Questions — WJEC (504 questions)

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WJEC Unit 2 2018 June Q05
6 marks Easy -1.2
A baker is aware that the pH of his sourdough, \(y\), and the hydration, \(x\), affect the taste and texture of the final product. The hydration is measured in ml of water per 100 g of flour (ml/100 g). The baker researches how the pH of his sourdough changes as the hydration changes. The results of his research are shown in the diagram below. \includegraphics{figure_5}
  1. Describe the relationship between pH and hydration. [2]
  2. The equation of the regression line for \(y\) on \(x\) is $$y = 5.4 - 0.02x.$$
    1. Interpret the gradient and intercept of the regression line in this context.
    2. Estimate the pH of the sourdough when the hydration is 20 ml/100 g. Comment on the reliability of this estimate. [4]
WJEC Unit 2 2018 June Q06
10 marks Moderate -0.8
Basel is a keen learner of languages. He finds a website on which a large number of language tutors offer their services. Basel records the cost, in dollars, of a one hour lesson from a random sample of tutors. He puts the data into a computer program which gives the following summary statistics. Cost per 1 hour lesson Min. :10.0 1st Qu. :16.0 Median :17.2 Mean :19.8 3rd Qu. :21.0 Max. :40.0
  1. Showing all calculations, comment on any outliers for the cost of a one hour lesson with a language tutor. [4]
  2. Describe the skewness of the data and explain what it means in this context. [2]
Dafydd is also a keen learner of languages. He takes his own random sample of the cost, in dollars, for a one hour lesson. He produces the following box plot. \includegraphics{figure_6}
    1. What will happen to the mean if the outlier is removed?
    2. What will happen to the median if the outlier is removed? [2]
  2. Compare and contrast the distributions of the cost of one hour language lessons for Dafydd's sample and Basel's sample. [2]
WJEC Unit 2 2018 June Q07
3 marks Moderate -0.8
A particle moves along the horizontal \(x\)-axis so that its velocity \(v\) ms\(^{-1}\) at time \(t\) seconds is given by $$v = 6t^2 - 8t - 5.$$ At time \(t = 1\), the particle's displacement from the origin is \(-4\) m. Find an expression for the displacement of the particle at time \(t\) seconds. [3]
WJEC Unit 2 2018 June Q08
7 marks Moderate -0.8
The diagram shows two objects \(A\) and \(B\), of mass 3 kg and 5 kg respectively, connected by a light inextensible string passing over a light smooth pulley fixed at the end of a smooth horizontal surface. Object \(A\) lies on the horizontal surface and object \(B\) hangs freely below the pulley. \includegraphics{figure_8} Initially, \(B\) is supported so that the objects are at rest with the string just taut. Object \(B\) is then released.
  1. Find the magnitude of the acceleration of \(A\) and the tension in the string. [6]
  2. State briefly what effect a rough pulley would have on the tension in the string. [1]
WJEC Unit 2 2018 June Q09
6 marks Moderate -0.8
Three forces \(\mathbf{L}\), \(\mathbf{M}\) and \(\mathbf{N}\) are given by $$\mathbf{L} = 2\mathbf{i} + 5\mathbf{j},$$ $$\mathbf{M} = 3\mathbf{i} - 22\mathbf{j},$$ $$\mathbf{N} = 4\mathbf{i} - 23\mathbf{j}.$$ Find the magnitude and direction of the resultant of the three forces. [6]
WJEC Unit 2 2018 June Q10
7 marks Moderate -0.8
A person, of mass 68 kg, stands in a lift which is moving upwards with constant acceleration. The lift is of mass 770 kg and the tension in the lift cable is 8000 N.
  1. Determine the acceleration of the lift, giving your answer correct to two decimal places. [3]
  2. State whether the lift is getting faster, staying at the same speed or slowing down. [1]
  3. Calculate the magnitude of the reaction of the floor of the lift on the person. [3]
WJEC Unit 2 2018 June Q11
12 marks Moderate -0.8
A vehicle moves along a straight horizontal road. Points \(A\) and \(B\) lie on the road. As the vehicle passes point \(A\), it is moving with constant speed 15 ms\(^{-1}\). It travels with this constant speed for 2 minutes before a constant deceleration is applied for 12 seconds so that it comes to rest at point \(B\).
  1. Find the distance \(AB\). [3]
The vehicle then reverses with a constant acceleration of 2 ms\(^{-2}\) for 8 seconds, followed by a constant deceleration of 1·6 ms\(^{-2}\), coming to rest at the point \(C\), which is between \(A\) and \(B\).
