4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta)

5

1. Prove by induction that, if \(n\) is a positive integer, $$( \cos \theta + \mathrm { i } \sin \theta ) ^ { n } = \cos n \theta + \mathrm { i } \sin n \theta$$
2. Find the value of \(\left( \cos \frac { \pi } { 6 } + \mathrm { i } \sin \frac { \pi } { 6 } \right) ^ { 6 }\).
3. Show that $$( \cos \theta + \mathrm { i } \sin \theta ) ( 1 + \cos \theta - \mathrm { i } \sin \theta ) = 1 + \cos \theta + \mathrm { i } \sin \theta$$
4. Hence show that $$\left( 1 + \cos \frac { \pi } { 6 } + i \sin \frac { \pi } { 6 } \right) ^ { 6 } + \left( 1 + \cos \frac { \pi } { 6 } - i \sin \frac { \pi } { 6 } \right) ^ { 6 } = 0$$

6

1. Find the three roots of \(z ^ { 3 } = 1\), giving the non-real roots in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\), where \(- \pi < \theta \leqslant \pi\).
2. Given that \(\omega\) is one of the non-real roots of \(z ^ { 3 } = 1\), show that $$1 + \omega + \omega ^ { 2 } = 0$$
3. By using the result in part (b), or otherwise, show that:
   1. \(\frac { \omega } { \omega + 1 } = - \frac { 1 } { \omega }\);
   2. \(\frac { \omega ^ { 2 } } { \omega ^ { 2 } + 1 } = - \omega\);
   3. \(\left( \frac { \omega } { \omega + 1 } \right) ^ { k } + \left( \frac { \omega ^ { 2 } } { \omega ^ { 2 } + 1 } \right) ^ { k } = ( - 1 ) ^ { k } 2 \cos \frac { 2 } { 3 } k \pi\), where \(k\) is an integer.

5

1. Prove by induction that, if \(n\) is a positive integer, $$( \cos \theta + \mathrm { i } \sin \theta ) ^ { n } = \cos n \theta + \mathrm { i } \sin n \theta$$
2. Hence, given that $$z = \cos \theta + \mathrm { i } \sin \theta$$ show that $$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta$$
3. Given further that \(z + \frac { 1 } { z } = \sqrt { 2 }\), find the value of $$z ^ { 10 } + \frac { 1 } { z ^ { 10 } }$$

8 The complex number \(\omega\) is given by \(\omega = \cos \frac { 2 \pi } { 5 } + \mathrm { i } \sin \frac { 2 \pi } { 5 }\).

1. 1. Verify that \(\omega\) is a root of the equation \(z ^ { 5 } = 1\).
   2. Write down the three other non-real roots of \(z ^ { 5 } = 1\), in terms of \(\omega\).
2. 1. Show that \(1 + \omega + \omega ^ { 2 } + \omega ^ { 3 } + \omega ^ { 4 } = 0\).
   2. Hence show that \(\left( \omega + \frac { 1 } { \omega } \right) ^ { 2 } + \left( \omega + \frac { 1 } { \omega } \right) - 1 = 0\).
3. Hence show that \(\cos \frac { 2 \pi } { 5 } = \frac { \sqrt { 5 } - 1 } { 4 }\).

6 It is given that \(z = \mathrm { e } ^ { \mathrm { i } \theta }\).

1. 1. Show that $$z + \frac { 1 } { z } = 2 \cos \theta$$
   2. Find a similar expression for $$z ^ { 2 } + \frac { 1 } { z ^ { 2 } }$$ (2 marks)
   3. Hence show that $$z ^ { 2 } - z + 2 - \frac { 1 } { z } + \frac { 1 } { z ^ { 2 } } = 4 \cos ^ { 2 } \theta - 2 \cos \theta$$ (3 marks)
2. Hence solve the quartic equation $$z ^ { 4 } - z ^ { 3 } + 2 z ^ { 2 } - z + 1 = 0$$ giving the roots in the form \(a + \mathrm { i } b\).

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

1. Prove that \(w ^ { n }\) is also a root of the equation \(z ^ { 7 } = 1\) for any integer \(n\). 6
2. Prove that \(1 + w + w ^ { 2 } + w ^ { 3 } + w ^ { 4 } + w ^ { 5 } + w ^ { 6 } = 0\) 6
3. 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
4. Prove that $$\cos \frac { 2 \pi } { 7 } + \cos \frac { 4 \pi } { 7 } + \cos \frac { 6 \pi } { 7 } = - \frac { 1 } { 2 }$$

6 In this question you must show detailed reasoning.
You are given the complex number \(\omega = \cos \frac { 2 } { 5 } \pi + i \sin \frac { 2 } { 5 } \pi\) and the equation \(z ^ { 5 } = 1\).

