Questions — WJEC (504 questions)

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WJEC Further Unit 2 2019 June Q4
15 marks Standard +0.3
4. The continuous random variable, \(X\), has the following probability density function $$f ( x ) = \begin{cases} k x & \text { for } 0 \leqslant x < 1 \\ k x ^ { 3 } & \text { for } 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant. \includegraphics[max width=\textwidth, alt={}, center]{4ecf99c5-c4b3-41b7-a8df-a7c2ca7fcd6a-3_851_935_826_678}
  1. Show that \(k = \frac { 4 } { 17 }\).
  2. Determine \(\mathrm { E } ( X )\).
  3. Calculate \(\mathrm { E } ( 3 X - 1 )\) and \(\operatorname { Var } ( 3 X - 1 )\).
WJEC Further Unit 2 2019 June Q5
11 marks Standard +0.3
5. Chris is investigating the distribution of birth months for ice hockey players. He collects data for 869 randomly chosen National Hockey League (NHL) players. He decides to carry out a chi-squared test. Using a spreadsheet, he produces the following output.
ABcD
1Birth MonthObservedExpectedChi-Squared Contributions
2Jan-Mar259217.258.023302647
3Apr-June232217.251.001438435
4Jul-Sept200217.251.369677791
5Oct-Dec178217.257.091196778
6Total86986917.48561565
7
8p value
90.000561458
  1. By considering the output, state the null hypothesis that Chris is testing. State what conclusion Chris should reach and explain why. Chris now wonders if Premier League football players' birth months are distributed uniformly throughout the year. He collects the birth months of 75 randomly selected Premier League footballers. This information is shown in the table below.
    JanFebMarAprMayJunJulAugSepOctNovDec
    37114122665856
  2. Carry out the chi-squared goodness of fit test at the 10\% significance level that Chris should use to conduct his investigation.
WJEC Further Unit 2 2019 June Q6
6 marks Moderate -0.3
6. The University of Arizona surveyed a large number of households. One purpose of the survey was to determine if annual household income could be predicted from size of family home. The graph of Annual household income, \(y\), versus Size of family home, \(x\), is shown below. \includegraphics[max width=\textwidth, alt={}, center]{4ecf99c5-c4b3-41b7-a8df-a7c2ca7fcd6a-5_616_1257_566_365}
  1. State the limitations of using the regression line above with reference to the scatter diagram. The data for size of family homes between 2000 and 3000 square feet are shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{4ecf99c5-c4b3-41b7-a8df-a7c2ca7fcd6a-5_652_1244_1516_360} Summary statistics for these data are as follows. $$\begin{array} { r c c } \sum x = 93160 & \sum y = 3907142 & n = 37 \\ S _ { x x } = 2869673.03 & S _ { y y } = 44312797167 & S _ { x y } = 348512820 \cdot 6 \end{array}$$
  2. Calculate the equation of the least squares regression line to predict Annual household income from Size of family home for these data.
WJEC Further Unit 2 2019 June Q7
13 marks Moderate -0.5
7. An article published in a medical journal investigated sports injuries in adolescents' ball games: football, handball and basketball. In a study of 906 randomly selected adolescent players in the three ball games, 379 players incurred injuries over the course of one year of playing the sport. Rhian wants to test whether there is an association between the site of injury and the sport played. A summary of the injuries is shown in the table below.
