科目：进阶数学 Further Mathematics (9231)
年份：2025 冬季
试卷：Paper 24
总页数：14
章节：9231 Further Mathematics
1 It is given that $I_n = \int_0^1 x^n \sinh x \, dx$, where $n$ is a non-negative integer.
(a) Show that, for $n \geqslant 2$,
$$I_n = \cosh 1 - n \sinh 1 + n(n-1)I_{n-2}.$$
[3]
(b) Find the exact value of $I_2$.2 (a) Find the exact values of $x$ for which $\cosh 2x = 6 \sinh^2 x$, giving your answers in logarithmic form. [4] (b) Sketch the curves $C_1: y = \cosh 2x$ and $C_2: y = 6 \sinh^2 x$ on the same diagram. [2]
3 (a) Use de Moivre’s theorem to show that $$\sin 5\theta = 16 \sin^5\theta - 20 \sin^3\theta + 5 \sin\theta.$$ [4]
(b) Hence obtain the roots of the equation
$$32x^5 - 40x^3 + 10x - \sqrt{3} = 0$$
in the form $\sin q\pi$, where $q$ is a rational number. [4]The diagram shows the curve with equation $y = \frac{1}{2}(3^x)$ for $0 \leq x \leq 1$, together with a set of $N$ rectangles each of width $\frac{1}{N}$.
(a) By considering the sum of the areas of these rectangles, show that $\int_0^{1} \frac{1}{2}(3^x)dx < U_N$, where
$$U_N = \frac{3^{\frac{1}{N}}}{N\left(3^{\frac{1}{N}} - 1\right)}.$$
[4](b) Use a similar method to find, in terms of $N$, a lower bound $L_N$ for $\int_0^1 \frac{1}{2}(3^x)dx$.
(c) By simplifying $U_N - L_N$, show that $\lim_{N \to \infty} (U_N - L_N) = 0$.5 (a) Find the general solution of the differential equation
$$\frac{d^2y}{dx^2} + \frac{dy}{dx} + y = \sin 2x + \cos 2x.$$
[7](b) For large positive values of $x$ and for any initial conditions, show that the solution to part (a) can be approximated by $$y \approx R \sin(2x - \phi),$$ where $R$ and $\phi$ are positive constants to be determined. [3]
6 Find the particular solution of the differential equation
$$\frac{dy}{dx} + \frac{1+x}{1-2x-x^2}y = 1$$
given that $y = \pi$ when $x = 0$. Give your answer in an exact form. [10]
© UCLES 2025 9231/24/O/N/2511 © UCLES 2025 9231/24/O/N/25 [Turn over]
The curve $C$ has parametric equations
$$x = 8 \ln(\tan \frac{1}{2} t) - 3 \cot t - 3t, \quad y = 6 \ln(\tan \frac{1}{2} t) + 4 \cot t + 4t, \quad \text{for } \frac{1}{6} \pi \leq t \leq \frac{1}{3} \pi.$$
(a) (i) Show that $\frac{dx}{dt} = 8 \cos c t + 3 \cot^2 t$. [1]
(ii) Find $\frac{dy}{dt}$. [1]
(iii) Find the exact value of the length of $C$. [6](b) Show that
$$\frac{d^2y}{dx^2} = \frac{a \cot t \cos c t (\cos c^2 t + 1)}{(b \cos c t + c \cot^2 t)^n},$$
where $a, b, c$ and $n$ are integers to be determined. [5]8 (a) Find the set of values of $a$ for which the system of equations
$$5x + ay = 3,$$
$$20x - 5y = 2,$$
$$2x - 3y + z = 1,$$
has a unique solution and interpret this situation geometrically. [3]
(b) Given that $a = -\frac{5}{4}$, show that the system of equations in part (a) is inconsistent and interpret this situation geometrically. [3]The matrix $A$ is given by
$$A = \begin{pmatrix} 5 & 0 & 0 \\ 20 & -5 & 0 \\ 2 & -3 & 1 \end{pmatrix}.$$
(c) Find a matrix $P$ and a diagonal matrix $D$ such that $(A + 3I)^2 = PDP^{-1}$. [7]