科目：进阶数学 Further Mathematics (9231)
年份：2026 夏季
试卷：Paper 22
总页数：14
章节：9231 Further Mathematics
1. Find the first three terms in the Maclaurin’s series for $\sinh^{-1}(x+1)$ in the form $\ln a + bx + cx^2$, giving the exact values of the constants $a, b$ and $c$. [5]2 Let $I_n = \int_0^1 (1+t)^n \sin \pi t \, dt$, where $n$ is a non-negative integer.
(a) Show that, for $n \geqslant 2$, $\pi^2 I_n = \pi(2^n + 1) - n(n-1) I_{n-2}$. [4]
(b) Find the exact value of $I_2$. [3]3 (a) Using the substitution $u = \cosh x$, find $\int \sinh^3 x \mathrm{d}x$.
[4](b) Find the particular solution of the differential equation
$$\frac{dy}{dx} + y \coth x = \sinh^2 x,$$
given that $y = 0$ when $x = \ln \frac{1}{3}$. Give your answer in the form $y = f(x)$.
[7]4 The curve $C$ has equation
$(x - y)^5 + \ln(xy) = 1 + \ln2$.
(a) Show that, at the point $(-1, -2)$ on $C$, $\frac{dy}{dx} = \frac{8}{11}$. [3](b) Find the value of $\frac{d^2y}{dx^2}$ at the point $(-1, -2)$.
[5]The diagram shows the curve with equation $y = \frac{11}{10}x^3 + \frac{1}{10}$ for $0 \leqslant x \leqslant 1$, 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 $\int_0^1 \left( \frac{11}{10}x^3 + \frac{1}{10} \right) dx < U_n$, where $U_n = \frac{11}{40}\left(1 + \frac{1}{n}\right)^2 + \frac{1}{10}$.
[4](b) Use a similar method to find, in terms of $n$, a lower bound $L_n$ for $\int_0^1 \left(\frac{11}{10}x^3 + \frac{1}{10}\right)dx$. [4]
(c) Show that $\lim_{n \to \infty} (U_n - L_n) = 0$. [2]6 Find the particular solution of the differential equation
$$5 \frac{d^2 y}{dx^2} + 4 \frac{dy}{dx} + y = x^2 + 7x + 1,$$
given that, when $x = 0, y = 0$ and $\frac{dy}{dx} = 0$. [10]
© Cambridge University Press & Assessment 2026 9231/22/M/J/2611 © Cambridge University Press & Assessment 2026 9231/22/M/J/26 Turn over
7 (a) State the sum of the series $z + z^2 + z^3 + \ldots + z^n$, for $z \neq 1$. [1]
(b) Find all roots of the equation $1 + z + z^2 + \ldots + z^6 = 0$ in the form $e^{i\theta}$, where $q$ is rational. [2]
(c) Given instead that $z = e^{-1 + i\theta}$, use de Moivre’s theorem to show that
$$\sum_{m=1}^{\infty} e^{-m} \cos m\theta = \frac{e\cos \theta - 1}{e^2 - 2e\cos \theta + 1}.$$ [7]DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN © Cambridge University Press & Assessment 2026 9231/22/MJ/26 Turn over
(a) The constants $a$ and $b$ are such that the system of equations
$$-x - 2y + z = 1,$$
$$ax + y - z = 2,$$
$$by + 2z = 3$$
does not have a unique solution. Find $a$ in terms of $b$. [3]
The matrix $A$ is given by
$$A = \begin{pmatrix}
-1 & -2 & 1 \\
0 & 1 & -1 \\
0 & 0 & 2
\end{pmatrix}.$$
(b) Use the characteristic equation of $A$ to show that $A^4 = pA^2 + qI$, where $p$ and $q$ are integers to be determined. [4](c) Find a matrix $P$ and a diagonal matrix $D$ such that $A^7 = PDP^{-1}$.
[7]