Complex number arithmetic and simplification

5 The complex number 2 i is denoted by \(u\). The complex number with modulus 1 and argument \(\frac { 2 } { 3 } \pi\) is denoted by \(w\).

1. Find in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real, the complex numbers \(w , u w\) and \(\frac { u } { w }\).
2. Sketch an Argand diagram showing the points \(U , A\) and \(B\) representing the complex numbers \(u\), \(u w\) and \(\frac { u } { w }\) respectively.
3. Prove that triangle \(U A B\) is equilateral.

7 The complex number \(u\) is defined by \(u = \frac { \sqrt { 2 } - a \sqrt { 2 } \mathrm { i } } { 1 + 2 \mathrm { i } }\), where \(a\) is a positive integer.

1. Express \(u\) in terms of \(a\), in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
   It is now given that \(a = 3\).
2. Express \(u\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\), giving the exact values of \(r\) and \(\theta\).
3. Using your answer to part (b), find the two square roots of \(u\). Give your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\), giving the exact values of \(r\) and \(\theta\).

3 It is given that \(z = - \sqrt { 3 } + \mathrm { i }\).

1. Express \(z ^ { 2 }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
2. The complex number \(\omega\) is such that \(z ^ { 2 } \omega\) is real and \(\left| \frac { z ^ { 2 } } { \omega } \right| = 12\). Find the two possible values of \(\omega\), giving your answers in the form \(R \mathrm { e } ^ { \mathrm { i } \alpha }\), where \(R > 0\) and \(- \pi < \alpha \leqslant \pi\).

5 The complex numbers \(u\) and \(w\) are defined by \(u = 2 \mathrm { e } ^ { \frac { 1 } { 4 } \pi \mathrm { i } }\) and \(w = 3 \mathrm { e } ^ { \frac { 1 } { 3 } \pi \mathrm { i } }\).

1. Find \(\frac { u ^ { 2 } } { w }\), giving your answer in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Give the exact values of \(r\) and \(\theta\).
2. State the least positive integer \(n\) such that both \(\operatorname { Im } w ^ { n } = 0\) and \(\operatorname { Re } w ^ { n } > 0\).

2. $$z = 6 - 6 \sqrt { 3 } i$$

1. 1. Determine the modulus of \(z\)
   2. Show that the argument of \(z\) is \(- \frac { \pi } { 3 }\) Using de Moivre's theorem, and making your method clear,
2. determine, in simplest form, \(z ^ { 4 }\)
3. Determine the values of \(w\) such that \(w ^ { 2 } = z\), giving your answers in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real numbers.

9. $$z = 4 \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right) , \text { and } \boldsymbol { w } = 3 \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right)$$ Express zw in the form \(r ( \cos \theta + \mathrm { i } \sin \theta ) , r > 0 , - \pi < \theta < \pi\).

14. (a) Find the value of \(\lambda\) for which \(\lambda x \cos 3 x\) is a particular integral of the differential equation $$\frac { d ^ { 2 } y } { d x ^ { 2 } } + 9 y = - 12 \sin 3 x$$ (b) Hence find the general solution of this differential equation.(4) The particular solution of the differential equation for which \(\boldsymbol { y } = \mathbf { 1 }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathbf { 2 }\) at \(\boldsymbol { x } = \mathbf { 0 }\), is \(\boldsymbol { y } = \mathbf { g } ( \boldsymbol { x } )\).
(c) Find \(\mathrm { g } ( x )\).
(d) Sketch the graph of \(y = g ( x ) , 0 \leq x \leq \pi\).
(2) \section*{15.} \section*{Figure 1} Figure 1 shows a sketch of the cardioid \(C\) with equation \(r = a ( 1 + \cos \theta ) , - \pi < \theta \leq \pi\). Also shown are the tangents to \(C\) that are parallel and perpendicular to the initial line. These tangents form a rectangle WXYZ.
\includegraphics[max width=\textwidth, alt={}, center]{141c7b1b-4236-4433-84af-04fa9baa3d96-5_407_782_315_1142}
(a) Find the area of the finite region, shaded in Fig. 1, bounded by the curve \(C\).
(b) Find the polar coordinates of the points \(A\) and \(B\) where \(W Z\) touches the curve \(C\).
(c) Hence find the length of \(W X\). Given that the length of \(\boldsymbol { W } \boldsymbol { Z }\) is \(\frac { 3 \sqrt { 3 } a } { 2 }\),
(d) find the area of the rectangle \(W X Y Z\). A heart-shape is modelled by the cardioid \(C\), where \(\boldsymbol { a } = \mathbf { 1 0 ~ c m }\). The heart shape is cut from the rectangular card WXYZ, shown in Fig. 1.
(e) Find a numerical value for the area of card wasted in making this heart shape.
8. A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is defined by $$w = \frac { z + 1 } { i z - 1 } , \quad z \neq - i$$ where \(z = x + \mathrm { i } y , w = u + \mathrm { i } v\) and \(x , y , u\) and \(v\) are real.
\(T\) transforms the circle \(| z | = 1\) in the \(z\)-plane onto a straight line \(L\) in the \(w\)-plane.
(a) Find an equation of \(L\) giving your answer in terms of \(u\) and \(v\).
(b) Show that \(T\) transforms the line \(\operatorname { Im } z = 0\) in the \(z\)-plane onto a circle \(C\) in the \(w\)-plane, giving the centre and radius of this circle.
(c) On a single Argand diagram sketch \(L\) and \(C\). Question: Solve $$x ^ { 5 } = - ( 9 \sqrt { 3 } ) i$$

