Questions — AQA Paper 3 (141 questions)

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AQA Paper 3 2019 June Q8
12 marks Standard +0.3
A student is conducting an experiment in a laboratory to investigate how quickly liquids cool to room temperature. A beaker containing a hot liquid at an initial temperature of \(75°C\) cools so that the temperature, \(\theta °C\), of the liquid at time \(t\) minutes can be modelled by the equation $$\theta = 5(4 + \lambda e^{-kt})$$ where \(\lambda\) and \(k\) are constants. After 2 minutes the temperature falls to \(68°C\).
  1. Find the temperature of the liquid after 15 minutes. Give your answer to three significant figures. [7 marks]
    1. Find the room temperature of the laboratory, giving a reason for your answer. [2 marks]
    2. Find the time taken in minutes for the liquid to cool to \(1°C\) above the room temperature of the laboratory. [2 marks]
  3. Explain why the model might need to be changed if the experiment was conducted in a different place. [1 mark]
AQA Paper 3 2019 June Q9
15 marks Challenging +1.2
A curve has equation $$x^2y^2 + xy^4 = 12$$
  1. Prove that the curve does not intersect the coordinate axes. [2 marks]
    1. Show that \(\frac{dy}{dx} = -\frac{2xy + y^3}{2x^2 + 4xy^2}\) [5 marks]
    2. Prove that the curve has no stationary points. [4 marks]
    3. In the case when \(x > 0\), find the equation of the tangent to the curve when \(y = 1\) [4 marks]
AQA Paper 3 2019 June Q10
1 marks Easy -2.5
Which of the options below best describes the correlation shown in the diagram below? \includegraphics{figure_10} Tick \((\checkmark)\) one box. [1 mark] moderate positive strong positive moderate negative strong negative
AQA Paper 3 2019 June Q11
1 marks Easy -2.0
Lenny is one of a team of people interviewing shoppers in a town centre. He is asked to survey 50 women between the ages of 18 and 29 Identify the name of this type of sampling. Circle your answer. [1 mark] simple random \quad stratified \quad quota \quad systematic
AQA Paper 3 2019 June Q12
6 marks Moderate -0.3
Amelia decides to analyse the heights of members of her school rowing club. The heights of a random sample of 10 rowers are shown in the table below.
RowerJessNellLivNeveAnnToriMayaKathDarcyJen
Height (cm)162169172156146161159164157160
  1. Any value more than 2 standard deviations from the mean may be regarded as an outlier. Verify that Ann's height is an outlier. Fully justify your answer. [4 marks]
  2. Amelia thinks she may have written down Ann's height incorrectly. If Ann's height were discarded, state with a reason what, if any, difference this would make to the mean and standard deviation. [2 marks]
AQA Paper 3 2019 June Q13
10 marks Moderate -0.8
Patrick is practising his skateboarding skills. On each day, he has 30 attempts at performing a difficult trick. Every time he attempts the trick, there is a probability of 0.2 that he will fall off his skateboard. Assume that the number of times he falls off on any given day may be modelled by a binomial distribution.
    1. Find the mean number of times he falls off in a day. [1 mark]
    2. Find the variance of the number of times he falls off in a day. [1 mark]
    1. Find the probability that, on a particular day, he falls off exactly 10 times. [2 marks]
    2. Find the probability that, on a particular day, he falls off 5 or more times. [3 marks]
  3. Patrick has 30 attempts to perform the trick on each of 5 consecutive days.
    1. Calculate the probability that he will fall off his skateboard at least 5 times on each of the 5 days. [2 marks]
    2. Explain why it may be unrealistic to use the same value of 0.2 for the probability of falling off for all 5 days. [1 mark]
AQA Paper 3 2019 June Q14
7 marks Easy -1.3
A survey was conducted into the health of 120 teachers. The survey recorded whether or not they had suffered from a range of four health issues in the past year. In addition, their physical exercise level was categorised as low, medium or high. 50 teachers had a low exercise level, 40 teachers had a medium exercise level and 30 teachers had a high exercise level. The results of the survey are shown in the table below.
Low exerciseMedium exerciseHigh exercise
Back trouble14710
Stress38145
Depression921
Headache/Migraine455
  1. Find the probability that a randomly selected teacher:
    1. suffers from back trouble and has a high exercise level; [1 mark]
    2. suffers from depression. [2 marks]
    3. suffers from stress, given that they have a low exercise level. [2 marks]
  2. For teachers in the survey with a low exercise level, explain why the events 'suffers from back trouble' and 'suffers from stress' are not mutually exclusive. [2 marks]
AQA Paper 3 2019 June Q15
3 marks Moderate -0.8
Jamal, a farmer, claims that the larger the rainfall, the greater the yield of wheat from his farm. He decides to investigate his claim, at the 5\% level of significance. He measures the rainfall in centimetres and the yield in kilograms for a random sample of ten years. He correctly calculates the product moment correlation coefficient between rainfall and yield for his sample to be 0.567 The table below shows the critical values for correlation coefficients for a sample size of 10 for different significance levels, for both 1- and 2-tailed tests.
