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Edexcel S4 Q3
13 marks Standard +0.8
3. A certain vaccine is known to be only \(70 \%\) effective against a particular virus; thus \(30 \%\) of those vaccinated will actually catch the virus. In order to test whether or not a new and more expensive vaccine provides better protection against the same virus, a random sample of 30 people were chosen and given the new vaccine. If fewer than 6 people contracted the virus the new vaccine would be considered more effective than the current one.
  1. Write down suitable hypotheses for this test.
  2. Find the probability of making a Type I error.
  3. Find the power of this test if the new vaccine is
    1. \(80 \%\) effective,
    2. \(90 \%\) effective. An independent research organisation decided to test the new vaccine on a random sample of 50 people to see if it could be considered more than \(70 \%\) effective. They required the probability of a Type I error to be as close as possible to 0.05 .
  4. Find the critical region for this test.
  5. State the size of this critical region.
  6. Find the power of this test if the new vaccine is
    1. \(80 \%\) effective,
    2. \(90 \%\) effective.
  7. Give one advantage and one disadvantage of the second test.
Edexcel S4 Q4
14 marks Standard +0.8
4. Gill, a member of the accounts department in a large company, is studying the expenses claims of company employees. She assumes that the claims, in \(\pounds\), follow a normal distribution with mean \(\mu\) and variance \(\sigma ^ { 2 }\). As a first stage in her investigation she took the following random sample of 10 claims. $$30.85,99.75,142.73,223.16,75.43,28.57,53.90,81.43,68.62,43.45 .$$
  1. Find a 95\% confidence interval for \(\mu\). The chief accountant would like a \(95 \%\) confidence interval where the difference between the upper confidence limit and the lower confidence limit is less than 20 .
  2. Show that \(\frac { \sigma ^ { 2 } } { n } < 26.03\) (to 2 decimal places), where \(n\) is the size of the sample required to achieve this. Gill decides to use her original sample of 10 to obtain a value for \(\sigma ^ { 2 }\) so that the chance of her value being an underestimate is 0.01 .
  3. Find such a value for \(\sigma ^ { 2 }\).
  4. Use this value for \(\sigma ^ { 2 }\) to estimate the size of sample the chief accountant requires.
Edexcel S4 Q5
16 marks Standard +0.8
5. An educational researcher is testing the effectiveness of a new method of teaching a topic in mathematics. A random sample of 10 children were taught by the new method and a second random sample of 9 children, of similar age and ability, were taught by the conventional method. At the end of the teaching, the same test was given to both groups of children. The marks obtained by the two groups are summarised in the table below.
New methodConventional method
Mean \(( \bar { x } )\)82.378.2
Standard deviation \(( s )\)3.55.7
Number of students \(( n )\)109
  1. Stating your hypotheses clearly and using a \(5 \%\) level of significance, investigate whether or not
    1. the variance of the marks of children taught by the conventional method is greater than that of children taught by the new method,
    2. the mean score of children taught by the conventional method is lower than the mean score of those taught by the new method.
      [0pt] [In each case you should give full details of the calculation of the test statistics.]
  2. State any assumptions you made in order to carry out these tests.
  3. Find a 95\% confidence interval for the common variance of the marks of the two groups.
Edexcel S4 Q6
18 marks Standard +0.3
6. A statistics student is trying to estimate the probability, \(p\), of rolling a 6 with a particular die. The die is rolled 10 times and the random variable \(X _ { 1 }\) represents the number of sixes obtained. The random variable \(R _ { 1 } = \frac { X _ { 1 } } { 10 }\) is proposed as an estimator of \(p\).
  1. Show that \(R _ { 1 }\) is an unbiased estimator of \(p\). The student decided to roll the die again \(n\) times ( \(n > 10\) ) and the random variable \(X _ { 2 }\) represents the number of sixes in these \(n\) rolls. The random variable \(R _ { 2 } = \frac { X _ { 2 } } { n }\) and the random variable \(Y = \frac { 1 } { 2 } \left( R _ { 1 } + R _ { 2 } \right)\).
  2. Show that both \(R _ { 2 }\) and \(Y\) are unbiased estimators of \(p\).
  3. Find \(\operatorname { Var } \left( R _ { 2 } \right)\) and \(\operatorname { Var } ( Y )\).
  4. State giving a reason which of the 3 estimators \(R _ { 1 } , R _ { 2 }\) and \(Y\) are consistent estimators of \(p\).
  5. For the case \(n = 20\) state, giving a reason, which of the 3 estimators \(R _ { 1 } , R _ { 2 }\) and \(Y\) you would recommend. The student's teacher pointed out that a better estimator could be found based on the random variable \(X _ { 1 } + X _ { 2 }\).
