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

1 A ship \(A\) is steaming north at \(20 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Initially a ship \(B\) is at a distance 8 km due west of \(A\), and is steaming on a course such that it will take up a position 8 km directly ahead of \(A\) as quickly as possible.

1. Given that the maximum speed of B is \(35 \mathrm {~km} \mathrm {~h} ^ { - 1 }\), show that the bearing of this course is \(021 ^ { \circ }\), correct to the nearest degree.
2. Find the distance that \(A\) moves between the instants when \(B\) is due west of \(A\) and when \(B\) is due north of \(A\), giving your answer to the nearest kilometre.

2 A smooth uniform ball travelling along a smooth horizontal table collides with a second smooth uniform ball of the same mass and radius which is at rest on the table. At the moment of impact the line of centres makes an angle of \(30 ^ { \circ }\) with the direction in which the first ball is moving. If the coefficient of restitution between the balls is \(e\), show that

1. the component of the first ball's velocity, along the line of centres, after the impact is $$\frac { \sqrt { 3 } u } { 4 } ( 1 - e )$$
2. the first ball is deflected by the impact through an angle \(\theta\), where $$\tan \theta = \frac { ( 1 + e ) \sqrt { 3 } } { 5 - 3 e }$$

3 A light spring, of natural length 0.4 m and modulus of elasticity 6.4 N , has one end \(A\) attached to the ceiling of a room. A particle of mass \(m \mathrm {~kg}\) is attached to the free end of the spring and hangs in equilibrium. The particle is displaced vertically downwards and released from rest. In the subsequent motion the particle does not reach the ceiling and air resistance may be neglected.

1. Show that the particle oscillates in simple harmonic motion.
2. Given that the period of the motion is 1.12 s , find
   1. the value of \(m\), correct to 3 significant figures,
   2. the extension of the spring when the particle has a downwards acceleration of \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

4 A particle is projected with velocity \(V\), at an angle of elevation of \(60 ^ { \circ }\) to the horizontal, from a point on a plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The path of the particle is in a vertical plane through a line of greatest slope. If \(R _ { 1 }\) and \(R _ { 2 }\) are the respective ranges when the particle is projected up the plane and down the plane, show that $$R _ { 2 } = 2 R _ { 1 }$$

5 When a car of mass 990 kg moves at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a horizontal straight road, the power of its engine is 8.8 kW .

1. Find the magnitude of the resistance to the motion of the car at this speed.
2. Assuming that the resistance has magnitude \(k v ^ { 2 }\) newtons when the speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the value of the constant \(k\). The power of the engine is now increased to 22 kW and remains constant at this value.
3. Using the model in part (ii), show that $$\frac { \mathrm { d } v } { \mathrm {~d} x } = \frac { 20000 - v ^ { 3 } } { 900 v ^ { 2 } } .$$
4. Hence show that the car moves about 300 m as its speed increases from \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

6 A simple pendulum consists of a light inextensible string of length 1.5 m with a small bob of mass 0.2 kg at one end. When suspended from a fixed point and hanging at rest under gravity, the bob is given a horizontal speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it comes instantaneously to rest when the string makes an angle of 0.1 rad with the vertical. At time \(t\) seconds after projection the string makes an angle \(\theta\) with the vertical.

1. Show that, neglecting air resistance, $$\left( \frac { \mathrm { d } \theta } { \mathrm {~d} t } \right) ^ { 2 } = \frac { 40 } { 3 } \{ \cos \theta - \cos ( 0.1 ) \}$$
2. Find, correct to 2 significant figures,
   1. the value of \(u\),
   2. the tension in the string when \(\theta = 0.05 \mathrm { rad }\).
   3. By differentiating the above equation for \(\left( \frac { \mathrm { d } \theta } { \mathrm { d } t } \right) ^ { 2 }\), or otherwise, show that the motion of the bob can be modelled approximately by simple harmonic motion.
   4. Hence find the value of \(t\) at which the bob first comes instantaneously to rest.

