A Level Further Mathematics 9231 Paper 31, May/June 2025

A particle $P$ of mass $8\text{ kg}$ is moving in a straight horizontal line. At time $t\text{ s}$, $P$ has displacement $x\text{ m}$ from a fixed point $O$ on the line and velocity $v\text{ m}{\text{s}}^{-1}$. The only horizontal force acting on $P$ has magnitude $(x^3+4x)\text{ N}$ and acts in the direction $OP$. When $t=0$, $x=0$ and $v=1$.

A particle $P$ is projected with speed $24\text{ m}{\text{s}}^{-1}$ at an angle ${\theta }^{\circ }$ above the horizontal from a point $O$ on a horizontal plane, and moves freely under gravity. At a horizontal distance $35\text{ m}$ from $O$, there is a vertical wall of height $10\text{ m}$ which is perpendicular to the vertical plane of motion of $P$.

A rough horizontal disc, centre $O$, rotates with constant angular speed $\omega \text{ rad}{\text{s}}^{-1}$. A particle $P$ of mass $1.6\text{ kg}$ lies on the disc at a distance $1.5\text{ m}$ from $O$, and is attached to a point $A$ vertically above $O$ by a light elastic string. The string has natural length $2\text{ m}$, modulus of elasticity $32\text{ N}$ and makes an angle $\alpha$ with the vertical $OA$ (see diagram). Particle $P$ moves in a horizontal circle also at a constant angular speed $\omega \text{ rad}{\text{s}}^{-1}$. Particle $P$ is on the point of slipping in the direction $OP$. The coefficient of friction between the particle and the disc is $0.5$.

An object is formed by removing a solid hemisphere, radius $2r$, from a uniform solid cone, radius $3r$ and semi-vertical angle $\theta$, where $\tan \theta =\frac{1}{2}$. The axes of symmetry of the cone and the hemisphere coincide. The base of the cone and the base of the hemisphere are in the same plane as each other (see diagram).

One end of a light elastic string of natural length $0.5\text{ m}$ and modulus of elasticity $14\text{ N}$ is attached to a fixed point $A$ on a smooth plane. The plane makes an angle $\alpha$ to the horizontal, where $\tan \alpha =\frac{7}{24}$. A particle $P$ of mass $2\text{ kg}$ is attached to the other end of the string. The string lies along a line of greatest slope of the plane. The particle $P$ is initially held on the plane above the level of $A$, where $AP=0.8\text{ m}$. The particle $P$ is then released from rest. Find the maximum velocity of $P$ during the subsequent motion. [6]

Two uniform smooth spheres $A$ and $B$ of equal radii have masses $2m$ and $m$ respectively. Sphere $A$ is moving in a straight horizontal line with speed $u$, and sphere $B$ is stationary. Sphere $A$ collides directly with $B$, and they both then move in the same direction with speeds $v_A$ and $v_B$ respectively. After the collision, the kinetic energy of $B$ is $\frac{9}{2}$ times the kinetic energy of $A$.

A fixed hollow sphere has radius $a$ and centre $O$. The points $A$, $B$ and $C$ lie on the inner surface of the sphere with $OA$ and $BC$ horizontal. A portion of the sphere has been removed by a horizontal cut through points $B$ and $C$ at a vertical distance $ka$ above the centre of the sphere, where $k$ is a positive constant and $k<1$. The points $O$, $A$, $B$ and $C$ all lie in the same vertical plane. $OB$ makes an angle $\theta$ with the upward vertical through $O$ (see diagram). A particle $P$ of mass $m$ is free to move on the smooth inner surface of the sphere. The particle $P$ is projected vertically downwards from $A$ with speed $u$ and begins to move in a vertical circle.