Questions — AQA (3620 questions)

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AQA Paper 2 Specimen Q2
1 marks Easy -1.8
A zoologist is investigating the growth of a population of red squirrels in a forest. She uses the equation \(N = \frac{200}{1 + 9e^{-\frac{t}{5}}}\) as a model to predict the number of squirrels, \(N\), in the population \(t\) weeks after the start of the investigation. What is the size of the squirrel population at the start of the investigation? Circle your answer. [1 mark] \(5\) \(\quad\) \(20\) \(\quad\) \(40\) \(\quad\) \(200\)
AQA Paper 2 Specimen Q3
6 marks Moderate -0.3
A curve is defined by the parametric equations $$x = t^3 + 2, \quad y = t^2 - 1$$
  1. Find the gradient of the curve at the point where \(t = -2\) [4 marks]
  2. Find a Cartesian equation of the curve. [2 marks]
AQA Paper 2 Specimen Q4
6 marks Standard +0.3
The equation \(x^3 - 3x + 1 = 0\) has three real roots.
  1. Show that one of the roots lies between \(-2\) and \(-1\) [2 marks]
  2. Taking \(x_1 = -2\) as the first approximation to one of the roots, use the Newton-Raphson method to find \(x_2\), the second approximation. [3 marks]
  3. Explain why the Newton-Raphson method fails in the case when the first approximation is \(x_1 = -1\) [1 mark]
AQA Paper 2 Specimen Q5
9 marks Standard +0.3
  1. Determine a sequence of transformations which maps the graph of \(y = \cos \theta\) onto the graph of \(y = 3\cos \theta + 3\sin \theta\) Fully justify your answer. [6 marks]
  2. Hence or otherwise find the least value and greatest value of $$4 + (3\cos \theta + 3\sin \theta)^2$$ Fully justify your answer. [3 marks]
AQA Paper 2 Specimen Q6
5 marks Moderate -0.8
A curve \(C\) has equation \(y = x^2 - 4x + k\), where \(k\) is a constant. It crosses the \(x\)-axis at the points \((2 + \sqrt{5}, 0)\) and \((2 - \sqrt{5}, 0)\)
  1. Find the value of \(k\). [2 marks]
  2. Sketch the curve \(C\), labelling the exact values of all intersections with the axes. [3 marks]
AQA Paper 2 Specimen Q7
4 marks Standard +0.3
A student notices that when he adds two consecutive odd numbers together the answer always seems to be the difference between two square numbers. He claims that this will always be true. He attempts to prove his claim as follows: Step 1: Check first few cases \(3 + 5 = 8\) and \(8 = 3^2 - 1^2\) \(5 + 7 = 12\) and \(12 = 4^2 - 2^2\) \(7 + 9 = 16\) and \(16 = 5^2 - 3^2\) Step 2: Use pattern to predict and check a large example \(101 + 103 = 204\) subtract 1 and divide by 2 for the first number Add 1 and divide by two for the second number \(52^2 - 50^2 = 204\) it works! Step 3: Conclusion The first few cases work and there is a pattern, which can be used to predict larger numbers. Therefore, it must be true for all consecutive odd numbers.
  1. Explain what is wrong with the student's "proof". [1 mark]
  2. Prove that the student's claim is correct. [3 marks]
AQA Paper 2 Specimen Q8
8 marks Standard +0.8
A curve has equation \(y = 2x \cos 3x + (3x^2 - 4) \sin 3x\)
  1. Find \(\frac{dy}{dx}\), giving your answer in the form \((mx^2 + n) \cos 3x\), where \(m\) and \(n\) are integers. [4 marks]
  2. Show that the \(x\)-coordinates of the points of inflection of the curve satisfy the equation $$\cot 3x = \frac{9x^2 - 10}{6x}$$ [4 marks]
AQA Paper 2 Specimen Q9
10 marks Challenging +1.2
  1. Three consecutive terms in an arithmetic sequence are \(3e^{-q}\), \(5\), \(3e^q\) Find the possible values of \(p\). Give your answers in an exact form. [6 marks]
  2. Prove that there is no possible value of \(q\) for which \(3e^{-q}\), \(5\), \(3e^q\) are consecutive terms of a geometric sequence. [4 marks]
AQA Paper 2 Specimen Q10
1 marks Easy -1.8
A single force of magnitude 4 newtons acts on a particle of mass 50 grams. Find the magnitude of the acceleration of the particle. Circle your answer. [1 mark] \(12.5 \text{ m s}^{-2}\) \(\quad\) \(0.08 \text{ m s}^{-2}\) \(\quad\) \(0.0125 \text{ m s}^{-2}\) \(\quad\) \(80 \text{ m s}^{-2}\)
AQA Paper 2 Specimen Q11
2 marks Easy -1.2
A uniform rod, \(AB\), has length 3 metres and mass 24 kg. A particle of mass \(M\) kg is attached to the rod at \(A\). The rod is balanced in equilibrium on a support at \(C\), which is 0.8 metres from \(A\). \includegraphics{figure_11} Find the value of \(M\). [2 marks]
AQA Paper 2 Specimen Q12
