9709 Mathematics November 2025 Question Paper 41

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

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

Coplanar forces of magnitudes 45 N, 28 N, 72 N and 35 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 $m$ kg and 3 kg respectively, are connected by a light inextensible string. Particle $B$ is on a fixed plane at an angle of $\sin^{-1} 0.6$ to the horizontal ground; the string passes over a smooth pulley at the top of the plane and $A$ hangs vertically, 0.75 m above the ground. The system is released from rest; $B$ moves up the plane (not reaching the pulley) against a constant resistance of $\tfrac{10}{3}$ N, and the speed of $A$ immediately before it hits the ground is $2\text{ m s}^{-1}$. Use an energy method to find the value of $m$.

A particle $P$ of mass $m$ kg is in equilibrium on a rough plane inclined at an angle $\theta\degree$ to the horizontal. The equilibrium of $P$ is maintained by a force of magnitude $8mg$ N making an angle $\theta\degree$ 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 kg is released from rest from the top of a smooth plane inclined at $\sin^{-1} 0.2$ to the horizontal. $A$ collides 2 seconds later with a particle $B$, of mass 3 kg, moving up a line of greatest slope of the plane. The speed of $B$ immediately before the collision is $3.5\text{ m s}^{-1}$. Immediately after the collision, $B$ has a velocity of $0.5\text{ m s}^{-1}$ 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 $v\text{ m s}^{-1}$ of $P$, at time $t$ s after leaving $O$, is given by $v = \tfrac13(2t-3)(t-4)$.

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