科目：进阶数学 Further Mathematics (9231)
年份：2020 夏季
试卷：Paper 21
总页数：14
章节：9231 Further Mathematics
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1. Find the solution of the differential equation
$$\frac{dy}{dx} + 5y = e^{-7x}$$
for which $y = 0$ when $x = 0$. Give your answer in the form $y = f(x)$. [6]2 It is given that $y = 2^x$.
(a) By differentiating $\ln y$ with respect to $x$, show that $\frac{dy}{dx} = 2^x \ln 2$. [3]
(b) Write down $\frac{d^2 y}{dx^2}$. [1]
(c) Hence find the first three terms in the Maclaurin’s series for $2^x$. [3]3 (a) Find the roots of the equation $z^3 = -1 - i$, giving your answers in the form $re^{i\theta}$, where $r > 0$ and $0 \leqslant \theta < 2\pi$. [5]
Let $w = z_1^{3k} + z_2^{3k} + z_3^{3k}$, where $k$ is a positive integer and $z_1, z_2, z_3$ are the roots of $z^3 = -1 - i$.
(b) Express $w$ in the form $Re^{i\alpha}$, where $R > 0$, giving $R$ and $\alpha$ in terms of $k$. [3]The diagram shows the curve with equation $y = x^2$ 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 these rectangles, show that
$$\int_0^1 x^2 \mathrm{d}x < \frac{2n^2 + 3n + 1}{6n^2}.$$
[4](b) Use a similar method to find, in terms of $n$, a lower bound for $\int_0^1 x^2 \mathrm{d}x$. [4]5 The curves $C_1: y = \cosh x$ and $C_2: y = \sinh 2x$ intersect at the point where $x = a$. (a) Find the exact value of $a$, giving your answer in logarithmic form. [4] (b) Sketch $C_1$ and $C_2$ on the same diagram. [2]
(c) Find the exact value of the length of the arc of $C_1$ from $x = 0$ to $x = a$. [5]
6 The integral $I_n$, where $n$ is an integer, is defined by $I_n = \int_0^{\frac{1}{2}} \left(1-x^2\right)^{-\frac{1}{2}n} dx$.
(a) Find the exact value of $I_1$. [2]
(b) By considering $\frac{d}{dx}\left(x\left(1-x^2\right)^{-\frac{1}{2}n}\right)$, or otherwise, show that $nI_{n+2} = 2^{n-1}3^{-\frac{1}{2}n} + (n-1)I_n$. [5](c) Find the exact value of $I_5$ giving the answer in the form $k\sqrt{3}$, where $k$ is a rational number to be determined. [3]7 It is given that $x = t^3 y$ and
$$t^3 \frac{d^2 y}{d t^2} + (4 t^3 + 6 t^2) \frac{d y}{d t} + (13 t^3 + 12 t^2 + 6 t) y = 6 1 e^{\frac{1}{2} t}.$$
(a) Show that
$$\frac{d^2 x}{d t^2} + 4 \frac{d x}{d t} + 13 x = 6 1 e^{\frac{1}{2} t}.$$
[4](b) Find the general solution for $y$ in terms of $t$. [7]
8 (a) Find the values of $a$ for which the system of equations
$$3x + y + z = 0,$$
$$ax + 6y - z = 0,$$
$$ay - 2z = 0,$$
does not have a unique solution. [3]
The matrix $A$ is given by
$$A = \begin{pmatrix}
3 & 1 & 1 \\
0 & 6 & -1 \\
0 & 0 & -2
\end{pmatrix}.$$
(b) Use the characteristic equation of $A$ to find the inverse of $A^2$. [4](c) Find a matrix $P$ and a diagonal matrix $D$ such that $A^5 = PDP^{-1}$. [7]