Questions — WJEC (504 questions)

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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]
WJEC Further Unit 3 Specimen Q4
13 marks Challenging +1.3
Relative to a fixed origin \(O\), the position vector \(\mathbf{r}\) m at time \(t\) s of a particle \(P\), of mass 0.4 kg, is given by $$\mathbf{r} = e^{2t}\mathbf{i} + \sin(2t)\mathbf{j} + \cos(2t)\mathbf{k}.$$
  1. Show that the velocity vector \(\mathbf{v}\) and the position vector \(\mathbf{r}\) are never perpendicular to each other. [6]
  2. Given that the speed of \(P\) at time \(t\) is \(v\) ms\(^{-1}\), show that $$v^2 = 4e^{4t} + 4.$$ [2]
  3. Find the kinetic energy of \(P\) at time \(t\). [1]
  4. Calculate the work done by the force acting on \(P\) in the interval \(0 < t < 1\). [2]
  5. Determine an expression for the rate at which the force acting on \(P\) is working at time \(t\). [2]
WJEC Further Unit 3 Specimen Q5
6 marks Standard +0.3
A particle of mass \(m\) kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\). The particle is set in motion such that it moves in a horizontal circle of radius 2 m with constant speed 4.8 ms\(^{-1}\). Calculate the angle the string makes with the vertical. [6]
WJEC Further Unit 3 Specimen Q6
9 marks Challenging +1.2
\includegraphics{figure_6} A particle of mass 5 kg is attached to a string \(AB\) and a rod \(BC\) at the point \(B\). The string \(AB\) is light and elastic with modulus \(\lambda\) N and natural length 2 m. The rod \(BC\) is light and of length 2 m. The end \(A\) of the string is attached to a fixed point and the end \(C\) of the rod is attached to another fixed point such that \(A\) is vertically above \(C\) with \(AC = 2\) m. When the particle rests in equilibrium, \(AB\) makes an angle of 50° with the downward vertical.
  1. Determine, in terms of \(\lambda\), the tension in the string \(AB\). [3]
  2. Calculate, in terms of \(\lambda\), the energy stored in the string \(AB\). [2]
  3. Find, in terms of \(\lambda\), the thrust in the rod \(BC\). [4]
WJEC Further Unit 3 Specimen Q7
9 marks Challenging +1.2
A vehicle of mass 6000 kg is moving up a slope inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{6}{49}\). The vehicle's engine exerts a constant power of \(P\) W. The constant resistance to motion of the vehicle is \(R\) N. At the instant the vehicle is moving with velocity \(\frac{16}{5}\) ms\(^{-1}\), its acceleration is 2 ms\(^{-2}\). The maximum velocity of the vehicle is \(\frac{16}{3}\) ms\(^{-1}\). Determine the value of \(P\) and the value of \(R\). [9]
WJEC Further Unit 4 2019 June Q1
8 marks Standard +0.3
A complex number is defined by \(z = 3 + 4\mathrm{i}\).
  1. Express \(z\) in the form \(z = re^{i\theta}\), where \(-\pi \leqslant \theta \leqslant \pi\). [3]
    1. Find the Cartesian coordinates of the vertices of the triangle formed by the cube roots of \(z\) when plotted in an Argand diagram. Give your answers correct to two decimal places.
    2. Write down the geometrical name of the triangle. [5]
WJEC Further Unit 4 2019 June Q2
9 marks Challenging +1.2
  1. Show that \(3\sin x + 4\cos x - 2\) can be written as \(\frac{6t + 2 - 6t^2}{1 + t^2}\), where \(t = \tan\left(\frac{x}{2}\right)\). [2]
  2. Hence, find the general solution of the equation \(3\sin x + 4\cos x - 2 = 3\). [7]
WJEC Further Unit 4 2019 June Q3
8 marks Standard +0.3
  1. Determine whether or not the following set of equations $$\begin{pmatrix} 2 & -7 & 2 \\ 0 & 3 & -2 \\ -7 & 8 & 4 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} a \\ b \\ c \end{pmatrix}$$ has a unique solution, where \(a\), \(b\), \(c\) are constants. [3]
  2. Solve the set of equations \begin{align} x + 8y - 6z &= 5,
    2x + 4y + 6z &= -3,
    -5x - 4y + 9z &= -7. \end{align} Show all your working. [5]
WJEC Further Unit 4 2019 June Q4
16 marks Standard +0.3
  1. Given that \(y = \cot^{-1} x\), show that \(\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{-1}{x^2 + 1}\). [5]
  2. Express \(\frac{6x^2 - 10x - 9}{(2x + 3)(x^2 + 1)}\) in terms of partial fractions. [5]
  3. Hence find \(\int \frac{6x^2 - 8x - 6}{(2x + 3)(x^2 + 1)} \mathrm{d}x\). [5]
  4. Explain why \(\int_{-2}^{5} \frac{6x^2 - 8x - 6}{(2x + 3)(x^2 + 1)} \mathrm{d}x\) cannot be evaluated. [1]