A Level Mathematics 9709 Paper 42, May/June 2025

A crate is being pushed in a straight line along a horizontal surface by a force of magnitude $25\text{ N}$ inclined at ${20}^{\circ }$ above the horizontal. The crate moves a distance of $12\text{ m}$ in $8\text{ seconds}$ with constant speed.

Two particles $P$ and $Q$, of masses $0.2\text{ kg}$ and $0.1\text{ kg}$ respectively, are free to move in a straight line on a smooth horizontal plane. $P$ is projected towards $Q$ with speed $5{\text{ ms}}^{-1}$. At the same instant, $Q$ is projected away from $P$ with speed $2{\text{ ms}}^{-1}$. When $P$ collides with $Q$, the particles coalesce. Find the kinetic energy lost during the collision.

A particle $P$ of mass $m\text{ kg}$ is attached to one end of a light inextensible string of length $2.6\text{ m}$. The other end of the string is attached to a fixed point on a horizontal ceiling, and the string is taut. The particle is held in equilibrium by a force of magnitude $35\text{ N}$, acting in a vertical plane which is perpendicular to the ceiling and contains the string. The force acts in a direction perpendicular to the string (see diagram). The tension in the string is $T\text{ N}$ and the vertical distance of $P$ from the ceiling is $2.4\text{ m}$. Find, in either order, the value of $m$ and the value of $T$.

A car is travelling along a straight horizontal road. The car passes through a point $A$, on the road travelling at a speed of $15{\text{ ms}}^{-1}$, and then accelerates uniformly at $0.4{\text{ ms}}^{-2}$ for $30\text{ seconds}$. The car then moves at constant speed for $3T\text{ seconds}$, where $T<30$. The car then decelerates uniformly at $0.2{\text{ ms}}^{-2}$ and after a further $T\text{ seconds}$ passes through a point $B$ on the road.

One end of a light inextensible string is attached to a particle $A$ of mass $3\text{ kg}$. The other end of the string is attached to a particle $B$ of mass $4\text{ kg}$. Particle $A$ is in contact with a rough plane inclined at ${30}^{\circ }$ to the horizontal, and particle $B$ is in contact with a smooth horizontal plane. A second light inextensible string is attached to $B$. The other end of this second string is attached to a particle $C$ of mass $5\text{ kg}$ which hangs vertically. Both strings are taut and pass over small smooth pulleys that are fixed at the ends of the horizontal plane. The part of the string from $A$ to the pulley is parallel to a line of greatest slope of the inclined plane, and $A$, $B$ and $C$ are in the same vertical plane (see diagram). The system is released from rest. In the subsequent motion, $C$ moves vertically downwards with acceleration $2{\text{ ms}}^{-2}$, and neither $A$ nor $B$ reach a pulley.

A particle $P$ moves in a straight line and passes through the point $A$ at time $t=0$. The velocity $v{\text{ ms}}^{-1}$ of $P$ at time $t\text{ seconds}$ is given by $$v=(2t+1{)}^{\frac{3}{2}}-2t^2,\text{ where }0⩽t⩽3.$$

A particle $P$ of mass $3\text{ kg}$ is projected with a speed of $8{\text{ ms}}^{-1}$ up a line of greatest slope of a rough plane inclined at ${30}^{\circ }$ to the horizontal. $P$ is projected from a point $A$ on the plane and comes to instantaneous rest at a point $B$ on the plane. $P$ then slides back down the plane. The coefficient of friction between $P$ and the plane is $\frac{1}{12}\sqrt{3}$. Using an energy method throughout, find the speed of $P$ at the instant it returns to $A$.