科目：进阶数学 Further Mathematics (9231)
年份：2017 夏季
试卷：Paper 13
总页数：23
章节：9231 Further Mathematics
1 The roots of the cubic equation $x^3 + 2x^2 - 3 = 0$ are $\alpha, \beta$ and $\gamma$.
(i) By using the substitution $y = \frac{1}{x^2}$, find the cubic equation with roots $\frac{1}{\alpha^2}, \frac{1}{\beta^2}$ and $\frac{1}{\gamma^2}$. [3]
(ii) Hence find the value of $\frac{1}{\alpha^2} + \frac{1}{\beta^2} + \frac{1}{\gamma^2}$. [1]
(iii) Find also the value of $\frac{1}{\alpha^2\beta^2} + \frac{1}{\beta^2\gamma^2} + \frac{1}{\gamma^2\alpha^2}$. [1]2 (i) Verify that $\frac{2r+1}{r(r+1)(r+2)} = \frac{1}{2} \left\{ \frac{(2r+1)(2r+3)}{(r+1)(r+2)} - \frac{(2r-1)(2r+1)}{r(r+1)} \right\}$. [2]
(ii) Hence show that $\sum_{r=1}^{n} \frac{2r+1}{r(r+1)(r+2)} = \frac{1}{2} \left\{ \frac{(2n+1)(2n+3)}{(n+1)(n+2)} - \frac{3}{2} \right\}$. [2]
(iii) Deduce the value of $\sum_{r=1}^{\infty} \frac{2r+1}{r(r+1)(r+2)}$. [2]3 Prove, by mathematical induction, that $\sum_{r=1}^{n} r \ln\left(\frac{r+1}{r}\right) = \ln\left(\frac{(n+1)^n}{n!}\right)$ for all positive integers $n$. [6]4 A curve $C$ has equation $x^3 - 3xy + y^2 = 4$. Find the value of $\frac{d^2y}{dx^2}$ at the point (0, 2) of $C$. [7]5 A curve $C$ has parametric equations
$$x = \frac{2}{5}t^{\frac{5}{2}} - 2t^{\frac{1}{2}}, \quad y = \frac{4}{3}t^{\frac{3}{2}}, \quad \text{for } 1 \leq t \leq 4.$$
(i) Find the exact value of the arc length of $C$. [5](ii) Find also the exact value of the surface area generated when $C$ is rotated through $2\pi$ radians about the $x$-axis. [3]
6 Let $I_n$ denote $\int_0^2 (4+x^2)^{-n} \, dx$.
(i) Find $\frac{d}{dx}\left(x(4+x^2)^{-n}\right)$ and hence show that
$$8nI_{n+1} = (2n-1)I_n + 2 \times 8^{-n}.$$ [5](ii) Use the result for integrating $\frac{1}{x^2 + a^2}$ with respect to $x$, in the List of Formulae (MF10), to find the value of $I_1$ and deduce that
$$I_3 = \frac{3}{1024}\pi + \frac{1}{128}.$$
[5]7 (i) Use de Moivre’s theorem to prove that
$$\tan 4\theta = \frac{4\tan\theta - 4\tan^3\theta}{1 - 6\tan^2\theta + \tan^4\theta}.$$ [5](ii) Hence find the solutions of the equation $$t^4 - 4t^3 - 6t^2 + 4t + 1 = 0,$$ giving your answers in the form $\tan k\pi$, where $k$ is a rational number. [5]
8 Find the solution of the differential equation
$$\frac{d^2x}{dr^2} + 6 \frac{dx}{dt} + 9x = 18t^2 + 6t + 1,$$
given that, when $t = 0, x = 3$ and $\frac{dx}{dt} = 0$. [10]13 © UCLES 2017 9231/13/MJJ/17 [Turn over]
9 The plane $\Pi_1$ passes through the points (1, 2, 1) and (5, -2, 9) and is parallel to the vector $\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}$.
(i) Find the cartesian equation of $\Pi_1$.
