A Level Mathematics 9709 Paper 41, May/June 2025

Find the total time taken for the journey from $O$ to $P$.

On the given axes, sketch a displacement-time graph for the cyclist’s journey from $O$ to $P$, showing on your graph the distances travelled after $10\text{ s}$ and $40\text{ s}$.

On a horizontal section of the road, the power of the lorry’s engine is constant. There is a constant resistance to motion of $1600\text{ N}$. The steady speed which the lorry can maintain with the engine working at power $P\text{ W}$ is $30\text{ m}{\text{s}}^{-1}$. Find the value of $P$.

At an instant when the speed of the lorry is $16\text{ m}{\text{s}}^{-1}$, its engine is working at a power of $40\text{ kW}$. Find the acceleration of the lorry at this instant.

When the lorry has reached a speed of $20\text{ m}{\text{s}}^{-1}$, it begins to ascend a section of road inclined at an angle ${\alpha }^{\circ }$ to the horizontal. The engine now works at a power of $120\text{ kW}$. There is no change in the lorry’s speed as it ascends the hill. The constant resistance to motion remains $1600\text{ N}$. Find the value of $\alpha$.

When a particle $P$ of mass $m\text{ kg}$ has speed $u\text{ m}{\text{s}}^{-1}$, its momentum is $4\text{ N}\text{s}$ and its kinetic energy is $16\text{ J}$. Find the value of $m$ and the value of $u$.

P is now projected on a smooth horizontal surface with speed $v\text{ m}{\text{s}}^{-1}$ directly towards a particle $Q$ of mass $1.25\text{ kg}$ which is stationary. After $P$ and $Q$ collide, the velocity of $P$ is $w\text{ m}{\text{s}}^{-1}$ and the velocity of $Q$ is $2w\text{ m}{\text{s}}^{-1}$. The loss of kinetic energy in the collision is $25\text{ J}$. Find the value of $v$ and the value of $w$.

It is given that the plane $BC$ is smooth. Find the tension in the string and the acceleration of $Q$.

It is given instead that the plane $BC$ is rough. The work done against the frictional force when $Q$ moves $2\text{ m}$ down the plane is $1.8\text{ J}$. You should assume that $P$ does not reach the pulley and that $Q$ does not reach $C$. Use an energy method to find the speed of $Q$ when it has moved $2\text{ m}$ down the plane.

Find the time at which $X$ is instantaneously at rest, and hence find the total distance travelled by $X$ between $t=0$ and $t=16$.

A second particle $Y$ moves along another straight track, starting from a point $P$ at time $t=0$. The acceleration of $Y$ at time $t\text{ s}$ is $a\text{ m}{\text{s}}^{-2}$, where $a=0.8-0.6t$. The velocity of $Y$ when it leaves $P$ is $7.5\text{ m}{\text{s}}^{-1}$. When the velocity of $Y$ is $-9.6\text{ m}{\text{s}}^{-1}$, show that the displacement of $X$ from $O$ is equal to the displacement of $Y$ from $P$.