科目：进阶数学 Further Mathematics (9231)
年份：2015 夏季
试卷：Paper 12
总页数：3
章节：9231 Further Mathematics
1 Use the List of Formulae (MF10) to show that $\sum_{r=1}^{13}(3r^2 - 5r + 1)$ and $\sum_{r=0}^{9}(r^3 - 1)$ have the same numerical value. [4]
2 Find the value of the constant $k$ for which the system of equations
$$2x - 3y + 4z = 1,$$
$$3x - y = 2,$$
$$x + 2y + kz = 1,$$
does not have a unique solution. [2]
For this value of $k$, solve the system of equations. [4]
3 The sequence $a_1, a_2, a_3, \ldots$ is such that $a_1 > 5$ and $a_{n+1} = \frac{4a_n}{5} + \frac{5}{a_n}$ for every positive integer $n$. Prove by mathematical induction that $a_n > 5$ for every positive integer $n$. [5]
Prove also that $a_n > a_{n+1}$ for every positive integer $n$. [2]
4 The roots of the cubic equation $x^3 - 7x^2 + 2x - 3 = 0$ are $\alpha, \beta$ and $\gamma$. Find the values of
(i) $\fra7 Let $I_n = \int_0^{\frac{4\pi}{2}} x^n \sin x \, dx$, where $n$ is a non-negative integer. Show that
$$I_n = n\left(\frac{1}{2}\pi\right)^{n-1} - n(n-1)I_{n-2}, \quad \text{for } n \geq 2.$$
[5]
Find the exact value of $I_4$.
[4]
8 By considering $\sum_{r=1}^{n} z^{2r-1}$, where $z = \cos \theta + i \sin \theta$, show that, if $\sin \theta \neq 0$,
$$\sum_{r=1}^{n} \sin(2r-1)\theta = \frac{\sin^2 n\theta}{\sin \theta}.$$
[7]
Deduce that
$$\sum_{r=1}^{n}(2r-1)\cos\left[\frac{(2r-1)\pi}{2n}\right] = -\cosec\left(\frac{\pi}{2n}\right)\cot\left(\frac{\pi}{2n}\right).$$
[4]
9 The curve $C$ has parametric equations
$$x = 4t + 2t^2, \quad y = 4t - 2t^2, \quad \text{for } 0 \leq t \leq 4.$$
Find the arc length of $C$, giving your answer correct to 311 Answer only one of the following two alternatives.
EITHER
Show that the substitution $v = \frac{1}{y}$ reduces the differential equation
$$\frac{2}{y^3} \left( \frac{dy}{dx} \right)^2 - \frac{1}{y^2} \frac{d^2 y}{dx^2} - \frac{2}{y^2} \frac{dy}{dx} + \frac{5}{y} = 17 + 6x - 5x^2$$
to the differential equation
$$\frac{d^2 v}{dx^2} + 2 \frac{dv}{dx} + 5v = 17 + 6x - 5x^2.$$
[4]
Hence find $y$ in terms of $x$, given that when $x = 0, y = \frac{1}{2}$ and $\frac{dy}{dx} = -1.$
[10]
OR
The lines $l_1$ and $l_2$ have equations $r = 8i + 2j + 3k + \lambda(i - 2j)$ and $r = 5i + 3j - 14k + \mu(2j - 3k)$ 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 $l_2$. Find the position vector of the poi