科目：进阶数学 Further Mathematics (9231)
年份：2025 夏季
试卷：Paper 22
总页数：18
章节：9231 Further Mathematics
1. Find the roots of the equation $z^2 = 27 - 27i$, giving your answers in the form $re^{i\theta}$, where $r > 0$ and $-\pi \leq \theta < \pi$. [5]2 Let $I_n = \int_0^1 (1-x)^n \sinh x \, dx$, where $n$ is a non-negative integer.
(a) Show that, for $n \geqslant 2$, $I_n = -1 + n(n-1)I_{n-2}$. [4]
(b) Find the exact value of $I_2$. [3]3 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
$$\text{cosec}^5\theta = \frac{a}{\sin 5\theta + b\sin 3\theta + c\sin \theta},$$
where $a, b$ and $c$ are integers to be determined. [6]5 © UCLES 2025 9231/22/M/J/25 [Turn over]
The diagram shows the curve with equation $y = \frac{1}{\sqrt{x}} e^{\sqrt{x}}$ for $x \geqslant 1$, together with a set of $n-1$ rectangles of unit width.
(a) By considering the sum of the areas of these rectangles, show that
$$\sum_{r=1}^{n} \frac{1}{\sqrt{r}} e^{\sqrt{r}} < \left(2 + \frac{1}{\sqrt{n}}\right) e^{\sqrt{n}} - 2e.$$
[5](b) Use a similar method to find, in terms of $n$, a lower bound for $\sum_{r=1}^{n} \frac{1}{\sqrt{r}} e^{\sqrt{r}}$. [4]5 Find the particular solution of the differential equation
$$6 \frac{d^2x}{dr^2} + 3 \frac{dx}{dt} + 6x = e^{-t},$$
given that, when $t=0$, $x = \frac{dx}{dt} = 0$. [10]
© UCLES 2025 9231/22/M/J/259 © UCLES 2025 9231/22/M/J/25 [Turn over]
6 (a) Starting from the definitions of tanh and sech in terms of exponentials, prove that
$$1 - \tanh^2 u = \text{sech}^2 u.$$
[3]
(b) Show that $$\frac{d}{dt}(\text{sech}^{-1} t) = -\frac{1}{t\sqrt{1-t^2}}.$$
[4]It is given that
$$x = \tanh^{-1} t \quad \text{and} \quad y = t \text{sech}^{-1} t, \quad \text{for } 0 < t < 1.$$
(c) Show that $$\frac{dy}{dx} = -\sqrt{1-t^2} + (1-t^2)\text{sech}^{-1} t.$$ [4]
(d) Find $$\frac{d^2y}{dx^2}$$ in terms of $t.$ [4]7 Find the solution of the differential equation
$$\frac{dy}{dx} - \frac{x+5}{x^2 + 10x + 61}y = 1,$$
given that $y = 0$ when $x = 3$. Give your answer in an exact form. [10]
© UCLES 2025 9231/22/M/J/2513 © UCLES 2025 9231/22/M/J/25 [Turn over]
8 (a) It is given that $\lambda$ is an eigenvalue of the non-singular square matrix $A$, with corresponding eigenvector $e$.
Show that $e$ is an eigenvector of $A^3$ with corresponding eigenvalue $\lambda^3$.
[2]
The matrix $A$ is given by
$$A = \begin{pmatrix}
-1 & 3 & 4 \\
0 & 1 & 0 \\
0 & -2 & 5
\end{pmatrix}$$
(b) Show that the eigenvalues of $A$ are $-1, 1$ and $5$. [2](c) Find a matrix $P$ and a diagonal matrix $D$ such that $A - 2I = PDP^{-1}$.
(d) Use the characteristic equation of $A$ to show that $(A - 2I)^3 = aA^2 + bA + cI$ where $a, b$ and $c$ are constants to be determined.Additional page If you use the following page to complete the answer to any question, the question number must be clearly shown.
17 BLANK PAGE © UCLES 2025 9231/22/M/J/25
18 BLANK PAGE © UCLES 2025 9231/22/M/J/25
19 BLANK PAGE © UCLES 2025 9231/22/M/J/25