Questions — AQA AS Paper 2 (137 questions)

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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 AS Paper 2 2019 June Q9
9
  1. Find the exact coordinates of the turning points of \(C\).
    Determine the nature of each turning point.
    Fully justify your answer.
    9
  2. State the coordinates of the turning points of the curve $$y = \mathrm { f } ( x + 1 ) - 4$$
AQA AS Paper 2 2019 June Q10
10 As part of an experiment, Zena puts a bucket of hot water outside on a day when the outside temperature is \(0 ^ { \circ } \mathrm { C }\). She measures the temperature of the water after 10 minutes and after 20 minutes. Her results are shown below.
Time (minutes)1020
Temperature (degrees Celsius)3012
Zena models the relationship between \(\theta\), the temperature of the water in \({ } ^ { \circ } \mathrm { C }\), and \(t\), the time in minutes, by $$\theta = A \times 10 ^ { - k t }$$ where \(A\) and \(k\) are constants. 10
  1. Using \(t = 0\), explain how the value of \(A\) relates to the experiment. 10
  2. Show that $$\log _ { 10 } \theta = \log _ { 10 } A - k t$$ 10
  3. Using Zena's results, calculate the values of \(A\) and \(k\).
    10
  4. Zena states that the temperature of the water will be less than \(1 ^ { \circ } \mathrm { C }\) after 45 minutes. Determine whether the model supports this statement.
    10
  5. Explain why Zena's model is unlikely to accurately give the value of \(\theta\) after 45 minutes.
AQA AS Paper 2 2019 June Q11
11 A survey is undertaken to find out the most popular political party in London.
The first 1100 available people from London are surveyed.
Identify the name of this type of sampling.
Circle your answer.
simple random
opportunity
stratified
quota
AQA AS Paper 2 2019 June Q12
12 Manny is studying the price and number of pages of a random sample of books.
He calculates the value of the product moment correlation coefficient between the price and number of pages in each book as 1.05 Which of the following best describes the value 1.05 ?
Tick ( \(\checkmark\) ) one box.
definitely correct □
probably correct □
probably incorrect □
definitely incorrect □
\includegraphics[max width=\textwidth, alt={}, center]{b45dc98e-1699-47c9-9228-5abe0e5c9195-15_2488_1716_219_153}
AQA AS Paper 2 2019 June Q13
13 Denzel wants to buy a car with a propulsion type other than petrol or diesel.
He takes a sample, from the Large Data Set, of the CO2 emissions, in \(\mathrm { g } / \mathrm { km }\), of cars with one particular propulsion type. The sample is as follows $$\begin{array} { l l l l l l l l } 82 & 13 & 96 & 49 & 96 & 92 & 70 & 81 \end{array}$$ 13
  1. Using your knowledge of the Large Data Set, state which propulsion type this sample is for, giving a reason for your answer.
    13
  2. Calculate the mean of the sample.
    13
  3. Calculate the standard deviation of the sample.
    13
  4. Denzel claims that the value 13 is an outlier. 13
    1. Any value more than 2 standard deviations from the mean can be regarded as an outlier. Verify that Denzel's claim is correct.
      13
  6. (ii) State what effect, if any, removing the value 13 from the sample would have on the standard deviation.
AQA AS Paper 2 2019 June Q14
14 A probability distribution is given by $$\mathrm { P } ( X = x ) = c ( 4 - x ) , \text { for } x = 0,1,2,3$$ where \(c\) is a constant.
14
  1. Show that \(c = \frac { 1 } { 10 }\)
    14
  2. Calculate \(\mathrm { P } ( X \geq 1 )\)
AQA AS Paper 2 2019 June Q15
15 Two independent events, \(A\) and \(B\), are such that $$\begin{aligned} \mathrm { P } ( A ) & = 0.2
\mathrm { P } ( A \cup B ) & = 0.8 \end{aligned}$$ 15
    1. Find \(\mathrm { P } ( B )\)
      15
  2. (ii) Find \(\mathrm { P } ( A \cap B )\)
    15
  3. State, with a reason, whether or not the events \(A\) and \(B\) are mutually exclusive.
    \begin{center} \begin{tabular}{|l|l|} \hline \begin{tabular}{l}
AQA AS Paper 2 2019 June Q16
7 marks
16
16

  1. \end{tabular} &
    Andrea is the manager of a company which makes mobile phone chargers.
    In the past, she had found that \(12 \%\) of all chargers are faulty.
    Andrea decides to move the manufacture of chargers to a different factory.
    Andrea tests 60 of the new chargers and finds that 4 chargers are faulty.
    Investigate, at the \(10 \%\) level of significance, whether the proportion of faulty chargers has reduced.
    [7 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

