科目：进阶数学 Further Mathematics (9231)
年份：2021 夏季
试卷：Paper 21
总页数：14
章节：9231 Further Mathematics
1. (a) Given that $a$ is an integer, show that the system of equations $$ax + 3y + z = 14,$$ $$2x + y + 3z = 0,$$ $$-x + 2y - 5z = 17,$$ has a unique solution and interpret this situation geometrically. (b) Find the value of $a$ for which $x = 1, y = 4, z = -2$ is the solution to the system of equations in part (a).
2 The variables $x$ and $y$ are related by the differential equation
$$\frac{d^2y}{dx^2} + 3 \frac{dy}{dx} + 2y = 2x + 1.$$
(a) Find the general solution for $y$ in terms of $x$.
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(b) State an approximate solution for large positive values of $x$. [1]The diagram shows the curve with equation $y = x^3$ for $0 \leqslant x \leqslant 1$, together with a set of $n$ rectangles of width $\frac{1}{n}$.
(a) By considering the sum of the areas of these rectangles, show that $\int_0^1 x^3 \mathrm{d}x < U_n$, where
$$U_n = \left( \frac{n+1}{2n} \right)^2.$$
[4](b) Use a similar method to find, in terms of $n$, a lower bound $L_n$ for $\int_0^1 x^3 \, dx$.
(c) Find the least value of $n$ such that $U_n - L_n < 10^{-3}$.4 Find the solution of the differential equation
$$\sin \theta \frac{dy}{d\theta} + y = \tan \frac{1}{2}\theta,$$
where $0 < \theta < \pi$, given that $y = 1$ when $\theta = \frac{1}{2}\pi$. Give your answer in the form $y = f(\theta)$.
[You may use without proof the result that $$\int \cosec \theta d\theta = \ln \tan \frac{1}{2}\theta.$$]7 © UCLES 2021 9231/21/M/J/21 [Turn over]
5 (a) State the sum of the series $z + z^2 + z^3 + \ldots + z^n$, for $z \neq 1$.
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(b) Given that $z$ is an $n$th root of unity and $z \neq 1$, deduce that $1 + z + z^2 + \ldots + z^{n-1} = 0$.
[2]
(c) Given instead that $z = \frac{1}{3}(\cos \theta + i \sin \theta)$, use de Moivre's theorem to show that
$$\sum_{m=1}^{\infty} 3^{-m} \cos m\theta = \frac{3 \cos \theta - 1}{10 - 6 \cos \theta}$$
[7]9 © UCLES 2021 9231/21/M/J/21 [Turn over]
6 The matrix $A$ is given by
$$A = \begin{pmatrix} 5 & -\frac{22}{3} & 8 \\ 0 & -6 & 0 \\ 0 & 0 & 1 \end{pmatrix}.$$
(a) Find a matrix $P$ and a diagonal matrix $D$ such that $A^2 = PDP^{-1}$. [7](b) Use the characteristic equation of A to find $A^3$. [4]
7 (a) It is given that $y = \text{sech}^{-1}\left(x + \frac{1}{2}\right)$.
Express $\cosh y$ in terms of $x$ and hence show that $\sinh y \frac{\mathrm{d}y}{\mathrm{d}x} = -\frac{1}{\left(x + \frac{1}{2}\right)^2}$. [3]
(b) Find the first three terms in the Maclaurin’s series for $\text{sech}^{-1}\left(x + \frac{1}{2}\right)$ in the form
$$\ln a + bx + cx^2,$$
where $a, b$ and $c$ are constants to be determined. [7]Turn over
8 The curve $C$ has parametric equations
$$x = 2 \cosh t, \quad y = \frac{3}{2} t - \frac{1}{4} \sinh 2t, \text{ for } 0 \leq t \leq 1.$$
(a) Find $\frac{dx}{dt}$ and show that $\frac{dy}{dt} = 1 - \sinh^2 t$.
[3]
The area of the surface generated when $C$ is rotated through $2\pi$ radians about the $x$-axis is denoted by $A$.
(b) (i) Show that $A = \pi \int_0^1 \left( \frac{3}{2} t - \frac{1}{4} \sinh 2t \right)(1 + \cosh 2t) dt$.(ii) Hence find $A$ in terms of $\pi$, $\sinh 2$ and $\cosh 2$. [6]