科目：进阶数学 Further Mathematics (9231)
年份：2019 夏季
试卷：Paper 12
总页数：27
章节：9231 Further Mathematics
1 A curve $C$ has equation $\cos y = x$, for $-\pi < x < \pi$.
(i) Use implicit differentiation to show that
$$\frac{d^2y}{dx^2} = -\cot y\left(\frac{dy}{dx}\right)^2.$$
[4]
(ii) Hence find the exact value of $\frac{d^2y}{dx^2}$ at the point $\left(\frac{1}{2}, \frac{1}{3}\pi\right)$ on $C$.
[2]2 Let $u_n = \frac{4 \sin(n - \frac{1}{2}) \sin \frac{1}{2}}{\cos(2n - 1) + \cos 1}$.
(i) Using the formulae for $\cos P \pm \cos Q$ given in the List of Formulae MF10, show that
$$u_n = \frac{1}{\cos n} - \frac{1}{\cos(n - 1)}$$
[2]
(ii) Use the method of differences to find $\sum_{n=1}^{N} u_n$.
[2]
(iii) Explain why the infinite series $u_1 + u_2 + u_3 + \ldots$ does not converge.
[1]3 The lines $l_1$ and $l_2$ have equations $\mathbf{r} = 6\mathbf{i} + 2\mathbf{j} + 7\mathbf{k} + \lambda(\mathbf{i} + \mathbf{j})$ and $\mathbf{r} = 4\mathbf{i} + 4\mathbf{j} + \mu(-6\mathbf{j} + \mathbf{k})$ respectively. The point $P$ on $l_1$ and the point $Q$ on $l_2$ are such that $PQ$ is perpendicular to both $l_1$ and $l_2$. Find the position vectors of $P$ and $Q$. [8]5 © UCLES 2019 9231/12/M/J/19 [Turn over]
4 It is given that, for $n \geqslant 0$,
$$I_n = \int_0^1 x^n e^{x^2} dx.$$
(i) Show that $I_2 = \frac{1}{3}(e - 1)$. [2]
(ii) Show that, for $n \geqslant 3$,
$$3I_n = e - (n - 2)I_{n-3}$$. [3](iii) Hence find the exact value of $I_g$. [3]
5 A curve $C$ is defined parametrically by
$$x = \frac{2}{e^t + e^{-t}} \quad \text{and} \quad y = \frac{e^t - e^{-t}}{e^t + e^{-t}},$$
for $0 \leq t \leq 1$. The area of the surface generated when $C$ is rotated through $2\pi$ radians about the $x$-axis is denoted by $S$.
(i) Show that $S = 4\pi \int_0^1 \frac{e^t - e^{-t}}{\left(e^t + e^{-t}\right)^2} \, dt.$ [5](ii) Using the substitution $u = e^t + e^{-t}$, or otherwise, find $S$ in terms of $\pi$ and $e$. [3]6 The equation
$$x^3 - x + 1 = 0$$
has roots $\alpha, \beta, \gamma$.
(i) Use the relation $x = y^{\frac{1}{3}}$ to show that the equation
$$y^3 + 3y^2 + 2y + 1 = 0$$
has roots $\alpha^3, \beta^3, \gamma^3$. Hence write down the value of $\alpha^3 + \beta^3 + \gamma^3$. [3]Let $S_n = \alpha^n + \beta^n + \gamma^n$.
(ii) Find the value of $S_{-3}$. [2]
(iii) Show that $S_6 = 5$ and find the value of $S_9$. [4]7 Find the particular solution of the differential equation
$$10 \frac{d^2x}{dt^2} + 3 \frac{dx}{dt} - x = t + 2,$$
given that when $t = 0, x = 0$ and $\frac{dx}{dt} = 0$. [10]Turn over
8 (i) Prove by mathematical induction that, for $z \neq 1$ and all positive integers $n$,
$$1 + z + z^2 + \ldots + z^{n-1} = \frac{z^n - 1}{z - 1}.$$
[5](ii) By letting $z = \frac{1}{2}(\cos \theta + i \sin \theta)$, use de Moivre’s theorem to deduce that
$$\sum_{m=1}^{\infty} \left(\frac{1}{2}\right)^m \sin m\theta = \frac{2 \sin \theta}{5 - 4 \cos \theta}.$$
[5]9 It is given that $e$ is an eigenvector of the matrix $A$, with corresponding eigenvalue $\lambda$.