1. Show that \(\omega\) is a root of the equation.
2. Write down the other four roots of the equation.
3. Show that \(\omega + \omega ^ { 2 } + \omega ^ { 3 } + \omega ^ { 4 } = - 1\).
4. Hence show that \(\left( \omega + \frac { 1 } { \omega } \right) ^ { 2 } + \left( \omega + \frac { 1 } { \omega } \right) - 1 = 0\).
5. Hence determine the value of \(\cos \frac { 2 } { 5 } \pi\) in the form \(a + b \sqrt { c }\) where \(a , b\) and \(c\) are rational numbers to be found. Total Marks for Question Set 4: 38

2 In this question you must show detailed reasoning.
Solve the equation \(2 \cosh ^ { 2 } x + 5 \sinh x - 5 = 0\) giving each answer in the form \(\ln ( p + q \sqrt { r } )\) where \(p\) and \(q\) are rational numbers, and \(r\) is an integer, whose values are to be determined. You are given that the matrix \(\mathbf { A } = \left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & \frac { 2 a - a ^ { 2 } } { 3 } & 0 \\ 0 & 0 & 1 \end{array} \right)\), where \(a\) is a positive constant, represents the transformation R which is a reflection in 3-D.

1. State the plane of reflection of \(R\).
2. Determine the value of \(a\).
3. With reference to R explain why \(\mathbf { A } ^ { 2 } = \mathbf { I }\), the \(3 \times 3\) identity matrix.
   1. By using Euler's formula show that \(\cosh ( \mathrm { iz } ) = \cos z\).
   2. Hence, find, in logarithmic form, a root of the equation \(\cos z = 2\). [You may assume that \(\cos z = 2\) has complex roots.] A swing door is a door to a room which is closed when in equilibrium but which can be pushed open from either side and which can swing both ways, into or out of the room, and through the equilibrium position. The door is sprung so that when displaced from the equilibrium position it will swing back towards it. The extent to which the door is open at any time, \(t\) seconds, is measured by the angle at the hinge, \(\theta\), which the plane of the door makes with the plane of the equilibrium position. See the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{20816f61-154d-4491-9d2d-4c62687bf81e-03_317_954_497_255} In an initial model of the motion of a certain swing door it is suggested that \(\theta\) satisfies the following differential equation. $$4 \frac { \mathrm {~d} ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + 25 \theta = 0$$
   3. 1. Write down the general solution to (\textit{).
      2. With reference to the behaviour of your solution in part (a)(i) explain briefly why the model using (}) is unlikely to be realistic. In an improved model of the motion of the door an extra term is introduced to the differential equation so that it becomes $$4 \frac { \mathrm {~d} ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + \lambda \frac { \mathrm { d } \theta } { \mathrm {~d} t } + 25 \theta = 0$$ where \(\lambda\) is a positive constant.
   4. In the case where \(\lambda = 16\) the door is held open at an angle of 0.9 radians and then released from rest at time \(t = 0\).
      1. Find, in a real form, the general solution of ( \(\dagger\) ).
      2. Find the particular solution of ( \(\dagger\) ).
      3. With reference to the behaviour of your solution found in part (b)(ii) explain briefly how the extra term in ( \(\dagger\) ) improves the model.
      4. Find the value of \(\lambda\) for which the door is critically damped.

4 In this question you must show detailed reasoning.

1. By writing \(\sin \theta\) in terms of \(\mathrm { e } ^ { \mathrm { i } \theta }\) and \(\mathrm { e } ^ { - \mathrm { i } \theta }\) show that $$\sin ^ { 6 } \theta = \frac { 1 } { 32 } ( 10 - 15 \cos 2 \theta + 6 \cos 4 \theta - \cos 6 \theta ) .$$
2. Hence show that \(\sin \frac { 1 } { 8 } \pi = \frac { 1 } { 2 } \sqrt [ 6 ] { 20 - 14 \sqrt { 2 } }\).
   1. Use differentiation to find the first two non-zero terms of the Maclaurin expansion of \(\ln \left( \frac { 1 } { 2 } + \cos x \right)\).
   2. By considering the root of the equation \(\ln \left( \frac { 1 } { 2 } + \cos x \right) = 0\) deduce that \(\pi \approx 3 \sqrt { 3 \ln \left( \frac { 3 } { 2 } \right) }\).

