Questions — WJEC Unit 2 (16 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
WJEC Unit 2 2018 June Q11
11 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 \(15 \mathrm {~ms} ^ { - 1 }\). 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 \(B\).
a) Find the distance \(A B\). The vehicle then reverses with a constant acceleration of \(2 \mathrm {~ms} ^ { - 2 }\) for 8 seconds, followed by a constant deceleration of \(1.6 \mathrm {~ms} ^ { - 2 }\), coming to rest at the point \(C\), which is between \(A\) and \(B\).
b) Calculate the time it takes for the vehicle to reverse from \(B\) to \(C\).
c) Sketch a velocity-time graph for the motion of the vehicle.
d) Determine the distance \(A C\).
WJEC Unit 2 2022 June Q1
1 In this question, \(\mathbf { i }\) and \(\mathbf { j }\) represent unit vectors due east and due north respectively. Sarah is going for a walk. She leaves her house and walks directly to the shop. She then walks directly from the shop to the park. Relative to her house:
  • the shop has position vector \(\left( - \frac { 2 } { 3 } \mathbf { j } \right) \mathrm { km }\),
  • the park is 2 km away on a bearing of \(060 ^ { \circ }\).
    a) Show that the position vector of the park relative to the house is \(( \sqrt { 3 } \mathbf { i } + \mathbf { j } ) \mathrm { km }\).
    b) Determine the total distance walked by Sarah from her house to the park.
    c) By considering a modelling assumption you have made, explain why the answer you found in part (b) may not be the actual distance that Sarah walked.
\(\mathbf { 1 }\)\(\mathbf { 1 }\)
A particle \(P\) moves along the \(x\)-axis so that its velocity \(v \mathrm {~ms} ^ { - 1 }\) at time \(t\) seconds \(( t \geqslant 0 )\) is given by $$v = 3 t ^ { 2 } - 24 t + 36$$ a) Find the values of \(t\) when \(P\) is instantaneously at rest.
b) Calculate the total distance travelled by the particle \(P\) whilst its velocity is decreasing.
WJEC Unit 2 2024 June Q1
  1. 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?
    2. What type of sampling is this?
    3. How could the sampling process be improved?
    4. A baker sells \(3 \cdot 5\) birthday cakes per hour on average.
    5. State, in context, two assumptions you would have to make in order to model the number of birthday cakes sold using a Poisson distribution.
    6. 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.
      （c）Calculate the probability that，during a randomly selected 3 －hour period，the baker sells more than 10 birthday cakes．
      （d）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．
    7. Select one of the assumptions in part (a) and comment on its reasonableness.
    \section*{PLEASE DO NOT WRITE ON THIS PAGE}
WJEC Unit 2 2024 June Q3
  1. 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[max width=\textwidth, alt={}, center]{d9ef2033-bf8b-4aec-bc88-34dbc8b9c208-08_641_1050_477_511}
For these 100 students, participation in athletics and participation in climbing are independent events.
  1. Show that \(x = 10\) and find the value of \(y\).
  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.
WJEC Unit 2 2024 June Q4
4. 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. 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.
  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?
  4. Now suppose that the proportion of blue sweets is \(7 \%\). Find the probability of a Type II error. Interpret your answer in context.
WJEC Unit 2 2024 June Q5
4 marks
5. 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 \href{http://gov.uk}{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." \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Number of people (1000s) who received 2nd dose vaccinations daily in the UK, by report date} \includegraphics[alt={},max width=\textwidth]{d9ef2033-bf8b-4aec-bc88-34dbc8b9c208-12_531_1525_906_260}
\end{figure}
  1. By referring to the graph, explain how the quote could be misleading.
    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 number of 2nd dose vaccines (1000s)Midpoint \(x\)Frequency \(f\)Percentage
    \(0 \leqslant v < 100\)5023.3
    \(100 \leqslant v < 200\)150813.1
    \(200 \leqslant v < 300\)2501016.4
    \(300 \leqslant v < 400\)3501321.3
    \(400 \leqslant v < 500\)45026\(42 \cdot 6\)
    \(500 \leqslant v < 600\)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.
