Questions — AQA AS Paper 2 (143 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 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 2020 June Q7
2 marks Moderate -0.8
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. [2 marks]
AQA AS Paper 2 2020 June Q8
6 marks Moderate -0.3
  1. Using \(y = 2^{2x}\) as a substitution, show that $$16^x - 2^{(2x+3)} - 9 = 0$$ can be written as $$y^2 - 8y - 9 = 0$$ [2 marks]
  2. Hence, show that the equation $$16^x - 2^{(2x+3)} - 9 = 0$$ has \(x = \log_2 3\) as its only solution. Fully justify your answer. [4 marks]
AQA AS Paper 2 2020 June Q9
7 marks Moderate -0.3
    1. Find $$\int (4x - x^3) dx$$ [2 marks]
    2. Evaluate $$\int_{-2}^{2} (4x - x^3) dx$$ [1 mark]
  2. Using a sketch, explain why the integral in part (a)(ii) does not give the area enclosed between the curve \(y = 4x - x^3\) and the \(x\)-axis. [2 marks]
  3. Find the area enclosed between the curve \(y = 4x - x^3\) and the \(x\)-axis. [2 marks]
AQA AS Paper 2 2020 June Q10
8 marks Standard +0.3
A curve has gradient function $$\frac{dy}{dx} = 3x^2 - 12x + c$$ The curve has a turning point at \((-1, 1)\)
  1. Find the coordinates of the other turning point of the curve. Fully justify your answer. [6 marks]
  2. Find the set of values of \(x\) for which \(y\) is increasing. [2 marks]
AQA AS Paper 2 2020 June Q11
11 marks Moderate -0.8
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{figure_11} The path of the jet can be modelled by the equation $$y = -0.0125x^2 + 0.5x - 2.55$$ 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 \((a, 0)\)
    1. Find the value of \(a\). [3 marks]
    2. Find the horizontal distance from \(A\) to the point where the jet hits the ground. [1 mark]
  2. Calculate the maximum vertical height reached by the jet. [4 marks]
  3. 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. [3 marks]
AQA AS Paper 2 2020 June Q12
1 marks Easy -2.0
A student plots the scatter diagram below showing the mass in kilograms against the CO₂ emissions in grams per kilogram for a sample of cars in the Large Data Set. \includegraphics{figure_12} Their teacher tells them to remove an error to clean the data. Identify the data point which should be removed. Circle your answer below. [1 mark] \(A\) \quad \(B\) \quad \(C\) \quad \(D\)
AQA AS Paper 2 2020 June Q13
1 marks Easy -1.2
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. [1 mark] 9 \quad 12 \quad 18 \quad 72
AQA AS Paper 2 2020 June Q14
4 marks Easy -2.5
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 \quad Choose 100 people who work at the largest store Method B \quad Choose one person at random from each of the 100 stores Method C \quad 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.
  1. Give one disadvantage of using Method A compared with using Method B. [1 mark]
  2. Give one advantage of using Method B compared with using Method C. [1 mark]
    1. Identify the method of sampling used in Method C. [1 mark]
    2. Give a reason why Method C does not provide a random sample. [1 mark]
AQA AS Paper 2 2020 June Q15
3 marks Moderate -0.8
A random sample of ten CO₂ emissions was selected from the Large Data Set. The emissions in grams per kilogram were: 13 \quad 45 \quad 45 \quad 0 \quad 49 \quad 77 \quad 49 \quad 49 \quad 49 \quad 78
  1. Find the standard deviation of the sample. [1 mark]
  2. An environmentalist calculated the average CO₂ emissions for cars in the Large Data Set registered in 2002 and in 2016. The averages are listed below.
    Year of registration20022016
    Average CO₂ 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. [2 marks]
AQA AS Paper 2 2020 June Q16
4 marks Moderate -0.8
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.
  1. For a randomly chosen 14-day period find the probability that:
    1. Paula correctly solves exactly 8 puzzles [1 mark]
    2. Paula correctly solves at least 7 but not more than 11 puzzles. [2 marks]
  2. State one assumption that is necessary for the distribution used in part (a) to be valid. [1 mark]
AQA AS Paper 2 2020 June Q17
3 marks Easy -1.8
A game consists of spinning a circular wheel divided into numbered sectors as shown below. \includegraphics{figure_17} 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.
