Questions — AQA Paper 2 (140 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
AQA Paper 2 2018 June Q1
1 Which of these statements is correct? Tick one box. $$\begin{aligned} & x = 2 \Rightarrow x ^ { 2 } = 4
& x ^ { 2 } = 4 \Rightarrow x = 2
& x ^ { 2 } = 4 \Leftrightarrow x = 2
& x ^ { 2 } = 4 \Rightarrow x = - 2 \end{aligned}$$
AQA Paper 2 2018 June Q2
2 Find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 1 + 2 x ) ^ { 7 }\)
Circle your answer. 4242184
\includegraphics[max width=\textwidth, alt={}, center]{5e475c96-6bbb-44fb-a411-706dac20e2d6-03_1442_1168_219_365} Find the total shaded area. Circle your answer.
-68 60686128
AQA Paper 2 2018 June Q4
4 A curve, \(C\), has equation \(y = x ^ { 2 } - 6 x + k\), where \(k\) is a constant. The equation \(x ^ { 2 } - 6 x + k = 0\) has two distinct positive roots. 4
  1. Sketch \(C\) on the axes below.
    \includegraphics[max width=\textwidth, alt={}, center]{5e475c96-6bbb-44fb-a411-706dac20e2d6-04_1013_1016_534_513} 4
  2. Find the range of possible values for \(k\). Fully justify your answer.
    \begin{center} \begin{tabular}{ | l | } \hline
AQA Paper 2 2018 June Q5
2 marks
5 Prove that 23 is a prime number.
[0pt] [2 marks]
\end{tabular} \end{center}
AQA Paper 2 2018 June Q6
6 Find the coordinates of the stationary point of the curve with equation $$( x + y - 2 ) ^ { 2 } = \mathrm { e } ^ { y } - 1$$ \(7 \quad\) A function f has domain \(\mathbb { R }\) and range \(\{ y \in \mathbb { R } : y \geq \mathrm { e } \}\) The graph of \(y = \mathrm { f } ( x )\) is shown.
\includegraphics[max width=\textwidth, alt={}, center]{5e475c96-6bbb-44fb-a411-706dac20e2d6-08_922_1108_447_466} The gradient of the curve at the point \(( x , y )\) is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = ( x - 1 ) \mathrm { e } ^ { x }\)
Find an expression for \(\mathrm { f } ( x )\).
Fully justify your answer.
AQA Paper 2 2018 June Q8
8
  1. Determine a sequence of transformations which maps the graph of \(y = \sin x\) onto the graph of \(y = \sqrt { 3 } \sin x - 3 \cos x + 4\) Fully justify your answer.
    8
  2. (ii) Find the greatest value of \(\frac { 1 } { \sqrt { 3 } \sin x - 3 \cos x + 4 }\)
AQA Paper 2 2018 June Q9
9 A market trader notices that daily sales are dependent on two variables:
number of hours, \(t\), after the stall opens
total sales, \(x\), in pounds since the stall opened.
The trader models the rate of sales as directly proportional to \(\frac { 8 - t } { x }\)
After two hours the rate of sales is \(\pounds 72\) per hour and total sales are \(\pounds 336\)
9
  1. Show that $$x \frac { \mathrm {~d} x } { \mathrm {~d} t } = 4032 ( 8 - t )$$ 9
  2. Hence, show that $$x ^ { 2 } = 4032 t ( 16 - t )$$ \(\mathbf { 9 }\) (c) The stall opens at 09.30. 9
    1. The trader closes the stall when the rate of sales falls below \(\pounds 24\) per hour.
      Using the results in parts (a) and (b), calculate the earliest time that the trader closes the stall.
      9
  4. (ii) Explain why the model used by the trader is not valid at 09.30.
AQA Paper 2 2018 June Q10
10 A garden snail moves in a straight line from rest to \(1.28 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\), with a constant acceleration in 1.8 seconds. Find the acceleration of the snail. Circle your answer.
\(2.30 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
\(0.71 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
\(0.0071 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
\(0.023 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
AQA Paper 2 2018 June Q11
1 marks
11 A uniform rod, \(A B\), has length 4 metres.
The rod is resting on a support at its midpoint \(C\).
A particle of mass 4 kg is placed 0.6 metres to the left of \(C\).
Another particle of mass 1.5 kg is placed \(x\) metres to the right of \(C\), as shown.
\includegraphics[max width=\textwidth, alt={}, center]{5e475c96-6bbb-44fb-a411-706dac20e2d6-16_277_908_1521_568} The rod is balanced in equilibrium at \(C\).
Find \(x\). Circle your answer.
