Questions — AQA AS Paper 2 (137 questions)

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 2023 June Q17
3 marks
17 An archer is training for the Olympics.
Each of the archer's training sessions consists of 30 attempts to hit the centre of a target. The archer consistently hits the centre of the target with \(79 \%\) of their attempts.
It can be assumed that the number of times the centre of the target is hit in any training session can be modelled by a binomial distribution. 17
  1. Find the mean of the number of times that the archer hits the centre of the target during a training session. 17
  2. Find the probability that the archer hits the centre of the target exactly 22 times during a particular training session. 17
  3. Find the probability that the archer hits the centre of the target 18 times or less during a particular training session.
    [0pt] [1 mark] 17
  4. Find the probability that the archer hits the centre of the target more than 26 times in a training session.
    [0pt] [2 marks]
AQA AS Paper 2 2023 June Q18
18
  1. State the Null and Alternative hypotheses for this test. 18
  2. State, in context, the conclusion to this test. 18 It is believed that 25\% of the customers at a bakery buy a loaf of bread. is believed that \(25 \%\) of the customers at a bakery buy a loaf of bread.
    \includegraphics[max width=\textwidth, alt={}, center]{e3635007-2ad1-4b2a-b937-41fe90bb1111-27_2488_1719_219_150}
AQA AS Paper 2 2023 June Q19
19 A comparison of the masses (in kg ) of convertible cars was made using the Large Data Set. A sample of 20 masses was chosen from both the 2002 data and the 2016 data.
The masses of the 20 cars in each sample were used to create a box plot for each year. The box plots were labelled Box Plot A and Box Plot B as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{e3635007-2ad1-4b2a-b937-41fe90bb1111-28_1109_1660_751_191} 19
  1. Estimate the median of the masses from Box Plot A
    19
  2. It is claimed that Box Plot B must be incorrectly drawn.
    19
    1. Give a reason why this claim was made.
      19
  3. (ii) Comment on the validity of this claim.
    19
  4. It is claimed that Box Plot B must be from the 2002 data. Give a reason why this claim is correct.
    \includegraphics[max width=\textwidth, alt={}, center]{e3635007-2ad1-4b2a-b937-41fe90bb1111-30_2492_1721_217_150}
AQA AS Paper 2 2024 June Q1
1 Line \(L\) has equation $$5 y = 4 x + 6$$ Find the gradient of a line parallel to line \(L\)
Circle your answer.
\(- \frac { 5 } { 4 }\)
\(- \frac { 4 } { 5 }\)
\(\frac { 4 } { 5 }\)
\(\frac { 5 } { 4 }\)
AQA AS Paper 2 2024 June Q2
2 One of the equations below is true for all values of \(x\)
Identify the correct equation.
Tick \(( \checkmark )\) one box.
\(\cos ^ { 2 } x = - 1 - \sin ^ { 2 } x\) □
\(\cos ^ { 2 } x = - 1 + \sin ^ { 2 } x\)
\includegraphics[max width=\textwidth, alt={}, center]{f5e0d980-4c50-4735-aea7-1bdf448a58f7-02_113_113_1658_790}
\(\cos ^ { 2 } x = 1 - \sin ^ { 2 } x\)
\includegraphics[max width=\textwidth, alt={}, center]{f5e0d980-4c50-4735-aea7-1bdf448a58f7-02_113_113_1813_790}
\(\cos ^ { 2 } x = 1 + \sin ^ { 2 } x\) □
AQA AS Paper 2 2024 June Q3
4 marks
3 It is given that $$3 \log _ { a } x = \log _ { a } 72 - 2 \log _ { a } 3$$ Solve the equation to find the value of \(x\)
Fully justify your answer.
[0pt] [4 marks]
AQA AS Paper 2 2024 June Q4
4 Curve \(C\) has equation \(y = 8 \sin x\) 4
  1. Curve \(C\) is transformed onto curve \(C _ { 1 }\) by a translation of vector \(\left[ \begin{array} { l } 0
    4 \end{array} \right]\)
    Find the equation of \(C _ { 1 }\) 4
  2. \(\quad\) Curve \(C\) is transformed onto curve \(C _ { 2 }\) by a stretch of scale factor 4 in the \(y\) direction. Find the equation of \(C _ { 2 }\) 4
  3. Curve \(C\) is transformed onto curve \(C _ { 3 }\) by a stretch of scale factor 2 in the \(x\) direction. Find the equation of \(C _ { 3 }\)
AQA AS Paper 2 2024 June Q5
5 A student suggests that for any positive integer \(n\) the value of the expression $$4 n ^ { 2 } + 3$$ is always a prime number.
