Pre-U Pre-U 9794/3 (Pre-U Mathematics Paper 3) 2016 June

The following data refer to the annual rate of inflation and the annual percentage pay increase measured on 10 randomly chosen occasions. Show that, for these data, the product moment correlation coefficient between the rate of inflation and the annual pay increase is \(-0.679\), correct to 3 significant figures. [4]

The weights of pineapples on sale at a wholesaler are normally distributed with mean \(1.349\) kg and standard deviation \(0.236\) kg. Before going on sale the pineapples are classified as 'Small', 'Medium', 'Large' and 'Extra Large'.

1. A pineapple is classified as 'Small' if it weighs less than \(1.100\) kg. Find the probability that a randomly chosen pineapple will be classified as 'Small'. [5]
2. \(10\%\) of pineapples are classified as 'Extra Large'. Find the minimum weight required for a pineapple to be classified as 'Extra Large'. [3]

Chris plays for his local hockey club. In his first 20 games for the club, the mean number of goals per game he has scored is \(0.7\), with a standard deviation of \(0.9\). In the next 5 games he scores \(0, 1, 0, 2, 1\) goals.

1. Find the mean and standard deviation for the number of goals per game Chris has scored in all 25 games. [7]
2. A sponsor pays Chris £65 each time he plays for the club and a further £25 for each goal he scores. Find the mean and standard deviation of the amount per game he earns from the sponsor for all 25 games. [4]

A certain type of sweet is made in a variety of colours. \(20\%\) of the sweets made are blue. Sweets of the various colours are thoroughly mixed before being put into packets.

1. In a packet that contains 10 sweets, find the probability that the packet contains
   1. at most 3 blue sweets, [1]
   2. exactly 3 blue sweets, [2]
   3. at least 1 blue sweet. [2]
2. What is the smallest number of sweets that a packet should contain in order to be at least \(95\%\) certain of having at least 1 blue sweet? [3]

The letters of the word 'SEPARATE' are to be rearranged. Find the probability that, in a randomly chosen rearrangement, the two letters 'A' are not next to each other. [4]

\(A\) and \(B\) are independent events. \(P(A) = \frac{3}{4}\) and \(P(A \cap B) = \frac{1}{4}\). Find \(P(A' \cap B)\) and \(P(A' \cap B')\). [5]

A stone that weighs 15 kg is propelled across the ice in an ice rink with an initial speed of \(4 \text{ m s}^{-1}\). The coefficient of friction between the stone and the ice is \(0.017\). How far does the stone slide before it comes to rest? [5]

A particle is projected with speed \(U \text{ m s}^{-1}\) at an angle \(\theta\) above the horizontal, where \(\sin \theta = \frac{12}{13}\), and reaches its maximum height after \(2.4\) seconds.

1. Find \(U\) and the maximum height reached by the particle. [4]
2. Find the horizontal range of the particle. [4]

A particle of mass \(0.01\) kg is projected vertically upwards from a point \(G\) at ground level with speed \(165 \text{ m s}^{-1}\) and reaches a maximum height of \(1237.5\) m. Throughout its motion it experiences a constant resistance.

1. Find the acceleration of the particle as it ascends and hence the magnitude of the resistance. [4]
2. During its descent back to \(G\) the particle experiences the same constant resistance. Find the time taken for the descent. [4]

1. A particle \(A\) of mass \(m\) travelling with speed \(u\) on a smooth horizontal surface collides directly with a particle \(B\) of mass \(3m\) travelling with speed \(\frac{2u}{5}\) in the opposite direction. After the collision, \(A\) travels at speed \(\frac{2u}{5}\) and \(B\) travels at speed \(\frac{4u}{15}\), both in the same direction as \(B\) before the collision. Find \(A\) and the coefficient of restitution between the two particles. [4]
2. A particle of mass 3 kg moving with velocity \((2\mathbf{i} + 3\mathbf{j} - 2\mathbf{k}) \text{ m s}^{-1}\) receives an impulse of \((6\mathbf{i} - 6\mathbf{j} - 9\mathbf{k})\) N s. Find the velocity of the particle after the impulse. [3]

\includegraphics{figure_11} The diagram shows a particle, \(A\), of mass \(m_1\) at rest on a rough slope at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{3}{5}\). Particle \(A\) is connected by a light inextensible string to another particle, \(B\), of mass \(m_2\). The string passes over a smooth peg at the top of the slope and particle \(B\) is hanging freely.

1. In the case when \(m_2 = \frac{1}{4}m_1\), particle \(A\) is on the point of sliding down the slope.
   1. Draw a fully labelled diagram to show all the forces acting on the particles. [2]
   2. Find the coefficient of friction between \(A\) and the slope. [6]
2. In the case when \(m_2 = m_1\), find the acceleration of the particles. [4]