科目：进阶数学 Further Mathematics (9231)
年份：2025 夏季
试卷：Paper 24
总页数：18
章节：9231 Further Mathematics
1. (a) Find the values of $k$ for which the system of equations $$x + 2y + 3z = 1,$$ $$kx + 5y + 6z = 2,$$ $$7x + 2ky + 9z = 3,$$ does not have a unique solution. (b) Given that $k = 1$, show that the system of equations in part (a) is consistent. Interpret this situation geometrically.
2. Find the exact value of $\int_{1}^{\frac{5}{2}} \frac{1}{\sqrt{x^2 - 2x + 5}} dx$, giving your answer in logarithmic form. [6]3 Find the particular solution of the differential equation
$$\frac{d^2y}{dx^2} + 4\frac{dy}{dx} + 5y = 13e^{3x}$$
given that $y = 1$ and $\frac{dy}{dx} = 0$ when $x = 0$. [10]5 © UCLES 2025 9231/24/M/J/25 [Turn over]
4 A curve has parametric equations
$$x = t^3 - t^2 + t - 1 \quad \text{and} \quad y = te^t.$$
(a) Show that 1 is the only real value of $t$ for which $x = 0$. [1]
(b) Show that $$\frac{dy}{dx} = \frac{(t+1)e^t}{3t^2 - 2t + 1}.$$ [3](c) Find the Maclaurin’s series for $y$ up to and including the term in $x^2$. [6]
5 (a) Use de Moivre’s theorem to show that $$\sin 7\theta = -64 \sin^7\theta + 112 \sin^5\theta - 56 \sin^3\theta + 7 \sin\theta.$$ [5]
(b) Hence find all roots of the equation $$64x^6 - 112x^4 + 56x^2 - 7 = 0$$ in the form $\sin q\pi$, where $q$ is a rational number. [3]
6 Find the solution of the differential equation
$$x \frac{dy}{dx} - y = 2x^2 \tan^{-1} x$$
for which $y = \frac{1}{2}\pi$ when $x = 1$. Give your answer in the form $y = f(x)$. [9]11 © UCLES 2025 9231/24/M/J/25 [Turn over]
7 The matrix $A$ is given by
$$A = \begin{pmatrix} 1 & 7 & 11 \\ 0 & 2 & 5 \\ 0 & 0 & -3 \end{pmatrix}.$$
(a) Find a matrix $P$ and a diagonal matrix $D$ such that $A^6 = PDP^{-1}$. [7](b) Use the characteristic equation of $A$ to show that $$A^6 = aA^2 + bA + cI,$$ where $a, b$ and $c$ are integers to be determined. [4]
8 The curve $C$ has equation $y = \tanh x$ for $x \geqslant 0$.
(a) Sketch $C$ and state the equation of the asymptote. [2]
(b) By considering a suitable set of $N$ rectangles of unit width, use your sketch to show that
$$\sum_{r=1}^{N} \tanh r > \ln(\cosh N).$$ [3](c) The arc of $C$ joining the point where $x = 0$ to the point where $x = \frac{1}{2} \ln 3$ is rotated through one complete revolution about the $x$-axis. The area of the surface generated is denoted by $S$.
(i) Use the substitution $u = \sqrt{1 + \text{sech}^4 x}$ to show that
$$S = \pi \int_{\frac{5}{4}}^{\sqrt{2}} \frac{u^2}{u^2 - 1} \, \mathrm{d}u.$$
[7](ii) Find the exact value of $\pi \int_{\frac{5}{4}}^{\sqrt{2}} \frac{u^2}{u^2 - 1} du$. You need not simplify your answer. [3]Additional page If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
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19 BLANK PAGE © UCLES 2025 9231/24/M/J/25