4. $$z = - 8 + ( 8 \sqrt { } 3 ) i$$

1. Find the modulus of \(z\) and the argument of \(z\). Using de Moivre's theorem,
2. find \(z ^ { 3 }\),
3. find the values of \(w\) such that \(w ^ { 4 } = z\), giving your answers in the form \(a + \mathrm { i } b\), where \(a , b \in \mathbb { R }\).

2. $$z = - 2 + ( 2 \sqrt { 3 } ) \mathrm { i }$$

1. Find the modulus and the argument of \(z\). Using de Moivre's theorem,
2. find \(z ^ { 6 }\), simplifying your answer,
3. find the values of \(w\) such that \(w ^ { 4 } = z ^ { 3 }\), giving your answers in the form \(a + \mathrm { i } b\) where \(a , b \in \mathbb { R }\).

4. $$z = - 8 + ( 8 \sqrt { } 3 ) \mathrm { i }$$

1. Find the modulus of \(z\) and the argument of \(z\). Using de Moivre's theorem,
2. find \(z ^ { 3 }\),
3. find the values of \(w\) such that \(w ^ { 4 } = z\), giving your answers in the form \(a + \mathrm { i } b\), where \(a , b \in \mathbb { R }\).

4 The integrals \(C\) and \(S\) are defined by $$C = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \mathrm { e } ^ { 2 x } \cos 3 x \mathrm {~d} x \quad \text { and } \quad S = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \mathrm { e } ^ { 2 x } \sin 3 x \mathrm {~d} x$$ By considering \(C + \mathrm { i } S\) as a single integral, show that $$C = - \frac { 1 } { 13 } \left( 2 + 3 \mathrm { e } ^ { \pi } \right) ,$$ and obtain a similar expression for \(S\).
(You may assume that the standard result for \(\int \mathrm { e } ^ { k x } \mathrm {~d} x\) remains true when \(k\) is a complex constant, so that \(\left. \int \mathrm { e } ^ { ( a + \mathrm { i } b ) x } \mathrm {~d} x = \frac { 1 } { a + \mathrm { i } b } \mathrm { e } ^ { ( a + \mathrm { i } b ) x } .\right)\)

2

1. Given that \(z _ { 1 } = 2 \mathrm { e } ^ { \frac { 1 } { 6 } \pi \mathrm { i } }\) and \(z _ { 2 } = 3 \mathrm { e } ^ { \frac { 1 } { 4 } \pi \mathrm { i } }\), express \(z _ { 1 } z _ { 2 }\) and \(\frac { z _ { 1 } } { z _ { 2 } }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).
2. Given that \(w = 2 \left( \cos \frac { 1 } { 8 } \pi + \mathrm { i } \sin \frac { 1 } { 8 } \pi \right)\), express \(w ^ { - 5 }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).

