科目：进阶数学 Further Mathematics (9231)
年份：2020 冬季
试卷：Paper 22
总页数：14
章节：9231 Further Mathematics
1. Find the Maclaurin’s series for $\tan\left(x + \frac{1}{4}\pi\right)$ up to and including the term in $x^2$.
2. A curve has equation $y = \cosh x$, for $0 \leqslant x \leqslant \frac{1}{2}$.
Find, in terms of $\pi$ and $e$, the area of the surface generated when the curve is rotated through $2\pi$ radians about the $x$-axis.3 Find all the roots of the equation $(w+1)^6 = 1$, giving your answers in the form $x+iy$ where $x$ and $y$ are real and exact. [4]
4 Find the solution of the differential equation
$$x \frac{dy}{dx} + 2y = e^x$$
for which $y = 3$ when $x = 1$. Give your answer in the form $y = f(x)$. [8]5 The curve $C$ has equation
$$y^2 + (xy + 1)^2 = 5.$$
(a) Show that, at the point $(1, 1)$ on $C$, $\frac{dy}{dx} = -\frac{2}{3}$.
(b) Find the value of $\frac{d^2 y}{dx^2}$ at the point $(1, 1)$.
[Turn over]6 Find the particular solution of the differential equation
$$\frac{d^2x}{dt^2} + 8 \frac{dx}{dt} + 15x = 102 \cos 3t,$$
given that, when $t=0$, $x=1$ and $\frac{dx}{dt}=0$. [11]7 © UCLES 2020 9231/22/O/N/20 [Turn over]
7 (a) Show that $\sum_{r=1}^{n} z^{2r} = \frac{z^{2n+1}-z}{z-z^{-1}}$, for $z \neq 0, 1, -1$. [2](b) By letting $z = \cos\theta + i\sin\theta$, show that, if $\sin\theta \neq 0$,
$$1 + 2\sum_{r=1}^{n} \cos(2r\theta) = \frac{\sin(2n+1)\theta}{\sin\theta}.$$
[5]The diagram shows the curve $y = \frac{1}{\sqrt{x^2 + x + 1}}$ for $x \geqslant 0$, together with a set of $n$ rectangles of unit width. By considering the sum of the areas of these rectangles, show that
$$\sum_{r=1}^{n} \frac{1}{\sqrt{r^2 + r + 1}} < \ln\left(\frac{1}{3} + \frac{2}{3}n + \frac{2}{3}\sqrt{n^2 + n + 1}\right).$$
[10]11 © UCLES 2020 9231/22/O/N/20 [Turn over]
9 It is given that $a$ is a positive constant. (a) Show that the system of equations $$ax + (2a+5)y + (a+1)z = 1,$$ $$-4y = 2,$$ $$3y - z = 3,$$ has a unique solution and interpret this situation geometrically. [3]
The matrix $A$ is given by
$$A = \begin{pmatrix} a & 2a+5 & a+1 \\ 0 & -4 & 0 \\ 0 & 3 & -1 \end{pmatrix}.$$
(b) Show that the eigenvalues of $A$ are $a, -1$ and $-4$. [2]
(c) Find a matrix $P$ such that
$$A = P \begin{pmatrix} a & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -4 \end{pmatrix} P^{-1}.$$ [5]© UCLES 2020 9231/22/O/N/20
(d) Use the characteristic equation of $A$ to find $A^{-1}$. [6]