Questions — Pre-U Pre-U 9794/3 (125 questions)

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Pre-U Pre-U 9794/3 2015 June Q6
4 marks Moderate -0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{9ddae838-2639-4952-bbc0-3944a81e5762-3_401_1224_1315_456} The diagram shows a barge being towed along a canal by a force of 240 N at an angle of \(25 ^ { \circ }\) to its direction of motion. A force, \(F \mathrm {~N}\), perpendicular to the direction of motion, is applied to the barge to keep it moving in the direction shown.
  1. Find the magnitude of \(F\).
  2. The mass of the barge is 1100 kg and there is a resistance force of 100 N parallel to the direction of motion. Find the acceleration of the barge.
Pre-U Pre-U 9794/3 2015 June Q7
8 marks Standard +0.3
7 A particle is projected from the origin with initial speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal. After 2 seconds the particle is at a point which is 18 m horizontally from the origin and 4 m above it.
  1. Show that \(\tan \theta = \frac { 4 } { 3 }\) and find \(u\).
  2. Find the horizontal range of the particle.
Pre-U Pre-U 9794/3 2015 June Q8
5 marks Moderate -0.8
8 A tram travels from stop \(A\) to stop \(B\), a distance of 300 m . First the tram starts from rest at \(A\) and accelerates uniformly at \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 16 seconds. Then it travels at a constant speed and finally it slows down uniformly at \(1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) coming to rest at \(B\).
  1. Sketch the velocity-time graph for the journey of the tram from \(A\) to \(B\).
  2. Find the speed of the tram and the distance travelled at the end of the first 16 seconds.
  3. Show that the journey from \(A\) to \(B\) takes 49.5 seconds.
Pre-U Pre-U 9794/3 2015 June Q9
7 marks Moderate -0.3
9 A particle of mass 0.5 kg moving on a smooth horizontal plane with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) collides directly with another particle of mass \(k \mathrm {~kg}\) (where \(k\) is a constant) which is at rest. After the collision the first particle comes to rest but the second particle moves off with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find \(v\) in terms of \(k\) and \(u\).
  2. The coefficient of restitution between the two particles is \(e\). Find \(e\) in terms of \(k\) only.
  3. Show that \(k \geqslant \frac { 1 } { 2 }\).
Pre-U Pre-U 9794/3 2015 June Q10
10 marks Standard +0.3
10 A particle is projected up a long smooth slope at a speed of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The slope is at an angle \(\theta\) to the horizontal where \(\sin \theta = \frac { 1 } { 25 }\). After 2 seconds it passes a mark on the slope. Find the total time taken from the moment of projection until it passes the mark again and the total distance travelled in that time. {www.cie.org.uk} after the live examination series. }
Pre-U Pre-U 9794/3 2016 Specimen Q1
5 marks Moderate -0.8
1 The times for a motorist to travel from home to work are normally distributed with a mean of 24 minutes and a standard deviation of 4 minutes. Find the probability that a particular trip from home to work takes
  1. more than 27 minutes,
  2. between 20 and 25 minutes.
Pre-U Pre-U 9794/3 2016 Specimen Q2
11 marks Moderate -0.8
2
  1. A music club has 200 members. 75 members play the piano, 130 members like Elgar, and 30 members do not play the piano, nor do they like Elgar.
    1. Calculate the probability that a member chosen at random plays the piano but does not like Elgar.
    2. Calculate the probability that a member chosen at random plays the piano given that this member likes Elgar.
  2. The music club is organising a concert. The programme is to consist of 7 pieces of music which are to be selected from 9 classical pieces and 6 modern pieces. Find the number of different concert programmes than can be produced if
    1. there are no restrictions,
    2. the programme must consist of 5 classical pieces and 2 modern pieces,
    3. there are to be more modern pieces than classical pieces.
Pre-U Pre-U 9794/3 2016 Specimen Q3
5 marks Easy -1.2
3 The table shows fuel economy figures in miles per gallon (mpg) for some new cars.
CarABCDEFGHIJKLMNO
Mpg574034331117302731203524262332
  1. Find the median and quartiles for the mpg of these fifteen cars.
  2. Use the values in part (i) to identify any cars for which the mpg is an outlier.
Pre-U Pre-U 9794/3 2016 Specimen Q4
6 marks Moderate -0.8
4 A survey into left-handedness found that 13\% of the population of the world are left-handed.
  1. State the assumptions necessary for it to be appropriate to model the number of left-handed children in a class of 20 children using the binomial distribution \(\mathrm { B } ( 20,0.13 )\).
