Questions Paper 2 (413 questions)

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AQA Paper 2 2024 June Q8
7 marks Moderate -0.3
A zookeeper models the median mass of infant monkeys born at their zoo, up to the age of 2 years, by the formula $$y = a + b \log_{10} x$$ where \(y\) is the median mass in kilograms, \(x\) is age in months and \(a\) and \(b\) are constants. The zookeeper uses the data shown below to determine the values of \(a\) and \(b\).
Age in months (\(x\))324
Median mass (\(y\))6.412
  1. The zookeeper uses the data for monkeys aged 3 months to write the correct equation $$6.4 = a + b \log_{10} 3$$
    1. Use the data for monkeys aged 24 months to write a second equation. [1 mark]
    2. Show that $$b = \frac{5.6}{\log_{10} 8}$$ [3 marks]
    3. Find the value of \(a\). Give your answer to two decimal places. [1 mark]
  2. Use a suitable value for \(x\) to determine whether the model can be used to predict the median mass of monkeys less than one week old. [2 marks]
AQA Paper 2 2024 June Q9
13 marks Standard +0.3
    1. Find the binomial expansion of \((1 + 3x)^{-1}\) up to and including the term in \(x^2\) [2 marks]
    2. Show that the first three terms in the binomial expansion of $$\frac{1}{2 - 3x}$$ form a geometric sequence and state the common ratio. [5 marks]
  2. It is given that $$\frac{36x}{(1 + 3x)(2 - 3x)} = \frac{P}{(2 - 3x)} + \frac{Q}{(1 + 3x)}$$ where \(P\) and \(Q\) are integers. Find the value of \(P\) and the value of \(Q\) [3 marks]
    1. Using your answers to parts (a) and (b), find the binomial expansion of $$\frac{12x}{(1 + 3x)(2 - 3x)}$$ up to and including the term in \(x^2\) [2 marks]
    2. Find the range of values of \(x\) for which the binomial expansion of $$\frac{12x}{(1 + 3x)(2 - 3x)}$$ is valid. [1 mark]
AQA Paper 2 2024 June Q10
4 marks Standard +0.3
The function f is defined by $$f(x) = x^2 + 2 \cos x \text{ for } -\pi \leq x \leq \pi$$ Determine whether the curve with equation \(y = f(x)\) has a point of inflection at the point where \(x = 0\) Fully justify your answer. [4 marks]
AQA Paper 2 2024 June Q11
6 marks Moderate -0.8
  1. A student states that 3 is the smallest value of \(k\) in the interval \(3 < k < 4\) Explain the error in the student's statement. [1 mark]
  2. The student's teacher says there is no smallest value of \(k\) in the interval \(3 < k < 4\) The teacher gives the following correct proof: Step 1: Assume there is a smallest number in the interval \(3 < k < 4\) and let this smallest number be \(x\) Step 2: let \(y = \frac{3 + x}{2}\) Step 3: \(3 < y < x\) which is a contradiction. Step 4: Therefore, there is no smallest number in interval \(3 < k < 4\)
    1. Explain the contradiction stated in Step 3 [1 mark]
    2. Prove that there is no largest value of \(k\) in the interval \(3 < k < 4\) [4 marks]
AQA Paper 2 2024 June Q12
1 marks Easy -2.0
Two constant forces act on a particle, of mass 2 kilograms, so that it moves forward in a straight line. The two forces are: • a forward driving force of 10 newtons • a resistance force of 4 newtons. Find the acceleration of the particle. Circle your answer. [1 mark] \(2 \text{ m s}^{-2}\) \(\quad\) \(3 \text{ m s}^{-2}\) \(\quad\) \(5 \text{ m s}^{-2}\) \(\quad\) \(12 \text{ m s}^{-2}\)
AQA Paper 2 2024 June Q13
1 marks Easy -1.8
A car starting from rest moves forward in a straight line. The motion of the car is modelled by the velocity–time graph below: \includegraphics{figure_13} One of the following assumptions about the motion of the car is implied by the graph. Identify this assumption. [1 mark] Tick \((\checkmark)\) one box. The car never accelerates. The acceleration of the car is always positive. The acceleration of the car can change instantaneously. The acceleration of the car is never constant.
