4.02b Express complex numbers: cartesian and modulus-argument forms

9 The complex number \(1 + i \sqrt { } 3\) is denoted by \(u\).

1. Express \(u\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Hence, or otherwise, find the modulus and argument of \(u ^ { 2 }\) and \(u ^ { 3 }\).
2. Show that \(u\) is a root of the equation \(z ^ { 2 } - 2 z + 4 = 0\), and state the other root of this equation.
3. Sketch an Argand diagram showing the points representing the complex numbers \(i\) and \(u\). Shade the region whose points represent every complex number \(z\) satisfying both the inequalities $$| z - \mathrm { i } | \leqslant 1 \quad \text { and } \quad \arg z \geqslant \arg u .$$

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.

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 \(z ^ { 3 } = - 1\).

3

1. Solve the equation \(z ^ { 2 } - 2 \mathrm { i } z - 5 = 0\), giving your answers in the form \(x + \mathrm { i } y\) where \(x\) and \(y\) are real.
2. Find the modulus and argument of each root.
3. Sketch an Argand diagram showing the points representing the roots.

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

1. Find the modulus and argument of \(u\).
2. Sketch an Argand diagram showing the points representing the complex numbers 1, i and \(u\). Shade the region whose points represent the complex numbers \(z\) which satisfy both the inequalities \(| z - 1 | \leqslant | z - \mathrm { i } |\) and \(| z - u | \leqslant 1\).
3. Using your diagram, calculate the value of \(| z |\) for the point in this region for which \(\arg z\) is least.

8 The variable complex number \(z\) is given by $$z = 1 + \cos 2 \theta + i \sin 2 \theta$$ where \(\theta\) takes all values in the interval \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\).

1. Show that the modulus of \(z\) is \(2 \cos \theta\) and the argument of \(z\) is \(\theta\).
2. Prove that the real part of \(\frac { 1 } { z }\) is constant.

8 The complex number \(u\) is defined by \(u = \frac { 6 - 3 \mathrm { i } } { 1 + 2 \mathrm { i } }\).

1. Showing all your working, find the modulus of \(u\) and show that the argument of \(u\) is \(- \frac { 1 } { 2 } \pi\).
2. For complex numbers \(z\) satisfying \(\arg ( z - u ) = \frac { 1 } { 4 } \pi\), find the least possible value of \(| z |\).
3. For complex numbers \(z\) satisfying \(| z - ( 1 + \mathrm { i } ) u | = 1\), find the greatest possible value of \(| z |\).

7

1. The complex number \(u\) is defined by \(u = \frac { 5 } { a + 2 \mathrm { i } }\), where the constant \(a\) is real.
   1. Express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
   2. Find the value of \(a\) for which \(\arg \left( u ^ { * } \right) = \frac { 3 } { 4 } \pi\), where \(u ^ { * }\) denotes the complex conjugate of \(u\).
2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) which satisfy both the inequalities \(| z | < 2\) and \(| z | < | z - 2 - 2 \mathrm { i } |\).

7

1. Find the roots of the equation $$z ^ { 2 } + ( 2 \sqrt { } 3 ) z + 4 = 0$$ giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. State the modulus and argument of each root.
3. Showing all your working, verify that each root also satisfies the equation $$z ^ { 6 } = - 64$$

8 Let \(\mathrm { f } ( x ) = \frac { 7 x + 4 } { ( 2 x + 1 ) ( x + 1 ) ^ { 2 } }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence show that \(\int _ { 0 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = 2 + \ln \frac { 5 } { 3 }\).

7 The number of insects in a population \(t\) days after the start of observations is denoted by \(N\). The variation in the number of insects is modelled by a differential equation of the form $$\frac { \mathrm { d } N } { \mathrm {~d} t } = k N \cos ( 0.02 t )$$ where \(k\) is a constant and \(N\) is taken to be a continuous variable. It is given that \(N = 125\) when \(t = 0\).

