(a) Use de Moivre's theorem to show that
\(\sec 5 \theta=\frac{\sec ^{5} \theta}{5 \sec ^{4} \theta-20 \sec ^{2} \theta+16} .\)
(b) Hence, obtain the roots of the equation
\(\sqrt{3} x^{5}-10 x^{4}+40 x^{2}-32=0\)
in the form \(\sec (q \pi)\), where \(q\) is rational.

Find the roots of the equation \(z^{3}=27-27 \mathrm{i}\), giving your answers in the form \(r \mathrm{e}^{\mathrm{i} \theta}\), where \(r>0\) and \(-\pi \leqslant \theta<\pi\).

(a) Show that \(\sum_{r=1}^{n} z^{4 r}=\frac{z^{4 n+2}-z^{2}}{z^{2}-z^{-2}}\), for \(z^{2} \neq z^{-2}\).
(b) By letting \(z=\cos \theta+\mathrm{i} \sin \theta\), show that, if \(\sin 2 \theta \neq 0\),
\(\sum_{r=1}^{n} \sin (4 r \theta)=\frac{\cos 2 \theta-\cos (4 n+2) \theta}{2 \sin 2 \theta} .\)

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

By considering the binomial expansions of \(\left(z+\frac{1}{z}\right)^{4}\) and \(\left(z-\frac{1}{z}\right)^{4}\), where \(z=\cos \theta+\mathrm{i} \sin \theta\), use de Moivre's theorem to show that
\(\cot ^{4} \theta=\frac{\cos 4 \theta+a \cos 2 \theta+b}{\cos 4 \theta-a \cos 2 \theta+b}\)
where \(a\) and \(b\) are integers to be determined.

(a) By considering the binomial expansion of \(\left(z+z^{-1}\right)^{4}\), where \(z=\cos \theta+\mathrm{i} \sin \theta\), use de Moivre's theorem to show that \(\cos ^{4} \theta=\frac{1}{8}(\cos 4 \theta+4 \cos 2 \theta+3)\).
(b) Use the substitution \(x=\sin \theta\) to find the exact value of \(\int_{0}^{\frac{1}{2}}\left(1-x^{2}\right)^{\frac{3}{2}} \mathrm{~d} x\).

(a) Use de Moivre's theorem to show that
\(\cot 6 \theta=\frac{\cot ^{6} \theta-15 \cot ^{4} \theta+15 \cot ^{2} \theta-1}{6 \cot ^{5} \theta-20 \cot ^{3} \theta+6 \cot \theta} .\)
(b) Hence obtain the roots of the equation
\(x^{6}-6 x^{5}-15 x^{4}+20 x^{3}+15 x^{2}-6 x-1=0\)
in the form \(\cot (q \pi)\), where \(q\) is a rational number.

(a) By considering the binomial expansion of \(\left(z+\frac{1}{z}\right)^{7}\), where \(z=\cos \theta+\mathrm{i} \sin \theta\), use de Moivre's theorem to show that
\(\cos ^{7} \theta=a \cos 7 \theta+b \cos 5 \theta+c \cos 3 \theta+d \cos \theta\)
where \(a, b, c\) and \(d\) are constants to be determined.

Let \(I_{n}=\int_{0}^{\frac{1}{4} \pi} \cos ^{n} \theta \mathrm{~d} \theta\).
(b) Show that
\(n I_{n}=2^{-\frac{1}{2} n}+(n-1) I_{n-2}\)
(c) Using the results given in parts (a) and (b), find the exact value of \(I_{9}\).

Find the roots of the equation \(z^{3}=7 \sqrt{3}-7 \mathrm{i}\), giving your answers in the form \(r \mathrm{e}^{\mathrm{i} \theta}\), where \(r\gt 0\) and \(-\pi \leqslant \theta\lt \pi\).

(a) Write down the fourth roots of unity.

(b) Use de Moivre's theorem to show that
\(\cos 4 \theta=8 \cos ^{4} \theta-8 \cos ^{2} \theta+1\)
(c) Hence obtain the real roots of the equation
\(16\left(8 x^{4}-8 x^{2}+1\right)^{4}-9=0\)
in the form \(\cos (q \pi)\), where \(q\) is a rational number.

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

Let \(w=z_{1}^{3 k}+z_{2}^{3 k}+z_{3}^{3 k}\), where \(k\) is a positive integer and \(z_{1}, z_{2}, z_{3}\) are the roots of \(z^{3}=-1-\mathrm{i}\).
(b) Express \(w\) in the form \(R \mathrm{e}^{\mathrm{i} \alpha}\), where \(R>0\), giving \(R\) and \(\alpha\) in terms of \(k\).