  6. Find a suitable estimator and explain why it is better than \(R _ { 1 } , R _ { 2 }\) and \(Y\). END
AQA M1 2005 January Q1
7 marks Moderate -0.8
1 A train travels along a straight horizontal track. It is travelling at a speed of \(12 \mathrm {~ms} ^ { - 1 }\) when it begins to accelerate uniformly. It reaches a speed of \(40 \mathrm {~ms} ^ { - 1 }\) after accelerating for 100 seconds.
    1. Show that the acceleration of the train is \(0.28 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    2. Find the distance that the train travelled in the 100 seconds.
  2. The mass of the train is 200 tonnes and a resistance force of 40000 N acts on the train. Find the magnitude of the driving force produced by the engine that acts on the train as it accelerates.
AQA M1 2005 January Q2
7 marks Moderate -0.8
2 A particle, \(A\), of mass 12 kg is moving on a smooth horizontal surface with velocity \(\left[ \begin{array} { l } 4 \\ 7 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). It then collides and coalesces with a second particle, \(B\), of mass 4 kg .
  1. If before the collision the velocity of \(B\) was \(\left[ \begin{array} { l } 2 \\ 3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\), find the velocity of the combined particle after the collision.
  2. If after the collision the velocity of the combined particle is \(\left[ \begin{array} { l } 1 \\ 4 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\), find the velocity of \(B\) before the collision.
AQA M1 2005 January Q3
11 marks Moderate -0.3
3 The diagram shows a rope that is attached to a box of mass 25 kg , which is being pulled along rough horizontal ground. The rope is at an angle of \(30 ^ { \circ }\) to the ground. The tension in the rope is 40 N . The box accelerates at \(0.1 \mathrm {~ms} ^ { - 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{eb1f2470-aeeb-4b1d-a6c0-bdeb7048edd5-3_214_729_504_644}
  1. Draw a diagram to show all of the forces acting on the box.
  2. Show that the magnitude of the friction force acting on the box is 32.1 N , correct to three significant figures.
  3. Show that the magnitude of the normal reaction force that the ground exerts on the box is 225 N .
  4. Find the coefficient of friction between the box and the ground.
  5. State what would happen to the magnitude of the friction force if the angle between the rope and the horizontal were increased. Give a reason for your answer.
AQA M1 2005 January Q4
8 marks Moderate -0.8
4 Two particles are connected by a string, which passes over a pulley. Model the string as light and inextensible. The particles have masses of 2 kg and 5 kg . The particles are released from rest. \includegraphics[max width=\textwidth, alt={}, center]{eb1f2470-aeeb-4b1d-a6c0-bdeb7048edd5-3_392_209_1685_909}
  1. State one modelling assumption that you should make about the pulley in order to determine the acceleration of the particles.
  2. By forming an equation of motion for each particle, show that the magnitude of the acceleration of each particle is \(4.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  3. Find the tension in the string.
AQA M1 2005 January Q5
7 marks Moderate -0.8
5 Two ropes are attached to a load of mass 500 kg . The ropes make angles of \(30 ^ { \circ }\) and \(45 ^ { \circ }\) to the vertical, as shown in the diagram. The tensions in these ropes are \(T _ { 1 }\) and \(T _ { 2 }\) newtons. The load is also supported by a vertical spring. \includegraphics[max width=\textwidth, alt={}, center]{eb1f2470-aeeb-4b1d-a6c0-bdeb7048edd5-4_533_565_507_744} The system is in equilibrium and \(T _ { 1 } = 200\).
  1. Show that \(T _ { 2 } = 141\), correct to three significant figures.
  2. Find the force that the spring exerts on the load.
AQA M1 2005 January Q6
7 marks Standard +0.3
6 A motor boat can travel at a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the water. It is used to cross a river in which the current flows at \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resultant velocity of the boat makes an angle of \(60 ^ { \circ }\) to the river bank, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{eb1f2470-aeeb-4b1d-a6c0-bdeb7048edd5-4_561_1339_1692_350} The angle between the direction in which the boat is travelling relative to the water and the resultant velocity is \(\alpha\).
  1. Show that \(\alpha = 16.8 ^ { \circ }\), correct to three significant figures.
  2. Find the magnitude of the resultant velocity.
AQA M1 2005 January Q7
12 marks Moderate -0.3
7 The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. A yacht moves with a constant acceleration. At time \(t\) seconds the position vector of the yacht is \(\mathbf { r }\) metres. When \(t = 0\) the velocity of the yacht is \(( 2 \mathbf { i } - \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), and when \(t = 10\) the velocity of the yacht is \(( - \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
  1. Find the acceleration of the yacht.
  2. When \(t = 0\) the yacht is 20 metres due east of the origin. Find an expression for \(\mathbf { r }\) in terms of \(t\).
