科目：进阶数学 Further Mathematics (9231)
年份：2012 冬季
试卷：Paper 11
总页数：3
章节：9231 Further Mathematics
1. Find the cartesian equation corresponding to the polar equation $r = (\sqrt{2}) \sec(\theta - \frac{1}{4}\pi)$. [3]
Sketch the the graph of $r = (\sqrt{2}) \sec(\theta - \frac{1}{4}\pi)$, for $-\frac{1}{4}\pi < \theta < \frac{3}{4}\pi$, indicating clearly the polar coordinates of the intersection with the initial line. [2]
2. The curve $C$ has equation $y = 2x^{\frac{1}{2}}$ for $0 \leq x \leq 4$. Find
(i) the mean value of $y$ with respect to $x$ for $0 \leq x \leq 4$, [3]
(ii) the $y$-coordinate of the centroid of the region enclosed by $C$, the line $x = 4$ and the $x$-axis. [3]
3. Find the general solution of the differential equation
$$\frac{d^2x}{dt^2} + 4 \frac{dx}{dt} + 13x = 26t^2 + 3t + 13.$$ [6]
4. Let $f(r) = r(r+1)(r+2)$. Show that
$$f(r) - f(r-1) = 7 The curve $C$ has equation
$$y = \lambda x + \frac{x}{x - 2},$$
where $\lambda$ is a non-zero constant. Find the equations of the asymptotes of $C$. [3]
Show that $C$ has no turning points if $\lambda < 0$. [3]
Sketch $C$ in the case $\lambda = -1$, stating the coordinates of the intersections with the axes. [3]
8 The curve $C$ has parametric equations
$$x = \frac{1}{3} t^3 - \ln t, \quad y = \frac{4}{3} t^2,$$
for $1 \leq t \leq 3$. Find the arc length of $C$. [6]
Find also the area of the surface generated when $C$ is rotated through $2\pi$ radians about the $x$-axis. [4]
9 The plane $\Pi$ has equation
$$\mathbf{r} = 2\mathbf{i} + 3\mathbf{j} - \mathbf{k} + \lambda (\mathbf{i} - 2\mathbf{j} + 2\mathbf{k}) + \mu (3\mathbf{i} + \mathbf{j} - 2\mathbf{k}).$$
11 Answer only one of the following two alternatives.
EITHER
The roots of the equation $x^4 - 3x^2 + 5x - 2 = 0$ are $\alpha, \beta, \gamma, \delta$, and $\alpha^n + \beta^n + \gamma^n + \delta^n$ is denoted by $S_n$. Show that
$$S_{n+4} - 3S_{n+2} + 5S_{n+1} - 2S_n = 0.$$
[2]
Find the values of
(i) $S_2$ and $S_4$.
[3]
(ii) $S_3$ and $S_5$.
[6]
Hence find the value of
$$\alpha^2(\beta^3 + \gamma^3 + \delta^3) + \beta^2(\gamma^3 + \delta^3 + \alpha^3) + \gamma^2(\delta^3 + \alpha^3 + \beta^3) + \delta^2(\alpha^3 + \beta^3 + \gamma^3).$$
[3]
OR
The linear transformation $T: \mathbb{R}^4 \rightarrow \mathbb{R}^3$ is represented by the matrix $M$, where
$$M = \begin{pmatrix}
2 & 1 & -1 & 4 \\
3 & 4 & 6 & 1 \\
-1 & 2 & 8 & -7
\end{pmatrix}.$$