1. Solve the differential equation, obtaining a relation between \(N , k\) and \(t\).
2. Given also that \(N = 166\) when \(t = 30\), find the value of \(k\).
3. Obtain an expression for \(N\) in terms of \(t\), and find the least value of \(N\) predicted by this model.

10 The complex number \(w\) is given by \(w = - \frac { 1 } { 2 } + \mathrm { i } \frac { \sqrt { } 3 } { 2 }\).

1. Find the modulus and argument of \(w\).
2. The complex number \(z\) has modulus \(R\) and argument \(\theta\), where \(- \frac { 1 } { 3 } \pi < \theta < \frac { 1 } { 3 } \pi\). State the modulus and argument of \(w z\) and the modulus and argument of \(\frac { z } { w }\).
3. Hence explain why, in an Argand diagram, the points representing \(z , w z\) and \(\frac { z } { w }\) are the vertices of an equilateral triangle.
4. In an Argand diagram, the vertices of an equilateral triangle lie on a circle with centre at the origin. One of the vertices represents the complex number \(4 + 2 \mathrm { i }\). Find the complex numbers represented by the other two vertices. Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.

6 The complex number \(z\) is given by $$z = ( \sqrt { } 3 ) + \mathrm { i } .$$

1. Find the modulus and argument of \(z\).
2. The complex conjugate of \(z\) is denoted by \(z ^ { * }\). Showing your working, express in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real,
   1. \(2 z + z ^ { * }\),
   2. \(\frac { \mathrm { i } z ^ { * } } { z }\).
   3. On a sketch of an Argand diagram with origin \(O\), show the points \(A\) and \(B\) representing the complex numbers \(z\) and \(\mathrm { i } z ^ { * }\) respectively. Prove that angle \(A O B = \frac { 1 } { 6 } \pi\).

6 The complex number \(w\) is defined by \(w = - 1 + \mathrm { i }\).

1. Find the modulus and argument of \(w ^ { 2 }\) and \(w ^ { 3 }\), showing your working.
2. The points in an Argand diagram representing \(w\) and \(w ^ { 2 }\) are the ends of a diameter of a circle. Find the equation of the circle, giving your answer in the form \(| z - ( a + b \mathrm { i } ) | = k\).

9 The complex number \(1 + ( \sqrt { } 2 ) \mathrm { i }\) is denoted by \(u\). The polynomial \(x ^ { 4 } + x ^ { 2 } + 2 x + 6\) is denoted by \(\mathrm { p } ( x )\).

1. Showing your working, verify that \(u\) is a root of the equation \(\mathrm { p } ( x ) = 0\), and write down a second complex root of the equation.
2. Find the other two roots of the equation \(\mathrm { p } ( x ) = 0\).

10

1. Without using a calculator, solve the equation \(\mathrm { i } w ^ { 2 } = ( 2 - 2 \mathrm { i } ) ^ { 2 }\).
2. 1. Sketch an Argand diagram showing the region \(R\) consisting of points representing the complex numbers \(z\) where $$| z - 4 - 4 i | \leqslant 2$$
   2. For the complex numbers represented by points in the region \(R\), it is given that $$p \leqslant | z | \leqslant q \quad \text { and } \quad \alpha \leqslant \arg z \leqslant \beta$$ Find the values of \(p , q , \alpha\) and \(\beta\), giving your answers correct to 3 significant figures.

7 Throughout this question the use of a calculator is not permitted.
The complex number \(1 - ( \sqrt { } 3 ) \mathrm { i }\) is denoted by \(u\).

1. Find the modulus and argument of \(u\).
2. Show that \(u ^ { 3 } + 8 = 0\).
3. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying both the inequalities \(| z - u | \leqslant 2\) and \(\operatorname { Re } z \geqslant 2\), where \(\operatorname { Re } z\) denotes the real part of \(z\).
   [0pt] [4] \(8 \quad\) Let \(\mathrm { f } ( x ) = \frac { 8 x ^ { 2 } + 9 x + 8 } { ( 1 - x ) ( 2 x + 3 ) ^ { 2 } }\).