Prove that the student's statement is false by finding a counter example.
Fully justify your answer.
AQA AS Paper 2 2024 June Q6
3 marks
6 In the expansion of \(( 3 + a x ) ^ { n }\), where \(a\) and \(n\) are integers, the coefficient of \(x ^ { 2 }\) is 4860 6
  1. Show that $$3 ^ { n } a ^ { 2 } n ( n - 1 ) = 87480$$ [3 marks]
    6
  2. The constant term in the expansion is 729 The coefficient of \(x\) in the expansion is negative. 6
    1. Verify that \(n = 6\)
      6
  3. (ii) Find the value of \(a\)
AQA AS Paper 2 2024 June Q7
4 marks
7
  1. Find the equation of the perpendicular bisector of \(A B\)
    7
  2. \(\quad\) A circle passes through the points \(A\) and \(B\) A diameter of the circle lies along the \(x\)-axis.
    Find the equation of the circle.
    [0pt] [4 marks]
AQA AS Paper 2 2024 June Q8
5 marks
8 Prove that the graph of the curve with equation $$y = x ^ { 3 } + 15 x - \frac { 18 } { x }$$ has no stationary points.
[0pt] [5 marks]
\includegraphics[max width=\textwidth, alt={}, center]{f5e0d980-4c50-4735-aea7-1bdf448a58f7-11_2491_1753_173_123}
AQA AS Paper 2 2024 June Q9
6 marks
9 A curve has equation $$y = x - a \sqrt { x } + b$$ where \(a\) and \(b\) are constants. The curve intersects the line \(y = 2\) at points with coordinates \(( 1,2 )\) and \(( 9,2 )\), as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{f5e0d980-4c50-4735-aea7-1bdf448a58f7-12_696_807_641_605} 9
  1. Show that \(a\) has the value 4 and find the value of \(b\)
    9
  2. On the diagram, the region enclosed between the curve and the line \(y = 2\) is shaded. Show that the area of this shaded region is \(\frac { 16 } { 3 }\) Fully justify your answer.
    [0pt] [6 marks]
AQA AS Paper 2 2024 June Q10
4 marks
10
  1. (ii) Using the graph, estimate the value of the constant \(a\) and the value of the constant \(k\) [4 marks]
    \hline \end{tabular} \end{center} 10
    1. Show that \(\frac { \mathrm { d } F } { \mathrm {~d} t } = k F\)
      10
  2. (ii) Using the model, estimate the rate at which the number of followers is increasing 5 days after the song is released.
    10
  3. The singer claims that 30 days after the song is released, the account will have more than a billion followers. Comment on the singer's claim.
AQA AS Paper 2 2024 June Q11
11 The table below shows the daily salt intake, \(x\) grams, and the daily Vitamin C intake, \(y\) milligrams, for a group of 10 adults.
AdultABCDEFGHIJ
\(\boldsymbol { x }\)5.36.23.610.42.49.4657.111.2
\(y\)9014588481144480955541
A scatter diagram of the data is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{f5e0d980-4c50-4735-aea7-1bdf448a58f7-17_675_1150_1110_431} One of the adults is an outlier. Identify the letter of the adult that is the outlier.
Circle your answer below.
A
B
E
J Which one of the following is not a measure of spread?
Circle your answer.
median
range
standard deviation
variance
AQA AS Paper 2 2024 June Q13
13 The headteacher of a school wishes to collect the opinions of the students on a new timetable structure. To do this, a random sample of size 50 , stratified by year group, will be selected.
The school has a total of 720 students. The number of students in each of the year groups at this school is shown below.
Year group10111213
Number of students200240150130
13
  1. Find the number of students from each year group that should be selected in the stratified random sample.
    13
  2. State one advantage of using a stratified random sample.
AQA AS Paper 2 2024 June Q14
14 The discrete random variables \(X\) and \(Y\) can be modelled by the distributions $$\begin{gathered} X \sim \mathrm {~B} ( 40 , p )
Y \sim \mathrm {~B} ( 25,0.6 ) \end{gathered}$$ It is given that the mean of \(X\) is equal to the variance of \(Y\) 14
  1. Find the value of \(p\)
    14
  2. \(\quad\) Find \(\mathrm { P } ( Y = 17 )\)
AQA AS Paper 2 2024 June Q15
3 marks
15 The number of flowers which grow on a certain type of plant can be modelled by the discrete random variable \(X\) The probability distribution of \(X\) is given in the table below.