2

1. Express \(\frac { \sqrt { 3 } + \mathrm { i } } { \sqrt { 3 } - \mathrm { i } }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).
2. Hence find the smallest positive value of \(n\) for which \(\left( \frac { \sqrt { 3 } + \mathrm { i } } { \sqrt { 3 } - \mathrm { i } } \right) ^ { n }\) is real and positive.

2

1. Show that \(\left( z ^ { n } - \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( z ^ { n } - \mathrm { e } ^ { - \mathrm { i } \theta } \right) \equiv z ^ { 2 n } - ( 2 \cos \theta ) z ^ { n } + 1\).
2. Express \(z ^ { 4 } - z ^ { 2 } + 1\) as the product of four factors of the form \(\left( z - e ^ { \mathrm { i } \alpha } \right)\), where \(0 \leqslant \alpha < 2 \pi\).

4 The complex numbers 0,3 and \(3 \mathrm { e } ^ { \frac { 1 } { 3 } \pi \mathrm { i } }\) are represented in an Argand diagram by the points \(O , A\) and \(B\) respectively.

1. Sketch the triangle \(O A B\) and show that it is equilateral.
2. Hence express \(3 - 3 e ^ { \frac { 1 } { 3 } \pi i }\) in polar form.
3. Hence find \(\left( 3 - 3 \mathrm { e } ^ { \frac { 1 } { 3 } \pi \mathrm { i } } \right) ^ { 5 }\), giving your answer in the form \(a + b \sqrt { 3 } \mathrm { i }\) where \(a\) and \(b\) are rational numbers.

4 In an Argand diagram, the complex numbers \(0 , z\) and \(z \mathrm { e } ^ { \frac { 1 } { 6 } \mathrm { i } \pi }\) are represented by the points \(O , A\) and \(B\) respectively.

1. Sketch a possible Argand diagram showing the triangle \(O A B\). Show that the triangle is isosceles and state the size of angle \(A O B\). The complex numbers \(1 + \mathrm { i }\) and \(5 + 2 \mathrm { i }\) are represented by the points \(C\) and \(D\) respectively. The complex number \(w\) is represented by the point \(E\), such that \(C D = C E\) and angle \(D C E = \frac { 1 } { 6 } \pi\).
2. Calculate the possible values of \(w\), giving your answers exactly in the form \(a + b \mathrm { i }\).

2 It is given that \(z = \mathrm { e } ^ { \mathrm { i } \theta }\), where \(0 < \theta < 2 \pi\), and \(w = \frac { 1 + z } { 1 - z }\).

1. Prove that \(w = \mathrm { i } \cot \frac { 1 } { 2 } \theta\).
2. Sketch separate Argand diagrams to show the locus of \(z\) and the locus of \(w\). You should show the direction in which each locus is described when \(\theta\) increases in the interval \(0 < \theta < 2 \pi\).

3 \end{array} \right)$$

1. Find the acute angle between \(\Pi\) and \(l\).
2. Find the coordinates of the point of intersection of \(\Pi\) and \(l\).
3. \(S\) is the point \(( 4,5 , - 5 )\). Find the shortest distance from \(S\) to \(\Pi\). 2 The complex number \(2 + \mathrm { i }\) is denoted by \(z\).
4. Show that \(z ^ { 2 } = 3 + 4 \mathrm { i }\).
5. Plot the following on the Argand diagram in the Printed Answer Booklet.
   • \(z\)
   • \(z ^ { 2 }\)
   • State the relationship between \(\left| z ^ { 2 } \right|\) and \(| z |\).
   • State the relationship between \(\arg \left( z ^ { 2 } \right)\) and \(\arg ( z )\).
   3 In this question you must show detailed reasoning. Use the formula \(\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )\) to evaluate \(121 ^ { 2 } + 122 ^ { 2 } + 123 ^ { 2 } + \ldots + 300 ^ { 2 }\).

