A Level Mathematics 9709 Paper 43, Oct/Nov 2025

The diagram shows the velocity-time graph for the motion of an athlete. The athlete runs in a straight line from point $A$ to point $B$, runs back to point $A$ and finishes at point $B$. The graph consists of four straight line segments.

The engine of a motorcycle can generate a maximum power of $105\text{ kW}$. The mass of the motorcycle and its rider is $400\text{ kg}$. The total resistance to the motion of the motorcycle and its rider is $cv\text{ N}$, where $v{\text{ ms}}^{-1}$ is the motorcyclist's speed and $c$ is a constant. The motorcyclist travels along a straight horizontal road under maximum engine power. When the motorcyclist's speed is $35{\text{ ms}}^{-1}$, his acceleration is $0.5{\text{ ms}}^{-2}$.

A block of mass $4\text{ kg}$ is pulled along a rough horizontal road by a constant force of magnitude $25\text{ N}$ acting at an angle of ${36}^{\circ }$ above the horizontal. The block moves in a straight line passing through two points $A$ and $B$ on the road, where $AB=120\text{ m}$. The coefficient of friction between the block and the road is $0.4$.

A particle $P$ moves in a straight line. At time $t\text{ s}$ after passing through a point $O$ on the line, the displacement of $P$ from $O$ is $s\text{ m}$, where $s=0.01t^3-0.3t^2-2.07t$.

Particles $A$ and $B$, of masses $2\text{ kg}$ and $6\text{ kg}$ respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley and the particles hang vertically below the pulley. Both particles are initially held at rest at a height of $3.2\text{ m}$ above horizontal ground (see diagram). Particle $A$ is projected vertically downwards with a speed of $1.2{\text{ ms}}^{-1}$.

Particle $P$ of mass $m\text{ kg}$ and particle $Q$ of mass $2\text{ kg}$ are free to move on a smooth horizontal plane. $P$ and $Q$ are moving directly towards each other with speeds $u{\text{ ms}}^{-1}$ and $3u{\text{ ms}}^{-1}$ respectively. $P$ and $Q$ collide and the direction of motion of each particle is reversed by the collision. Immediately after the collision the speed of $P$ is $\frac{u}{3}{\text{ ms}}^{-1}$.

A particle $P$ of mass $6\text{ kg}$ lies on a rough plane inclined at an acute angle $\theta$ to the horizontal. A horizontal force of magnitude $12\text{ N}$ acts on $P$ as shown in the diagram. The line of action of this horizontal force lies in a vertical plane which contains the line of greatest slope of the plane that passes through $P$. The coefficient of friction between $P$ and the plane is $\mu$. $P$ is in equilibrium and on the point of sliding down the plane.