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AQA Paper 3 2019 June Q17
12 marks Standard +0.8
17 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.
17
  1. Find the mean and standard deviation of \(X\).
    1. Find \(\mathrm { P } ( X \neq 35 )\) 17
  3. (ii) Find \(\mathrm { P } ( X < 35 )\) 17
  4. 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. \includegraphics[max width=\textwidth, alt={}, center]{c8a41c47-bbda-4e91-a7a2-d0bcf6a46f25-28_2496_1719_214_148}
AQA Paper 3 2020 June Q1
1 marks Easy -1.8
1 Given that $$\int _ { 0 } ^ { 10 } \mathrm { f } ( x ) \mathrm { d } x = 7$$ deduce the value of $$\int _ { 0 } ^ { 10 } ( \mathrm { f } ( x ) + 1 ) \mathrm { d } x$$ Circle your answer.
-3
7
8
17
AQA Paper 3 2020 June Q2
1 marks Easy -1.2
2 Given that $$6 \cos \theta + 8 \sin \theta \equiv R \cos ( \theta + \alpha )$$ find the value of \(R\). Circle your answer. 681014
AQA Paper 3 2020 June Q3
1 marks Easy -2.0
3 Determine which one of these graphs does not represent \(y\) as a function of \(x\). Tick \(( \checkmark )\) one box. \includegraphics[max width=\textwidth, alt={}, center]{076ea8e9-9295-46d2-b5f9-b27fa969129e-03_2246_974_443_495}
AQA Paper 3 2020 June Q4
7 marks Standard +0.3
4
  1. Use the factor theorem to prove that \(x - 6\) is a factor of \(\mathrm { p } ( x )\). \(4 \quad \mathrm { p } ( x ) = 4 x ^ { 3 } - 15 x ^ { 2 } - 48 x - 36\) 4
    1. Prove that the graph of \(y = \mathrm { p } ( x )\) intersects the \(x\)-axis at exactly one point.
      4
  3. (ii) State the coordinates of this point of intersection.
AQA Paper 3 2020 June Q5
9 marks Moderate -0.3
5 The number of radioactive atoms, \(N\), in a sample of a sodium isotope after time \(t\) hours can be modelled by $$N = N _ { 0 } \mathrm { e } ^ { - k t }$$ 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.
5
  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.
    [0pt] [5 marks]
    5
  2. Find the percentage of the atoms remaining after the first week. Give your answer to two significant figures.
    5
  3. Explain why the model can only provide an estimate for the number of remaining atoms.
    5
  4. Explain why the model is invalid in the long run.
AQA Paper 3 2020 June Q8
12 marks Easy -1.8
8
17 2 Given that $$6 \cos \theta + 8 \sin \theta \equiv R \cos ( \theta + \alpha )$$ find the value of \(R\). Circle your answer. 681014 3 Determine which one of these graphs does not represent \(y\) as a function of \(x\). Tick \(( \checkmark )\) one box. \includegraphics[max width=\textwidth, alt={}, center]{076ea8e9-9295-46d2-b5f9-b27fa969129e-03_2246_974_443_495} 4
  1. Use the factor theorem to prove that \(x - 6\) is a factor of \(\mathrm { p } ( x )\). \(4 \quad \mathrm { p } ( x ) = 4 x ^ { 3 } - 15 x ^ { 2 } - 48 x - 36\) 4
    1. Prove that the graph of \(y = \mathrm { p } ( x )\) intersects the \(x\)-axis at exactly one point.
      4
  3. (ii) State the coordinates of this point of intersection.
AQA Paper 3 2020 June Q9
5 marks Standard +0.3
9
  1. For \(\cos \theta \neq 0\), prove that $$\operatorname { cosec } 2 \theta + \cot 2 \theta = \cot \theta$$ 9
  2. Explain why $$\cot \theta \neq \operatorname { cosec } 2 \theta + \cot 2 \theta$$ when \(\cos \theta = 0\)
AQA Paper 3 2020 June Q10
1 marks Easy -1.8
10 The probabilities of events \(A , B\) and \(C\) are related, as shown in the Venn diagram below. \(\varepsilon\) \includegraphics[max width=\textwidth, alt={}, center]{076ea8e9-9295-46d2-b5f9-b27fa969129e-15_620_1200_799_443} Find the value of \(x\). Circle your answer. \(0.11 \quad 0.46 \quad 0.54 \quad 0.89\)
AQA Paper 3 2020 June Q11
1 marks Easy -1.8
11 The table below shows the temperature on Mount Everest on the first day of each month.
MonthJanFebMarAprMayJunJulAugSepOctNovDec
Temperature \(\left( { } ^ { \circ } \mathbf { C } \right)\)- 17- 16- 14- 9- 2265- 3- 4- 11- 18
Calculate the standard deviation of these temperatures.