[0pt] [1 mark]
\(1.8 \mathrm {~m} \quad 1.5 \mathrm {~m} \quad 1.75 \mathrm {~m} \quad 1.6 \mathrm {~m}\)
AQA Paper 2 2018 June Q12
12 The graph below shows the velocity of an object moving in a straight line over a 20 second journey.
\includegraphics[max width=\textwidth, alt={}, center]{5e475c96-6bbb-44fb-a411-706dac20e2d6-17_579_1682_406_169} 12
  1. Find the maximum magnitude of the acceleration of the object. 12
  2. The object is at its starting position at times \(0 , t _ { 1 }\) and \(t _ { 2 }\) seconds.
    Find \(t _ { 1 }\) and \(t _ { 2 }\)
AQA Paper 2 2018 June Q13
13 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A boy attempts to move a wooden crate of mass 20 kg along horizontal ground. The coefficient of friction between the crate and the ground is 0.85 13
  1. The boy applies a horizontal force of 150 N . Show that the crate remains stationary.
    13
  2. Instead, the boy uses a handle to pull the crate forward. He exerts a force of 150 N , at an angle of \(15 ^ { \circ }\) above the horizontal, as shown in the diagram.
    \includegraphics[max width=\textwidth, alt={}, center]{5e475c96-6bbb-44fb-a411-706dac20e2d6-19_244_915_408_561} Determine whether the crate remains stationary.
    Fully justify your answer.
AQA Paper 2 2018 June Q14
14 A quadrilateral has vertices \(A , B , C\) and \(D\) with position vectors given by $$\overrightarrow { O A } = \left[ \begin{array} { l } 3
5
1 \end{array} \right] , \overrightarrow { O B } = \left[ \begin{array} { r } - 1
2
7 \end{array} \right] , \overrightarrow { O C } = \left[ \begin{array} { l } 0
7
6 \end{array} \right] \text { and } \overrightarrow { O D } = \left[ \begin{array} { r } 4
10
0 \end{array} \right]$$ 14
  1. Write down the vector \(\overrightarrow { A B }\) 14
  2. Show that \(A B C D\) is a parallelogram, but not a rhombus.
AQA Paper 2 2018 June Q15
15 A driver is road-testing two minibuses, \(A\) and \(B\), for a taxi company. The performance of each minibus along a straight track is compared.
A flag is dropped to indicate the start of the test.
Each minibus starts from rest.
The acceleration in \(\mathrm { ms } ^ { - 2 }\) of each minibus is modelled as a function of time, \(t\) seconds, after the flag is dropped: The acceleration of \(\mathrm { A } = 0.138 t ^ { 2 }\)
The acceleration of \(\mathrm { B } = 0.024 t ^ { 3 }\)
15
  1. Find the time taken for A to travel 100 metres.
    Give your answer to four significant figures.
    15
  2. The company decides to buy the minibus which travels 100 metres in the shortest time. Determine which minibus should be bought.
    15
  3. The models assume that both minibuses start moving immediately when \(t = 0\) In light of this, explain why the company may, in reality, make the wrong decision.
    A particle is projected with an initial speed \(u\), at an angle of \(35 ^ { \circ }\) above the horizontal.
    It lands at a point 10 metres vertically below its starting position.
    The particle takes 1.5 seconds to reach the highest point of its trajectory.
AQA Paper 2 2018 June Q16
16
  1. \(\quad\) Find \(u\). 16 In this question use \(g = 9.81 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) In this question use \(\boldsymbol { g } = 9.81 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) 16
  2. Find the total time that the particle is in flight.
AQA Paper 2 2018 June Q17
17 A buggy is pulling a roller-skater, in a straight line along a horizontal road, by means of a connecting rope as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{5e475c96-6bbb-44fb-a411-706dac20e2d6-24_240_1006_402_516} The combined mass of the buggy and driver is 410 kg
A driving force of 300 N and a total resistance force of 140 N act on the buggy.
The mass of the roller-skater is 72 kg
A total resistance force of \(R\) newtons acts on the roller-skater.
The buggy and the roller-skater have an acceleration of \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
17
    1. Find \(R\).
      17
    		2. 17
    3. (ii) Find the tension in the rope.
      4. 17
    4. The roller-skater releases the rope at a point \(A\), when she reaches a speed of \(6 \mathrm {~ms} ^ { - 1 }\) She continues to move forward, experiencing the same resistance force.
      The driver notices a change in motion of the buggy, and brings it to rest at a distance of 20 m from \(A\). 17
      1. Determine whether the roller-skater will stop before reaching the stationary buggy.
        Fully justify your answer.
        17
    6. (ii) Explain the change in motion that the driver noticed.
AQA Paper 2 2019 June Q1
1 Identify the graph of \(y = 1 - | x + 2 |\) from the options below.
Tick ( \(\checkmark\) ) one box.
\includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-02_389_526_845_500}
B
\includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-02_362_442_1279_525}
\includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-02_113_116_977_1107}
C
\includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-02_496_704_1688_523}
D
\includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-02_474_686_2211_534}
AQA Paper 2 2019 June Q2
2 Simplify \(\sqrt { a ^ { \frac { 2 } { 3 } } \times a ^ { \frac { 2 } { 5 } } }\)
Circle your answer.
\(a ^ { \frac { 2 } { 15 } }\)
\(a ^ { \frac { 4 } { 15 } }\)
\(a ^ { \frac { 8 } { 15 } }\)
\(a ^ { \frac { 16 } { 15 } }\)
AQA Paper 2 2019 June Q3
3 Each of these functions has domain \(x \in \mathbb { R }\)
Which function does not have an inverse?
