Pre-U Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) 2020 Specimen

1 The discrete random variable X has probability generating function \(\mathrm { G } _ { X } ( t )\) given by $$G _ { X } ( t ) = a t \left( t + \frac { 1 } { t } \right) ^ { 3 }$$ where \(a\) is a constant.

1. Find, in either order, the value of \(a\) and the set of values that \(X\) can take.
2. Find the value of \(\mathrm { E } ( X )\).

4 The independent random variables \(X\) and \(Y\) have normal distributions where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 3 \mu , 4 \sigma ^ { 2 } \right)\). Two random samples each of size \(n\) are taken, one from each of these normal populations.

1. Show that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\) provided that \(a + 3 b = 1\), where \(a\) and \(b\) are constants and \(\bar { X }\) and \(\bar { Y }\) are the respective sample means. In the remainder of the question assume that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\).
2. Show that \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\) can be written as \(\frac { \sigma ^ { 2 } } { n } \left( 1 - 6 b + 13 b ^ { 2 } \right)\).
3. The value of the constant \(b\) can be varied. Find the value of \(b\) that gives the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\), and hence find the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\) in terms of \(\sigma\) and \(n\).

5 The random variable \(X\) has probability density function \(\mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = \begin{cases} k e ^ { - k x } & x \geqslant 0 \\ 0 & x < 0 \end{cases}$$ 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 \(\operatorname { Var } ( X )\).
3. Show that the moment generating function of \(- X\) is \(k ( k + t ) ^ { - 1 }\).
4. \(X _ { 1 }\) and \(X _ { 2 }\) are two independent observations of \(X\). Use the moment generating function of \(X _ { 1 } - X _ { 2 }\) to find the value of \(\mathrm { E } \left[ \left( X _ { 1 } - X _ { 2 } \right) ^ { 2 } \right]\).

6 The lengths of time, in years, that sales representatives for a certain company keep their company cars may be modelled by the distribution with probability density function \(\mathrm { f } ( x )\), where $$f ( x ) = \left\{ \begin{array} { c c } \frac { 4 } { 27 } x ^ { 2 } ( 3 - x ) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise. } \end{array} \right.$$

1. Draw a sketch of this probability density function.
2. Calculate the mean and the mode of \(X\).
3. Comment briefly on the values obtained in part (b) in relation to the sketch in part (a).
4. Show that the lower quartile \(\mathrm { Q } _ { 1 }\) of \(X\) satisfies the equation \(\mathrm { Q } _ { 1 } { } ^ { 4 } - 4 \mathrm { Q } _ { 1 } { } ^ { 3 } + 6.75 = 0\), and use an appropriate numerical method to find the value of \(\mathrm { Q } _ { 1 }\) correct to 2 decimal places, showing full details of your method.

8 \includegraphics[max width=\textwidth, alt={}, center]{f4acd242-eb78-4124-bfa2-fdecaa188690-4_614_741_1548_662} 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. Find the tension in the string in terms of \(m , l\) and \(\omega\).
2. Show that \(\omega ^ { 2 } h = g\).
3. Deduce an expression in terms of \(g\) and \(h\) for the time taken for \(P\) to complete one full circle during its motion.

9 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]{f4acd242-eb78-4124-bfa2-fdecaa188690-5_657_659_392_705} Given that the value of the coefficient of friction between the rod and the ground is 0.2 , determine whether the rod slips.

10 A cyclist and her bicycle 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. Given that the steady speed at which the cyclist can move is \(10 \mathrm {~ms} ^ { - 1 }\), show that \(k = \frac { 3 } { 4 }\).
2. 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 \(7 \mathrm {~ms} ^ { - 1 }\).

11 \includegraphics[max width=\textwidth, alt={}, center]{f4acd242-eb78-4124-bfa2-fdecaa188690-6_438_953_264_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 particle is projected from the origin with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal.

1. Prove that the equation of its trajectory is $$y = x \tan \alpha - \frac { x ^ { 2 } } { 80 } \left( l + \tan ^ { 2 } \alpha \right) .$$
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 of \(30 ^ { \circ }\) to the horizontal. The line \(l\), with equation \(y = x \tan 30 ^ { \circ }\), is a line of greatest slope in the plane. The particle is projected from the origin with speed \(20 \mathrm {~ms} ^ { - 1 }\) from a point on the plane, in the vertical plane containing \(l\). By considering the intersection of \(l\) with the curve \(y = 20 - \frac { x ^ { 2 } } { 80 }\), find the maximum range up this inclined plane.

13 Two light strings, each of natural length \(l\) and modulus of elasticity \(6 m g\), are attached at their ends to a particle \(P\) of mass \(m\). The other ends of the strings are attached to two fixed points \(A\) and \(B\), which are at a distance \(6 l\) apart on a smooth horizontal table. Initially \(P\) is at rest at the mid-point of \(A B\). The particle is now given a horizontal impulse in the direction perpendicular to \(A B\). At time \(t\) the displacement of \(P\) from the line \(A B\) is \(x\).

1. Show that the tension in each string is \(\frac { 6 m g } { l } \left( \sqrt { 9 l ^ { 2 } + x ^ { 2 } } - l \right)\).
2. Show that $$\ddot { x } = - \frac { 12 g x } { l } \left( 1 - \frac { 1 } { \sqrt { 9 l ^ { 2 } + x ^ { 2 } } } \right)$$
3. Given that throughout the motion \(\frac { x ^ { 2 } } { l ^ { 2 } }\) is small enough to be negligible, show that the equation of motion is approximately $$\ddot { x } = - \frac { 8 g x } { l } .$$
4. Given that the initial speed of \(P\) is \(\sqrt { \frac { g l } { 200 } }\), find the time taken for the particle to travel a distance of \(\frac { 1 } { 80 } l\).