |$ and find $\arg \omega$ as a rational multiple of $\pi$.
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]

Pre-U Pre-U 9795/1 2020 Specimen Q11

8 marks Challenging +1.8

11

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 $\omega = 16 ( - 1 + \mathrm { i } \sqrt { 3 } )$.
State the value of $| \omega |$ and find $\arg \omega$ as a rational multiple of $\pi$.
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.

Pre-U Pre-U 9795/1 Specimen Q10

24 marks Challenging +1.8

10

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 $\omega = 16 ( - 1 + \mathrm { i } \sqrt { 3 } )$.
State the value of $| \omega |$ and find $\arg \omega$ as a rational multiple of $\pi$.
(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.

CAIE FP1 2005 November Q1

4 marks Standard +0.8

Write down the fifth roots of unity. Hence, or otherwise, find all the roots of the equation $$z^5 = -16 + (16\sqrt{3})i,$$ giving each root in the form $re^{i\theta}$. [4]

CAIE FP1 2019 November Q9

11 marks Challenging +1.8

Use de Moivre's theorem to show that $$\sec 6\theta = \frac{\sec^6 \theta}{32 - 48\sec^2 \theta + 18\sec^4 \theta - \sec^6 \theta}.$$ [6]
Hence obtain the roots of the equation $$3t^6 - 36t^4 + 96t^2 - 64 = 0$$ in the form $\sec q\pi$, where $q$ is rational. [5]

CAIE Further Paper 2 2021 November Q4

10 marks Challenging +1.8

Write down all the roots of the equation $x^5 - 1 = 0$. [2]
Use de Moivre's theorem to show that $\cos 4\theta = 8\cos^4 \theta - 8\cos^2 \theta + 1$. [4]
Use the results of parts (a) and (b) to express each real root of the equation $$8x^9 - 8x^7 + x^5 - 8x^4 + 8x^2 - 1 = 0$$ in the form $\cos k\pi$, where $k$ is a rational number. [4]

CAIE Further Paper 2 2023 November Q2

5 marks Standard +0.3

Find the roots of the equation $(z + 5i)^3 = 4 + 4\sqrt{3}i$, giving your answers in the form $r\cos\theta + ir\sin\theta - 5)$, where $r > 0$ and $0 \leq \theta < 2\pi$. [5]

CAIE Further Paper 2 2024 November Q4

10 marks Challenging +1.8

Use de Moivre's theorem to show that $$\cot 6\theta = \frac{\cot^4 \theta - 15\cot^4 \theta + 15\cot^2 \theta - 1}{6\cot^5 \theta - 20\cot^3 \theta + 6\cot \theta}.$$ [6]
Hence obtain the roots of the equation $$x^6 - 6x^5 - 15x^4 + 20x^3 + 15x^2 - 6x - 1 = 0$$ in the form $\cot(q\pi)$, where $q$ is a rational number. [4]

Edexcel FP1 Q11

7 marks Moderate -0.8

Using that 3 is the real root of the cubic equation $x^3 - 27 = 0$, show that the complex roots of the cubic satisfy the quadratic equation $x^2 + 3x + 9 = 0$. [2]
Hence, or otherwise, find the three cube roots of 27, giving your answers in the form $a + ib$, where $a, b \in \mathbb{R}$. [3]
Show these roots on an Argand diagram. [2]

Edexcel FP1 Q45