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AQA Paper 2 Specimen Q14
7 marks Moderate -0.3
14 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 800 kg .
\includegraphics[max width=\textwidth, alt={}, center]{a57b0526-cf9c-44d6-a349-cac392f85a70-22_645_1374_699_479} 14
  1. Find the total distance travelled in the 20 seconds.
    14
  2. Find the distance of the front of the train from the point \(A\) at the end of the 20 seconds.
    [0pt] [1 mark]
    14
  3. Find the maximum magnitude of the resultant force acting on the train.
    [0pt] [2 marks]
    14
  4. Explain why, in reality, the graph may not be an accurate model of the motion of the train.
AQA Paper 2 Specimen Q15
11 marks Standard +0.3
15 At time \(t = 0\), a parachutist jumps out of an airplane that is travelling horizontally.
The velocity, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), of the parachutist at time \(t\) seconds is given by: $$\mathbf { v } = \left( 40 \mathrm { e } ^ { - 0.2 t } \right) \mathbf { i } + 50 \left( \mathrm { e } ^ { - 0.2 t } - 1 \right) \mathbf { j }$$ 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. 15
  1. Find an expression for the position vector of the parachutist at time \(t\).
    [0pt] [4 marks] 15
  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.
    [0pt] [4 marks]
    15
  3. Carefully, explaining the steps that you take, deduce the value of \(g\) used in the formulation of this model.
AQA Paper 2 Specimen Q16
12 marks Moderate -0.3
16 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
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 \(40 ^ { \circ }\) to the horizontal.
The tension in the string is 50 newtons.
\includegraphics[max width=\textwidth, alt={}, center]{a57b0526-cf9c-44d6-a349-cac392f85a70-26_334_862_884_575} The coefficient of friction between the box and the board is \(\mu\)
Model the box as a particle.
16
  1. Show that \(\mu = 0.83\)
    [0pt] [4 marks] Question 16 continues on the next page 16
  2. One end of the board is lifted up so that the board is now inclined at an angle of \(5 ^ { \circ }\) to the horizontal. The box is pulled up the inclined board.
    The string remains at an angle of \(40 ^ { \circ }\) to the board.
    The tension in the string is increased so that the box accelerates up the board at \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
    \includegraphics[max width=\textwidth, alt={}, center]{a57b0526-cf9c-44d6-a349-cac392f85a70-28_385_858_778_577} 16
    1. Draw a diagram to show the forces acting on the box as it moves. 16
  4. (ii) Find the tension in the string as the box accelerates up the slope at \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    [0pt] [7 marks]
AQA Paper 2 Specimen Q20
8 marks Moderate -0.3
20
40
200 3 A curve is defined by the parametric equations $$x = t ^ { 3 } + 2 , \quad y = t ^ { 2 } - 1$$ 3
  1. Find the gradient of the curve at the point where \(t = - 2\)
    [0pt] [4 marks]
    3
  2. Find a Cartesian equation of the curve.
    4 The equation \(x ^ { 3 } - 3 x + 1 = 0\) has three real roots. 4
  3. Show that one of the roots lies between - 2 and - 1
    4
  4. 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.
    [0pt] [3 marks]
    4
  5. Explain why the Newton-Raphson method fails in the case when the first approximation is \(x _ { 1 } = - 1\)
    [0pt] [1 mark]
AQA Paper 3 2018 June Q2
1 marks Easy -2.0
2 A curve has equation \(y = x ^ { 5 } + 4 x ^ { 3 } + 7 x + q\) where \(q\) is a positive constant.
Find the gradient of the curve at the point where \(x = 0\)
Circle your answer.
0
4
7
\(q\)
AQA Paper 3 2018 June Q4
3 marks Easy -1.8
4
7
\(q\) 3 The line \(L\) has equation \(2 x + 3 y = 7\)
Which one of the following is perpendicular to \(L\) ?
Tick one box. $$\begin{aligned} & 2 x - 3 y = 7 \\ & 3 x + 2 y = - 7 \\ & 2 x + 3 y = - \frac { 1 } { 7 } \\ & 3 x - 2 y = 7 \end{aligned}$$ □
□
□
□ 4 Sketch the graph of \(y = | 2 x + a |\), where \(a\) is a positive constant. Show clearly where the graph intersects the axes.
