4.02f Convert between forms: cartesian and modulus-argument

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 \(| z - ( a + b \mathrm { i } ) | = k\).

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.

9

1. It is given that \(( 1 + 3 \mathrm { i } ) w = 2 + 4 \mathrm { i }\). Showing all necessary working, prove that the exact value of \(\left| w ^ { 2 } \right|\) is 2 and find \(\arg \left( w ^ { 2 } \right)\) correct to 3 significant figures.
2. On a single Argand diagram sketch the loci \(| z | = 5\) and \(| z - 5 | = | z |\). Hence determine the complex numbers represented by points common to both loci, giving each answer in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\).

8

1. Showing all necessary working, express the complex number \(\frac { 2 + 3 \mathrm { i } } { 1 - 2 \mathrm { i } }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Give the values of \(r\) and \(\theta\) correct to 3 significant figures.
2. On an Argand diagram sketch the locus of points representing complex numbers \(z\) satisfying the equation \(| z - 3 + 2 i | = 1\). Find the least value of \(| z |\) for points on this locus, giving your answer in an exact form.

10

1. The complex number \(u\) is defined by \(u = \frac { 3 \mathrm { i } } { a + 2 \mathrm { i } }\), where \(a\) is real.
   1. Express \(u\) in the Cartesian form \(x + \mathrm { i } y\), where \(x\) and \(y\) are in terms of \(a\).
   2. Find the exact value of \(a\) for which \(\arg u ^ { * } = \frac { 1 } { 3 } \pi\).
2. 1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 2 \mathbf { i } | \leqslant | z - 1 - \mathbf { i } |\) and \(| z - 2 - \mathbf { i } | \leqslant 2\).
   2. Calculate the least value of \(\arg z\) for points in this region.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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 \(\theta\).

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.

9 The complex numbers \(z\) and \(\omega\) are defined by \(z = 1 - i\) and \(\omega = - 3 + 3 \sqrt { 3 } i\).

1. Express \(z \omega\) in the form \(\mathrm { a } + \mathrm { bi }\), where \(a\) and \(b\) are real and in exact surd form.
2. Express \(z\) and \(\omega\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Give the exact values of \(r\) and \(\theta\) in each case.
3. On an Argand diagram, the points representing \(\omega\) and \(z \omega\) are \(A\) and \(B\) respectively. Prove that \(O A B\) is an isosceles right-angled triangle, where \(O\) is the origin.
4. Using your answers to part (b), prove that \(\tan \frac { 5 } { 12 } \pi = \frac { \sqrt { 3 } + 1 } { \sqrt { 3 } - 1 }\).

10 The complex number \(1 + 2 \mathrm { i }\) is denoted by \(u\). The polynomial \(2 x ^ { 3 } + a x ^ { 2 } + 4 x + b\), where \(a\) and \(b\) are real constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(u\) is a root of the equation \(\mathrm { p } ( x ) = 0\).

1. Find the values of \(a\) and \(b\).
2. State a second complex root of this equation.
3. Find the real factors of \(\mathrm { p } ( x )\).
4. 1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - u | \leqslant \sqrt { 5 }\) and \(\arg z \leqslant \frac { 1 } { 4 } \pi\).
   2. Find the least value of \(\operatorname { Im } z\) for points in the shaded region. Give your answer in an exact form.
      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 defined by \(u = \frac { 3 + 2 \mathrm { i } } { a - 5 \mathrm { i } }\), where \(a\) is real.

1. Express \(u\) in the Cartesian form \(x + \mathrm { i } y\), where \(x\) and \(y\) are in terms of \(a\).
2. Given that \(\arg u = \frac { 1 } { 4 } \pi\), find the value of \(a\).

8 It is given that \(\frac { 2 + 3 a \mathrm { i } } { a + 2 \mathrm { i } } = \lambda ( 2 - \mathrm { i } )\), where \(a\) and \(\lambda\) are real constants.

1. Show that \(3 a ^ { 2 } + 4 a - 4 = 0\).
2. Hence find the possible values of \(a\) and the corresponding values of \(\lambda\).

5. (i) Given that $$\frac { 2 z + 3 } { z + 5 - 2 i } = 1 + i$$ find \(z\), giving your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants.
(ii) Given that $$w = ( 3 + \lambda \mathrm { i } ) ( 2 + \mathrm { i } )$$ where \(\lambda\) is a real constant, and that $$| w | = 15$$ find the possible values of \(\lambda\).

4. $$z _ { 1 } = 3 \mathrm { i } \text { and } z _ { 2 } = \frac { 6 } { 1 + \mathrm { i } \sqrt { 3 } }$$

1. Express \(z _ { 2 }\) in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real numbers.
2. Find the modulus and the argument of \(z _ { 2 }\), giving the argument in radians in terms of \(\pi\).
3. Show the three points representing \(z _ { 1 } , z _ { 2 }\) and \(\left( z _ { 1 } + z _ { 2 } \right)\) respectively, on a single Argand diagram.

4. $$z = \frac { 4 } { 1 + \mathrm { i } }$$ Find, in the form \(a + \mathrm { i } b\) where \(a , b \in \mathbb { R }\)

1. \(Z\)
2. \(z ^ { 2 }\) Given that \(z\) is a complex root of the quadratic equation \(x ^ { 2 } + p x + q = 0\), where \(p\) and \(q\) are real integers,
3. find the value of \(p\) and the value of \(q\).

4. (i) The complex number \(w\) is given by $$w = \frac { p - 4 \mathrm { i } } { 2 - 3 \mathrm { i } }$$ where \(p\) is a real constant.

