Questions — AQA (3620 questions)

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AQA Paper 3 Specimen Q9
3 marks Moderate -0.8
A survey has found that, of the 2400 households in Growmore, 1680 eat home-grown fruit and vegetables.
  1. Using the binomial distribution, find the probability that, out of a random sample of 25 households in Growmore, exactly 22 eat home-grown fruit and vegetables. [2 marks]
  2. Give a reason why you would not expect your calculation in part (a) to be valid for the 25 households in Gifford Terrace, a residential road in Growmore. [1 mark]
AQA Paper 3 Specimen Q10
7 marks Moderate -0.8
Shona calculated four correlation coefficients using data from the Large Data Set. In each case she calculated the correlation coefficient between the masses of the cars and the CO₂ emissions for varying sample sizes. A summary of these calculations, labelled A to D, are listed in the table below.
Sample sizeCorrelation coefficient
A38270.088
B37350.246
C240.400
D1250-1.183
Shona would like to use calculation A to test whether there is evidence of positive correlation between mass and CO₂ emissions. She finds the critical value for a one-tailed test at the 5% level for a sample of size 3827 is 0.027
    1. State appropriate hypotheses for Shona to use in her test. [1 mark]
    2. Determine if there is sufficient evidence to reject the null hypothesis. Fully justify your answer. [1 mark]
  2. Shona's teacher tells her to remove calculation D from the table as it is incorrect. Explain how the teacher knew it was incorrect. [1 mark]
  3. Before performing calculation B, Shona cleaned the data. She removed all cars from the Large Data Set that had incorrect masses. Using your knowledge of the large data set, explain what was incorrect about the masses which were removed from the calculation. [1 mark]
  4. Apart from CO2 and CO emissions, state one other type of emission that Shona could investigate using the Large Data Set. [1 mark]
  5. Wesley claims that calculation C shows that a heavier car causes higher CO2 emissions. Give two reasons why Wesley's claim may be incorrect. [2 marks]
AQA Paper 3 Specimen Q11
3 marks Moderate -0.8
Terence owns a local shop. His shop has three checkouts, at least one of which is always staffed. A regular customer observed that the probability distribution for \(N\), the number of checkouts that are staffed at any given time during the spring, is $$P(N = n) = \begin{cases} \frac{3}{4}\left(\frac{1}{4}\right)^{n-1} & \text{for } n = 1, 2 \\ k & \text{for } n = 3 \end{cases}$$
  1. Find the value of \(k\). [1 mark]
  2. Find the probability that a customer, visiting Terence's shop during the spring, will find at least 2 checkouts staffed. [2 marks]
AQA Paper 3 Specimen Q12
10 marks Standard +0.8
During the 2006 Christmas holiday, John, a maths teacher, realised that he had fallen ill during 65% of the Christmas holidays since he had started teaching. In January 2007, he increased his weekly exercise to try to improve his health. For the next 7 years, he only fell ill during 2 Christmas holidays.
  1. Using a binomial distribution, investigate, at the 5% level of significance, whether there is evidence that John's rate of illness during the Christmas holidays had decreased since increasing his weekly exercise. [6 marks]
  2. State two assumptions, regarding illness during the Christmas holidays, that are necessary for the distribution you have used in part (a) to be valid. For each assumption, comment, in context, on whether it is likely to be correct. [4 marks]
AQA Paper 3 Specimen Q13
8 marks Moderate -0.8
In the South West region of England, 100 households were randomly selected and, for each household, the weekly expenditure, \(£X\), per person on food and drink was recorded. The maximum amount recorded was £40.48 and the minimum amount recorded was £22.00 The results are summarised below, where \(\bar{x}\) denotes the sample mean. $$\sum x = 3046.14 \quad\quad \sum (x - \bar{x})^2 = 1746.29$$
    1. Find the mean of \(X\) Find the standard deviation of \(X\) [2 marks]
    2. Using your results from part (a)(i) and other information given, explain why the normal distribution can be used to model \(X\). [2 marks]
    3. Find the probability that a household in the South West spends less than £25.00 on food and drink per person per week. [1 mark]
  2. For households in the North West of England, the weekly expenditure, \(£Y\), per person on food and drink can be modelled by a normal distribution with mean £29.55 It is known that \(P(Y < 30) = 0.55\) Find the standard deviation of \(Y\), giving your answer to one decimal place. [3 marks]
AQA Paper 3 Specimen Q14
11 marks Standard +0.3
A survey during 2013 investigated mean expenditure on bread and on alcohol. The 2013 survey obtained information from 12 144 adults. The survey revealed that the mean expenditure per adult per week on bread was 127p.
  1. For 2012, it is known that the expenditure per adult per week on bread had mean 123p, and a standard deviation of 70p.