\includegraphics[max width=\textwidth, alt={}, center]{d9149857-5f94-4fa5-a6d8-550c0c07fefb-03_1001_1002_450_520}
AQA Paper 3 2018 June Q5
3 marks Moderate -0.5
5 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.
AQA Paper 3 2018 June Q7
5 marks Easy -1.8
7
\(q\) 3 The line \(L\) has equation \(2 x + 3 y = 7\)
Which one of the following is perpendicular to \(L\) ?
Tick one box. $$\begin{aligned} & 2 x - 3 y = 7 \\ & 3 x + 2 y = - 7 \\ & 2 x + 3 y = - \frac { 1 } { 7 } \\ & 3 x - 2 y = 7 \end{aligned}$$ □
□
□
□ 4 Sketch the graph of \(y = | 2 x + a |\), where \(a\) is a positive constant. Show clearly where the graph intersects the axes.
\includegraphics[max width=\textwidth, alt={}, center]{d9149857-5f94-4fa5-a6d8-550c0c07fefb-03_1001_1002_450_520} 5 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.
6 (b) Use the quotient rule to show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { x - 2 } { ( 2 x - 2 ) ^ { \frac { 3 } { 2 } } }\) 6 (a) State the maximum possible domain of f .
\(6 \quad\) A function f is defined by \(\mathrm { f } ( x ) = \frac { x } { \sqrt { 2 x - 2 } }\) $$\begin{gathered} \text { Do not write } \\ \text { outside the } \\ \text { box } \end{gathered}$$ 6 (a)
6 (c) Show that the graph of \(y = \mathrm { f } ( x )\) has exactly one point of inflection.
6 (d) Write down the values of \(x\) for which the graph of \(y = \mathrm { f } ( x )\) is convex.
7 (a) Given that \(\log _ { a } y = 2 \log _ { a } 7 + \log _ { a } 4 + \frac { 1 } { 2 }\), find \(y\) in terms of \(a\).
7 (b) When asked to solve the equation $$2 \log _ { a } x = \log _ { a } 9 - \log _ { a } 4$$ a student gives the following solution: $$\begin{aligned} & 2 \log _ { a } x = \log _ { a } 9 - \log _ { a } 4 \\ & \Rightarrow 2 \log _ { a } x = \log _ { a } \frac { 9 } { 4 } \\ & \Rightarrow \log _ { a } x ^ { 2 } = \log _ { a } \frac { 9 } { 4 } \\ & \Rightarrow x ^ { 2 } = \frac { 9 } { 4 } \\ & \therefore x = \frac { 3 } { 2 } \text { or } - \frac { 3 } { 2 } \end{aligned}$$ Explain what is wrong with the student's solution.
AQA Paper 3 2018 June Q8
9 marks Standard +0.3
8
  1. Prove the identity \(\frac { \sin 2 x } { 1 + \tan ^ { 2 } x } \equiv 2 \sin x \cos ^ { 3 } x\) 8
  2. Hence find \(\int \frac { 4 \sin 4 \theta } { 1 + \tan ^ { 2 } 2 \theta } \mathrm {~d} \theta\)
AQA Paper 3 2018 June Q9
7 marks Standard +0.3
9 Helen is creating a mosaic pattern by placing square tiles next to each other along a straight line.
\includegraphics[max width=\textwidth, alt={}, center]{d9149857-5f94-4fa5-a6d8-550c0c07fefb-10_364_1300_406_370} The area of each tile is half the area of the previous tile, and the sides of the largest tile have length \(w\) centimetres. 9
  1. Find, in terms of \(w\), the length of the sides of the second largest tile. 9
  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 \(3.5 w\).
    \(\mathbf { 9 }\) (c) 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.
    [0pt] [2 marks]
AQA Paper 3 2018 June Q11
1 marks Easy -1.8
11 The table below shows the probability distribution for a discrete random variable \(X\).
\(\boldsymbol { x }\)12345
\(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)\(k\)\(2 k\)\(4 k\)\(2 k\)\(k\)
Find the value of \(k\). Circle your answer.
