科目：进阶数学 Further Mathematics (9231)
年份：2019 夏季
试卷：Paper 13
总页数：23
章节：9231 Further Mathematics
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1 Prove by mathematical induction that $3^{3n} - 1$ is divisible by 13 for every positive integer $n$. [5]2 The curve $C$ has polar equation $r^2 = \ln(1 + \theta)$, for $0 \leqslant \theta \leqslant 2\pi$. (i) Sketch $C$. (ii) Using the substitution $u = 1 + \theta$, or otherwise, find the area of the region bounded by $C$ and the initial line, leaving your answer in an exact form. [5]
3 (i) Write down the fifth roots of unity. [2]
(ii) Find all the roots of the equation
$$z^{10} + z^5 + 1 = 0,$$
giving each root in the form $e^{i\theta}$. [5]4 (i) Use the method of differences to show that $\sum_{r=1}^{N} \frac{1}{(3r+1)(3r-2)} = \frac{1}{3} - \frac{1}{3(3N+1)}$. [4](ii) Find the limit, as $N \to \infty$, of $\sum_{r=N+1}^{N^2} \frac{N}{(3r+1)(3r-2)}$.5 The linear transformation $T : \mathbb{R}^4 \rightarrow \mathbb{R}^4$ is represented by the matrix $\mathbf{M}$, where
$$\mathbf{M} = \begin{pmatrix}
1 & 2 & 0 & 4 \\
5 & 2 & 1 & -3 \\
4 & 0 & 1 & -7 \\
-2 & 4 & -1 & \alpha
\end{pmatrix}.$$
It is given that the rank of $\mathbf{M}$ is 2.
(i) Find the value of $\alpha$ and state a basis for the range space of $T$. [4](ii) Obtain a basis for the null space of T. [4]
6 The curve $C$ has equation
$$y = \frac{x^2}{kx - 1},$$
where $k$ is a positive constant.
(i) Obtain the equations of the asymptotes of $C$. [3]
(ii) Find the coordinates of the stationary points of $C$. [3](iii) Sketch C. [3]
7 The line $l_1$ passes through the points $A(-3, 1, 4)$ and $B(-1, 5, 9)$. The line $l_2$ passes through the points $C(-2, 6, 5)$ and $D(-1, 7, 5)$. (i) Find the shortest distance between the lines $l_1$ and $l_2$. [5]
(ii) Find the acute angle between the line $l_2$ and the plane containing $A, B$ and $D$. [5]
8 Find the particular solution of the differential equation
$$9 \frac{d^2x}{dt^2} + 6 \frac{dx}{dt} + x = 50 \sin t,$$
given that when $t = 0, x = 0$ and $\frac{dx}{dt} = 0$. [10]Turn over
9 A cubic equation $x^3 + bx^2 + cx + d = 0$ has real roots $\alpha, \beta$ and $\gamma$ such that
$$\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} = -\frac{5}{12},$$
$$\alpha\beta\gamma = -12,$$
$$\alpha^3 + \beta^3 + \gamma^3 = 90.$$
(i) Find the values of $c$ and $d$. [3]
(ii) Express $\alpha^2 + \beta^2 + \gamma^2$ in terms of $b$. [2]
(iii) Show that $b^3 - 15b + 126 = 0$. [4](iv) Given that $3 + \sqrt{12}$ is a root of $y^3 - 15y + 126 = 0$, deduce the value of $b$. [2]10 Let $I_n = \int_{\frac{1}{2}\pi}^{\frac{1}{4}\pi} \cot^n x \, dx$, where $n \geqslant 0$.
(i) By considering $\frac{d}{dx}(\cot^{n+1} x)$, or otherwise, show that
$$I_{n+2} = \frac{1}{n+1} - I_n.$$
[5]The curve $C$ has equation $y = \cot x$, for $\frac{1}{4}\pi \leqslant x \leqslant \frac{1}{2}\pi$.
(ii) Find, in an exact form, the $y$-coordinate of the centroid of the region enclosed by $C$, the line $x = \frac{1}{4}\pi$ and the $x$-axis. [6]11 Answer only one of the following two alternatives.
EITHER
A $3 \times 3$ matrix $\mathbf{A}$ has distinct eigenvalues 2, 1, 3, with corresponding eigenvectors
$$\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}, \quad \begin{pmatrix} -1 \\ 0 \\ b \end{pmatrix}, \quad \begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix}$$
respectively, where $b$ is a positive constant.
(i) Find $\mathbf{A}$ in terms of $b$. [9](ii) Find $A^{-1}\begin{pmatrix} 0 \\ 2 \\ -2 \end{pmatrix}$.
(iii) It is given that
$$A^n\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 4 \\ 4 \\ 0 \end{pmatrix}$$ and
$$A^n\begin{pmatrix} -1 \\ 0 \\ b \end{pmatrix} = \begin{pmatrix} -1 \\ 0 \\ b^{-1} \end{pmatrix}.$$
Find the values of $n$ and $b$. [3]OR
The positive variables $y$ and $t$ are related by
$$y = a^t,$$
where $a$ is a positive constant.
(i) (a) By differentiating $\ln y$ with respect to $t$, show that $$\frac{dy}{dt} = a^t \ln a.$$ [3]
(b) Write down $$\frac{d^2y}{dt^2}.$$ [1]
(ii) Determine the set of values of $a$ for which the infinite series
$$y + \frac{dy}{dt} + \frac{d^2y}{dt^2} + \frac{d^3y}{dt^2} + \dots$$
is convergent. [3]A curve has parametric equations
$$x = t^a, \quad y = a^t.$$
(iii) Find $\frac{d^2y}{dx^2}$ in terms of $a$ and $t$, and show that, when $t = 2$,
$$\frac{d^2y}{dx^2} = 2^{1-2a}(1 - a + 2\ln a)\ln a.$$
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