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AQA S2 2010 June Q6
18 marks Standard +0.3
  1. The number of strokes, \(R\), taken by the members of Duffers Golf Club to complete the first hole may be modelled by the following discrete probability distribution.
    \(r\)\(\leqslant 2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(\geqslant 9\)
    \(\mathrm{P}(R = r)\)\(0\)\(0.1\)\(0.2\)\(0.3\)\(0.25\)\(0.1\)\(0.05\)\(0\)
    1. Determine the probability that a member, selected at random, takes at least \(5\) strokes to complete the first hole. [1 mark]
    2. Calculate \(\mathrm{E}(R)\). [2 marks]
    3. Show that \(\mathrm{Var}(R) = 1.66\). [4 marks]
  2. The number of strokes, \(S\), taken by the members of Duffers Golf Club to complete the second hole may be modelled by the following discrete probability distribution.
    \(s\)\(\leqslant 2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(\geqslant 9\)
    \(\mathrm{P}(S = s)\)\(0\)\(0.15\)\(0.4\)\(0.3\)\(0.1\)\(0.03\)\(0.02\)\(0\)
    Assuming that \(R\) and \(S\) are independent:
    1. show that \(\mathrm{P}(R + S \leqslant 8) = 0.24\); [5 marks]
    2. calculate the probability that, when \(5\) members are selected at random, at least \(4\) of them complete the first two holes in fewer than \(9\) strokes; [3 marks]
    3. calculate \(\mathrm{P}(R = 4 \mid R + S \leqslant 8)\). [3 marks]
AQA S2 2010 June Q7
12 marks Standard +0.3
The random variable \(X\) has probability density function defined by $$f(x) = \begin{cases} \frac{1}{2} & 0 \leqslant x \leqslant 1 \\ \frac{1}{18}(x - 4)^2 & 1 \leqslant x \leqslant 4 \\ 0 & \text{otherwise} \end{cases}$$
  1. State values for the median and the lower quartile of \(X\). [2 marks]
  2. Show that, for \(1 \leqslant x \leqslant 4\), the cumulative distribution function, \(\mathrm{F}(x)\), of \(X\) is given by $$\mathrm{F}(x) = 1 + \frac{1}{54}(x - 4)^3$$ (You may assume that \(\int (x - 4)^2 \, dx = \frac{1}{3}(x - 4)^3 + c\).) [4 marks]
  3. Determine \(\mathrm{P}(2 \leqslant X \leqslant 3)\). [2 marks]
    1. Show that \(q\), the upper quartile of \(X\), satisfies the equation \((q - 4)^3 = -13.5\). [3 marks]
    2. Hence evaluate \(q\) to three decimal places. [1 mark]
AQA S2 2016 June Q1
13 marks Standard +0.3
The water in a pond contains three different species of a spherical green algae: Volvox globator, at an average rate of 4.5 spheres per 1 cm³; Volvox aureus, at an average rate of 2.3 spheres per 1 cm³; Volvox tertius, at an average rate of 1.2 spheres per 1 cm³. Individual Volvox spheres may be considered to occur randomly and independently of all other Volvox spheres. Random samples of water are collected from this pond. Find the probability that:
  1. a 1 cm³ sample contains no more than 5 Volvox globator spheres; [1 mark]
  2. a 1 cm³ sample contains at least 2 Volvox aureus spheres; [3 marks]
  3. a 5 cm³ sample contains more than 8 but fewer than 12 Volvox tertius spheres; [3 marks]
  4. a 0.1 cm³ sample contains a total of exactly 2 Volvox spheres; [3 marks]
  5. a 1 cm³ sample contains at least 1 sphere of each of the three different species of algae. [3 marks]
AQA S2 2016 June Q2
4 marks Moderate -0.3
A normally distributed variable, \(X\), has unknown mean \(\mu\) and unknown standard deviation \(\sigma\). A sample of 10 values of \(X\) was taken. From these 10 values, a 95% confidence interval for \(\mu\) was calculated to be $$(30.47, 32.93)$$ Use this confidence interval to find unbiased estimates for \(\mu\) and \(\sigma^2\). [4 marks]
AQA S2 2016 June Q3
13 marks Moderate -0.8
Members of a library may borrow up to 6 books. Past experience has shown that the number of books borrowed, \(X\), follows the distribution shown in the table.
