科目：进阶数学 Further Mathematics (9231)
年份：2013 夏季
试卷：Paper 13
总页数：3
章节：9231 Further Mathematics
1. Let $f(r) = r!(r - 1)$. Simplify $f(r + 1) - f(r)$ and hence find $\sum_{r=n+1}^{2n} r!(r^2 + 1)$. [5]
2. The roots of the equation $x^4 - 4x^2 + 3x - 2 = 0$ are $\alpha, \beta, \gamma$ and $\delta$; the sum $\alpha^n + \beta^n + \gamma^n + \delta^n$ is denoted by $S_n$. By using the relation $y = x^2$, or otherwise, show that $\alpha^2, \beta^2, \gamma^2$ and $\delta^2$ are the roots of the equation
$$y^4 - 8y^3 + 12y^2 + 7y + 4 = 0.$$ [3]
State the value of $S_2$ and hence show that
$$S_8 = 8S_6 - 12S_4 - 72.$$ [3]
3. Prove by mathematical induction that, for every positive integer $n,$
$$\frac{d^n}{dx^n} (e^x \sin x) = (\sqrt{2})^n e^x \sin(x + \frac{1}{4} n\pi).$$ [7]
4. Show that $\frac{dy}{dx} = -\frac{4}{3}$ at the point $A(1, -2)$ on the curve with equation
$$y^3 -7 Find the value of the constant $\lambda$ such that $\lambda xe^{-x}$ is a particular integral of the differential equation
$$\frac{d^2y}{dx^2} + 5 \frac{dy}{dx} + 4y = 6e^{-x}.$$
[4]
Find the solution of the differential equation for which $y = 2$ and $\frac{dy}{dx} = 3$ when $x = 0$. [6]
8 The curve $C$ has parametric equations $x = \frac{3}{2}t^2, y = t^3$, for $0 \leqslant t \leqslant 2$. Find the arc length of $C$. [4]
Find the coordinates of the centroid of the region enclosed by $C$, the $x$-axis and the line $x = 6$. [7]
9 The square matrix $A$ has an eigenvalue $\lambda$ with corresponding eigenvector $e$. The non-singular matrix $M$ is of the same order as $A$. Show that $Me$ is an eigenvector of the matrix $B$, where $B = MAM^{-1}$, and that $\lambda$ is the c11 Answer only one of the following two alternatives.
EITHER
The line $l_1$ passes through the point $A$ whose position vector is $4\mathbf{i} + 7\mathbf{j} - \mathbf{k}$ and is parallel to the vector $3\mathbf{i} + 2\mathbf{j} - \mathbf{k}$. The line $l_2$ passes through the point $B$ whose position vector is $\mathbf{i} + 7\mathbf{j} + 11\mathbf{k}$ and is parallel to the vector $\mathbf{i} - 6\mathbf{j} - 2\mathbf{k}$. The points $P$ on $l_1$ and $Q$ on $l_2$ are such that $PQ$ is perpendicular to both $l_1$ and $l_2$. Find the position vectors of $P$ and $Q$.
[8]
Find the shortest distance between the line through $A$ and $B$ and the line through $P$ and $Q$, giving your answer correct to 3 significant figures.
[6]
OR
Show the cube roots of 1 on an Argand diagram.