    1. Show that when \(t = 20\) the yacht is due north of the origin.
    2. Find the speed of the yacht when \(t = 20\).
AQA M1 2005 January Q8
16 marks Moderate -0.8
8 A football is placed on a horizontal surface. It is then kicked, so that it has an initial velocity of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(40 ^ { \circ }\) above the horizontal.
  1. State two modelling assumptions that it would be appropriate to make when considering the motion of the football.
    1. Find the time that it takes for the ball to reach its maximum height.
    2. Hence show that the maximum height of the ball is 3.04 metres, correct to three significant figures.
  3. After the ball has reached its maximum height, it hits the bar of a goal at a height of 2.44 metres. Find the horizontal distance of the goal from the point where the ball was kicked.
AQA M1 2007 January Q1
6 marks Moderate -0.8
1 Two particles \(A\) and \(B\) have masses of 3 kg and 2 kg respectively. They are moving along a straight horizontal line towards each other. Each particle is moving with a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when they collide. \includegraphics[max width=\textwidth, alt={}, center]{965a176a-848c-478d-a748-80fc9dfe4399-2_225_579_676_660}
  1. If the particles coalesce during the collision to form a single particle, find the speed of the combined particle after the collision.
  2. If, after the collision, \(A\) moves in the same direction as before the collision with speed \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the speed of \(B\) after the collision.
AQA M1 2007 January Q2
10 marks Standard +0.3
2 A lift rises vertically from rest with a constant acceleration.
After 4 seconds, it is moving upwards with a velocity of \(2 \mathrm {~ms} ^ { - 1 }\).
It then moves with a constant velocity for 5 seconds.
The lift then slows down uniformly, coming to rest after it has been moving for a total of 12 seconds.
  1. Sketch a velocity-time graph for the motion of the lift.
  2. Calculate the total distance travelled by the lift.
  3. The lift is raised by a single vertical cable. The mass of the lift is 300 kg . Find the maximum tension in the cable during this motion.
AQA M1 2007 January Q3
6 marks Moderate -0.8
3 The diagram shows three forces which act in the same plane and are in equilibrium. \includegraphics[max width=\textwidth, alt={}, center]{965a176a-848c-478d-a748-80fc9dfe4399-3_419_516_383_761}
  1. Find \(F\).
  2. Find \(\alpha\).
AQA M1 2007 January Q4
13 marks Moderate -0.3
4 The diagram shows a block, of mass 13 kg , on a rough horizontal surface. It is attached by a string that passes over a smooth peg to a sphere of mass 7 kg , as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{965a176a-848c-478d-a748-80fc9dfe4399-3_323_974_1256_575} The system is released from rest, and after 4 seconds the block and the sphere both have speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and the block has not reached the peg.
  1. State two assumptions that you should make about the string in order to model the motion of the sphere and the block.
  2. Show that the acceleration of the sphere is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  3. Find the tension in the string.
  4. Find the coefficient of friction between the block and the surface.
AQA M1 2007 January Q5
9 marks Moderate -0.8
5 A girl in a boat is rowing across a river, in which the water is flowing at \(0.1 \mathrm {~ms} ^ { - 1 }\). The velocity of the boat relative to the water is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is perpendicular to the bank, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{965a176a-848c-478d-a748-80fc9dfe4399-4_314_1152_468_450}
  1. Find the magnitude of the resultant velocity of the boat.
  2. Find the acute angle between the resultant velocity and the bank.
  3. The width of the river is 15 metres.
    1. Find the time that it takes the boat to cross the river.
    2. Find the total distance travelled by the boat as it crosses the river.
AQA M1 2007 January Q6
9 marks Moderate -0.8
6 A trolley, of mass 100 kg , rolls at a constant speed along a straight line down a slope inclined at an angle of \(4 ^ { \circ }\) to the horizontal. Assume that a constant resistance force, of magnitude \(P\) newtons, acts on the trolley as it moves. Model the trolley as a particle.
  1. Draw a diagram to show the forces acting on the trolley.
  2. Show that \(P = 68.4 \mathrm {~N}\), correct to three significant figures.
    1. Find the acceleration of the trolley if it rolls down a slope inclined at \(5 ^ { \circ }\) to the horizontal and experiences the same constant force of magnitude \(P\) that you found in part (b).
