Edexcel A-Level Maths, Statistics and Mechanics, June 2019 Question Walkthroughs
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Statistics June 2019, Q1Three bags, A, B and C, each contain 1 red marble and some green marbles. Bag A contains 1 red marble and 9 green marbles only Bag B contains 1 red marble and 4 green marbles only Bag C contains 1 red marble and 2 green marbles only Sasha selects at random one marble from bag A. If he selects a red marble, he stops selecting. If the marble is green, he continues by selecting at random one marble from Bag B.
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If he selects a red marble, he stops selecting. If the marble is green, he continues by selecting at random one marble from bag C.
a. Draw a tree diagram to represent this information.
b. Find the probability that Sasha selects 3 green marbles.
c. Find the probability that Sasha selects at least 1 marble of each colour.
d. Given that Sasha selects a red marble, find the probability that he selects it from bag B.
a. Draw a tree diagram to represent this information.
b. Find the probability that Sasha selects 3 green marbles.
c. Find the probability that Sasha selects at least 1 marble of each colour.
d. Given that Sasha selects a red marble, find the probability that he selects it from bag B.
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Statistics June 2019, Q2The partially completed box plot in Figure 1 shows the distribution of daily mean air temperatures using the data from the large data set for Beijing in 2015. An outlier is defined as a value more than 1.5 x IQR below Q1 or more than 1.5 x IQR above Q2 The three lowest air temperatures in the data set are 7.6C, 8.1C and 9.1C. The highest air temperature in the data set is 32.5C. Complete the box plot in Figure 1 showing clearly any outliers.
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Using your knowledge of the large data set, suggest from which month the two outliers are likely to have come.
Using the date from the large data set, Simon produced the following summary statistics for the daily mean air temperature, xC, for Beijing in 2015. Show that, to 3 significant figures, the standard deviation is 5.19C.
Simon decides to model the air temperatures with the random variable T N(22.6, 5.19^2)
Using Simon's model, calculate the 10th to 90th interpercentile range.
Simon wants to model another variable from the large data set for Beijing using a normal distruibution. State two variables from the large data set for Beijing that are not suitable to be modelled by a normal distribution. Give a reason for each answer.
Using the date from the large data set, Simon produced the following summary statistics for the daily mean air temperature, xC, for Beijing in 2015. Show that, to 3 significant figures, the standard deviation is 5.19C.
Simon decides to model the air temperatures with the random variable T N(22.6, 5.19^2)
Using Simon's model, calculate the 10th to 90th interpercentile range.
Simon wants to model another variable from the large data set for Beijing using a normal distruibution. State two variables from the large data set for Beijing that are not suitable to be modelled by a normal distribution. Give a reason for each answer.
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Statistics June 2019, Q3Barbara is investigating the relationship between average income (GCD per capita), x US dollars, and average annual carbon dioxide (C02) emissions, y tonnes, for different countries. She takes a random sample of 24 countries and finds the product moment correlation coefficient between annual CO2 emissions and average income to be 0.446.
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a. Stating your hypotheses clearly, test, at the 5% level of significance, whether or not the product moment correlation coefficient for all countries is greater than zero.
Barbara believes that a non-linear model would be a better fit to the data. She codes the data using the code m = log10x and c = log10y and obtains the model c = -1.82 + 0.89m. The product moment correlation coefficient between c and m is found to be 0.882.
b. Explain how this value supports Barbara's belief.
c. Show that the relationship between y and x can be written in the form y = ax^n where a and n are constants to be found.
Barbara believes that a non-linear model would be a better fit to the data. She codes the data using the code m = log10x and c = log10y and obtains the model c = -1.82 + 0.89m. The product moment correlation coefficient between c and m is found to be 0.882.
b. Explain how this value supports Barbara's belief.
c. Show that the relationship between y and x can be written in the form y = ax^n where a and n are constants to be found.
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Statistics June 2019, Q4Magali is studying the mean total cloud cover, in oktas, for Leuchars in 1987 using data from the large data set. The daily mean total cloud cover for all 184 days from the large data set is summarised in the table below. One of the 184 days is selected at random.
a. Find the probability that it has a daily mean total cloud over of 6 or greater. Magali is investigating whether the daily mean total cloud cover can be modelled |
using a binomial distribution. She uses the random variable X to denote the daily mean total cloud cover and believes that X - B(8, 0.76). Using Magali's model,
bi. find P(X larger than or equal to 6
ii. find, to 1 decimal place, the expected number of days in a sample of 184 days with a daily mean total cloud cover of 7.
c. Explain whether or not your answers to part b support the use of Magali's model.
There were 28 days that had a daily mean total cloud cover of 8. For these 28 days the daily mean total cloud cover for the following day is shown in the table below,
d. Find the proportion of these days when the daily mean total cloud cover was 6 or greater.
e. Comment on Magali's model in light of your answer to part d.
bi. find P(X larger than or equal to 6
ii. find, to 1 decimal place, the expected number of days in a sample of 184 days with a daily mean total cloud cover of 7.
c. Explain whether or not your answers to part b support the use of Magali's model.
