科目：进阶数学 Further Mathematics (9231)
年份：2018 冬季
试卷：Paper 13
总页数：27
章节：9231 Further Mathematics
1 The vectors $a, b, c$ and $d$ in $\mathbb{R}^3$ are given by
$$a = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}, \quad b = \begin{pmatrix} 2 \\ 9 \\ 0 \end{pmatrix}, \quad c = \begin{pmatrix} 3 \\ 3 \\ 4 \end{pmatrix} \text{ and } d = \begin{pmatrix} 0 \\ -8 \\ 3 \end{pmatrix}.$$
(i) Show that $\{a, b, c\}$ is a basis for $\mathbb{R}^3$. [3]
(ii) Express $d$ in terms of $a, b$ and $c$. [2]2 The roots of the equation $$x^3 + px^2 + qx + r = 0$$ are $\alpha, 2\alpha, 4\alpha$, where $p, q, r$ and $\alpha$ are non-zero real constants. (i) Show that $$2p\alpha + q = 0.$$ [4] (ii) Show that $$p^3r - q^3 = 0.$$ [2]
3 The sequence of positive numbers $u_1, u_2, u_3, \ldots$ is such that $u_1 < 3$ and, for $n \geqslant 1$,
$$u_{n+1} = \frac{4u_n + 9}{u_n + 4}.$$
(i) By considering $3 - u_{n+1}$, or otherwise, prove by mathematical induction that $u_n < 3$ for all positive integers $n$. [5](ii) Show that $u_{n+1} > u_n$ for $n \geqslant 1$.4 A curve is defined parametrically by
$$x = t - \frac{1}{2} \sin 2t \quad \text{and} \quad y = \sin^2 t.$$
The arc of the curve joining the point where $t = 0$ to the point where $t = \pi$ is rotated through one complete revolution about the $x$-axis. The area of the surface generated is denoted by $S$.
(i) Show that
$$S = a\pi \int_0^\pi \sin^3 t \, dt,$$
where the constant $a$ is to be found. [5](ii) Using the result $\sin 3t = 3\sin t - 4\sin^3 t$, find the exact value of $S$. [3]
5 It is given that $\lambda$ is an eigenvalue of the matrix $A$ with $e$ as a corresponding eigenvector, and $\mu$ is an eigenvalue of the matrix $B$ for which $e$ is also a corresponding eigenvector.
(i) Show that $\lambda + \mu$ is an eigenvalue of the matrix $A + B$ with $e$ as a corresponding eigenvector. [2]
The matrix $A$, given by
$$A = \begin{pmatrix}
2 & 0 & 1 \\
-1 & 2 & 3 \\
1 & 0 & 2
\end{pmatrix}$$
has $\begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}$, $\begin{pmatrix} 1 \\ 4 \\ -1 \end{pmatrix}$ and $\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$ as eigenvectors.
(ii) Find the corresponding eigenvalues. [3]The matrix $B$ has eigenvalues 4, 5 and 1 with corresponding eigenvectors $\begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 4 \\ -1 \end{pmatrix}$ and $\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$ respectively.
(iii) Find a matrix $P$ and a diagonal matrix $D$ such that $(A + B)^3 = PDP^{-1}$. [3]6 The curve $C$ has equation
$$y = \frac{x^2 + ax - 1}{x + 1},$$
where $a$ is constant and $a > 1$.
(i) Find the equations of the asymptotes of $C$. [3]
(ii) Show that $C$ intersects the $x$-axis twice. [1](iii) Justifying your answer, find the number of stationary points on $C$. (iv) Sketch $C$, stating the coordinates of its point of intersection with the $y$-axis.
