9
0
\end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { l }
3
3
4
\end{array} \right) \quad \text { and } \quad \mathbf { d } = \left( \begin{array} { r }
0
- 8
3
\end{array} \right) .$$
- Show that \(\{ \mathbf { a } , \mathbf { b } , \mathbf { c } \}\) is a basis for \(\mathbb { R } ^ { 3 }\).
- Express \(\mathbf { d }\) in terms of \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\).
2 The roots of the equation
$$x ^ { 3 } + p x ^ { 2 } + q x + r = 0$$
are \(\alpha , 2 \alpha , 4 \alpha\), where \(p , q , r\) and \(\alpha\) are non-zero real constants. - Show that
$$2 p \alpha + q = 0$$
- Show that
$$p ^ { 3 } r - q ^ { 3 } = 0$$
3 The sequence of positive numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is such that \(u _ { 1 } < 3\) and, for \(n \geqslant 1\),
$$u _ { n + 1 } = \frac { 4 u _ { n } + 9 } { u _ { n } + 4 }$$
- By considering \(3 - u _ { n + 1 }\), or otherwise, prove by mathematical induction that \(u _ { n } < 3\) for all positive integers \(n\).
- Show that \(u _ { n + 1 } > u _ { n }\) for \(n \geqslant 1\).
4 A curve is defined parametrically by
$$x = t - \frac { 1 } { 2 } \sin 2 t \quad \text { and } \quad y = \sin ^ { 2 } t$$
The arc of the curve joining the point where \(t = 0\) to the point where \(t = \pi\) is rotated through one complete revolution about the \(x\)-axis. The area of the surface generated is denoted by \(S\). - Show that
$$S = a \pi \int _ { 0 } ^ { \pi } \sin ^ { 3 } t \mathrm {~d} t$$
where the constant \(a\) is to be found.
- Using the result \(\sin 3 t = 3 \sin t - 4 \sin ^ { 3 } t\), find the exact value of \(S\).
5 It is given that \(\lambda\) is an eigenvalue of the matrix \(\mathbf { A }\) with \(\mathbf { e }\) as a corresponding eigenvector, and \(\mu\) is an eigenvalue of the matrix \(\mathbf { B }\) for which \(\mathbf { e }\) is also a corresponding eigenvector. - Show that \(\lambda + \mu\) is an eigenvalue of the matrix \(\mathbf { A } + \mathbf { B }\) with \(\mathbf { e }\) as a corresponding eigenvector.
The matrix \(\mathbf { A }\), given by
$$\mathbf { A } = \left( \begin{array} { r r r }
2 & 0 & 1
- 1 & 2 & 3
1 & 0 & 2
\end{array} \right)$$
has \(\left( \begin{array} { l } 1
2
1 \end{array} \right) , \left( \begin{array} { r } 1
4
- 1 \end{array} \right)\) and \(\left( \begin{array} { l } 0
1
0 \end{array} \right)\) as eigenvectors. - Find the corresponding eigenvalues.
The matrix \(\mathbf { B }\) has eigenvalues 4, 5 and 1 with corresponding eigenvectors \(\left( \begin{array} { l } 1
2
1 \end{array} \right) , \left( \begin{array} { r } 1
4
- 1 \end{array} \right)\) and \(\left( \begin{array} { l } 0
1
0 \end{array} \right)\) respectively. - Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(( \mathbf { A } + \mathbf { B } ) ^ { 3 } = \mathbf { P D P } ^ { - 1 }\).
6 The curve \(C\) has equation
$$y = \frac { x ^ { 2 } + a x - 1 } { x + 1 }$$
where \(a\) is constant and \(a > 1\). - Find the equations of the asymptotes of \(C\).
- Show that \(C\) intersects the \(x\)-axis twice.
- Justifying your answer, find the number of stationary points on \(C\).
- Sketch \(C\), stating the coordinates of its point of intersection with the \(y\)-axis.
7
- Use de Moivre's theorem to show that
$$\sin 8 \theta = 8 \sin \theta \cos \theta \left( 1 - 10 \sin ^ { 2 } \theta + 24 \sin ^ { 4 } \theta - 16 \sin ^ { 6 } \theta \right) .$$
- Use the equation \(\frac { \sin 8 \theta } { \sin 2 \theta } = 0\) to find the roots of
$$16 x ^ { 6 } - 24 x ^ { 4 } + 10 x ^ { 2 } - 1 = 0$$
in the form \(\sin k \pi\), where \(k\) is rational.
8 The plane \(\Pi _ { 1 }\) has equation
$$\mathbf { r } = \left( \begin{array} { l }
5
1
0
\end{array} \right) + s \left( \begin{array} { r }
- 4
1
3
\end{array} \right) + t \left( \begin{array} { l }
0
1
2
\end{array} \right)$$ - Find a cartesian equation of \(\Pi _ { 1 }\).
The plane \(\Pi _ { 2 }\) has equation \(3 x + y - z = 3\). - Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in degrees.
- Find an equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\). [5]
9 The curve \(C\) has polar equation
$$r = 5 \sqrt { } ( \cot \theta ) ,$$
where \(0.01 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). - Find the area of the finite region bounded by \(C\) and the line \(\theta = 0.01\), showing full working. Give your answer correct to 1 decimal place.
Let \(P\) be the point on \(C\) where \(\theta = 0.01\). - Find the distance of \(P\) from the initial line, giving your answer correct to 1 decimal place.
- Find the maximum distance of \(C\) from the initial line.
- Sketch \(C\).