科目：进阶数学 Further Mathematics (9231)
年份：2026 夏季
试卷：Paper 24
总页数：14
章节：9231 Further Mathematics
1. (a) Find the set of values of $k$ for which the system of equations
$$3x + ky + 7z = 1,$$
$$8x + y + 2z = 2,$$
$$8x + 2y + 4z = 3,$$
has a unique solution and interpret this situation geometrically. [4]
(b) Given that $k = \frac{7}{2}$, show that the system of equations in part 1(a) is inconsistent and interpret this situation geometrically. [2]2 Let $A = \begin{pmatrix} a & -1 \\ 5 & 1 \end{pmatrix}$, where $a$ is a constant. It is given that 2 is an eigenvalue of $A$.
(a) Find $a$ and the other eigenvalue of $A$. [4](b) Find a matrix $P$ and a diagonal matrix $D$ such that $A^5 = PDP^{-1}$.
[5]3 Find the particular solution of the differential equation
$$\frac{dy}{dx} + y = e^{\frac{1}{2}x},$$
given that $y = 3$ when $x = 0$. Give your answer in the form $y = f(x)$. [7]
© Cambridge University Press & Assessment 2026 9231/24/M/J/26 [Turn over]4 The curve $C$ has parametric equations
$$x = 2 \cosh 2t + 3t, \quad y = \frac{3}{2} \cosh 2t - 4t, \quad \text{for } 0 \leq t \leq 1.$$
(a) Find, in terms of $e$, the length of $C$. [5]
$$\text{length of } C = \sqrt{(2 \cosh 2t + 3t)^2 + (\frac{3}{2} \cosh 2t - 4t)^2}$$(b) Show that
$$\frac{d^2y}{dx^2} = \frac{a \cosh 2t}{(b \sinh 2t + c)^n},$$
where $a, b, c$ and $n$ are integers to be determined. [5]The diagram shows the curve $y = \frac{1}{\sqrt{4x^2 + 4x + 3}}$ for $x \geqslant 0$, together with a set of $n$ rectangles of unit width. By considering the sum of the areas of these rectangles, show that
$$\sum_{r=1}^{n} \frac{1}{\sqrt{4r^2 + 4r + 3}} < \frac{1}{2} \ln \left( \frac{1 + 2n + \sqrt{4n^2 + 4n + 3}}{1 + \sqrt{3}} \right).$$
[9]9 © Cambridge University Press & Assessment 2026 9231/24/M/J/26 Turn over
6 Let $I_n = \int_{-\frac{1}{2}}^{\frac{1}{2}}\mathrm{e}^x\cos^n x\mathrm{d}x$, where $n$ is a non-negative integer.
(a) Show that, for $n \geqslant 2$,
$$I_n = n(n-1)\int_{-\frac{1}{2}}^{\frac{1}{2}}\mathrm{e}^x\cos^{n-2}x\sin^2x\mathrm{d}x - nI_n,$$
and deduce the reduction formula
$$(n^2+1)I_n = n(n-1)I_{n-2}.$$
[6](b) Find the exact value of $I_4$. [4]
7 (a) Use de Moivre’s theorem to show that
$$\sec 7\theta = \frac{\sec^7\theta}{64 - 112\sec^2\theta + 56\sec^4\theta - 7\sec^6\theta}.$$
[6](b) Hence obtain the roots of the equation $$x^7 - 14x^6 + 112x^4 - 224x^2 + 128 = 0$$ in the form sec $q\pi$, where $q$ is rational. [5]
8 It is given that $v = xy^3$ and
$$xy^2 \frac{d^2y}{dx^2} + 2xy \left( \frac{dy}{dx} \right)^2 + (3x+2)y^2 \frac{dy}{dx} + (x+1)y^3 = x+5.$$
(a) Show that
$$\frac{1}{3} \frac{d^2v}{dx^2} + \frac{dv}{dx} + v = x+5.$$
[4]
(b) Find the general solution for $y$ in terms of $x$. [7](c) Deduce that, for large positive values of $x$ and for any initial conditions,
$$y \approx \sqrt[3]{a + \frac{b}{x}}.$$
where the constants $a$ and $b$ are to be determined. [2]