\multirow{2}{*}{}Site of injury
Observed valuesShoulder/ ArmHand/ FingersThigh/ LegKneeAnkleFootOtherTotal
\multirow{3}{*}{Sport}Football834536513612191
Handball14266154266115
Basketball428442211073
Total265755551154328379
  1. Calculate the values of \(A , B , C\) in the tables below.
    \multirow{2}{*}{}Site of injury
    Expected valuesShoulder/ ArmHand/ FingersThigh/ LegKneeAnkleFootOther
    \multirow{3}{*}{sodod}Football13.102928.725627.717727.717757.955121.670214.1108
    Handball7.889217.295516.688716.6887A13.04758.4960
    Basketball5.007910.978910.593710.593722.15048.28235.3931
    \multirow{2}{*}{}\multirow[b]{2}{*}{Chi-Squared Contributions}Site of injury
    Shoulder/ ArmHand/ FingersThigh/ LegKneeAnkleFootOther
    \multirow{3}{*}{sodoct}Football1.9873223.03890\(10 \cdot 77575\)2.47484\(B\)9.475860.31575
    Handball4.733334.38079C0.170871.446903.806640.73331
    Basketball0.2028626.388654.104004.104000.001026.403063.93521
  2. Given that the test statistic, \(X ^ { 2 }\), is 116.16, carry out the significance test at the \(5 \%\) level.
  3. Which site of injury most affects the conclusion of this test? Comment on your answer. Rhian also analyses the data on the type of contact that caused the injuries and the sport in which they occur, shown in the table below.
    Observed valuesBallOpponentSurfaceNoneTotal
    Football17681792194
    Handball23341938114
    Basketball2817121471
    Total6811948144379
    The chi-squared test statistic is 46.0937 . Rhian notes that this value is smaller than 116.16 , the test statistic in part (b). She concludes that there is weaker evidence for association in this case than there was in part (b).
  4. State Rhian's misconception and explain what she should consider instead. \section*{END OF PAPER}
WJEC Further Unit 2 2022 June Q1
7 marks Easy -1.8
  1. The probability distribution for the prize money, \(\pounds X\) per ticket, in a local fundraising lottery is shown below.
\(x\)021001000
\(\mathrm { P } ( X = x )\)0.90.09\(p\)0.0001
  1. Calculate the value of \(p\).
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
    1. What is the minimum lottery ticket price that the organiser should set in order to make a profit in the long run?
    2. Suggest why, in practice, people would be prepared to pay more than this minimum price.
WJEC Further Unit 2 2022 June Q2
11 marks Standard +0.3
2. An economist suggested the rate of unemployment and the rate of wage inflation are independent. Amy sets about investigating this suggestion. She collects unemployment data and wage inflation data from a random sample of regions in the UK and decides that it is appropriate to carry out a significance test on Pearson's product moment correlation coefficient. Amy's summary statistics for percentage unemployment, \(x\), and percentage wage inflation, \(y\), are shown below. $$\begin{array} { l l l } \sum x = 62 \cdot 8 & \sum y = 19 \cdot 4 & n = 10 \\ \sum x ^ { 2 } = 413 \cdot 44 & \sum y ^ { 2 } = 46 \cdot 16 & \sum x y = 113 \cdot 16 \end{array}$$
  1. Calculate Pearson's product moment correlation coefficient for these data.
  2. Carry out Amy's test at the \(5 \%\) level of significance and state whether the economist's suggestion is reasonable. Amy also collects unemployment data and wage inflation data from a random sample of 10 regions in Spain and calculates Pearson's product moment correlation coefficient to be - 0.2525 .
  3. Should this change Amy's opinion on the economist's suggestion above? What could she do to improve her investigation?
  4. What assumption has Amy made in deciding that it is appropriate to carry out a significance test on Pearson's product moment correlation coefficient?
WJEC Further Unit 2 2022 June Q3
11 marks Standard +0.3
3. Two basketball players, Steph and Klay, score baskets at random at a rate of \(2 \cdot 1\) and \(1 \cdot 9\) respectively per quarter of a game. Assume that baskets are scored independently, and that Steph and Klay each play all four quarters of the game.
  1. Stating the model that you are using, find the probability that they will score a combined total of exactly 20 baskets in a randomly selected game.
  2. A quarter of a game lasts 12 minutes.
    1. State the distribution of the time between baskets for Steph. Give the mean and standard deviation of this distribution.
    2. Given that Klay scores at the end of the third minute in a quarter of a game, find the probability that Klay doesn't score for the rest of the quarter.
  3. When practising, Klay misses \(4 \%\) of the free throws he takes. One week he takes 530 free throws. Calculate the probability that he misses more than 25 free throws.
