科目：进阶数学 Further Mathematics (9231)
年份：2022 夏季
试卷：Paper 23
总页数：18
章节：9231 Further Mathematics
1. Find the roots of the equation $z^3 = 7\sqrt{3} - 7i$, giving your answers in the form $re^{i\theta}$, where $r > 0$ and $-\pi \leq \theta < \pi$. [5]2 (a) Find the coefficient of $x^2$ in the Maclaurin’s series for $-\ln \cos x$.
(b) Find the length of the arc of the curve with equation $y = -\ln \cos x$ from the point where $x = 0$ to the point where $x = \frac{1}{4}\pi$.3 The matrix $A$ is given by
$$A = \begin{pmatrix} 6 & -9 & 5 \\ 5 & -8 & 5 \\ 1 & -1 & 2 \end{pmatrix}.$$
(a) Find the eigenvalues of $A$. [4](b) Use the characteristic equation of A to show that $A^{-1} = pA^2 + qI$, where $p$ and $q$ are constants to be determined. [3]4 It is given that
$$x = -t + \tan^{-1} t \quad \text{and} \quad y = t + \sinh^{-1} t.$$
(a) Show that $$\frac{dy}{dx} = -\frac{t^2 + 1 + \sqrt{t^2 + 1}}{t^2}.$$ [4](b) Find the value of $\frac{d^2y}{dx^2}$ when $t = \frac{3}{4}$. [5]5 Find the solution of the differential equation
$$x(x+7)\frac{dy}{dx} + 7y = x$$
for which $y = 7$ when $x = 1$. Give your answer in the form $y = f(x)$. [9]9 © UCLES 2022 9231/23/M/J/22 [Turn over]
The diagram shows the curve with equation $y = \ln(1+x)$ for $0 \leq x \leq 1$, together with a set of $n$ rectangles each of width $\frac{1}{n}$.
(a) By considering the sum of the areas of these rectangles, show that $\int_{0}^{1} \ln(1+x) \, dx < U_n$, where
$$U_n = \frac{1}{n} \ln \frac{(2n)!}{n!} - \ln n.$$
[4](b) Use a similar method to find, in terms of $n$, a lower bound $L_n$ for $\int_0^1 \ln(1+x) \, dx$. [4]
(c) By simplifying $U_n - L_n$, show that $\lim_{n \to \infty} (U_n - L_n) = 0$. [2]7 The variables $x$ and $y$ are related by the differential equation
$$4 \frac{d^2 y}{dx^2} - y = 3.$$
It is given that, when $x = 0$, $y = -3$ and $\frac{dy}{dx} = 2$.
(a) Find $y$ in terms of $x$. [8](b) Deduce the exact value of $x$ for which $y = 0$. Give your answer in logarithmic form. [3]
8 (a) Find $\int \sin \theta \cos^n \theta \mathrm{d}\theta$, where $n \neq -1$.
Let $I_{m,n} = \int_0^{\frac{1}{2}\pi} \sin^m \theta \cos^n \theta \mathrm{d}\theta$.
(b) Show that, for $m \geqslant 2$ and $n \geqslant 0$,
$$I_{m,n} = \frac{m-1}{m+n} I_{m-2,n}.$$
[2]
[5](c) By considering the binomial expansion of $\left(z + \frac{1}{z}\right)^5$, where $z = \cos\theta + i\sin\theta$, use de Moivre’s theorem to show that
$$\cos^5\theta = a\cos 5\theta + b\cos 3\theta + c\cos\theta,$$
where $a, b$ and $c$ are constants to be determined. [5](d) Using the results given in parts (b) and (c), find the exact value of $I_{2.5}$.Additional page If you use the following page to complete the answer to any question, the question number must be clearly shown.
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