\(x\)0123456
P(X = x)00.190.260.200.130.070.15
  1. Find the probability that a member borrows more than 3 books. [1 mark]
  2. Assume that the numbers of books borrowed by two particular members are independent. Find the probability that one of these members borrows more than 3 books and the other borrows fewer than 3 books. [3 marks]
  3. Show that the mean of \(X\) is 3.08, and calculate the variance of \(X\). [4 marks]
  4. One of the library staff notices that the values of the mean and the variance of \(X\) are similar and suggests that a Poisson distribution could be used to model \(X\). Without further calculations, give two reasons why a Poisson distribution would not be suitable to model \(X\). [2 marks]
  5. The library introduces a fee of 10 pence for each book borrowed. Assuming that the probabilities do not change, calculate:
    1. the mean amount that will be paid by a member;
    2. the standard deviation of the amount that will be paid by a member.
    [3 marks]
AQA S2 2016 June Q4
7 marks Moderate -0.8
A digital thermometer measures temperatures in degrees Celsius. The thermometer rounds down the actual temperature to one decimal place, so that, for example, 36.23 and 36.28 are both shown as 36.2. The error, \(X\) °C, resulting from this rounding down can be modelled by a rectangular distribution with the following probability density function. $$f(x) = \begin{cases} k & 0 \leqslant x \leqslant 0.1 \\ 0 & \text{otherwise} \end{cases}$$
  1. State the value of \(k\). [1 mark]
  2. Find the probability that the error resulting from this rounding down is greater than 0.03 °C. [1 mark]
    1. State the value for E(\(X\)).
    2. Use integration to find the value for E(\(X^2\)).
    3. Hence find the value for the standard deviation of \(X\).
    [5 marks]
AQA S2 2016 June Q5
13 marks Standard +0.3
A car manufacturer keeps a record of how many of the new cars that it has sold experience mechanical problems during the first year. The manufacturer also records whether the cars have a petrol engine or a diesel engine. Data for a random sample of 250 cars are shown in the table.
Problems during first 3 monthsProblems during first year but after first 3 monthsNo problems during first yearTotal
Petrol engine1035170215
Diesel engine482335
Total1443193250
  1. Use a \(\chi^2\)-test to investigate, at the 10% significance level, whether there is an association between the mechanical problems experienced by a new car from this manufacturer and the type of engine. [11 marks]
  2. Arisa is planning to buy a new car from this manufacturer. She would prefer to buy a car with a diesel engine, but a friend has told her that cars with diesel engines experience more mechanical problems. Based on your answer to part (a), state, with a reason, the advice that you would give to Arisa. [2 marks]
AQA S2 2016 June Q6
16 marks Standard +0.3
Gerald is a scientist who studies sand lizards. He believes that sand lizards on islands are, on average, shorter than those on the mainland. The population of sand lizards on the mainland has a mean length of 18.2 cm and a standard deviation of 1.8 cm. Gerald visited three islands, A, B and C, and measured the length, \(X\) centimetres, of each of a sample of \(n\) sand lizards on each island. The samples may be regarded as random. The data are shown in the table.
Island\(\sum x\)\(n\)
A1384.578
B116.97
C394.620
  1. Carry out a hypothesis test to investigate whether the data from Island A provide support for Gerald's belief at the 2% significance level. Assume that the standard deviation of the lengths of sand lizards on Island A is 1.8 cm. [7 marks]
  2. For Island B, it is also given that $$\sum(x - \bar{x})^2 = 22.64$$
    1. Construct a 95% confidence interval for \(\mu_B\), where \(\mu_B\) centimetres is the mean length of sand lizards on Island B. Assume that the lengths of sand lizards on Island B are normally distributed with unknown standard deviation.
    2. Comment on whether your confidence interval provides support for Gerald's belief.
    [7 marks]
  3. Comment on whether the data from Island C provide support for Gerald's belief. [2 marks]
AQA S2 2016 June Q7
9 marks Standard +0.3
The continuous random variable \(X\) has a cumulative distribution function F(\(x\)), where $$\text{F}(x) = \begin{cases} 0 & x < 1 \\ \frac{1}{4}(x - 1) & 1 \leqslant x < 4 \\ \frac{1}{16}(12x - x^2 - 20) & 4 \leqslant x \leqslant 6 \\ 1 & x > 6 \end{cases}$$
  1. Sketch the probability density function, f(\(x\)), on the grid below. [5 marks]
  2. Find the mean value of \(X\). [4 marks]
AQA S3 2016 June Q1
8 marks Standard +0.3
In advance of a referendum on independence, the regional assembly of an eastern province of a particular country carried out an opinion poll to assess the strength of the 'Yes' vote. Of the 480 men polled, 264 indicated that they intended to vote 'Yes', and of the 500 women polled, 220 indicated that they intended to vote 'Yes'.