4 marks Moderate -0.8
A particle moves on a straight line with a constant acceleration, \(a\) m s\(^{-2}\). The initial velocity of the particle is \(U\) m s\(^{-1}\). After \(T\) seconds the particle has velocity \(V\) m s\(^{-1}\). This information is shown on the velocity-time graph. \includegraphics{figure_12} The displacement, \(S\) metres, of the particle from its initial position at time \(T\) seconds is given by the formula $$S = \frac{1}{2}(U + V)T$$
  1. By considering the gradient of the graph, or otherwise, write down a formula for \(a\) in terms of \(U\), \(V\) and \(T\). [1 mark]
  2. Hence show that \(V^2 = U^2 + 2aS\) [3 marks]
AQA Paper 2 Specimen Q13
5 marks Moderate -0.8
The three forces \(\mathbf{F_1}\), \(\mathbf{F_2}\) and \(\mathbf{F_3}\) are acting on a particle. \(\mathbf{F_1} = (25\mathbf{i} + 12\mathbf{j})\) N \(\mathbf{F_2} = (-7\mathbf{i} + 5\mathbf{j})\) N \(\mathbf{F_3} = (15\mathbf{i} - 28\mathbf{j})\) N The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal and vertical respectively. The resultant of these three forces is \(\mathbf{F}\) newtons.
  • The fourth force, \(\mathbf{F_4}\), is applied to the particle so that the four forces are in equilibrium. Find \(\mathbf{F_4}\), giving your answer in terms of \(\mathbf{i}\) and \(\mathbf{j}\). [1 mark]
    •
    AQA Paper 2 Specimen Q14
    7 marks Moderate -0.3
    The graph below models the velocity of a small train as it moves on a straight track for 20 seconds. The front of the train is at the point \(A\) when \(t = 0\) The mass of the train is 800kg. \includegraphics{figure_14}
    1. Find the total distance travelled in the 20 seconds. [3 marks]
    2. Find the distance of the front of the train from the point \(A\) at the end of the 20 seconds. [1 mark]
    3. Find the maximum magnitude of the resultant force acting on the train. [2 marks]
    4. Explain why, in reality, the graph may not be an accurate model of the motion of the train. [1 mark]
    AQA Paper 2 Specimen Q15
    11 marks Standard +0.8
    At time \(t = 0\), a parachutist jumps out of an airplane that is travelling horizontally. The velocity, \(\mathbf{v}\) m s\(^{-1}\), of the parachutist at time \(t\) seconds is given by: $$\mathbf{v} = (40e^{-0.2t})\mathbf{i} + 50(e^{-0.2t} - 1)\mathbf{j}$$ The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal and vertical respectively. Assume that the parachutist is at the origin when \(t = 0\) Model the parachutist as a particle.
    1. Find an expression for the position vector of the parachutist at time \(t\). [4 marks]
    2. The parachutist opens her parachute when she has travelled 100 metres horizontally. Find the vertical displacement of the parachutist from the origin when she opens her parachute. [4 marks]
    3. Carefully, explaining the steps that you take, deduce the value of \(g\) used in the formulation of this model. [3 marks]
    AQA Paper 2 Specimen Q16
    12 marks Standard +0.3
    In this question use \(g = 9.8\) m s\(^{-2}\). The diagram shows a box, of mass 8.0 kg, being pulled by a string so that the box moves at a constant speed along a rough horizontal wooden board. The string is at an angle of 40° to the horizontal. The tension in the string is 50 newtons. \includegraphics{figure_16a} The coefficient of friction between the box and the board is \(\mu\) Model the box as a particle.
    1. Show that \(\mu = 0.83\) [4 marks]
    2. One end of the board is lifted up so that the board is now inclined at an angle of 5° to the horizontal. The box is pulled up the inclined board. The string remains at an angle of 40° to the board. The tension in the string is increased so that the box accelerates up the board at 3 m s\(^{-2}\) \includegraphics{figure_16b}
      1. Draw a diagram to show the forces acting on the box as it moves. [1 mark]
      2. Find the tension in the string as the box accelerates up the slope at 3 m s\(^{-2}\). [7 marks]
    AQA Paper 2 Specimen Q17
    8 marks Standard +0.3
    In this question use \(g = 9.81\) m s\(^{-2}\). A ball is projected from the origin. After 2.5 seconds, the ball lands at the point with position vector \((40\mathbf{i} - 10\mathbf{j})\) metres. The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal and vertical respectively. Assume that there are no resistance forces acting on the ball.