    1. Carry out a hypothesis test, at the 5% significance level, to investigate whether the mean expenditure per adult per week on bread changed from 2012 to 2013. Assume that the survey data is a random sample taken from a normal distribution. [5 marks]
    2. Calculate the greatest and least values for the sample mean expenditure on bread per adult per week for 2013 that would have resulted in acceptance of the null hypothesis for the test you carried out in part (a)(i). Give your answers to two decimal places. [2 marks]
  2. The 2013 survey revealed that the mean expenditure per adult, per week on alcohol was 324p. The mean expenditure per adult per week on alcohol for 2009 was 307p. A test was carried out on the following hypotheses relating to mean expenditure per adult per week on alcohol in 2013. \(H_0 : \mu = 307\) \(H_1 : \mu \neq 307\) This test resulted in the null hypothesis, \(H_0\), being rejected. State, with a reason, whether the test result supports the following statements:
    1. the mean UK expenditure on alcohol per adult per week increased by 17p from 2009 to 2013; [2 marks]
    2. the mean UK consumption of alcohol per adult per week changed from 2009 to 2013. [2 marks]
AQA Paper 3 Specimen Q15
6 marks Standard +0.8
A sample of 200 households was obtained from a small town. Each household was asked to complete a questionnaire about their purchases of takeaway food. \(A\) is the event that a household regularly purchases Indian takeaway food. \(B\) is the event that a household regularly purchases Chinese takeaway food. It was observed that \(P(B|A) = 0.25\) and \(P(A|B) = 0.1\) Of these households, 122 indicated that they did not regularly purchase Indian or Chinese takeaway food. A household is selected at random from those in the sample. Find the probability that the household regularly purchases both Indian and Chinese takeaway food. [6 marks]
AQA Further AS Paper 1 2018 June Q1
1 marks Easy -1.8
\(z = 3 - i\) Determine the value of \(zz*\) Circle your answer. [1 mark] \(10\) \(\qquad\) \(\sqrt{10}\) \(\qquad\) \(10 - 2i\) \(\qquad\) \(10 + 2i\)
AQA Further AS Paper 1 2018 June Q2
1 marks Easy -1.8
Three matrices \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\) are given by $$\mathbf{A} = \begin{pmatrix} 5 & 2 & -3 \\ 0 & 7 & 6 \\ 4 & 1 & 0 \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 1 & 0 \\ 3 & -5 \\ -2 & 6 \end{pmatrix} \quad \text{and } \mathbf{C} = \begin{pmatrix} 6 & 4 & 3 \\ 1 & 2 & 0 \end{pmatrix}$$ Which of the following **cannot** be calculated? Circle your answer. [1 mark] \(\mathbf{AB}\) \(\qquad\) \(\mathbf{AC}\) \(\qquad\) \(\mathbf{BC}\) \(\qquad\) \(\mathbf{A}^2\)
AQA Further AS Paper 1 2018 June Q3
1 marks Moderate -0.8
Which of the following functions has the fourth term \(-\frac{1}{720}x^6\) in its Maclaurin series expansion? Circle your answer. [1 mark] \(\sin x\) \(\qquad\) \(\cos x\) \(\qquad\) \(e^x\) \(\qquad\) \(\ln(1 + x)\)
AQA Further AS Paper 1 2018 June Q4
2 marks Standard +0.3
Sketch the graph given by the polar equation $$r = \frac{a}{\cos \theta}$$ where \(a\) is a positive constant. [2 marks] \includegraphics{figure_4}
AQA Further AS Paper 1 2018 June Q5
3 marks Standard +0.3
Describe fully the transformation given by the matrix \(\begin{pmatrix} -\frac{1}{2} & -\frac{\sqrt{3}}{2} & 0 \\ \frac{\sqrt{3}}{2} & -\frac{1}{2} & 0 \\ 0 & 0 & 1 \end{pmatrix}\) [3 marks]
AQA Further AS Paper 1 2018 June Q6
3 marks Standard +0.8
  1. Matthew is finding a formula for the inverse function \(\text{arsinh } x\). He writes his steps as follows: Let \(y = \sinh x\) \(y = \frac{1}{2}(e^x - e^{-x})\) \(2y = e^x - e^{-x}\) \(0 = e^x - 2y - e^{-x}\) \(0 = (e^x)^2 - 2ye^x - 1\) \(0 = (e^x - y)^2 - y^2 - 1\) \(y^2 + 1 = (e^x - y)^2\) \(\pm \sqrt{y^2 + 1} = e^x - y\) \(y + \sqrt{y^2 + 1} = e^x\) To find the inverse function, swap \(x\) and \(y\): \(x + \sqrt{x^2 + 1} = e^y\) \(\ln\left(x + \sqrt{x^2 + 1}\right) = y\) \(\text{arsinh } x = \ln\left(x + \sqrt{x^2 + 1}\right)\) Identify, and explain, the error in Matthew's proof. [2 marks]
  2. Solve \(\ln\left(x + \sqrt{x^2 + 1}\right) = 3\) [1 mark]
AQA Further AS Paper 1 2018 June Q7
2 marks Moderate -0.3
Find two invariant points under the transformation given by \(\begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}\) [2 marks]
AQA Further AS Paper 1 2018 June Q8
5 marks Standard +0.8
\(2 - 3i\) is one root of the equation $$z^3 + mz + 52 = 0$$ where \(m\) is real.
