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WJEC Further Unit 2 Specimen Q5
10 marks Standard +0.3
The manager of a hockey team studies last season's results and puts forward the theory that the number of goals scored per match by her team can be modelled by a Poisson distribution with mean 2.0. The number of goals scored during the season are summarised below.
Goals scored01234 or more
Frequency61115108
  1. State suitable hypotheses to carry out a goodness of fit test. [1]
  2. Carry out a \(\chi^2\) goodness of fit test on this data set, using a 5% level of significance and draw a conclusion in context. [9]
WJEC Further Unit 2 Specimen Q6
10 marks Standard +0.3
Customers arrive at a shop such that the number of arrivals in a time interval of \(t\) minutes follows a Poisson distribution with mean \(0.5t\).
  1. Find the probability that exactly 5 customers arrive between 11 a.m. and 11.15 a.m. [3]
  2. A customer arrives at exactly 11 a.m.
    1. Let the next customer arrive at \(T\) minutes past 11 a.m. Show that $$P(T > t) = e^{-0.5t}.$$
    2. Hence find the probability density function, \(f(t)\), of \(T\).
    3. Hence, giving a reason, write down the mean and the standard deviation of the time between the arrivals of successive customers. [7]
WJEC Further Unit 2 Specimen Q7
12 marks Moderate -0.5
The Pew Research Center's Internet Project offers scholars access to raw data sets from their research. One of the Pew Research Center's projects was on teenagers and technology. A random sample of American families was selected to complete a questionnaire. For each of their children, between and including the ages of 13 and 15, parents of these families were asked: Do you know your child's password for any of [his/her] social media accounts? Responses to this question were received from 493 families. The table below provides a summary of their responses.
Age (years)\multirow{2}{*}{Total}
\cline{2-4} Parent know password131415
Yes767567218
No66103106275
Total142178173493
  1. A test for significance is to be undertaken to see whether there is an association between whether a parent knows any of their child's social media passwords and the age of the child.
    1. Clearly state the null and alternative hypotheses.
    2. Obtain the expected value that is missing from the table below, indicating clearly how it is calculated from the data values given in the table above.
    Expected values:
    Age (years)
    \cline{2-4} Parent knows password131415
    Yes62.7978.7176.50
    No99.2996.50
    1. Obtain the two chi-squared contributions that are missing from the table below.
    Chi-squared contributions:
    Age (years)
    \cline{2-4} Parent knows password131415
    Yes0.1751.180
    No2.2030.935
    The following output was obtained from the statistical package that was used to undertake the analysis: Pearson chi-squared (2) = 7.409 \quad \(p\)-value = 0.0305
    1. Indicate how the degrees of freedom have been calculated for the chi-squared statistic.
    2. Interpret the output obtained from the statistical test in terms of the initial hypotheses. [10]
  2. Comment on the nature of the association observed, based on the contributions to the test statistic calculated in (a). [2]
WJEC Further Unit 3 2018 June Q1
13 marks Standard +0.8
Two objects, \(A\) of mass 18 kg and \(B\) of mass 7 kg, are moving in the same straight line on a smooth horizontal surface. Initially, they are moving with the same speed of \(4\text{ ms}^{-1}\) and in the same direction. Object \(B\) collides with a vertical wall which is perpendicular to its direction of motion and rebounds with a speed of \(3\text{ ms}^{-1}\). Subsequently, the two objects \(A\) and \(B\) collide directly. The coefficient of restitution between the two objects is \(\frac{5}{7}\).
  1. Find the coefficient of restitution between \(B\) and the wall. [1]
  2. Determine the speed of \(A\) and the speed of \(B\) immediately after the two objects collide. [7]
  3. Calculate the impulse exerted by \(A\) on \(B\) due to the collision and clearly state its units. [2]
  4. Find the loss in energy due to the collision between \(A\) and \(B\). [2]
  5. State the direction of motion of \(A\) relative to the wall after the collision with \(B\). [1]
WJEC Further Unit 3 2018 June Q2
10 marks Challenging +1.2
A car of mass 750 kg is moving on a slope inclined at an angle \(\theta\) to the horizontal, where \(\sin\theta = 0.1\). When the car's engine is working at a constant power \(PW\), the car can travel at maximum speeds of \(14\text{ ms}^{-1}\) up the slope and \(28\text{ ms}^{-1}\) down the slope. In each case, the resistance to motion experienced by the car is proportional to the square of its speed. Find the value of \(P\) and determine the resistance to the motion of the car when its speed is \(10.5\text{ ms}^{-1}\). [10]
WJEC Further Unit 3 2018 June Q3
10 marks Challenging +1.8
A light elastic string of natural length \(1.5\) m and modulus of elasticity \(490\) N has one end attached to a fixed point \(A\) and the other end attached to a particle \(P\) of mass \(30\) kg. Initially, \(P\) is held at rest vertically below \(A\) such that the distance \(AP\) is \(0.6\) m. It is then allowed to fall vertically.