WJEC Further Unit 2 2022 June Q4
12 marks Standard +0.3
4. The continuous random variable \(R\) has probability density function \(f ( r )\) given by $$f ( r ) = \begin{cases} k r ( b - r ) & \text { for } 1 \leqslant r \leqslant 4 , \\ 0 & \text { otherwise } , \end{cases}$$ where \(k\) and \(b\) are positive constants.
  1. Explain why \(b \geqslant 4\).
  2. Given that \(b = 4\),
    1. show that \(k = \frac { 1 } { 9 }\),
    2. find an expression for \(F ( r )\), valid for \(1 \leqslant r \leqslant 4\), where \(F\) denotes the cumulative distribution function of \(R\),
    3. find the probability that \(R\) lies between 2 and 3 .
WJEC Further Unit 2 2022 June Q5
11 marks Standard +0.3
5. John has a game that involves throwing a set of three identical, cubical dice with faces numbered 1 to 6 . He wishes to investigate whether these dice are fair in terms of the number of sixes obtained when they are thrown. John throws the set of three dice 1100 times and records the number of sixes obtained for each throw. The results are shown in the table below.
Number of sixes0123
Frequency6253848110
Using these results, conduct a goodness of fit test and draw an appropriate conclusion.
WJEC Further Unit 2 2022 June Q6
11 marks Standard +0.3
6. An online survey on the use of social media asked the following question: \begin{displayquote} "Do you use any form of social media?" \end{displayquote} The results for a total of 1953 respondents are shown in the table below.
Age in years
Use social media18-2930-4950-6465 or olderTotal
Yes3104123481961266
No42116196333687
Total3525285445291953
To test whether there is a relationship between social media use and age, a significance test is carried out at the \(5 \%\) level.
  1. State the null and alternative hypotheses.
  2. Show how the expected frequency \(228 \cdot 18\) is calculated in the table below.
    Expected valuesAge in years
    Use social media18-2930-4950-6465 or older
    Yes\(228 \cdot 18\)\(342 \cdot 27\)352.64342.92
    No123.82185.73191.36186.08
  3. Determine the value of \(s\) in the table below.
    Chi-squared contributionsAge in years
    Use social media18-2930-4950-6465 or older
    Yes29.34\(s\)0.0662.94
    No54.0726-180.11115.99
  4. Complete the significance test, showing all your working.
  5. A student, analysing these data on a spreadsheet, obtains the following output. \includegraphics[max width=\textwidth, alt={}, center]{77fd7ad7-f5a3-4947-afc6-e5ef45bef7a8-5_202_1271_445_415} Explain why the student must have made an error in calculating the \(p\)-value.
WJEC Further Unit 2 2022 June Q7
7 marks Moderate -0.3
7. Data from a large dataset shows the percentage of children enrolled in secondary education and the percentage of the adult population who are literate. The following graphs show data from 30 randomly selected regions from each of the Arab World, Africa and Asia. In each case, the least squares regression line of '\% Literacy' on '\% Enrolled in Secondary Education' is shown. \includegraphics[max width=\textwidth, alt={}, center]{77fd7ad7-f5a3-4947-afc6-e5ef45bef7a8-6_682_1200_584_395} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Africa} \includegraphics[alt={},max width=\textwidth]{77fd7ad7-f5a3-4947-afc6-e5ef45bef7a8-6_623_1191_1548_397}
\end{figure} \includegraphics[max width=\textwidth, alt={}, center]{77fd7ad7-f5a3-4947-afc6-e5ef45bef7a8-7_665_1200_331_434}
  1. Calculate the equation of the least squares regression line of '\% Literacy' ( \(y\) ) on '\% Enrolled in Secondary Education' ( \(x\) ) for Asia, given the following summary statistics. $$\begin{array} { l l l } \sum x = 2850.836 & \sum y = 2738.656 & S _ { x x } = 88.42142 \\ S _ { y y } = 204.733 & S _ { x y } = 96.60984 & n = 30 \end{array}$$
  2. The Arab World, Africa and Asia each contain a region where \(70 \%\) are enrolled in secondary education. The three regression lines are used to estimate the corresponding \% Literacy. Which of these estimates is likely to be the most reliable? Clearly explain your reasoning. \section*{END OF PAPER}
WJEC Further Unit 2 2024 June Q1
14 marks Standard +0.3
  1. Dave and Llinos like to go fishing. When they go fishing, on average, Dave catches 4.3 fish per day and Llinos catches 3.8 fish per day. A day of fishing is assumed to be 8 hours.
