科目：进阶数学 Further Mathematics (9231)
年份：2011 冬季
试卷：Paper 13
总页数：3
章节：9231 Further Mathematics
1 Verify that $\frac{1}{n^2} - \frac{1}{(n+1)^2} = \frac{2n+1}{n^2(n+1)^2}$. [1]
Let $S_N = \sum_{r=1}^{N} \frac{2r+1}{r^2(r+1)^2}$. Express $S_N$ in terms of $N$. [2]
Let $S = \lim_{N \to \infty} S_N$. Find the least value of $N$ such that $S - S_N < 10^{-16}$. [3]
2 Prove by mathematical induction that, for all positive integers $n$,
$$\frac{d^n}{dx^n}\left(\frac{1}{2x+3}\right) = (-1)^n \frac{n!2^n}{(2x+3)^{n+1}}.$$ [6]
3 The equation
$$x^3 + 5x^2 - 3x - 15 = 0$$
has roots $\alpha, \beta, \gamma$. Find the value of $\alpha^2 + \beta^2 + \gamma^2$. [3]
Hence show that the matrix $\begin{pmatrix} 1 & \alpha & \beta \\ \alpha & 1 & \gamma \\ \beta & \gamma & 1 \end{pmatrix}$ is singular. [4]
4 A curve has parametric equations
$$x = 2\sin 2t, \quad y = 3\cos 2t6. Find the general solution of the differential equation
$$\frac{d^2x}{dt^2} + 4\frac{dx}{dt} + 4x = \sin 2t.$$
Describe the behaviour of $x$ as $t \to \infty$, justifying your answer. [2]
7. Show that $$\frac{d}{dt}\left(t(1+t^3)^n\right) = (3n+1)(1+t^3)^n - 3n(1+t^3)^{n-1}.$$
Let $I_n = \int_0^1 (1+t^3)^n dt$. Using the above result, or otherwise, show that
$$(3n+1)I_n = 2^n + 3nI_{n-1}.$$
Hence evaluate $I_3$. [4]
8. The curve $C$ has polar equation $r = 1 + \sin \theta$ for $-\frac{1}{2}\pi \leq \theta \leq \frac{1}{2}\pi$. Draw a sketch of $C$. [2]
The area of the region enclosed by the initial line, the half-line $\theta = \frac{1}{2}\pi$, and the part of $C$ for which $\theta$ is positive, is denoted by $A_1$. The area of the region enclosed by the ini11 Answer only one of the following two alternatives.
EITHER
The curve $C$ has equation $y = \frac{1}{3}x^2(3 - x)$, for $0 \leqslant x \leqslant 3$. Find the mean value of $y$ with respect to $x$ over the interval $0 \leqslant x \leqslant 3$.
Show that
$$\frac{ds}{dx} = \frac{1}{2}\left(x^{-\frac{1}{2}} + x^{\frac{1}{2}}\right),$$
where $s$ denotes arc length, and find the arc length of $C$.
Find the area of the surface generated when $C$ is rotated through $2\pi$ radians about the $x$-axis.
OR
Find the eigenvalues and corresponding eigenvectors of the matrix $A$, where
$$A = \begin{pmatrix}
1 & 1 & 2 \\
0 & 2 & 2 \\
-1 & 1 & 3
\end{pmatrix}.$$
The linear transformation $T: \mathbb{R}^3 \to \mathbb{R}^3$ is defined by $x \mapsto Ax$. Let $e, f$ be two lin