  1. Construct an approximate 95\% confidence interval for the difference between the proportion of men who intend to vote 'Yes' and the proportion of women who intend to vote 'Yes'. [6 marks]
  2. Comment on a claim that, in the forthcoming referendum, the percentage of men voting 'Yes' will exceed the percentage of women voting 'Yes' by at least 2.5 per cent. Justify your answer. [2 marks]
AQA S3 2016 June Q2
15 marks Moderate -0.3
A plane flies regularly between airports D and T with an intermediate stop at airport M. The time of the plane's departure from, or arrival at, each airport is classified as either early, on time, or late. On 90\% of flights, the plane departs from D on time, and on 10\% of flights, it departs from D late. Of those flights that depart from D on time, 65\% then depart from M on time and 35\% depart from M late. Of those flights that depart from D late, 15\% then depart from M on time and 85\% depart from M late. Any flight that departs from M on time has probability 0.25 of arriving at T early, probability 0.60 of arriving at T on time and probability 0.15 of arriving at T late. Any flight that departs from M late has probability 0.10 of arriving at T early, probability 0.20 of arriving at T on time and probability 0.70 of arriving at T late.
  1. Represent this information by a tree diagram on which labels and percentages or probabilities are shown. [3 marks]
  2. Hence, or otherwise, calculate the probability that the plane:
    1. arrives at T on time;
    2. arrives at T on time, given that it departed from D on time;
    3. does not arrive at T late, given that it departed from D on time;
    4. does not arrive at T late, given that it departed from M on time.
    [8 marks]
  3. Three independent flights of the plane depart from D on time. Calculate the probability that two flights arrive at T on time and that one flight arrives at T early. [4 marks]
AQA S3 2016 June Q3
7 marks Standard +0.3
Car parking in a market town's high street was, until 31 May 2014, limited to one hour free of charge between 8 am and 6 pm. Records show that, during a period of 30 days prior to this date, a total of 315 penalty tickets were issued. Car parking in the high street later became limited to thirty minutes free of charge between 8 am and 6 pm. A subsequent investigation revealed that, during a period of 60 days from 1 October 2014, a total of 747 penalty tickets were issued. The daily numbers of penalty tickets issued may be modelled by independent Poisson distributions with means \(\lambda_A\) until 31 May 2014 and \(\lambda_B\) from 1 October 2014. Investigate, at the 1\% level of significance, a claim by traders on the high street that \(\lambda_B > \lambda_A\). [7 marks]
AQA S3 2016 June Q4
13 marks Standard +0.3
Ben is a fencing contractor who is often required to repair a garden fence by replacing a broken post between fence panels, as illustrated. \includegraphics{figure_4} The tasks involved are as follows. \(U\): detach the two fence panels from the broken post \(V\): remove the broken post \(W\): insert a new post \(X\): attach the two fence panels to the new post The mean and the standard deviation of the time, in minutes, for each of these tasks are shown in the table.
TaskMeanStandard deviation
\(U\)155
\(V\)4015
\(W\)7520
\(X\)2010
The random variables \(U\), \(V\), \(W\) and \(X\) are pairwise independent, except for \(V\) and \(W\) for which \(\rho_{VW} = 0.25\).
  1. Determine values for the mean and the variance of:
    1. \(R = U + X\);
    2. \(F = V + W\);
    3. \(T = R + F\);
    4. \(D = W - V\).
    [8 marks]
  2. Assuming that each of \(R\), \(F\), \(T\) and \(D\) is approximately normally distributed, determine the probability that:
    1. the total time taken by Ben to repair a garden fence is less than 3 hours;
    2. the time taken by Ben to insert a new post is at least 1 hour more than the time taken by him to remove the broken post.
    [5 marks]
AQA S3 2016 June Q5
10 marks Standard +0.3
  1. The random variable \(X\), which has distribution \(\mathrm{N}(\mu_X, \sigma^2)\), is independent of the random variable \(Y\), which has distribution \(\mathrm{N}(\mu_Y, \sigma^2)\). In order to test \(\mathrm{H_0}: \mu_X = 1.5\mu_Y\), samples of size \(n\) are taken on each of \(X\) and \(Y\) and the random variable \(D\) is defined as $$D = \overline{X} - 1.5\overline{Y}$$ State the distribution of \(D\) assuming that \(\mathrm{H_0}\) is true. [4 marks]
  2. A machine that fills bags with rice delivers weights that are normally distributed with a standard deviation of 4.5 grams. The machine fills two sizes of bags: large and extra-large. The mean weight of rice in a random sample of 50 large bags is 1509 grams. The mean weight of rice in an independent random sample of 50 extra-large bags is 2261 grams. Test, at the 5\% level of significance, the claim that, on average, the rice in an extra-large bag is \(1\frac{1}{3}\) times as heavy as that in a large bag. [6 marks]
AQA S3 2016 June Q6
22 marks Standard +0.3
  1. The discrete random variable \(X\) has probability distribution given by $$\mathrm{P}(X = x) = \begin{cases} \frac{e^{-\lambda}\lambda^x}{x!} & x = 0, 1, 2, \ldots \\ 0 & \text{otherwise} \end{cases}$$ Show that \(\mathrm{E}(X) = \lambda\) and that \(\mathrm{Var}(X) = \lambda\). [7 marks]