AQA Paper 2 2024 June Q14
3 marks Moderate -0.8
The displacement, \(r\) metres, of a particle at time \(t\) seconds is $$r = 6t - 2t^2$$
  1. Find the value of \(r\) when \(t = 4\) [1 mark]
  2. Determine the range of values of \(t\) for which the displacement is positive. [2 marks]
AQA Paper 2 2024 June Q15
4 marks Standard +0.3
Two forces, \(\mathbf{F_1}\) and \(\mathbf{F_2}\), are acting on a particle of mass 3 kilograms. It is given that $$\mathbf{F_1} = \begin{pmatrix} a \\ 23 \end{pmatrix} \text{ newtons and } \mathbf{F_2} = \begin{pmatrix} 4 \\ b \end{pmatrix} \text{ newtons}$$ where \(a\) and \(b\) are constants. The particle has an acceleration of \(\begin{pmatrix} 4b \\ a \end{pmatrix}\) m s\(^{-2}\) Find the value of \(a\) and the value of \(b\) [4 marks]
AQA Paper 2 2024 June Q16
4 marks Moderate -0.8
In this question use \(g = 9.8\) m s\(^{-2}\) An apple tree stands on horizontal ground. An apple hangs, at rest, from a branch of the tree. A second apple also hangs, at rest, from a different branch of the tree. The vertical distance between the two apples is \(d\) centimetres. At the same instant both apples begin to fall freely under gravity. The first apple hits the ground after 0.5 seconds. The second apple hits the ground 0.1 seconds later. Show that \(d\) is approximately 54 [4 marks]
AQA Paper 2 2024 June Q17
4 marks Standard +0.3
A uniform rod is resting on two fixed supports at points \(A\) and \(B\). \(A\) lies at a distance \(x\) metres from one end of the rod. \(B\) lies at a distance \((x + 0.1)\) metres from the other end of the rod. The rod has length \(2L\) metres and mass \(m\) kilograms. The rod lies horizontally in equilibrium as shown in the diagram below. \includegraphics{figure_17} The reaction force of the support on the rod at \(B\) is twice the reaction force of the support on the rod at \(A\). Show that $$L - x = k$$ where \(k\) is a constant to be found. [4 marks]
AQA Paper 2 2024 June Q18
7 marks Standard +0.3
A particle is moving in a straight line through the origin \(O\) The displacement of the particle, \(r\) metres, from \(O\), at time \(t\) seconds is given by $$r = p + 2t - qe^{-0.2t}$$ where \(p\) and \(q\) are constants. When \(t = 3\), the acceleration of the particle is \(-1.8\) m s\(^{-2}\)
  1. Show that \(q \approx 82\) [5 marks]
  2. The particle has an initial displacement of 5 metres. Find the value of \(p\) Give your answer to two significant figures. [2 marks]
AQA Paper 2 2024 June Q19
8 marks Standard +0.3
In this question use \(g = 9.8\) m s\(^{-2}\) A toy shoots balls upwards with an initial velocity of 7 m s\(^{-1}\) The advertisement for this toy claims the balls can reach a maximum height of 2.5 metres from the ground.
  1. Suppose that the toy shoots the balls vertically upwards.
    1. Verify the claim in the advertisement. [2 marks]
    2. State two modelling assumptions you have made in verifying this claim. [2 marks]
  2. In fact the toy shoots the balls anywhere between 0 and 11 degrees from the vertical. The range of maximum heights, \(h\) metres, above the ground which can be reached by the balls may be expressed as $$k \leq h \leq 2.5$$ Find the value of \(k\) [4 marks]
AQA Paper 2 2024 June Q20
9 marks Standard +0.3
Two particles \(P\) and \(Q\) are moving in separate straight lines across a smooth horizontal surface. \(P\) moves with constant velocity \((3\mathbf{i} + 4\mathbf{j})\) m s\(^{-1}\) \(Q\) moves from position vector \((5\mathbf{i} - 7\mathbf{j})\) metres to position vector \((14\mathbf{i} + 5\mathbf{j})\) metres during a 3 second period.
  1. Show that \(P\) and \(Q\) move along parallel lines. [3 marks]
  2. Stevie says Q is also moving with a constant velocity of \((3\mathbf{i} + 4\mathbf{j})\) m s\(^{-1}\) Explain why Stevie may be incorrect. [1 mark]
  3. A third particle \(R\) is moving with a constant speed of 4 m s\(^{-1}\), in a straight line, across the same surface. \(P\) and \(R\) move along lines that intersect at a fixed point \(X\) It is given that: • \(P\) passes through \(X\) exactly 2 seconds after \(R\) passes through \(X\) • \(P\) and \(R\) are exactly 13 metres apart 3 seconds after \(R\) passes through \(X\) Show that \(P\) and \(R\) move along perpendicular lines. [5 marks]
AQA Paper 2 2024 June Q21
9 marks Standard +0.3
Two heavy boxes, \(M\) and \(N\), are connected securely by a length of rope. The mass of \(M\) is 50 kilograms. The mass of \(N\) is 80 kilograms. \(M\) is placed near the bottom of a rough slope. The slope is inclined at 60° above the horizontal. The rope is passed over a smooth pulley at the top end of the slope so that \(N\) hangs with the rope vertical. The boxes are initially held in this position, with the rope taut and running parallel to the line of greatest slope, as shown in the diagram below. \includegraphics{figure_21} When the boxes are released, \(M\) moves up the slope as \(N\) descends, with acceleration \(a\) m s\(^{-2}\) The tension in the rope is \(T\) newtons.
  1. Explain why the equation of motion for \(N\) is $$80g - T = 80a$$ [1 mark]
  2. Show that the normal reaction force between \(M\) and the slope is \(25g\) newtons. [1 mark]
  3. The coefficient of friction, \(\mu\), between the slope and \(M\) is such that \(0 \leq \mu \leq 1\) Show that $$a \geq \frac{(11 - 5\sqrt{3})g}{26}$$ [6 marks]
  4. State one modelling assumption you have made throughout this question. [1 mark]
AQA Paper 2 Specimen Q1
1 marks Easy -1.8
State the values of \(|x|\) for which the binomial expansion of \((3 + 2x)^{-4}\) is valid. Circle your answer. [1 mark] \(|x| < \frac{2}{3}\) \(\quad\) \(|x| < 1\) \(\quad\) \(|x| < \frac{3}{2}\) \(\quad\) \(|x| < 3\)
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]