    1. (i) Calculate the probability that they will catch fewer than 2 fish in total on a randomly selected half-day of fishing.
      (ii) Justify any distribution you have used in answering (a)(i).
    2. On a randomly selected day, Dave starts fishing at 7 am. Given that Dave has not caught a fish by 11 am,
      1. find the expected time he catches his first fish,
      2. calculate the probability that he will not catch a fish by 3 pm .
    3. On average, only \(2 \%\) of the fish that Llinos catches are trout. Over the course of a year, she catches 950 fish. Calculate the probability that at least 30 of these fish are trout. [3]
      [0pt] she catches 950 fish. Calculate the probability that at least 30 of these fish are trout. [3]
    4. State, with a reason, a distribution, including any parameters, that could approximate the distribution used in part (c).
    PLEASE DO NOT WRITE ON THIS PAGE
WJEC Further Unit 2 2024 June Q2
13 marks Standard +0.3
2. Emlyn aims to produce podcast episodes that are a standard length of time, which he calls the 'target time'. The time, \(X\) minutes, above or below the target time, which he calls the 'allowed time', can be modelled by the following cumulative distribution function. $$F ( x ) = \begin{cases} 0 & x < - 2 \\ \frac { x + 2 } { 5 } & - 2 \leqslant x < 1 \\ \frac { x ^ { 2 } - x + 3 } { 5 } & 1 \leqslant x \leqslant 2 \\ 1 & x > 2 \end{cases}$$
  1. Calculate the upper quartile for the 'allowed time'.
  2. Find \(f ( x )\), the probability density function, for all values of \(x\).
    1. Calculate the mean 'allowed time'.
    2. Interpret your answer in context.
WJEC Further Unit 2 2024 June Q3
12 marks Standard +0.3
  1. A company makes bags. The table below shows the number of bags sold on a random sample of 50 days. A manager believes that the number of bags sold per day can be modelled by the Poisson distribution with mean \(2 \cdot 2\).
Number of
bags sold
012345 or more
Frequency71011967
  1. Carry out a chi-squared goodness of fit test, using a \(10 \%\) significance level.
  2. A chi-squared goodness of fit test for the Poisson distribution with mean \(2 \cdot 5\) is conducted. This uses the same number of degrees of freedom as part (a) and gives a test statistic of 1.53 . State, with a reason, which of these two Poisson models is a better fit for the data.
WJEC Further Unit 2 2024 June Q4
12 marks Standard +0.8
4. An author poses the following question: Does using cash for transactions affect people's financial behaviour?
She collects data on 'Cash transactions as a \% of all transactions' and 'Household debt as a \(\%\) of net disposable income' from a random sample of 25 countries. The table below shows the data she collected. There are missing values, \(p\) and \(q\), for Malta and Denmark respectively.
