科目：进阶数学 Further Mathematics (9231)
年份：2012 夏季
试卷：Paper 13
总页数：3
章节：9231 Further Mathematics
1. Find the sum of the first $n$ terms of the series
$$\frac{1}{1 \times 3} + \frac{1}{2 \times 4} + \frac{1}{3 \times 5} + \ldots$$
and deduce the sum to infinity. [5]
2. For the sequence $u_1, u_2, u_3, \ldots$, it is given that $u_1 = 1$ and $u_{r+1} = \frac{3u_r - 2}{4}$ for all $r$. Prove by mathematical induction that $u_n = 4\left(\frac{3}{4}\right)^n - 2$, for all positive integers $n$. [5]
3. The curve $C$ has equation
$$xy + (x + y)^3 = 1.$$
Show that $\frac{dy}{dx} = -\frac{3}{4}$ at the point $A(1, 0)$ on $C$. [3]
Find the value of $\frac{d^2y}{dx^2}$ at $A$. [5]
4. Let
$$I_n = \int_1^e x^2(\ln x)^n dx,$$
for $n \geqslant 0$. Show that, for all $n \geqslant 1,$
$$I_n = \frac{1}{3}e^3 - \frac{1}{3}nI_{n-1}.$$ [4]
Find the exact value of $I_3$. [4]
5. The mat6 The curve $C$ has equation $y = \frac{x^2}{x-2}$. Find the equations of the asymptotes of $C$. [3]
Find the coordinates of the turning points on $C$. [3]
Draw a sketch of $C$. [3]
7 Expand $\left(z + \frac{1}{z}\right)^4 \left(z - \frac{1}{z}\right)^2$ and, by substituting $z = \cos \theta + i \sin \theta$, find integers $p, q, r, s$ such that $64 \sin^2 \theta \cos^4 \theta = p + q \cos 2\theta + r \cos 4\theta + s \cos 6\theta$. [6]
Using the substitution $x = 2 \cos \theta$, show that $\int_1^2 x^4 \sqrt{(4-x^2)} \mathrm{d}x = \frac{4}{3} \pi + \sqrt{3}$. [4]
8 The cubic equation $x^3 - x^2 - 3x - 10 = 0$ has roots $\alpha, \beta, \gamma$.
(i) Let $u = -\alpha + \beta + \gamma$. Show that $u + 2\alpha = 1$, and hence find a cubic equation having roots $-\alpha + \bet11 Answer only one of the following two alternatives.
EITHER
The curve $C$ has cartesian equation
$$\left(x^2 + y^2\right)^2 = a^2 \left(x^2 - y^2\right),$$
where $a$ is a positive constant. Show that $C$ has polar equation
$$r^2 = a^2 \cos 2\theta.$$
[2]
Sketch $C$ for $-\pi < \theta \leq \pi$.
[2]
Find the area of the sector between $\theta = -\frac{1}{4}\pi$ and $\theta = \frac{1}{4}\pi$.
[3]
Find the polar coordinates of all points of $C$ where the tangent is parallel to the initial line.
[7]
OR
Show that the substitution $y = xz$ reduces the differential equation
$$\frac{1}{x} \frac{d^2y}{dx^2} + \left(\frac{6}{x} - \frac{2}{x^2}\right) \frac{dy}{dx} + \left(\frac{9}{x} - \frac{6}{x^2} + \frac{2}{x^3}\right) y = 169 \sin 2x$$
to