  2. Assuming that this binomial model is appropriate, calculate the probability that fewer than \(13 \%\) of the 20 children are left-handed.
Pre-U Pre-U 9794/3 2016 Specimen Q5
11 marks Moderate -0.8
5 James plays an arcade game. Each time he plays, he puts a \(\pounds 1\) coin in the slot to start the game. The possible outcomes of each game are as follows: James loses the game with a probability of 0.7 and the machine pays out nothing, James draws the game with a probability of 0.25 and the machine pays out a \(\pounds 1\) coin, James wins the game with a probability of 0.05 and the machine pays out ten \(\pounds 1\) coins. The outcomes can be modelled by a random variable \(X\) representing the number of \(\pounds 1\) coins gained at the end of a game.
  1. Construct a probability distribution table for \(X\).
  2. Show that \(\mathrm { E } ( X ) = - 0.25\) and find \(\operatorname { Var } ( X )\). James starts off with \(10 \pounds 1\) coins and decides to play exactly 10 games.
  3. Find the expected number of \(\pounds 1\) coins that James will have at the end of his 10 games.
  4. Find the probability that after his 10 games James will have at least \(10 \pounds 1\) coins left.
Pre-U Pre-U 9794/3 2016 Specimen Q6
6 marks Easy -1.3
6 \includegraphics[max width=\textwidth, alt={}, center]{01bd6354-3514-4dad-901b-7ecbe155b2c7-4_572_672_456_701} The diagram shows two horizontal forces \(\mathbf { P }\) and \(\mathbf { Q }\) acting at the origin \(O\) of rectangular coordinates \(O x y\). The components of \(\mathbf { P }\) in the \(x\) - and \(y\)-directions are 12 N and 17 N respectively. The components of \(\mathbf { Q }\) in the \(x\) - and \(y\)-directions are - 5 N and 7 N respectively.
  1. Write down the components, in the \(x\) - and \(y\)-directions, of the resultant of \(\mathbf { P }\) and \(\mathbf { Q }\).
  2. Hence, or otherwise, calculate the magnitude of this resultant and the angle the resultant makes with the positive \(x\)-axis.
Pre-U Pre-U 9794/3 2016 Specimen Q7
6 marks Moderate -0.3
7 A particle travels along a straight line. Its velocity \(v \mathrm {~ms} ^ { - 1 }\) after \(t\) seconds is given by $$v = t ^ { 3 } - 9 t ^ { 2 } + 20 t$$ When \(t = 0\), the particle is at rest at \(P\).
  1. Find the times, other than \(t = 0\), at which the particle is at rest.
  2. Find the displacement of the particle from \(P\) when \(t = 2\).
Pre-U Pre-U 9794/3 2016 Specimen Q8
6 marks Standard +0.3
8 Two trucks, \(S\) and \(T\), of masses 8000 kg and 10000 kg respectively, are pulled along a straight, horizontal track by a constant, horizontal force of \(P\) N. A resistive force of 600 N acts on \(S\) and a resistive force of 450 N acts on \(T\). The coupling between the trucks is light and horizontal (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{01bd6354-3514-4dad-901b-7ecbe155b2c7-5_213_1095_429_479} The acceleration of the system is \(0.3 \mathrm {~ms} ^ { - 2 }\) in the direction of the pulling force of magnitude \(P\).
  1. Calculate the value of \(P\). Truck \(S\) is now subjected to an extra resistive force of 1800 N . The pulling force, \(P\), does not change.
  2. Calculate the new acceleration of the trucks.
  3. Calculate the force in the coupling between the trucks.
Pre-U Pre-U 9794/3 2016 Specimen Q9
10 marks Standard +0.8
9 \includegraphics[max width=\textwidth, alt={}, center]{01bd6354-3514-4dad-901b-7ecbe155b2c7-5_118_851_1265_607} Three particles \(A , B\) and \(C\), having masses of \(1 \mathrm {~kg} , 2 \mathrm {~kg}\) and 5 kg respectively, are placed 1 metre apart in a straight line on a smooth horizontal plane (see diagram). The particles \(B\) and \(C\) are initially at rest and \(A\) is moving towards \(B\) with speed \(14 \mathrm {~ms} ^ { - 1 }\). The coefficient of restitution between each pair of particles is 0.5 .
