Pre-U Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) 2017 June

1

1. Explain the meaning of the term ' \(95 \%\) confidence interval'.
2. The values of five independent observations of a normally distributed random variable are as follows. $$\begin{array} { l l l l l } 35.2 & 38.2 & 39.7 & 41.6 & 43.9 \end{array}$$ Obtain a 95\% confidence interval for the population mean.

2 A discrete random variable \(X\) has the following probability distribution.

1. Write down the probability generating function of \(X\).
2. \(T\) is the sum of ten independent observations of \(X\). Use the probability generating function of \(T\) to find
   1. \(\mathrm { E } ( T )\),
   2. \(\mathrm { P } ( T = 8 )\).

3 In a random sample of 100 voters from a constituency, 32 said that they would support the Cyan Party.

1. Find an approximate \(99 \%\) confidence interval for the proportion of voters in the constituency who would support the Cyan Party.
2. Using the given sample proportion, estimate the smallest size of sample needed for the width of a \(99 \%\) confidence interval to be less than 0.04 .

4 A continuous random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \begin{cases} a & - 1 \leqslant x < 0 \\ a \left( 1 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.

1. Find the value of \(a\).
2. Find the cumulative distribution function of \(X\).
3. Determine whether the upper quartile is greater than or less than 0.25 .

5 The number of calls to a car breakdown service during any one hour of the day is modelled by the distribution \(\operatorname { Po } ( 20 )\).

1. Find the probability that in a randomly chosen 12 -minute period there are at least 7 calls to the service.
2. Find the period of time, correct to the nearest second, for which the probability that no calls are made to the service is 0.6 .
3. Use a suitable approximation to find the probability that, in a randomly chosen 3-hour period, there are no more than 65 calls to the service.

6 The random variable \(X\) has a uniform distribution on the interval \([ - 1,1 ]\), so that its probability density function is given by $$f ( x ) = \begin{cases} \frac { 1 } { 2 } & - 1 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$

1. Show from the definition of the moment generating function that the moment generating function of \(X\) is \(\frac { \sinh t } { t }\).
2. By using the series expansion of \(\sinh t\), find the variance of \(X\) and the value of \(\mathrm { E } \left( X ^ { 4 } \right)\).

7 The total mass of a can of pears is the sum of three independent random variables: the mass of pears, the mass of juice, and the mass of the container. The mass in grams of pears in a can has the distribution \(\mathrm { N } ( 300,400 )\). The mass in grams of juice has the distribution \(\mathrm { N } ( 200,60 )\). The mass in grams of the container has the distribution \(\mathrm { N } ( 70,10 )\).

1. Find the probability that the total mass of a randomly chosen can is less than 530 g .
2. Find the probability that the mass of the container of a randomly chosen can is more than one eighth of the total mass of the can.

8 A horizontal turntable rotates about a vertical axis. Starting from rest, it accelerates uniformly to an angular velocity of \(8.4 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in 2 s .

1. Find the angular acceleration of the turntable.
2. A particle rests on the turntable at a distance of 0.15 m from the axis. Find the radial and transverse components of the acceleration of the particle when the angular velocity is \(1.2 \mathrm { rad } \mathrm { s } ^ { - 1 }\). Find also the magnitude of the acceleration at this instant.

9 A particle is projected from a point \(O\) on horizontal ground with speed \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal.

1. Write down the equation of the trajectory, in terms of \(\tan \theta\).
2. The particle passes through a point whose horizontal and vertical distances from \(O\) are 72 m and \(y \mathrm {~m}\) respectively. By considering the equation of the trajectory as a quadratic equation in \(\tan \theta\), or otherwise, find the greatest possible value of \(y\).

10 The engine of a lorry of mass 4000 kg works at a constant rate of 75 kW . Resistance to motion is modelled by a constant resistive force. On a horizontal road the lorry travels at a constant speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the work done by the engine in travelling for 1 minute on the horizontal road.
2. The lorry travels at a constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a slope of angle \(\sin ^ { - 1 } 0.05\) to the horizontal. Find the value of \(v\).

11 \includegraphics[max width=\textwidth, alt={}, center]{22640c3b-792f-4003-a4f8-78220efd73b0-4_280_1002_1722_568} A non-uniform \(\operatorname { rod } A B\) of mass 1.6 kg and length 1.25 m has its centre of mass at \(G\) where \(A G = 0.4 \mathrm {~m}\). The rod rests on a rough horizontal table. A force \(P \mathrm {~N}\) is applied at \(B\), acting at an angle \(\alpha\) above the horizontal, such that the rod is in equilibrium but about to rotate about \(A\) (see diagram).

1. Assume that the rod is in contact with the table only at \(A\). By taking moments about \(A\), show that \(P \sin \alpha = 5.12\).
2. The coefficient of friction between the rod and the table is \(\frac { 6 } { 17 }\). Show that \(P \leqslant 6.4\).

12 A particle of mass 0.6 kg is projected vertically upwards from horizontal ground, with initial speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The upwards velocity at any instant is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and the displacement is \(x \mathrm {~m}\). Air resistance is modelled by a force \(0.024 v ^ { 2 } \mathrm {~N}\) acting downwards.

1. Show that \(v\) and \(x\) satisfy the differential equation $$v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 10 - 0.04 v ^ { 2 } .$$
2. Find the value of \(u\) if the maximum height reached is 50 m .

13 A fairground game consists of a small ball fixed to one end of a light inextensible string of length 0.6 m . The other end of the string is fixed to a point \(O\). A small bell is fixed 0.6 m vertically above \(O\). Initially the ball hangs vertically in equilibrium. The object of the game is to project the ball with an initial horizontal velocity \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) so that it moves in a vertical circle and hits the bell.

1. Find the smallest possible value of \(u\) for which the ball hits the bell.
2. Given, instead, that the value of \(u\) is 5 , find the angle made by the string with the upward vertical at the moment when the string becomes slack.

14 \includegraphics[max width=\textwidth, alt={}, center]{22640c3b-792f-4003-a4f8-78220efd73b0-5_86_1589_1297_278} A particle of mass 0.05 kg is attached to two identical light elastic strings, each of natural length 1.2 m and modulus of elasticity 0.6 N . The other ends of the strings are attached to points \(A\) and \(E\) on a smooth horizontal table. The distance \(A E\) is 2 m and points \(B , C\) and \(D\) lie between \(A\) and \(E\) so that \(A B = 0.7 \mathrm {~m} , B C = 0.1 \mathrm {~m} , C D = 0.4 \mathrm {~m}\) and \(D E = 0.8 \mathrm {~m}\) (see diagram). Initially the particle is held at \(B\) and it is then released. In the subsequent motion the displacement of the particle from \(C\), in the direction of \(A\), is denoted by \(x \mathrm {~m}\).

1. Find the equation of motion for the particle when it is between \(B\) and \(C\).
2. Find the velocity of the particle when it is at \(C\).
3. Find the total time that elapses before the particle first returns to \(B\).