1. Express \(w\) in the form \(a + b i\), where \(a\) and \(b\) are real constants. Give your answer in its simplest form in terms of \(p\). Given that \(\arg w = \frac { \pi } { 4 }\)
2. find the value of \(p\).
   (ii) The complex number \(z\) is given by $$z = ( 1 - \lambda i ) ( 4 + 3 i )$$ where \(\lambda\) is a real constant. Given that $$| z | = 45$$ find the possible values of \(\lambda\).
   Give your answers as exact values in their simplest form.
   II

8. The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { z + 3 \mathrm { i } } { 1 + \mathrm { i } z } , \quad z \neq \mathrm { i }$$ The transformation \(T\) maps the circle \(| z | = 1\) in the \(z\)-plane onto the line \(l\) in the \(w\)-plane.

1. Find a cartesian equation of the line \(l\). The circle \(| z - a - b \mathrm { i } | = c\) in the \(z\)-plane is mapped by \(T\) onto the circle \(| w | = 5\) in the \(w\)-plane.
2. Find the exact values of the real constants \(a\), \(b\) and \(c\).
   

   END

2. A transformation from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { 1 - \mathrm { i } z } { z } , \quad z \neq 0$$ The transformation maps points on the real axis in the \(z\)-plane onto the line \(l\) in the \(w\)-plane.
Find an equation of the line \(l\).

4 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. \(2 z + 5 z ^ { * }\),
2. \(( z - \mathrm { i } ) ^ { 2 }\),
3. \(\frac { 3 } { z }\).

1 The complex number \(a + \mathrm { i } b\) is denoted by \(z\). Given that \(| z | = 4\) and \(\arg z = \frac { 1 } { 3 } \pi\), find \(a\) and \(b\).

2 It is given that the set of complex numbers of the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) for \(- \pi < \theta \leqslant \pi\) and \(r > 0\), under multiplication, forms a group.

1. Write down the inverse of \(5 \mathrm { e } ^ { \frac { 1 } { 3 } \pi \mathrm { i } }\).
2. Prove the closure property for the group.
3. \(Z\) denotes the element \(\mathrm { e } ^ { \mathrm { i } \gamma }\), where \(\frac { 1 } { 2 } \pi < \gamma < \pi\). Express \(Z ^ { 2 }\) in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\), where \(- \pi < \theta < 0\).

8 Two complex numbers, \(\alpha\) and \(\beta\), are given by \(\alpha = \sqrt { 3 } + \mathrm { j }\) and \(\beta = 3 \mathrm { j }\).

1. Find the modulus and argument of \(\alpha\) and \(\beta\).
2. Find \(\alpha \beta\) and \(\frac { \beta } { \alpha }\), giving your answers in the form \(a + b \mathrm { j }\), showing your working.
3. Plot \(\alpha , \beta , \alpha \beta\) and \(\frac { \beta } { \alpha }\) on a single Argand diagram.

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

1. Express \(\frac { z + w } { w }\) in the form \(a + b \mathrm { j }\).
2. Express \(w\) in modulus-argument form.
3. Show \(w\) on an Argand diagram, indicating its modulus and argument.

2 You are given that \(z _ { 1 }\) and \(z _ { 2 }\) are complex numbers. \(z _ { 1 } = 3 + 3 \sqrt { 3 } \mathrm { j }\), and \(z _ { 2 }\) has modulus 5 and argument \(\frac { \pi } { 3 }\).

1. Find the modulus and argument of \(z _ { 1 }\), giving your answers exactly.
2. Express \(z _ { 2 }\) in the form \(a + b \mathrm { j }\), where \(a\) and \(b\) are to be given exactly.
3. Explain why, when plotted on an Argand diagram, \(z _ { 1 } , z _ { 2 }\) and the origin lie on a straight line.

8 Abdoallah wants to write the complex number \(- 1 + \mathrm { i } \sqrt { 3 }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\) where \(r \geq 0\) and \(- \pi < \theta \leq \pi\) Here is his method: $$\begin{array} { r l r l } r & = \sqrt { ( - 1 ) ^ { 2 } + ( \sqrt { 3 } ) ^ { 2 } } & & \tan \theta = \frac { \sqrt { 3 } } { - 1 } \\ & = \sqrt { 1 + 3 } & & \Rightarrow \\ & = \sqrt { 4 } & & \tan \theta = - \sqrt { 3 } \\ & = 2 & & \theta = \tan ^ { - 1 } ( - \sqrt { 3 } ) \\ & & \theta = - \frac { \pi } { 3 } \\ & - 1 + i \sqrt { 3 } = 2 \left( \cos \left( - \frac { \pi } { 3 } \right) + i \sin \left( - \frac { \pi } { 3 } \right) \right) \end{array}$$ There is an error in Abdoallah's method. 8

1. Show that Abdoallah's answer is wrong by writing $$2 \left( \cos \left( - \frac { \pi } { 3 } \right) + i \sin \left( - \frac { \pi } { 3 } \right) \right)$$ in the form \(x + \mathrm { i } y\) Simplify your answer.
   8
2. Explain the error in Abdoallah's method.
   8
3. Express \(- 1 + \mathrm { i } \sqrt { 3 }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\) 8
4. Write down the complex conjugate of \(- 1 + i \sqrt { 3 }\)

3 The complex number \(z\) satisfies the equation \(5 ( z - \mathrm { i } ) = ( - 1 + 2 \mathrm { i } ) z ^ { * }\).
Determine \(z\), giving your answer in the form \(\mathrm { a } + \mathrm { bi }\), where \(a\) and \(b\) are real.