科目：进阶数学 Further Mathematics (9231)
年份：2025 冬季
试卷：Paper 22
总页数：14
章节：9231 Further Mathematics
1 The curve $C$ has polar equation $r = e^{\frac{4\theta}{3}}$ for $0 \leqslant \theta \leqslant \pi$.
Find the length of $C$. Give your answer in an exact form. [4]2. Find the roots of the equation $z^4 = 8 - 8i\sqrt{3}$, giving your answers in the form $re^{i\theta}$, where $r > 0$ and $-\pi \leq \theta < \pi$. [5]3 The variables $x$ and $y$ are such that $y = 2$ when $x = -1$ and
$$x^2 + (x + y)^3 = 3.$$
(a) Show that $\frac{dy}{dx} = \frac{1}{4}$ when $x = -1$. [3](b) Find the value of $\frac{d^2y}{dx^2}$ when $x = -1$. [6]4 (a) By considering the binomial expansion of $\left(z + \frac{1}{z}\right)^6$, where $z = \cos\theta + i\sin\theta$, use de Moivre’s theorem to show that
$$\cos^6\theta = a\cos 6\theta + b\cos 4\theta + c\cos 2\theta + d,$$
where $a, b, c$ and $d$ are constants to be determined. [5](b) Find the exact value of $\int_{0}^{\frac{1}{2}\pi} \cos^6 2x \, dx$. [3]5 (a) Find the general solution of the differential equation
$$\frac{d^2y}{dx^2} + 4 \frac{dy}{dx} + 3y = 5 \cos x.$$
[7](b) For large positive values of $x$ and for any initial conditions, show that the solution to part (a) can be approximated by $$y \approx R \sin(x + \phi),$$ where the constants $R$ and $\phi$ are to be determined. [3]
6 (a) Use the substitution $x = \frac{1}{2}\sqrt{2} \sinh u$ to find $\int \frac{1}{\sqrt{2x^2 + 1}} \, dx$.
The diagram shows the curve with equation $y = \frac{1}{\sqrt{2x^2 + 1}}$ for $0 \leq x \leq 1$, together with a set of $n$ rectangles of width $\frac{1}{n}$.
(b) By considering the sum of the areas of these rectangles, show that
$$\sum_{r=1}^{n} \frac{1}{\sqrt{2r^2 + n^2}} < \frac{1}{2} \sqrt{2} \ln(\sqrt{2} + \sqrt{3}).$$(c) Use a similar method to find, in terms of $n$, a lower bound for $\sum_{r=1}^{n} \frac{1}{\sqrt{2r^2+n^2}}$.
(d) Deduce the exact value of $\lim_{n \to \infty} \sum_{r=1}^{n} \frac{1}{\sqrt{2r^2+n^2}}$.7 (a) Show that $\frac{d}{dx}\left(\frac{1}{2}x\sqrt{4-x^2}+2\sin^{-1}\left(\frac{1}{2}x\right)\right)=\sqrt{4-x^2}$.
(b) Find the solution of the differential equation
$$2\frac{dy}{dx}+\frac{y}{2+x}=2\sqrt{2-x}$$
for which $y=\frac{1}{2}$ when $x=1$. Give your answer in an exact form. [8]13 © UCLES 2025 9231/22/O/N/25 [Turn over]
8 (a) Find the values of $k$ for which the system of equations
$$x - y + 2kz = 1,$$
$$kx + y + 2z = 2,$$
$$2x - y + z = 3,$$
does not have a unique solution. [3]
(b) Given that $k = -\frac{1}{2}$, show that the system of equations in part (a) is inconsistent. Interpret this situation geometrically. [3]
(c) Given instead that $k = -1$, show that the system of equations in part (a) is also inconsistent. Interpret this situation geometrically. [4]The matrix $A$ is given by
$$A = \begin{pmatrix}
1 & -1 & -2 \\
-1 & 1 & 2 \\
2 & -1 & 1
\end{pmatrix}.$$
(d) Use the characteristic equation of $A$ to show that $A^4 = pA^2 + qA$ where $p$ and $q$ are integers to be determined.
[5]