科目：进阶数学 Further Mathematics (9231)
年份：2020 冬季
试卷：Paper 21
总页数：14
章节：9231 Further Mathematics
1 (a) By differentiating $e^{-x^2}$, find the Maclaurin’s series for $e^{-x^2}$ up to and including the term in $x^2$.
(b) Deduce an approximation to $\int_0^{\frac{1}{2}} e^{-x^2} dx$, giving your answer as a rational fraction in its lowest terms.2 The variables $x$ and $y$ are related by the differential equation
$$9 \frac{d^2 y}{dx^2} + 6 \frac{dy}{dx} + y = 3x^2 + 30x.$$
(a) Find the general solution for $y$ in terms of $x$. [6]
(b) State an approximate solution for large positive values of $x$. [1]3 (a) Show that the system of equations $$x - 2y - 4z = 1,$$ $$x - 2y + kz = 1,$$ $$-x + 2y + 2z = 1,$$ where $k$ is a constant, does not have a unique solution. [2] (b) Given that $k = -4$, show that the system of equations in part (a) is consistent. Interpret this situation geometrically. [3]
(c) Given instead that $k = -2$, show that the system of equations in part (a) is inconsistent. Interpret this situation geometrically. (d) For the case where $k \neq -2$ and $k \neq -4$, show that the system of equations in part (a) is inconsistent. Interpret this situation geometrically.
The diagram shows the curve with equation $y = 1 - x^3$ for $0 \leqslant x \leqslant 1$, together with a set of $n$ rectangles of width $\frac{1}{n}$.
(a) By considering the sum of the areas of the rectangles, show that
$$\int_0^1 (1 - x^3) dx \leqslant \frac{3n^2 + 2n - 1}{4n^2}.$$
[4](b) Use a similar method to find, in terms of $n$, a lower bound for $\int_0^1 (1-x^3)dx$. [4]
5 It is given that
$$x = \sinh^{-1} t, \quad y = \cos^{-1} t,$$
where $-1 < t < 1$.
(a) By differentiating $\cos y$ with respect to $t$, show that $$\frac{dy}{dt} = -\frac{1}{\sqrt{1-t^2}}.$$ [4](b) Find $\frac{d^2y}{dx^2}$ in terms of $t$, simplifying your answer. [5]6 (a) Use de Moivre’s theorem to show that $\sin^4\theta = \frac{1}{8}(\cos 4\theta - 4\cos 2\theta + 3)$. [5](b) Find the solution of the differential equation
$$\frac{dy}{d\theta} + y \cot \theta = \sin^3 \theta$$
for which $y = 0$ when $\theta = \frac{1}{2}\pi$. [6]7 The matrix $P$ is given by
$$P = \begin{pmatrix} 1 & 4 & 2 \\ 0 & -1 & 1 \\ 0 & 0 & 2 \end{pmatrix}.$$
(a) State the eigenvalues of $P$. [1]
(b) Use the characteristic equation of $P$ to find $P^{-1}$. [4]The $3 \times 3$ matrix $\mathbf{A}$ has distinct eigenvalues $b, -1, 1$ with corresponding eigenvectors
$$\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \quad \begin{pmatrix} 4 \\ -1 \\ 0 \end{pmatrix}, \quad \begin{pmatrix} 2 \\ 1 \\ 2 \end{pmatrix},$$
respectively.
(c) Find $\mathbf{A}$ in terms of $b$. [4]8 (a) Sketch the graph of $y = \coth x$ for $x > 0$ and state the equations of the asymptotes. [2] (b) Starting from the definitions of $\coth x$ and $\cosech x$ in terms of exponentials, prove that $$\coth^2 x - \cosech^2 x = 1.$$ [3]
The curve $C$ has equation $y = \ln \coth\left(\frac{1}{2}x\right)$ for $x > 0$.
(c) Show that $\frac{\mathrm{d}y}{\mathrm{d}x} = -\text{cosech} x$. [3]
(d) It is given that the arc length of $C$ from $x = a$ to $x = 2a$ is $\ln 4$, where $a$ is a positive constant. Show that $\cosh a = 2$ and find, in logarithmic form, the exact value of $a$. [7]