科目：进阶数学 Further Mathematics (9231)
年份：2011 夏季
试卷：Paper 12
总页数：3
章节：9231 Further Mathematics
1. Express $\frac{1}{(2r+1)(2r+3)}$ in partial fractions and hence use the method of differences to find
$$\sum_{r=1}^{n} \frac{1}{(2r+1)(2r+3)}$$
[4]
Deduce the value of
$$\sum_{r=1}^{\infty} \frac{1}{(2r+1)(2r+3)}$$
[1]
2. The roots of the equation
$$x^3 + px^2 + qx + r = 0$$
are $\frac{\beta}{k}, \beta, k\beta$, where $p, q, r, k$ and $\beta$ are non-zero real constants. Show that $\beta = -\frac{q}{p}$.
[4]
Deduce that $rp^3 = q^3$.
[2]
3. The linear transformation $T: \mathbb{R}^4 \rightarrow \mathbb{R}^4$ is represented by the matrix $M = \begin{pmatrix}
1 & 3 & -2 & 4 \\
5 & 15 & -9 & 19 \\
-2 & -6 & 3 & -7 \\
3 & 9 & -5 & 11
\end{pmatrix}$.
(i) Find the rank of $M$.
[3]
(ii) Obtain a basis for the null space of $T$.
[3]
4. It 7 The variables $x$ and $y$ are related by the differential equation
$$y^2 \frac{d^2 y}{dx^2} + 2y^2 \frac{dy}{dx} + 2y \left( \frac{dy}{dx} \right)^2 - 5y^3 = 8e^{-x}.$$
Given that $v = y^3$, show that
$$\frac{d^2 v}{dx^2} + 2 \frac{dv}{dx} - 15v = 24e^{-x}.$$
Hence find the general solution for $y$ in terms of $x$. [7]
8 Find the eigenvalues and corresponding eigenvectors of the matrix $A = \begin{pmatrix} 4 & -1 & 1 \\ -1 & 0 & -3 \\ 1 & -3 & 0 \end{pmatrix}$. [8]
Find a non-singular matrix $P$ and a diagonal matrix $D$ such that $A^5 = PDP^{-1}$. [3]
9 The curve $C$ has equation $y = x^{\frac{1}{2}}$. Find the coordinates of the centroid of the region bounded by $C$, the lines $x = 1, x = 4$ and the $x$-axis. [7]
Show that the length of the arc of $C$ from the p11 Answer only one of the following two alternatives.
EITHER
Use de Moivre’s theorem to prove that
$$\tan 3\theta = \frac{3\tan\theta - \tan^3\theta}{1 - 3\tan^2\theta}.$$
[6]
State the exact values of $\theta$, between 0 and $\pi$, that satisfy $\tan 3\theta = 1$.
[2]
Express each root of the equation $t^3 - 3t^2 - 3t + 1 = 0$ in the form $\tan(k\pi)$, where $k$ is a positive rational number.
[3]
For each of these values of $k$, find the exact value of $\tan(k\pi)$.
[3]
OR
The curve $C$ has equation
$$y = \frac{x^2 + \lambda x - 6\lambda^2}{x + 3},$$
where $\lambda$ is a constant such that $\lambda \neq 1$ and $\lambda \neq -\frac{3}{2}$.
(i) Find $\frac{dy}{dx}$ and deduce that if $C$ has two stationary points then $-\frac{3}{2} < \lam