科目：进阶数学 Further Mathematics (9231)
年份：2022 夏季
试卷：Paper 22
总页数：14
章节：9231 Further Mathematics
1 The curve $C$ has polar equation $r = e^{\frac{3}{\theta}}$ for $0 \leqslant \theta \leqslant \alpha$.
Given that the length of $C$ is $s$, find $\alpha$ in terms of $s$. [5]2 (a) Starting from the definitions of cosh and sinh in terms of exponentials, prove that $$\cosh 2x = 2 \sinh^2 x + 1.$$ [3] (b) Find the set of values of $k$ for which $\cosh 2x = k \sinh x$ has two distinct real roots. [5]
3 The variables $t$ and $x$ are related by the differential equation
$$\frac{d^2x}{dt^2} + \frac{dx}{dt} + x = t^2 + 1.$$
(a) Find the general solution for $x$ in terms of $t$. [6](b) Deduce an approximate value of $\frac{d^2x}{dt^2}$ for large positive values of $t$. [2]4 The diagram shows the curve with equation $y = 2^x$ for $0 \leq x \leq 1$, together with a set of $N$ rectangles each of width $\frac{1}{N}$.
(a) By considering the sum of the areas of these rectangles, show that $\int_0^1 2^x \mathrm{d}x < U_N$, where
$$U_N = \frac{2^{\frac{1}{N}}}{N\left(2^{\frac{1}{N}} - 1\right)}.$$
[4](b) Use a similar method to find, in terms of $N$, a lower bound $L_N$ for $\int_0^1 2^x \mathrm{d}x$. [4]
(c) Find the least value of $N$ such that $U_N - L_N < 10^{-4}$. [2]5 The variables $x$ and $y$ are such that $y = 0$ when $x = 0$ and
$(x + 1)y + (x + y + 1)^3 = 1.$
(a) Show that $\frac{dy}{dx} = -\frac{3}{4}$ when $x = 0$. [3]
(b) Find the Maclaurin’s series for $y$ up to and including the term in $x^2$. [7]9 © UCLES 2022 9231/22/M/J/22 [Turn over]
6 Use the substitution $y = vx$ to find the solution of the differential equation
$$x \frac{dy}{dx} = y + \sqrt{9x^2 + y^2}$$
for which $y = 0$ when $x = 1$. Give your answer in the form $y = f(x)$, where $f(x)$ is a polynomial in $x$.
[10]11 © UCLES 2022 9231/22/M/J/22 [Turn over]
7 (a) Use de Moivre’s theorem to show that
$$\text{cosec } 7\theta = \frac{\text{cosec}^7\theta}{7 \text{cosec}^6\theta - 56 \text{cosec}^4\theta + 112 \text{cosec}^2\theta - 64}.$$
[6](b) Hence obtain the roots of the equation $$x^7 - 14x^6 + 112x^4 - 224x^2 + 128 = 0$$ in the form $\cos c\pi$, where $q$ is rational. [5]
8 (a) Find the value of $a$ for which the system of equations
$$3x + ay = 0,$$
$$5x - y = 0,$$
$$x + 3y + 2z = 0,$$
does not have a unique solution. [2]
The matrix $A$ is given by
$$A = \begin{pmatrix} 3 & 0 & 0 \\ 5 & -1 & 0 \\ 1 & 3 & 2 \end{pmatrix}.$$
(b) Find a matrix $P$ and a diagonal matrix $D$ such that $A^2 = PDP^{-1}$. [7](c) Use the characteristic equation of $A$ to show that
$$(\mathbf{A} + 6\mathbf{I})^2 = \mathbf{A}^4(\mathbf{A} + b\mathbf{I})^2,$$
where $b$ is an integer to be determined. [4]