(i) Show that $e$ is an eigenvector of $A^2$, with corresponding eigenvalue $\lambda^2$. [2]
The matrices $A$ and $B$ are given by
$$A = \begin{pmatrix} n & 1 & 3 \\ 0 & 2n & 0 \\ 0 & 0 & 3n \end{pmatrix}$$ and $$B = (A + nI)^2,$$
where $I$ is the $3 \times 3$ identity matrix and $n$ is a non-zero integer.
(ii) Find, in terms of $n$, a non-singular matrix $P$ and a diagonal matrix $D$ such that $B = PDP^{-1}$. [8]Turn over
10 The curves $C_1$ and $C_2$ have equations
$$y = \frac{ax}{x+5} \quad \text{and} \quad y = \frac{x^2 + (a+10)x + 5a + 26}{x+5}$$
respectively, where $a$ is a constant and $a > 2$.
(i) Find the equations of the asymptotes of $C_1$. [2]
(ii) Find the equation of the oblique asymptote of $C_2$. [2]
(iii) Show that $C_1$ and $C_2$ do not intersect. [2](iv) Find the coordinates of the stationary points of $C_2$. (v) Sketch $C_1$ and $C_2$ on a single diagram. [You do not need to calculate the coordinates of any points where $C_2$ crosses the axes.]
11 Answer only one of the following two alternatives.
EITHER
The curve $C_1$ has polar equation $r^2 = 2\theta$, for $0 \leqslant \theta \leqslant \frac{1}{2}\pi$.
(i) The point on $C_1$ furthest from the line $\theta = \frac{1}{2}\pi$ is denoted by $P$. Show that, at $P$,
$$2\theta \tan \theta = 1$$
and verify that this equation has a root between 0.6 and 0.7. [5]The curve $C_2$ has polar equation $r^2 = \theta \sec^2 \theta$, for $0 \leqslant \theta < \frac{1}{2}\pi$. The curves $C_1$ and $C_2$ intersect at the pole, denoted by $O$, and at another point $Q$.
(ii) Find the exact value of $\theta$ at $Q$. [2]
(iii) The diagram below shows the curve $C_2$. Sketch $C_1$ on this diagram. [2](iv) Find, in exact form, the area of the region OPQ enclosed by $C_1$ and $C_2$. [5]
OR
The linear transformation $T : \mathbb{R}^4 \rightarrow \mathbb{R}^4$ is represented by the matrix
$$M = \begin{pmatrix}
-1 & 2 & 3 & 4 \\
1 & 0 & 1 & -1 \\
1 & -2 & -3 & a \\
1 & 2 & 5 & 2
\end{pmatrix}.$$
(i) For $a \neq -4$, the range space of $T$ is denoted by $V$.
(a) Find the dimension of $V$ and show that
$$\begin{pmatrix}
-1 \\
1 \\
1 \\
1
\end{pmatrix}, \quad \begin{pmatrix}
2 \\
0 \\
-2 \\
2
\end{pmatrix}$$ and $$\begin{pmatrix}
4 \\
-1 \\
a \\
2
\end{pmatrix}$$
form a basis for $V$. [5](b) Show that if $\begin{pmatrix} x \\ y \\ z \\ t \end{pmatrix}$ belongs to $V$ then $x + 2y = t$. [4](ii) For $a = -4$, find the general solution of
$$\mathbf{Mx} = \begin{pmatrix}
-1 \\
1 \\
1 \\
1
\end{pmatrix}.$$
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