科目：进阶数学 Further Mathematics (9231)
年份：2017 冬季
试卷：Paper 13
总页数：23
章节：9231 Further Mathematics
1. Find $\sum_{r=1}^{n}(4r-3)(4r+1)$, giving your answer in its simplest form. [4]2 Find the general solution of the differential equation
$$\frac{d^2x}{dt^2} + 2\frac{dx}{dt} + 5x = 4 - 5t^2.$$
[6]3 (i) Show that $\frac{d^{n+1}}{dx^{n+1}}(x^{n+1}\ln x) = \frac{d^n}{dx^n}(x^n + (n+1)x^n\ln x)$. [2]
(ii) Prove by mathematical induction that, for all positive integers $n,$
$$\frac{d^n}{dx^n}(x^n\ln x) = n!\left(\ln x + 1 + \frac{1}{2} + \ldots + \frac{1}{n}\right).$$ [5]4 The cubic equation $2x^3 - 3x^2 + 4x - 10 = 0$ has roots $\alpha, \beta$ and $\gamma$. (i) Find the value of $(\alpha + 1)(\beta + 1)(\gamma + 1)$. [4] (ii) Find the value of $(\beta + \gamma)(\gamma + \alpha)(\alpha + \beta)$. [4]
5 The curve $C$ has equation $2x^3 + 3x^2y - 3y^3 - 16 = 0$.
(i) Find the coordinates of the point $A$ on $C$ at which $\frac{dy}{dx} = 0$ and $x \neq 0$. [5](ii) Find the value of $\frac{d^2y}{dx^2}$ at $A$. [3]6 The points $A, B$ and $C$ have position vectors $2i - j + k$, $3i + 4j - k$ and $-i + 2j + 4k$ respectively. (i) Find the area of the triangle $ABC$. [4]
(ii) Find the perpendicular distance of the point $A$ from the line $BC$. [3] (iii) Find the cartesian equation of the plane through $A, B$ and $C$. [2]
7 The linear transformation $T : \mathbb{R}^4 \rightarrow \mathbb{R}^4$ is represented by the matrix $A$, where
$$A = \begin{pmatrix}
1 & -1 & -2 & 3 \\
5 & -3 & -4 & 25 \\
6 & -4 & -6 & 28 \\
7 & -5 & -8 & 31
\end{pmatrix}.$$
(i) Find the rank of $A$ and a basis for the null space of $T$. [7](ii) Find the matrix product $A \begin{pmatrix} -1 \\ 1 \\ -1 \\ 1 \end{pmatrix}$ and hence find the general solution of the equation $Ax = \begin{pmatrix} 3 \\ 21 \\ 24 \\ 27 \end{pmatrix}$.
[3]8 Let $I_n = \int_0^{\frac{1}{2}\pi} \sec^n x \, dx$ for $n > 0$.
(i) Find the value of $I_2$.
(ii) Show that, for $n > 2$,
$(n-1)I_n = 2^{\frac{1}{2}n-1} + (n-2)I_{n-2}$.(iii) The curve $C$ has equation $y = \sec^3 x$ for $0 \leqslant x \leqslant \frac{1}{4}\pi$. The region $R$ is bounded by $C$, the $x$-axis, the $y$-axis and the line $x = \frac{1}{4}\pi$. Find the volume of revolution generated when $R$ is rotated through $2\pi$ radians about the $x$-axis. [4]9 The curve $C$ has equation
$$y = \frac{3x - 9}{(x - 2)(x + 1)}$$
(i) Find the equations of the asymptotes of $C$. [2]
(ii) Show that there is no point on $C$ for which $\frac{1}{3} < y < 3$. [4](iii) Find the coordinates of the turning points of $C$. (iv) Sketch $C$.
10 (i) Use de Moivre’s theorem to show that $$\sin 5\theta = 5 \sin \theta - 20 \sin^3 \theta + 16 \sin^5 \theta.$$ [5]
(ii) Hence explain why the roots of the equation $16x^4 - 20x^2 + 5 = 0$ are $x = \pm \sin \frac{1}{5}\pi$ and $x = \pm \sin \frac{2}{5}\pi$.
(iii) Without using a calculator, find the exact values of
$$\sin \frac{1}{5}\pi \sin \frac{2}{5}\pi \sin \frac{3}{5}\pi \sin \frac{4}{5}\pi \text{ and } \sin^2(\frac{1}{5}\pi) + \sin^2(\frac{2}{5}\pi).$$11 Answer only one of the following two alternatives.
EITHER
(i) The vector $\mathbf{e}$ is an eigenvector of the matrix $\mathbf{A}$, with corresponding eigenvalue $\lambda$, and is also an eigenvector of the matrix $\mathbf{B}$, with corresponding eigenvalue $\mu$. Show that $\mathbf{e}$ is an eigenvector of the matrix $\mathbf{AB}$ with corresponding eigenvalue $\lambda \mu$.
[3]
(ii) Find the eigenvalues and corresponding eigenvectors of the matrix $\mathbf{A}$, where
$$\mathbf{A} = \begin{pmatrix}
0 & 1 & 3 \\
3 & 2 & -3 \\
1 & 1 & 2
\end{pmatrix}$$
[6]19 © UCLES 2017 9231/13/O/N/17 [Turn over]
(iii) The matrix $B$, where
$$B = \begin{pmatrix} 3 & 6 & 1 \\ 1 & -2 & -1 \\ 6 & 6 & -2 \end{pmatrix},$$
has eigenvectors $\begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix}$, $\begin{pmatrix} 1 \\ -1 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}$. Find the eigenvalues of the matrix $AB$, and state corresponding eigenvectors.
[4]OR
The polar equation of a curve $C$ is $r = a(1 + \cos \theta)$ for $0 \leq \theta < 2\pi$, where $a$ is a positive constant.
(i) Sketch $C$. [2]
(ii) Show that the cartesian equation of $C$ is
$$x^2 + y^2 = a(x + \sqrt{(x^2 + y^2)})$$. [2](iii) Find the area of the sector of $C$ between $\theta = 0$ and $\theta = \frac{1}{4}\pi$. [4](iv) Find the arc length of $C$ between the point where $\theta = 0$ and the point where $\theta = \frac{1}{3}\pi$. [5]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