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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
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. Calculate the probability that they will catch fewer than 2 fish in total on a randomly selected half-day of fishing.
    2. Justify any distribution you have used in answering (a)(i).
    (b) 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 .
    (c) 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]
  2. 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 \(B\).
      [0pt] [4]
  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.
      \section*{PLEASE DO NOT WRITE ON THIS PAGE}
WJEC Further Unit 3 2024 June Q2
10 marks Standard +0.3
  1. The diagram below shows a light spring of natural length 1.2 m and modulus of elasticity 84 N . One end of the spring \(A\) is fixed and the other end is attached to an object \(P\) of mass 4 kg . \includegraphics[max width=\textwidth, alt={}, center]{ae23a093-1419-4be4-8285-951650dc5a35-06_542_451_466_808}
Initially, \(P\) is held at rest with the spring stretched to a total length of 2.2 m and \(A P\) vertical.
  1. Show that the elastic energy stored in the spring is 35 J .
  2. The object \(P\) is then released. Find the speed of \(P\) at the instant when the elastic energy in the spring is reduced to \(5 \cdot 6 \mathrm {~J}\).
WJEC Further Unit 3 2024 June Q3
5 marks Moderate -0.8
3. Three forces \(( 4 \mathbf { i } - 7 \mathbf { j } + 9 \mathbf { k } ) \mathrm { N } , ( 5 \mathbf { i } + 3 \mathbf { j } - 8 \mathbf { k } ) \mathrm { N }\) and \(( - 2 \mathbf { i } + 6 \mathbf { j } - 11 \mathbf { k } ) \mathrm { N }\) act on a particle.
  1. Find the resultant \(\mathbf { R }\) of the three forces.
  2. The points \(A\) and \(B\) have position vectors \(( 3 \mathbf { i } + 4 \mathbf { j } - 12 \mathbf { k } ) \mathrm { m }\) and \(( a \mathbf { i } + 7 \mathbf { j } - 10 \mathbf { k } ) \mathrm { m }\) respectively, where \(a\) is a constant. The work done by \(\mathbf { R }\) in moving the particle from \(A\) to \(B\) is 21 J . Calculate the value of \(a\).
    \section*{PLEASE DO NOT WRITE ON THIS PAGE}
WJEC Further Unit 3 2024 June Q4
7 marks Moderate -0.3
  1. The diagram below shows a particle \(P\), of mass 5 kg , attached to one end of a light inextensible string of length 3 m . The other end is fixed at a point \(A\). The particle \(P\) is moving in a horizonal circle with centre \(C\), where the point \(C\) is vertically below \(A\). The string is inclined at an angle \(\theta\) to the downward vertical, where \(\tan \theta = \frac { 20 } { 21 }\). \includegraphics[max width=\textwidth, alt={}, center]{ae23a093-1419-4be4-8285-951650dc5a35-10_725_796_639_628}
Find the speed of the particle.
WJEC Further Unit 3 2024 June Q5
9 marks Standard +0.8
5. A particle of mass 2 kg is moving under the action of a force \(\mathbf { F N }\) which, at time \(t\) seconds, is given by $$\mathbf { F } = 4 t \mathbf { i } - \sqrt { t } \mathbf { j } + 6 \mathbf { k }$$ When \(t = 1\), the velocity of the particle is \(\left( 3 \mathbf { i } - \frac { 1 } { 3 } \mathbf { j } - \mathbf { k } \right) \mathrm { ms } ^ { - 1 }\).
  1. Find an expression for the velocity vector of the particle at time \(t \mathrm {~s}\).
  2. Determine the values of \(t\) when the particle is moving in a direction perpendicular to the vector \(( - \mathbf { i } + 3 \mathbf { k } )\).
WJEC Further Unit 3 2024 June Q6
10 marks Challenging +1.2
6. A slope is inclined at an angle of \(5 ^ { \circ }\) to the horizontal. A car, of mass 1500 kg , has an engine that is working at a constant rate of \(P \mathrm {~W}\). The resistance to motion of the car is constant at 4500 N . When the car is moving up the slope, its acceleration is \(a \mathrm {~ms} ^ { - 2 }\) at the instant when its speed is \(10 \mathrm {~ms} ^ { - 1 }\). When the car is moving down the slope, its deceleration is \(a \mathrm {~ms} ^ { - 2 }\) at the instant when its speed is \(20 \mathrm {~ms} ^ { - 1 }\). Determine the value of \(P\) and the value of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{ae23a093-1419-4be4-8285-951650dc5a35-14_87_1609_635_267}
WJEC Further Unit 3 2024 June Q7
15 marks Standard +0.3
7. One end of a light rod of length \(\frac { 5 } { 7 } \mathrm {~m}\) is attached to a fixed point \(O\) and the other end is attached to a particle \(P\), of mass \(m \mathrm {~kg}\). The particle \(P\) is projected from the point \(A\), which is vertically below \(O\), with a horizontal speed of \(u \mathrm {~ms} ^ { - 1 }\) so that it moves in a vertical circle with centre \(O\). When the rod \(O P\) is inclined at an angle \(\theta\) to the downward vertical, the speed of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\) and the tension in the rod is \(T \mathrm {~N}\). \includegraphics[max width=\textwidth, alt={}, center]{ae23a093-1419-4be4-8285-951650dc5a35-16_629_593_646_735}
  1. Show that $$v ^ { 2 } = u ^ { 2 } - 14 + 14 \cos \theta$$
  2. Hence determine the least possible value of \(u ^ { 2 }\) for the particle to reach the highest point of the circle.
  3. Given that \(u ^ { 2 } = 32 \cdot 2\),
    1. find, in terms of \(m\) and \(\theta\), an expression for \(T\),
    2. calculate the range of values of \(\theta\) such that the rod is exerting a thrust.
      State whether your answer to (c)(ii) would be different if the mass of the particle was reduced. Give a reason for your answer. Additional page, if required. Write the question number(s) in the left-hand margin. only