科目：进阶数学 Further Mathematics (9231)
年份：2023 夏季
试卷：Paper 23
总页数：14
章节：9231 Further Mathematics
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1. (a) Find the Maclaurin series for $\sin^{-1} x$ up to and including the term in $x^3$.
[5]
(b) Deduce an approximation to $\int_0^{\frac{1}{\sqrt{1-u^2}}} \frac{1}{\sqrt{1-u^2}} du$, giving your answer as a fraction.
[1]2 The variables $x$ and $y$ are related by the differential equation
$$6 \frac{d^2 x}{dt^2} + 5 \frac{dx}{dt} + x = t^2 + 10t + 13.$$
(a) Find the general solution for $x$ in terms of $t$.
[6]
(b) State an approximate solution for large positive values of $t$. [1]3 By considering the binomial expansions of $\left(z + \frac{1}{z}\right)^4$ and $\left(z - \frac{1}{z}\right)^4$, where $z = \cos\theta + i\sin\theta$, use de Moivre’s theorem to show that
$$\cot^4\theta = \frac{\cos 4\theta + a\cos 2\theta + b}{\cos 4\theta - a\cos 2\theta + b}.$$
where $a$ and $b$ are integers to be determined. [7]4 The curve $C$ has equation
$$4y^3 + (x+y)^6 = 109.$$
(a) Show that, at the point $(-4, 3)$ on $C$, $\frac{dy}{dx} = \frac{1}{17}$. [3](b) Find the value of $\frac{d^2y}{dx^2}$ at the point $(-4, 3)$. [5]5 (a) Starting from the definitions of cosh and sinh in terms of exponentials, prove that
$$2 \cosh^2 x = \cosh 2x + 1.$$
[3]
(b) Find the solution of the differential equation
$$\frac{dy}{dx} + 2y \tanh x = 1$$
for which $y = 1$ when $x = 0$. Give your answer in the form $y = f(x).$
[8]9 © UCLES 2023 9231/23/M/J/23 [Turn over]
The diagram shows the curve with equation $y = (1-x)^2$ for $0 \leq x \leq 1$, together with a set of $n$ rectangles of width $\frac{1}{n}$.
(a) By considering the sum of the areas of these rectangles, show that $\int_0^1 (1-x)^2 dx < U_n$, where
$$U_n = \frac{2n^2 + 3n + 1}{6n^2}.$$
[5](b) Use a similar method to find, in terms of $n$, a lower bound $L_n$ for $\int_0^1 (1-x)^2 \mathrm{d}x$. [4]
(c) Show that $\lim_{n \to \infty} (U_n - L_n) = 0$. [2]7 The integral $I_n$, where $n$ is an integer, is defined by $I_n = \int_0^{\frac{4}{3}} (1+x^2)^{\frac{1}{2}n} \, dx$.
(a) Find the exact value of $I_{-1}$ giving your answer in the form $\ln a$, where $a$ is an integer to be determined. [2]
(b) By considering $\frac{d}{dx}\left(x(1+x^2)\right)^{\frac{1}{2}n}$, or otherwise, show that
$(n+1)I_n = nI_{n-2} + \frac{4}{3}\left(\frac{5}{3}\right)^n$. [5](c) A curve has equation $y = x^2$, for $0 \leqslant x \leqslant \frac{2}{3}$. The arc length of the curve is denoted by $s$.
Use the substitution $u = 2x$ to show that $s = \frac{1}{2} I_1$ and find the exact value of $s$. [4]8 The matrix $A$ is given by
$$A = \begin{pmatrix} a & -6a & 2a+2 \\ 0 & 1-a & 0 \\ 0 & 2-a & -1 \end{pmatrix}$$
where $a$ is a constant with $a \neq 0$ and $a \neq 1$.
(a) Show that the equation $A \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$ has a unique solution and interpret this situation geometrically.
[3]
(b) Show that the eigenvalues of $A$ are $a, 1-a$ and $-1$. [2](c) Find a matrix $P$ and a diagonal matrix $D$ such that $A^4 = PDP^{-1}$. [6]
(d) Use the characteristic equation of $A$ to find $A^4$ in terms of $A$ and $a$. [3]