1

1. 1. Given that \(\mathrm { f } ( x ) = \arctan x\), write down an expression for \(\mathrm { f } ^ { \prime } ( x )\). Assuming that \(x\) is small, use a binomial expansion to express \(\mathrm { f } ^ { \prime } ( x )\) in ascending powers of \(x\) as far as the term in \(x ^ { 4 }\).
   2. Hence express \(\arctan x\) in ascending powers of \(x\) as far as the term in \(x ^ { 5 }\).
2. Find, in exact form, the value of the following integral. $$\int _ { 0 } ^ { \frac { 3 } { 4 } } \frac { 1 } { \sqrt { 3 - 4 x ^ { 2 } } } \mathrm {~d} x$$
3. A curve has polar equation \(r = \frac { a } { \sqrt { \theta } }\) where \(a > 0\).
   1. Sketch the curve for \(\frac { \pi } { 4 } \leqslant \theta \leqslant 2 \pi\).
   2. State what happens to \(r\) as \(\theta\) tends to zero.
   3. Find the area of the region enclosed by the part of the curve sketched in part (i) and the lines \(\theta = \frac { \pi } { 4 }\) and \(\theta = 2 \pi\). Give your answer in an exact simplified form.
      1. (i) Express \(2 \sin \frac { 1 } { 2 } \theta \left( \sin \frac { 1 } { 2 } \theta - \mathrm { j } \cos \frac { 1 } { 2 } \theta \right)\) in terms of \(z\) where \(z = \cos \theta + \mathrm { j } \sin \theta\).
         (ii) The series \(C\) and \(S\) are defined as follows. $$\begin{aligned} C & = 1 - \binom { n } { 1 } \cos \theta + \binom { n } { 2 } \cos 2 \theta - \ldots + ( - 1 ) ^ { n } \binom { n } { n } \cos n \theta \\ S & = - \binom { n } { 1 } \sin \theta + \binom { n } { 2 } \sin 2 \theta - \ldots + ( - 1 ) ^ { n } \binom { n } { n } \sin n \theta \end{aligned}$$ Show that $$C + \mathrm { j } S = \left\{ - 2 \mathrm { j } \sin \frac { 1 } { 2 } \theta \left( \cos \frac { 1 } { 2 } \theta + \mathrm { j } \sin \frac { 1 } { 2 } \theta \right) \right\} ^ { n } .$$ Hence show that, for even values of \(n\), $$\frac { C } { S } = \cot \left( \frac { 1 } { 2 } n \theta \right)$$
      2. Write the complex number \(z = \sqrt { 6 } + \mathrm { j } \sqrt { 2 }\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\), expressing \(r\) and \(\theta\) as simply as possible. Hence find the cube roots of \(z\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\). Show the points representing \(z\) and its cube roots on an Argand diagram.
         1. Find the eigenvalues and eigenvectors of the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { l l } \frac { 1 } { 2 } & \frac { 1 } { 2 } \\ \frac { 2 } { 3 } & \frac { 1 } { 3 } \end{array} \right)$$ Hence express \(\mathbf { M }\) in the form \(\mathbf { P D P } ^ { - 1 }\) where \(\mathbf { D }\) is a diagonal matrix.
         2. Write down an equation for \(\mathbf { M } ^ { n }\) in terms of the matrices \(\mathbf { P }\) and \(\mathbf { D }\). Hence obtain expressions for the elements of \(\mathbf { M } ^ { n }\).
            Show that \(\mathbf { M } ^ { n }\) tends to a limit as \(n\) tends to infinity. Find that limit.
         3. Express \(\mathbf { M } ^ { - 1 }\) in terms of the matrices \(\mathbf { P }\) and \(\mathbf { D }\). Hence determine whether or not \(\left( \mathbf { M } ^ { - 1 } \right) ^ { n }\) tends to a limit as \(n\) tends to infinity. Section B (18 marks)
            1. Given that \(y = \cosh x\), use the definition of \(\cosh x\) in terms of exponential functions to prove that $$x = \pm \ln \left( y + \sqrt { y ^ { 2 } - 1 } \right) .$$
            2. Solve the equation $$\cosh x + \cosh 2 x = 5$$ giving the roots in an exact logarithmic form.
            3. Sketch the curve with equation \(y = \cosh x + \cosh 2 x\). Show on your sketch the line \(y = 5\). Find the area of the finite region bounded by the curve and the line \(y = 5\). Give your answer in an exact form that does not involve hyperbolic functions. \section*{END OF QUESTION PAPER}

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

1. 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 $$\cos \frac { 2 } { 5 } \pi \quad \text { and } \quad \cos \frac { 4 } { 5 } \pi$$

5 Let \(z = \cos \theta + \mathrm { i } \sin \theta\).

1. Prove the result \(z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta\).
2. Use this result to express \(\sin ^ { 5 } \theta\) in the form \(A \sin 5 \theta + B \sin 3 \theta + C \sin \theta\), for constants \(A , B\) and \(C\) to be determined.

