科目：进阶数学 Further Mathematics (9231)
年份：2018 夏季
试卷：Paper 13
总页数：27
章节：9231 Further Mathematics
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1 The variables $x$ and $y$ are such that $y = -1$ when $x = 0$ and
$$\left( x + \frac{dy}{dx} \right)^3 = y^2 + x.$$
(i) Find the value of $\frac{dy}{dx}$ when $x = 0$. [1]
(ii) Find also the value of $\frac{d^2y}{dx^2}$ when $x = 0$. [4]2 (i) Verify that
$$\frac{n(e-1) + e}{n(n+1)e^{n+1}} = \frac{1}{ne^n} - \frac{1}{(n+1)e^{n+1}}.$$
[1]
Let $S_N = \sum_{n=1}^{N} \frac{n(e-1) + e}{n(n+1)e^{n+1}}$.
(ii) Express $S_N$ in terms of $N$ and $e$. [2]Let $S = \lim_{N \to \infty} S_N$.
(iii) Find the least value of $N$ such that $(N + 1)(S - S_N) < 10^{-3}$. [3]3 (i) Use de Moivre’s theorem to show that $$\cos 4\theta = \cos^4\theta - 6\cos^2\theta \sin^2\theta + \sin^4\theta.$$ [3]
(ii) Hence find all the roots of the equation $$x^4 - 6x^2 + 1 = 0$$ in the form $\tan q\pi$, where $q$ is a positive rational number. [5]
4 The curve $C$ has equation
$$y = \frac{x^2 + 7x + 6}{x - 2}.$$
(i) Find the coordinates of the points of intersection of $C$ with the axes. [2]
(ii) Find the equation of each of the asymptotes of $C$. [3](iii) Sketch C.
5 It is given that $e$ is an eigenvector of the matrix $A$ with corresponding eigenvalue $\lambda$.
(i) Show that $e$ is an eigenvector of $A^3$ and state the corresponding eigenvalue. [3]
It is given that
$$A = \begin{pmatrix} 2 & 0 \\ -1 & 3 \end{pmatrix}.$$
(ii) Find a matrix $P$ and a diagonal matrix $D$ such that
$$A^3 + I = PDP^{-1},$$
where $I$ is the $2 \times 2$ identity matrix. [5]11 © UCLES 2018 9231/13/M/J/18 [Turn over]
6 The equation $$9x^3 - 9x^2 + x - 2 = 0$$ has roots $\alpha, \beta, \gamma$. (i) Use the substitution $y = 3x - 1$ to show that $3\alpha - 1, 3\beta - 1, 3\gamma - 1$ are the roots of the equation $$y^3 - 2y - 7 = 0.$$ [2] The sum $(3\alpha - 1)^n + (3\beta - 1)^n + (3\gamma - 1)^n$ is denoted by $S_n$. (ii) Find the value of $S_3$. [2]
(iii) Find the value of $S_{-2}$.7 The lines $l_1$ and $l_2$ have vector equations
$$\mathbf{r} = a\mathbf{i} + 9\mathbf{j} + 13\mathbf{k} + \lambda(\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}) \quad \text{and} \quad \mathbf{r} = -3\mathbf{i} + 7\mathbf{j} - 2\mathbf{k} + \mu(-\mathbf{i} + 2\mathbf{j} - 3\mathbf{k})$$
respectively. It is given that $l_1$ and $l_2$ intersect.
(i) Find the value of the constant $a$.
[3]
The point $P$ has position vector $3\mathbf{i} + \mathbf{j} + 6\mathbf{k}$.
(ii) Find the perpendicular distance from $P$ to the plane containing $l_1$ and $l_2$. [4](iii) Find the perpendicular distance from $P$ to $l_2$. [4]
8 The curves $C_1$ and $C_2$ have polar equations, for $0 \leqslant \theta \leqslant \pi$, as follows: $$C_1: r = a,$$ $$C_2: r = 2a|\cos \theta|,$$ where $a$ is a positive constant. The curves intersect at the points $P_1$ and $P_2$. (i) Find the polar coordinates of $P_1$ and $P_2$. [2] (ii) In a single diagram, sketch $C_1, C_2$ and their line of symmetry. [3]
(iii) The region $R$ enclosed by $C_1$ and $C_2$ is bounded by the arcs $OP_1, P_1P_2$ and $P_2O$, where $O$ is the pole. Find the area of $R$, giving your answer in exact form. [5]
9 For the sequence $u_1, u_2, u_3, \ldots$, it is given that $u_1 = 8$ and
$$u_{r+1} = \frac{5u_r - 3}{4}$$
for all $r$.