  1. Calculate the time it takes for the vehicle to reverse from \(B\) to \(C\). [4]
  2. Sketch a velocity-time graph for the motion of the vehicle. [3]
  3. Determine the distance \(AC\). [2]
WJEC Unit 2 2024 June Q1
3 marks Easy -2.0
An exercise gym opens at 6:00 a.m. every day. The manager decides to use a questionnaire to gather the opinions of the gym members. The first 30 members arriving at the gym on a particular morning are asked to complete the questionnaire.
  1. What is the intended population in this context? [1]
  2. What type of sampling is this? [1]
  3. How could the sampling process be improved? [1]
WJEC Unit 2 2024 June Q2
10 marks Moderate -0.8
A baker sells 3-5 birthday cakes per hour on average.
  1. State, in context, two assumptions you would have to make in order to model the number of birthday cakes sold using a Poisson distribution. [1]
  2. Using a Poisson distribution and showing your calculation, find the probability that exactly 2 birthday cakes are sold in a randomly selected 1-hour period. [2]
  3. Calculate the probability that, during a randomly selected 3-hour period, the baker sells more than 10 birthday cakes. [3]
  4. The baker sells a birthday cake at 9:30 a.m. Calculate the probability that the baker will sell the next birthday cake before 10:00 a.m. [3]
  5. Select one of the assumptions in part (a) and comment on its reasonableness. [1]
WJEC Unit 2 2024 June Q3
8 marks Moderate -0.3
The following Venn diagram shows the participation of 100 students in three activities, \(A\), \(B\), and \(C\), which represent athletics, baseball and climbing respectively. \includegraphics{figure_3} For these 100 students, participation in athletics and participation in climbing are independent events.
  1. Show that \(x = 10\) and find the value of \(y\). [5]
  2. Two students are selected at random, one after the other without replacement. Find the probability that the first student does athletics and the second student does only climbing. [3]
WJEC Unit 2 2024 June Q4
11 marks Standard +0.3
A company produces sweets of varying colours. The company claims that the proportion of blue sweets is 13·6%. A consumer believes that the true proportion is less than this. In order to test this belief, the consumer collects a random sample of 80 sweets.
  1. State suitable hypotheses for the test. [1]
    1. Determine the critical region if the test is to be carried out at a significance level as close as possible to, but not exceeding, 5%.
    2. Given that there are 6 blue sweets in the sample of 80, complete the significance test. [5]
  3. Suppose the proportion of blue sweets claimed by the company is correct. The consumer conducts the sampling and testing process on a further 20 occasions, using the sample size of 80 each time. What is the expected number of these occasions on which the consumer would reach the incorrect conclusion? [2]
  4. Now suppose that the proportion of blue sweets is 7%. Find the probability of a Type II error. Interpret your answer in context. [3]
WJEC Unit 2 2024 June Q5
8 marks Easy -1.2
In March 2020, the coronavirus pandemic caused major disruption to the lives of individuals across the world. A newspaper published the following graph from the gov.uk website, along with an article which included the following excerpt. "The daily number of vaccines administered continues to fall. In order to get control of the virus, we need the number of people receiving a second dose of the vaccine to keep rocketing. The fear is it will start to drop off soon, which will leave many people still unprotected." \includegraphics{figure_5}
  1. By referring to the graph, explain how the quote could be misleading. [1]
The daily numbers of second dose vaccines, in thousands, over the period April 1st 2021 to May 31st 2021 are shown in the table below.
Daily numberMidpointFrequencyPercentage
of 2nd dose\(x\)\(f\)
vaccines
(1000s)
\(0 \leqslant v < 100\)5023·3
\(100 \leqslant v < 200\)150813·1
\(200 \leqslant v < 300\)2501016·4
\(300 \leqslant v < 400\)3501321·3
\(400 \leqslant v < 500\)4502642·6
\(500 \leqslant v < 600\)55023·3
Total61100
    1. Calculate estimates of the mean and standard deviation for the daily number of second dose vaccines given over this period. You may use \(\sum x^2 f = 8272500\). [4]
    2. Comment on the skewness of these data. [1]
  2. Give a possible reason for the pattern observed in this graph. [1]
  3. State, with a reason, whether or not you think the data for April 15th to April 18th are incorrect. [1]
WJEC Unit 2 2024 June Q6
4 marks Easy -1.2
A ship \(S\) is moving with constant velocity \((4\mathbf{i} - 7\mathbf{j})\text{ms}^{-1}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors due east and due north respectively. Find the speed and direction of \(S\), giving the direction as a three-figure bearing, correct to the nearest degree. [4]
WJEC Unit 2 2024 June Q7
4 marks Moderate -0.8
The diagram below shows a forklift truck being used to raise two boxes, \(P\) and \(Q\), vertically. Box \(Q\) rests on horizontal forks and box \(P\) rests on top of box \(Q\). Box \(P\) has mass 25 kg and box \(Q\) has mass 55 kg. \includegraphics{figure_7}