4 Let \(z = \cos \theta + \mathrm { i } \sin \theta\).

1. Use de Moivre's theorem to prove that \(z ^ { n } + z ^ { - n } = 2 \cos n \theta\).
2. Deduce the identity \(\cos ^ { 5 } \theta = \frac { 1 } { 16 } ( \cos 5 \theta + 5 \cos 3 \theta + 10 \cos \theta )\).

11 The complex number \(z\) is defined as \(z = \cos \theta + \mathrm { i } \sin \theta\).

1. Show that \(z ^ { n } + z ^ { - n } = 2 \cos n \theta\).
2. By expanding \(\left( z + z ^ { - 1 } \right) ^ { 5 }\), show that \(16 \cos ^ { 5 } \theta = \cos 5 \theta + 5 \cos 3 \theta + 10 \cos \theta\).
3. Hence find \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { 5 } \theta \mathrm {~d} \theta\).
4. Sketch the graphs of \(\mathrm { f } ( \theta ) = \sin ^ { 5 } \theta\) and \(\mathrm { f } ( \theta ) = \cos ^ { 5 } \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\), and hence give the value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { 5 } \theta \mathrm {~d} \theta$$

Given that \(z = e^{i\theta}\) and \(n\) is a positive integer, show that $$z^n + \frac{1}{z^n} = 2 \cos n\theta \quad \text{and} \quad z^n - \frac{1}{z^n} = 2i \sin n\theta.$$ [2] Hence express \(\sin^6 \theta\) in the form $$p \cos 6\theta + q \cos 4\theta + r \cos 2\theta + s,$$ where the constants \(p\), \(q\), \(r\), \(s\) are to be determined. [4] Hence find the mean value of \(\sin^6 \theta\) with respect to \(\theta\) over the interval \(0 \leq \theta \leq \frac{1}{4}\pi\). [5]

Write down an expression in terms of \(z\) and \(N\) for the sum of the series $$\sum_{n=1}^N 2^{-n}z^n.$$ [2] Use de Moivre's theorem to deduce that $$\sum_{n=1}^{10} 2^{-n}\sin\left(\frac{1}{10}n\pi\right) = \frac{1025\sin\left(\frac{1}{10}\pi\right)}{2560 - 2048\cos\left(\frac{1}{10}\pi\right)}.$$ [6]

1. State the sum of the series \(1 + z + z^2 + \ldots + z^{n-1}\), for \(z \neq 1\). [1]
2. By letting \(z = \cos\theta + i\sin\theta\), where \(\cos\theta \neq 1\), show that $$1 + \cos\theta + \cos 2\theta + \ldots + \cos(n-1)\theta = \frac{1}{2}\left(1 - \cos n\theta + \frac{\sin n\theta \sin\theta}{1 - \cos\theta}\right).$$ [7]

\includegraphics{figure_8} The diagram shows the curve with equation \(y = \cos x\) for \(0 \leq x \leq 1\), together with a set of \(n\) rectangles of width \(\frac{1}{n}\).

1. By considering the sum of the areas of these rectangles, show that $$\int_0^1 \cos x\,dx < \frac{1}{2n}\left(1 - \cos 1 + \frac{\sin 1\sin\frac{1}{n}}{1 - \cos\frac{1}{n}}\right).$$ [4]
2. Use a similar method to find, in terms of \(n\), a lower bound for \(\int_0^1 \cos x\,dx\). [3]

The complex number \(z = e^{i\theta}\), where \(\theta\) is real.

1. Use de Moivre's theorem to show that $$z^n + \frac{1}{z^n} = 2\cos n\theta$$ where \(n\) is a positive integer. [2]
2. Show that $$\cos^n \theta = \frac{1}{16}(\cos 5\theta + 5\cos 3\theta + 10\cos \theta)$$ [5]
3. Hence find all the solutions of $$\cos 5\theta + 5\cos 3\theta + 12\cos \theta = 0$$ in the interval \(0 \leq \theta < 2\pi\). [4]

Prove that \(\sinh(i\pi - \theta) = \sinh \theta\). [4]

The integrals \(C\) and \(S\) are defined by $$C = \int_0^{2\pi} e^{3x} \cos 3x \, dx \quad \text{and} \quad S = \int_0^{2\pi} e^{3x} \sin 3x \, dx.$$ By considering \(C + iS\) as a single integral, show that $$C = \frac{1}{13}(2 + 3e^\pi),$$ and obtain a similar expression for \(S\). [8] (You may assume that the standard result for \(\int e^{kx} dx\) remains true when \(k\) is a complex constant, so that \(\int e^{(a+ib)x} dx = \frac{1}{a+ib} e^{(a+ib)x}\).)