科目：进阶数学 Further Mathematics (9231)
年份：2013 夏季
试卷：Paper 12
总页数：3
章节：9231 Further Mathematics
1. Find the area of the region enclosed by the curve with polar equation $r = 2(1 + \cos \theta)$, for $0 \leq \theta < 2\pi$.
2. Prove by mathematical induction that $5^{2n} - 1$ is divisible by 8 for every positive integer $n$.
3. The cubic equation $x^3 - 2x^2 - 3x + 4 = 0$ has roots $\alpha, \beta, \gamma$. Given that $c = \alpha + \beta + \gamma$, state the value of $c$.
Use the substitution $y = c - x$ to find a cubic equation whose roots are $\alpha + \beta, \beta + \gamma, \gamma + \alpha$.
Find a cubic equation whose roots are $\frac{1}{\alpha + \beta}, \frac{1}{\beta + \gamma}, \frac{1}{\gamma + \alpha}$.
Hence evaluate $\frac{1}{(\alpha + \beta)^2} + \frac{1}{(\beta + \gamma)^2} + \frac{1}{(\gamma + \alpha)^2}$.
4. Let $I_n = \int_0^1 \frac{1}{(1+x^2)^7 By considering the binomial expansion of $\left(z - \frac{1}{z}\right)^6$, where $z = \cos \theta + i \sin \theta$, express $\sin^6 \theta$ in the form
$$\frac{1}{32}(p + q \cos 2\theta + r \cos 4\theta + s \cos 6\theta),$$
where $p, q, r$ and $s$ are integers to be determined. [6]
Hence find the exact value of $\int_0^{\frac{1}{2}\pi} \sin^6 \theta \mathrm{d}\theta$. [4]
8 The linear transformations $T_1 : \mathbb{R}^4 \rightarrow \mathbb{R}^4$ and $T_2 : \mathbb{R}^4 \rightarrow \mathbb{R}^4$ are represented by the matrices $M_1$ and $M_2$ respectively, where
$$M_1 = \begin{pmatrix}
1 & -2 & 3 & 5 \\
3 & -4 & 17 & 33 \\
5 & -9 & 20 & 36 \\
4 & -7 & 16 & 29
\end{pmatrix}$$
and
$$M_2 = \begin{pmatrix}
1 & -2 & 0 & -3 \\
2 & -1 & 0 & 0 \\
4 & -7 & 1 & -9 \\
610 The curve $C$ has equation $y = \frac{2x^2 - 3x - 2}{x^2 - 2x + 1}$. State the equations of the asymptotes of $C$. [2]
Show that $y \leqslant \frac{25}{12}$ at all points of $C$. [4]
Find the coordinates of any stationary points of $C$. [3]
Sketch $C$, stating the coordinates of any intersections of $C$ with the coordinate axes and the asymptotes. [4]
11 Answer only one of the following two alternatives.
EITHER
The curve $C$ has equation $y = 2\sec x$, for $0 \leqslant x \leqslant \frac{1}{4}\pi$. Show that the arc length $s$ of $C$ is given by
$$s = \int_0^{\frac{1}{4}\pi} (2\sec^2 x - 1) \, dx.$$ [4]
Find the exact value of $s$. [2]
The surface area generated when $C$ is rotated through $2\pi$ radians about the $x$-axis is denoted by $S$. Show that
(i) $S