Questions — AQA AS Paper 2 (137 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 AS Paper 2 2020 June Q19
19 It is known from historical data that 15\% of the residents of a town buy the local weekly newspaper, 'Local News'. A new free weekly paper is introduced into the town.
The owners of 'Local News' are interested to know whether the introduction of the free newspaper has changed the proportion of residents who buy their paper. In a random sample of 50 residents of the town taken after the free newspaper was introduced, it was found that 3 of them purchased 'Local News' regularly. Investigate, at the \(5 \%\) significance level, whether this sample provides evidence that the proportion of local residents who buy 'Local News' has changed.
\includegraphics[max width=\textwidth, alt={}, center]{7cca79eb-fd09-4ec0-8a1d-a7a38ca73f7a-27_2492_1721_217_150}
\includegraphics[max width=\textwidth, alt={}]{7cca79eb-fd09-4ec0-8a1d-a7a38ca73f7a-32_2486_1719_221_150}
AQA AS Paper 2 2021 June Q1
1 Express as a single power of \(a\) $$\frac { a ^ { 2 } } { \sqrt { a } }$$ where \(a \neq 0\) Circle your answer.
\(a ^ { 1 }\)
\(a ^ { \frac { 3 } { 2 } }\)
\(a ^ { \frac { 5 } { 2 } }\)
\(a ^ { 4 }\)
AQA AS Paper 2 2021 June Q2
2 One of the diagrams below shows the graph of \(y = \sin \left( x + 90 ^ { \circ } \right)\) for \(0 ^ { \circ } \leq x \leq 360 ^ { \circ }\) Identify the correct graph. Tick ( \(\checkmark\) ) one box.
\includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-03_451_465_568_497}
\includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-03_124_154_724_1073}
\includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-03_458_472_1105_495}
□
\includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-03_453_468_1647_497}
\includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-03_117_132_1809_1091}
\includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-03_461_479_2183_488}
AQA AS Paper 2 2021 June Q3
3 marks
3 It is given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \sqrt { x }$$ Find an expression for \(y\).
[0pt] [3 marks]
L
AQA AS Paper 2 2021 June Q4
2 marks
4
  1. Find the binomial expansion of \(( 1 - 2 x ) ^ { 5 }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\) 4
  2. Find the first two non-zero terms in the expansion of $$( 1 - 2 x ) ^ { 5 } + ( 1 + 5 x ) ^ { 2 }$$ in ascending powers of \(x\).
    4
  3. Hence, use an appropriate value of \(x\) to obtain an approximation for \(0.998 ^ { 5 } + 1.005 ^ { 2 }\) [2 marks]
    \(5 A B C\) is a triangle. The point \(D\) lies on \(A C\).
    \(A B = 8 \mathrm {~cm} , B C = B D = 7 \mathrm {~cm}\) and angle \(A = 60 ^ { \circ }\) as shown in the diagram.
    \includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-06_604_978_486_532}
AQA AS Paper 2 2021 June Q5
5
  1. Using the cosine rule, find the length of \(A C\).
    5
  2. Hence, state the length of \(A D\).
AQA AS Paper 2 2021 June Q6
6 Find the solution to $$5 ^ { ( 2 x + 4 ) } = 9$$ giving your answer in the form \(a + \log _ { 5 } b\), where \(a\) and \(b\) are integers.
AQA AS Paper 2 2021 June Q7
7 The diagram below shows the graph of the curve that has equation \(y = x ^ { 2 } - 3 x + 2\) along with two shaded regions.
\includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-08_646_711_408_667} 7
  1. State the coordinates of the points \(A , B\) and \(C\).
    7
  2. Katy is asked by her teacher to find the total area of the two shaded regions.
    Katy uses her calculator to find \(\int _ { 0 } ^ { 2 } \left( x ^ { 2 } - 3 x + 2 \right) \mathrm { d } x\) and gets the answer \(\frac { 2 } { 3 }\)
    Katy's teacher says that her answer is incorrect.
    7
    1. Show that the total area of the two shaded regions is 1
      Fully justify your answer.
      7
  4. (ii) Explain why Katy's method was not valid.
AQA AS Paper 2 2021 June Q8
4 marks
8 It is given that \(y = 3 x - 5 x ^ { 2 }\) Use differentiation from first principles to find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
[0pt] [4 marks]
LIH
AQA AS Paper 2 2021 June Q9
9
  1. Express \(n ^ { 3 } - n\) as a product of three factors. 9
  2. Given that \(n\) is a positive integer, prove that \(n ^ { 3 } - n\) is a multiple of 6
AQA AS Paper 2 2021 June Q10
10 A square sheet of metal has edges 30 cm long. Four squares each with edge \(x \mathrm {~cm}\), where \(x < 15\), are removed from the corners of the sheet. The four rectangular sections are bent upwards to form an open-topped box, as shown in the diagrams.
\includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-12_392_460_630_347}
\includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-12_387_437_635_872}
\includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-12_282_380_703_1318} 10
  1. Show that the capacity, \(C \mathrm {~cm} ^ { 3 }\), of the box is given by $$C = 900 x - 120 x ^ { 2 } + 4 x ^ { 3 }$$ 10
  2. Find the maximum capacity of the box. Fully justify your answer.
AQA AS Paper 2 2021 June Q11
11 A circle \(C\) has centre \(( 0,10 )\) and radius \(\sqrt { 20 }\) A line \(L\) has equation \(y = m x\)
11
    1. Show that the \(x\)-coordinate of any point of intersection of \(L\) and \(C\) satisfies the equation $$\left( 1 + m ^ { 2 } \right) x ^ { 2 } - 20 m x + 80 = 0$$ 11
  2. (ii) Find the values of \(m\) for which the equation in part (a)(i) has equal roots.
    11
  3. Two lines are drawn from the origin which are tangents to \(C\). Find the coordinates of the points of contact between the tangents and \(C\).
AQA AS Paper 2 2021 June Q12
1 marks
12 The table below shows the total monthly rainfall (in mm ) in England and Wales in a sample of six years. The sample of six years was taken from a data set covering every year from 1768 to 2018.
JanFebMarAprMayJunJulAugSepOctNovDec
1768109.2129.112.885.646.1148.7121.991.6136.8119.4142.5103.6
181898.065.8134.7135.655.931.250.421.0115.675.8112.046.8
186899.962.271.161.436.716.520.0106.790.295.661.4185.6
191891.261.636.763.358.530.9110.062.9189.569.166.3122.5
196885.847.659.568.878.794.0107.872.2148.199.069.684.2
2018104.552.8115.191.451.916.539.676.767.075.8104.9116.0
Deduce the sampling method most likely to have been used to collect this sample. Circle your answer.
[0pt] [1 mark] Opportunity
Simple Random
Stratified
Systematic
AQA AS Paper 2 2021 June Q13
13 The diagram below shows the probability distribution for a discrete random variable \(Y\).
\includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-17_816_1338_356_351} Find \(\mathrm { P } ( 0 < Y \leq 3 )\).
Circle your answer. \(0.40 \quad 0.42 \quad 0.58 \quad 0.66\)
AQA AS Paper 2 2021 June Q14
14 The random variable \(T\) follows a binomial distribution where $$T \sim \mathrm {~B} ( 16,0.3 )$$ The mean of \(T\) is denoted by \(\mu\).
14
  1. \(\quad\) Find \(\mathrm { P } ( T \leq \mu )\).
    14
  2. Find the variance of \(T\).
    \includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-19_2488_1716_219_153}
AQA AS Paper 2 2021 June Q15
1 marks
15
The number of hours of sunshine and the daily maximum temperature were recorded over a 9-day period in June at an English seaside town. A scatter diagram representing the recorded data is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-20_872_1511_488_264} One of the points on the scatter diagram is an error. 15
    1. Write down the letter that identifies this point.
      15
  2. (ii) Suggest one possible action that could be taken to deal with this error.
    15
  3. It is claimed that the scatter diagram proves that longer hours of sunshine cause
    higher maximum daily temperatures. Comment on the validity of this claim.
    [0pt] [1 mark]
AQA AS Paper 2 2021 June Q16
16 An analysis was carried out using the Large Data Set to compare the \(\mathrm { CO } _ { 2 }\) emissions (in g/km) from 2002 and 2016. The summary statistics for the \(\mathrm { CO } _ { 2 }\) emissions, \(X\), for all cars registered as owned by either females or males is given in the table below.
\cline { 2 - 3 } \multicolumn{1}{c|}{}\(\mathbf { 2 0 0 2 }\)\(\mathbf { 2 0 1 6 }\)
\(\sum \boldsymbol { x }\)207901142103
Sample size12151144
16
  1. Find the reduction in the mean of the \(\mathrm { CO } _ { 2 }\) emissions in 2016 compared to the mean of the CO2 emissions in 2002.
