Pre-U Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) 2010 June

1 A lorry moves along a straight horizontal road. The engine of the lorry produces a constant power of 80 kW . The mass of the lorry is 10 tonnes and the resistance to motion is constant at 4000 N .

1. Express the driving force of the lorry in terms of its velocity and hence, using Newton's second law, write down a differential equation which connects the velocity of the lorry and the time for which it has been moving.
2. Hence find the time taken, in seconds, for the lorry to accelerate from \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

2 At 1200 hours an aircraft, \(A\), sets out to intercept a second aircraft, \(B\), which is 200 km away on a bearing of \(300 ^ { \circ }\) and is flying due east at \(600 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Both aircraft are at the same altitude and continue to fly horizontally.

1. (a) Find the bearing on which \(A\) should fly when travelling at \(800 \mathrm {~km} \mathrm {~h} ^ { - 1 }\).
   (b) Find the time at which \(A\) intercepts \(B\) in this case.
2. Find the least steady speed at which \(A\) can fly to intercept \(B\).

3 A particle is projected at an angle \(\theta\) above the horizontal from the foot of a plane which is inclined at \(45 ^ { \circ }\) to the horizontal. Subsequently the particle impacts on the plane at a higher point.

1. Prove that the angle at which the particle strikes the plane is \(\phi\), where $$\tan \phi = \frac { \tan \theta - 1 } { 3 - \tan \theta }$$
2. Find the angle to the horizontal at which the particle would have to be projected if it were to strike the plane horizontally.

4 One end of a light elastic string of natural length 0.2 m and modulus of elasticity 100 N is attached to a fixed point \(A\). The other end is attached to a particle of mass 5 kg . The particle moves with angular speed \(\omega\) radians per second in a horizontal circle with the centre vertically below \(A\). The string makes an angle \(\theta\) with the vertical.

1. By considering the horizontal component of the tension in the string, show that the tension in the string is \(( 1 + 5 x ) \omega ^ { 2 } \mathrm {~N}\), where \(x\) is the extension, in metres, of the string.
2. (a) By considering vertical forces and also Hooke's law, deduce that \(\cos \theta = \frac { 1 } { 10 x }\).
   (b) Show that \(\omega > \frac { 10 \sqrt { 3 } } { 3 }\).
3. When the value of \(\omega\) is \(5 \sqrt { 2 }\), find the radius of the circular motion.

5 A particle of mass \(m\) is attached by a light elastic string of natural length \(l\) and modulus of elasticity \(\lambda\) to a fixed point \(A\), from which it is allowed to fall freely. The particle first comes to rest, instantaneously, at \(B\), where \(A B = 2 l\). Prove that

1. \(\lambda = 4 m g\),
2. while the string is taut, \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - \frac { 4 g } { l } x\), where \(x\) is the displacement from the equilibrium position at time \(t\),
3. the time taken between the first occasion when the string becomes taut and the next occasion when it becomes slack is $$\left[ \frac { 1 } { 2 } \pi + \sin ^ { - 1 } \left( \frac { 1 } { 3 } \right) \right] \sqrt { \frac { l } { g } }$$

6 Two smooth spheres, \(A\) and \(B\), have masses \(m\) and \(2 m\) respectively and equal radii. Sphere \(B\) is at rest on a smooth horizontal floor. Sphere \(A\) is projected with speed \(u\) along the floor in a direction parallel to a smooth vertical wall and strikes \(B\) obliquely. Subsequently \(B\) strikes the wall at an angle \(\alpha\) with the wall. The coefficient of restitution between \(A\) and \(B\) and between \(B\) and the wall is 0.5. After \(B\) has struck the wall, \(A\) and \(B\) are moving parallel to each other.

1. Write down a momentum equation and a restitution equation along the line of centres for the impact between \(A\) and \(B\). Hence find the components of velocity of \(A\) and \(B\) in this direction after this first impact.
2. Find the value of \(\alpha\), giving your answer in degrees.