(i) Prove by mathematical induction that
$$u_n = 4\left(\frac{5}{4}\right)^n + 3,$$
for all positive integers $n$. [5](ii) Deduce the set of values of $x$ for which the infinite series
$$ (u_1 - 3)x + (u_2 - 3)x^2 + \ldots + (u_r - 3)x^r + \ldots $$ is convergent. [2]
(iii) Use the result given in part (i) to find surds $a$ and $b$ such that
$$ \sum_{n=1}^{N} \ln(u_n - 3) = N^2 \ln a + N \ln b. $$ [3]10 It is given that $t \neq 0$ and
$$t \frac{d^2x}{dt^2} + 2 \frac{dx}{dt} + 9tx = 3t^2 + 1.$$
(i) Show that if $y = tx$ then
$$\frac{d^2y}{dt^2} + 9y = 3t^2 + 1.$$ [3]
(ii) Find $x$ in terms of $t$, given that $x = \frac{1}{9}\pi$ and $\frac{dx}{dt} = \frac{2}{3}$ when $t = \frac{1}{3}\pi$. [9]21 © UCLES 2018 9231/13/M/J/18 [Turn over]
11 Answer only one of the following two alternatives.
EITHER
(i) Show that
$$\int_{\frac{1}{2}\pi}^{\frac{1}{2}\pi} e^x \cos x \, dx = \frac{1}{2}\left(e^{\frac{1}{2}\pi} + e^{-\frac{1}{2}\pi}\right).$$
[4](ii) It is given that, for $n \geqslant 0$,
$$I_n = \int_{-\frac{1}{2}}^{\frac{1}{2}} e^{2x} \cos^n x \, dx.$$
Show that, for $n \geqslant 2$,
$$4I_n = n(n-1) \int_{-\frac{1}{2}}^{\frac{1}{2}} e^{2x} \sin^2 x \cos^{n-2} x \, dx - nI_n.$$
and deduce the reduction formula
$$(n^2 + 4)I_n = n(n-1)I_{n-2}.$$
[6](iii) Using the result in part (i) and the reduction formula in part (ii), find the $y$-coordinate of the centroid of the region bounded by the $x$-axis and the arc of the curve $y = e^x \cos x$ from $x = -\frac{1}{2}\pi$ to $x = \frac{1}{2}\pi$. Give your answer correct to 3 significant figures. [4]OR
Let $V$ be the subspace of $\mathbb{R}^4$ spanned by
$$v_1 = \begin{pmatrix} 1 \\ 2 \\ 0 \\ 2 \end{pmatrix}, \quad v_2 = \begin{pmatrix} -2 \\ -5 \\ 5 \\ 6 \end{pmatrix}, \quad v_3 = \begin{pmatrix} 0 \\ -3 \\ 15 \\ 18 \end{pmatrix} \text{ and } v_4 = \begin{pmatrix} 0 \\ -2 \\ 10 \\ 8 \end{pmatrix}.$$
(i) Show that the dimension of $V$ is 3. [3](ii) Express $v_4$ as a linear combination of $v_1, v_2$ and $v_3$.
(iii) Write down a basis for $V$.
Let $\mathbf{M} = \begin{pmatrix}
1 & -2 & 0 & 0 \\
2 & -5 & -3 & -2 \\
0 & 5 & 15 & 10 \\
2 & 6 & 18 & 8
\end{pmatrix}$.
(iv) Find the general solution of $\mathbf{Mx} = v_1 + v_2$.The set of elements of $\mathbb{R}^4$ which are not solutions of $Mx = v_1 + v_2$ is denoted by $W$.
(v) State, with a reason, whether $W$ is a vector space. [2]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. 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 an