    \hline \end{tabular} \end{center} 16
  2. State, in context, two assumptions that are necessary for the distribution that you have used in part (a) to be valid.
AQA AS Paper 2 2020 June Q1
1 Identify the expression below that is equivalent to \(\mathrm { e } ^ { \frac { - 2 } { 5 } }\)
Circle your answer. $$\frac { 1 } { \sqrt [ 5 ] { e ^ { 2 } } } \quad - \sqrt { e ^ { 5 } } \quad - \sqrt [ 5 ] { e ^ { 2 } } \quad \frac { 1 } { \sqrt { e ^ { 5 } } }$$
AQA AS Paper 2 2020 June Q2
1 marks
2 It is given that \(y = \frac { 1 } { x }\) and \(x < - 1\)
Determine which statement below fully describes the possible values of \(y\).
Tick \(( \checkmark )\) one box.
[0pt] [1 mark] $$\begin{array} { l l } - \infty < y < - 1 & \square
y > - 1 & \square
- 1 < y < 0 & \square
y < 0 & \end{array}$$
AQA AS Paper 2 2020 June Q3
3 marks
3 It is given that $$y = 3 x ^ { 4 } + \frac { 2 } { x } - \frac { x } { 4 } + 1$$ Find an expression for \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
[0pt] [3 marks]
AQA AS Paper 2 2020 June Q5
5 Joseph is expanding \(( 2 - 3 x ) ^ { 7 }\) in ascending powers of \(x\). He states that the coefficient of the fourth term is 15120
Joseph's teacher comments that his answer is almost correct.
Using a suitable calculation, explain the teacher's comment.
AQA AS Paper 2 2020 June Q6
6 A circle has equation $$x ^ { 2 } + y ^ { 2 } + 10 x - 4 y - 71 = 0$$ 6
  1. Find the centre of the circle.
    6
  2. Hence, find the equation of the tangent to the circle at the point (1, 10), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
AQA AS Paper 2 2020 June Q7
7 The population of a country was 3.6 million in 1989. It grew exponentially to reach 6 million in 2019.
Estimate the population of the country in 2049 if the exponential growth continues unchanged.
AQA AS Paper 2 2020 June Q8
8
  1. Using \(y = 2 ^ { 2 x }\) as a substitution, show that $$16 ^ { x } - 2 ^ { ( 2 x + 3 ) } - 9 = 0$$ can be written as $$y ^ { 2 } - 8 y - 9 = 0$$ 8
  2. Hence, show that the equation $$16 ^ { x } - 2 ^ { ( 2 x + 3 ) } - 9 = 0$$ has \(x = \log _ { 2 } 3\) as its only solution.
    Fully justify your answer.
AQA AS Paper 2 2020 June Q9
2 marks
9
    1. Find $$\int \left( 4 x - x ^ { 3 } \right) d x$$ 9
  2. (ii) Evaluate $$\int _ { - 2 } ^ { 2 } \left( 4 x - x ^ { 3 } \right) \mathrm { d } x$$
    9
  3. Using a sketch, explain why the integral in part (a)(ii) does not give the area enclosed
    4. between the curve \(y = 4 x - x ^ { 3 }\) and the \(x\)-axis.
    between the curve \(y = 4 x - x ^ { 3 }\) and the \(x\)-axis. [2 marks] 9
  4. Find the area enclosed between the curve \(y = 4 x - x ^ { 3 }\) and the \(x\)-axis.
AQA AS Paper 2 2020 June Q10
10 A curve has gradient function $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 12 x + c$$ The curve has a turning point at ( \(- 1,1\) )
10
  1. Find the coordinates of the other turning point of the curve.
    Fully justify your answer.
    10
  2. Find the set of values of \(x\) for which \(y\) is increasing.
AQA AS Paper 2 2020 June Q11
11 A fire crew is tackling a grass fire on horizontal ground. The crew directs a single jet of water which flows continuously from point \(A\).
\includegraphics[max width=\textwidth, alt={}, center]{7cca79eb-fd09-4ec0-8a1d-a7a38ca73f7a-14_505_967_450_539} The path of the jet can be modelled by the equation $$y = - 0.0125 x ^ { 2 } + 0.5 x - 2.55$$ where \(x\) metres is the horizontal distance of the jet from the fire truck at \(O\) and \(y\) metres is the height of the jet above the ground. The coordinates of point \(A\) are ( \(a , 0\) )
11
    1. Find the value of \(a\).
      11
  2. (ii) Find the horizontal distance from \(\boldsymbol { A }\) to the point where the jet hits the ground.
    11
  3. Calculate the maximum vertical height reached by the jet.
    11
  4. A vertical wall is located 11 metres horizontally from \(A\) in the direction of the jet. The height of the wall is 2.3 metres. Using the model, determine whether the jet passes over the wall, stating any necessary modelling assumption.
AQA AS Paper 2 2020 June Q12
1 marks
12 A student plots the scatter diagram below showing the mass in kilograms against the \(\mathrm { CO } _ { 2 }\) emissions in grams per kilogram for a sample of cars in the Large Data Set.
\includegraphics[max width=\textwidth, alt={}, center]{7cca79eb-fd09-4ec0-8a1d-a7a38ca73f7a-16_947_1445_794_296} Their teacher tells them to remove an error to clean the data.
Identify the data point which should be removed.
Circle your answer below.
[0pt] [1 mark]
A
B
C
D
AQA AS Paper 2 2020 June Q13
13 The random variable \(X\) is such that \(X \sim B \left( n , \frac { 1 } { 3 } \right)\)
The standard deviation of \(X\) is 4 Find the value of \(n\). Circle your answer.
9121812 Turn over for the next question
AQA AS Paper 2 2020 June Q14
14 A retail company has 5200 employees in 100 stores throughout the United Kingdom. The company recently introduced a new reward scheme for its staff.
The management team wanted to sample the staff to find out their opinions of the new scheme. Three possible sampling methods were suggested:
Method A Choose 100 people who work at the largest store
Method B Choose one person at random from each of the 100 stores
Method C List all employees in alphabetical order and assign each a number from 1 to 5200 Choose a random number between 1 and 52
Choose this person and every 52nd person on the list thereafter. 14
  1. Give one disadvantage of using Method A compared with using Method B.
    14
  2. Give one advantage of using Method B compared with using Method C.
    14
    1. Identify the method of sampling used in Method C .
      14
  4. (ii) Give a reason why Method C does not provide a random sample.
AQA AS Paper 2 2020 June Q15
15 A random sample of ten \(\mathrm { CO } _ { 2 }\) emissions was selected from the Large Data Set. The emissions in grams per kilogram were: $$\begin{array} { l l l l l l l l l l } 13 & 45 & 45 & 0 & 49 & 77 & 49 & 49 & 49 & 78 \end{array}$$ 15
  1. Find the standard deviation of the sample. 15
  2. An environmentalist calculated the average \(\mathrm { CO } _ { 2 }\) emissions for cars in the Large Data Set registered in 2002 and in 2016. The averages are listed below.
    Year of registration20022016
    Average \(\mathbf { C O } _ { \mathbf { 2 } }\) emission171.2120.4
    The environmentalist claims that the average CO2 emissions for 2002 and 2016 combined is 145.8 Determine whether this claim is correct.
    Fully justify your answer.
    \begin{center} \begin{tabular}{|l|l|} \hline & \begin{tabular}{l}
AQA AS Paper 2 2020 June Q16
3 marks
16
A mathematical puzzle is published every day in a newspaper.
Over a long period of time Paula is able to solve the puzzle correctly \(60 \%\) of the time.
16
  1. For a randomly chosen 14-day period find the probability that:
    16