\(\frac { 1 } { 2 }\)\(\frac { 1 } { 4 }\)\(\frac { 1 } { 10 }\)1
AQA Paper 3 2018 June Q14
6 marks Moderate -0.3
14 A teacher in a college asks her mathematics students what other subjects they are studying. She finds that, of her 24 students:
12 study physics
8 study geography
4 study geography and physics
14
  1. A student is chosen at random from the class.
    Determine whether the event 'the student studies physics' and the event 'the student studies geography' are independent.
    14
  2. It is known that for the whole college:
    the probability of a student studying mathematics is \(\frac { 1 } { 5 }\)
    the probability of a student studying biology is \(\frac { 1 } { 6 }\)
    the probability of a student studying biology given that they study mathematics is \(\frac { 3 } { 8 }\)
    Calculate the probability that a student studies mathematics or biology or both.
AQA Paper 3 2018 June Q15
7 marks Moderate -0.5
15 (e) State two necessary assumptions in context so that the distribution stated in part (a) is valid.
AQA Paper 3 2018 June Q16
12 marks Standard +0.3
16 A survey of 120 adults found that the volume, \(X\) litres per person, of carbonated drinks they consumed in a week had the following results: $$\sum x = 165.6 \quad \sum x ^ { 2 } = 261.8$$ 16
    1. Calculate the mean of \(X\).
      16
  2. (ii) Calculate the standard deviation of \(X\).
    16
  3. Assuming that \(X\) can be modelled by a normal distribution find
    16
    1. \(\mathrm { P } ( 0.5 < X < 1.5 )\)
      16
  5. (ii) \(\mathrm { P } ( X = 1 )\) 16
  6. Determine with a reason, whether a normal distribution is suitable to model this data. [2 marks]
    16
  7. It is known that the volume, \(Y\) litres per person, of energy drinks consumed in a week may be modelled by a normal distribution with standard deviation 0.21 Given that \(\mathrm { P } ( Y > 0.75 ) = 0.10\), find the value of \(\mu\), correct to three significant figures. [4 marks]
AQA Paper 3 2018 June Q17
12 marks Standard +0.3
17 Suzanne is a member of a sports club. For each sport she competes in, she wins half of the matches.
17
  1. After buying a new tennis racket Suzanne plays 10 matches and wins 7 of them.
    Investigate, at the \(10 \%\) level of significance, whether Suzanne's new racket has made a difference to the probability of her winning a match. 17
  2. After buying a new squash racket, Suzanne plays 20 matches. Find the minimum number of matches she must win for her to conclude, at the \(10 \%\) level of significance, that the new racket has improved her performance.
AQA Paper 3 2018 June Q18
8 marks Standard +0.3
18 In a region of England, the government decides to use an advertising campaign to encourage people to eat more healthily. Before the campaign, the mean consumption of chocolate per person per week was known to be 66.5 g , with a standard deviation of 21.2 g 18
  1. After the campaign, the first 750 available people from this region were surveyed to find out their average consumption of chocolate. 18
    1. State the sampling method used to collect the survey. 18
  3. (ii) Explain why this sample should not be used to conduct a hypothesis test.
    18
  4. A second sample of 750 people revealed that the mean consumption of chocolate per person per week was 65.4 g Investigate, at the \(10 \%\) level of significance, whether the advertising campaign has decreased the mean consumption of chocolate per person per week. Assume that an appropriate sampling method was used and that the consumption of chocolate is normally distributed with an unchanged standard deviation.
    \includegraphics[max width=\textwidth, alt={}, center]{d9149857-5f94-4fa5-a6d8-550c0c07fefb-26_2488_1719_219_150}
    \includegraphics[max width=\textwidth, alt={}, center]{d9149857-5f94-4fa5-a6d8-550c0c07fefb-27_2492_1721_217_150}
    \includegraphics[max width=\textwidth, alt={}, center]{d9149857-5f94-4fa5-a6d8-550c0c07fefb-28_2496_1719_214_150}
AQA Paper 3 2019 June Q2
1 marks Easy -1.8
2 Find the value of \(\frac { 100 ! } { 98 ! \times 3 ! }\)
Circle your answer. $$\begin{array} { l l l l } \frac { 50 } { 147 } & 1650 & 3300 & 161700 \end{array}$$
AQA Paper 3 2019 June Q3
1 marks Moderate -0.5
3
Given \(u _ { 1 } = 1\), determine which one of the formulae below defines an increasing sequence for \(n \geq 1\) Circle your answer.