  1. Calculate the distance \(AP\) when \(P\) is instantaneously at rest for the first time, giving your answer correct to 2 decimal places. [8]
  2. Estimate the distance \(AP\) when \(P\) is instantaneously at rest for the second time and clearly state one assumption that you have made in making your estimate. [2]
WJEC Further Unit 3 2018 June Q4
11 marks Standard +0.3
The position vector \(\mathbf{x}\) metres at time \(t\) seconds of an object of mass 3 kg may be modelled by $$\mathbf{x} = 3\sin t \mathbf{i} - 4\cos 2t \mathbf{j} + 5\sin t \mathbf{k}.$$
  1. Find an expression for the velocity vector \(\mathbf{v}\text{ ms}^{-1}\) at time \(t\) seconds and determine the least value of \(t\) when the object is instantaneously at rest. [7]
  2. Write down the momentum vector at time \(t\) seconds. [1]
  3. Find, in vector form, an expression for the force acting on the object at time \(t\) seconds. [3]
WJEC Further Unit 3 2018 June Q5
15 marks Challenging +1.8
A particle \(P\), of mass \(m\) kg, is attached to one end of a light inextensible string of length \(l\) m. The other end of the string is attached to a fixed point \(O\). Initially, \(P\) is held at rest with the string just taut and making an angle of 60° with the downward vertical. It is then given a velocity \(u\text{ ms}^{-1}\) perpendicular to the string in a downward direction.
    1. When the string makes an angle \(\theta\) with the downward vertical, the velocity of the particle is \(v\) and the tension in the string is \(T\). Find an expression for \(T\) in terms of \(m\), \(l\), \(v^2\) and \(\theta\).
    2. Given that \(P\) describes complete circles in the subsequent motion, show that \(u^2 > 4lg\). [10]
  2. Given that now \(u^2 = 3lg\), find the position of the string when circular motion ceases. Briefly describe the motion of \(P\) after circular motion has ceased. [3]
  3. The string is replaced by a light rigid rod. Given that \(P\) describes complete circles in the subsequent motion, show that \(u^2 > klg\), where \(k\) is to be determined. [2]
WJEC Further Unit 3 2018 June Q6
11 marks Challenging +1.2
A vehicle of mass 1200 kg is moving with a constant speed of \(40\text{ ms}^{-1}\) around a horizontal circular path which is on a test track banked at an angle of 60° to the horizontal. There is no tendency to sideslip at this speed. The vehicle is modelled as a particle.
  1. Calculate the normal reaction of the track on the vehicle. [3]
  2. Determine
    1. the radius of the circular path,
    2. the angular speed of the vehicle and clearly state its units. [6]
  3. What further assumption have you made in your solution to (b)? Briefly explain what effect this assumption has on the radius of the circular path. [2]
WJEC Further Unit 3 2022 June Q1
4 marks Moderate -0.8
A particle of mass 1.2 kg is attached to one end of a light inextensible string of length 2 m. The other end of the string is fixed to a point O on a smooth horizontal surface. With the string taut, the particle moves on the surface with constant speed \(8\text{ ms}^{-1}\) in a horizontal circle with centre O.
  1. Find the angular velocity of the particle about O. [2]
  2. Calculate the tension in the string. [2]
WJEC Further Unit 3 2022 June Q2
11 marks Moderate -0.3
The diagram below shows a woman standing at the end of a diving platform. She is about to dive into the water below. The woman has mass 60 kg and she may be modelled as a particle positioned at the end of the platform which is 10 m above the water. \includegraphics{figure_2} When the woman dives, she projects herself from the platform with a speed of \(7.8\text{ ms}^{-1}\).
