科目：进阶数学 Further Mathematics (9231)
年份：2023 夏季
试卷：Paper 21
总页数：14
章节：9231 Further Mathematics
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1. (a) Show that the system of equations $$x + 2y + 3z = 1,$$ $$4x + 5y + 6z = 1,$$ $$7x + 8y + 9z = 1,$$ does not have a unique solution. [2] (b) Show that the system of equations in part (a) is consistent. Interpret this situation geometrically. [3]
2 Use the substitution $z = x + y$ to find the solution of the differential equation
$$\frac{dy}{dx} = \frac{1 + 3x + 3y}{3x + 3y - 1}$$
for which $y = 0$ when $x = 1$. Give your answer in the form $a \ln(x+y) + b(x-y) + c = 0$, where $a, b$ and $c$ are constants to be determined. [7]3 (a) By considering the binomial expansion of $(z + z^{-1})^4$, where $z = \cos \theta + i \sin \theta$, use de Moivre's theorem to show that $\cos^4 \theta = \frac{1}{8}(\cos 4\theta + 4\cos 2\theta + 3)$. [5]
(b) Use the substitution $x = \sin \theta$ to find the exact value of $\int_0^{\frac{\pi}{2}} (1 - x^2)^{\frac{3}{2}} dx$. [3]4 The integral $I_n$ is defined by $I_n = \int_0^1 (1+x^5)^n \, dx$.
(a) By considering $\frac{d}{dx}\left(x(1+x^5)^n\right)$, or otherwise, show that
$(5n+1)I_n = 2^n + 5nI_{n-1}$. [5](b) Find the exact value of $I_3$. [4]
5 The matrix $A$ is given by
$$A = \begin{pmatrix} 18 & 5 & -11 \\ 8 & 6 & -4 \\ 32 & 10 & -20 \end{pmatrix}.$$
(a) Show that the characteristic equation of $A$ is $\lambda^3 - 4\lambda^2 - 20\lambda + 48 = 0$ and hence find the eigenvalues of $A$. [4](b) Find a matrix $P$ and a diagonal matrix $D$ such that $A^5 = PDP^{-1}$. [6]6 Find the particular solution of the differential equation
$$\frac{d^2x}{dt^2} - 12 \frac{dx}{dt} + 36x = 37 \sin t,$$
given that, when $t = 0$, $x = \frac{dx}{dt} = 0$. [11]11 © UCLES 2023 9231/21/M/J/23 [Turn over]
7 (a) Use the substitution $u = x^2 - 1$ to find $\int \frac{x}{\sqrt{x^2 - 1}} dx$.
The diagram shows the curve with equation $y = \cosh^{-1}x$ together with a set of $(N-1)$ rectangles of unit width.
(b) By considering the sum of the areas of these rectangles, show that
$$\sum_{r=2}^{N} \ln(r + \sqrt{r^2 - 1}) > N \ln(N + \sqrt{N^2 - 1}) - \sqrt{N^2 - 1}$$(c) Use a similar method to find, in terms of $N$, an upper bound for $\sum_{r=2}^{N} \ln\left(r + \sqrt{r^2 - 1}\right)$. [3]8 (a) Starting from the definitions of sech and tanh in terms of exponentials, prove that
$$1 - \text{sech}^2 t = \tanh^2 t.$$
[3]
The curve $C$ has parametric equations
$$x = \frac{1}{2} \tanh^2 t + \ln \text{sech} t, \quad y = 1 + \tanh^4 t, \quad \text{for } t > 0.$$
(b) Show that $\frac{dy}{dx} = -4 \text{sech}^2 t.$
[5](c) Find the coordinates of the point on $C$ with $\frac{d^2y}{dx^2} = -\frac{9}{2}$, giving your answer in the form $(a + \ln b, c)$ where $a, b$ and $c$ are rational numbers. [6]