科目：进阶数学 Further Mathematics (9231)
年份：2014 冬季
试卷：Paper 11
总页数：3
章节：9231 Further Mathematics
1. Given that
$$u_k = \frac{1}{\sqrt{(2k-1)}} - \frac{1}{\sqrt{(2k+1)}},$$
express $\sum_{k=13}^{n} u_k$ in terms of $n$. [4]
Deduce the value of $\sum_{k=13}^{\infty} u_k$. [1]
2. A curve $C$ has parametric equations
$$x = e^t \cos t, \quad y = e^t \sin t, \quad \text{for } 0 \leq t \leq \frac{1}{2}\pi.$$
Find the arc length of $C$. [6]
3. It is given that $u_r = r \times r!$ for $r = 1, 2, 3, \ldots$. Let $S_n = u_1 + u_2 + u_3 + \ldots + u_n$. Write down the values of
$$2! - S_1, \quad 3! - S_2, \quad 4! - S_3, \quad 5! - S_4.$$ [2]
Conjecture a formula for $S_n$. [1]
Prove, by mathematical induction, a formula for $S_n$, for all positive integers $n$. [4]
4. A curve $C$ has equation $y = \frac{2x^2 + x - 1}{x - 1}$. Find the equations of the asymptotes of $C$. [37 Let $I_n = \int_0^1 (1-x)^n e^x dx$. Show that, for all positive integers $n$,
$$I_n = n I_{n-1} - 1.$$
[3]
Find the exact value of $I_4$.
[4]
By considering the area of the region enclosed by the $x$-axis, the $y$-axis and the curve with equation $y = (1-x)^4 e^x$ in the interval $0 \leqslant x \leqslant 1$, show that
$$\frac{65}{24} < e < \frac{11}{4}.$$
[3]
8 A circle has polar equation $r = a$, for $0 \leqslant \theta < 2\pi$, and a cardioid has polar equation $r = a(1-\cos \theta)$, for $0 \leqslant \theta < 2\pi$, where $a$ is a positive constant. Draw sketches of the circle and the cardioid on the same diagram.
[3]
Write down the polar coordinates of the points of intersection of the circle and the cardioid.
[2]
Show that the area of the11 Answer only one of the following two alternatives.
EITHER
The roots of the quartic equation $x^4 + 4x^3 + 2x^2 - 4x + 1 = 0$ are $\alpha, \beta, \gamma$ and $\delta$. Find the values of
(i) $\alpha + \beta + \gamma + \delta$, [1]
(ii) $\alpha^2 + \beta^2 + \gamma^2 + \delta^2$, [2]
(iii) $\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} + \frac{1}{\delta}$, [2]
(iv) $\frac{\alpha}{\beta\gamma\delta} + \frac{\beta}{\alpha\gamma\delta} + \frac{\gamma}{\alpha\beta\delta} + \frac{\delta}{\alpha\beta\gamma}$. [2]
Using the substitution $y = x + 1$, find a quartic equation in $y$. Solve this quartic equation and hence find the roots of the equation $x^4 + 4x^3 + 2x^2 - 4x + 1 = 0$. [7]
OR
The square matrix $A$ has $\lambda$ as an eigenvalue with $e$ as a corres