科目：进阶数学 Further Mathematics (9231)
年份：2008 冬季
试卷：Paper 1
总页数：3
章节：9231 Further Mathematics
1. The curve $C$ is defined parametrically by
$$x = t^4 - 4 \ln t, \quad y = 4t^2.$$
Show that the length of the arc of $C$ from the point where $t = 2$ to the point where $t = 4$ is
$$240 + 4 \ln 2.$$
2. Let $y = e^x$. Find the mean value of $y$ with respect to $x$ over the interval $0 \leqslant x \leqslant 2$. [2]
Show that the mean value of $x$ with respect to $y$ over the interval $1 \leqslant y \leqslant e^2$ is $\frac{e^2 + 1}{e^2 - 1}$.
3. The curve $C$ has polar equation
$$r = \left(\frac{1}{2}\pi - \theta\right)^2,$$
where $0 \leqslant \theta \leqslant \frac{1}{2}\pi$. Draw a sketch of $C$. [3]
Find the area of the region bounded by $C$ and the initial line, leaving your answer in terms of $\pi$. [3]
4. The matrix $A$ has $\lambda$ as an eigenvalue with $e$ as a co7 Let $I_n = \int_0^1 \frac{1}{(1+x^4)^n} dx$. By considering $\frac{d}{dx}\left(\frac{x}{(1+x^4)^n}\right)$, show that
$$4nI_{n+1} = \frac{1}{2^n} + (4n-1)I_n.$$
Given that $I_1 = 0.86697$, correct to 5 decimal places, find $I_3$.
8 Find $y$ in terms of $t$, given that
$$5\frac{d^2y}{dt^2} + 6\frac{dy}{dt} + 5y = 15 + 12t + 5t^2,$$
and that $y = \frac{dy}{dt} = 0$ when $t = 0$.
9 Use induction to prove that
$$\sum_{n=1}^{N} \frac{4n+1}{n(n+1)(2n-1)(2n+1)} = 1 - \frac{1}{(N+1)(2N+1)}.$$
Show that
$$\sum_{n=N+1}^{2N} \frac{4n+1}{n(n+1)(2n-1)(2n+1)} < \frac{3}{8N^2}.$$
10 Use de Moivre's theorem to express $\cos 8\theta$ as a polynomial in $\cos \theta$.
Hence
(i) express $\cos 8\theta$ as a polynomial in $\sin \theta$,
(ii) find the exact value of
12 Answer only one of the following two alternatives.
EITHER
The curve $C$ has equation
$$y = \frac{(x-2)(x-a)}{(x-1)(x-3)},$$
where $a$ is a constant not equal to 1, 2 or 3.
(i) Write down the equations of the asymptotes of $C$. [2]
(ii) Show that $C$ meets the asymptote parallel to the $x$-axis at the point where $x = \frac{2a-3}{a-2}$. [2]
(iii) Show that the $x$-coordinates of any stationary points on $C$ satisfy
$$(a-2)x^2 + (6-4a)x + (5a-6) = 0,$$
and hence find the set of values of $a$ for which $C$ has stationary points. [6]
(iv) Sketch the graph of $C$ for
(a) $a > 3,$
(b) $2 < a < 3$. [4]
OR
The roots of the equation
$$x^4 - 5x^2 + 2x - 1 = 0$$
are $\alpha, \beta, \gamma, \delta$. Let $S_n = \alpha^n + \beta^n + \gamma^n + \delta^n$.
(i) Show th