(i) By considering the binomial expansion of \(\left(z+\frac{1}{z}\right)^{6}\), where \(z=\cos \theta+\mathrm{i} \sin \theta\), express \(\cos ^{6} \theta\) in the form
\(\frac{1}{32}(p+q \cos 2 \theta+r \cos 4 \theta+s \cos 6 \theta),\)
where \(p, q, r\) and \(s\) are integers to be determined.

(ii) Hence find the exact value of
\(\int_{-\frac{1}{2} \pi}^{\frac{1}{2} \pi} \cos ^{6}\left(\frac{1}{2} x\right) \mathrm{d} x\)

(i) Find the value of \(k\) for which the set of linear equations
\(\begin{aligned} x+3 y+k z & =4 \\ 4 x-2 y-10 z & =-5 \\ x+y+2 z & =1 \end{aligned}\)
has no unique solution.

(ii) For this value of \(k\), find the set of possible solutions, giving your answer in the form
\(\left(\begin{array}{l} x \\ y \\ z \end{array}\right)=\mathbf{a}+t \mathbf{b},\)
where \(\mathbf{a}\) and \(\mathbf{b}\) are vectors and \(t\) is a scalar.

(i) Use de Moivre's theorem to prove that
\(\tan 4 \theta=\frac{4 \tan \theta-4 \tan ^{3} \theta}{1-6 \tan ^{2} \theta+\tan ^{4} \theta}\)

(ii) Hence find the solutions of the equation
\(t^{4}-4 t^{3}-6 t^{2}+4 t+1=0\)
giving your answers in the form \(\tan k \pi\), where \(k\) is a rational number.

Use de Moivre's theorem to show that
\(\tan 5 \theta=\frac{5 t-10 t^{3}+t^{5}}{1-10 t^{2}+5 t^{4}}\)
where \(t=\tan \theta\).

Deduce that the roots of the equation \(t^{4}-10 t^{2}+5=0\) are \(\pm \tan \frac{1}{5} \pi\) and \(\pm \tan \frac{2}{5} \pi\).

Hence show that \(\tan \frac{1}{5} \pi \tan \frac{2}{5} \pi=\sqrt{ } 5\).

Using de Moivre's theorem, show that
\(\tan 5 \theta=\frac{5 \tan \theta-10 \tan ^{3} \theta+\tan ^{5} \theta}{1-10 \tan ^{2} \theta+5 \tan ^{4} \theta} .\)

Hence show that the equation \(x^{2}-10 x+5=0\) has roots \(\tan ^{2}\left(\frac{1}{5} \pi\right)\) and \(\tan ^{2}\left(\frac{2}{5} \pi\right)\).

Deduce a quadratic equation, with integer coefficients, having roots \(\sec ^{2}\left(\frac{1}{5} \pi\right)\) and \(\sec ^{2}\left(\frac{2}{5} \pi\right)\).
[Question 11 is printed on the next page.]

Let \(z=\cos \theta+\mathrm{i} \sin \theta\). Use the binomial expansion of \((1+z)^{n}\), where \(n\) is a positive integer, to show that
\(\binom{n}{1} \cos \theta+\binom{n}{2} \cos 2 \theta+\ldots+\binom{n}{n} \cos n \theta=2^{n} \cos ^{n}\left(\frac{1}{2} \theta\right) \cos \left(\frac{1}{2} n \theta\right)-1 .\)

Find
\(\binom{n}{1} \sin \theta+\binom{n}{2} \sin 2 \theta+\ldots+\binom{n}{n} \sin n \theta .\)

Let \(z=\cos \theta+\mathrm{i} \sin \theta\). Show that
\(z^{n}+\frac{1}{z^{n}}=2 \cos n \theta \quad \text { and } \quad z^{n}-\frac{1}{z^{n}}=2 \mathrm{i} \sin n \theta .\)

By considering \(\left(z-\frac{1}{z}\right)^{4}\left(z+\frac{1}{z}\right)^{2}\), show that
\(\sin ^{4} \theta \cos ^{2} \theta=\frac{1}{32}(\cos 6 \theta-2 \cos 4 \theta-\cos 2 \theta+2) .\)

Hence find the exact value of \(\int_{0}^{\frac{1}{4} \pi} \sin ^{4} \theta \cos ^{2} \theta d \theta\).
[Question 11 is printed on the next page.]

Use de Moivre's theorem to show that \(\cos ^{4} \theta=\frac{1}{8}(\cos 4 \theta+4 \cos 2 \theta+3)\).

Find the corresponding expression for \(\sin ^{4} \theta\) in terms of \(\cos 4 \theta\) and \(\cos 2 \theta\).

Hence find the exact value of \(\int_{0}^{\frac{1}{8} \pi}\left(\cos ^{4} \theta+\sin ^{4} \theta\right) \mathrm{d} \theta\).

Use de Moivre's theorem to express \(\cot 7 \theta\) in terms of \(\cot \theta\).