Circle your answer.
-6.75
5.82
8.24
67.85 \begin{center} \begin{tabular}{|l|l|l|l|} \hline \multicolumn{4}{|c|}{\multirow[t]{2}{*}{\begin{tabular}{l}
AQA Paper 3 2020 June Q12
4 marks Easy -1.2
12
12
The box plot below summarises the \(\mathrm { CO } _ { 2 }\) emissions, in \(\mathrm { g } / \mathrm { km }\), for cars in the Large Data Set from the London and North West regions.
London
39
119142168
346
North West
AQA Paper 3 2020 June Q13
4 marks Easy -1.8
13
12

  1. Using the box plot, give one comparison of central tendency and one comparison of spread for the two regions.
    [0pt] [2 marks]
    Comparison of central tendency
    Comparison of spread \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \end{tabular}}}
    \hline & & &
    \hline \end{tabular} \end{center} 12
  2. Jaspal, an environmental researcher, used all of the data in the Large Data Set to produce a statistical comparison of the \(\mathrm { CO } _ { 2 }\) and CO emissions in regions of England. Using your knowledge of the Large Data Set, give two reasons why his conclusions may be invalid.
AQA Paper 3 2020 June Q14
7 marks Standard +0.3
14 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.
4.253.904.153.954.204.15
5.003.854.254.053.803.95
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. \includegraphics[max width=\textwidth, alt={}, center]{076ea8e9-9295-46d2-b5f9-b27fa969129e-21_2488_1728_219_141}
AQA Paper 3 2020 June Q15
5 marks Easy -1.8
15 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.
15
  1. Calculate how many of each age group should be included in his sample.
    15
  2. Explain how he could collect the random sample of members under 30 years old.
AQA Paper 3 2020 June Q16
4 marks Standard +0.3
16 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
[0pt] [4 marks]
AQA Paper 3 2020 June Q17
8 marks Moderate -0.3
17 The lifetime of Zaple smartphone batteries, \(X\) hours, is normally distributed with mean 8 hours and standard deviation 1.5 hours. 17
    1. Find \(\mathrm { P } ( X \neq 8 )\) 17
  2. (ii) Find \(\mathrm { P } ( 6 < X < 10 )\) 17
  3. Determine the lifetime exceeded by \(90 \%\) of Zaple smartphone batteries.
    17
  4. 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.
AQA Paper 3 2020 June Q18
14 marks Moderate -0.3
18
  1. Find the probability that:
    18
    1. a box contains exactly 5 shirts with a colour defect
      18
  3. (ii) a box contains fewer than 15 shirts with a sewing defect
    18
  4. (iii) a box contains at least 20 shirts which do not have a fabric defect.
    Question 18 continues on the next page 18
  5. Tiana wants to investigate the proportion, \(p\), of defective shirts with a fabric defect. She wishes to test the hypotheses $$\begin{aligned} & \mathrm { H } _ { 0 } : p = 0.3 \\ & \mathrm { H } _ { 1 } : p < 0.3 \end{aligned}$$ She takes a random sample of 60 shirts with a defect and finds that \(x\) of them have a fabric defect. 18
    1. Using a \(5 \%\) level of significance, find the critical region for \(x\).
      18
  7. (ii) In her sample she finds 13 shirts with a fabric defect. Complete the test stating her conclusion in context.
AQA Paper 3 2021 June Q1
1 marks Easy -2.0
1 The graph of \(y = \arccos x\) is shown. \includegraphics[max width=\textwidth, alt={}, center]{0abb6343-017d-4629-8a2d-cfb405dc2d14-02_599_856_758_591} State the coordinates of the end point \(P\).
Circle your answer.
