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 2018 June Q1
1 marks Easy -1.8
Given that \(\frac{dy}{dx} = \frac{1}{6x^2}\), find \(y\). Circle your answer. \(\frac{-1}{3x^3} + c\) \quad \(\frac{1}{2x^3} + c\) \quad \(\frac{-1}{6x} + c\) \quad \(\frac{-1}{3x} + c\) [1 mark]
AQA AS Paper 2 2018 June Q2
1 marks Easy -1.8
Figure 1 shows \(y = f(x)\). \includegraphics{figure_1} Which figure below shows \(y = f(2x)\)? Tick one box. \includegraphics{figure_2} \quad \includegraphics{figure_3} \quad \includegraphics{figure_4} \quad \includegraphics{figure_5} [1 mark]
AQA AS Paper 2 2018 June Q3
2 marks Easy -1.2
Express as a single logarithm \(2\log_a 6 - \log_a 3\) [2 marks]
AQA AS Paper 2 2018 June Q4
4 marks Moderate -0.3
Solve the equation \(\tan^2 2\theta - 3 = 0\) giving all the solutions for \(0° \leq \theta \leq 360°\) [4 marks]
AQA AS Paper 2 2018 June Q5
4 marks Standard +0.3
\(f'(x) = \left(2x - \frac{3}{x}\right)^2\) and \(f(3) = 2\) Find \(f(x)\). [4 marks]
AQA AS Paper 2 2018 June Q6
6 marks Standard +0.3
Points \(A(-7, -7)\), \(B(8, -1)\), \(C(4, 9)\) and \(D(-11, 3)\) are the vertices of a quadrilateral \(ABCD\).
  1. Prove that \(ABCD\) is a rectangle. [4 marks]
  2. Find the area of \(ABCD\). [2 marks]
AQA AS Paper 2 2018 June Q7
6 marks Moderate -0.8
  1. Express \(2x^2 - 5x + k\) in the form \(a(x - b)^2 + c\) [3 marks]
  2. Find the values of \(k\) for which the curve \(y = 2x^2 - 5x + k\) does not intersect the line \(y = 3\) [3 marks]
AQA AS Paper 2 2018 June Q8
4 marks Moderate -0.3
A circle of radius 6 passes through the points \((0, 0)\) and \((0, 10)\).
  1. Sketch the two possible positions of the circle. [1 mark]
  2. Find the equations of the two circles. [3 marks]
AQA AS Paper 2 2018 June Q9
3 marks Standard +0.3
It is given that \(\cos 15° = \frac{1}{2}\sqrt{2 + \sqrt{3}}\) and \(\sin 15° = \frac{1}{2}\sqrt{2 - \sqrt{3}}\) Show that \(\tan^2 15°\) can be written in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are integers. Fully justify your answer. [3 marks]
AQA AS Paper 2 2018 June Q10
5 marks Standard +0.3
In the binomial expansion of \((1 + x)^n\), where \(n \geq 4\), the coefficient of \(x^4\) is \(\frac{1}{2}\) times the sum of the coefficients of \(x^2\) and \(x^3\) Find the value of \(n\). [5 marks]
AQA AS Paper 2 2018 June Q11
9 marks Standard +0.8
Rakti makes open-topped cylindrical planters out of thin sheets of galvanised steel. She bends a rectangle of steel to make an open cylinder and welds the joint. She then welds this cylinder to the circumference of a circular base. \includegraphics{figure_11} The planter must have a capacity of \(8000\text{cm}^3\) Welding is time consuming, so Rakti wants the total length of weld to be a minimum. Calculate the radius, \(r\), and height, \(h\), of a planter which requires the minimum total length of weld. Fully justify your answers, giving them to an appropriate degree of accuracy. [9 marks]
AQA AS Paper 2 2018 June Q12
8 marks Standard +0.3
Trees in a forest may be affected by one of two types of fungal disease, but not by both. The number of trees affected by disease A, \(n_A\), can be modelled by the formula $$n_A = ae^{0.1t}$$ where \(t\) is the time in years after 1 January 2017. The number of trees affected by disease B, \(n_B\), can be modelled by the formula $$n_B = be^{0.2t}$$ On 1 January 2017 a total of 290 trees were affected by a fungal disease. On 1 January 2018 a total of 331 trees were affected by a fungal disease.
  1. Show that \(b = 90\), to the nearest integer, and find the value of \(a\). [3 marks]
  2. Estimate the total number of trees that will be affected by a fungal disease on 1 January 2020. [1 mark]
  3. Find the year in which the number of trees affected by disease B will first exceed the number affected by disease A. [3 marks]
  4. Comment on the long-term accuracy of the model. [1 mark]
AQA AS Paper 2 2018 June Q13
1 marks Easy -1.8
The table below shows the probability distribution for a discrete random variable \(X\).
