AQA Paper 3 (Paper 3) 2020 June

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

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
2. 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.

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.

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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
2. 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.

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 \(\cos \theta = 0\)

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.
\(0.11 \quad 0.46 \quad 0.54 \quad 0.89\)

11 The table below shows the temperature on Mount Everest on the first day of each month. 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}

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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

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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.

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. 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}

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.
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1. Calculate how many of each age group should be included in his sample.
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2. Explain how he could collect the random sample of members under 30 years old.

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]

17 The lifetime of Zaple smartphone batteries, \(X\) hours, is normally distributed with mean 8 hours and standard deviation 1.5 hours. 17

1. 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.

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1. Find the probability that:
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2. 1. a box contains exactly 5 shirts with a colour defect
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3. (ii) a box contains fewer than 15 shirts with a sewing defect
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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
6. 1. Using a \(5 \%\) level of significance, find the critical region for \(x\).
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7. (ii) In her sample she finds 13 shirts with a fabric defect. Complete the test stating her conclusion in context.