科目：进阶数学 Further Mathematics (9231)
年份：2026 夏季
试卷：Paper 23
总页数：18
章节：9231 Further Mathematics
1. Find the roots of the equation $w^5 + 32 = 0$, giving your answers in the form $re^{i\theta}$, where $r > 0$ and $-\pi \leq \theta < \pi$. [5]2 A curve has equation $y = \sqrt{x}$, for $0 \leqslant x \leqslant 2$. Find, in terms of $\pi$, the area of the surface generated when the curve is rotated through $2\pi$ radians about the $x$-axis.
[5]3 Let $I_n = \int_0^a (\ln(x+1))^n dx$, where $n$ is a non-negative integer and $a > 0$.
(a) By considering the derivative of $(x+1)(\ln(x+1))^n$, or otherwise, show that for $n \geqslant 1$
$$I_n = (a+1)(\ln(a+1))^n - nI_{n-1}.$$
[3](b) Find $I_2$ in terms of $a$. [3]
4. Find the particular solution of the differential equation
$$49 \frac{d^2x}{dt^2} + 14 \frac{dx}{dt} + x = 196e^{\frac{3}{2}t},$$
given that, when $t = 0$, $x = 7$ and $\frac{dx}{dt} = 0$. [10]
© Cambridge University Press & Assessment 2026 9231/23/M/J/267 © Cambridge University Press & Assessment 2026 9231/23/M/J/26 Turn over
(a) Starting from the definitions of sech and tanh in terms of exponentials, prove that
$$1 - \text{sech}^2 t = \tanh^2 t.$$
[3]
It is given that
$$x = t \sinh t \quad \text{and} \quad y = t \cosh t.$$
(b) Show that
$$\frac{dy}{dx} = \frac{t \tanh t + 1}{t + \tanh t}.$$
[3](c) Show that $\frac{d^2y}{dx^2} = \frac{(t^2 - 2)\text{sech}^n t}{(t + \tanh t)^n}$, where $n$ is an integer to be determined. [6]6 (a) Show that an appropriate integrating factor for
$$\left(1 - x^2\right) \frac{dv}{dx} + y = \left(1 - x^2\right) \sqrt{1 - x}$$
is $\sqrt{\frac{1+x}{1-x}}$.
[4](b) Hence find the solution of the differential equation
$$\left(1 - x^2\right) \frac{dy}{dx} + y = \left(1 - x^2\right) \sqrt{1 - x}$$
for which $y = 1$ when $x = 0$. [6]7 (a) Use the substitution $u = 1 - x^2$ to find $\int \frac{x}{\sqrt{1-x^2}} dx$.
The diagram shows the curve with equation for $y = \sin^{-1}x$ for $0 \leqslant x \leqslant 1$, together with a set of $n$ rectangles of width $\frac{1}{n}$.
(b) By considering the sum of the areas of these rectangles, show that
$$\sum_{r=1}^{n} \frac{1}{n} \sin^{-1}\left(\frac{r}{n}\right) \geqslant \frac{1}{2} \pi - 1.$$(c) Use a similar method to find, in terms of $n$, an upper bound for $\sum_{r=1}^{n} \frac{1}{n} \sin^{-1}\left(\frac{r}{n}\right)$.
(d) Deduce the exact value of $\lim_{n \to \infty} \sum_{r=1}^{n} \frac{1}{n} \sin^{-1}\left(\frac{r}{n}\right)$.8 (a) The constants $a$ and $b$ are such that the system of equations
$$6x - 12y - 12z = 1,$$
$$ax + 4y + z = 2,$$
$$2x + by - 8z = 3,$$
does not have a unique solution. Find $a$ in terms of $b$. [3]
(b) Given that $a = -\frac{1}{4}$ and $b = -32$, show that the system of equations in part (a) is inconsistent. Interpret this situation geometrically. [3]The matrix $A$ is given by
$$A = \begin{pmatrix}
6 & -12 & -12 \\
-1 & 4 & 1 \\
2 & -8 & -8
\end{pmatrix},$$
(c) Show that $\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}$ is an eigenvector of $A$ and state the corresponding eigenvalue. [2]
(d) Find a matrix $P$ and a diagonal matrix $D$ such that $A = PDP^{-1}$. [7]16 © Cambridge University Press & Assessment 2026 9231/23/M/J/26
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