\(x\)01234 or more
P(X = x)0.350.25\(k\)0.140.1
Find the value of \(k\). Circle your answer. 0.14 \quad 0.16 \quad 0.18 \quad 1 [1 mark]
AQA AS Paper 2 2018 June Q14
1 marks Easy -1.8
Given that \(\sum x = 364\), \(\sum x^2 = 19412\), \(n = 10\), find \(\sigma\), the standard deviation of \(X\). Circle your answer. 24.8 \quad 44.1 \quad 616.2 \quad 1941.2 [1 mark]
AQA AS Paper 2 2018 June Q15
6 marks Moderate -0.8
Nicola, a darts player, is practising hitting the bullseye. She knows from previous experience that she has a probability of 0.3 of hitting the bullseye with each dart. Nicola throws eight practice darts.
  1. Using a binomial distribution, calculate the probability that she will hit the bullseye three or more times. [2 marks]
  2. Nicola throws eight practice darts on three different occasions. Calculate the probability that she will hit the bullseye three or more times on all three occasions. [2 marks]
  3. State two assumptions that are necessary for the distribution you have used in part (a) to be valid. [2 marks]
AQA AS Paper 2 2018 June Q16
4 marks Easy -1.8
Kevin is the Principal of a college. He wishes to investigate types of transport used by students to travel to college. There are 3200 students in the college and Kevin decides to survey 60 of them. Describe how he could obtain a simple random sample of size 60 from the 3200 students. [4 marks]
AQA AS Paper 2 2018 June Q17
2 marks Easy -1.8
The table below is an extract from the Large Data Set, showing the purchased quantities of fats and oils for the South East of England in 2014.
DescriptionPurchased quantity
Butter42
Soft margarine16
Olive oil17
Other vegetable and salad oils28
Kim claims that more olive oil was purchased in the South East than soft margarine. Explain why Kim may be incorrect. [2 marks]
AQA AS Paper 2 2018 June Q18
6 marks Easy -1.2
Jennie is a piano teacher who teaches nine pupils. She records how many hours per week they practice the piano along with their most recent practical exam score.
StudentPractice (hours per week)Practical exam score (out of 100)
Donovan5064
Vazquez671
Higgins355
Begum2.547
Collins180
Coldbridge461
Nedbalek4.565
Carter883
White1192
[diagram]
  1. Identify two possible outliers by name, giving a possible explanation for the position on the scatter diagram of each outlier. [4 marks]
  2. Jennie discards the two outliers.
    1. Describe the correlation shown by the scatter diagram for the remaining points. [1 mark]
    2. Interpret this correlation in the context of the question. [1 mark]
AQA AS Paper 2 2018 June Q19
7 marks Moderate -0.3
Martin grows cucumbers from seed. In the past, he has found that 70% of all seeds successfully germinate and grow into cucumber plants. He decides to try out a new brand of seed. The producer of this brand claims that these seeds are more likely to successfully germinate than other brands of seeds. Martin sows 20 of this new brand of seed and 18 successfully germinate. Carry out a hypothesis test at the 5% level of significance to investigate the producer's claim. [7 marks]
AQA AS Paper 2 2020 June Q1
11 marks
Identify the expression below that is equivalent to \(e^{-\frac{2}{5}}\) Circle your answer. [1 mark] \(\frac{1}{\sqrt[5]{e^2}}\) \quad \(-\sqrt{e^5}\) \quad \(-\sqrt[5]{e^2}\) \quad \(\frac{1}{\sqrt{e^5}}\)
AQA AS Paper 2 2020 June Q2
1 marks Easy -1.8
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. [1 mark] \(-\infty < y < -1\) \(y > -1\) \(-1 < y < 0\) \(y < 0\)
AQA AS Paper 2 2020 June Q3
3 marks Moderate -0.8
It is given that $$y = 3x^4 + \frac{2}{x} - \frac{x}{4} + 1$$ Find an expression for \(\frac{d^2y}{dx^2}\) [3 marks]
AQA AS Paper 2 2020 June Q4
4 marks Standard +0.3
Find all the solutions of $$9 \sin^2 x - 6 \sin x + \cos^2 x = 0$$ where \(0° \leq x \leq 180°\) Give your solutions to the nearest degree. Fully justify your answer. [4 marks]
AQA AS Paper 2 2020 June Q5
4 marks Standard +0.3
Joseph is expanding \((2 - 3x)^7\) 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. [4 marks]
AQA AS Paper 2 2020 June Q6
6 marks Moderate -0.3
A circle has equation $$x^2 + y^2 + 10x - 4y - 71 = 0$$
  1. Find the centre of the circle. [2 marks]
  2. Hence, find the equation of the tangent to the circle at the point \((1, 10)\), giving your answer in the form \(ax + by + c = 0\) where \(a\), \(b\) and \(c\) are integers. [4 marks]