CountryCash transactions as a \% of all transactions \(\boldsymbol { x }\)Household debt as a \% of net disposable income \(\boldsymbol { y }\)CountryCash transactions as a \% of all transactions \(\boldsymbol { x }\)Household debt as a \% of net disposable income \(\boldsymbol { y }\)
Malta92\(p\)France68120
Mexico90-14Luxembourg64177
Greece88107Belgium63113
Spain87110Finland54137
Italy8687Estonia4882
Austria8591The Netherlands45247
Portugal81131UK42147
Slovenia8056Australia37214
Germany8095USA32109
Ireland79154Sweden20187
Slovakia7874South Korea14182
Lithuania7546Denmark\(q\)261
Latvia7143
The summary statistics and scatter diagram below are for the other 23 countries. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Household debt versus Cash transactions} \includegraphics[alt={},max width=\textwidth]{1538fa56-5b61-40ec-bb02-cf1ed9da5eb0-13_664_1296_511_379}
\end{figure} $$\begin{gathered} \sum x = 1467 \sum y = 2695 \sum x ^ { 2 } = 105073 \quad S _ { x x } = 11503 \cdot 91304 \quad S _ { y y } = 78669 \cdot 30435 \\ \sum y ^ { 2 } = 394453 \sum x y = 152999 \quad S _ { x y } = - 18895 \cdot 13043 \end{gathered}$$
  1. Using the summary statistics for the 23 countries, calculate and interpret Pearson's product moment correlation coefficient.
  2. Calculate the equation of the least squares regression line of Household debt as a \% of net disposable income \(( y )\) on Cash transactions as a \% of all transactions ( \(x\) ). The regression line \(x\) on \(y\) is given below. $$x = - 0 \cdot 24 y + 91 \cdot 92$$
  3. By selecting the appropriate regression line in each case, estimate the values of \(p\) and \(q\) in the table.
  4. Comment on the reliability of your answers in part (c).
  5. Interpret the negative value of \(y\) for Mexico.
WJEC Further Unit 2 2024 June Q5
12 marks Moderate -0.5
5. Lily is interested in the relationship between the way in which students learned Welsh and their attitude towards the Welsh language. Students were categorised as having learned Welsh in one of three ways:
  • from one Welsh-speaking parent/carer at home,
  • from two Welsh-speaking parents/carers at home,
  • at school only, for those with no Welsh-speaking parents/carers at home.
The students were asked to rate their attitude towards the Welsh language from 'Very negative' to 'Very positive'. The following data for a random sample of 253 students were collected as part of a project.
Learned Welsh
AttitudeFrom two parents/carersFrom one parent/carerAt school onlyTotal
Very negative2143046
Slightly negative4202145
Neutral1217837
Slightly positive21191151
Very positive25212874
Total649198253
Lily intends to carry out a chi-squared test for independence at the \(5 \%\) level. She produces the following tables which are incomplete.
Expected FrequenciesLearned Welsh
AttitudeFrom two parents/carersFrom one parent/carerAt school only
Very negative11.6416.5517.82
Slightly negative11.3816.1917.43
Neutral9.3613.3114.33
Slightly positive12.9018.3419.75
Very positiveF26.6228.66
Chi-Squared ContributionsLearned Welsh
AttitudeFrom two parents/carersFrom one parent/carerAt school only
Very negative7.980.398.33
Slightly negative\(4 \cdot 79\)0.900.73
Neutral\(0 \cdot 74\)1.02G
Slightly positive5.080.023.88
Very positive2.111.190.02
Total20.703.52H
  1. Calculate the values of \(F , G\) and \(H\).
  2. Carry out Lily's chi-squared test for independence at the \(5 \%\) level.
  3. By referring to the figures in the tables on pages 16 and 17, give two comments on the relationship between the way students learned Welsh and their attitude towards the Welsh language.
WJEC Further Unit 2 2024 June Q6
7 marks Challenging +1.2
6. Penelope makes 8 cakes per week. Each cake costs \(\pounds 20\) to make and sells for \(\pounds 60\). She always sells at least 5 cakes per week. Any cakes left at the end of the week are donated to a food bank. The probability that 5 cakes are sold in a week is \(0 \cdot 3\). She is twice as likely to sell 6 cakes in a week as she is to sell 7 cakes in a week. The expected profit per week is \(\pounds 206\). Construct a probability distribution for the weekly profit.