[0pt] [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
2 Simplify fully $$\frac { ( x + 3 ) ( 6 - 2 x ) } { ( x - 3 ) ( 3 + x ) } \quad \text { for } x \neq \pm 3$$ Circle your answer.
-2
2 \(\frac { ( 6 - 2 x ) } { ( x - 3 ) }\) \(\frac { ( 2 x - 6 ) } { ( x - 3 ) }\) \(3 \mathrm { f } ( x ) = 3 x ^ { 2 }\) Obtain \(\lim _ { h \rightarrow 0 } \frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\) Circle your answer. $$\frac { 3 h ^ { 2 } } { h } \quad x ^ { 3 } \quad \frac { 3 ( x + h ) ^ { 2 } - 3 x ^ { 2 } } { h } \quad 6 x$$
AQA Paper 3 2021 June Q4
5 marks Moderate -0.8
4
  1. Show that the first three terms, in descending powers of \(x\), of the expansion of $$( 2 x - 3 ) ^ { 10 }$$ are given by $$1024 x ^ { 10 } + p x ^ { 9 } + q x ^ { 8 }$$ where \(p\) and \(q\) are integers to be found.
    4
  2. Find the constant term in the expansion of $$\left( 2 x - \frac { 3 } { x } \right) ^ { 10 }$$
AQA Paper 3 2021 June Q5
19 marks Standard +0.8
5 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 \(\pounds 1.80\) 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[max width=\textwidth, alt={}, center]{0abb6343-017d-4629-8a2d-cfb405dc2d14-05_544_579_676_730} 5
    1. Find the area of this flowerbed.
      \section*{Question 5 continues on the next page} 5
  2. (ii) Find the cost of the edging strip required for this flowerbed.
    5
  3. A flowerbed is to be made with an area of \(20 \mathrm {~m} ^ { 2 }\) 5
    1. Show that the cost, \(\pounds C\), 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.
      5
  5. (ii) Hence, show that the minimum cost of the edging strip for this flowerbed occurs when \(r \approx 4.5\) Fully justify your answer.
    [0pt] [5 marks]
AQA Paper 3 2021 June Q6
4 marks Standard +0.8
6 Given that \(x > 0\) and \(x \neq 25\), fully simplify $$\frac { 10 + 5 x - 2 x ^ { \frac { 1 } { 2 } } - x ^ { \frac { 3 } { 2 } } } { 5 - \sqrt { x } }$$ Fully justify your answer.
[0pt] [4 marks] \includegraphics[max width=\textwidth, alt={}, center]{0abb6343-017d-4629-8a2d-cfb405dc2d14-09_2488_1716_219_153}
AQA Paper 3 2021 June Q7
10 marks Moderate -0.8
7 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. 7
  1. \(\quad\) Find \(W _ { 2 }\) 7
  2. Explain why $$W _ { n } = A \times 0.98 ^ { n - 1 }$$ and state the value of \(A\).
    7
  3. Find the increase in the water in the bucket 15 minutes after the rain stops. Give your answer to the nearest millilitre. 7
  4. Assuming it does not start to rain again, find the maximum amount of water in the bucket.
    7
  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).
AQA Paper 3 2021 June Q8
6 marks Standard +0.3
8 Given that $$\int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 3 } } x \cos x d x = a \pi + b$$ find the exact value of \(a\) and the exact value of \(b\). Fully justify your answer.
[0pt] [6 marks]
AQA Paper 3 2021 June Q9
9 marks Moderate -0.8
9 A function f is defined for all real values of \(x\) as $$f ( x ) = x ^ { 4 } + 5 x ^ { 3 }$$ The function has exactly two stationary points when \(x = 0\) and \(x = - \frac { 15 } { 4 }\) 9
    1. Find \(\mathrm { f } ^ { \prime \prime } ( x )\) 9
  2. (ii) Determine the nature of the stationary points.
    Fully justify your answer.
    9
  3. State the range of values of \(x\) for which $$f ( x ) = x ^ { 4 } + 5 x ^ { 3 }$$ is an increasing function.
    9
  4. A second function g is defined for all real values of \(x\) as $$\mathrm { g } ( x ) = x ^ { 4 } - 5 x ^ { 3 }$$ 9
    1. State the single transformation which maps f onto g .
      9
  6. (ii) State the range of values of \(x\) for which g is an increasing function.