科目：进阶数学 Further Mathematics (9231)
年份：2011 夏季
试卷：Paper 13
总页数：3
章节：9231 Further Mathematics
1. Find $2^2 + 4^2 + \ldots + (2n)^2$. [2]
Hence find $1^2 - 2^2 + 3^2 - 4^2 + \ldots - (2n)^2$, simplifying your answer. [3]
2. Let $A = \begin{pmatrix} 2 & 3 \\ 0 & 1 \end{pmatrix}$. Prove by mathematical induction that, for every positive integer $n$,
$A^n = \begin{pmatrix} 2^n & 3(2^n - 1) \\ 0 & 1 \end{pmatrix}$. [5]
3. Find a cubic equation with roots $\alpha, \beta$ and $\gamma$, given that
$\alpha + \beta + \gamma = -6$, $\alpha^2 + \beta^2 + \gamma^2 = 38$, $\alpha\beta\gamma = 30$. [3]
Hence find the numerical values of the roots. [3]
4. The curve $C$ has equation
$2xy^2 + 3x^2y = 1$.
Show that, at the point $A(-1, 1)$ on $C$, $\frac{dy}{dx} = -4$. [3]
Find the value of $\frac{d^2y}{dx^2}$ at $A$. [5]
5. Let
$I_n = \int_0^{\frac{1}{4}\p7 A curve $C$ has parametric equations $x = e^t \cos t, y = e^t \sin t$, for $0 \leq t \leq \pi$. Find the arc length of $C$.
Find the area of the surface generated when $C$ is rotated through $2\pi$ radians about the $x$-axis. [7]
8 Find the general solution of the differential equation
$$\frac{d^2x}{dt^2} + 2 \frac{dx}{dt} + 5x = 10 \sin t.$$ [6]
Find the particular solution, given that $x = 5$ and $\frac{dx}{dt} = 2$ when $t = 0$. [4]
State an approximate solution for large positive values of $t$. [1]
9 The curve $C$ with equation
$$y = \frac{ax^2 + bx + c}{x - 1},$$
where $a, b$ and $c$ are constants, has two asymptotes. It is given that $y = 2x - 5$ is one of these asymptotes.
(i) State the equation of the other asymptote. [1]
(ii) Find the value of $a$ a11 Answer only one of the following two alternatives.
EITHER
A $3 \times 3$ matrix $\mathbf{A}$ has eigenvalues $-1, 1, 2$, with corresponding eigenvectors
$$\begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix}, \quad \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix}, \quad \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix},$$
respectively. Find
(i) the matrix $\mathbf{A}$,
(ii) $\mathbf{A}^{2n}$, where $n$ is a positive integer.
[14]
OR
Determine the rank of the matrix
$$\mathbf{A} = \begin{pmatrix} 1 & -1 & -1 & 1 \\ 2 & -1 & -4 & 3 \\ 3 & -3 & -2 & 2 \\ 5 & -4 & -6 & 5 \end{pmatrix}.$$
[3]
Show that if
$$\mathbf{Ax} = p \begin{pmatrix} 1 \\ 2 \\ 3 \\ 5 \end{pmatrix} + q \begin{pmatrix} -1 \\ -1 \\ -3 \\ -4 \end{pmatrix} + r \begin{pmatrix} -1 \\ -4 \\ -2 \\ -6 \end{pmatrix}