      A second graph in the article shows the number of people receiving a third dose of the vaccine. This graph has a repeated pattern of rising then falling. An extract is shown below. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Number of people (millions) who received 3rd dose vaccinations daily in the UK, by report date} \includegraphics[alt={},max width=\textwidth]{d9ef2033-bf8b-4aec-bc88-34dbc8b9c208-14_556_1115_571_466}
      \end{figure}
  3. Give a possible reason for the pattern observed in this graph.
    Another extract shows the number of people who received the third dose of the vaccine between 27th March 2022 and 25th April 2022. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Number of people (1000s) who received 3rd dose vaccinations daily in the UK, by report date} \includegraphics[alt={},max width=\textwidth]{d9ef2033-bf8b-4aec-bc88-34dbc8b9c208-15_537_1246_571_406}
    \end{figure}
  4. State, with a reason, whether or not you think the data for April 15th to April 18th are incorrect.
WJEC Unit 2 2024 June Q6
2 marks
  1. A ship \(S\) is moving with constant velocity \(( 4 \mathbf { i } - 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), 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.
  2. 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[max width=\textwidth, alt={}, center]{d9ef2033-bf8b-4aec-bc88-34dbc8b9c208-17_504_814_504_589}
    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.
    2. Calculate the reaction of box \(Q\) on box \(P\) when they are moving vertically upwards with constant speed.
    3. A particle, of mass 4 kg , moves in a straight line under the action of a single force \(F \mathrm {~N}\), whose magnitude at time \(t\) seconds is given by
    $$F = 12 \sqrt { t } - 32 \text { for } t \geqslant 0 .$$
  3. Find the acceleration of the particle when \(t = 9\).
  4. Given that the particle has velocity \(- 1 \mathrm {~ms} ^ { - 1 }\) when \(t = 4\), find an expression for the velocity of the particle at \(t \mathrm {~s}\).
  5. Determine whether the speed of the particle is increasing or decreasing when \(t = 9\). [2]
WJEC Unit 2 2024 June Q9
9. The diagram below shows an object \(A\), of mass \(2 m \mathrm {~kg}\), lying on a horizontal table. It is connected to another object \(B\), of mass \(m \mathrm {~kg}\), 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[max width=\textwidth, alt={}, center]{d9ef2033-bf8b-4aec-bc88-34dbc8b9c208-20_589_871_593_605} Object \(A\) is then released.
  1. When object \(B\) has moved downwards a vertical distance of 0.4 m , its speed is \(1.2 \mathrm {~ms} ^ { - 1 }\). Use a formula for motion in a straight line with constant acceleration to show that the magnitude of the acceleration of \(B\) is \(1.8 \mathrm {~ms} ^ { - 2 }\).
  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.
  3. What assumption did the word 'inextensible' in the description of the string enable you to make in your solution?
    \section*{PLEASE DO NOT WRITE ON THIS PAGE}
WJEC Unit 2 2024 June Q10
1 marks
  1. 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 \(26 \mathrm {~ms} ^ { - 1 }\). The car continues at \(26 \mathrm {~ms} ^ { - 1 }\) for 1 minute, before decelerating at a constant rate of \(0.75 \mathrm {~ms} ^ { - 2 }\) until it passes the point \(B\).
    1. Show that the car travels 156 m while it is accelerating.
      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 \(B\).
    3. Sketch a displacement-time graph for the motion of the car between \(A\) and \(B\).
    Additional page, if required. Write the question number(s) in the left-hand margin. Additional page, if required. Write the question number(s) in the left-hand margin. \section*{PLEASE DO NOT WRITE ON THIS PAGE}
WJEC Unit 2 Specimen Q1
  1. The events \(\mathrm { A } , B\) 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. \(A , B\) are independent,
    3. \(\quad A \subset B\).
    4. 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,
    5. state hypotheses to be used to resolve this difference of opinion.
    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?
WJEC Unit 2 Specimen Q3
3. 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.
