科目：进阶数学 Further Mathematics (9231)
年份：2025 夏季
试卷：Paper 34
总页数：14
章节：9231 Further Mathematics
1. A light spring of natural length $a$ and modulus of elasticity $20mg$ is placed so that it stands vertically on a horizontal plane. The lower end of the spring is fixed to the plane. A particle of mass $m$ is attached to the upper end of the spring.
The particle is pushed vertically downwards until the length of the spring is $\frac{3}{5}a$. The system is then released from rest.
Find the maximum extension of the spring in the subsequent motion. [5]2 A particle $P$ of mass $m$ kg moves along a horizontal straight line against a resistive force of magnitude $2mv^3$ N, where $v$ ms$^{-1}$ is the velocity of $P$ at time $t$ s. When $t = 0$, $v = 1$.
(a) Find an expression for $v$ in terms of $t$.
[4]
(b) Find the displacement of $P$ from its initial position when $t = 6$.3 A particle $P$ is projected with speed $u$ at an angle $\alpha$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements of $P$ from $O$ at any subsequent time $t$ are denoted by $x$ and $y$ respectively.
(a) Derive the equation of the trajectory of $P$ in the form
$$y = x \tan \alpha - \frac{gx^2}{2u^2} \sec^2 \alpha.$$
[4]It is given that $u = 20\sqrt{2} \text{ ms}^{-1}$ and that the particle $P$ passes through the point where $x = 64 \text{ m}$ and $y = 8 \text{ m}$.
(b) Find the possible values of $\tan \alpha$. [3]A hollow cone with a smooth inner surface is fixed with its vertex $O$ downwards. The semi-vertical angle of the cone is $\alpha$, where $\tan \alpha = \frac{3}{4}$. A light inextensible string has a particle $A$ of mass $m$ attached to one end and a particle $B$ of mass $m$ attached to the other end. The string passes through a small hole in the cone at $O$. Particle $B$ hangs in equilibrium below $O$. Particle $A$ is on the inner surface of the cone at a height $h$ above the level of $O$ and moves in horizontal circles with constant angular speed $\omega$ (see diagram).
Find $\omega$ in terms of $g$ and $h$. [6]7 © UCLES 2025 9231/34/M/J/25 [Turn over]
ABCD is a uniform square lamina of side 6a. Points E and F are on DC and AB respectively and are such that DE = FB = h. The quadrilateral BCEF is removed from the square lamina (see diagram).
(a) Show that the distance of the centre of mass of the resulting lamina AFED from AD is $\frac{h^2 - 6ah + 36a^2}{18a}$ and find a corresponding expression for the distance of the centre of mass from AB.
[5]When the lamina $AFED$ is suspended from the point $D$, the edge $DA$ makes an angle $\theta$ with the downward vertical, where $\tan \theta = \frac{7}{15}$.
(b) Find, in terms of $a$, the two possible values of $h$. [3]Two uniform smooth spheres $A$ and $B$ of equal radii have masses $5m$ and $4m$ respectively. Sphere $A$ is moving with speed $u$ on a horizontal surface when it collides with sphere $B$ which is at rest. Immediately before the collision, $A$’s direction of motion makes an angle $\theta$ with the line of centres (see diagram). The coefficient of restitution between the spheres is $e$.
(a) Show that the speed of $B$ after the collision is $\frac{5}{9} u(1 + e)\cos \theta$. [3]After the collision the kinetic energy of $A$ is equal to the kinetic energy of $B$.
(b) Given that $\tan \theta = \frac{2}{3}$, find the value of $e$. [6]7 A particle $P$ of mass $m$ is attached to one end of a light inextensible string of length $a$. The other end of the string is attached to a fixed point $O$. The particle moves in complete vertical circles with centre $O$, with the string taut. When the string makes an angle $\theta$ with the downward vertical through $O$ the speed of $P$ is $\sqrt{4ag}$. The ratio of the greatest and least tensions in the string during the motion is $11:1$.
Find the value of $\cos \theta$. [8]13 © UCLES 2025 9231/34/MJ/25
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.
15 BLANK PAGE © UCLES 2025 9231/34/M/J/25