    2. Make one criticism of the assumption that the resistance force on the trolley is constant.
AQA M1 2007 January Q7
10 marks Moderate -0.8
7 A golf ball is struck from a point on horizontal ground so that it has an initial velocity of \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(40 ^ { \circ }\) above the horizontal. Assume that the golf ball is a particle and its weight is the only force that acts on it once it is moving.
  1. Find the maximum height of the golf ball.
  2. After it has reached its maximum height, the golf ball descends but hits a tree at a point which is at a height of 6 metres above ground level. \includegraphics[max width=\textwidth, alt={}, center]{965a176a-848c-478d-a748-80fc9dfe4399-5_289_1358_813_335} \begin{displayquote} Find the time that it takes for the ball to travel from the point where it was struck to the tree. \end{displayquote}
AQA M1 2007 January Q8
12 marks Standard +0.3
8 A particle is initially at the origin, where it has velocity \(( 5 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). It moves with a constant acceleration \(\mathbf { a } \mathrm { ms } ^ { - 2 }\) for 10 seconds to the point with position vector \(75 \mathbf { i }\) metres.
  1. Show that \(\mathbf { a } = 0.5 \mathbf { i } + 0.4 \mathbf { j }\).
  2. Find the position vector of the particle 8 seconds after it has left the origin.
  3. Find the position vector of the particle when it is travelling parallel to the unit vector \(\mathbf { i }\).
AQA M1 2008 January Q1
6 marks Moderate -0.8
1 A crane is used to lift a crate, of mass 70 kg , vertically upwards. As the crate is lifted, it accelerates uniformly from rest, rising 8 metres in 5 seconds.
  1. Show that the acceleration of the crate is \(0.64 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. The crate is attached to the crane by a single cable. Assume that there is no resistance to the motion of the crate. Find the tension in the cable.
  3. Calculate the average speed of the crate during these 5 seconds.
AQA M1 2008 January Q2
4 marks Moderate -0.8
2 The velocity of a ship, relative to the water in which it is moving, is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) due north. The water is moving due east with a speed of \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resultant velocity of the ship has magnitude \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find \(U\).
  2. Find the direction of the resultant velocity of the ship. Give your answer as a bearing to the nearest degree.
AQA M1 2008 January Q3
6 marks Moderate -0.8
3 A particle, of mass 4 kg , is suspended in equilibrium by two light strings, \(A P\) and \(B P\). The string \(A P\) makes an angle of \(30 ^ { \circ }\) to the horizontal and the other string, \(B P\), is horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{217f0e3e-9d1b-41f1-8299-f56d073fbbeb-2_231_757_1841_639}
  1. Draw and label a diagram to show the forces acting on the particle.
  2. Show that the tension in the string \(A P\) is 78.4 N .
  3. Find the tension in the horizontal string \(B P\).
AQA M1 2008 January Q4
7 marks Moderate -0.3
4 Two particles, \(A\) and \(B\), are moving on a horizontal plane when they collide and coalesce to form a single particle. The mass of \(A\) is 5 kg and the mass of \(B\) is 15 kg . Before the collision, the velocity of \(A\) is \(\left[ \begin{array} { c } 2 U \\ U \end{array} \right] \mathrm { ms } ^ { - 1 }\) and the velocity of \(B\) is \(\left[ \begin{array} { c } V \\ - 1 \end{array} \right] \mathrm { ms } ^ { - 1 }\). After the collision, the velocity of the combined particle is \(\left[ \begin{array} { l } V \\ 0 \end{array} \right] \mathrm { ms } ^ { - 1 }\).
  1. Find:
    1. \(U\);
    2. \(V\).
  2. Find the speed of \(A\) before the collision.
AQA M1 2008 January Q5
16 marks Standard +0.3
5 A puck, of mass 0.2 kg , is placed on a slope inclined at \(20 ^ { \circ }\) above the horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{217f0e3e-9d1b-41f1-8299-f56d073fbbeb-3_280_773_1249_623} The puck is hit so that initially it moves with a velocity of \(4 \mathrm {~ms} ^ { - 1 }\) directly up the slope.
  1. A simple model assumes that the surface of the slope is smooth.
    1. Show that the acceleration of the puck up the slope is \(- 3.35 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), correct to three significant figures.
    2. Find the distance that the puck will travel before it comes to rest.
    3. What will happen to the puck after it comes to rest? Explain why.
  2. A revised model assumes that the surface is rough and that the coefficient of friction between the puck and the surface is 0.5 .
    1. Show that the magnitude of the friction force acting on the puck during this motion is 0.921 N , correct to three significant figures.
    2. Find the acceleration of the puck up the slope.
    3. What will happen to the puck after it comes to rest in this case? Explain why.