科目：进阶数学 Further Mathematics (9231)
年份：2026 夏季
试卷：Paper 34
总页数：14
章节：9231 Further Mathematics
BLANK PAGE © Cambridge University Press & Assessment 2026 9231/34/M/J/26
1. One end of a light inextensible string of length $a$ is attached to a fixed point $O$. The other end of the string is attached to a particle of mass $m$. The string is taut and makes an angle $\theta$ with the downward vertical through $O$. The particle moves in a horizontal circle with speed $2\sqrt{ag}$.
Find the value of $\cos \theta$. [4]A uniform lamina $AEFGD$ is formed by removing two triangles from a square $ABCD$ of side length $a$. Triangle $FCG$ is isosceles and the two equal sides each have length $\frac{1}{2}a$. The angle between $EF$ and $BF$ is $\alpha$, where $\tan \alpha = \frac{1}{2}$ (see diagram).
The centre of mass of the lamina is at a distance $\overline{x}$ from $AD$ and at a distance $\overline{y}$ from $AE$.
(a) Show that $\overline{x} = \frac{5}{12}a$ and find a corresponding expression for $\overline{y}$ in terms of $a$. [5](b) The lamina is freely suspended from the point $G$ and hangs in equilibrium. The angle between $DA$ and the downward vertical through $G$ is $\theta$. Find the value of $\theta$. [2]
3 A particle of mass $m$ is attached to a fixed point $O$ by a light inextensible string of length $a$. The particle moves in complete vertical circles about $O$. Points $A$ and $B$ are on the path of the particle. $OA$ makes an angle of 60° with the upward vertical through $O$. The speed of the particle at $A$ is $u$ and the tension in the string at $A$ is $T_A$. The point $B$ is vertically below $O$. The speed of the particle at $B$ is $2u$ and the tension in the string at $B$ is $T_B$. (a) Show that $T_B = kT_A$, where $k$ is a constant to be determined. [6]
(b) Find the loss in kinetic energy of the particle between its lowest and highest points. Give your answer in terms of $m, g$ and $a$. [1]
4 Two particles $A$ and $B$ have masses $km$ and $m$ respectively. Particle $A$ is at rest on a smooth horizontal surface and is exactly halfway between two fixed parallel smooth vertical walls. Particle $B$ is moving on the surface with speed $u$ in a direction perpendicular to the walls, and collides directly with $A$. The coefficient of restitution between the two particles is 0.2. As a result of the collision, the direction of motion of $B$ is reversed. (a) Show that $k > 5$. [3] © Cambridge University Press & Assessment 2026 9231/34/M/J/26
(b) Particle $B$ then collides with one of the vertical walls and rebounds. The coefficient of restitution between $B$ and the wall is 0.8. It then collides again with particle $A$ at the instant when $A$ collides with the other vertical wall. Find the value of $k$. [3]
5 A particle $P$ of mass $m$ is moving in a straight horizontal line under a variable force $m(v^2 - ku^2)$, where $v$ is the velocity of $P$ at time $t$, $u$ is the initial velocity of $P$, and $k$ is a positive constant. It is given that $u > 0$ and $k > 1$. (a) Explain why, initially, the acceleration and velocity of $P$ have opposite directions. [1] (b) Find, in terms of $k$, the distance travelled by $P$ until it first comes to instantaneous rest. [4]
(c) Find, in terms of $k$ and $u$, the time taken for $P$ to first come to instantaneous rest. [4]
The points $A$ and $B$ lie on the line of greatest slope of a smooth inclined plane, with $A$ above $B$. The plane is inclined to the horizontal at an acute angle $\alpha$.
At time $t = 0$, a particle $P$ is projected from $A$ with speed $u$ at an angle $2\alpha$ to the plane and moves freely under gravity (see diagram). At the same time, a second particle $Q$ is released from rest at $A$. Particle $Q$ moves down the plane along the line of greatest slope. The two particles move in the same vertical plane. At time $t = T$, the two particles collide at point $B$. The length of $AB$ is $d$.
(a) By considering the motion of particle $Q$, show that $T = \sqrt{\frac{2d}{g \sin \alpha}}$. [2](b) Find the value of $\alpha$. [5]
7 One end of a light elastic string of natural length $8a$ and modulus of elasticity $6mg$ is attached to a fixed point $O$ on a rough plane. The plane is inclined at an angle $\theta$ to the horizontal where $\sin \theta = \frac{5}{13}$. A particle $P$ of mass $m$ is attached to the other end of the string. The string lies along a line of greatest slope of the plane, with $P$ below the level of $O$. The particle is held at a point such that the extension of the string is $3a$.
When $P$ is released from rest, its acceleration is $\frac{5}{4}g$ in the direction $PO$.
(a) Find the coefficient of friction between the particle and the plane. [4](b) In its subsequent motion, $P$ comes to rest at the point $C$. Find, in terms of $a$, the distance $OC$. [6]