9 The complex number \(3 - \mathrm { i }\) is denoted by \(u\). Its complex conjugate is denoted by \(u ^ { * }\).

1. On an Argand diagram with origin \(O\), show the points \(A , B\) and \(C\) representing the complex numbers \(u , u ^ { * }\) and \(u ^ { * } - u\) respectively. What type of quadrilateral is \(O A B C\) ?
2. Showing your working and without using a calculator, express \(\frac { u ^ { * } } { u }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
3. By considering the argument of \(\frac { u ^ { * } } { u }\), prove that $$\tan ^ { - 1 } \left( \frac { 3 } { 4 } \right) = 2 \tan ^ { - 1 } \left( \frac { 1 } { 3 } \right) .$$ \includegraphics[max width=\textwidth, alt={}, center]{d4a7604c-9e2c-47ef-a496-8697bc88fdd4-18_360_758_260_689} The diagram shows the curve \(y = \frac { x ^ { 2 } } { 1 + x ^ { 3 } }\) for \(x \geqslant 0\), and its maximum point \(M\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = p\).
4. Find the exact value of the \(x\)-coordinate of \(M\).
5. Calculate the value of \(p\) for which the area of \(R\) is equal to 1 . Give your answer correct to 3 significant figures.

11 The complex number \(z\) is defined by \(z = \frac { 5 a - 2 \mathrm { i } } { 3 + a \mathrm { i } }\), where \(a\) is an integer. It is given that \(\arg z = - \frac { 1 } { 4 } \pi\).

1. Find the value of \(a\) and hence express \(z\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. Express \(z ^ { 3 }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Give the simplified exact values of \(r\) and \(\theta\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 The complex number \(u\) is given by \(u = - 1 - \mathrm { i } \sqrt { 3 }\).

1. Express \(u\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Give the exact values of \(r\) and \(\theta\).
   The complex number \(v\) is given by \(v = 5 \left( \cos \frac { 1 } { 6 } \pi + \mathrm { i } \sin \frac { 1 } { 6 } \pi \right)\).
2. Express the complex number \(\frac { \mathrm { v } } { \mathrm { u } }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).

1 Find the roots of the equation \(z ^ { 3 } = - 108 \sqrt { 3 } + 108\) i, giving your answers in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(0 < \theta < 2 \pi\).

7. (i) Given that $$\frac { 2 w - 3 } { 10 } = \frac { 4 + 7 i } { 4 - 3 i }$$ find \(w\), giving your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants. You must show your working.
(ii) Given that $$z = ( 2 + \lambda i ) ( 5 + i )$$ where \(\lambda\) is a real constant, and that $$\arg z = \frac { \pi } { 4 }$$ find the value of \(\lambda\).
□

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.

6. The complex number \(z\) is defined by $$z = - \lambda + 3 i \quad \text { where } \lambda \text { is a positive real constant }$$ Given that the modulus of \(z\) is 5

1. write down the value of \(\lambda\)
2. determine the argument of \(z\), giving your answer in radians to one decimal place. In part (c) you must show detailed reasoning.
   Solutions relying on calculator technology are not acceptable.
3. Express in the form \(a + \mathrm { i } b\) where \(a\) and \(b\) are real,
   1. \(\frac { z + 3 i } { 2 - 4 i }\)
   2. \(\mathrm { Z } ^ { 2 }\)
4. Show on a single Argand diagram the points \(A\), \(B\), \(C\) and \(D\) that represent the complex numbers $$z , z ^ { * } , \frac { z + 3 i } { 2 - 4 i } \text { and } z ^ { 2 }$$

1. Given that

$$\frac { 3 z - 1 } { 2 } = \frac { \lambda + 5 i } { \lambda - 4 i }$$ where \(\lambda\) is a real constant,

1. determine \(z\), giving your answer in the form \(x + y i\), where \(x\) and \(y\) are real and in terms of \(\lambda\). Given also that \(\arg z = \frac { \pi } { 4 }\)
2. find the possible values of \(\lambda\).