Additional page, if required. number 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 Further Unit 3 2019 June Q1
8 marks Standard +0.3
  1. The diagram shows a spring of natural length 0.15 m enclosed in a smooth horizontal tube. One end of the spring \(A\) is fixed and the other end \(B\) is compressed against a ball of mass \(0 \cdot 1 \mathrm {~kg}\). \includegraphics[max width=\textwidth, alt={}, center]{b430aa50-27e3-46f7-afef-7b8e75d46e1f-2_241_714_639_632}
Initially, the ball is held in equilibrium by a force of 21 N so that the compressed length of the spring is \(\frac { 2 } { 5 }\) of its natural length.
  1. Calculate the modulus of elasticity of the spring.
  2. The ball is released by removing the force. Determine the speed of the ball when the spring reaches its natural length. Give your answer correct to two significant figures.
WJEC Further Unit 3 2019 June Q2
10 marks Standard +0.3
2. A particle of mass 0.5 kg is moving under the action of a single force \(\mathbf { F N }\) so that its velocity \(\mathrm { v } \mathrm { ms } ^ { - 1 }\) at time \(t\) seconds is given by $$\mathbf { v } = 3 t ^ { 2 } \mathbf { i } - 8 t \mathbf { j } + 2 \mathrm { e } ^ { - t } \mathbf { k }$$
  1. Find an expression for the acceleration of the particle at time \(t \mathrm {~s}\).
  2. Determine an expression for F.v at time \(t \mathrm {~s}\).
  3. Find the kinetic energy of the particle at time \(t \mathrm {~s}\).
  4. Describe the relationship between the kinetic energy of a particle and the rate of working of the force acting on the particle. Verify this relationship using your answers to part (b) and part (c).
WJEC Further Unit 3 2019 June Q3
10 marks Standard +0.8
3. The position vectors \(\mathbf { r } _ { A }\) and \(\mathbf { r } _ { B }\), in kilometres, of two small aeroplanes \(A\) and \(B\) relative to a fixed point \(O\) are given by $$\begin{aligned} & \mathbf { r } _ { A } = ( 60 \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k } ) + ( 168 \mathbf { i } + 132 \mathbf { j } ) t \\ & \mathbf { r } _ { B } = ( 62 \mathbf { i } + 3 \mathbf { k } ) + ( 160 \mathbf { i } + p \mathbf { j } + q \mathbf { k } ) t \end{aligned}$$ where \(t\) denotes the time in hours after 9:00 a.m. and \(p , q\) are constants.
The aeroplanes \(A\) and \(B\) are on course to collide.
  1. Show that \(p = 140\) and \(q = 4\).
  2. Find an expression for the square of the distance between \(A\) and \(B\) at time \(t\) hours after 9:00 a.m.
  3. Both aeroplanes have systems that activate an alarm if they come within 600 m of each other. Determine the time when the alarms are first activated.
WJEC Further Unit 3 2019 June Q4
9 marks Standard +0.3
4. A car of mass 1200 kg has an engine that is capable of producing a maximum power of 80 kW . When in motion, the car experiences a constant resistive force of 2000 N .
  1. Calculate the maximum possible speed of the car when travelling on a straight horizontal road.
  2. The car travels up a slope inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 20 }\). If the car's engine is working at \(80 \%\) capacity, calculate the acceleration of the car at the instant when its speed is \(20 \mathrm {~ms} ^ { - 1 }\).
  3. Explain why the assumption of a constant resistive force may be unrealistic.
WJEC Further Unit 3 2019 June Q5
8 marks Moderate -0.5
5. The diagram shows a fairground ride that consists of a number of seats suspended by chains that swing out as the centre rotates. \includegraphics[max width=\textwidth, alt={}, center]{b430aa50-27e3-46f7-afef-7b8e75d46e1f-4_711_718_466_678} When the ride rotates at a constant angular speed of \(\omega = 1.4 \mathrm { rads } ^ { - 1 }\), the seats move in a horizontal circle with each chain making an angle \(\theta\) with the vertical. Each of the seats and the chains may be modelled as light. Assume that all chains have the same length and are inextensible. When a man of mass 75 kg occupies a seat, the tension in the chain is \(490 \sqrt { 3 } \mathrm {~N}\).
