CAIE Further Paper 3 (Further Paper 3) 2024 June

3 At time \(t = 0\) seconds, a particle \(P\) is projected with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(60 ^ { \circ }\) above the horizontal from a point \(O\). In the subsequent motion \(P\) moves freely under gravity. The direction of motion of \(P\) when \(t = 5\) is perpendicular to its direction of motion when \(t = 15\). Find the value of \(u\).
\includegraphics[max width=\textwidth, alt={}, center]{c1a3340d-158d-4c37-9577-96074e59ac3d-08_419_876_255_596} A ring of weight \(W\), with radius \(a\) and centre \(O\), is at rest on a rough surface that is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac { 1 } { 2 }\). The plane of the ring is perpendicular to the inclined surface and parallel to a line of greatest slope of the surface. The point \(P\) on the circumference of the ring is such that \(O P\) is parallel to the surface. A light inextensible string is attached to \(P\) and to the point \(Q\), which is on the surface, such that \(P Q\) is horizontal (see diagram). The points \(O , P\) and \(Q\) are in the same vertical plane. The system is in limiting equilibrium and the coefficient of friction between the ring and the surface is \(\mu\).

1. Find, in terms of \(W\), the tension in the string \(P Q\).
2. Find the value of \(\mu\).

5 Two particles \(A\) and \(B\) of masses \(m\) and \(k m\) respectively are connected by a light inextensible string of length \(a\). The particles are placed on a rough horizontal circular turntable with the string taut and lying along a radius of the turntable. Particle \(A\) is at a distance \(a\) from the centre of the turntable and particle \(B\) is at a distance \(2 a\) from the centre of the turntable. The coefficient of friction between each particle and the turntable is \(\frac { 1 } { 5 }\).
When the turntable is made to rotate with angular speed \(\frac { 2 } { 5 } \sqrt { \frac { \mathrm {~g} } { \mathrm { a } } }\), the system is in limiting equilibrium.

1. Find the tension in the string, in terms of \(m\) and \(g\).
2. Find the value of \(k\).

6 A particle \(P\) of mass 2 kg moving on a horizontal straight line has displacement \(x \mathrm {~m}\) from a fixed point \(O\) on the line and velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\). The only horizontal force acting on \(P\) has magnitude \(\frac { 1 } { 10 } ( 2 \mathrm { v } - 1 ) ^ { 2 } \mathrm { e } ^ { - \mathrm { t } } \mathrm { N }\) and acts towards \(O\). When \(t = 0 , x = 1\) and \(v = 3\).

1. Find an expression for \(v\) in terms of \(t\).
   \includegraphics[max width=\textwidth, alt={}, center]{c1a3340d-158d-4c37-9577-96074e59ac3d-12_69_1569_466_328}
2. Find an expression for \(x\) in terms of \(t\).