A Level Mathematics 9709 Paper 41, Oct/Nov 2025

A car of mass $900\text{ kg}$ is moving along a straight horizontal road against a constant resistance to motion of $350\text{ N}$. At an instant when the car is moving at $15{\text{ ms}}^{-1}$ its acceleration is $0.25{\text{ ms}}^{-2}$.

Two particles, $P$ and $Q$, of masses $3\text{ kg}$ and $5\text{ kg}$ respectively, are at rest on a smooth horizontal plane. $P$ is projected at a speed of $4{\text{ ms}}^{-1}$ directly towards $Q$. After $P$ and $Q$ collide, $P$ has speed $1{\text{ ms}}^{-1}$.

Coplanar forces of magnitudes $45\text{ N}$, $28\text{ N}$, $72\text{ N}$ and $35\text{ N}$ act at a point in the directions shown in the diagram. Find, in either order, the magnitude and direction of the resultant force.

Two particles, $A$ and $B$, of masses $m\text{ kg}$ and $3\text{ kg}$ respectively, are connected by a light inextensible string. Particle $B$ is on a fixed plane which is at an angle of ${\sin }^{-1}0.6$ to the horizontal ground. The string passes over a fixed smooth pulley at the top of the plane. Particle $A$ hangs vertically below the pulley and is $0.75\text{ m}$ above the ground (see diagram). The system is released from rest. In the subsequent motion $B$ moves up a line of greatest slope of the plane and does not reach the pulley. As $B$ moves up the plane there is a constant resistance to its motion of magnitude $\frac{10}{3}\text{ N}$. The speed of $A$ immediately before it hits the ground is $2{\text{ ms}}^{-1}$. Use an energy method to find the value of $m$.

A particle $P$ of mass $m\text{ kg}$ is in equilibrium on a rough plane inclined at an angle ${\theta }^{\circ }$ to the horizontal. The equilibrium of $P$ is maintained by a force of magnitude $8mg\text{ N}$ making an angle ${\theta }^{\circ }$ with a line of greatest slope (see diagram). The coefficient of friction between $P$ and the plane is $0.5$ and $P$ is on the point of slipping down the plane. Find the value of $\theta$.

A particle $A$ of mass $2.5\text{ kg}$ is released from rest from the top of a smooth plane, which makes an angle of ${\sin }^{-1}0.2$ with the horizontal. The particle $A$ collides $2\text{ seconds}$ later with a particle $B$, of mass $3\text{ kg}$, which is moving up a line of greatest slope of the plane. The speed of $B$ immediately before the collision is $3.5{\text{ ms}}^{-1}$. Immediately after the collision, $B$ has a velocity of $0.5{\text{ ms}}^{-1}$ down the plane. Find the distance $A$ moves up the plane after the collision.

A particle $P$ starts from a point $O$ and moves in a straight line. The velocity $v{\text{ ms}}^{-1}$ of $P$, at time $t\text{ s}$ after leaving $O$, is given by $v=\frac{1}{3}(2t-3)(t-4)$.

A block $A$ of mass $2\text{ kg}$ and a particle $B$ of mass $0.5\text{ kg}$ are connected by a light inextensible string inclined at ${15}^{\circ }$ to the horizontal. They are pulled across a horizontal surface with acceleration $1.2{\text{ ms}}^{-2}$ by a force of magnitude $6\text{ N}$, applied to $A$, acting at ${20}^{\circ }$ above the horizontal as shown in the diagram. The string and the force applied to $A$ are in the same vertical plane. The contact between $B$ and the surface is smooth and the contact between $A$ and the surface is rough.