科目：进阶数学 Further Mathematics (9231)
年份：2024 夏季
试卷：Paper 21
总页数：18
章节：9231 Further Mathematics
1. Find the roots of the equation $z^3 = -108\sqrt{3} + 108i$, giving your answers in the form $r(\cos \theta + i\sin \theta)$, where $r > 0$ and $0 < \theta < 2\pi$. [5]2 Find the Maclaurin’s series for $e^{1+x^2} + e^{1-x}$ up to and including the term in $x^2$. [4]3 It is given that
$$x = \sin^{-1} t \quad \text{and} \quad y = t \cos^{-1} t, \quad \text{for } 0 \leq t < 1.$$
(a) Show that $\frac{dy}{dx} = -t + \sqrt{1 - t^2} \cos^{-1} t$. [3](b) Find $\frac{d^2y}{dx^2}$ in terms of $t$. [4]4 It is given that, for $n \geqslant 0$, $I_n = \int_0^{\ln 3} \text{sech}^n x \, dx$.
(a) Show that, for $n \geqslant 2$,
$$\left(n-1\right) I_n = \left(\frac{3}{5}\right)^{n-2} \left(\frac{4}{5}\right) + \left(n-2\right) I_{n-2}.$$
[You may use the result that $\frac{d}{dx}(\text{sech} x) = -\tanh x \text{sech} x$.](b) Find the value of $I_4$.
The diagram shows the curve with equation $y = 2x - x^2$ 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 $\int_0^1 (2x - x^2) dx < U_n$, where $U_n = \left(1 + \frac{1}{n}\right)\left(\frac{2}{3} - \frac{1}{6n}\right)$. [5](b) Use a similar method to find, in terms of $n$, a lower bound $L_n$ for $\int_0^1 (2x - x^2) \mathrm{d}x$. [4]
(c) Show that $\lim_{n \to \infty} (U_n - L_n) = 0$. [2]6 (a) Show that $(\cosh x + \sinh x)^{\frac{1}{2}} = e^{\frac{1}{2}x}$.
(b) Find the particular solution of the differential equation
$$\frac{d^2y}{dx^2} + \frac{dy}{dx} + 3y = 5(\cosh x + \sinh x)^{\frac{1}{2}},$$
given that, when $x = 0$, $y = 1$ and $\frac{dy}{dx} = \frac{4}{3}$.11 © UCLES 2024 9231/21/M/J/24 [Turn over]
7 (a) Use the substitution $u = 1 + x^2$ to find
$$\int \frac{x}{\sqrt{1+x^2}} \, dx.$$
[2]
(b) Find the solution of the differential equation
$$x \frac{\mathrm{d}y}{\mathrm{d}x} - y = x^2 \sinh^{-1} x,$$
given that $y = 1$ when $x = 1$. Give your answer in the form $y = f(x).$
[10]Turn over
8 (a) Find the set of values of $a$ for which the system of equations $$6x + ay = 3,$$ $$2x - y = 1,$$ $$x + 5y + 4z = 2$$ has a unique solution. [2] (b) Show that the system of equations in part (a) is consistent for all values of $a$. [3]
The matrix $A$ is given by
$$A = \begin{pmatrix} 6 & 0 & 0 \\ 2 & -1 & 0 \\ 1 & 5 & 4 \end{pmatrix}.$$
(c) Find a matrix $P$ and a diagonal matrix $D$ such that $(14A + 24I)^2 = PDP^{-1}$. [7](d) Use the characteristic equation of $A$ to show that $$\left(14A + 24I\right)^2 = A^4\left(A + bI\right)^2,$$ where $b$ is an integer to be determined. [4]
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