7 (i) Use de Moivre's theorem to show that $$\sin 8\theta = 8 \sin \theta \cos \theta (1 - 10 \sin^2 \theta + 24 \sin^4 \theta - 16 \sin^6 \theta).$$ [6]
(ii) Use the equation $\frac{\sin 8\theta}{\sin 2\theta} = 0$ to find the roots of
$$16x^6 - 24x^4 + 10x^2 - 1 = 0$$
in the form $\sin k\pi$, where $k$ is rational. [4]8 The plane $\Pi_1$ has equation
$$\mathbf{r} = \begin{pmatrix} 5 \\ 1 \\ 0 \end{pmatrix} + s \begin{pmatrix} -4 \\ 1 \\ 3 \end{pmatrix} + t \begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix}.$$
(i) Find a cartesian equation of $\Pi_1$.
The plane $\Pi_2$ has equation $3x + y - z = 3$.
(ii) Find the acute angle between $\Pi_1$ and $\Pi_2$, giving your answer in degrees.(iii) Find an equation of the line of intersection of $\Pi_1$ and $\Pi_2$, giving your answer in the form $r = a + \lambda b$. [5]
9 The curve $C$ has polar equation
$$r = 5\sqrt{(\cot \theta)},$$
where $0.01 \leq \theta \leq \frac{1}{2}\pi$.
(i) Find the area of the finite region bounded by $C$ and the line $\theta = 0.01$, showing full working. Give your answer correct to 1 decimal place. [3]
Let $P$ be the point on $C$ where $\theta = 0.01$.
(ii) Find the distance of $P$ from the initial line, giving your answer correct to 1 decimal place. [2](iii) Find the maximum distance of $C$ from the initial line. [3] (iv) Sketch $C$. [2]
10 (i) Find the particular solution of the differential equation
$$\frac{d^2x}{dt^2} + 2 \frac{dx}{dt} + 10x = 37 \sin 3t,$$
given that $x = 3$ and $\frac{dx}{dt} = 0$ when $t = 0$. [10](ii) Show that, for large positive values of $t$ and for any initial conditions,
$$x \approx \sqrt{37} \sin(3t - \phi),$$
where the constant $\phi$ is such that $\tan \phi = 6$. [3]11 Answer only one of the following two alternatives.
EITHER
(i) By considering $(2r + 1)^2 - (2r - 1)^2$, use the method of differences to prove that
$$\sum_{r=1}^{n} r = \frac{1}{2}n(n+1).$$
[3](ii) By considering $(2r + 1)^4 - (2r - 1)^4$, use the method of differences and the result given in part (i) to prove that
$$\sum_{r=1}^{n} r^3 = \frac{1}{4} n^2 (n + 1)^2.$$
[5]The sums $S$ and $T$ are defined as follows:
$$S = 1^3 + 2^3 + 3^3 + 4^3 + \ldots + (2N)^3 + (2N + 1)^3,$$
$$T = 1^3 + 3^3 + 5^3 + 7^3 + \ldots + (2N - 1)^3 + (2N + 1)^3.$$
(iii) Use the result given in part (ii) to show that $S = (2N + 1)^2(N + 1)^2$. [1]
(iv) Hence, or otherwise, find an expression in terms of $N$ for $T$, factorising your answer as far as possible. [2]
(v) Deduce the value of $\frac{S}{T}$ as $N \to \infty$. [2]OR
The curve $C$ has equation
$$x^2 + 2xy = y^3 - 2.$$
(i) Show that $A(-1, 1)$ is the only point on $C$ with $x$-coordinate equal to $-1$. [2]
For $n \geqslant 1$, let $A_n$ denote the value of $\frac{d^n y}{dx^n}$ at the point $A(-1, 1)$.
(ii) Show that $A_1 = 0$. [3](iii) Show that $A_2 = \frac{2}{5}$.Let $I_n = \int_{-1}^{0} x^n \frac{\mathrm{d}^n y}{\mathrm{d}x^n} \mathrm{d}x$.
(iv) Show that for $n \geqslant 2$,
$$I_n = (-1)^{n+1} A_{n-1} - n I_{n-1}.$$
[3](v) Deduce the value of $I_3$ in terms of $I_1$. [2]
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