A Level Further Mathematics 9231 Paper 33, May/June 2023

A particle $P$ of mass $m$ is attached to one end of a light inextensible string of length $a$. The other end of the string is attached to a fixed point $O$. The particle $P$ is held at the point $A$, where $OA$ makes an angle $\alpha$ with the downward vertical through $O$, and with the string taut. The particle $P$ is projected perpendicular to $OA$ in an upwards direction with speed $\sqrt{3ag}$. It then starts to move along a circular path in a vertical plane. The string goes slack when $P$ is at $B$, where $OB$ makes an angle $\theta$ with the upward vertical. Given that $\cos \alpha =\frac{4}{5}$, find the value of $\cos \theta$.

One end of a light elastic string, of natural length $a$ and modulus of elasticity $\lambda mg$, is attached to a fixed point $O$. The string lies on a smooth horizontal surface. A particle $P$ of mass $m$ is attached to the other end of the string. The particle $P$ is projected in the direction $OP$. When the length of the string is $\frac{4}{3}a$, the speed of $P$ is $\sqrt{2ag}$. When the length of the string is $\frac{5}{3}a$, the speed of $P$ is $\frac{1}{2}\sqrt{2ag}$. Find the value of $\lambda$.

A uniform lamina is in the form of a triangle $ABC$, with $AC=8a$, $BC=6a$ and angle $ACB={90}^{\circ }$. The point $D$ on $AC$ is such that $AD=3a$. The point $E$ on $CB$ is such that $CE=x$ (see diagram). The triangle $CDE$ is removed from the lamina.

Two identical smooth uniform spheres $A$ and $B$ each have mass $m$. The two spheres are moving on a smooth horizontal surface when they collide with speeds $u$ and $2u$ respectively. Immediately before the collision, $A$’s direction of motion makes an angle of ${30}^{\circ }$ with the line of centres, and $B$’s direction of motion is perpendicular to the line of centres (see diagram). After the collision, $A$ and $B$ are moving in the same direction. The coefficient of restitution between the spheres is $e$.

One end of a light elastic string, of natural length $12a$ and modulus of elasticity $kmg$, is attached to a fixed point $O$. The other end of the string is attached to a particle of mass $m$. The particle moves with constant speed $\frac{3}{2}\sqrt{3ag}$ in a horizontal circle with centre at a distance $12a$ below $O$. The string is inclined at an angle $\theta$ to the downward vertical through $O$.

A particle of mass $m\text{ kg}$ falls vertically under gravity, from rest. At time $t\text{ s}$, $P$ has fallen $x\text{ m}$ and has velocity $v\text{ m}{\text{ s}}^{-1}$. The only forces acting on $P$ are its weight and a resistance of magnitude $kmgv\text{ N}$, where $k$ is a constant.

The points $O$ and $P$ are on a horizontal plane, a distance $8\text{ m}$ apart. A ball is thrown from $O$ with speed $u\text{ m}{\text{ s}}^{-1}$ at an angle $\theta$ above the horizontal, where $\tan \theta =\frac{4}{3}$. At the same instant, a model aircraft is launched with speed $5\text{ m}{\text{ s}}^{-1}$ parallel to the horizontal plane from a point $4\text{ m}$ vertically above $P$. The model aircraft moves in the same vertical plane as the ball and in the same horizontal direction as the ball. The model aircraft moves horizontally with a constant speed of $5\text{ m}{\text{ s}}^{-1}$. After $T\text{ s}$, the ball and the model aircraft collide.