Questions — AQA AS Paper 2 (143 questions)

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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 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
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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 PURE 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 PURE S1 S2 S3 S4 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 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 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
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AQA AS Paper 2 2019 June Q1
1 marks Easy -1.8
1 Find the gradient of the curve \(y = \mathrm { e } ^ { - 3 x }\) at the point where it crosses the \(y\)-axis. Circle your answer. \(\begin{array} { l l l } - 3 & - 1 & 1 \end{array}\)
AQA AS Paper 2 2019 June Q2
1 marks Easy -1.3
2 Find the centre of the circle \(x ^ { 2 } + y ^ { 2 } + 4 x - 6 y = 12\) Tick ( \(\checkmark\) ) one box.
(-2, -3) □
(-2, 3) □ \(( 2 , - 3 )\) □ \(( 2,3 )\) □
AQA AS Paper 2 2019 June Q3
2 marks Moderate -0.8
3 It is given that \(\sin \theta = - 0.1\) and \(180 ^ { \circ } < \theta < 270 ^ { \circ }\) Find the exact value of \(\cos \theta\)
AQA AS Paper 2 2019 June Q4
4 marks Moderate -0.8
4 Show that, for \(x > 0\) $$\log _ { 10 } \frac { x ^ { 4 } } { 100 } + \log _ { 10 } 9 x - \log _ { 10 } x ^ { 3 } \equiv 2 \left( - 1 + \log _ { 10 } 3 x \right)$$
AQA AS Paper 2 2019 June Q5
4 marks Moderate -0.3
5 A triangular prism has a cross section \(A B C\) as shown in the diagram below. Angle \(A B C = 25 ^ { \circ }\) Angle \(A C B = 30 ^ { \circ }\) \(B C = 40\) millimetres. The length of the prism is 300 millimetres.
Calculate the volume of the prism, giving your answer to three significant figures.
AQA AS Paper 2 2019 June Q6
5 marks Moderate -0.3
6 A curve has equation \(y = \frac { 2 } { x \sqrt { x } }\) \includegraphics[max width=\textwidth, alt={}, center]{b45dc98e-1699-47c9-9228-5abe0e5c9195-05_508_549_420_744} The region enclosed between the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = a\) has area 3 units. Given that \(a > 1\), find the value of \(a\).
Fully justify your answer.
AQA AS Paper 2 2019 June Q7
6 marks Moderate -0.3
7 The points \(A ( a , 3 )\) and \(B ( 10,6 )\) lie on a circle. \(A B\) is a diameter of the circle and passes through the point ( 2,4 )
The circle has equation $$( x - c ) ^ { 2 } + ( y - d ) ^ { 2 } = e$$ where \(c , d\) and \(e\) are rational numbers. Find the values of \(a , c , d\) and \(e\).
AQA AS Paper 2 2019 June Q8
10 marks Standard +0.3
8 A curve has equation $$y = x ^ { 3 } + p x ^ { 2 } + q x - 45$$ The curve passes through point \(R ( 2,3 )\) The gradient of the curve at \(R\) is 8
8
  1. Find the value of \(p\) and the value of \(q\).
    8
  2. Calculate the area enclosed between the normal to the curve at \(R\) and the coordinate 8 (b) axes. \(9 \quad\) A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = ( x - 2 ) ( x - 3 ) ^ { 2 }$$
AQA AS Paper 2 2019 June Q9
10 marks Moderate -0.3
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 marks Moderate -0.3
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
1 marks Easy -2.0
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
1 marks Easy -1.8
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
6 marks Easy -1.2
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 (d) (i) Any value more than 2 standard deviations from the mean can be regarded as an outlier. Verify that Denzel's claim is correct.
    13 (d) (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
4 marks Easy -1.2
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
6 marks Moderate -0.3
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
      1. (ii) Find \(\mathrm { P } ( A \cap B )\) 15
    2. State, with a reason, whether or not the events \(A\) and \(B\) are mutually exclusive.
AQA AS Paper 2 2019 June Q16
9 marks Moderate -0.3
16
16
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] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
16
  • 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 2021 June Q1
    1 marks Easy -1.8
    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
    1 marks Easy -1.3
    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 Easy -1.8
    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
    6 marks Moderate -0.8
    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
    4 marks Easy -1.2
    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
    3 marks Easy -1.2
    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
    8 marks Moderate -0.3
    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 (b) (i) Show that the total area of the two shaded regions is 1
      Fully justify your answer.
      7 (b) (ii) Explain why Katy's method was not valid.
    AQA AS Paper 2 2021 June Q8
    4 marks Moderate -0.3
    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
    4 marks Standard +0.3
    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