科目：进阶数学 Further Mathematics (9231)
年份：2012 冬季
试卷：Paper 13
总页数：3
章节：9231 Further Mathematics
1. Show that $\sum_{r=n+1}^{2n} r^2 = \frac{1}{6}n(2n+1)(7n+1)$.
2. Find the set of values of $a$ for which the system of equations
$$ax + y + 2z = 0,$$
$$3x - 2y = 4,$$
$$3x - 4y - 6az = 14,$$
has a unique solution.
3. Let $S_N = \frac{1}{2!} + \frac{2}{3!} + \frac{3}{4!} + \ldots + \frac{N}{(N+1)!}$. Prove by mathematical induction that, for all positive integers $N,$
$$S_N = 1 - \frac{1}{(N+1)!}$$.
4. The points $A, B$ and $C$ have position vectors $\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}, 2\mathbf{i} + 4\mathbf{j} + 5\mathbf{k}$ and $2\mathbf{i} + 3\mathbf{j} + 4\mathbf{k}$ respectively. Find $\overrightarrow{AB} \times \overrightarrow{AC}$.
5. Deduce, in either order, the exact value of
(i) the area of the triangle $ABC$,
(ii) the perpendicular dist8 Let $z = \cos \theta + i \sin \theta$. Show that
$$1 + z = 2 \cos \frac{1}{2} \theta (\cos \frac{1}{2} \theta + i \sin \frac{1}{2} \theta).$$
By considering $(1 + z)^n$, where $n$ is a positive integer, deduce the sum of the series
$$\binom{n}{1} \sin \theta + \binom{n}{2} \sin 2\theta + \ldots + \binom{n}{n} \sin n\theta.$$
[6]
9 The curve $C$ has equation $y = \frac{x^2 - 3x + 3}{x - 2}$. Find the equations of the asymptotes of $C$. [3]
Show that there are no points on $C$ for which $-1 < y < 3$. [4]
Find the coordinates of the turning points of $C$. [3]
Sketch $C$. [2]
10 The curve $C$ has equation $x^3 + y^3 = 3xy$, for $x > 0$ and $y > 0$. Find a relationship between $x$ and $y$ when $\frac{dy}{dx} = 0$. [4]
Find the exact coordinates of the turning poi12 Answer only one of the following two alternatives.
EITHER
The vector $\mathbf{e}$ is an eigenvector of each of the $n \times n$ matrices $\mathbf{A}$ and $\mathbf{B}$, with corresponding eigenvalues $\lambda$ and $\mu$ respectively. Prove that $\mathbf{e}$ is an eigenvector of the matrix $\mathbf{AB}$ with eigenvalue $\lambda \mu$.
It is given that the matrix $\mathbf{A}$, where
$$\mathbf{A} = \begin{pmatrix}
3 & 2 & 2 \\
-2 & -2 & -2 \\
1 & 2 & 2
\end{pmatrix},$$
has eigenvectors $\begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}$. Find the corresponding eigenvalues.
Given that 2 is also an eigenvalue of $\mathbf{A}$, find a corresponding eigenvector.
The matrix $\mathbf{B}$, where
$$\mathbf{B} = \begin{pmatrix}
-1