科目：进阶数学 Further Mathematics (9231)
年份：2022 冬季
试卷：Paper 22
总页数：14
章节：9231 Further Mathematics
1. (a) Find the set of values of $k$ for which the system of equations $$x + 2y + 3z = 1,$$ $$kx + 4y + 6z = 0,$$ $$7x + 8y + 9z = 3,$$ has a unique solution. [3] (b) Interpret the situation geometrically in the case where the system of equations does not have a unique solution. [2]
2 A curve has equation
$(x+1)y + y^2 = 2$.
(a) Show that $\frac{dy}{dx} = -\frac{2}{3}$ at the point $(0, -2)$. [3]
(b) Find the value of $\frac{d^2y}{dx^2}$ at the point $(0, -2)$. [4]3 (a) A curve has equation $y = e^x + \frac{1}{4}e^{-x}$, for $0 \leqslant x \leqslant 1$. 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. [6](b) Using standard results from the list of formulae (MF19), or otherwise, find the Maclaurin’s series for $e^x + \frac{1}{4}e^{-x}$ up to and including the term in $x^2$. [2]4 Find the solution of the differential equation
$(4t^2 - 1)\frac{dx}{dt} + 4x = 4t^2 - 1$
for which $x = 3$ when $t = 1$. Give your answer in the form $x = f(t)$. [9]7 © UCLES 2022 9231/22/O/N/22 [Turn over]
5 (a) Write down the fourth roots of unity. [1] (b) Use de Moivre’s theorem to show that $$\cos 4\theta = 8\cos^4\theta - 8\cos^2\theta + 1.$$ [4]
(c) Hence obtain the real roots of the equation $$16(8x^4 - 8x^2 + 1)^4 - 9 = 0$$ in the form $\cos(q\pi)$, where $q$ is a rational number. [5]
The diagram shows the curve $y = \frac{1}{\sqrt{x^2 + 2x}}$ for $x > 0$, together with a set of $(n-1)$ rectangles of unit width.
By considering the sum of the areas of these rectangles, show that
$$\sum_{r=1}^{n} \frac{1}{\sqrt{r^2 + 2r}} < \ln\left(n+1 + \sqrt{n^2 + 2n}\right) + \frac{1}{3}\sqrt{3} - \ln(2 + \sqrt{3}).$$
[10]11 © UCLES 2022 9231/22/O/N/22 [Turn over]
7 (a) It is given that $\lambda$ is an eigenvalue of the non-singular square matrix $A$, with corresponding eigenvector $e$.
Show that $\lambda^{-1}$ is an eigenvalue of $A^{-1}$ for which $e$ is a corresponding eigenvector. [2]
The matrix $A$ is given by
$$A = \begin{pmatrix}
2 & 0 & 3 \\
15 & -4 & 3 \\
3 & 0 & 2
\end{pmatrix}.$$
(b) Given that $-1$ is an eigenvalue of $A$, find a corresponding eigenvector. [2]
(c) It is also given that $\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}$ are eigenvectors of $A$. Find the corresponding eigenvalues. [2](d) Hence find a matrix $P$ and a diagonal matrix $D$ such that $A^{-1} = PDP^{-1}$.
(e) Use the characteristic equation of $A$ to show that $A^{-1} = pA^2 + qI$, where $p$ and $q$ are rational numbers to be determined.8 It is given that $y = \cosh u$, where $u > 0$, and
$$\sqrt{\cosh^2 u - 1} \left( \frac{d^2 u}{dx^2} + \frac{du}{dx} \right) + \cosh u \left( \frac{du}{dx} \right)^2 - 2 \cosh u = 4e^{-x}.$$
(a) Show that
$$\frac{d^2 y}{dx^2} + \frac{dy}{dx} - 2y = 4e^{-x}.$$
[4]
(b) Find $u$ in terms of $x$, given that, when $x = 0, u = \ln 3$ and $\frac{du}{dx} = 3$. [10]15 © UCLES 2022 9231/22/O/N/22