  2. Phil stands at the bridge for 20 minutes. Determine the probability that he sees exactly 6 cars arrive.
  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 .
WJEC Unit 2 Specimen Q4
4. 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.
  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'.
    2. Interpret the correlation between 'Number of calories' and 'Carbohydrate' in this context.
      \includegraphics[max width=\textwidth, alt={}, center]{dfe44f43-5e4d-4b8b-a581-f7889abc5cda-3_810_1154_1315_539}
  3. The equation of the regression line for this dataset is: $$\text { 'Number of calories' } = 12.4 + 2.9 \times \text { 'Carbohydrate' }$$
    1. Interpret the gradient of the regression line in this context.
    2. Explain why it is reasonable for the regression line to have a non-zero intercept in this context.
WJEC Unit 2 Specimen Q5
5. Gareth has a keen interest in pop music. He recently read the following claim in a music magazine. \section*{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.
  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[max width=\textwidth, alt={}, center]{dfe44f43-5e4d-4b8b-a581-f7889abc5cda-4_860_1435_1343_379} State two corrections that Gareth needs to make to the histogram so that it accurately represents the data in the table.
  3. Gareth also produced a box plot of the lengths of singles. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Length of single for top 50 UK Official Singles Chart 17 June 2016} \includegraphics[alt={},max width=\textwidth]{dfe44f43-5e4d-4b8b-a581-f7889abc5cda-5_504_812_406_644}
    \end{figure} 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?
  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)
      \multirow{2}{*}{Length of single}NMeanStandard deviationMinimumLower quartileMedianUpper quartileMaximum
      493.570.3932.773.263.603.894.38
    2. State, with a reason, whether these statistics support the magazine's claim.
  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)
    \multirow{2}{*}{Length of single}NMeanStandard 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.
WJEC Unit 2 Specimen Q6
  1. A small object, of mass 0.02 kg , is dropped from rest from the top of a building which is 160 m high.
    1. Calculate the speed of the object as it hits the ground.
    2. Determine the time taken for the object to reach the ground.
    3. State one assumption you have made in your solution.
    4. 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[max width=\textwidth, alt={}, center]{dfe44f43-5e4d-4b8b-a581-f7889abc5cda-6_515_238_1023_868}
    5. Calculate the magnitude of the acceleration of \(A\) and the tension in the string.
    6. What assumption does the word 'light' in the description of the string enable you to make in your solution?
    7. A particle \(P\), of mass 3 kg , moves along the horizontal \(x\)-axis under the action of a resultant force \(F \mathrm {~N}\). Its velocity \(v \mathrm {~ms} ^ { - 1 }\) at time \(t\) seconds is given by
    $$v = 12 t - 3 t ^ { 2 }$$
  2. 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 \(t \mathrm {~s}\).
  3. Find an expression for the acceleration of the particle at time \(t \mathrm {~s}\).
WJEC Unit 2 Specimen Q9
9. A truck of mass 180 kg runs on smooth horizontal rails. A light inextensible rope is attached to the front of the truck. The rope runs parallel to the rails until it passes over a light smooth pulley. The rest of the rope hangs down a vertical shaft. When the truck is required to move, a load of \(M \mathrm {~kg}\) is attached to the end of the rope in the shaft and the brakes are then released.
  1. Find the tension in the rope when the truck and the load move with an acceleration of magnitude \(0.8 \mathrm {~ms} ^ { - 2 }\) and calculate the corresponding value of \(M\).
  2. In addition to the assumptions given in the question, write down one further assumption that you have made in your solution to this problem and explain how that assumption affects both of your answers.
WJEC Unit 2 Specimen Q10
10. Two forces \(\mathbf { F }\) and \(\mathbf { G }\) acting on an object are such that $$\begin{aligned} & \mathbf { F } = \mathbf { i } - 8 \mathbf { j }
& \mathbf { G } = 3 \mathbf { i } + 11 \mathbf { j } \end{aligned}$$ The object has a mass of 3 kg . Calculate the magnitude and direction of the acceleration of the object.