  1. Find the kinetic energy of the woman when she leaves the platform. [2]
  2. Initially, the situation is modelled ignoring air resistance. By using conservation of energy, show that the model predicts that the woman enters the water with an approximate speed of \(16\text{ ms}^{-1}\). [6]
  3. Suppose that this model is refined to include air resistance so that the speed with which the woman enters the water is now predicted to be \(13\text{ ms}^{-1}\). Determine the amount of energy lost to air resistance according to the refined model. [3]
WJEC Further Unit 3 2022 June Q3
10 marks Standard +0.3
Two spheres \(A\) and \(B\), of equal radii, are moving towards each other on a smooth horizontal surface and collide directly. Sphere \(A\) has mass \(4m\) kg and sphere \(B\) has mass \(3m\) kg. Just before the collision, \(A\) has speed \(9\text{ ms}^{-1}\) and \(B\) has speed \(3.5\text{ ms}^{-1}\). Immediately after the collision, \(A\) has speed \(1.5\text{ ms}^{-1}\) in the direction of its original motion.
  1. Show that the speed of \(B\) immediately after the collision is \(6.5\text{ ms}^{-1}\). [3]
  2. Calculate the coefficient of restitution between \(A\) and \(B\). [3]
  3. Given that the magnitude of the impulse exerted by \(B\) on \(A\) is 36 Ns, find the value of \(m\). [3]
  4. Give a reason why it is not necessary to model the spheres as particles in this question. [1]
WJEC Further Unit 3 2022 June Q4
9 marks Standard +0.3
A particle \(P\) of mass 0.5 kg is in equilibrium under the action of three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\). $$\mathbf{F}_1 = (9\mathbf{i} + 6\mathbf{j} - 12\mathbf{k})\text{N} \quad \text{and} \quad \mathbf{F}_2 = (6\mathbf{i} - 7\mathbf{j} + 3\mathbf{k})\text{N}.$$
  1. Find the force \(\mathbf{F}_3\). [2]
  2. Forces \(\mathbf{F}_2\) and \(\mathbf{F}_3\) are removed so that \(P\) moves in a straight line \(AB\) under the action of the single force \(\mathbf{F}_1\). The points \(A\) and \(B\) have position vectors \((2\mathbf{i} - 9\mathbf{j} + 7\mathbf{k})\) m and \((8\mathbf{i} - 5\mathbf{j} - \mathbf{k})\) m respectively. The particle \(P\) is initially at rest at \(A\).
    1. Verify that \(\mathbf{F}_1\) acts parallel to the vector \(\overrightarrow{AB}\).
    2. Find the work done by the force \(\mathbf{F}_1\) as the particle moves from \(A\) to \(B\).
    3. By using the work-energy principle, find the speed of \(P\) as it reaches \(B\). [7]
WJEC Further Unit 3 2022 June Q5
14 marks Challenging +1.2
One end of a light elastic string, of natural length 2.5 m and modulus of elasticity \(30g\) N, is fixed to a point O. A particle \(P\), of mass 2 kg, is attached to the other end of the string. Initially, \(P\) is held at rest at the point O. It is then released and allowed to fall under gravity.
  1. Show that, while the string is taut, $$v^2 = g(5 + 2x - 6x^2),$$ where \(v\text{ ms}^{-1}\) denotes the velocity of the particle when the extension in the string is \(x\) m. [6]
  2. Calculate the maximum extension of the string. [3]
    1. Find the extension of the string when \(P\) attains its maximum speed.
    2. Hence determine the maximum speed of \(P\). [5]
WJEC Further Unit 3 2022 June Q6
10 marks Standard +0.8
A vehicle of mass 3500 kg is moving up a slope inclined at an angle \(\alpha\) to the horizontal. When the vehicle is travelling at a velocity of \(v\text{ ms}^{-1}\), the resistance to motion can be modelled by a variable force of magnitude \(40v\) N.