  2. In light-weight chain, faults occur randomly and independently, and at a constant average rate of 0.075 per metre.
    1. Calculate the probability that there are no faults in a 10-metre length of this chain. [2 marks]
    2. Use a distributional approximation to estimate the probability that, in a 500-metre reel of light-weight chain, there are:
      1. fewer than 30 faults;
      2. at least 35 faults but at most 45 faults.
      [7 marks]
  3. As part of an investigation into the quality of a new design of medium-weight chain, a sample of fifty 10-metre lengths was selected. Subsequent analysis revealed a total of 49 faults. Assuming that faults occur randomly and independently, and at a constant average rate, construct an approximate 98\% confidence interval for the average number of faults per metre. [6 marks]
AQA M2 2014 June Q1
8 marks Moderate -0.8
An eagle has caught a salmon of mass 3 kg to take to its nest. When the eagle is flying with speed \(8 \text{ m s}^{-1}\), it drops the salmon. The salmon falls a vertical distance of 13 metres back into the sea. The salmon is to be modelled as a particle. The salmon's weight is the only force that acts on it as it falls to the sea.
  1. Calculate the kinetic energy of the salmon when it is dropped by the eagle. [2 marks]
  2. Calculate the potential energy lost by the salmon as it falls to the sea. [2 marks]
    1. Find the kinetic energy of the salmon when it reaches the sea. [2 marks]
    2. Hence find the speed of the salmon when it reaches the sea. [2 marks]
AQA M2 2014 June Q2
10 marks Standard +0.3
A particle has mass 6 kg. A single force \((24e^{-2t}\mathbf{i} - 12t^3\mathbf{j})\) newtons acts on the particle at time \(t\) seconds. No other forces act on the particle.
  1. Find the acceleration of the particle at time \(t\). [2 marks]
  2. At time \(t = 0\), the velocity of the particle is \((-7\mathbf{i} - 4\mathbf{j}) \text{ m s}^{-1}\). Find the velocity of the particle at time \(t\). [4 marks]
  3. Find the speed of the particle when \(t = 0.5\). [4 marks]
AQA M2 2014 June Q3
5 marks Moderate -0.8
Four tools are attached to a board. The board is to be modelled as a uniform lamina and the four tools as four particles. The diagram shows the lamina, the four particles \(A\), \(B\), \(C\) and \(D\), and the \(x\) and \(y\) axes. \includegraphics{figure_3} The lamina has mass 5 kg and its centre of mass is at the point \((7, 6)\). Particle \(A\) has mass 4 kg and is at the point \((11, 2)\). Particle \(B\) has mass 3 kg and is at the point \((3, 6)\). Particle \(C\) has mass 7 kg and is at the point \((5, 9)\). Particle \(D\) has mass 1 kg and is at the point \((1, 4)\). Find the coordinates of the centre of mass of the system of board and tools. [5 marks]
AQA M2 2014 June Q4
9 marks Standard +0.3
A particle, of mass 0.8 kg, is attached to one end of a light inextensible string. The other end of the string is attached to the fixed point \(O\). The particle is set in motion, so that it moves in a horizontal circle at constant speed, with the string at an angle of \(35°\) to the vertical. The centre of this circle is vertically below \(O\), as shown in the diagram. \includegraphics{figure_4} The particle moves in a horizontal circle and completes 20 revolutions each minute.