7 The number of goals scored by a hockey team in an interval of time of length \(t\) minutes follows a Poisson distribution with mean \(\frac { 1 } { 24 } t\). The random variable \(T\) is defined as the length of time, in minutes, between successive goals.

1. (a) Show that \(\mathrm { P } ( T < t ) = 1 - \mathrm { e } ^ { - \frac { 1 } { 24 } t }\) for \(t \geqslant 0\).
   (b) Hence find the probability density function of \(T\).
2. Find the exact value of the interquartile range of \(T\).

8 Two groups of Year 12 pupils, one at each of schools \(A\) and \(B\), are given the same mathematics test. The scores, \(x\) and \(y\), of pupils at schools \(A\) and \(B\) respectively are summarised as follows.

1. Assuming that the two groups are random samples from independent normal populations with means \(\mu _ { A }\) and \(\mu _ { B }\) respectively and a common, but unknown, variance, construct a \(98 \%\) confidence interval for \(\mu _ { A } - \mu _ { B }\).
2. Comment, with a reason, on any difference in ability between the two schools.

9

1. Two independent discrete random variables \(X\) and \(Y\) follow Poisson distributions with means \(\lambda\) and \(\mu\) respectively. Prove that the discrete random variable \(Z = X + Y\) follows a Poisson distribution with mean \(\lambda + \mu\). A garage has a white limousine and a green limousine for hire. Demands to hire the white limousine occur at a constant mean rate of 3 per week and demands to hire the green limousine occur at a constant mean rate of 2 per week. Demands for hire are received independently and randomly.
2. Calculate the probability that in a period of two weeks
   1. no demands for hire are received, giving your answer to 3 significant figures,
   2. seven demands for hire are received.
   3. Find the least value of \(n\) such that the probability of at least \(n\) demands for hire in a period of three weeks is less than 0.005 .

10 A box contains a large number, \(n\), of identical dice, which are thought to be biased. The probability that one of these dice will show a six in a single roll is \(p\). The \(n\) dice are rolled many times and the number of sixes obtained in each trial is recorded. In \(4.01 \%\) of these trials 56 or more dice showed a six. In \(10.56 \%\) of these trials 37 or fewer dice showed a six. Using a suitable normal approximation, find the values of \(n\) and \(p\).

11 The thickness of a randomly chosen paperback book is \(P \mathrm {~cm}\) and the thickness of a randomly chosen hardback is \(H \mathrm {~cm}\), where \(P\) and \(H\) have distributions \(\mathrm { N } ( 2.0,0.75 )\) and \(\mathrm { N } ( 5.0,2.25 )\) respectively. When more than one book is selected, any book is selected independently of all other books.

1. Calculate the probability that a randomly chosen hardback is more than 1 cm thicker than a randomly chosen paperback.
2. Calculate the probability that 2 paperbacks and 4 hardbacks, randomly chosen, have a combined thickness of less than 20 cm .
3. Find the probability that a randomly chosen hardback is more than twice the thickness of a randomly chosen paperback.

12 Two players, \(A\) and \(B\), are taking turns to shoot at a basket with a basketball. The winner of this game is the first player to score a basket. The probability that \(A\) scores a basket with any shot is \(\frac { 1 } { 4 }\) and the probability that \(B\) scores a basket with any shot is \(\frac { 1 } { 5 }\). Each shot is independent of all other shots. \(A\) shoots first.

1. Find
   1. the probability that \(B\) wins with his first shot,
   2. the probability that \(A\) wins with his second shot,
   3. the probability that \(A\) wins the game.
   4. \(R\) is the total number of shots taken by \(A\) and \(B\) up to and including the shot that scores a basket.
      (a) Show that the probability generating function of \(R\) is given by $$\mathrm { G } ( t ) = \frac { 5 t + 3 t ^ { 2 } } { 4 \left( 5 - 3 t ^ { 2 } \right) }$$ (b) Hence find \(\mathrm { E } ( R )\).