  1. Find the velocity of \(B\) immediately after the first impact and show that \(A\) comes to rest.
  2. Show that \(B\) reversed direction after the impact with \(C\).
  3. Find the distances between \(B\) and \(C\) at the instant that \(B\) collides with \(A\) for the second time.
Pre-U Pre-U 9794/3 2016 Specimen Q10
11 marks Challenging +1.2
10 \includegraphics[max width=\textwidth, alt={}, center]{01bd6354-3514-4dad-901b-7ecbe155b2c7-6_490_661_267_703} Particles \(A\) and \(B\) of masses \(2 m\) and \(m\), respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley \(P\). The particle \(A\) rests in equilibrium on a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\alpha \leqslant 45 ^ { \circ }\) and \(B\) is above the plane. The vertical plane defined by \(A P B\) contains a line of greatest slope of the plane, and \(P A\) is inclined at angle \(2 \alpha\) to the horizontal (see diagram).
  1. Show that the normal reaction \(R\) between \(A\) and the plane is \(m g ( 2 \cos \alpha - \sin \alpha )\).
  2. Show that \(R \geqslant \frac { 1 } { 2 } m g \sqrt { 2 }\). The coefficient of friction between \(A\) and the plane is \(\mu\). The particle is about to slip down the plane.
  3. Show that \(0.5 < \tan \alpha \leqslant 1\).
  4. Express \(\mu\) as a function of \(\tan \alpha\) and deduce its maximum value as \(\alpha\) varies.
Pre-U Pre-U 9794/3 2016 Specimen Q6
7 marks Easy -1.2
6 \includegraphics[max width=\textwidth, alt={}, center]{b18b1bc5-bf26-4161-b5a5-764b00e97bea-4_572_672_456_701} The diagram shows two horizontal forces \(\mathbf { P }\) and \(\mathbf { Q }\) acting at the origin \(O\) of rectangular coordinates \(O x y\). The components of \(\mathbf { P }\) in the \(x\) - and \(y\)-directions are 12 N and 17 N respectively. The components of \(\mathbf { Q }\) in the \(x\) - and \(y\)-directions are - 5 N and 7 N respectively.
  1. Write down the components, in the \(x\) - and \(y\)-directions, of the resultant of \(\mathbf { P }\) and \(\mathbf { Q }\).
  2. Hence, or otherwise, calculate the magnitude of this resultant and the angle the resultant makes with the positive \(x\)-axis.
Pre-U Pre-U 9794/3 2016 Specimen Q8
6 marks Moderate -0.3
8 Two trucks, \(S\) and \(T\), of masses 8000 kg and 10000 kg respectively, are pulled along a straight, horizontal track by a constant, horizontal force of \(P\) N. A resistive force of 600 N acts on \(S\) and a resistive force of 450 N acts on \(T\). The coupling between the trucks is light and horizontal (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{b18b1bc5-bf26-4161-b5a5-764b00e97bea-5_215_1095_427_479} The acceleration of the system is \(0.3 \mathrm {~ms} ^ { - 2 }\) in the direction of the pulling force of magnitude \(P\).
  1. Calculate the value of \(P\). Truck \(S\) is now subjected to an extra resistive force of 1800 N . The pulling force, \(P\), does not change.
  2. Calculate the new acceleration of the trucks.
  3. Calculate the force in the coupling between the trucks.
Pre-U Pre-U 9794/3 2016 Specimen Q9
10 marks Standard +0.8
9 \includegraphics[max width=\textwidth, alt={}, center]{b18b1bc5-bf26-4161-b5a5-764b00e97bea-5_118_851_1265_607} Three particles \(A , B\) and \(C\), having masses of \(1 \mathrm {~kg} , 2 \mathrm {~kg}\) and 5 kg respectively, are placed 1 metre apart in a straight line on a smooth horizontal plane (see diagram). The particles \(B\) and \(C\) are initially at rest and \(A\) is moving towards \(B\) with speed \(14 \mathrm {~ms} ^ { - 1 }\). The coefficient of restitution between each pair of particles is 0.5 .