1-tailed test significance level5\%2.5\%1\%0.5\%
2-tailed test significance level10\%5\%2\%1\%
Critical value0.5490.6320.7160.765
Determine what Jamal's conclusion to his investigation should be, justifying your answer. [3 marks]
AQA Paper 3 2019 June Q16
10 marks Moderate -0.3
  1. The graph below shows the amount of salt, in grams, purchased per person per week in England between 2001–02 and 2014, based upon the Large Data Set. \includegraphics{figure_16a} Meera and Gemma are arguing about what this graph shows. Meera believes that the amount of salt consumed by people decreased greatly during this period. Gemma says that this is not the case. Using your knowledge of the Large Data Set, give two reasons why Gemma may be correct. [2 marks]
  2. It is known that the mean amount of sugar purchased per person in England in 2014 was 78.9 grams, with a standard deviation of 25.0 grams. In 2018, a sample of 918 people had a mean of 80.4 grams of sugar purchased per person. Investigate, at the 5\% level of significance, whether the mean amount of sugar purchased per person in England has changed between 2014 and 2018. Assume that the survey data is a random sample taken from a normal distribution and that the standard deviation has remained the same. [6 marks]
  3. Another test is performed to determine whether the mean amount of fat purchased per person has changed between 2014 and 2018. At the 10\% significance level, the null hypothesis is rejected. With reference to the 10\% significance level, explain why it is not necessarily true that there has been a change. [2 marks]
AQA Paper 3 2019 June Q17
12 marks Standard +0.3
Elizabeth's Bakery makes brownies. It is known that the mass, \(X\) grams, of a brownie may be modelled by a normal distribution. 10\% of the brownies have a mass less than 30 grams. 80\% of the brownies have a mass greater than 32.5 grams.
  1. Find the mean and standard deviation of \(X\). [7 marks]
    1. Find P\((X \neq 35)\) [1 mark]
    2. Find P\((X < 35)\) [2 marks]
  3. Brownies are baked in batches of 13. Calculate the probability that, in a batch of brownies, no more than 3 brownies are less than 35 grams. You may assume that the masses of brownies are independent of each other. [2 marks]
AQA Paper 3 2020 June Q1
1 marks Easy -1.8
Given that $$\int_0^{10} f(x) \, dx = 7$$ deduce the value of $$\int_0^{10} \left( f(x) + 1 \right) dx$$ Circle your answer. [1 mark] \(-3\) \quad \(7\) \quad \(8\) \quad \(17\)
AQA Paper 3 2020 June Q2
1 marks Easy -1.2
Given that $$6 \cos \theta + 8 \sin \theta \equiv R \cos (\theta + \alpha)$$ find the value of \(R\). Circle your answer. [1 mark] \(6\) \quad \(8\) \quad \(10\) \quad \(14\)
AQA Paper 3 2020 June Q3
1 marks Easy -2.5
Determine which one of these graphs does not represent \(y\) as a function of \(x\). Tick (\(\checkmark\)) one box. [1 mark] \includegraphics{figure_3}
AQA Paper 3 2020 June Q4
7 marks Standard +0.3
\(p(x) = 4x^3 - 15x^2 - 48x - 36\)
  1. Use the factor theorem to prove that \(x - 6\) is a factor of \(p(x)\). [2 marks]
    1. Prove that the graph of \(y = p(x)\) intersects the \(x\)-axis at exactly one point. [4 marks]
    2. State the coordinates of this point of intersection. [1 mark]
AQA Paper 3 2020 June Q5
9 marks Moderate -0.3
The number of radioactive atoms, \(N\), in a sample of a sodium isotope after time \(t\) hours can be modelled by $$N = N_0 e^{-kt}$$ where \(N_0\) is the initial number of radioactive atoms in the sample and \(k\) is a positive constant. The model remains valid for large numbers of atoms.