[0pt] [1 mark]
\(u _ { n + 1 } = 1 + \frac { 1 } { u _ { n } } \quad u _ { n } = 2 - 0.9 ^ { n - 1 } \quad u _ { n + 1 } = - 1 + 0.5 u _ { n } \quad u _ { n } = 0.9 ^ { n - 1 }\)
AQA Paper 3 2019 June Q4
3 marks Moderate -0.8
4 Sketch the region defined by the inequalities $$y \leq ( 1 - 2 x ) ( x + 3 ) \text { and } y - x \leq 3$$ Clearly indicate your region by shading it in and labelling it \(R\).
\includegraphics[max width=\textwidth, alt={}, center]{c8a41c47-bbda-4e91-a7a2-d0bcf6a46f25-03_1000_1004_833_518}
AQA Paper 3 2019 June Q5
5 marks Standard +0.3
5 A circle has equation \(x ^ { 2 } + y ^ { 2 } - 6 x - 8 y = 264\)
\(A B\) is a chord of the circle. The angle at the centre of the circle, subtended by \(A B\), is 0.9 radians, as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{c8a41c47-bbda-4e91-a7a2-d0bcf6a46f25-04_563_600_584_719} Find the area of the minor segment shaded on the diagram.
Give your answer to three significant figures.
AQA Paper 3 2019 June Q6
4 marks Standard +0.3
6 The three sides of a right-angled triangle have lengths \(a , b\) and \(c\), where \(a , b , c \in \mathbb { Z }\) 6
  1. State an example where \(a , b\) and \(c\) are all even.
    6
  2. Prove that it is not possible for all of \(a , b\) and \(c\) to be odd.
AQA Paper 3 2019 June Q7
8 marks Moderate -0.8
7
  1. Express \(\frac { 4 x + 3 } { ( x - 1 ) ^ { 2 } }\) in the form \(\frac { A } { x - 1 } + \frac { B } { ( x - 1 ) ^ { 2 } }\) 7
  2. Show that $$\int _ { 3 } ^ { 4 } \frac { 4 x + 3 } { ( x - 1 ) ^ { 2 } } \mathrm {~d} x = p + \ln q$$ where \(p\) and \(q\) are rational numbers.
AQA Paper 3 2019 June Q8
12 marks Moderate -0.3
8 A student is conducting an experiment in a laboratory to investigate how quickly liquids cool to room temperature. A beaker containing a hot liquid at an initial temperature of \(75 ^ { \circ } \mathrm { C }\) cools so that the temperature, \(\theta ^ { \circ } \mathrm { C }\), of the liquid at time \(t\) minutes can be modelled by the equation $$\theta = 5 \left( 4 + \lambda \mathrm { e } ^ { - k t } \right)$$ where \(\lambda\) and \(k\) are constants. After 2 minutes the temperature falls to \(68 ^ { \circ } \mathrm { C }\).
8
  1. Find the temperature of the liquid after 15 minutes.
    Give your answer to three significant figures.
    8
    1. Find the room temperature of the laboratory, giving a reason for your answer.
      8
  3. (ii) Find the time taken in minutes for the liquid to cool to \(1 ^ { \circ } \mathrm { C }\) above the room temperature of the laboratory.
    8
  4. Explain why the model might need to be changed if the experiment was conducted in a different place.
AQA Paper 3 2019 June Q9
15 marks Standard +0.3
9 A curve has equation $$x ^ { 2 } y ^ { 2 } + x y ^ { 4 } = 12$$ 9
  1. Prove that the curve does not intersect the coordinate axes.
    9
    1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 2 x y + y ^ { 3 } } { 2 x ^ { 2 } + 4 x y ^ { 2 } }\)
      9
  3. (ii) Prove that the curve has no stationary points.
    9
  4. (iii) In the case when \(x > 0\), find the equation of the tangent to the curve when \(y = 1\)
AQA Paper 3 2019 June Q10
1 marks Easy -1.8
10 Which of the options below best describes the correlation shown in the diagram below?
\includegraphics[max width=\textwidth, alt={}, center]{c8a41c47-bbda-4e91-a7a2-d0bcf6a46f25-12_750_1246_847_395} Tick \(( \checkmark )\) one box.
moderate positive □
strong positive □
moderate negative □
strong negative □