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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]
AQA Paper 3 2020 June Q16
4 marks Moderate -0.8
An educational expert found that the correlation coefficient between the hours of revision and the scores achieved by 25 students in their A-level exams was 0.379 Her data came from a bivariate normal distribution. Carry out a hypothesis test at the 1\% significance level to determine if there is a positive correlation between the hours of revision and the scores achieved by students in their A-level exams. The critical value of the correlation coefficient is 0.4622 [4 marks]
AQA Paper 3 2020 June Q17
8 marks Moderate -0.8
The lifetime of Zaple smartphone batteries, \(X\) hours, is normally distributed with mean 8 hours and standard deviation 1.5 hours.
    1. Find P(\(X \neq 8\)) [1 mark]
    2. Find P(\(6 < X < 10\)) [1 mark]
  2. Determine the lifetime exceeded by 90\% of Zaple smartphone batteries. [2 marks]
  3. A different smartphone, Kaphone, has its battery's lifetime, \(Y\) hours, modelled by a normal distribution with mean 7 hours and standard deviation \(\sigma\). 25\% of randomly selected Kaphone batteries last less than 5 hours. Find the value of \(\sigma\), correct to three significant figures. [4 marks]
AQA Paper 3 2020 June Q18
14 marks Standard +0.3
Tiana is a quality controller in a clothes factory. She checks for four possible types of defects in shirts. Of the shirts with defects, the proportion of each type of defect is as shown in the table below.
Type of defectColourFabricSewingSizing
Probability0.250.300.400.05
Shirts with defects are packed in boxes of 30 at random.
  1. Find the probability that:
    1. a box contains exactly 5 shirts with a colour defect [2 marks]
    2. a box contains fewer than 15 shirts with a sewing defect [2 marks]
    3. a box contains at least 20 shirts which do not have a fabric defect. [3 marks]
  2. Tiana wants to investigate the proportion, \(p\), of defective shirts with a fabric defect. She wishes to test the hypotheses H\(_0\): \(p = 0.3\) H\(_1\): \(p < 0.3\) She takes a random sample of 60 shirts with a defect and finds that \(x\) of them have a fabric defect.
    1. Using a 5\% level of significance, find the critical region for \(x\). [5 marks]
    2. In her sample she finds 13 shirts with a fabric defect. Complete the test stating her conclusion in context. [2 marks]
AQA Paper 3 2021 June Q1
1 marks Easy -2.0
The graph of \(y = \arccos x\) is shown. \includegraphics{figure_1} State the coordinates of the end point \(P\). Circle your answer. [1 mark] \((-\pi, 1)\) \quad \((-1, \pi)\) \quad \(\left(-\frac{\pi}{2}, 1\right)\) \quad \(\left(-1, \frac{\pi}{2}\right)\)
AQA Paper 3 2021 June Q2
1 marks Easy -1.8
Simplify fully $$\frac{(x + 3)(6 - 2x)}{(x - 3)(3 + x)} \quad \text{for } x \neq \pm 3$$ Circle your answer. [1 mark] \(-2\) \quad \(2\) \quad \(\frac{(6 - 2x)}{(x - 3)}\) \quad \(\frac{(2x - 6)}{(x - 3)}\)
AQA Paper 3 2021 June Q3
1 marks Easy -1.8
\(f(x) = 3x^2\) Obtain \(\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\) Circle your answer. [1 mark] \(\frac{3h^2}{h}\) \quad \(x^3\) \quad \(\frac{3(x + h)^2 - 3x^2}{h}\) \quad \(6x\)
AQA Paper 3 2021 June Q4
5 marks Moderate -0.8
  1. Show that the first three terms, in descending powers of \(x\), of the expansion of $$(2x - 3)^{10}$$ are given by $$1024x^{10} + px^9 + qx^8$$ where \(p\) and \(q\) are integers to be found. [3 marks]
  2. Find the constant term in the expansion of $$\left(2x - \frac{3}{x}\right)^{10}$$ [2 marks]
AQA Paper 3 2021 June Q5
13 marks Moderate -0.8
A gardener is creating flowerbeds in the shape of sectors of circles. The gardener uses an edging strip around the perimeter of each of the flowerbeds. The cost of the edging strip is £1.80 per metre and can be purchased for any length. One of the flowerbeds has a radius of 5 metres and an angle at the centre of 0.7 radians as shown in the diagram below. \includegraphics{figure_5}
    1. Find the area of this flowerbed. [2 marks]
    2. Find the cost of the edging strip required for this flowerbed. [3 marks]
  2. A flowerbed is to be made with an area of 20 m²
    1. Show that the cost, £\(C\), of the edging strip required for this flowerbed is given by $$C = \frac{18}{5}\left(\frac{20}{r} + r\right)$$ where \(r\) is the radius measured in metres. [3 marks]
    2. Hence, show that the minimum cost of the edging strip for this flowerbed occurs when \(r \approx 4.5\) Fully justify your answer. [5 marks]
AQA Paper 3 2021 June Q6
4 marks Standard +0.3
Given that \(x > 0\) and \(x \neq 25\), fully simplify $$\frac{10 + 5x - 2x^{\frac{1}{2}} - x^{\frac{3}{2}}}{5 - \sqrt{x}}$$ Fully justify your answer. [4 marks]
AQA Paper 3 2021 June Q7
10 marks Moderate -0.8
A building has a leaking roof and, while it is raining, water drips into a 12 litre bucket. When the rain stops, the bucket is one third full. Water continues to drip into the bucket from a puddle on the roof. In the first minute after the rain stops, 30 millilitres of water drips into the bucket. In each subsequent minute, the amount of water that drips into the bucket reduces by 2%. During the \(n\)th minute after the rain stops, the volume of water that drips into the bucket is \(W_n\) millilitres.
