A Level Further Mathematics 9231 Paper 33, May/June 2026

A particle $P$ of mass $m$ is moving in a horizontal circle with constant angular speed $\omega$ on the smooth inner surface of a hemispherical shell of radius $a$. The plane of the circular motion is at a height $h=\frac{1}{3}a$ above the lowest point on the shell. The angle between the vertical and the normal reaction of the surface on $P$ is $\theta$. Find $\omega$ in terms of $a$ and $g$.

A particle $P$ of mass $m$ is moving in a straight horizontal line towards a fixed point $A$ on the line. The particle is subject to a resistive force that is inversely proportional to the square of the distance between $P$ and $A$. Initially, the distance between $P$ and $A$ is $d$ and the speed of $P$ is $u$. When the distance between $P$ and $A$ is $\frac{1}{2}d$, the particle has lost $\frac{3}{4}$ of its kinetic energy. Find, in terms of $d$, the minimum distance of $P$ from $A$.

A fixed smooth solid sphere has centre $O$ and radius $a$. A particle of mass $m$ is projected perpendicular to $OA$ in a downward direction with speed $u$ from the point $A$ on the outer surface of the sphere, where $OA$ makes an angle $\alpha$ with the upward vertical through $O$ (see diagram). The particle moves in part of a vertical circle on the surface of the sphere. It loses contact with the sphere at the point $B$, where $OB$ makes an angle $\beta$ with the upward vertical through $O$. The speed of the particle at the point $B$ is $v$. Given that the ratio of $u$ to $v$ is $2:3$ find the ratio $\cos \alpha :\cos \beta$.

An object consists of a uniform solid cube of side length $d$, and two uniform solid square-based pyramids. The cube and the pyramids are made of the same material. One pyramid, of base length $d$ and height $h$, is fixed to the top face of the cube so that the base of this pyramid and the face of the cube coincide. Point $A$ is at the top vertex of this pyramid. Point $B$ is in the centre of the bottom face of the cube, vertically below $A$. The second pyramid, of base length $d$ and height $kh$, is fixed to a vertical face of the cube so that the base of this pyramid and the vertical face of the cube coincide (see diagram, which shows a cross-section, including the axes of symmetry).

Two uniform smooth spheres $A$ and $B$ of equal radii and equal masses $m$ are moving with speeds $u_A$ and $u_B$ respectively on a smooth horizontal surface when they collide. Immediately before the collision, the direction of motion of $A$ makes an angle $\alpha$ with the line of centres, and the direction of motion of $B$ makes an angle $\beta$ with the line of centres (see diagram). Immediately after the collision, the direction of motion of $A$ makes an angle $\theta$ with the line of centres, and the direction of motion of $B$ makes an angle $\varphi$ with the line of centres. It is given that $\cos \alpha =\cos \varphi =\frac{4}{5}$ and $\cos \beta =\frac{24}{25}$ and $\cos \theta =\frac{3}{5}$.

A smooth plane is inclined at an angle $\theta$ to the horizontal, where $\sin \theta =\frac{4}{5}$. One end of a light elastic spring, with natural length $10a$ and modulus of elasticity $1.6mg$, is attached to a fixed point $O$ on the plane. The other end of the spring is attached to a particle $P$ of mass $0.2m$. The point $O$ and the particle $P$ lie on the same line of greatest slope of the plane, with $P$ above $O$. The particle $P$ is at rest.

At time $t=0\mathrm{s}$, a particle $P$ is projected from a point $O$ on horizontal ground. The initial velocity of $P$ is $25{\mathrm{m}\mathrm{s}}^{-1}$ at an angle $\alpha$ above the horizontal, where $\sin \alpha =\frac{4}{5}$. At time $t=T\mathrm{s}$, a second particle $Q$ is projected vertically upwards from the horizontal ground directly below the point where $P$ will reach its greatest height. The initial velocity of $Q$ is also $25{\mathrm{m}\mathrm{s}}^{-1}$. Both particles move freely under gravity. The two particles collide when $Q$ is travelling upwards and at the point where $P$ reaches its greatest height.