科目：进阶数学 Further Mathematics (9231)
年份：2025 夏季
试卷：Paper 23
总页数：14
章节：9231 Further Mathematics
1. Find the Maclaurin’s series for $e^{\frac{1}{x^2}}$ up to and including the term in $x^2$. [5]2 (a) Starting from the definitions of tanh and sech in terms of exponentials, prove that
$$\tanh^2 t + \text{sech}^2 t = 1.$$
[3]
(b) The curve $C$ has parametric equations
$$x = \ln(\cosh t), \quad y = \tan^{-1}(\sinh t), \quad \text{for } 0 \leq t \leq 1.$$
Find the length of $C$.
[5]3 The curve $C$ has equation
$$9y^2 - 3\sinh^{-1}(xy) = 1 - 3\ln 3.$$
(a) Show that, at the point $(4, \frac{1}{3})$ on $C$, $\frac{dy}{dx} = -\frac{1}{2}$. [4](b) Find the value of $\frac{d^2y}{dx^2}$ at the point $(4, \frac{1}{3})$. [5]4 Find the particular solution of the differential equation
$$\frac{d^2x}{dt^2} + \frac{dx}{dt} - 2x = 2t^2 + t - 1,$$
given that, when $t = 0$, $x = \frac{dx}{dt} = 0$. [10]
© UCLES 2025 9231/23/M/J/257 © UCLES 2025 9231/23/M/J/25 [Turn over]
5 (a) Use de Moivre’s theorem to show that
$$\sec 5\theta = \frac{\sec^5\theta}{5\sec^4\theta - 20\sec^2\theta + 16}.$$
[6](b) Hence, obtain the roots of the equation
$$\sqrt{3}x^5 - 10x^4 + 40x^2 - 32 = 0$$
in the form sec($q\pi$), where $q$ is rational. [4]The diagram shows the curve with equation $y = \frac{1}{x^2 + 1}$ 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
$$\sum_{r=1}^{n} \frac{n}{n^2 + r^2} < \frac{1}{4}\pi.$$
[5](b) Use a similar method to find a lower bound for $\sum_{r=1}^{n} \frac{n}{n^2 + r^2}$. Give your answer in terms of $n$ and $\pi$.
(c) Deduce the exact value of $\lim_{n \to \infty} \sum_{r=1}^{n} \frac{n}{n^2 + r^2}$.7 Find the solution of the differential equation
$$\frac{dy}{dx} - \frac{2x+6}{x^2+6x+5}y = 4,$$
given that $y=0$ when $x=0$. Give your answer in an exact form. [9]
© UCLES 2025 9231/23/M/J/2513 © UCLES 2025 9231/23/M/J/25 [Turn over]
8 (a) Find the values of $a$ for which the system of equations
$$\frac{3}{2}x + 3y + 8z = 1,$$
$$ax + 3y + 4z = 2,$$
$$ay - z = 3,$$
does not have a unique solution. [3]
The matrix $A$ is given by
$$A = \begin{pmatrix}
\frac{3}{2} & 3 & 8 \\
0 & 3 & 4 \\
0 & 0 & -1
\end{pmatrix}.$$
(b) Given that $B = A^{-1}$, use the characteristic equation of $A$ to show that $B^2 = pI + qA$, where $p$ and $q$ are constants to be determined. [4](c) Find a matrix $P$ and a diagonal matrix $D$ such that $A^{-1} = PDP^{-1}$. [7]