  1. When the boxes are moving upwards with uniform acceleration, the reaction of the horizontal forks on box \(Q\) is 820 N. Calculate the magnitude of the acceleration. [3]
  2. Calculate the reaction of box \(Q\) on box \(P\) when they are moving vertically upwards with constant speed. [1]
WJEC Unit 2 2024 June Q8
7 marks Moderate -0.3
A particle, of mass 4 kg, moves in a straight line under the action of a single force \(F\) N, whose magnitude at time \(t\) seconds is given by $$F = 12\sqrt{t} - 32 \quad \text{for} \quad t \geqslant 0.$$
  1. Find the acceleration of the particle when \(t = 9\). [2]
  2. Given that the particle has velocity \(-1\text{ms}^{-1}\) when \(t = 4\), find an expression for the velocity of the particle at \(t\) s. [3]
  3. Determine whether the speed of the particle is increasing or decreasing when \(t = 9\). [2]
WJEC Unit 2 2024 June Q9
9 marks Moderate -0.3
The diagram below shows an object \(A\), of mass \(2m\) kg, lying on a horizontal table. It is connected to another object \(B\), of mass \(m\) kg, by a light inextensible string, which passes over a smooth pulley \(P\), fixed at the edge of the table. Initially, object \(A\) is held at rest so that object \(B\) hangs freely with the string taut. \includegraphics{figure_9} Object \(A\) is then released.
  1. When object \(B\) has moved downwards a vertical distance of 0·4 m, its speed is 1·2 ms\(^{-1}\). Use a formula for motion in a straight line with constant acceleration to show that the magnitude of the acceleration of \(B\) is 1·8 ms\(^{-2}\). [2]
  2. During the motion, object \(A\) experiences a constant resistive force of 22 N. Find the value of \(m\) and hence determine the tension in the string. [6]
  3. What assumption did the word 'inextensible' in the description of the string enable you to make in your solution? [1]
WJEC Unit 2 2024 June Q10
11 marks Moderate -0.8
A car, starting from rest at a point \(A\), travels along a straight horizontal road towards a point \(B\). The distance between points \(A\) and \(B\) is 1·9 km. Initially, the car accelerates uniformly for 12 seconds until it reaches a speed of 26 ms\(^{-1}\). The car continues at 26 ms\(^{-1}\) for 1 minute, before decelerating at a constant rate of 0·75 ms\(^{-2}\) until it passes the point \(B\).
  1. Show that the car travels 156 m while it is accelerating. [2]
    1. Work out the distance travelled by the car while travelling at a constant speed. [1]
    2. Hence calculate the length of time for which the car is decelerating until it passes the point \(B\). [5]
  3. Sketch a displacement-time graph for the motion of the car between \(A\) and \(B\). [3]
WJEC Unit 2 Specimen Q1
6 marks Moderate -0.8
The events \(A, B\) are such that \(P(A) = 0.2, P(B) = 0.3\). Determine the value of \(P(A \cup B)\) when
  1. \(A,B\) are mutually exclusive, [2]
  2. \(A,B\) are independent, [3]
  3. \(A \subset B\). [1]
WJEC Unit 2 Specimen Q2
9 marks Standard +0.3
Dewi, a candidate in an election, believes that 45% of the electorate intend to vote for him. His agent, however, believes that the support for him is less than this. Given that \(p\) denotes the proportion of the electorate intending to vote for Dewi,
  1. state hypotheses to be used to resolve this difference of opinion. [1]
They decide to question a random sample of 60 electors. They define the critical region to be \(X \leq 20\), where \(X\) denotes the number in the sample intending to vote for Dewi.
    1. Determine the significance level of this critical region.
    2. If in fact \(p\) is actually 0.35, calculate the probability of a Type II error.
    3. Explain in context the meaning of a Type II error.
    4. Explain briefly why this test is unsatisfactory. How could it be improved while keeping approximately the same significance level? [8]
WJEC Unit 2 Specimen Q3
7 marks Moderate -0.8
Cars arrive at random at a toll bridge at a mean rate of 15 per hour.
  1. Explain briefly why the Poisson distribution could be used to model the number of cars arriving in a particular time interval. [1]
  2. Phil stands at the bridge for 20 minutes. Determine the probability that he sees exactly 6 cars arrive. [3]
  3. Using the statistical tables provided, find the time interval (in minutes) for which the probability of more than 10 cars arriving is approximately 0.3. [3]
WJEC Unit 2 Specimen Q4
7 marks Easy -1.3
A researcher wishes to investigate the relationship between the amount of carbohydrate and the number of calories in different fruits. He compiles a list of 90 different fruits, e.g. apricots, kiwi fruits, raspberries. As he does not have enough time to collect data for each of the 90 different fruits, he decides to select a simple random sample of 14 different fruits from the list. For each fruit selected, he then uses a dieting website to find the number of calories (kcal) and the amount of carbohydrate (g) in a typical adult portion (e.g. a whole apple, a bunch of 10 grapes, half a cup of strawberries). He enters these data into a spreadsheet for analysis.
