科目：进阶数学 Further Mathematics (9231)
年份：2018 冬季
试卷：Paper 12
总页数：27
章节：9231 Further Mathematics
1 The roots of the cubic equation $$x^3 - 5x^2 + 13x - 4 = 0$$ are $\alpha, \beta, \gamma$. (i) Find the value of $\alpha^2 + \beta^2 + \gamma^2$. [3] (ii) Find the value of $\alpha^3 + \beta^3 + \gamma^3$. [2]
2 It is given that
$$\mathbf{A} = \begin{pmatrix} 2 & 3 & 1 \\ 0 & -2 & 1 \\ 0 & 0 & 1 \end{pmatrix}.$$
(i) Find the eigenvalue of $\mathbf{A}$ corresponding to the eigenvector $\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$. [1]
(ii) Write down the negative eigenvalue of $\mathbf{A}$ and find a corresponding eigenvector. [3]
(iii) Find an eigenvalue and a corresponding eigenvector of the matrix $\mathbf{A} + \mathbf{A}^6$. [2]3 The curve $C$ has polar equation $r = a \cos 3\theta$, for $-\frac{1}{6}\pi \leq \theta \leq \frac{1}{6}\pi$, where $a$ is a positive constant.
(i) Sketch $C$.
(ii) Find the area of the region enclosed by $C$, showing full working. [3](iii) Using the identity $\cos 3\theta \equiv 4\cos^3\theta - 3\cos\theta$, find a cartesian equation of $C$. [3]
4 (i) Find the general solution of the differential equation
$$\frac{d^2x}{dt^2} + 2 \frac{dx}{dt} + x = 4 \sin t.$$
[7](ii) State an approximate solution for large positive values of $t$. [1]
5 The linear transformation $T : \mathbb{R}^4 \rightarrow \mathbb{R}^4$ is represented by the matrix $M$, where
$$M = \begin{pmatrix}
3 & 2 & 0 & 1 \\
6 & 5 & -1 & 3 \\
9 & 8 & -2 & 5 \\
-3 & -2 & 0 & -1
\end{pmatrix}.$$
(i) Find the rank of $M$. [3]
Let $K$ be the null space of $T$.
(ii) Find a basis for $K$. [3](iii) Find the general solution of
$$\mathbf{Mx} = \begin{pmatrix} 2 \\ 5 \\ 8 \\ -2 \end{pmatrix}.$$
[3]6 It is given that $y = \mathrm{e}^{x}u$, where $u$ is a function of $x$. The $r$th derivatives $\frac{\mathrm{d}^{r}y}{\mathrm{d}x^{r}}$ and $\frac{\mathrm{d}^{r}u}{\mathrm{d}x^{r}}$ are denoted by $y^{(r)}$ and $u^{(r)}$ respectively. Prove by mathematical induction that, for all positive integers $n$,
$$y^{(n)} = \mathrm{e}^{x}\left(\binom{n}{0}u + \binom{n}{1}u^{(1)} + \binom{n}{2}u^{(2)} + \ldots + \binom{n}{n}u^{(n)}\right).$$
[You may use without proof the result $\binom{k}{r} + \binom{k}{r-1} = \binom{k+1}{r}$.]11 © UCLES 2018 9231/12/O/N/18 [Turn over]
7 Let
$$S_N = \sum_{r=1}^{N} (3r + 1)(3r + 4) \quad \text{and} \quad T_N = \sum_{r=1}^{N} \frac{1}{(3r + 1)(3r + 4)}.$$
(i) Use standard results from the List of Formulae (MF10) to show that
$$S_N = N(3N^2 + 12N + 13).$$
[3]
(ii) Use the method of differences to show that
$$T_N = \frac{1}{12} - \frac{1}{3(3N + 4)}.$$
[3](iii) Deduce that $\frac{S_N}{T_N}$ is an integer. [2]
(iv) Find $\lim_{N \to \infty} \frac{S_N}{N^3 T_N}$. [2]8 (i) By considering the binomial expansion of $\left(z + \frac{1}{z}\right)^6$, where $z = \cos \theta + i \sin \theta$, express $\cos^6 \theta$ 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](ii) Hence find the exact value of
$$\int_{-\frac{1}{2}}^{\frac{1}{2}} \cos^6\left(\frac{1}{2}x\right) dx.$$
[4]9 The curve $C$ has equation
$$y = \frac{5x^2 + 5x + 1}{x^2 + x + 1}.$$
(i) Find the equation of the asymptote of $C$. [2]
(ii) Show that, for all real values of $x$, $-\frac{1}{3} \leq y < 5$. [4](iii) Find the coordinates of any stationary points of $C$. (iv) Sketch $C$, stating the coordinates of any intersections with the $y$-axis.
10 The position vectors of the points $A, B, C, D$ are
$$\mathbf{i} + \mathbf{j} + 3\mathbf{k}, \quad 3\mathbf{i} + 4\mathbf{j} + 5\mathbf{k}, \quad -\mathbf{i} + 3\mathbf{k}, \quad m\mathbf{j} + 4\mathbf{k},$$
respectively, where $m$ is a constant.
(i) Show that the lines $AB$ and $CD$ are parallel when $m = \frac{3}{2}$. [1]
(ii) Given that $m \neq \frac{3}{2}$ find the shortest distance between the lines $AB$ and $CD$. [5](iii) When $m = 2$, find the acute angle between the planes $ABC$ and $ABD$, giving your answer in degrees. [6]
11 Answer only one of the following two alternatives.
EITHER
The curve $C$ is defined parametrically by
$$x = 18t - t^2 \quad \text{and} \quad y = 8t^{\frac{3}{2}},$$
where $0 < t \leq 4$.
(i) Show that at all points of $C$,
$$\frac{d^2y}{dx^2} = \frac{3(9 + t)}{2t^2(9 - t)^3}.$$
[4](ii) Show that the mean value of $\frac{d^2y}{dx^2}$ with respect to $x$ over the interval $0 < x \leq 56$ is $\frac{3}{70}$. [4](iii) Find the area of the surface generated when $C$ is rotated through $2\pi$ radians about the $x$-axis, showing full working. [6]
OR
Let $I_n = \int_1^{\sqrt{2}} (x^2 - 1)^n \, dx$.
(i) Show that, for $n \geqslant 1$,
$(2n + 1)I_n = \sqrt{2} - 2nI_{n-1}$. [5](ii) Using the substitution $x = \sec \theta$, show that
$$I_n = \int_0^{\frac{1}{4}\pi} \tan^{2n+1}\theta \sec \theta \mathrm{d}\theta.$$
[4](iii) Deduce the exact value of
$$\int_{0}^{\frac{4\pi}{\sin^7 \theta}} \frac{\cos^8 \theta}{d\theta}.$$
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