科目：进阶数学 Further Mathematics (9231)
年份：2023 冬季
试卷：Paper 21
总页数：14
章节：9231 Further Mathematics
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1 Show that the system of equations $$14x - 4y + 6z = 5,$$ $$x + y + kz = 3,$$ $$-21x + 6y - 9z = 14,$$ where $k$ is a constant, does not have a unique solution and interpret this situation geometrically. [4]
2. Find the roots of the equation $(z + 5i)^3 = 4 + 4\sqrt{3}i$, giving your answers in the form $r \cos \theta + i(r \sin \theta - 5)$, where $r > 0$ and $0 < \theta < 2\pi$. [5]3 Find the first three terms in the Maclaurin's series for $\tanh^{-1}\left(\frac{1}{2}e^x\right)$ in the form $\frac{1}{2}\ln a + bx + cx^2$, giving the exact values of the constants $a, b$ and $c$. [6]4 Find the particular solution of the differential equation
$$\frac{d^2y}{dx^2} + 2\frac{dy}{dx} + 3y = 27x^2,$$
given that, when $x = 0$, $y = 2$ and $\frac{dy}{dx} = -8.$ [10]7 © UCLES 2023 9231/21/O/N/23 [Turn over]
5 The curve $C$ has parametric equations
$$x = \frac{2}{3}t^{\frac{3}{2}} - 2t^{\frac{1}{2}}, \quad y = 2t + 5, \quad \text{for } 0 < t \leqslant 3.$$
(a) Find the exact length of $C$. [5](b) Find the set of values of $t$ for which $\frac{d^2y}{dx^2} > 0$. [5]6 (a) Starting from the definitions of cosh and sinh in terms of exponentials, prove that $$\sinh 2x = 2 \sinh x \cosh x.$$ [3] (b) Using the substitution $u = \sinh x$, find $\int \sinh^2 2x \cosh x \, dx.$ [4]
(c) Find the particular solution of the differential equation
$$\frac{dy}{dx} + y \tanh x = \sinh^2 2x,$$
given that $y = 4$ when $x = 0$. Give your answer in the form $y = f(x)$. [7]7 The matrix $A$ is given by
$$A = \begin{pmatrix}
-6 & 2 & 13 \\
0 & -2 & 5 \\
0 & 0 & 8
\end{pmatrix}.$$
(a) Find a matrix $P$ and a diagonal matrix $D$ such that $A^{-1} = PDP^{-1}$. [7](b) Use the characteristic equation of $A$ to find $A^{-1}$. [4]8 (a) State the sum of the series $1 + z + z^2 + \ldots + z^{n-1}$, for $z \neq 1$.
(b) By letting $z = \cos\theta + i\sin\theta$, where $\cos\theta \neq 1$, show that
$$1 + \cos\theta + \cos 2\theta + \ldots + \cos(n-1)\theta = \frac{1}{2}\left(1 - \cos n\theta + \frac{\sin n\theta \sin\theta}{1 - \cos\theta}\right).$$The diagram shows the curve with equation $y = \cos x$ for $0 \leq x \leq 1$, together with a set of $n$ rectangles of width $\frac{1}{n}$.
(c) By considering the sum of the areas of these rectangles, show that
$$\int_{0}^{1} \cos x dx < \frac{1}{2n} \left( 1 - \cos 1 + \frac{\sin 1 \sin \frac{1}{n}}{1 - \cos \frac{1}{n}} \right).$$
(d) Use a similar method to find, in terms of $n$, a lower bound for $\int_{0}^{1} \cos x dx$.