科目：进阶数学 Further Mathematics (9231)
年份：2026 夏季
试卷：Paper 31
总页数：14
章节：9231 Further Mathematics
An object consists of a uniform solid hemisphere of radius $3r$ attached to a uniform solid cylinder of radius $r$ and height $x$. The hemisphere and the cylinder are made of the same material. The axes of symmetry of the hemisphere and the cylinder coincide (see diagram). (a) Find, in terms of $r$ and $x$, the distance of the centre of mass of the object from the base of the cylinder. [4]
(b) The point $A$ lies on the circumference of the base of the hemisphere. When the object is freely suspended from point $A$, the base of the hemisphere makes an angle $\theta$ with the downward vertical through $A$. It is given that $x = r$. Find the value of $\theta$. [3]
The point $B$ is vertically below the point $A$. One end of a light inextensible string is attached to $A$, and the other end is attached to $B$. A particle $P$ of mass 7 kg is attached to the string so that the length $AP$ is 15 m. The particle $P$ moves at a constant angular speed of 4 radians per second in a horizontal circle with centre between $A$ and $B$ at a distance 9 m vertically below $A$ (see diagram). The string has length 28 m, and both sections of the string are taut. The tensions in the sections of the string $AP$ and $BP$ are $T_A$ and $T_B$ respectively. Find the value of $T_A$ and the value of $T_B$. [5]
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3 A particle $P$ of mass 6.4 kg moves along a straight horizontal line. At time $t$ s, the speed of $P$ is $v \text{ m/s}^{-1}$ and the distance of $P$ from its starting position is $x \text{ m}$. The forces acting on $P$ are a constant driving force of magnitude 8 N and a resistive force of magnitude $0.5v^2 \text{ N}$. Initially, $P$ is at rest.
(a) Find the speed of $P$ when it has travelled 0.8 m. [5](b) Find an expression for $t$ in terms of $v$. [4]
4 A particle $P$ is moving on a fixed smooth horizontal surface with speed $u$. The particle collides with a fixed smooth vertical barrier. Immediately before the collision, the direction of motion of $P$ makes an angle $\alpha$ with the barrier, where $\tan \alpha = \frac{12}{5}$.
The kinetic energy of $P$ after the collision is $\frac{4}{13}$ times its kinetic energy before the collision.
(a) Find the coefficient of restitution between $P$ and the barrier. [4](b) A second smooth vertical barrier is fixed to the surface and to the first barrier. The angle between the two barriers is 120°. Particle $P$ subsequently collides with the second barrier and rebounds immediately after with speed $w$ at an angle $\theta$ with the second barrier. The coefficient of restitution is the same as in the first collision. Find the value of $\theta$ correct to 3 significant figures. [4]
5 A particle of mass $m$ is attached to a fixed point $O$ by a light inextensible string of length $a$. The particle initially hangs directly below $O$ at the point $A$. The particle is then projected horizontally with speed $2\sqrt{ag}$, and begins to move in a vertical circle.
(a) Find an expression for the tension in the string when it has turned through an angle $\theta$. Give your answer in terms of $m, g$ and $\theta$.
[4](b) Find, in terms of $a$, the maximum height above the horizontal through $A$ that is reached by the particle. [5]
6 A light elastic string has natural length 2.5 m and modulus of elasticity 15 N. One end of the string is attached to the fixed point $O$, and the other end is attached to a particle $P$ of mass 1.5 kg. The particle $P$ is held at $O$ and then released from rest so that it begins to fall vertically. In its subsequent motion, $P$ moves with velocity $v$ m s$^{-1}$ when the extension of the string is $x$ m. The maximum extension of the string during this motion is $E$ m.
(a) Show that, for $0 < x < E,$
$$v^2 = -4(x^2 - 5x - 12.5).$$
[3](b) Find the value of $E$. (c) Find the maximum speed of $P$.
7 The points $O$ and $A$ lie on a horizontal plane 10 m apart. The point $A$ lies at the foot of another plane which is inclined at an angle of $\alpha$ above the horizontal such that $\tan \alpha = \frac{4}{3}$. The point $B$ is on the inclined plane at a distance of 5 m from $A$ measured along a line of greatest slope. The points $O, A$ and $B$ are in the same vertical plane and angle $OAB$ is obtuse.
A particle is projected from $O$ at an angle $\theta$ to the horizontal with initial speed $u$ m s$^{-1}$ towards the inclined plane. The particle moves freely under gravity. The particle strikes the inclined plane at the point $B$ when its direction of motion is perpendicular to the inclined plane.
Find, in either order, the value of $u$ and the value of $\theta$. [6]DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN © Cambridge University Press & Assessment 2026 9231/31/M/J/26