  1. Show that \(\theta = 30 ^ { \circ }\).
  2. Calculate the length of each chain.
WJEC Further Unit 3 2019 June Q6
13 marks Standard +0.8
6. The diagram shows a rollercoaster at an amusement park where a car is projected from a launch point \(O\) so that it performs a loop before instantaneously coming to rest at point \(C\). The car then performs the same journey in reverse. \includegraphics[max width=\textwidth, alt={}, center]{b430aa50-27e3-46f7-afef-7b8e75d46e1f-5_677_1733_552_166} The loop section is modelled by considering the track to be a vertical circle of radius 10 m and the car as a particle of mass \(m\) kg moving on the inside surface of the circular loop. You may assume that the track is smooth. At point \(A\), which is the lowest point of the circle, the car has velocity \(u \mathrm {~ms} ^ { - 1 }\) such that \(u ^ { 2 } = 60 g\). When the car is at point \(B\) the radius makes an angle \(\theta\) with the downward vertical.
  1. Find, in terms of \(\theta\) and \(g\), an expression for \(v ^ { 2 }\), where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of the car at \(B\).
  2. Show that \(R \mathrm {~N}\), the reaction of the track on the car at \(B\), is given by $$R = m g ( 4 + 3 \cos \theta ) .$$
  3. Explain why the expression for \(R\) in part (b) shows that the car will perform a complete loop.
  4. This model predicts that the car will stop at \(C\) at a vertical height of 30 m above \(A\). However, after the car has completed the loop, the track becomes rough and the car only reaches a point \(D\) at a vertical height of 28 m above \(A\). The resistance to motion of the car beyond the loop is of constant magnitude \(\frac { m g } { 32 } \mathrm {~N}\). Calculate the length of the rough track between \(A\) and \(D\).
WJEC Further Unit 3 2019 June Q7
12 marks Standard +0.3
7. Three spheres \(A , B , C\), of equal radii and each of mass \(m \mathrm {~kg}\), lie at rest on a smooth horizontal surface such that their centres are in a straight line with \(B\) between \(A\) and \(C\). The coefficient of restitution between \(A\) and \(B\) is \(e\). Sphere \(A\) is projected towards \(B\) with speed \(u \mathrm {~ms} ^ { - 1 }\) so that it collides with \(B\).
  1. Find expressions, in terms of \(e\) and \(u\), for the speed of \(A\) and the speed of \(B\) after they collide. You are now given that \(e = \frac { 1 } { 2 }\).
  2. Find, in terms of \(m\) and \(u\), the loss in kinetic energy due to the collision between \(A\) and \(B\).
  3. After the collision between \(A\) and \(B\), sphere \(B\) then collides with \(C\). The coefficient of restitution between \(B\) and \(C\) is \(e _ { 1 }\). Show that there will be no further collisions if \(e _ { 1 } \leqslant \frac { 1 } { 3 }\).
WJEC Further Unit 3 2024 June Q1
14 marks Standard +0.3
1. Two particles \(A\) and \(B\), of masses 2 kg and 5 kg respectively, are moving in the same direction along a smooth horizontal surface when they collide directly. Before the collision, \(B\) is moving with speed \(1.2 \mathrm {~ms} ^ { - 1 }\) and, immediately after the collision, its speed is \(3.8 \mathrm {~ms} ^ { - 1 }\). The coefficient of restitution between the particles \(A\) and \(B\) is 0.3 .
    1. Find the impulse exerted by \(A\) on \(B\).
    2. Given that the particles \(A\) and \(B\) were in contact for 0.08 seconds, find the average force between \(A\) and \(B\).
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  2. Calculate the speed of \(A\) before and after the collision.
  3. After the collision between \(A\) and \(B\), particle \(B\) continues to move with speed \(3.8 \mathrm {~ms} ^ { - 1 }\) until it collides directly with a stationary particle \(C\) of mass 4 kg . When \(B\) and \(C\) collide, they coalesce to form a single particle.
    1. Write down the coefficient of restitution between \(B\) and \(C\).
    2. Determine the speed of the combined particle after the collision.
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