科目：进阶数学 Further Mathematics (9231)
年份：2018 夏季
试卷：Paper 12
总页数：27
章节：9231 Further Mathematics
1 The curve $C$ is defined parametrically by
$$x = e^t - t, \quad y = 4e^{\frac{3}{2}t}.$$
Find the length of the arc of $C$ from the point where $t = 0$ to the point where $t = 3$. [5]2 It is given that $f(n) = 2^{3n} + 8^{n-1}$. By simplifying $f(k) + f(k+1)$, or otherwise, prove by mathematical induction that $f(n)$ is divisible by 9 for every positive integer $n$. [6]3 The curve $C$ has polar equation $r = \cos 2\theta$, for $-\frac{1}{4}\pi \leq \theta \leq \frac{1}{4}\pi$.
(i) Sketch $C$.
(ii) Find the area of the region enclosed by $C$, showing full working. [3](iii) Find a cartesian equation of $C$. [3]
4 It is given that the equation
$$x^3 - 21x^2 + kx - 216 = 0,$$
where $k$ is a constant, has real roots $a, ar$ and $ar^{-1}$.
(i) Find the numerical values of the roots. [6](ii) Deduce the value of $k$. [2]
5 Let $S_n = \sum_{r=1}^{n} (-1)^{r-1} r^2$.
(i) Use the standard result for $\sum_{r=1}^{n} r^2$ given in the List of Formulae (MF10) to show that
$$S_{2n} = -n(2n + 1).$$
[4](ii) State the value of $\lim_{n \to \infty} \frac{S_{2n}}{n^2}$ and find $\lim_{n \to \infty} \frac{S_{2n+1}}{n^2}$.6 The curve $C$ has equation
$$y = \frac{x^2 + b}{x + b},$$
where $b$ is a positive constant.
(i) Find the equations of the asymptotes of $C$. [3]
(ii) Show that $C$ does not intersect the $x$-axis. [1](iii) Justifying your answer, find the number of stationary points on $C$. (iv) Sketch $C$. Your sketch should indicate the coordinates of any points of intersection with the y-axis. You do not need to find the coordinates of any stationary points.
7 Find the particular solution of the differential equation
$$49 \frac{d^2 y}{dx^2} + 14 \frac{dy}{dx} + y = 49x + 735,$$
given that when $x = 0, y = 0$ and $\frac{dy}{dx} = 0$. [10]Turn over
8 The linear transformation $T : \mathbb{R}^4 \rightarrow \mathbb{R}^3$ is represented by the matrix $M$, where
$$M = \begin{pmatrix}
1 & 2 & \alpha & -1 \\
2 & 6 & -3 & -3 \\
3 & 10 & -6 & -5
\end{pmatrix}$$
and $\alpha$ is a constant. When $\alpha \neq 0$ the null space of $T$ is denoted by $K_1$.
(i) Find a basis for $K_1$. [5]When $\alpha = 0$ the null space of $T$ is denoted by $K_2$. (ii) Find a basis for $K_2$. [3] (iii) Determine, justifying your answer, whether $K_1$ is a subspace of $K_2$. [2]
9 (i) Using the substitution $u = \tan x$, or otherwise, find $\int \sec^2 x \tan^2 x \, dx$.
It is given that, for $n \geqslant 0$,
$$I_n = \int_0^{\frac{1}{4}\pi} \sec^n x \tan^2 x \, dx.$$
(ii) Using the result that $\frac{d}{dx}(\sec x) = \tan x \sec x$, show that, for $n \geqslant 2$,
$$(n + 1)I_n = (\sqrt{2})^{n-2} + (n - 2)I_{n-2}.$$(iii) Hence find the mean value of $\sec^4 x \tan^2 x$ with respect to $x$ over the interval $0 \leqslant x \leqslant \frac{1}{4}\pi$, giving your answer in exact form.
[3]10 The line $l_1$ is parallel to the vector $a\mathbf{i} - \mathbf{j} + \mathbf{k}$, where $a$ is a constant, and passes through the point whose position vector is $9\mathbf{j} + 2\mathbf{k}$. The line $l_2$ is parallel to the vector $-a\mathbf{i} + 2\mathbf{j} + 4\mathbf{k}$ and passes through the point whose position vector is $-6\mathbf{i} - 5\mathbf{j} + 10\mathbf{k}$.
(i) It is given that $l_1$ and $l_2$ intersect.
(a) Show that $a = -\frac{6}{13}$. [3](b) Find a cartesian equation of the plane containing $l_1$ and $l_2$.
(ii) Given instead that the perpendicular distance between $I_1$ and $I_2$ is $3\sqrt{30}$, find the value of $a$. [5]11 Answer only one of the following two alternatives.
EITHER
(i) Show that if $z = e^{i\theta}$ and $z \neq -1$ then
$$\frac{z-1}{z+1} = i \tan \frac{1}{2}\theta.$$
[3](ii) Hence, or otherwise, show that if $z$ is a cube root of unity then
$$\frac{z^3 - 1}{z^3 + 1} + \frac{z^2 - 1}{z^2 + 1} + \frac{z - 1}{z + 1} = 0.$$
[5](iii) Hence write down three roots of the equation $$(z^3 - 1)(z^2 + 1)(z + 1) + (z^2 - 1)(z^3 + 1)(z + 1) + (z - 1)(z^3 + 1)(z^2 + 1) = 0$$ and find the other three roots. Give your answers in an exact form. [6]
OR
It is given that $\mathbf{e}$ is an eigenvector of the matrix $\mathbf{A}$, with corresponding eigenvalue $\lambda$.
(i) Write down another eigenvector of $\mathbf{A}$ corresponding to $\lambda$. [1]
(ii) Write down an eigenvector and corresponding eigenvalue of $\mathbf{A}^n$, where $n$ is a positive integer. [2]
Let $\mathbf{A} = \begin{pmatrix}
3 & 0 & 0 \\
2 & 7 & 0 \\
4 & 8 & 1
\end{pmatrix}$.
(iii) Find a matrix $\mathbf{P}$ and a diagonal matrix $\mathbf{D}$ such that $\mathbf{A}^n = \mathbf{PDP}^{-1}$. [7]25 © UCLES 2018 9231/12/M/J/18 [Turn over]
(iv) Determine the set of values of the real constant $k$ such that
$$\sum_{n=1}^{\infty} k^n \left( A^n - kA^{n+1} \right) = kA.$$
[4]Additional Page If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
BLANK PAGE Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after the live examination series. Cambridge International Examinations is part o