  1. It takes 15.9 hours for half of the sodium atoms to decay. Determine the number of days required for at least 90\% of the number of atoms in the original sample to decay. [5 marks]
  2. Find the percentage of the atoms remaining after the first week. Give your answer to two significant figures. [2 marks]
  3. Explain why the model can only provide an estimate for the number of remaining atoms. [1 mark]
  4. Explain why the model is invalid in the long run. [1 mark]
AQA Paper 3 2020 June Q6
7 marks Moderate -0.3
The graph of \(y = f(x)\) is shown below. \includegraphics{figure_6}
  1. Sketch the graph of \(y = f(-x)\) [2 marks]
  2. Sketch the graph of \(y = 2f(x) - 4\) [2 marks]
  3. Sketch the graph of \(y = f'(x)\) [3 marks]
AQA Paper 3 2020 June Q7
7 marks Moderate -0.8
  1. Using \({}^n C_r = \frac{n!}{r!(n-r)!}\) show that \({}^n C_2 = \frac{n(n-1)}{2}\) [2 marks]
    1. Show that the equation $$2 \times {}^n C_4 = 51 \times {}^n C_2$$ simplifies to $$n^2 - 5n - 300 = 0$$ [3 marks]
    2. Hence, solve the equation $$2 \times {}^n C_4 = 51 \times {}^n C_2$$ [2 marks]
AQA Paper 3 2020 June Q8
12 marks Standard +0.3
The sum to infinity of a geometric series is 96 The first term of the series is less than 30 The second term of the series is 18
  1. Find the first term and common ratio of the series. [5 marks]
    1. Show that the \(n\)th term of the series, \(u_n\), can be written as $$u_n = \frac{3^n}{2^{2n-5}}$$ [4 marks]
    2. Hence show that $$\log_3 u_n = n(1 - 2\log_3 2) + 5\log_3 2$$ [3 marks]
AQA Paper 3 2020 June Q9
5 marks Standard +0.3
  1. For \(\cos \theta \neq 0\), prove that $$\cosec 2\theta + \cot 2\theta = \cot \theta$$ [4 marks]
  2. Explain why $$\cot \theta \neq \cosec 2\theta + \cot 2\theta$$ when \(\cos \theta = 0\) [1 mark]
AQA Paper 3 2020 June Q10
1 marks Easy -1.8
The probabilities of events \(A\), \(B\) and \(C\) are related, as shown in the Venn diagram below. \includegraphics{figure_10} Find the value of \(x\). Circle your answer. [1 mark] \(0.11\) \quad \(0.46\) \quad \(0.54\) \quad \(0.89\)
AQA Paper 3 2020 June Q11
1 marks Easy -1.8
The table below shows the temperature on Mount Everest on the first day of each month.
MonthJanFebMarAprMayJunJulAugSepOctNovDec
Temperature (\(^\circ\)C)\(-17\)\(-16\)\(-14\)\(-9\)\(-2\)\(2\)\(6\)\(5\)\(-3\)\(-4\)\(-11\)\(-18\)
Calculate the standard deviation of these temperatures. Circle your answer. [1 mark] \(-6.75\) \quad \(5.82\) \quad \(8.24\) \quad \(67.85\)
AQA Paper 3 2020 June Q12
4 marks Easy -1.8
The box plot below summarises the CO\(_2\) emissions, in g/km, for cars in the Large Data Set from the London and North West regions. \includegraphics{figure_12}
  1. Using the box plot, give one comparison of central tendency and one comparison of spread for the two regions. [2 marks]
  2. Jaspal, an environmental researcher, used all of the data in the Large Data Set to produce a statistical comparison of the CO\(_2\) and CO emissions in regions of England. Using your knowledge of the Large Data Set, give two reasons why his conclusions may be invalid. [2 marks]
AQA Paper 3 2020 June Q13
6 marks Easy -1.3
Diedre is a head teacher in a school which provides primary, secondary and sixth-form education. There are 200 teachers in her school. The number of teachers in each level of education along with their gender is shown in the table below.
PrimarySecondarySixth-form
Male92423
Female358524
  1. A teacher is selected at random. Find the probability that:
    1. the teacher is female [1 mark]
    2. the teacher is not a sixth-form teacher. [1 mark]
  2. Given that a randomly chosen teacher is male, find the probability that this teacher is not a primary teacher. [2 marks]
  3. Diedre wants to select three different teachers at random to be part of a school project. Calculate the probability that all three chosen are secondary teachers. [2 marks]
AQA Paper 3 2020 June Q14
7 marks Moderate -0.3
It is known that a hospital has a mean waiting time of 4 hours for its Accident and Emergency (A\&E) patients. After some new initiatives were introduced, a random sample of 12 patients from the hospital's A\&E Department had the following waiting times, in hours. \(4.25\) \quad \(3.90\) \quad \(4.15\) \quad \(3.95\) \quad \(4.20\) \quad \(4.15\) \(5.00\) \quad \(3.85\) \quad \(4.25\) \quad \(4.05\) \quad \(3.80\) \quad \(3.95\) Carry out a hypothesis test at the 10\% significance level to investigate whether the mean waiting time at this hospital's A\&E department has changed. You may assume that the waiting times are normally distributed with standard deviation 0.8 hours. [7 marks]
AQA Paper 3 2020 June Q15
5 marks Easy -1.3
A political party is holding an election to choose a new leader. A statistician within the party decides to sample 70 party members to find their opinions of the leadership candidates. There are 4735 members under 30 years old and 8565 members 30 years old and over. The statistician wants to use a sample of 70 party members in the survey. He decides to use a random stratified sample.
  1. Calculate how many of each age group should be included in his sample. [2 marks]
  2. Explain how he could collect the random sample of members under 30 years old. [3 marks]