科目：进阶数学 Further Mathematics (9231)
年份：2013 冬季
试卷：Paper 13
总页数：3
章节：9231 Further Mathematics
1. Express $\frac{1}{r(r+1)(r-1)}$ in partial fractions.
Find
$$\sum_{r=2}^{n} \frac{1}{r(r+1)(r-1)}$$
State the value of
$$\sum_{r=2}^{\infty} \frac{1}{r(r+1)(r-1)}$$
2. Show that the matrix $\begin{pmatrix} 1 & 4 & 2 \\ 3 & 0 & -2 \\ 3 & -3 & -4 \end{pmatrix}$ has no inverse.
Solve the system of equations
$$x + 4y + 2z = 0,$$
$$3x - 2z = 4,$$
$$3x - 3y - 4z = 5.$$
3. Find the general solution of the differential equation
$$\frac{d^2y}{dx^2} + 2\frac{dy}{dx} + 4y = 4x^2 + 8.$$
4. A curve has parametric equations
$$x = 2\theta - \sin 2\theta, \quad y = 1 - \cos 2\theta, \quad \text{for } -3\pi \leq \theta \leq 3\pi.$$
Show that
$$\frac{dy}{dx} = \cot \theta,$$
except for certain values of $\theta$, which should be stated.
Find the value of $\f6 [In this question you may use, without proof, the formula $\int \sec x \, dx = \ln(\sec x + \tan x) + \text{const.}$]
(a) Let $y = \sec x$. Find the mean value of $y$ with respect to $x$ over the interval $\frac{1}{6}\pi \leq x \leq \frac{1}{3}\pi$.
(b) The curve $C$ has equation $y = -\ln(\cos x)$, for $0 \leq x \leq \frac{1}{3}\pi$. Find the arc length of $C$.
7 The curve $C$ has equation
$$y = \frac{2x^2 + 5x - 1}{x + 2}.$$
Find the equations of the asymptotes of $C$.
Show that $\frac{dy}{dx} > 2$ at all points on $C$.
Sketch $C$.
8 The points $A, B, C$ have position vectors
$$4\text{i} + 5\text{j} + 6\text{k}, \quad 5\text{i} + 7\text{j} + 8\text{k}, \quad 2\text{i} + 6\text{j} + 4\text{k},$$
respectively, relative to the origin $O$. Find a cartesian equ11 Answer only one of the following two alternatives.
EITHER
Let $I_n = \int_0^1 (1+x^2)^n \, dx$. Show that, for all integers $n$,
$$(2n+1)I_n = 2nI_{n-1} + 2^n.$$
[5]
Evaluate $I_0$ and hence find $I_3$.
[4]
Given that $I_{-1} = \frac{1}{4}\pi$, find $I_{-3}$.
[5]
OR
The vector $\mathbf{e}$ is an eigenvector of each of the $3 \times 3$ matrices $\mathbf{A}$ and $\mathbf{B}$, with corresponding eigenvalues $\lambda$ and $\mu$ respectively. Justifying your answer, state an eigenvalue of $\mathbf{A} + \mathbf{B}$.
[3]
The matrix $\mathbf{A}$, where
$$\mathbf{A} = \begin{pmatrix}
6 & -1 & -6 \\
1 & 0 & -2 \\
3 & -1 & -3
\end{pmatrix},$$
has eigenvectors $\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} -1 \\ 1 \\ 1 \end{pmatrix}