  1. Find the other roots. [3 marks]
  2. Determine the value of \(m\). [2 marks]
AQA Further AS Paper 1 2018 June Q9
6 marks Standard +0.3
  1. Sketch the graph of \(y^2 = 4x\) [1 mark] \includegraphics{figure_9a}
  2. Ben is using a 3D printer to make a plastic bowl which holds exactly \(1000\text{cm}^3\) of water. Ben models the bowl as a region which is rotated through \(2\pi\) radians about the \(x\)-axis. He uses the finite region enclosed by the lines \(x = d\) and \(y = 0\) and the curve with equation \(y^2 = 4x\) for \(y \geq 0\)
    1. Find the depth of the bowl to the nearest millimetre. [4 marks]
    2. What assumption has Ben made about the bowl? [1 mark]
AQA Further AS Paper 1 2018 June Q10
8 marks Standard +0.8
  1. Prove by induction that, for all integers \(n \geq 1\), $$\sum_{r=1}^{n} r^3 = \frac{1}{4}n^2(n + 1)^2$$ [4 marks]
  2. Hence show that $$\sum_{r=1}^{2n} r(r - 1)(r + 1) = n(n + 1)(2n - 1)(2n + 1)$$ [4 marks]
AQA Further AS Paper 1 2018 June Q11
3 marks Challenging +1.2
Four finite regions \(A\), \(B\), \(C\) and \(D\) are enclosed by the curve with equation $$y = x^3 - 7x^2 + 11x + 6$$ and the lines \(y = k\), \(x = 1\) and \(x = 4\), as shown in the diagram below. \includegraphics{figure_11} The areas of \(B\) and \(C\) are equal. Find the value of \(k\). [3 marks]
AQA Further AS Paper 1 2018 June Q12
6 marks Standard +0.3
  1. Show that the matrix \(\begin{pmatrix} 5 - k & 2 \\ k^3 + 1 & k \end{pmatrix}\) is singular when \(k = 1\). [1 mark]
  2. Find the values of \(k\) for which the matrix \(\begin{pmatrix} 5 - k & 2 \\ k^3 + 1 & k \end{pmatrix}\) has a negative determinant. Fully justify your answer. [5 marks]
AQA Further AS Paper 1 2018 June Q13
9 marks Challenging +1.2
The graph of the rational function \(y = f(x)\) intersects the \(x\)-axis exactly once at \((-3, 0)\) The graph has exactly two asymptotes, \(y = 2\) and \(x = -1\)
  1. Find \(f(x)\) [2 marks]
  2. Sketch the graph of the function. [3 marks] \includegraphics{figure_13b}
  3. Find the range of values of \(x\) for which \(f(x) \leq 5\) [4 marks]
AQA Further AS Paper 1 2018 June Q14
7 marks Challenging +1.2
  1. Sketch, on the Argand diagram below, the locus of points satisfying the equation $$|z - 3| = 2$$ [1 mark] \includegraphics{figure_14a}
  2. There is a unique complex number \(w\) that satisfies both $$|w - 3| = 2 \quad \text{and} \quad \arg(w + 1) = \alpha$$ where \(\alpha\) is a constant such that \(0 < \alpha < \pi\)
    1. Find the value of \(\alpha\). [2 marks]
    2. Express \(w\) in the form \(r(\cos \theta + i \sin \theta)\). Give each of \(r\) and \(\theta\) to two significant figures. [4 marks]
AQA Further AS Paper 1 2018 June Q15
4 marks Standard +0.3
  1. Show that $$\frac{1}{r + 2} - \frac{1}{r + 3} = \frac{1}{(r + 2)(r + 3)}$$ [1 mark]
  2. Use the method of differences to show that $$\sum_{r=1}^{n} \frac{1}{(r + 2)(r + 3)} = \frac{n}{3(n + 3)}$$ [3 marks]
AQA Further AS Paper 1 2018 June Q16
3 marks Standard +0.8
Two matrices \(\mathbf{A}\) and \(\mathbf{B}\) satisfy the equation $$\mathbf{AB} = \mathbf{I} + 2\mathbf{A}$$ where \(\mathbf{I}\) is the identity matrix and \(\mathbf{B} = \begin{pmatrix} 3 & -2 \\ -4 & 8 \end{pmatrix}\) Find \(\mathbf{A}\). [3 marks]
AQA Further AS Paper 1 2018 June Q17
4 marks Standard +0.8
Find the exact solution to the equation $$\sinh \theta(\sinh \theta + \cosh \theta) = 1$$ [4 marks]
AQA Further AS Paper 1 2018 June Q18
4 marks Challenging +1.8
\(\alpha\), \(\beta\) and \(\gamma\) are the real roots of the cubic equation $$x^3 + mx^2 + nx + 2 = 0$$ By considering \((\alpha - \beta)^2 + (\gamma - \alpha)^2 + (\beta - \gamma)^2\), prove that $$m^2 \geq 3n$$ [4 marks]