    1. Paula correctly solves exactly 8 puzzles
      [0pt] [1 mark]
      16
  3. (ii)
    Paula correctly solves at least 7 but not more than 11 puzzles.
    [0pt] [2 marks]
    16

  4. \end{tabular}
    \hline \end{tabular} \end{center}
AQA AS Paper 2 2020 June Q17
17 A game consists of spinning a circular wheel divided into numbered sectors as shown below.
\includegraphics[max width=\textwidth, alt={}, center]{7cca79eb-fd09-4ec0-8a1d-a7a38ca73f7a-22_764_963_404_541} On each spin the score, \(X\), is the value shown in the sector that the arrow points to when the spinner stops. The probability of the arrow pointing at a sector is proportional to the angle subtended at the centre by that sector. 17
  1. Show that \(\mathrm { P } ( X = 1 ) = \frac { 5 } { 18 }\)
    17
  2. Complete the probability distribution for \(X\) in the table below.
    \(\boldsymbol { x }\)1
    \(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)\(\frac { 5 } { 18 }\)
AQA AS Paper 2 2020 June Q18
18
  1. Bag A contains 7 blue discs, 4 red discs and 1 yellow disc. Two discs are drawn at random from bag A without replacement.
    Find the probability that exactly one of the discs is blue.
    18
  2. Bag A contains 7 blue discs, 4 red discs and 1 yellow disc.