  1. Find the velocity of \(B\) immediately after the first impact and show that \(A\) comes to rest.
  2. Show that \(B\) reversed direction after the impact with \(C\).
  3. Find the distances between \(B\) and \(C\) at the instant that \(B\) collides with \(A\) for the second time.
Pre-U Pre-U 9794/3 2016 Specimen Q10
12 marks Challenging +1.2
10 \includegraphics[max width=\textwidth, alt={}, center]{b18b1bc5-bf26-4161-b5a5-764b00e97bea-6_490_661_267_703} Particles \(A\) and \(B\) of masses \(2 m\) and \(m\), respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley \(P\). The particle \(A\) rests in equilibrium on a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\alpha \leqslant 45 ^ { \circ }\) and \(B\) is above the plane. The vertical plane defined by \(A P B\) contains a line of greatest slope of the plane, and \(P A\) is inclined at angle \(2 \alpha\) to the horizontal (see diagram).
  1. Show that the normal reaction \(R\) between \(A\) and the plane is \(m g ( 2 \cos \alpha - \sin \alpha )\).
  2. Show that \(R \geqslant \frac { 1 } { 2 } m g \sqrt { 2 }\). The coefficient of friction between \(A\) and the plane is \(\mu\). The particle is about to slip down the plane.
  3. Show that \(0.5 < \tan \alpha \leqslant 1\).
  4. Express \(\mu\) as a function of \(\tan \alpha\) and deduce its maximum value as \(\alpha\) varies.
Pre-U Pre-U 9794/3 2017 June Q1
5 marks Moderate -0.8
1 Levels of nitrogen dioxide in the atmosphere are being monitored at the side of a road in a busy city centre. A sample of 18 measurements taken (in suitable units) is as follows. $$\begin{array} { l l l l l l l l l l l l l l l l l l } 83 & 44 & 95 & 92 & 98 & 63 & 69 & 76 & 19 & 91 & 70 & 91 & 74 & 65 & 62 & 70 & 95 & 108 \end{array}$$
  1. Find the mean and standard deviation of the sample.
  2. Hence identify, with justification, any possible outliers.
Pre-U Pre-U 9794/3 2017 June Q2
9 marks Moderate -0.8
2 The table shows the turnover, in millions of pounds, of a small company at 3-year intervals over a period of 15 years, starting in 2000.
Year since 200003691215
Turnover ( \(\pounds\) millions)2.302.943.373.974.936.13
  1. For the information in the table find the equation of the least squares regression line of \(y\) on \(x\), where \(x\) is the year since 2000 and \(y\) is the turnover in millions of pounds.
  2. Use the equation of the regression line to calculate the residual for 2009.
  3. Use the equation of the regression line to estimate the turnover in 2024, and explain why it is inadvisable to rely on this estimate.
Pre-U Pre-U 9794/3 2017 June Q3
8 marks Standard +0.3
3 The probability distribution of the discrete random variable \(X\) is defined as follows. $$\mathrm { P } ( X = x ) = k ( 2 + x ) ( 5 - x ) \quad \text { for } x = 0,1,2,3,4$$
  1. Show that \(k = \frac { 1 } { 50 }\).
  2. Find the variance of \(X\).
  3. Find \(\mathrm { P } ( X = 4 \mid X > 0 )\).
Pre-U Pre-U 9794/3 2017 June Q4
9 marks Moderate -0.3
4 The letters of the word 'STATISTICS' are to be rearranged.
  1. How many distinct arrangements are there?
  2. How many of the arrangements start and end with the letter S ?
  3. What is the probability that, in a randomly chosen arrangement, the S's are all together?
Pre-U Pre-U 9794/3 2017 June Q5
9 marks Standard +0.3
5 The random variable \(X\) has a geometric distribution: \(X \sim \operatorname { Geo } ( p )\).
  1. Show that \(\mathrm { P } ( X > n ) = q ^ { n }\), where \(q = 1 - p\). You are given that \(\mathrm { P } ( X \geqslant 4 ) = 0.216\).
  2. Use the result given in part (i) to find the value of \(p\) and \(\mathrm { P } ( X \leqslant 8 )\).
  3. Write down \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
Pre-U Pre-U 9794/3 2017 June Q6
11 marks Moderate -0.3
6 A crate, which has a mass of 220 kg , is being lowered on the end of a cable onto the back of a lorry.
  1. Draw a diagram to show the forces acting on the crate. The crate is lowered in three stages.
    Stage 1 It starts from rest and accelerates at \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it reaches a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    Stage 2 It descends at a constant speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    Stage 3 It decelerates at \(0.75 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and eventually comes to rest.
  2. Find the tension in the cable in each of the three stages.
  3. Sketch the velocity-time graph for the complete downward motion of the crate.
  4. The crate is lowered 15 m altogether. By considering your velocity-time graph, find the total time taken.