科目：进阶数学 Further Mathematics (9231)
年份：2023 冬季
试卷：Paper 22
总页数：14
章节：9231 Further Mathematics
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1. Find the Maclaurin’s series for $\ln(x+2) + \ln(x^2+5)$ up to and including the term in $x^2$. [5]
2 It is given that
$$x = 1 + \frac{1}{t} \quad \text{and} \quad y = te^t.$$
(a) Show that $\frac{dy}{dx} = -e^t(t^3 + t^2)$.
(b) Find $\frac{d^2y}{dx^2}$ in terms of $t$.3 (a) Use de Moivre’s theorem to show that
$$\cos 5\theta = 16 \cos^5\theta - 20 \cos^3\theta + 5 \cos\theta.$$
[4]
(b) Hence obtain the roots of the equation
$$32x^5 - 40x^3 + 10x - \sqrt{2} = 0$$
in the form $\cos(q\pi)$, where $q$ is a rational number.
[4]4 Find the solution of the differential equation
$$\frac{dy}{dx} + 3y = \sin x$$
for which $y = 1$ when $x = 0$. Give your answer in the form $y = f(x)$. [9]7 © UCLES 2023 9231/22/O/N/23 [Turn over]
The diagram shows part of the curve $y = x \text{sech}^2 x$ and its maximum point $M$.
(a) Show that, at $M$,
$$2x \tanh x - 1 = 0$$
and verify that this equation has a root between 0.7 and 0.8. [4](b) By considering a suitable set of rectangles, use the diagram to show that
$$\sum_{r=2}^{n} r \text{sech}^2 r < n \tanh n + \ln \text{sech} n - \tanh 1 - \ln \text{sech} 1.$$
[6]6 The matrix $P$ is given by
$$P = \begin{pmatrix} 1 & -1 & 1 \\ 0 & 2 & 1 \\ 0 & 0 & -1 \end{pmatrix}.$$
(a) State the eigenvalues of $P$. [1]
(b) Use the characteristic equation of $P$ to find $P^{-1}$. [4]The $3 \times 3$ matrix $A$ has distinct non-zero eigenvalues $a, \frac{1}{2}, 2$ with corresponding eigenvectors
$$\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \quad \begin{pmatrix} -1 \\ 2 \\ 0 \end{pmatrix}, \quad \begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix},$$
respectively.
(c) Find $A^{-1}$ in terms of $a$. [5]7 (a) Starting from the definitions of cosh and sinh in terms of exponentials, prove that
$$2 \sinh^2 A = \cosh 2A - 1.$$
[3]
(b) A curve has equation $y = x^2$, for $0 \leqslant x \leqslant \frac{2}{3}$. The area of the surface generated when the curve is rotated through $2\pi$ radians about the $x$-axis is denoted by $S$.
Use the substitution $x = \frac{1}{2} \sinh u$ to show that $S = \frac{1}{32} \pi \left( \frac{820}{81} - \ln 3 \right).$
[9]Turn over
8 It is given that $v = y^4$ and
$$y^3 \frac{d^2y}{dx^2} + 3y^2 \left( \frac{dy}{dx} \right)^2 + y^3 \frac{dy}{dx} + y^4 = e^{-2x}.$$
(a) Show that
$$\frac{d^2v}{dx^2} + \frac{dv}{dx} + 4v = 4e^{-2x}.$$
[4](b) Find $y$ in terms of $x$, given that, when $x = 0$, $y = 1$ and $\frac{dy}{dx} = -\frac{3}{8}$. [10]