    1. Find the speed of the ball when it is at a height of 3 metres above its initial position. [6 marks]
    2. State the speed of the ball when it is at its maximum height. [1 mark]
    3. Explain why the answer you found in part (b) may not be the actual speed of the ball when it is at its maximum height. [1 mark]
    AQA Paper 3 2018 June Q1
    1 marks Easy -2.0
    A circle has equation \((x - 4)^2 + (y + 4)^2 = 9\) What is the area of the circle? Circle your answer. [1 mark] \(3\pi\) \quad \(9\pi\) \quad \(16\pi\) \quad \(81\pi\)
    AQA Paper 3 2018 June Q2
    1 marks Easy -1.8
    A curve has equation \(y = x^5 + 4x^3 + 7x + q\) where \(q\) is a positive constant. Find the gradient of the curve at the point where \(x = 0\) Circle your answer. [1 mark] \(0\) \quad \(4\) \quad \(7\) \quad \(q\)
    AQA Paper 3 2018 June Q3
    1 marks Easy -1.8
    The line \(L\) has equation \(2x + 3y = 7\) Which one of the following is perpendicular to \(L\)? Tick one box. [1 mark] \(2x - 3y = 7\) \(3x + 2y = -7\) \(2x + 3y = -\frac{1}{7}\) \(3x - 2y = 7\)
    AQA Paper 3 2018 June Q4
    3 marks Easy -1.2
    Sketch the graph of \(y = |2x + a|\), where \(a\) is a positive constant. Show clearly where the graph intersects the axes. [3 marks] \includegraphics{figure_4}
    AQA Paper 3 2018 June Q5
    3 marks Moderate -0.3
    Show that, for small values of \(x\), the graph of \(y = 5 + 4\sin\frac{x}{2} + 12\tan\frac{x}{3}\) can be approximated by a straight line. [3 marks]
    AQA Paper 3 2018 June Q6
    13 marks Standard +0.8
    A function \(f\) is defined by \(f(x) = \frac{x}{\sqrt{2x - 2}}\)
    1. State the maximum possible domain of \(f\). [2 marks]
    2. Use the quotient rule to show that \(f'(x) = \frac{x - 2}{(2x - 2)^{\frac{3}{2}}}\). [3 marks]
    3. Show that the graph of \(y = f(x)\) has exactly one point of inflection. [7 marks]
    4. Write down the values of \(x\) for which the graph of \(y = f(x)\) is convex. [1 mark]
    AQA Paper 3 2018 June Q7
    5 marks Moderate -0.8
    1. Given that \(\log_a y = 2\log_a 7 + \log_a 4 + \frac{1}{2}\), find \(y\) in terms of \(a\). [4 marks]
    2. When asked to solve the equation $$2\log_a x = \log_a 9 - \log_a 4$$ a student gives the following solution: \(2\log_a x = \log_a 9 - \log_a 4\) \(\Rightarrow 2\log_a x = \log_a \frac{9}{4}\) \(\Rightarrow \log_a x^2 = \log_a \frac{9}{4}\) \(\Rightarrow x^2 = \frac{9}{4}\) \(\therefore x = \frac{3}{2}\) or \(-\frac{3}{2}\) Explain what is wrong with the student's solution. [1 mark]
    AQA Paper 3 2018 June Q8
    9 marks Standard +0.3
    1. Prove the identity \(\frac{\sin 2x}{1 + \tan^2 x} = 2\sin x \cos^3 x\) [3 marks]
    2. Hence find \(\int \frac{4\sin 4\theta}{1 + \tan^2 2\theta} d\theta\) [6 marks]
    AQA Paper 3 2018 June Q9
    7 marks Standard +0.3
    Helen is creating a mosaic pattern by placing square tiles next to each other along a straight line. \includegraphics{figure_9} The area of each tile is half the area of the previous tile, and the sides of the largest tile have length \(w\) centimetres.
    1. Find, in terms of \(w\), the length of the sides of the second largest tile. [1 mark]
    2. Assume the tiles are in contact with adjacent tiles, but do not overlap. Show that, no matter how many tiles are in the pattern, the total length of the series of tiles will be less than \(3.5w\). [4 marks]
    3. Helen decides the pattern will look better if she leaves a 3 millimetre gap between adjacent tiles. Explain how you could refine the model used in part (b) to account for the 3 millimetre gap, and state how the total length of the series of tiles will be affected. [2 marks]