科目：进阶数学 Further Mathematics (9231)
年份：2025 冬季
试卷：Paper 21
总页数：18
章节：9231 Further Mathematics
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1. Find the values of $k$ for which the system of equations $$x - y + z = 0,$$ $$x + ky + 3z = 0,$$ $$x + 2y + kz = 0,$$ does not have a unique solution. [3]
The curve $C$ has parametric equations
$$x = \sinh t \quad \text{and} \quad y = t + \cosh t.$$
(a) Find $\frac{dy}{dx}$ in terms of $t$.
[2]
(b) Show that $\frac{d^2y}{dx^2} = \frac{1 - \sinh t}{\cosh^3 t}$.
[4](c) Find the Maclaurin’s series for $y$ in terms of $x$ up to and including the term in $x^2$. [2]
3 The integral $I_n$ is defined by $I_n = \int_0^1 (1+x^2)^n \, dx$.
(a) By considering $\frac{d}{dx} \left( x(1+x^2)^n \right)$, or otherwise, show that
$(2n+1)I_n = 2^n + 2nI_{n-1}$. [5](b) Find the exact value of $I_{-2}$.4 Find the particular solution of the differential equation
$$5 \frac{d^2 y}{dx^2} + 2 \frac{dy}{dx} + y = x^2 + 5x + 3,$$
given that, when $x = 0$, $y = \frac{dy}{dx} = 0$. [10]
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The diagram shows the curve with equation $y = \frac{1}{3}x^3 + x$ 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 $\int_0^1 \left( \frac{1}{3}x^3 + x \right) dx < U_n$, where
$$U_n = \frac{1}{12} \left( 1 + \frac{1}{n} \right) \left( 7 + \frac{1}{n} \right).$$
[5](b) Use a similar method to find, in terms of $n$, a lower bound $L_n$ for $\int_0^1 \left( \frac{1}{3}x^3 + x \right) dx$. [4]
(c) Show that $\lim_{n \to \infty} (U_n - L_n) = 0$. [2]6 The matrix $P$ has non-zero eigenvalues and is given by
$$P = \begin{pmatrix}
a & 1 & 1 \\
0 & 2a & -1 \\
0 & 0 & -3a
\end{pmatrix}.$$
(a) State, in terms of $a$, the eigenvalues of $P$. [1]
(b) (i) Find $P^2$ in terms of $a$. [1]
(ii) Use the characteristic equation of $P$ to find $P^{-1}$ in terms of $a$. [3]The $3 \times 3$ matrix $A$ has eigenvalues 1, 2, 3 with corresponding eigenvectors
$$\begin{pmatrix} a \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ 2a \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ -1 \\ -3a \end{pmatrix},$$
respectively.
(c) Find $A$ in terms of $a$. [5]7 (a) Show that $\frac{d}{dx}(\tanh^{-1}x) = \frac{1}{1-x^2}$.
(b) Find the solution of the differential equation
$$x \frac{dy}{dx} - y = x^2 \tanh^{-1}x,$$
for $0 < x < 1$, given that $y = 0$ when $x = \frac{1}{2}$. Give your answer in an exact form. [9]15 © UCLES 2025 9231/21/O/N/25 [Turn over]
(a) State the sum of the series $z + z^2 + \ldots + z^n$, for $z \neq 1$.
(b) By letting $z = \frac{1}{2}(\cos \theta + i \sin \theta)$ use de Moivre’s theorem to deduce that
$$\sum_{m=1}^{n} \left(\frac{1}{2}\right)^m \sin m\theta = \frac{\left(\frac{1}{2}\right)^{n+2} \sin n\theta - \left(\frac{1}{2}\right)^{n+1} \sin(n+1)\theta + \frac{1}{2} \sin \theta}{\frac{5}{4} - \cos \theta}.$$(c) Use the result in (b) to find $\sum_{m=1}^{n} \left(\frac{1}{2}\right)^m m \cos m\theta$ in terms of $n$ and $\theta$. [You do not need to simplify your answer.]
(d) Hence find $\sum_{m=1}^{\infty} \left(\frac{1}{2}\right)^m m \cos m\theta$ in terms of $\cos\theta$. [You may assume that $\frac{n}{2^n} \to 0$ as $n \to \infty$.]Additional page If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
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