1

1. The Argand diagram below shows the two points which represent two complex numbers, \(z _ { 1 }\) and \(z _ { 2 }\).
   \includegraphics[max width=\textwidth, alt={}, center]{e792797e-6d20-4fc3-9733-43db10f764d7-2_371_689_429_360} On the copy of the diagram in the Printed Answer Booklet
   • draw an appropriate shape to illustrate the geometrical effect of adding \(z _ { 1 }\) and \(z _ { 2 }\),
   • indicate with a cross \(( \times )\) the location of the point representing the complex number \(z _ { 1 } + z _ { 2 }\).
   • You are given that \(\arg z _ { 3 } = \frac { 1 } { 4 } \pi\) and \(\arg z _ { 4 } = \frac { 3 } { 8 } \pi\).
   In each part, sketch and label the points representing the numbers \(z _ { 3 } , z _ { 4 }\) and \(z _ { 3 } z _ { 4 }\) on the diagram in the Printed Answer Booklet. You should join each point to the origin with a straight line.
   (i) \(\left| z _ { 3 } \right| = 1.5\) and \(\left| z _ { 4 } \right| = 1.2\)
   (ii) \(\left| z _ { 3 } \right| = 0.7\) and \(\left| z _ { 4 } \right| = 0.5\)

9

1. By using Euler's formula show that \(\cosh ( \mathrm { i } z ) = \cos z\).
2. Hence, find, in logarithmic form, a root of the equation \(\cos z = 2\). [You may assume that \(\cos z = 2\) has complex roots.]

3 The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by $$z _ { 1 } = \frac { 1 + \mathrm { i } } { 1 - \mathrm { i } } \quad \text { and } \quad z _ { 2 } = \frac { 1 } { 2 } + \frac { \sqrt { 3 } } { 2 } \mathrm { i }$$

1. Show that \(z _ { 1 } = \mathrm { i }\).
2. Show that \(\left| z _ { 1 } \right| = \left| z _ { 2 } \right|\).
3. Express both \(z _ { 1 }\) and \(z _ { 2 }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
4. Draw an Argand diagram to show the points representing \(z _ { 1 } , z _ { 2 }\) and \(z _ { 1 } + z _ { 2 }\).
5. Use your Argand diagram to show that $$\tan \frac { 5 } { 12 } \pi = 2 + \sqrt { 3 }$$

2 Simplify $$\frac { \cos \left( \frac { 6 \pi } { 13 } \right) + i \sin \left( \frac { 6 \pi } { 13 } \right) } { \cos \left( \frac { 2 \pi } { 13 } \right) - i \sin \left( \frac { 2 \pi } { 13 } \right) }$$ Tick ( \(\checkmark\) ) one box. $$\begin{array} { l l } \cos \left( \frac { 8 \pi } { 13 } \right) + i \sin \left( \frac { 8 \pi } { 13 } \right) & \square \\ \cos \left( \frac { 8 \pi } { 13 } \right) - i \sin \left( \frac { 8 \pi } { 13 } \right) & \square \\ \cos \left( \frac { 4 \pi } { 13 } \right) + i \sin \left( \frac { 4 \pi } { 13 } \right) & \square \\ \cos \left( \frac { 4 \pi } { 13 } \right) - i \sin \left( \frac { 4 \pi } { 13 } \right) & \square \end{array}$$

2 The complex number \(z = e ^ { \frac { i \pi } { 3 } }\)
Which one of the following is a real number?
Circle your answer.
[0pt] [1 mark]
\(z ^ { 4 }\)
\(z ^ { 5 }\)
\(z ^ { 6 }\)
\(z ^ { 7 }\)

6 The set \(S\) consists of the following four complex numbers.
\(\begin{array} { l l l l } \sqrt { 3 } + \mathrm { i } & - \sqrt { 3 } - \mathrm { i } & 1 - \mathrm { i } \sqrt { 3 } & - 1 + \mathrm { i } \sqrt { 3 } \end{array}\)
For \(z _ { 1 } , z _ { 2 } \in S\), the binary operation \(\bigcirc\) is defined by \(z _ { 1 } \bigcirc z _ { 2 } = \frac { 1 } { 4 } ( 1 + i \sqrt { 3 } ) z _ { 1 } z _ { 2 }\).

1. 1. Complete the Cayley table for \(( S , \bigcirc )\) given in the Printed Answer Booklet.
   2. Verify that ( \(S , \bigcirc\) ) is a group.
   3. State the order of each element of \(( S , \bigcirc )\).
2. Write down the only proper subgroup of ( \(S , \bigcirc\) ).
3. 1. Explain why ( \(S , \bigcirc\) ) is a cyclic group.
   2. List all possible generators of \(( S , \bigcirc )\).