    16
  2. It is claimed that the move to more electric and gas/petrol powered cars has caused the reduction in the mean \(\mathrm { CO } _ { 2 }\) emissions found in part (a). Using your knowledge of the Large Data Set, state whether you agree with this claim.
    Give a reason for your answer.
    16
  3. There are 3827 data values in the Large Data Set. It is claimed that the data in the table above must have been summarised incorrectly.
    16
    1. Explain why this claim is being made. 16
  5. (ii) State whether this claim is correct.
    Give a reason for your answer.
AQA AS Paper 2 2021 June Q17
17 The number of toilets in each of a random sample of 200 properties from a town was recorded. Four types of properties were included: terraced, semi-detached, detached and apartment. The data is summarised in the table below.
\multirow{2}{*}{}Number of toilets
OneTwoThree
Terraced20104
Semi-Detached185016
Detached12108
Apartment22300
One of the properties is selected at random.
\(A\) is the event 'the property has exactly two toilets'.
\(B\) is the event 'the property is detached'.
17
    1. Find \(\mathrm { P } ( A )\). 17
  2. (ii) Find \(\mathrm { P } \left( A ^ { \prime } \cap B \right)\). 17
  3. (iii) Find \(\mathrm { P } ( A \cup B )\).
    17
  4. Determine whether events \(A\) and \(B\) are independent.
    Fully justify your answer.
    17
  5. Using the table, write down two events, other than event \(\boldsymbol { A }\) and event \(\boldsymbol { B }\), which are mutually exclusive. Event 1 \(\_\_\_\_\) \section*{Event 2}
AQA AS Paper 2 2021 June Q18
1 marks
18 It is known from previous data that 14\% of the visitors to a particular cookery website are under 30 years of age. To encourage more visitors under 30 years of age a large advertising campaign took place to target this age group. To test whether the campaign was effective, a sample of 60 visitors to the website was selected. It was found that 15 of the visitors were under 30 years of age. 18
  1. Explain why a one-tailed hypothesis test should be used to decide whether the sample provides evidence that the campaign was effective. 18
  2. Carry out the hypothesis test at the \(5 \%\) level of significance to investigate whether the sample provides evidence that the proportion of visitors under 30 years of age has increased.
    18
  3. State one necessary assumption about the sample for the distribution used in part (b) to be valid.
    [0pt] [1 mark]
AQA AS Paper 2 2022 June Q1
1 Find \(\int 12 x ^ { 3 } \mathrm {~d} x\)
Circle your answer.
\(36 x ^ { 2 } + c\)
\(3 x ^ { 4 } + c\)
\(3 x ^ { 2 } + c\)
\(36 x ^ { 4 } + c\)
AQA AS Paper 2 2022 June Q2
1 marks
2 Given that $$\cos \left( \theta - 20 ^ { \circ } \right) = \cos 60 ^ { \circ }$$ which one of the following is a possible value for \(\theta\) ?
Circle your answer.
[0pt] [1 mark]
\(40 ^ { \circ }\)
\(140 ^ { \circ }\)
\(280 ^ { \circ }\)
\(320 ^ { \circ }\)
AQA AS Paper 2 2022 June Q3
3 A curve has equation \(y = k \sqrt { x }\) where \(k\) is a constant. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(( 4,2 k )\) on the curve, giving your answer as an expression in terms of \(k\).
AQA AS Paper 2 2022 June Q4
4 marks
4 The equation \(9 x ^ { 2 } + 4 x + p ^ { 2 } = 0\) has no real solutions for \(x\). Find the set of possible values of \(p\).
Fully justify your answer.
[0pt] [4 marks]
AQA AS Paper 2 2022 June Q5
5 Kaya is investigating the function $$f ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } - 12 x + 45$$ Kaya makes two statements.
Statement 1: \(\mathrm { f } ( 3 ) = 0\)
Statement 2: this shows that ( \(x + 3\) ) must be a factor of \(\mathrm { f } ( x )\).
5
  1. State, with a reason, whether each of Kaya's statements is correct. Statement 1: \(\_\_\_\_\)
    Statement 2: \(\_\_\_\_\)
    5
  2. Fully factorise f (x).
AQA AS Paper 2 2022 June Q6
6 An on-line science website states:
'To find a dog's equivalent human age in years, multiply the natural logarithm of the dog's age in years by 16 then add 31.' 6
  1. Calculate the equivalent age to the nearest human year of a dog aged 5 years. 6
  2. A dog's equivalent age in human years is 40 years. Find the dog's actual age to the nearest month.
    6
  3. Explain why the behaviour of the natural logarithm for values close to zero means that the formula given on the website cannot be true for very young dogs.