科目：进阶数学 Further Mathematics (9231)
年份：2024 夏季
试卷：Paper 23
总页数：18
章节：9231 Further Mathematics
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1. Find the exact value of $\int_{2}^{\frac{2}{2}} \frac{1}{\sqrt{4x-x^2-1}} dx$. [5]2 The curve $C$ has parametric equations
$$x = \cosh t, \quad y = \sinh t, \quad \text{for } 0 < t \leqslant \frac{3}{5}.$$
The length of $C$ is denoted by $s$.
(a) Show that $s = \int_{0}^{\frac{3}{5}} \sqrt{\cosh 2t} \mathrm{d}t.$ [4](b) By finding the Maclaurin’s series for $\sqrt{\cosh 2t}$ up to and including the term in $t^2$, deduce an approximation to $s$.
[5]3 The curve $C$ has equation
$$x^3 + 2xy + 8y^3 = -12.$$
(a) Show that, at the point $(-2, -1)$ on $C$, $\frac{dy}{dx} = -\frac{1}{2}$. [3](b) Find the value of $\frac{d^2y}{dx^2}$ at the point $(-2, -1)$. [5]The diagram shows the curve with equation $y = x^{-2}$ for $2 \leqslant x \leqslant N$ together with a set of $(N-2)$ rectangles of unit width.
(a) By considering the sum of the areas of these rectangles, show that
$$\sum_{r=1}^{N} \frac{1}{r^2} > \frac{3}{2} - \frac{1}{N} + \frac{1}{N^2}.$$
[5](b) Use a similar method to find, in terms of $N$, an upper bound for $\sum_{r=1}^{N} \frac{1}{r^2}$.
(c) Deduce lower and upper bounds for $\sum_{r=1}^{\infty} \frac{1}{r^2}$.5 (a) Find the general solution of the differential equation
$$\frac{d^2x}{dt^2} + 10 \frac{dx}{dt} + 25x = 338 \sin t.$$
[7](b) Show that, for large positive values of $t$ and for any initial conditions, $$x \approx R \sin(t - \phi),$$ where the constants $R$ and $\phi$ are to be determined. [3]
6 (a) Show that $\sum_{r=1}^{n} z^{4r} = \frac{z^{4n+2}-z^2}{z^2-z^{-2}}$, for $z^2 \neq z^{-2}$.
(b) By letting $z = \cos\theta + i\sin\theta$, show that, if $\sin 2\theta \neq 0$,
$$\sum_{r=1}^{n} \sin(4r\theta) = \frac{\cos 2\theta - \cos(4n+2)\theta}{2\sin 2\theta}.$$13 © UCLES 2024 9231/23/M/J/24 Turn over
7 (a) Show that
$$\frac{d}{dx}\left(\frac{x}{2}\sqrt{x^2-9} - \frac{9}{2}\cosh^{-1}\frac{x}{3}\right) = \sqrt{x^2-9}.$$
[3]
(b) Find the solution of the differential equation
$$x\frac{dy}{dx} - y = x^2\sqrt{x^2-9},$$
given that $y = 1$ when $x = 3$. Give your answer in the form $y = f(x)$.
[9]15
8 The planes $\Pi_1$ and $\Pi_2$ do not intersect and are both perpendicular to $\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}$. The line $l$ intersects $\Pi_1$ at the point $(1, 6, 0)$ and intersects $\Pi_2$ at the point $(3, -6, 0)$.
(a) Find Cartesian equations of $\Pi_1$ and $\Pi_2$.
[3]
(b) Express the vector equation of $l$ in the form $\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \mathbf{a} + \lambda \mathbf{b}$, where $\mathbf{a}$ and $\mathbf{b}$ are vectors to be determined, and hence show that for points on $l$, $\frac{1}{2}x + \frac{1}{12}y = 1$ and $z = 0$. [2]The matrix $A$ is given by
$$A = \begin{pmatrix}
1 & 2 & 3 \\
1 & 2 & 3 \\
\frac{1}{2} & \frac{1}{12} & 0
\end{pmatrix}.$$
(c) Show that the characteristic equation of $A$ is $-\lambda^3 + 3\lambda^2 + \frac{7}{4}\lambda = 0$ and hence find the eigenvalues of $A$. [3](d) Find a matrix $P$ and a diagonal matrix $D$ such that $A^n = PDP^{-1}$, where $n$ is a positive integer. [6]Additional page If you use the following page to complete the answer to any question, the question number must be clearly shown.