科目：进阶数学 Further Mathematics (9231)
年份：2013 冬季
试卷：Paper 12
总页数：3
章节：9231 Further Mathematics
1. The curve $C$ has polar equation $r = 2e^{\theta}$, for $\frac{1}{6}\pi \leq \theta \leq \frac{1}{2}\pi$. Find
(i) the area of the region bounded by the half-lines $\theta = \frac{1}{6}\pi, \theta = \frac{1}{2}\pi$ and $C$, [2]
(ii) the length of $C$. [3]
2. The cubic equation $x^3 - px - q = 0$, where $p$ and $q$ are constants, has roots $\alpha, \beta, \gamma$. Show that
(i) $\alpha^2 + \beta^2 + \gamma^2 = 2p$, [1]
(ii) $\alpha^3 + \beta^3 + \gamma^3 = 3q$, [2]
(iii) $6(\alpha^5 + \beta^5 + \gamma^5) = 5(\alpha^3 + \beta^3 + \gamma^3)(\alpha^2 + \beta^2 + \gamma^2)$. [3]
3. It is given that
$$S_n = \sum_{r=1}^{n} u_r = 2n^2 + n.$$
Write down the values of $S_1, S_2, S_3, S_4$. Express $u_r$ in terms of $r$, justifying your answer. [4]
Find
$$6 The linear transformation $T: \mathbb{R}^4 \rightarrow \mathbb{R}^4$ is represented by the matrix $M$, where
$$M = \begin{pmatrix}
1 & -3 & -1 & 2 \\
4 & -10 & 0 & 2 \\
1 & -1 & 3 & -4 \\
5 & -12 & 1 & 1
\end{pmatrix}.$$
Find, in either order, the rank of $M$ and a basis for the null space $K$ of $T$. [6]
Evaluate
$$M \begin{pmatrix}
1 \\
-2 \\
-3 \\
-4
\end{pmatrix},$$
and hence show that every solution of
$$Mx = \begin{pmatrix}
2 \\
16 \\
10 \\
22
\end{pmatrix}$$
has the form
$$x = \begin{pmatrix}
1 \\
-2 \\
-3 \\
-4
\end{pmatrix} + \lambda e_1 + \mu e_2,$$
where $\lambda$ and $\mu$ are real numbers and $\{e_1, e_2\}$ is a basis for $K$. [3]
7 The square matrix $A$ has $\lambda$ as an eigenvalue with $e$ as a corresponding eigenvector. 10 The curve $C$ has equation
$$y = \frac{px^2 + 4x + 1}{x + 1},$$
where $p$ is a positive constant and $p \neq 3$.
(i) Obtain the equations of the asymptotes of $C$. [3]
(ii) Find the value of $p$ for which the $x$-axis is a tangent to $C$, and sketch $C$ in this case. [4]
(iii) For the case $p = 1$, show that $C$ has no turning points, and sketch $C$, giving the exact coordinates of the points of intersection of $C$ with the $x$-axis. [5]
11 Answer only one of the following two alternatives.
EITHER
State the fifth roots of unity in the form $\cos \theta + i \sin \theta$, where $-\pi < \theta \leq \pi$. [2]
Simplify
$$(x - [\cos \frac{2}{5}\pi + i \sin \frac{2}{5}\pi])(x - [\cos \frac{2}{5}\pi - i \sin \frac{2}{5}\pi]).$$ [2]
Hence find the real factors of