科目：进阶数学 Further Mathematics (9231)
年份：2009 夏季
试卷：Paper 1
总页数：7
章节：9231 Further Mathematics
1. The equation
$$x^4 - x^3 - 1 = 0$$
has roots $\alpha, \beta, \gamma, \delta$. By using the substitution $y = x^3$, or by any other method, find the exact value of $\alpha^6 + \beta^6 + \gamma^6 + \delta^6$.
2. Verify that, for all positive values of $n$,
$$\frac{1}{(n+2)(2n+3)} - \frac{1}{(n+3)(2n+5)} = \frac{4n+9}{(n+2)(n+3)(2n+3)(2n+5)}.$$
For the series
$$\sum_{n=1}^{N} \frac{4n+9}{(n+2)(n+3)(2n+3)(2n+5)},$$
find
(i) the sum to $N$ terms,
(ii) the sum to infinity.
3. The equation of a curve is $y = \lambda x^2$, where $\lambda > 0$. The region bounded by the curve, the $x$-axis and the line $x = a$, where $a > 0$, is denoted by $R$. The $y$-coordinate of the centroid of $R$ is $a$. Show that $\lambda = \frac{10}{3a}$.
4. A curve has equation
$$y = \frac{1}{3}x^3 + 16 A curve has equation
$(x + y)(x^2 + y^2) = 1$.
Find the values of $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$ at the point $(0, 1)$. [7]
7 Let
$$I_n = \int_0^1 t^n e^{-t} dt$$,
where $n \geqslant 0$. Show that, for all $n \geqslant 1$,
$$I_n = n I_{n-1} - e^{-1}$$. [3]
Hence prove by induction that, for all positive integers $n$,
$$I_n < n!$$. [5]
8 Find the general solution of the differential equation
$$4 \frac{d^2y}{dx^2} + 4 \frac{dy}{dx} + 65y = 65x^2 + 8x + 73$$. [6]
Show that, whatever the initial conditions, $\frac{y}{x^2} \to 1$ as $x \to \infty$. [2]
9 The matrix
$$A = \begin{pmatrix}
3 & 1 & 4 \\
1 & 5 & -1 \\
2 & 1 & 5
\end{pmatrix}$$
has eigenvalues 1, 5, 7. Find a set of corresponding eigenvectors. [5]
Find a matrix $P$ and a dia11 The line $l_1$ is parallel to the vector $4\mathbf{j} - \mathbf{k}$ and passes through the point $A$ whose position vector is $2\mathbf{i} + \mathbf{j} + 4\mathbf{k}$. The variable line $l_2$ is parallel to the vector $\mathbf{i} - (2\sin t)\mathbf{j}$, where $0 \leq t < 2\pi$, and passes through the point $B$ whose position vector is $\mathbf{i} + 2\mathbf{j} + 4\mathbf{k}$. The points $P$ and $Q$ are on $l_1$ and $l_2$, respectively, and $PQ$ is perpendicular to both $l_1$ and $l_2$.
(i) Find the length of $PQ$ in terms of $t$. [5]
(ii) Hence find the values of $t$ for which $l_1$ and $l_2$ intersect. [2]
(iii) For the case $t = \frac{1}{4}\pi$, find the perpendicular distance from $A$ to the plane $BPQ$, giving your answer correct to 3 decimal places. [5]12 Answer only one of the following two alternatives.
EITHER
By considering $\sum_{k=0}^{n-1} (1 + i \tan \theta)^k$, show that
$$\sum_{k=0}^{n-1} \cos k \theta \sec^k \theta = \cot \theta \sin n \theta \sec^n \theta,$$
provided $\theta$ is not an integer multiple of $\frac{1}{2}\pi$. [7]
Hence or otherwise show that
$$\sum_{k=0}^{n-1} 2^k \cos\left(\frac{1}{2} k \pi\right) = \frac{2^n}{\sqrt{3}} \sin\left(\frac{1}{3} n \pi\right).$$ [2]
Given that $0 < x < 1$, show that
$$\sum_{k=0}^{n-1} \frac{\cos(k \cos^{-1} x)}{x^k} = \frac{\sin(n \cos^{-1} x)}{x^{n-1} \sqrt{(1-x^2)}}.$$ [4]
OR
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 $M_1$ and $M_2$, respectiBLANK PAGE
BLANK PAGE
BLANK PAGE Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.