科目：进阶数学 Further Mathematics (9231)
年份：2012 夏季
试卷：Paper 12
总页数：3
章节：9231 Further Mathematics
1 The roots of the cubic equation $x^3 - 7x^2 + 2x - 3 = 0$ are $\alpha, \beta, \gamma$. Find the values of
(i) $\alpha^2 + \beta^2 + \gamma^2$, [2]
(ii) $\alpha^3 + \beta^3 + \gamma^3$. [3]
2 Prove, by mathematical induction, that, for integers $n \geqslant 2$,
$$4^n > 2^n + 3^n.$$ [5]
3 Given that $f(r) = \frac{1}{(r+1)(r+2)}$, show that
$$f(r-1) - f(r) = \frac{2}{r(r+1)(r+2)}.$$ [2]
Hence find $\sum_{r=1}^{n} \frac{1}{r(r+1)(r+2)}$. [3]
Deduce the value of $\sum_{r=1}^{\infty} \frac{1}{r(r+1)(r+2)}$. [1]
4 The curve $C$ has polar equation $r = 2 + 2\cos \theta$, for $0 \leqslant \theta \leqslant \pi$. Sketch the graph of $C$. [2]
Find the area of the region $R$ enclosed by $C$ and the initial line. [4]
The half-line $\theta = \frac{1}{2}\pi$ divides $R$ into two parts. 7 The linear transformations $T_1: \mathbb{R}^4 \rightarrow \mathbb{R}^4$ and $T_2: \mathbb{R}^4 \rightarrow \mathbb{R}^4$ are represented by the matrices
$$\mathbf{M}_1 = \begin{pmatrix} 1 & 1 & 1 & 4 \\ 2 & 1 & 4 & 11 \\ 3 & 4 & 1 & 9 \\ 4 & -3 & 18 & 37 \end{pmatrix}$$ and $$\mathbf{M}_2 = \begin{pmatrix} 1 & 1 & 1 & -1 \\ 2 & 3 & 0 & 1 \\ 3 & 4 & 1 & 0 \\ 4 & 5 & 2 & 0 \end{pmatrix},$$ respectively. The null space of $T_1$ is denoted by $K_1$ and the null space of $T_2$ is denoted by $K_2$. Show that the dimension of $K_1$ is 2 and that the dimension of $K_2$ is 1.
Find the basis of $K_1$ which has the form $\left\{ \begin{pmatrix} p \\ q \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} r \\ s \\ 0 \\ 1 \end{pmatrix} \right\}$ and show that $K_2$ is a subspace of $K_1$.
8 Find the par11 Answer only one of the following two alternatives.
EITHER
Show that
$$\int_{0}^{\pi} e^{x} \sin x \, dx = \frac{1 + e^{\pi}}{2}.$$
Given that
$$I_{n} = \int_{0}^{\pi} e^{x} \sin^{n} x \, dx,$$
show that, for $n \geqslant 2$,
$$I_{n} = n(n-1) \int_{0}^{\pi} e^{x} \cos^{2} x \sin^{n-2} x \, dx - n I_{n},$$
and deduce that
$$\left(n^{2}+1\right) I_{n} = n(n-1) I_{n-2}.$$
A curve has equation $y = e^{x} \sin^{5} x$. Find, in an exact form, the mean value of $y$ over the interval $0 \leqslant x \leqslant \pi.$
OR
The position vectors of the points $A, B, C, D$ are
$$2i + 4j - 3k, \quad -2i + 5j - 4k, \quad i + 4j + k, \quad i + 5j + mk,$$
respectively, where $m$ is an integer. It is given that the shortest distance between the line through $A$ and $B$ and the li