Use the equation \(\cot 7 \theta=0\) to show that the roots of the equation
\(x^{6}-21 x^{4}+35 x^{2}-7=0\)
are \(\cot \left(\frac{1}{14} k \pi\right)\) for \(k=1,3,5,9,11,13\), and deduce that
\(\cot ^{2}\left(\frac{1}{14} \pi\right) \cot ^{2}\left(\frac{3}{14} \pi\right) \cot ^{2}\left(\frac{5}{14} \pi\right)=7\)

By considering the binomial expansion of \(\left(z-\frac{1}{z}\right)^{6}\), where \(z=\cos \theta+\mathrm{i} \sin \theta\), express \(\sin ^{6} \theta\) in the form
\(\frac{1}{32}(p+q \cos 2 \theta+r \cos 4 \theta+s \cos 6 \theta),\)
where \(p, q, r\) and \(s\) are integers to be determined.

Hence find the exact value of \(\int_{0}^{\frac{1}{4} \pi} \sin ^{6} \theta \mathrm{~d} \theta\).

Answer only one of the following two alternatives.

EITHER

The line \(l_{1}\) passes through the point \(A\) whose position vector is \(4 \mathbf{i}+7 \mathbf{j}-\mathbf{k}\) and is parallel to the vector \(3 \mathbf{i}+2 \mathbf{j}-\mathbf{k}\). The line \(l_{2}\) passes through the point \(B\) whose position vector is \(\mathbf{i}+7 \mathbf{j}+11 \mathbf{k}\) and is parallel to the vector \(\mathbf{i}-6 \mathbf{j}-2 \mathbf{k}\). The points \(P\) on \(l_{1}\) and \(Q\) on \(l_{2}\) are such that \(P Q\) is perpendicular to both \(l_{1}\) and \(l_{2}\). Find the position vectors of \(P\) and \(Q\).

Find the shortest distance between the line through \(A\) and \(B\) and the line through \(P\) and \(Q\), giving your answer correct to 3 significant figures.

OR

Show the cube roots of 1 on an Argand diagram.

Show that the two non-real cube roots can be expressed in the form \(\omega\) and \(\omega^{2}\), and find these cube roots in exact cartesian form \(x+\mathrm{i} y\).

Evaluate the determinant
\(\left|\begin{array}{ccc} 1 & 3 \omega & 2 \omega^{2} \\ 3 \omega^{2} & 2 & \omega \\ 2 \omega & \omega^{2} & 3 \end{array}\right| .\)

It is given that \(z=(4\sqrt{3})\left(\cos \frac{4}{3} \pi+i \sin \frac{4}{3} \pi\right)-4\left(\cos \frac{11}{6} \pi+i \sin \frac{11}{6} \pi\right)\). Express \(z\) in the form \(r(\cos \theta+\mathrm{i} \sin \theta)\), giving exact values for \(r\) and \(\theta\).

Hence find the cube roots of \(z\) in the form \(r(\cos \theta+\mathrm{i} \sin \theta)\).

Answer only one of the following two alternatives.
EITHER
The position vectors of the points \(A, B, C, D\) are
\(7 \mathbf{i}+4 \mathbf{j}-\mathbf{k}, \quad 3 \mathbf{i}+5 \mathbf{j}-2 \mathbf{k}, \quad 2 \mathbf{i}+6 \mathbf{j}+3 \mathbf{k}, \quad 2 \mathbf{i}+7 \mathbf{j}+\lambda \mathbf{k}\)
respectively. It is given that the shortest distance between the line \(A B\) and the line \(C D\) is 3 .
(i) Show that \(\lambda^{2}-5 \lambda+4=0\).

(ii) Find the acute angle between the planes through \(A, B, D\) corresponding to the values of \(\lambda\) satisfying the equation in part (i).

OR
The linear transformation \(\mathrm{T}: \mathbb{R}^{4} \rightarrow \mathbb{R}^{4}\) is represented by the matrix
\(\left(\begin{array}{rrrr} 1 & 2 & -1 & -1 \\ 1 & 3 & -1 & 0 \\ 1 & 0 & 3 & 1 \\ 0 & 3 & -4 & -1 \end{array}\right) .\)

The range space of T is denoted by \(V\).
(i) Determine the dimension of \(V\).

(ii) Show that the vectors \(\left(\begin{array}{l}1 \\ 1 \\ 1 \\ 0\end{array}\right),\left(\begin{array}{l}2 \\ 3 \\ 0 \\ 3\end{array}\right),\left(\begin{array}{r}-1 \\ -1 \\ 3 \\ -4\end{array}\right)\) are linearly independent.

(iii) Write down a basis of \(V\).

The set of elements of \(\mathbb{R}^{4}\) which do not belong to \(V\) is denoted by \(W\).
(iv) State, with a reason, whether \(W\) is a vector space.