There were 28 days that had a daily mean total cloud cover of 8. For these 28 days the daily mean total cloud cover for the following day is shown in the table below,
d. Find the proportion of these days when the daily mean total cloud cover was 6 or greater.
e. Comment on Magali's model in light of your answer to part d.
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Statistics June 2019, Q5A machine puts liquid into bottles of perfume. The amount of liquid put into each bottle, D ml, follows a normal distribution with mean 25 ml. Given that 15% of bottles contain less than 24.63 ml
a. find, to 2 decimal places, the value of k such that P(24.63 D k) = 0.45 A random sample of 200 bottles is taken. |
b. Using a normal approximation, find the probability that fewer than half of these bottles contain between 24.63 ml and k ml.
The machine is adjusted so that the standard deviation of the liquid put into the bottles is now 0.16 ml. Following the adjustments, Hannah believes that the mean amount of liquid put into each bottle is less than 25 ml. She takes a random sample of 20 bottles and finds the mean amount of liquid to be 24.94 ml.
c. Test Hannah's belief at the 5% level of significance. You should state your hypotheses clearly.
The machine is adjusted so that the standard deviation of the liquid put into the bottles is now 0.16 ml. Following the adjustments, Hannah believes that the mean amount of liquid put into each bottle is less than 25 ml. She takes a random sample of 20 bottles and finds the mean amount of liquid to be 24.94 ml.
c. Test Hannah's belief at the 5% level of significance. You should state your hypotheses clearly.
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Mechanics June 2019, Q1At time t seconds, where t >= 0, a particle, P, moves so that its velocity v ms-1 is given by v = 6ti - 5t^3/2 j
When t = o, the position vector of P is (-20i+20j)m. a. Find the acceleration of P when t = 4. b. Find the position vector of P when t = 4. |
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Mechanics June 2019, Q2A particle, P, moves with constant acceleration (2i - 3j) ms^-2.
At time t = 0, the particle is at the point A and is moving with velocity (-i + 4j) ms^-1. At time t = T seconds, P is moving in the direction of vector (3i - 4j) a. Find the value of T. At time t = 4 seconds, P is at the point B. b. Find the distance AB. |
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Mechanics June 2019, Q3Two blocks, A and B, of masses 2m and 3m respectively, are attached to the ends of a light string. Initially A is held at rest on a fixed rough plane. The plane is inclined at angle alpha to the horizontal ground, where tan alpha = 5/12
The string passes over a small smooth pulley, P, fixed at the top of the plane. The part of the string from A to P is parallel to a line of greatest slope of the plane. |
Block B hangs freely below P, as shown in figure 1. The coefficient of friction between A and the plane is 2/3. The blocks are released from rest with the string taut and A moves up the plane. The tension in the string immediately after the blocks are released is T. The blocks are modelled as particles and the string is modelled as being inextensible.
a. Show that T = 12mg/5
After B reaches the ground, A continues to move up the plane until it comes to rest before reaching P.
b. Determine whether A will remain at rest, carefully justifying your answer.
c. Suggest two refinements to the model that would make it more realistic.
a. Show that T = 12mg/5
After B reaches the ground, A continues to move up the plane until it comes to rest before reaching P.
b. Determine whether A will remain at rest, carefully justifying your answer.
c. Suggest two refinements to the model that would make it more realistic.
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Mechanics June 2019, Q4A ramp, AB, of length 8m and mass 20 kg, rests in equilibrium with the end A on rough horizontal ground. The ramp rests on a smooth cylindrical drum which is partly under the ground. The drum is fixed with its axis at the same horizontal level as A. The point of contact between the ramp and the drum is C, where AC = 5 m, as shown in Figure 2. The ramp is resting in a vertical plane which is perpendicular to the axis of the drum,
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at an angle theta to the horizontal, where tan theta = 7/24. The ramp is modelled as a uniform rod.
a. Explain why the reaction from the drum on the ramp at point C acts in a direction which is perpendicular to the ramp.
b. Find the magnitude of the resultant force acting on the ramp at A.
The ramp is still in equilibrium in the position shown in Figure 2 but the ramp is not now modelled as being uniform. Given that the centre of the ramp is assumed to be closer to A than to B,
c. state how this would affect the magnitude of the normal reaction between the ramp and the drum at C.
a. Explain why the reaction from the drum on the ramp at point C acts in a direction which is perpendicular to the ramp.
b. Find the magnitude of the resultant force acting on the ramp at A.
The ramp is still in equilibrium in the position shown in Figure 2 but the ramp is not now modelled as being uniform. Given that the centre of the ramp is assumed to be closer to A than to B,
c. state how this would affect the magnitude of the normal reaction between the ramp and the drum at C.
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Mechanics June 2019, Q5The points A and B lie 50m apart on horizontal ground. At time t=0 two small balls, P and Q, are projected in the vertical plane containing AB.
Ball P is projected from A with speed 20ms-1 at 30 degrees to AB. Ball Q is projected from B with speed ums-1 at angle theta to BA, as shown in Figure 3. At time t=2 seconds P and Q collide. Until they collide, the balls are modelled as particles moving freely under gravity. a. Find the velocity of P at the instant before it collides with Q. b. Find i. the size of angle theta ii. the value of u. c. State one limitation of the model, other than air resistance, that could affect the accuracy of your answers. |