科目：进阶数学 Further Mathematics (9231)
年份：2016 夏季
试卷：Paper 13
总页数：3
章节：9231 Further Mathematics
1. Verify that $\frac{1}{(3r+1)(3r+4)} = \frac{1}{3}\left(\frac{1}{3r+1} - \frac{1}{3r+4}\right)$. [1]
Let $S_N$ denote $\sum_{r=1}^{N} \frac{1}{(3r+1)(3r+4)}$ and let $S$ denote $\sum_{r=1}^{\infty} \frac{1}{(3r+1)(3r+4)}$. Find the least value of $N$ such that $S - S_N < \frac{1}{10000}$. [5]
2. It is given that a diagonal of a polygon is a line joining two non-adjacent vertices. Prove, by mathematical induction, that an $n$-sided polygon has $\frac{1}{2}n(n-3)$ diagonals, where $n \geqslant 3$. [6]
3. Find the two values of the constant $k$ for which the equations
$$kx + y + z = 2,$$
$$x + ky + z = -1,$$
$$x + y + kz = -1,$$
have no unique solution. [4]
Show that, for one of these values of $k$, the equations have no solution, and solve the equations for the othe7 A curve has polar equation $r = \frac{1}{1 - \cos \theta}$, for $0 < \theta < 2\pi$. Find, in the form $y^2 = f(x)$, the cartesian equation of the curve.
Hence sketch the curve, and shade the region whose area is given by $\frac{1}{2} \int_{\frac{1}{2}\pi}^{\frac{3}{2}\pi} \frac{1}{(1 - \cos \theta)^2} d\theta$.
Using the cartesian equation of the curve, find the area of this region.
8 The cubic equation
$z^3 - z^2 - z - 5 = 0$
has roots $\alpha, \beta$ and $\gamma$. Show that the value of $\alpha^3 + \beta^3 + \gamma^3$ is 19.
Find the value of $\alpha^4 + \beta^4 + \gamma^4$.
Show that the cubic equation with roots $\frac{\alpha-1}{\alpha}, \frac{\beta-1}{\beta}$ and $\frac{\gamma-1}{\gamma}$ may be found using the substitution $z = \frac{1}{1-x}$ and find this equ11 Answer only one of the following two alternatives.
EITHER
It is given that 1 and 4 are eigenvalues of the matrix $A$, where
$$A = \begin{pmatrix} 1 & -3 & -3 \\ -8 & 6 & -3 \\ 8 & -2 & 7 \end{pmatrix}.$$
Find eigenvectors corresponding to each of these eigenvalues. [3]
Given further that $\begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix}$ is an eigenvector of $A$, find the corresponding eigenvalue. [2]
Write down matrices $P$ and $D$ such that $P^{-1}AP = D$, where $D$ is a diagonal matrix, and find $P^{-1}$. [5]
Write down a matrix $C$ such that $C^2 = D$, and deduce a matrix $B$ such that $B^2 = A$. [4]
OR
The position vectors of the points $A, B, C, D$ are
$$a = 2i + \lambda j - 3k, \quad b = 6i + 3j - 2k, \quad c = i + 2j - k, \quad d = i + 7j + 4k$$
respec