Questions Paper 3 - A-Level Maths

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AQA Paper 3 2019 June Q7

Express $\frac { 4 x + 3 } { ( x - 1 ) ^ { 2 } }$ in the form $\frac { A } { x - 1 } + \frac { B } { ( x - 1 ) ^ { 2 } }$ 7
Show that $$\int _ { 3 } ^ { 4 } \frac { 4 x + 3 } { ( x - 1 ) ^ { 2 } } \mathrm {~d} x = p + \ln q$$ where $p$ and $q$ are rational numbers.

AQA Paper 3 2019 June Q8

8 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 $75 ^ { \circ } \mathrm { C }$ cools so that the temperature, $\theta ^ { \circ } \mathrm { C }$, of the liquid at time $t$ minutes can be modelled by the equation $$\theta = 5 \left( 4 + \lambda \mathrm { e } ^ { - k t } \right)$$ where $\lambda$ and $k$ are constants. After 2 minutes the temperature falls to $68 ^ { \circ } \mathrm { C }$.
8

Find the temperature of the liquid after 15 minutes.
Give your answer to three significant figures.
8
1. Find the room temperature of the laboratory, giving a reason for your answer.
  8
(ii) Find the time taken in minutes for the liquid to cool to $1 ^ { \circ } \mathrm { C }$ above the room temperature of the laboratory.
8
Explain why the model might need to be changed if the experiment was conducted in a different place.

AQA Paper 3 2019 June Q9

9 A curve has equation $$x ^ { 2 } y ^ { 2 } + x y ^ { 4 } = 12$$ 9

Prove that the curve does not intersect the coordinate axes.
9
1. Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 2 x y + y ^ { 3 } } { 2 x ^ { 2 } + 4 x y ^ { 2 } }$
  9
(ii) Prove that the curve has no stationary points.
9
(iii) In the case when $x > 0$, find the equation of the tangent to the curve when $y = 1$

Rower	Jess	Nell	Liv	Neve	Ann	Tori	Maya	Kath	Darcy	Jen
Height (cm)	162	169	172	156	146	161	159	164	157	160

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.
12
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.

AQA Paper 3 2019 June Q14

14 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 exercise	Medium exercise	High exercise
Back trouble	14	7	10
Stress	38	14	5
Depression	9	2	1
Headache/Migraine	4	5	5

Find the probability that a randomly selected teacher:
14
1. suffers from back trouble and has a high exercise level;
  14
(ii) suffers from depression.
14
(iii) suffers from stress, given that they have a low exercise level.
14
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.

AQA Paper 3 2019 June Q15

15 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 level	$5 \%$	$2.5 \%$	$1 \%$	$0.5 \%$
2-tailed test significance level	$10 \%$	$5 \%$	$2 \%$	$1 \%$
Critical value	0.549	0.632	0.716	0.765

Determine what Jamal's conclusion to his investigation should be, justifying your answer.
\includegraphics[max width=\textwidth, alt={}, center]{c8a41c47-bbda-4e91-a7a2-d0bcf6a46f25-21_2488_1716_219_153}

AQA Paper 3 2019 June Q16

4 marks

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[max width=\textwidth, alt={}, center]{c8a41c47-bbda-4e91-a7a2-d0bcf6a46f25-22_821_1349_406_347} 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.
[0pt] [2 marks]
16
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.
16
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.
[0pt] [2 marks]
\includegraphics[max width=\textwidth, alt={}, center]{c8a41c47-bbda-4e91-a7a2-d0bcf6a46f25-25_2488_1716_219_153}

AQA Paper 3 2019 June Q17

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

Find the mean and standard deviation of $X$.
1. Find $\mathrm { P } ( X \neq 35 )$
  17
(ii) Find $\mathrm { P } ( X < 35 )$
17
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 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

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

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

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
(ii) State the coordinates of this point of intersection.

AQA Paper 3 2020 June Q5

5 marks

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

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
Find the percentage of the atoms remaining after the first week. Give your answer to two significant figures.
5
Explain why the model can only provide an estimate for the number of remaining atoms.
5
Explain why the model is invalid in the long run.

AQA Paper 3 2020 June Q8

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

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
(ii) State the coordinates of this point of intersection.

AQA Paper 3 2020 June Q9

For $\cos \theta \neq 0$, prove that $$\operatorname { cosec } 2 \theta + \cot 2 \theta = \cot \theta$$ 9
Explain why $$\cot \theta \neq \operatorname { cosec } 2 \theta + \cot 2 \theta$$ when $\cos \theta = 0$

AQA Paper 3 2020 June Q10

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

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

Month	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
Temperature $\left( { } ^ { \circ } \mathbf { C } \right)$	- 17	- 16	- 14	- 9	- 2	2	6	5	- 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

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

2 marks

13
12

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
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

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.

4.25	3.90	4.15	3.95	4.20	4.15
5.00	3.85	4.25	4.05	3.80	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.
\includegraphics[max width=\textwidth, alt={}, center]{076ea8e9-9295-46d2-b5f9-b27fa969129e-21_2488_1728_219_141}

AQA Paper 3 2020 June Q15

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

Calculate how many of each age group should be included in his sample.
15
Explain how he could collect the random sample of members under 30 years old.

AQA Paper 3 2020 June Q16

4 marks

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

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
(ii) Find $\mathrm { P } ( 6 < X < 10 )$
17
Determine the lifetime exceeded by $90 \%$ of Zaple smartphone batteries.
17
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.

1-tailed test significance level	\(5 \%\)	\(2.5 \%\)	\(1 \%\)	\(0.5 \%\)
2-tailed test significance level	\(10 \%\)	\(5 \%\)	\(2 \%\)	\(1 \%\)
Critical value	0.549	0.632	0.716	0.765

Month	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
Temperature \(\left( { } ^ { \circ } \mathbf { C } \right)\)	- 17	- 16	- 14	- 9	- 2	2	6	5	- 3	- 4	- 11	- 18

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