Direct nth roots of complex numbers

8

1. Express in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\) :
   1. \(\quad 4 ( 1 + i \sqrt { 3 } )\);
   2. \(4 ( 1 - i \sqrt { 3 } )\).
2. The complex number \(z\) satisfies the equation $$\left( z ^ { 3 } - 4 \right) ^ { 2 } = - 48$$ Show that \(z ^ { 3 } = 4 \pm 4 \sqrt { 3 } \mathrm { i }\).
3. 1. Solve the equation $$\left( z ^ { 3 } - 4 \right) ^ { 2 } = - 48$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
   2. Illustrate the roots on an Argand diagram.
4. 1. Explain why the sum of the roots of the equation $$\left( z ^ { 3 } - 4 \right) ^ { 2 } = - 48$$ is zero.
   2. Deduce that \(\cos \frac { \pi } { 9 } + \cos \frac { 3 \pi } { 9 } + \cos \frac { 5 \pi } { 9 } + \cos \frac { 7 \pi } { 9 } = \frac { 1 } { 2 }\).

8

1. Express \(- 4 + 4 \sqrt { 3 } \mathrm { i }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
2. 1. Solve the equation \(z ^ { 3 } = - 4 + 4 \sqrt { 3 } \mathrm { i }\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
   2. The roots of the equation \(z ^ { 3 } = - 4 + 4 \sqrt { 3 } \mathrm { i }\) are represented by the points \(P , Q\) and \(R\) on an Argand diagram. Find the area of the triangle \(P Q R\), giving your answer in the form \(k \sqrt { 3 }\), where \(k\) is an integer.
3. By considering the roots of the equation \(z ^ { 3 } = - 4 + 4 \sqrt { 3 } \mathrm { i }\), show that $$\cos \frac { 2 \pi } { 9 } + \cos \frac { 4 \pi } { 9 } + \cos \frac { 8 \pi } { 9 } = 0$$

1

1. Express - 9 i in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
   [0pt] [2 marks]
2. Solve the equation \(z ^ { 4 } + 9 \mathrm { i } = 0\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
   [0pt] [5 marks]

5

1. Find the modulus of the complex number \(- 4 \sqrt { 3 } + 4 \mathrm { i }\), giving your answer as an integer.
2. The locus of points, \(L\), satisfies the equation \(| z + 4 \sqrt { 3 } - 4 \mathrm { i } | = 4\).
   1. Sketch the locus \(L\) on the Argand diagram below.
   2. The complex number \(w\) lies on \(L\) so that \(- \pi < \arg w \leqslant \pi\). Find the least possible value of arg \(w\), giving your answer in terms of \(\pi\).
3. Solve the equation \(z ^ { 3 } = - 4 \sqrt { 3 } + 4 \mathrm { i }\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
   [0pt] [5 marks]

11 An Argand diagram with the point A representing a complex number \(z _ { 1 }\) is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{b57a2590-84e8-4998-9633-902db861f23a-4_716_778_932_239} The complex numbers \(z _ { 2 }\) and \(z _ { 3 }\) are \(z _ { 1 } \mathrm { e } ^ { \frac { 2 } { 3 } \mathrm { i } \pi }\) and \(z _ { 1 } \mathrm { e } ^ { \frac { 4 } { 3 } \mathrm { i } \pi }\) respectively.

1. 1. On the copy of the Argand diagram in the Printed Answer Booklet, mark the points B and C representing the complex numbers \(z _ { 2 }\) and \(z _ { 3 }\).
   2. Show that \(z _ { 1 } + z _ { 2 } + z _ { 3 } = 0\).
2. Given now that \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) are roots of the equation \(z ^ { 3 } = 8 \mathrm { i }\), find these three roots, giving your answers in the form \(\mathrm { a } + \mathrm { ib }\), where \(a\) and \(b\) are real and exact.

5

1. In this question you must show detailed reasoning.
   Determine the sixth roots of - 64 , expressed in \(r \mathrm { e } ^ { \mathrm { i } \theta }\) form.
2. Represent the roots on an Argand diagram.

10

1. Show on an Argand diagram the points representing the three cube roots of unity.
2. 1. Find the exact roots of the equation \(z ^ { 3 } - 1 = \sqrt { 3 } \mathrm { i }\), expressing them in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta < \pi\).
   2. The points representing the cube roots of unity form a triangle \(\Delta _ { 1 }\). The points representing the roots of the equation \(z ^ { 3 } - 1 = \sqrt { 3 } \mathrm { i }\) form a triangle \(\Delta _ { 2 }\). State a sequence of two transformations that maps \(\Delta _ { 1 }\) onto \(\Delta _ { 2 }\).
   3. The three roots in part (b)(i) are \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\). By simplifying \(z _ { 1 } + z _ { 2 } + z _ { 3 }\), verify that the sum of these roots is zero.
   4. Hence show that \(\sin 20 ^ { \circ } + \sin 140 ^ { \circ } = \sin 100 ^ { \circ }\).

7. Find the cube roots of the complex number \(z = 11 - 2 \mathrm { i }\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and correct to three decimal places.

1. (a) Express the three cube roots of \(5 + 10 \mathrm { i }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(0 \leqslant \theta < 2 \pi\).
   …………………………………………………………………………………………………………………………………………………..
   (b) The three cube roots of \(5 + 10 \mathrm { i }\) are plotted in an Argand diagram. The points are joined by straight lines to form a triangle. Find the area of this triangle, giving your answer correct to two significant figures.
   
2. The function \(f\) is defined by \(f ( x ) = \cosh \left( \frac { x } { 2 } \right)\).
   (a) State the Maclaurin series expansion for \(\cosh \left( \frac { x } { 2 } \right)\) up to and including the term in \(x ^ { 4 }\).
   

Another function \(g\) is defined by \(g ( x ) = x ^ { 2 } - 2\). The diagram below shows parts of the graphs of \(y = f ( x )\) and \(y = g ( x )\).
\includegraphics[max width=\textwidth, alt={}, center]{7316672a-ae33-4f5b-9c59-51ef43af8ff1-04_894_940_1471_552}
(b) The two graphs intersect at the point \(A\), as shown in the diagram. Use your answer from part (a) to find an approximation for the \(x\)-coordinate of \(A\), giving your answer correct to two decimal places.
(c) Using your answer to part (b), find an approximation for the area of the shaded region enclosed by the two graphs, the \(x\)-axis and the \(y\)-axis.
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8 In this question you must show detailed reasoning. The complex number \(- 4 + i \sqrt { 48 }\) is denoted by \(z\).

1. 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.
2. Write down the angles that any lines of symmetry of triangle \(A B C\) make with the positive real axis, justifying your answer.

1

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

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:

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

9

1. Solve the equation \(z ^ { 3 } = \sqrt { 2 } - \sqrt { 6 } \mathrm { i }\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(r > 0\) and \(0 \leq \theta < 2 \pi\) 9
2. The transformation represented by the matrix \(\mathbf { M } = \left[ \begin{array} { l l } 5 & 1 \\ 1 & 3 \end{array} \right]\) acts on the points on an Argand Diagram which represent the roots of the equation in part (a). Find the exact area of the shape formed by joining the transformed points.