Pre-U Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) 2016 Specimen

1 The random variable \(X\) has probability density function \(\mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = \begin{cases} k \mathrm { e } ^ { - k x } & x \geqslant 0 \\ 0 & x < 0 \end{cases}$$ and \(k\) is a positive constant.

1. Show that the moment generating function of \(X\) is \(\mathrm { M } _ { X } ( t ) = k ( k - t ) ^ { - 1 } , t < k\).
2. Use the moment generating function to find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

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1. The random variable \(X\) has the distribution \(\operatorname { Po } ( \lambda )\). Prove that the probability generating function, \(\mathrm { G } _ { X } ( t )\), is given by $$\mathrm { G } _ { X } ( t ) = \mathrm { e } ^ { \lambda ( t - 1 ) }$$
2. The independent random variables \(X\) and \(Y\) have distributions \(\operatorname { Po } ( \lambda )\) and \(\operatorname { Po } ( \mu )\) respectively. Use probability generating functions to show that the distribution of \(X + Y\) is \(\operatorname { Po } ( \lambda + \mu )\).
3. Given that \(X \sim \operatorname { Po } ( 1.5 )\) and \(Y \sim \operatorname { Po } ( 2.5 )\), find \(\mathrm { P } ( X \leqslant 2 \mid X + Y = 4 )\).

7 A cyclist and her machine have a combined mass of 90 kg and she is riding along a straight horizontal road. She is working at a constant power of 75 W . At time \(t\) seconds her speed is \(v \mathrm {~ms} ^ { - 1 }\) and the resistance to motion is \(k v \mathrm {~N}\), where \(k\) is a constant.

1. If the cyclist's maximum steady speed is \(10 \mathrm {~ms} ^ { - 1 }\), show that \(k = \frac { 3 } { 4 }\).
2. Use Newton's second law to show that $$\frac { 25 } { v } - \frac { v } { 4 } = 30 \frac { \mathrm {~d} v } { \mathrm {~d} t } .$$
3. Find the time taken for the cyclist to accelerate from a speed of \(3 \mathrm {~ms} ^ { - 1 }\) to a speed of \(7 \mathrm {~ms} ^ { - 1 }\).

8 The diagram shows a uniform rod \(A B\) of length 40 cm and mass 2 kg placed with the end \(A\) resting against a smooth vertical wall and the end \(B\) on rough horizontal ground. The angle between \(A B\) and the horizontal is \(60 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{a19fab61-da1c-4803-9dbc-38d618a0c58e-4_657_655_1128_705}

1. Given that the value of the coefficient of friction between the rod and the ground is 0.2 , determine whether the rod slips.
2. Explain why it is impossible for the rod to be in equilibrium with one end on smooth horizontal ground and the other against a rough vertical wall.

9 \includegraphics[max width=\textwidth, alt={}, center]{a19fab61-da1c-4803-9dbc-38d618a0c58e-5_671_817_255_623} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point \(A\). The particle moves with constant angular speed \(\omega\) in a horizontal circle whose centre is at a distance \(h\) vertically below \(A\) (see diagram).

1. Show that however fast the particle travels \(A P\) will never become horizontal, and that the tension in the string is always greater than the weight of the particle.
2. Find the tension in the string in terms of \(m , l\) and \(\omega\).
3. Show that \(\omega ^ { 2 } h = g\) and calculate \(\omega\) when \(h\) is 0.5 m .

10 \includegraphics[max width=\textwidth, alt={}, center]{a19fab61-da1c-4803-9dbc-38d618a0c58e-5_435_951_1528_557} A smooth sphere \(P\) of mass \(3 m\) is at rest on a smooth horizontal table. A second smooth sphere \(Q\) of mass \(m\) and the same radius as \(P\) is moving along the table towards \(P\) and strikes it obliquely (see diagram). After the collision, the directions of motion of the two spheres are perpendicular.

1. Find the coefficient of restitution.
2. Given that one-sixth of the original kinetic energy is lost as a result of the collision, find the angle between the initial direction of motion of \(Q\) and the line of centres.

12 A projectile is launched from the origin with speed \(20 \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal.

1. Prove that the equation of its trajectory is $$y = x \tan \alpha - \frac { x ^ { 2 } } { 80 } \left( 1 + \tan ^ { 2 } \alpha \right) .$$
2. Regarding the equation of the trajectory as a quadratic equation in \(\tan \alpha\), show that \(\tan \alpha\) has real values provided that $$y \leqslant 20 - \frac { x ^ { 2 } } { 80 } .$$
3. A plane is inclined at an angle \(\beta\) to the horizontal. The line \(l\), with equation \(y = x \tan \beta\), is a line of greatest slope in the plane. A particle is projected from a point on the plane, in the vertical plane containing \(l\). By considering the intersection of \(l\) with the bounding parabola \(y = 20 - \frac { x ^ { 2 } } { 80 }\), deduce that the maximum range up, or down, this inclined plane is \(\frac { 40 } { 1 + \sin \beta }\), or \(\frac { 40 } { 1 - \sin \beta }\), respectively.