科目：进阶数学 Further Mathematics (9231)
年份：2021 夏季
试卷：Paper 23
总页数：14
章节：9231 Further Mathematics
1. (a) Find $a$ and $b$ such that
$$z^8 - iz^5 - z^3 + i = (z^5 - a)(z^3 - b).$$
[1]
(b) Hence find the roots of
$$z^8 - iz^5 - z^3 + i = 0,$$
giving your answers in the form $re^{i\theta}$, where $r > 0$ and $0 \leqslant \theta < 2\pi$. [6]2 Find the Maclaurin’s series for $\ln \cosh x$ up to and including the term in $x^4$. [7]
The diagram shows the curve $y = \frac{x}{2x^2 - 1}$ for $x \geqslant 1$, together with a set of $N-1$ rectangles of unit width.
(a) By considering the sum of the areas of these rectangles, show that
$$\sum_{r=1}^{N} \frac{r}{2r^2 - 1} < \frac{1}{4} \ln(2N^2 - 1) + 1.$$
[7](b) Use a similar method to find, in terms of $N$, a lower bound for $\sum_{r=1}^{N} \frac{r}{2r^2-1}$. [3]4 By considering the binomial expansions of $\left(z + \frac{1}{z}\right)^5$ and $\left(z - \frac{1}{z}\right)^5$, where $z = \cos\theta + i\sin\theta$, use de Moivre’s theorem to show that
$$\tan^5\theta = \frac{\sin 5\theta - a\sin 3\theta + b\sin \theta}{\cos 5\theta + a\cos 3\theta + b\cos \theta},$$
where $a$ and $b$ are integers to be determined. [7]7 © UCLES 2021 9231/23/M/J/21 [Turn over]
5 The variables $x$ and $y$ are related by the differential equation
$$\frac{d^2 y}{dx^2} - 2 \frac{dy}{dx} - 3y = 4e^{-x}.$$
(a) Find the value of the constant $k$ such that $y = kxe^{-x}$ is a particular integral of the differential equation.
(b) Find the solution of the differential equation for which $y = \frac{dy}{dx} = \frac{1}{2}$ when $x = 0$.9 © UCLES 2021 9231/23/M/J/21 [Turn over]
6 (a) Starting from the definitions of sinh and cosh in terms of exponentials, prove that
$$2 \sinh^2 x = \cosh 2x - 1.$$
[3]
(b) Find the solution to the differential equation
$$\frac{dy}{dx} + y \coth x = 4 \sinh x$$
for which $y = 1$ when $x = \ln 3.$
[7]11 © UCLES 2021 9231/23/M/J/21 [Turn over]
7 The integral $I_n$, where $n$ is an integer, is defined by $I_n = \int_0^{\frac{3}{2}} \left(4 + x^2\right)^{-\frac{1}{2}} n \, dx$.
(a) Find the exact value of $I_1$, expressing your answer in logarithmic form. [3]
(b) By considering $\frac{d}{dx} \left(x \left(4 + x^2\right)^{-\frac{1}{2}}\right)$, or otherwise, show that
$$4n I_{n+2} = \frac{3}{2} \left(\frac{2}{5}\right)^n + (n-1) I_n.$$ [5](c) Find the value of $I_5$. [3]
8 (a) Find the value of $a$ for which the system of equations
$$
\begin{align*}
13x + 18y - 28z &= 0, \\
-4x - ay + 8z &= 0, \\
2x + 6y - 5z &= 0,
\end{align*}
$$
does not have a unique solution. [2]
The matrix $A$ is given by
$$
A = \begin{pmatrix}
13 & 18 & -28 \\
-4 & -1 & 8 \\
2 & 6 & -5
\end{pmatrix}.
$$
(b) Find the eigenvalue of $A$ corresponding to the eigenvector $\begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix}$. [1]
(c) Find a matrix $P$ and a diagonal matrix $D$ such that $A = PDP^{-1}$. [8](d) Use the characteristic equation of $A$ to find $A^{-1}$ in terms of $A$. [2]