7 The time taken for me to walk from my house to the bus stop has a normal distribution with mean 10 minutes and standard deviation 1.5 minutes. The arrival time of the bus is normally distributed with mean 0900 and standard deviation 1 minute. If the bus arrives early it does not wait. I leave home at 0845 . Find, correct to 3 decimal places, the probability that I catch the bus.

8 Specimens of rain were collected at random from the north and south sides of an island and analysed for sulphur content. The results (in suitable units) are given below. Assume that the sulphur contents have normal distributions with population means \(\mu _ { N }\) and \(\mu _ { S }\) and a common, but unknown, variance.

1. Calculate a symmetric \(95 \%\) confidence interval for the difference in population mean sulphur contents of the rain on the north and south sides of the island, \(\mu _ { S } - \mu _ { N }\).
2. Comment on a claim that the mean sulphur content is the same on both sides of the island.

9 The service time, \(X\) minutes, for each customer in a post office is modelled by the probability density function given by $$\mathrm { f } ( x ) = \begin{cases} 0.2 \mathrm { e } ^ { - 0.2 x } & x > 0 , \\ 0 & \text { otherwise } . \end{cases}$$ Two customers begin to be served independently at the same instant. The larger of the two service times is \(T\) minutes.

1. By considering the probability that both customers have been served in less than \(t\) minutes, show that the cumulative distribution function of \(T\) is given by $$\mathrm { G } ( t ) = \begin{cases} 1 - 2 \mathrm { e } ^ { - 0.2 t } + \mathrm { e } ^ { - 0.4 t } & t > 0 \\ 0 & \text { otherwise } \end{cases}$$
2. Find the probability that both customers are served within 10 minutes.
3. Find the value of the interquartile range of \(T\).

10

1. \(X , Y\) and \(Z\) are independent random variables having Poisson distributions with means \(\lambda , \mu\) and \(\lambda + \mu\) respectively. Find \(\mathrm { P } ( X = 0\) and \(Y = 2 ) , \mathrm { P } ( X = 1\) and \(Y = 1 )\) and \(\mathrm { P } ( X = 2\) and \(Y = 0 )\). Hence verify that \(\mathrm { P } ( X + Y = 2 ) = \mathrm { P } ( Z = 2 )\).
2. In an office the male absence rate, i.e. the number of working days lost each month due to the absence of male employees, has a Poisson distribution with mean 4.5. In the same office the female absence rate has an independent Poisson distribution with mean 4.1. Calculate the probability that
   1. during a particular month both the male absence rate and the female absence rate are equal to 3,
   2. during a particular month the total of the male and female absence rates is equal to 6,
   3. during a particular month the male and female absence rates were each equal to 3 , given that the total of the male and female absence rates was equal to 6 .

11 The time, \(T\) years, before a particular type of washing machine breaks down may be taken to have probability density function f given by $$\mathrm { f } ( t ) = \begin{cases} a t \mathrm { e } ^ { - b t } & t > 0 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are positive constants. It may be assumed that, if \(n\) is a positive integer, $$\int _ { 0 } ^ { \infty } t ^ { n } \mathrm { e } ^ { - b t } \mathrm {~d} t = \frac { n ! } { b ^ { n + 1 } }$$

1. Records show that the mean of \(T\) is 1.5 . Show that \(b = \frac { 4 } { 3 }\) and find the value of \(a\).
2. Find \(\operatorname { Var } ( T )\).
3. Calculate \(\mathrm { P } ( T < 1.5 )\). State, giving a reason, whether this value indicates that the median of \(T\) is smaller than the mean of \(T\) or greater than the mean of \(T\).

12 A game is played in which the number of points scored, \(X\), has the probability distribution given in the following table.

1. Write down the probability generating function of \(X\).
2. Use this generating function to find the mean and variance of \(X\).
3. The game is played 4 times (independently) and the total number of points scored is denoted by \(Y\). Show that the probability generating function of \(Y\) can be written in the form $$\frac { \left( a + t ^ { 2 } \right) ^ { 8 } } { b t ^ { 4 } }$$ where \(a\) and \(b\) are constants.
4. Hence find \(\mathrm { P } ( Y < 0 )\).