Circle your answer. $$\mathrm { f } ( x ) = x ^ { 3 } \quad \mathrm { f } ( x ) = 2 x + 1 \quad \mathrm { f } ( x ) = x ^ { 2 } \quad \mathrm { f } ( x ) = \mathrm { e } ^ { x }$$
AQA Paper 2 2019 June Q5
5 Solve the differential equation $$\frac { \mathrm { d } t } { \mathrm {~d} x } = \frac { \ln x } { x ^ { 2 } t } \quad \text { for } x > 0$$ given \(x = 1\) when \(t = 2\)
Write your answer in the form \(t ^ { 2 } = \mathrm { f } ( x )\)
AQA Paper 2 2019 June Q6
6 A curve has equation $$y = a \sin x + b \cos x$$ where \(a\) and \(b\) are constants. The maximum value of \(y\) is 4 and the curve passes through the point \(\left( \frac { \pi } { 3 } , 2 \sqrt { 3 } \right)\) as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-06_746_696_719_671} Find the exact values of \(a\) and \(b\).
AQA Paper 2 2019 June Q7
7
  1. Sketch the graph of any cubic function that has both three distinct real roots and a positive coefficient of \(x ^ { 3 }\)
    \includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-08_997_1004_406_518} 7
  2. The function \(\mathrm { f } ( x )\) is defined by $$\mathrm { f } ( x ) = x ^ { 3 } + 3 p x ^ { 2 } + q$$ where \(p\) and \(q\) are constants and \(p > 0\)
    7
    1. Show that there is a turning point where the curve crosses the \(y\)-axis.
      7
  4. (ii) The equation \(\mathrm { f } ( x ) = 0\) has three distinct real roots. By considering the positions of the turning points find, in terms of \(p\), the range of possible values of \(q\).
AQA Paper 2 2019 June Q8
8 Theresa bought a house on 2 January 1970 for \(\pounds 8000\). The house was valued by a local estate agent on the same date every 10 years up to 2010. The valuations are shown in the following table.
Year19701980199020002010
Valuation price\(\pounds 8000\)\(\pounds 19000\)\(\pounds 36000\)\(\pounds 82000\)\(\pounds 205000\)
The valuation price of the house can be modelled by the equation $$V = p q ^ { t }$$ where \(V\) pounds is the valuation price \(t\) years after 2 January 1970 and \(p\) and \(q\) are constants. 8
  1. Show that \(V = p q ^ { t }\) can be written as \(\log _ { 10 } V = \log _ { 10 } p + t \log _ { 10 } q\)
    8
  2. The values in the table of \(\log _ { 10 } V\) against \(t\) have been plotted and a line of best fit has been drawn on the graph below.
    \(t\)010203040
    \(\log _ { 10 } V\)3.904.284.564.915.31
    \includegraphics[max width=\textwidth, alt={}]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-11_1927_1207_625_338}
    Using the given line of best fit, find estimates for the values of \(p\) and \(q\). Give your answers correct to three significant figures.
    8
  3. Determine the year in which Theresa's house will first be worth half a million pounds. 8
  4. Explain whether your answer to part (c) is likely to be reliable.
AQA Paper 2 2019 June Q9
2 marks
9
  1. Show that the first two terms of the binomial expansion of \(\sqrt { 4 - 2 x ^ { 2 } }\) are $$2 - \frac { x ^ { 2 } } { 2 }$$ 9
  2. State the range of values of \(x\) for which the expansion found in part (a) is valid.
    [0pt] [2 marks]
    9
  3. Hence, find an approximation for $$\int _ { 0 } ^ { 0.4 } \sqrt { \cos x } \mathrm {~d} x$$ giving your answer to five decimal places.
    Fully justify your answer.
    9
  4. A student decides to use this method to find an approximation for $$\int _ { 0 } ^ { 1.4 } \sqrt { \cos x } \mathrm {~d} x$$ Explain why this may not be a suitable method.
AQA Paper 2 2019 June Q10
10 The diagram below shows a velocity-time graph for a particle moving with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds.
\includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-16_684_1653_847_182} Which statement is correct? Tick ( \(\checkmark\) ) one box. The particle was stationary for \(9 \leq t \leq 12\) □ The particle was decelerating for \(12 \leq t \leq 20\) □ The particle had a displacement of zero when \(t = 6\) □ The particle's speed when \(t = 4\) was \(- 12 \mathrm {~ms} ^ { - 1 }\) □
AQA Paper 2 2019 June Q11
11 A wooden crate rests on a rough horizontal surface.
The coefficient of friction between the crate and the surface is 0.6 A forward force acts on the crate, parallel to the surface.
When this force is 600 N , the crate is on the point of moving.
Find the weight of the crate.
Circle your answer. \(1000 \mathrm {~N} 360 \mathrm {~N} \quad 100 \mathrm {~kg} \quad 36 \mathrm {~kg}\)