(v) Show that if the vector \(\left(\begin{array}{l}x \\ y \\ z \\ t\end{array}\right)\) belongs to \(W\) then \(y-z-t \neq 0\).

Use de Moivre's theorem to express \(\cos 8 \theta\) as a polynomial in \(\cos \theta\).

Hence
(i) express \(\cos 8 \theta\) as a polynomial in \(\sin \theta\),

(ii) find the exact value of
\(4 x^{4}-8 x^{3}+5 x^{2}-x\)
where \(x=\cos ^{2}\left(\frac{1}{8} \pi\right)\).

Answer only one of the following two alternatives.
EITHER

Use de Moivre's theorem to prove that
\(\tan 3 \theta=\frac{3 \tan \theta-\tan ^{3} \theta}{1-3 \tan ^{2} \theta} .\)

State the exact values of \(\theta\), between 0 and \(\pi\), that satisfy \(\tan 3 \theta=1\).

Express each root of the equation \(t^{3}-3 t^{2}-3 t+1=0\) in the form \(\tan (k \pi)\), where \(k\) is a positive rational number.

For each of these values of \(k\), find the exact value of \(\tan (k \pi)\).

OR

The curve \(C\) has equation
\(y=\frac{x^{2}+\lambda x-6 \lambda^{2}}{x+3},\)
where \(\lambda\) is a constant such that \(\lambda \neq 1\) and \(\lambda \neq-\frac{3}{2}\).
(i) Find \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) and deduce that if \(C\) has two stationary points then \(-\frac{3}{2}\lt \lambda\lt 1\).

(ii) Find the equations of the asymptotes of \(C\).

(iii) Draw a sketch of \(C\) for the case \(0\lt \lambda\lt 1\).

(iv) Draw a sketch of \(C\) for the case \(\lambda\gt 3\).

Expand \(\left(z+\frac{1}{z}\right)^{4}\left(z-\frac{1}{z}\right)^{2}\) and, by substituting \(z=\cos \theta+\mathrm{i} \sin \theta\), find integers \(p, q, r, s\) such that
\(64 \sin ^{2} \theta \cos ^{4} \theta=p+q \cos 2 \theta+r \cos 4 \theta+s \cos 6 \theta\)

Using the substitution \(x=2 \cos \theta\), show that
\(\int_{1}^{2} x^{4} \sqrt{ }\left(4-x^{2}\right) \mathrm{d} x=\frac{4}{3} \pi+\sqrt{ } 3\)

Find the set of values of \(a\) for which the system of equations
\(\begin{aligned} x-2 y-2 z & =-7 \\ 2 x+(a-9) y-10 z & =-11 \\ 3 x-6 y+2 a z & =-29 \end{aligned}\)
has a unique solution.

Show that the system has no solution in the case \(a=-3\).

Given that \(a=5\),
(i) show that the number of solutions is infinite,

(ii) find the solution for which \(x+y+z=2\).

Use de Moivre's theorem to show that
\(\tan 5 \theta=\frac{5 t-10 t^{3}+t^{5}}{1-10 t^{2}+5 t^{4}}\)
where \(t=\tan \theta\).

Deduce that the roots of the equation \(t^{4}-10 t^{2}+5=0\) are \(\pm \tan \frac{1}{5} \pi\) and \(\pm \tan \frac{2}{5} \pi\).

Hence show that \(\tan \frac{1}{5} \pi \tan \frac{2}{5} \pi=\sqrt{ } 5\).

Find the set of values of \(a\) for which the system of equations
\(\begin{aligned} a x+y+2 z & =0 \\ 3 x-2 y & =4 \\ 3 x-4 y-6 a z & =14 \end{aligned}\)
has a unique solution.

Find the set of values of \(a\) for which the system of equations
\(\begin{aligned} x+4 y+12 z & =5 \\ 2 x+a y+12 z & =a-1, \\ 3 x+12 y+2 a z & =10, \end{aligned}\)
has a unique solution.

Show that the system does not have any solution in the case \(a=18\).

Given that \(a=8\), show that the number of solutions is infinite and find the solution for which \(x+y+z=1\).

Use de Moivre's theorem to express \(\cos ^{4} \theta\) in the form
\(a \cos 4 \theta+b \cos 2 \theta+c\)
where \(a, b, c\) are constants to be found.

Hence evaluate
\(\int_{0}^{\frac{1}{4} \pi} \cos ^{4} \theta \mathrm{~d} \theta\)
leaving your answer in terms of \(\pi\).

Use de Moivre's theorem to express \(\sin ^{6} \theta\) in the form
\(a+b \cos 2 \theta+c \cos 4 \theta+d \cos 6 \theta,\)
where \(a, b, c, d\) are constants to be found.

Hence evaluate
\(\int_{0}^{\frac{1}{4} \pi} \sin ^{6} 2 x \mathrm{~d} x\)
leaving your answer in terms of \(\pi\).