科目：进阶数学 Further Mathematics (9231)
年份：2016 冬季
试卷：Paper 12
总页数：3
章节：9231 Further Mathematics
1 Use the method of differences to find $\sum_{r=1}^{n} \frac{1}{(2r)^2 - 1}$.
Deduce the value of $\sum_{r=1}^{\infty} \frac{1}{(2r)^2 - 1}$.
2 Find the cubic equation with roots $\alpha, \beta$ and $\gamma$ such that
$$\alpha + \beta + \gamma = 3,$$
$$\alpha^2 + \beta^2 + \gamma^2 = 1,$$
$$\alpha^3 + \beta^3 + \gamma^3 = -30,$$
giving your answer in the form $x^3 + px^2 + qx + r = 0$, where $p, q$ and $r$ are integers to be found.
3 Find a matrix $A$ whose eigenvalues are $-1, 1, 2$ and for which corresponding eigenvectors are
$$\begin{pmatrix}
1 \\
0 \\
0
\end{pmatrix}, \quad \begin{pmatrix}
1 \\
1 \\
0
\end{pmatrix}, \quad \begin{pmatrix}
0 \\
1 \\
1
\end{pmatrix},$$
respectively.
4 Using factorials, show that
$$\binom{n}{r-1} + \binom{n}{r} = \binom{n+1}6. Find the general solution of the differential equation
$$\frac{d^2x}{dt^2} + 7 \frac{dx}{dt} + 10x = 116 \sin 2t.$$
State an approximate solution for large positive values of $t$.
[1]
7. The curve $C$ has equation $y = e^{-2x}$. Find, giving your answers correct to 3 significant figures,
(i) the mean value of $\frac{dy}{dx}$ over the interval $0 \leq x \leq 2$,
(ii) the coordinates of the centroid of the region bounded by $C$, $x = 0$, $x = 2$ and $y = 0$.
[2]
[9]
8. A curve $C$ has equation $x^2 + 4xy - y^2 + 20 = 0$. Show that, at stationary points on $C$, $x = -2y$.
[3]
Find the coordinates of the stationary points on $C$, and determine their nature by considering the value of $\frac{d^2y}{dx^2}$ at the stationary points.
[8]
9. Evalua11 Answer only one of the following two alternatives.
EITHER
The lines $l_1$ and $l_2$ have equations
$$\mathbf{r} = 6\mathbf{i} - 3\mathbf{j} + s(3\mathbf{i} - 4\mathbf{j} - 2\mathbf{k}) \quad \text{and} \quad \mathbf{r} = 2\mathbf{i} - \mathbf{j} - 4\mathbf{k} + t(\mathbf{i} - 3\mathbf{j} - \mathbf{k})$$
respectively. The point $P$ on $l_1$ and the point $Q$ on $l_2$ are such that $PQ$ is perpendicular to both $l_1$ and $l_2$. Show that the position vector of $P$ is $3\mathbf{i} + \mathbf{j} + 2\mathbf{k}$ and find the position vector of $Q$. [7]
Find, in the form $\mathbf{r} = \mathbf{a} + \lambda \mathbf{b} + \mu \mathbf{c}$, an equation of the plane $\Pi$ which passes through $P$ and is perpendicular to $l_1$. [3]
The plane $\Pi$ meets the plane $\mathbf{r} = p\mathb