科目：进阶数学 Further Mathematics (9231)
年份：2022 冬季
试卷：Paper 23
总页数：14
章节：9231 Further Mathematics
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1. Find the Maclaurin’s series for $\ln(1 + e^x)$ up to and including the term in $x^2$. [5]
2 (a) Show that the system of equations $$x - y + 2z = 4,$$ $$x - y - 3z = a,$$ $$x - y + 7z = 13,$$ where $a$ is a constant, does not have a unique solution. [2] (b) Given that $a = -5$, show that the system of equations in part (a) is consistent. Interpret this situation geometrically. [3] (c) Given instead that $a \neq -5$, show that the system of equations in part (a) is inconsistent. Interpret this situation geometrically. [2]
3 The curve $C$ has parametric equations
$$x = e^t - \frac{1}{3} t^3, \quad y = 4e^{\frac{1}{2}t}(t-2), \quad \text{for } 0 \leq t \leq 2.$$
Find, in terms of $e$, the length of $C$. [6]4 (a) Starting from the definitions of cosh and sinh in terms of exponentials, prove that
$$\cosh^2 x - \sinh^2 x = 1.$$
[3]
(b) Show that $$\frac{d}{dx}\left(\tan^{-1}(\sinh x)\right) = \text{sech}x.$$
[3](c) Sketch the graph of $y = \text{sech}x$, stating the equation of the asymptote. [2]
(d) By considering a suitable set of $n$ rectangles of unit width, use your sketch to show that
$$\sum_{r=1}^{n} \text{sech}r < \tan^{-1}(\sinh n).$$ [3]
(e) Hence state an upper bound, in terms of $\pi$, for $$\sum_{r=1}^{\infty} \text{sech}r.$$ [1]5 Find the particular solution of the differential equation
$$2 \frac{d^2 y}{dx^2} + 2 \frac{dy}{dx} + y = 4x^2 + 3x + 3,$$
given that, when $x = 0$, $y = \frac{dy}{dx} = 0$. [10]9 © UCLES 2022 9231/23/O/N/22 [Turn over]
6 The matrix $A$ is given by
$$A = \begin{pmatrix} 2 & -3 & -7 \\ 0 & 5 & 7 \\ 0 & 0 & -2 \end{pmatrix}.$$
(a) Find a matrix $P$ and a diagonal matrix $D$ such that $A^5 = PDP^{-1}$. [7](b) Use the characteristic equation of $A$ to show that $$A^4 = aA^2 + bI,$$ where $a$ and $b$ are integers to be determined. [4]
7 (a) State the sum of the series $1 + w + w^2 + w^3 + \ldots + w^{n-1}$, for $w \neq 1$. [1]
(b) Show that $(1 + i \tan \theta)^k = \sec^k \theta (\cos k\theta + i \sin k\theta)$, where $\theta$ is not an integer multiple of $\frac{1}{2}\pi$. [2]
(c) By considering $\sum_{k=0}^{n-1} (1 + i \tan \theta)^k$, show that $\sum_{k=0}^{n-1} \sec^k \theta \sin k\theta = \cot \theta (1 - \sec^n \theta \cos n\theta)$, provided $\theta$ is not an integer multiple of $\frac{1}{2}\pi$. [5](d) Hence find $\sum_{k=0}^{6m-1} 2^k \sin\left(\frac{1}{3} k\pi\right)$ in terms of $m$. [2]8 (a) Use the substitution $u = 1 - (\theta - 1)^2$ to find
$$\int \frac{\theta - 1}{\sqrt{1 - (\theta - 1)^2}} \mathrm{d}\theta.$$
[3]
(b) Find the solution of the differential equation
$$\theta \frac{\mathrm{d}y}{\mathrm{d}\theta} - y = \theta^2 \sin^{-1}(\theta - 1),$$
where $0 < \theta < 2$, given that $y = 1$ when $\theta = 1$. Give your answer in the form $y = f(\theta)$.
[11]15 © UCLES 2022 9231/23/O/N/22