A Level Mathematics 9709 Paper 42, May/June 2026

A car of mass $960\text{ kg}$ is moving on a straight horizontal road. There is a constant force of magnitude $620\text{ N}$ resisting the motion of the car.

Two particles, $A$ and $B$, of masses $2.5\text{ kg}$ and $3.5\text{ kg}$ respectively are at rest on a straight smooth horizontal track. Particle $B$ is situated $9\text{ m}$ from a vertical wall which is fixed at right angles to the track. Particle $A$ is projected directly towards $B$ with a speed of $3{\text{ ms}}^{-1}$. In the subsequent motion, $A$ collides and coalesces with $B$ to form particle $C$. Particle $C$ then collides directly with the wall and rebounds. The collision of $C$ with the wall reduces the speed of $C$ by $25\mathrm{\%}$.

Two particles, $A$ and $B$, of masses $0.5\text{ kg}$ and $0.8\text{ kg}$ respectively are attached to the ends of a light inextensible string. The string passes over a small smooth pulley fixed at the end of a rough horizontal plane and to the top of a smooth inclined plane. Particle $A$ is held on the horizontal plane, while $B$ lies on the inclined plane, which makes an angle of ${55}^{\circ }$ with the horizontal. The string is in the same vertical plane as a line of greatest slope of the inclined plane (see diagram). Particle $A$ is released from rest with the string taut. Particle $B$ moves $0.3\text{ m}$ down the inclined plane in $0.4\text{ s}$.

A particle $P$ is projected from a point $A$ with speed $4{\text{ ms}}^{-1}$ up a line of greatest slope of a rough plane inclined at an angle $\theta$ to the horizontal, where $\tan \theta =\frac{4}{3}$. $P$ comes to instantaneous rest at a point $B$ on the plane (see diagram). The coefficient of friction between $P$ and the plane is $\frac{1}{3}$.

The points $A$ and $B$ are at the same vertical height $h\text{ m}$ above horizontal ground. A particle $P$ is released from rest from $A$. One second later, a particle $Q$ is projected vertically downwards from $B$ with speed $18{\text{ ms}}^{-1}$. Given that $P$ and $Q$ reach the ground at the same time, find the value of $h$.

A crate of weight $W\text{ N}$ is at rest on rough horizontal ground. The crate is pulled at a constant acceleration of $1.5{\text{ ms}}^{-2}$ along the ground in a straight line by a light rope. The rope is inclined at an angle of $\theta$ to the horizontal, where $\theta ={\sin }^{-1}\frac{5}{13}$. The tension in the rope is $26\text{ N}$ (see diagram). The coefficient of friction between the crate and the ground is $0.25$.

A particle $P$ moves in a straight line. The velocity $v{\text{ ms}}^{-1}$ of $P$ at time $t\text{ s}$, where $t⩾0$, is given by $$v=k_1(4t+11{)}^{\frac{1}{2}}-\frac{1}{4}(2t+1{)}^2+k_2,$$ where $k_1$ and $k_2$ are constants. When $t=1.25$ the deceleration of $P$ is $0.5{\text{ ms}}^{-2}$.