  1. Explain how the random number function on a calculator could be used to select this sample of 14 different fruits. [3]
  2. The scatter graph represents 'Number of calories' against 'Carbohydrate' for the sample of 14 different fruits.
    1. Describe the correlation between 'Number of calories' and 'Carbohydrate'. [1]
    2. Interpret the correlation between 'Number of calories' and 'Carbohydrate' in this context. [1]
    \includegraphics{figure_1}
  3. The equation of the regression line for this dataset is: 'Number of calories' = 12.4 + 2.9 × 'Carbohydrate'
    1. Interpret the gradient of the regression line in this context. [1]
    2. Explain why it is reasonable for the regression line to have a non-zero intercept in this context. [1]
WJEC Unit 2 Specimen Q5
12 marks Easy -1.2
Gareth has a keen interest in pop music. He recently read the following claim in a music magazine. In the pop industry most songs on the radio are not longer than three minutes.
  1. He decided to investigate this claim by recording the lengths of the top 50 singles in the UK Official Singles Chart for the week beginning 17 June 2016. (A 'single' in this context is one digital audio track.) Comment on the suitability of this sample to investigate the magazine's claim. [1]
  2. Gareth recorded the data in the table below.
    Length of singles for top 50 UK Official Chart singles, 17 June 2016
    2.5-(3.0)3.0-(3.5)3.5-(4.0)4.0-(4.5)4.5-(5.0)5.0-(5.5)5.5-(6.0)6.0-(6.5)6.5-(7.0)7.0-(7.5)
    317227000001
    He used these data to produce a graph of the distributions of the lengths of singles \includegraphics{figure_2} State two corrections that Gareth needs to make to the histogram so that it accurately represents the data in the table. [2]
  3. Gareth also produced a box plot of the lengths of singles. \includegraphics{figure_3} He sees that there is one obvious outlier.
    1. What will happen to the mean if the outlier is removed?
    2. What will happen to the standard deviation if the outlier is removed? [2]
  4. Gareth decided to remove the outlier. He then produced a table of summary statistics.
    1. Use the appropriate statistics from the table to show, by calculation, that the maximum value for the length of a single is not an outlier.
      Summary statistics
      Length of single for top 50 UK Official Singles Chart (minutes)
      Length of singleNMeanStandard deviationMinimumLower quartileMedianUpper quartileMaximum
      493.570.3932.773.263.603.894.38
    2. State, with a reason, whether these statistics support the magazine's claim. [4]
  5. Gareth also calculated summary statistics for the lengths of 30 singles selected at random from his personal collection.
    Summary statistics
    Length of single for Gareth's random sample of 30 singles (minutes)
    Length of singleNMeanStandard deviationMinimumLower quartileMedianUpper quartileMaximum
    303.130.3642.582.732.923.223.95
    Compare and contrast the distribution of lengths of singles in Gareth's personal collection with the distribution in the top 50 UK Official Singles Chart. [3]
WJEC Unit 2 Specimen Q6
7 marks Easy -1.2
A small object, of mass 0.02 kg, is dropped from rest from the top of a building which is160 m high.
  1. Calculate the speed of the object as it hits the ground. [3]
  2. Determine the time taken for the object to reach the ground. [3]
  3. State one assumption you have made in your solution. [1]
WJEC Unit 2 Specimen Q7
7 marks Moderate -0.8
The diagram below shows two particles \(A\) and \(B\), of mass 2 kg and 5 kg respectively, which are connected by a light inextensible string passing over a fixed smooth pulley. Initially, \(B\) is held at rest with the string just taut. It is then released. \includegraphics{figure_4}
  1. Calculate the magnitude of the acceleration of \(A\) and the tension in the string. [6]
  2. What assumption does the word 'light' in the description of the string enable you to make in your solution? [1]
WJEC Unit 2 Specimen Q8
5 marks Moderate -0.8
A particle \(P\), of mass 3 kg, moves along the horizontal \(x\)-axis under the action of a resultant force \(F\) N. Its velocity \(v\) ms\(^{-1}\) at time \(t\) seconds is given by $$v = 12t - 3t^2.$$
  1. Given that the particle is at the origin \(O\) when \(t = 1\), find an expression for the displacement of the particle from \(O\) at time \(t\) s. [3]
  2. Find an expression for the acceleration of the particle at time \(t\) s. [2]