  1. Given that \(\sin\alpha = \frac{3}{49}\), calculate the power developed by the engine at the instant when the speed of the vehicle is \(25\text{ ms}^{-1}\) and its deceleration is \(0.2\text{ ms}^{-2}\). [5]
  2. When the vehicle's engine is working at a constant rate of 40 kW, the maximum speed that can be maintained up the slope is \(20\text{ ms}^{-1}\). Find the value of \(\alpha\). Give your answer in degrees, correct to one significant figure. [5]
WJEC Further Unit 3 2022 June Q7
12 marks Challenging +1.2
The diagram below shows a particle \(P\), of mass 2.5 kg, attached by means of two light inextensible strings fixed at points \(A\) and \(B\). Point \(A\) is vertically above point \(B\). \(BP\) makes an angle of \(60°\) with the upward vertical and \(AP\) is inclined at an angle \(\theta\) to the downward vertical where \(\cos\theta = 0.8\). The particle \(P\) describes a horizontal circle with constant angular speed \(\omega\) radians per second about centre \(C\) with both strings taut. \includegraphics{figure_7} The tension in the string \(BP\) is 39.2 N.
  1. Calculate the tension in the string \(AP\). [4]
  2. Given that the length of the string \(AP\) is 1.5 m, find the value of \(\omega\). [5]
  3. Calculate the kinetic energy of \(P\). [3]
WJEC Further Unit 3 2023 June Q1
10 marks Standard +0.3
One end of a light elastic string, of natural length \(0.2\) m and modulus of elasticity \(5g\) N, is attached to a fixed point \(O\). The other end is attached to a particle of mass \(4\) kg. The particle hangs in equilibrium vertically below \(O\).
  1. Show that the extension of the string is \(0.16\) m. [2]
  2. The particle is pulled down vertically and held at rest so that the extension of the string is \(0.28\) m. The particle is then released. Determine the speed of the particle as it passes through the equilibrium position. [8]
WJEC Further Unit 3 2023 June Q2
11 marks Standard +0.3
At time \(t = 0\) seconds, a particle \(A\) has position vector \((6\mathbf{i} + 2\mathbf{j} - 8\mathbf{k})\) m relative to a fixed origin \(O\) and is moving with constant velocity \((3\mathbf{i} - \mathbf{j} + 4\mathbf{k})\) ms\(^{-1}\).
  1. Write down the position vector of particle \(A\) at time \(t\) seconds and hence find the distance \(OA\) when \(t = 5\). [4]
  2. The position vector, \(\mathbf{r}_B\) metres, of another particle \(B\) at time \(t\) seconds is given by $$\mathbf{r}_B = 3\sin\left(\frac{t}{2}\right)\mathbf{i} - 3\cos\left(\frac{t}{2}\right)\mathbf{j} + 5\mathbf{k}.$$
    1. Show that \(B\) is moving with constant speed.
    2. Determine the smallest value of \(t\) such that particles \(A\) and \(B\) are moving perpendicular to each other. [7]
WJEC Further Unit 3 2023 June Q3
10 marks Challenging +1.2
The diagram below shows a hollow cone, of base radius \(5\) m and height \(12\) m, which is fixed with its axis vertical and vertex \(V\) downwards. A particle \(P\), of mass \(M\) kg, moves in a horizontal circle with centre \(C\) on the smooth inner surface of the cone with constant speed \(v = 3\sqrt{g}\) ms\(^{-1}\). \includegraphics{figure_3}
  1. Show that the normal reaction of the surface of the cone on the particle is \(\frac{13Mg}{5}\) N. [4]
  2. Calculate the length of \(CP\) and hence determine the height of \(C\) above \(V\). [6]
WJEC Further Unit 3 2023 June Q4
13 marks Standard +0.3
Geraint is a cyclist competing in a race along the Taff Trail. The Taff Trail is a track that runs from Cardiff Bay to Brecon. The chart below shows the altitude (height above sea level) along the route. \includegraphics{figure_4} Geraint starts from rest at Cardiff Bay and has a speed of \(10\) ms\(^{-1}\) when he crosses the finish line in Brecon. Geraint and his bike have a total mass of \(80\) kg. The resistance to motion may be modelled by a constant force of magnitude \(16\) N.
  1. Given that \(1440\) kJ of energy is used in overcoming resistances during the race,
    1. find the length of the track,
    2. calculate the work done by Geraint. [8]
  2. The steepest section of the track may be modelled as a slope inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{2}{7}\). \includegraphics{figure_4b} Geraint is capable of producing a maximum power of \(250\) W. Find the maximum speed that Geraint can attain whilst travelling on this section of the track. [5]
WJEC Further Unit 3 2023 June Q5
11 marks Standard +0.8
The diagram below shows two spheres \(A\) and \(B\), of equal radii, moving in the same direction on a smooth horizontal surface. Sphere \(A\), of mass \(3\) kg, is moving with speed \(4\) ms\(^{-1}\) and sphere \(B\), of mass \(2\) kg, is moving with speed \(10\) ms\(^{-1}\). \includegraphics{figure_5} Sphere \(B\) is then given an impulse after which it moves in the opposite direction with speed \(6\) ms\(^{-1}\).