The plane $\Pi_2$ contains the lines
$$\mathbf{r} = 2\mathbf{i} - 3\mathbf{j} + \mathbf{k} + \lambda(\mathbf{i} - 2\mathbf{j} - \mathbf{k}) \quad \text{and} \quad \mathbf{r} = 2\mathbf{i} - 3\mathbf{j} + \mathbf{k} + \mu(2\mathbf{i} + 3\mathbf{j} - \mathbf{k}).$$
(ii) Find the cartesian equation of $\Pi_2$.(iii) Find the acute angle between $\Pi_1$ and $\Pi_2$. [3]
10 The matrix $A$ is given by
$$A = \begin{pmatrix} 6 & -8 & 7 \\ 7 & -9 & 7 \\ 6 & -6 & 5 \end{pmatrix}.$$
(i) Given that $\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$ is an eigenvector of $A$, find the corresponding eigenvalue. [2]
(ii) Given also that $-1$ is an eigenvalue of $A$, find a corresponding eigenvector. [2](iii) It is given that the determinant of $A$ is equal to the product of the eigenvalues of $A$. Use this result to find the third eigenvalue of $A$, and find also a corresponding eigenvector. [3]
(iv) Write down matrices $P$ and $D$ such that $P^{-1}AP = D$, where $D$ is a diagonal matrix, and hence find the matrix $A^n$ in terms of $n$, where $n$ is a positive integer. [6]11 Answer only one of the following two alternatives.
EITHER
A curve $C$ has polar equation $r = 2a \cos(2\theta + \frac{1}{2}\pi)$ for $0 \leq \theta < 2\pi$, where $a$ is a positive constant.
(i) Show that $r = -2a \sin 2\theta$ and sketch $C$.
[4]
(ii) Deduce that the cartesian equation of $C$ is
$(x^2 + y^2)^{\frac{3}{2}} = -4axy.$
[2](iii) Find the area of one loop of $C$. [5]
(iv) Show that, at the points (other than the pole) at which a tangent to $C$ is parallel to the initial line, $$2 \tan \theta = -\tan 2\theta.$$ [3]
OR
The matrix $A$, given by
$$A = \begin{pmatrix}
1 & -1 & 0 & 2 \\
3 & -1 & 4 & 0 \\
5 & -8 & -6 & 19 \\
-2 & 3 & 2 & -7
\end{pmatrix},$$
represents a transformation from $\mathbb{R}^4$ to $\mathbb{R}^4$.
(i) Find the rank of $A$ and show that $\left\{ \begin{pmatrix} 2 \\ 2 \\ -1 \\ 0 \end{pmatrix}, \left( \begin{array}{c} 1 \\ 3 \\ 0 \\ 1 \end{array} \right) \right\}$ is a basis for the null space of the transformation. [6](ii) Show that if
$$\mathbf{Ax} = p \begin{pmatrix} 1 \\ 3 \\ 5 \\ -2 \end{pmatrix} + q \begin{pmatrix} -1 \\ -1 \\ -8 \\ 3 \end{pmatrix},$$
where $p$ and $q$ are given real numbers, then
$$\mathbf{x} = \begin{pmatrix} p + 2\lambda + \mu \\ q + 2\lambda + 3\mu \\ -\lambda \\ \mu \end{pmatrix},$$
where $\lambda$ and $\mu$ are real numbers. [2](iii) Find the values of $p$ and $q$ such that
$$p \begin{pmatrix} 1 \\ 3 \\ 5 \\ -2 \end{pmatrix} + q \begin{pmatrix} -1 \\ -1 \\ -8 \\ 3 \end{pmatrix} = \begin{pmatrix} 3 \\ 7 \\ 18 \\ -7 \end{pmatrix}.$$
(iv) Find the solution of the equation $Ax = \begin{pmatrix} 3 \\ 7 \\ 18 \\ -7 \end{pmatrix}$ of the form $x = \begin{pmatrix} 4 \\ 9 \\ \alpha \\ \beta \end{pmatrix}$, where $\alpha$ and $\beta$ are positive integers to be found.
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