科目：进阶数学 Further Mathematics (9231)
年份：2008 夏季
试卷：Paper 1
总页数：7
章节：9231 Further Mathematics
1 The finite region enclosed by the line $y = kx$, where $k$ is a positive constant, the $x$-axis for $0 \leqslant x \leqslant h$, and the line $x = h$ is rotated through 1 complete revolution about the $x$-axis. Prove by integration that the centroid of the resulting cone is at a distance $\frac{2}{4}h$ from the origin $O$. [4]
[The volume of a cone of height $h$ and base radius $r$ is $\frac{1}{3}\pi r^2 h$.]
2 Given that
$$u_n = \ln\left(\frac{1+x^{n+1}}{1+x^n}\right).$$
where $x > -1$, find $\sum_{n=1}^{N} u_n$ in terms of $N$ and $x$. [3]
Find the sum to infinity of the series
$$u_1 + u_2 + u_3 + \ldots$$
when
(i) $-1 < x < 1$, [1]
(ii) $x = 1$. [1]
3 Show that if $\lambda$ is an eigenvalue of the square matrix $A$ with $e$ as a corresponding eigenvector,4 The curves $C_1$ and $C_2$ have polar equations
$$r = \theta + 2 \quad \text{and} \quad r = \theta^2$$
respectively, where $0 \leq \theta \leq \pi$.
(i) Find the polar coordinates of the point of intersection of $C_1$ and $C_2$.
(ii) Sketch $C_1$ and $C_2$ on the same diagram.
(iii) Find the area bounded by $C_1, C_2$ and the line $\theta = 0$.
5 The equation
$$x^3 + x - 1 = 0$$
has roots $\alpha, \beta, \gamma$. Show that the equation with roots $\alpha^3, \beta^3, \gamma^3$ is
$$y^3 - 3y^2 + 4y - 1 = 0.$$
Hence find the value of $\alpha^6 + \beta^6 + \gamma^6$.
6 The curve $C$ is defined parametrically by
$$x = 4t - t^2 \quad \text{and} \quad y = 1 - e^{-t},$$
where $0 \leq t < 2$. Show that at all points of $C$,
$$\frac{d^2y}{dx^2} = \frac{(t-8 (i) Given that
$$I_n = \int_0^{\frac{1}{2}\pi} t^n \sin t \, dt,$$
show that, for $n \geqslant 2$,
$$I_n = n\left(\frac{\pi}{2}\right)^{n-1} - n(n-1)I_{n-2}.$$
(ii) A curve $C$ in the $x-y$ plane is defined parametrically in terms of $t$. It is given that
$$\frac{dx}{dt} = t^4(1 - \cos 2t) \text{ and } \frac{dy}{dt} = t^4 \sin 2t.$$
Find the length of the arc of $C$ from the point where $t = 0$ to the point where $t = \frac{1}{2}\pi$. [5]
9 The curve $C$ has equation
$$y = \frac{x^2 - 2x + \lambda}{x + 1},$$
where $\lambda$ is a constant. Show that the equations of the asymptotes of $C$ are independent of $\lambda$. [3]
Find the value of $\lambda$ for which the $x$-axis is a tangent to $C$, and sketch $C$ in this case. [4]
Sketch $C$ in the case $\lambda 12 Answer only one of the following two alternatives.
EITHER
The position vectors of the points $A, B, C, D$ are
$$7\text{i} + 4\text{j} - \text{k}, \quad 3\text{i} + 5\text{j} - 2\text{k}, \quad 2\text{i} + 6\text{j} + 3\text{k}, \quad 2\text{i} + 7\text{j} + \lambda\text{k}$$
respectively. It is given that the shortest distance between the line $AB$ and the line $CD$ is 3.
(i) Show that $\lambda^2 - 5\lambda + 4 = 0$. [7]
(ii) Find the acute angle between the planes through $A, B, D$ corresponding to the values of $\lambda$ satisfying the equation in part (i). [7]
OR
The linear transformation $T: \mathbb{R}^4 \rightarrow \mathbb{R}^4$ is represented by the matrix
$$\begin{pmatrix}
1 & 2 & -1 & -1 \\
1 & 3 & -1 & 0 \\
1 & 0 & 3 & 1 \\
0 & 3 & -4 & -1
\end{pmatrixBLANK PAGE
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