科目：进阶数学 Further Mathematics (9231)
年份：2024 冬季
试卷：Paper 22
总页数：18
章节：9231 Further Mathematics
2 BLANK PAGE © UCLES 2024 9231/22/O/N/24
1. Find the value of $\int_{6}^{7} \frac{1}{\sqrt{(x-5)^2}-1} dx$, giving your answer in the form $\ln(a+\sqrt{b})$, where $a$ and $b$ are integers to be determined.
[4]2 The curve $C$ has equation
$$4y^2 + 4\ln(xy) = 1.$$
(a) Show that, at the point $\left(2, \frac{1}{2}\right)$ on $C$, $\frac{\mathrm{d}y}{\mathrm{d}x} = -\frac{1}{6}$. [3](b) Find the value of $\frac{d^2y}{dx^2}$ at the point $(2, \frac{1}{2})$.3 The curve $C$ has parametric equations
$$x = \frac{1}{2}e^{2t} - \frac{1}{3}t^3 - \frac{1}{2}, \quad y = 2e^t(t-1), \quad \text{for } 0 \leq t \leq 1.$$
Find the exact length of $C$. [7]7 © UCLES 2024 9231/22/O/N/24 [Turn over]
4 (a) Use de Moivre’s theorem to show that
$$\cot 6\theta = \frac{\cot^6\theta - 15\cot^4\theta + 15\cot^2\theta - 1}{6\cot^5\theta - 20\cot^3\theta + 6\cot\theta}.$$
[6](b) Hence obtain the roots of the equation $$x^6 - 6x^5 - 15x^4 + 20x^3 + 15x^2 - 6x - 1 = 0$$ in the form $\cot(q\pi)$, where $q$ is a rational number. [4]
5 Find the particular solution of the differential equation
$$3 \frac{d^2 y}{dx^2} + 2 \frac{dy}{dx} + y = x^2,$$
given that, when $x = 0$, $y = \frac{dy}{dx} = 0$. [10]11 © UCLES 2024 9231/22/O/N/24 [Turn over]
The diagram shows the curve with equation $y = e^{1-x}$ 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 $\int_{0}^{1} e^{1-x} dx < U_n$, where
$$U_n = \frac{e-1}{n(1-e^{-\frac{1}{n}})}$$
[4](b) Use a similar method to find, in terms of $n$, a lower bound $L_n$ for $\int_0^1 e^{1-x} dx$.
(c) Show that $\lim_{n \to \infty} (U_n - L_n) = 0$.(d) Use the Maclaurin’s series for $e^x$ given in the list of formulae (MF19) to find the first three terms of the series expansion of $z(1 - e^{-\frac{1}{z}})$, in ascending powers of $\frac{1}{z}$, and deduce the value of $\lim_{n \to \infty} (U_n)$.
[3]
7 (a) Show that $\frac{d}{dx}(\ln(\tanh x)) = 2 \cosh 2x$. [3](b) Find the solution of the differential equation
$$\sinh 2x \frac{dy}{dx} + 2y = \sinh 2x$$
for which $y = 5$ when $x = \ln 2$. Give your answer in an exact form. [7]8 The matrix $A$ is given by
$$A = \begin{pmatrix}
-2 & 0 & 0 \\
0 & 7 & 9 \\
4 & 1 & 7
\end{pmatrix}.$$
(a) Show that the characteristic equation of $A$ is $\lambda^3 - 12\lambda^2 + 12\lambda + 80 = 0$ and find the eigenvalues of $A$. [4](b) Use the characteristic equation of $A$ to show that $$A^4 = pA^2 + qA + rI,$$ where $p, q$ and $r$ are integers to be determined. [4]
(c) Find a matrix $P$ and a diagonal matrix $D$ such that $(A - 3I)^4 = PDP^{-1}$.
[6]Additional page If you use the following page to complete the answer to any question, the question number must be clearly shown.