  1. Find the angular speed of the particle in radians per second. [2 marks]
  2. Find the tension in the string. [3 marks]
  3. Find the radius of the horizontal circle. [4 marks]
AQA M2 2014 June Q5
7 marks Standard +0.3
A light inextensible string, of length \(a\), has one end attached to a fixed point \(O\). A particle, of mass \(m\), is attached to the other end of the string. The particle is moving in a vertical circle with centre \(O\). The point \(Q\) is the highest point of the particle's path. When the particle is at \(P\), vertically below \(O\), the string is taut and the particle is moving with speed \(7\sqrt{ag}\), as shown in the diagram. \includegraphics{figure_5}
  1. Find, in terms of \(g\) and \(a\), the speed of the particle at the point \(Q\). [4 marks]
  2. Find, in terms of \(g\) and \(m\), the tension in the string when the particle is at \(Q\). [3 marks]
AQA M2 2014 June Q6
13 marks Standard +0.8
A puck, of mass \(m\) kg, is moving in a straight line across smooth horizontal ice. At time \(t\) seconds, the puck has speed \(v \text{ m s}^{-1}\). As the puck moves, it experiences an air resistance force of magnitude \(0.3mv^3\) newtons, until it comes to rest. No other horizontal forces act on the puck. When \(t = 0\), the speed of the puck is \(8 \text{ m s}^{-1}\). Model the puck as a particle.
  1. Show that $$v = (4 - 0.2t)^{\frac{3}{2}}$$ [6 marks]
  2. Find the value of \(t\) when the puck comes to rest. [2 marks]
  3. Find the distance travelled by the puck as its speed decreases from \(8 \text{ m s}^{-1}\) to zero. [5 marks]
AQA M2 2014 June Q7
8 marks Standard +0.3
A uniform ladder \(AB\), of length 6 metres and mass 22 kg, rests with its foot, \(A\), on rough horizontal ground. The ladder rests against the top of a smooth vertical wall at the point \(C\), where the length \(AC\) is 5 metres. The vertical plane containing the ladder is perpendicular to the wall, and the angle between the ladder and the ground is \(60°\). A man, of mass 88 kg, is standing on the ladder. The man may be modelled as a particle at the point \(D\), where the length of \(AD\) is 4 metres. The ladder is on the point of slipping. \includegraphics{figure_7}
  1. Draw a diagram to show the forces acting on the ladder. [2 marks]
  2. Find the coefficient of friction between the ladder and the horizontal ground. [6 marks]
AQA M2 2014 June Q8
15 marks Challenging +1.2
An elastic string has natural length 1.5 metres and modulus of elasticity 120 newtons. One end of the string is attached to a fixed point, \(A\), on a rough plane inclined at \(20°\) to the horizontal. The other end of the elastic string is attached to a particle of mass 4 kg. The coefficient of friction between the particle and the plane is 0.8. The three points, \(A\), \(B\) and \(C\), lie on a line of greatest slope. The point \(C\) is \(x\) metres from \(A\), as shown in the diagram. The particle is released from rest at \(C\) and moves up the plane. \includegraphics{figure_8}
  1. Show that, as the particle moves up the plane, the frictional force acting on the particle is 29.5 N, correct to three significant figures. [3 marks]
  2. The particle comes to rest for an instant at \(B\), which is 2 metres from \(A\). The particle then starts to move back towards \(A\).
    1. Find \(x\). [8 marks]
    2. Find the acceleration of the particle as it starts to move back towards \(A\). [4 marks]
AQA M2 2016 June Q1
8 marks Moderate -0.8
A stone, of mass \(0.3\) kg, is thrown with a speed of \(8 \text{ m s}^{-1}\) from a point at a height of \(5\) metres above a horizontal surface.
  1. Calculate the initial kinetic energy of the stone. [2 marks]
    1. Find the kinetic energy of the stone when it hits the surface. [3 marks]
    2. Hence find the speed of the stone when it hits the surface. [2 marks]
    3. State one modelling assumption that you have made. [1 mark]
AQA M2 2016 June Q2
13 marks Standard +0.3
A particle moves in a horizontal plane under the action of a single force, \(\mathbf{F}\) newtons. The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are directed east and north respectively. At time \(t\) seconds, the velocity of the particle, \(\mathbf{v} \text{ m s}^{-1}\), is given by $$\mathbf{v} = (8t - t^4)\mathbf{i} + 6e^{-3t}\mathbf{j}$$
  1. Find an expression for the acceleration of the particle at time \(t\). [2 marks]
  2. The mass of the particle is \(2\) kg.
    1. Find an expression for the force \(\mathbf{F}\) acting on the particle at time \(t\). [2 marks]
    2. Find the magnitude of \(\mathbf{F}\) when \(t = 1\). [3 marks]
  3. Find the value of \(t\) when \(\mathbf{F}\) acts due south. [2 marks]
  4. When \(t = 0\), the particle is at the point with position vector \((3\mathbf{i} - 5\mathbf{j})\) metres. Find an expression for the position vector, \(\mathbf{r}\) metres, of the particle at time \(t\). [4 marks]