\(\boldsymbol { x }\)012345 or more
\(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)0.030.150.220.310.09\(p\)
15
  1. Find the value of \(p\)
    15
  2. Two plants of this type are randomly selected from a large batch received from a local garden centre. Find the probability that the two plants will produce a total of three flowers.
    [0pt] [3 marks]
    15
    1. State one assumption necessary for the calculation in part (b) to be valid. 15
  3. (ii) Comment on whether, in reality, this assumption is likely to be valid.
AQA AS Paper 2 2024 June Q16
2 marks
16
  1. (ii) &
    16
    where \(n\) is the total number of cars which had a measured hydrocarbon emission in the Large Data Set.
    16
  2. Find the mean of \(X\)
    [1 mark]
    16

  3. \hline &
    \hline \end{tabular} \end{center} 16
  • (ii) State one type of emission where more than 80\% of the data is known for cars in the entire UK Department for Transport Stock Vehicle Database.
    [0pt] [1 mark]
  • AQA AS Paper 2 2024 June Q17
    17 The proportion of vegans in a city is thought to be 8\% The owner of an organic food café in this city believes that the proportion of their customers who are vegan is greater than \(8 \%\) To test this belief, a random sample of 50 customers at the café were interviewed and it was found that 7 of them were vegan. Investigate, at the \(5 \%\) level, whether this sample supports the owner's belief.
    \includegraphics[max width=\textwidth, alt={}, center]{f5e0d980-4c50-4735-aea7-1bdf448a58f7-25_2498_1915_166_123}
    AQA AS Paper 2 Specimen Q1
    1 marks
    1 \(\mathrm { p } ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 3 x + a\), where \(a\) is a constant.
    Given that \(x - 3\) is a factor of \(\mathrm { p } ( x )\), find the value of \(a\)
    Circle your answer.
    [0pt] [1 mark]
    \(- 9 - 339\)
    AQA AS Paper 2 Specimen Q2
    1 marks
    2 The graph of \(y = \mathrm { f } ( x )\) is shown in Figure 1. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{f2bf5e19-98ba-4047-9023-3cfe20987e01-03_536_849_664_735}
    \end{figure} State the equation of the graph shown in Figure 2. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{f2bf5e19-98ba-4047-9023-3cfe20987e01-03_532_851_1530_733}
    \end{figure} Circle your answer.
    [0pt] [1 mark] $$y = \mathrm { f } ( 2 x ) \quad y = \mathrm { f } \left( \frac { x } { 2 } \right) \quad y = 2 \mathrm { f } ( x ) \quad y = \frac { 1 } { 2 } \mathrm { f } ( x )$$
    AQA AS Paper 2 Specimen Q3
    2 marks
    3 Find the value of \(\log _ { a } \left( a ^ { 3 } \right) + \log _ { a } \left( \frac { 1 } { a } \right)\)
    [0pt] [2 marks]
    AQA AS Paper 2 Specimen Q4
    3 marks
    4 Find the coordinates, in terms of \(a\), of the minimum point on the curve \(y = x ^ { 2 } - 5 x + a\), where \(a\) is a constant. Fully justify your answer.
    [0pt] [3 marks]
    AQA AS Paper 2 Specimen Q5
    4 marks
    5 The quadratic equation \(3 x ^ { 2 } + 4 x + ( 2 k - 1 ) = 0\) has real and distinct roots.
    Find the possible values of the constant \(k\)
    Fully justify your answer.
    [0pt] [4 marks]
    AQA AS Paper 2 Specimen Q6
    4 marks
    6 A curve has equation \(y = 6 x ^ { 2 } + \frac { 8 } { x ^ { 2 } }\) and is sketched below for \(x > 0\)
    \includegraphics[max width=\textwidth, alt={}, center]{f2bf5e19-98ba-4047-9023-3cfe20987e01-06_638_842_539_758} Find the area of the region bounded by the curve, the \(x\)-axis and the lines \(x = a\) and \(x = 2 a\), where \(a > 0\), giving your answer in terms of \(a\)
    [0pt] [4 marks]