科目：进阶数学 Further Mathematics (9231)
年份：2014 夏季
试卷：Paper 13
总页数：3
章节：9231 Further Mathematics
1. The vectors $a, b, c$ and $d$ in $\mathbb{R}^3$ are given by
$$a = \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}, \quad b = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \quad c = \begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix} \text{ and } d = \begin{pmatrix} 3 \\ -2 \\ 0 \end{pmatrix}.$$
Show that $\{a, b, c\}$ is a basis for $\mathbb{R}^3$. [3]
Express $d$ in terms of $a, b$ and $c$. [2]
2. Show that the difference between the squares of consecutive integers is an odd integer. [1]
Find the sum to $n$ terms of the series
$$\frac{3}{1^2 \times 2^2} + \frac{5}{2^2 \times 3^2} + \frac{7}{3^2 \times 4^2} + \ldots + \frac{2r+1}{r^2(r+1)^2} + \ldots$$
and deduce the sum to infinity of the series. [5]
3. It is given that $\phi(n) = 5^n(4n+1)-1$, for $n=1, 2, 3, \ldots$. Pr7 The curve $C$ has parametric equations
$$x = \sin t, \quad y = \sin 2t, \quad \text{for } 0 \leq t \leq \pi.$$
Find $\frac{d^2y}{dx^2}$ in terms of $t$. [5]
Hence, or otherwise, find the coordinates of the stationary points on $C$ and determine their nature. [5]
8 It is given that $\lambda$ is an eigenvalue of the non-singular square matrix $A$, with corresponding eigenvector $e$. Show that $\lambda^{-1}$ is an eigenvalue of $A^{-1}$ for which $e$ is a corresponding eigenvector. [2]
Deduce that $\lambda + \lambda^{-1}$ is an eigenvalue of $A + A^{-1}$. [1]
It is given that 1 is an eigenvalue of the matrix $A$, where
$$A = \begin{pmatrix} 2 & 0 & 1 \\ -1 & 2 & 3 \\ 1 & 0 & 2 \end{pmatrix}.$$
Find a corresponding eigenvector. [2]
It is also given that $\begin{pma11 Answer only one of the following two alternatives.
EITHER
Express $\frac{2x^2 - x + 5}{x^2 - 1}$ in the form $2 + \frac{A}{x - 1} + \frac{B}{x + 1}$, where $A$ and $B$ are integers to be found. [3]
The curve $C$ has equation $y = \frac{2x^2 - x + 5}{x^2 - 1}$. Show that there are two distinct values of $x$ for which $\frac{dy}{dx} = 0$. [4]
Sketch $C$, stating the equations of the asymptotes and giving the coordinates of any points of intersection with the coordinate axes and with the asymptotes. You do not need to find the coordinates of the turning points. [7]
OR
With respect to an origin $O$, the point $A$ has position vector $4i - 2j + 2k$ and the plane $\Pi_1$ has equation
$$r = (4 + \lambda + 3\mu)i + (-2 + 7\lambda + \mu)j + (2 + \lambda - \mu)k,$$
where $