  1. Calculate the magnitude and direction of the impulse exerted on \(B\). [3]
Sphere \(B\) continues to move with speed \(6\) ms\(^{-1}\) so that it collides directly with sphere \(A\). The kinetic energy lost due to the collision is \(45\) J.
  1. Calculate the speed of \(A\) and the speed of \(B\) immediately after the two spheres collide. State the direction in which each sphere is moving relative to its motion immediately before the collision. [8]
WJEC Further Unit 3 2023 June Q6
15 marks Challenging +1.8
The diagram shows a slide, \(ABC\), at a water park. The shape of the slide may be modelled by two circular arcs, \(AB\) and \(BC\), in the same vertical plane. Arc \(AB\) has radius \(7\) m and subtends an angle \(\alpha\) at its centre \(D\), where \(\cos \alpha = \frac{9}{10}\). Arc \(BC\) has radius \(5\) m and subtends an angle of \(45°\) at its centre, \(O\). The straight line \(DBO\) is vertical. \includegraphics{figure_6} Users of the slide are required to sit in a rubber ring and are released from rest at point \(A\). A girl decides to use the slide. The combined mass of the girl and the rubber ring is \(50\) kg.
  1. When the rubber ring is at a point \(P\) on the circular arc \(BC\), its speed is \(v\) ms\(^{-1}\) and \(OP\) makes an angle \(\theta\) with the upward vertical.
    1. Show that \(v^2 = 111.72 - 98\cos\theta\). [4]
    2. Find, in terms of \(\theta\), the reaction between the rubber ring and the slide at \(P\). [4]
    3. Show that, according to this model, the rubber ring loses contact with the slide before reaching \(C\). [3]
    4. In reality, there will be resistive forces opposing the motion of the rubber ring. Explain how this fact will affect your answer to (iii). [1]
  2. Show that the rubber ring will remain in contact with the slide along the arc \(AB\). [3]
WJEC Further Unit 3 Specimen Q1
12 marks Standard +0.3
By burning a charge, a cannon fires a cannon ball of mass 12 kg horizontally. As the cannon ball leaves the cannon, its speed is 600 ms\(^{-1}\). The recoiling part of the cannon has a mass of 1600 kg.
  1. Determine the speed of the recoiling part immediately after the cannon ball leaves the cannon. [3]
  2. Find the energy created by the burning of the charge. State any assumption you have made in your solution and briefly explain how the assumption affects your answer. [5]
  3. Calculate the constant force needed to bring the recoiling part to rest in 1.2 m. State, with a reason, whether your answer is an overestimate or an underestimate of the actual force required. [4]
WJEC Further Unit 3 Specimen Q2
12 marks Standard +0.8
A particle \(P\), of mass 3 kg, is attached to a fixed point \(O\) by a light inextensible string of length 4 m. Initially, particle \(P\) is held at rest at a point which is \(2\sqrt{3}\) m horizontally from \(O\). It is then released and allowed to fall under gravity.
  1. Show that the speed of \(P\) when it first begins to move in a circle is \(\sqrt{3g}\). [4]
  2. In the subsequent motion, when the string first makes an angle of 45° with the downwards vertical,
    1. calculate the speed \(v\) of \(P\),
    2. determine the tension in the string. [8]
WJEC Further Unit 3 Specimen Q3
9 marks Challenging +1.2
At time \(t = 0\) s, the position vector of an object \(A\) is \(\mathbf{i}\) m and the position vector of another object \(B\) is \(3\mathbf{i}\) m. The constant velocity vector of \(A\) is \(2\mathbf{i} + 5\mathbf{j} - 4k\) ms\(^{-1}\) and the constant velocity vector of \(B\) is \(\mathbf{i} + 3\mathbf{j} - 5k\) ms\(^{-1}\). Determine the value of \(t\) when \(A\) and \(B\) are closest together and find the least distance between \(A\) and \(B\). [9]