  1. Show that \(P(X = 1) = \frac{5}{18}\) [1 mark]
  2. Complete the probability distribution for \(X\) in the table below.
    \(x\)1
    \(P(X = x)\)\(\frac{5}{18}\)
    [2 marks]
AQA AS Paper 2 2020 June Q18
5 marks Moderate -0.8
  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. [2 marks]
  2. Bag A contains 7 blue discs, 4 red discs and 1 yellow disc. Bag B contains 3 blue discs and 6 red discs. A disc is drawn at random from Bag A and placed in Bag B. A disc is then drawn at random from Bag B. Find the probability that the disc drawn from Bag B is red. [3 marks]
AQA AS Paper 2 2020 June Q19
6 marks Moderate -0.3
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. [6 marks]
AQA AS Paper 2 2023 June Q1
1 marks Easy -1.8
Simplify \(\log_a 8^a\) Circle your answer. [1 mark] \(a^3\) \qquad \(2a\) \qquad \(3a\) \qquad \(8a\)
AQA AS Paper 2 2023 June Q2
1 marks Easy -1.8
It is given that \(\sin \theta = \frac{4}{5}\) and \(90° < \theta < 180°\) Find the value of \(\cos \theta\) Circle your answer. [1 mark] \(-\frac{3}{4}\) \qquad \(-\frac{3}{5}\) \qquad \(\frac{3}{5}\) \qquad \(\frac{3}{4}\)
AQA AS Paper 2 2023 June Q3
5 marks Moderate -0.8
  1. Find \(\int \left(2x^3 + \frac{8}{x^2}\right) dx\) [3 marks]
  2. A curve has gradient function \(\frac{dy}{dx} = 2x^3 + \frac{8}{x^2}\) The \(x\)-intercept of the curve is at the point \((2, 0)\) Find the equation of the curve. [2 marks]
AQA AS Paper 2 2023 June Q4
5 marks Moderate -0.3
Find the exact solution of the equation \(\ln(x + 1) + \ln(x - 1) = \ln 15 - 2\ln 7\) Fully justify your answer. [5 marks]
AQA AS Paper 2 2023 June Q5
4 marks Moderate -0.8
It is given that \(\sin 15° = \frac{\sqrt{6} - \sqrt{2}}{4}\) and \(\cos 15° = \frac{\sqrt{6} + \sqrt{2}}{4}\) Use these two expressions to show that \(\tan 15° = 2 - \sqrt{3}\) Fully justify your answer. [4 marks]
AQA AS Paper 2 2023 June Q6
5 marks Moderate -0.3
A curve has equation $$y = 2x^2 + px + 1$$ A line has equation $$y = 5x - 2$$ Find the set of values of \(p\) for which the line intersects the curve at two distinct points. Give your answer in exact form. [5 marks]
AQA AS Paper 2 2023 June Q7
3 marks Easy -1.3
The curve C has equation \(y = f(x)\) C has a maximum point at P with coordinates \((a, 2b)\) as shown in the diagram below. \includegraphics{figure_7}
  1. C is mapped by a single transformation onto curve \(C_1\) with equation \(y = f(x + 2)\) State the coordinates of the maximum point on curve \(C_1\) [1 mark]
  2. C is mapped by a single transformation onto curve \(C_2\) with equation \(y = 4f(x)\) State the coordinates of the maximum point on curve \(C_2\) [1 mark]
  3. C is mapped by a stretch in the \(x\)-direction onto curve \(C_3\) with equation \(y = f(3x)\) State the scale factor of the stretch. [1 mark]
AQA AS Paper 2 2023 June Q8
5 marks Standard +0.3
Prove that the sum of the cubes of two consecutive odd numbers is always a multiple of 4. [5 marks]
AQA AS Paper 2 2023 June Q9
6 marks Easy -1.2
A craft artist is producing items and selling them in a local market. The selling price, £P, of an item is inversely proportional to the number of items produced, \(n\)
  1. When \(n = 10\), \(P = 24\) Show that \(P = \frac{240}{n}\) [1 mark]
  2. The production cost, £C, of an item is inversely proportional to the square of the number of items produced, \(n\) When \(n = 10\), \(C = 60\) Find the set of values of \(n\) for which \(P > C\) [4 marks]
  3. Explain the significance to the craft artist of the range of values found in part (b). [1 mark]
AQA AS Paper 2 2023 June Q10
11 marks Standard +0.3
A piece of wire of length 66 cm is bent to form the five sides of a pentagon. The pentagon consists of three sides of a rectangle and two sides of an equilateral triangle. The sides of the rectangle measure \(x\) cm and \(y\) cm and the sides of the triangle measure \(x\) cm, as shown in the diagram below. \includegraphics{figure_10}
    1. You are given that \(\sin 60° = \frac{\sqrt{3}}{2}\) Explain why the area of the triangle is \(\frac{\sqrt{3}}{4}x^2\) [1 mark]
    2. Show that the area enclosed by the wire, \(A\) cm\(^2\), can be expressed by the formula $$A = 33x - \frac{1}{4}(6 - \sqrt{3})x^2$$ [3 marks]
  2. Use calculus to find the value of \(x\) for which the wire encloses the maximum area. Give your answer in the form \(p + q\sqrt{3}\), where \(p\) and \(q\) are integers. Fully justify your answer. [7 marks]
AQA AS Paper 2 2023 June Q11
7 marks Moderate -0.3
The line \(L_1\) has equation \(x + 7y - 41 = 0\) \(L_1\) is a tangent to the circle C at the point P(6, 5) The line \(L_2\) has equation \(y = x + 3\) \(L_2\) is a tangent to the circle C at the point Q(0, 3) The lines \(L_1\) and \(L_2\) and the circle C are shown in the diagram below. \includegraphics{figure_11}
  1. Show that the equation of the radius of the circle through P is \(y = 7x - 37\) [3 marks]
  2. Find the equation of C [4 marks]
AQA AS Paper 2 2023 June Q12
1 marks Easy -2.5
The mass of a bag of nuts produced by a company is known to have a mean of 40 grams and a standard deviation of 3 grams. The company produces five different flavours of nuts. The bags of nuts are packed in large boxes. Given the information above, identify the continuous variable from the options below. Tick (\(\checkmark\)) one box. [1 mark] The flavours of the bags of nuts The known standard deviation of the mass of a bag of nuts The mass of an individual bag of nuts The number of bags of nuts in a large box