  1. Find \(W_2\) [1 mark]
  2. Explain why $$W_n = A \times 0.98^{n-1}$$ and state the value of \(A\). [2 marks]
  3. Find the increase in the water in the bucket 15 minutes after the rain stops. Give your answer to the nearest millilitre. [2 marks]
  4. Assuming it does not start to rain again, find the maximum amount of water in the bucket. [3 marks]
  5. After several hours the water has stopped dripping. Give two reasons why the amount of water in the bucket is not as much as the answer found in part (d). [2 marks]
AQA Paper 3 2021 June Q8
6 marks Standard +0.3
Given that $$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} x \cos x \, dx = a\pi + b$$ find the exact value of \(a\) and the exact value of \(b\). Fully justify your answer. [6 marks]
AQA Paper 3 2021 June Q9
9 marks Standard +0.3
A function f is defined for all real values of \(x\) as $$f(x) = x^4 + 5x^3$$ The function has exactly two stationary points when \(x = 0\) and \(x = -\frac{15}{4}\)
    1. Find \(f''(x)\) [2 marks]
    2. Determine the nature of the stationary points. Fully justify your answer. [4 marks]
  2. State the range of values of \(x\) for which $$f(x) = x^4 + 5x^3$$ is an increasing function. [1 mark]
  3. A second function g is defined for all real values of \(x\) as $$g(x) = x^4 - 5x^3$$
    1. State the single transformation which maps f onto g. [1 mark]
    2. State the range of values of \(x\) for which g is an increasing function. [1 mark]
AQA Paper 3 2021 June Q10
1 marks Easy -2.5
Anke has collected data from 30 similar-sized cars to investigate any correlation between the age of the car and the current market value. She calculates the correlation coefficient. Which of the following statements best describes her answer of \(-1.2\)? Tick (\(\checkmark\)) one box. [1 mark] Definitely incorrect Probably incorrect Probably correct Definitely correct
AQA Paper 3 2021 June Q11
1 marks Easy -1.2
The random variable \(X\) is such that \(X \sim B(n, p)\) The mean value of \(X\) is 225 The variance of \(X\) is 144 Find \(p\). Circle your answer. [1 mark] 0.36 \quad 0.6 \quad 0.64 \quad 0.8
AQA Paper 3 2021 June Q12
3 marks Easy -1.8
An electoral register contains 8000 names. A researcher decides to select a systematic sample of 100 names from the register. Explain how the researcher should select such a sample. [3 marks]
AQA Paper 3 2021 June Q13
6 marks Moderate -0.8
The table below is an extract from the Large Data Set.
Propulsion TypeRegionEngine SizeMassCO₂Particulate Emissions
2London189615331540.04
2North West189614231460.029
2North West189613531380.025
2South West199815471590.026
2London189613881380.025
2South West189612141300.011
2South West189614801460.029
2South West189614131460.024
2South West249616951920.034
2South West142212511220.025
2South West199520751750.034
2London189612851400.036
2North West18960146
    1. Calculate the mean and standard deviation of CO₂ emissions in the table. [2 marks]
    2. Any value more than 2 standard deviations from the mean can be identified as an outlier. Determine, using this definition of an outlier, if there are any outliers in this sample of CO₂ emissions. Fully justify your answer. [2 marks]
  2. Maria claims that the last line in the table must contain two errors. Use your knowledge of the Large Data Set to comment on Maria's claim. [2 marks]