Questions Paper 3 - A-Level Maths

Show that the exact value of $\sin \theta$ is $\frac { 4 \sqrt { 14 } } { 15 }$
[0pt] [3 marks] 2
Write down the value of $\theta$ in radians to 3 significant figures.
2
Find the area of the garden that will be covered by the decking.

AQA Paper 3 Specimen Q3

16 marks

3 A circular ornamental garden pond, of radius 2 metres, has weed starting to grow and cover its surface. As the weed grows, it covers an area of $A$ square metres. A simple model assumes that the weed grows so that the rate of increase of its area is proportional to $A$. 3

Show that the area covered by the weed can be modelled by
where $B$ and $k$ are constants and $t$ is time in days since the weed was first noticed.
[0pt] [4 marks] $$A = B \mathrm { e } ^ { k t }$$ 3
When it was first noticed, the weed covered an area of $0.25 \mathrm {~m} ^ { 2 }$. Twenty days later the weed covered an area of $0.5 \mathrm {~m} ^ { 2 }$ 3
1. State the value of $B$.
  [0pt] [1 mark] 3
(ii) Show that the model for the area covered by the weed can be written as $$A = 2 ^ { \frac { t } { 20 } - 2 }$$ [4 marks]
Question 3 continues on the next page 3
(iii) How many days does it take for the weed to cover half of the surface of the pond?
[0pt] [2 marks]
3
State one limitation of the model.
3
Suggest one refinement that could be made to improve the model.
$4 \quad \int _ { 1 } ^ { 2 } x ^ { 3 } \ln ( 2 x ) \mathrm { d } x$ can be written in the form $p \ln 2 + q$, where $p$ and $q$ are rational numbers. Find $p$ and $q$.
[0pt] [5 marks]

AQA Paper 3 Specimen Q5

Find the first three terms, in ascending powers of $x$, in the binomial expansion of $( 1 + 6 x ) ^ { \frac { 1 } { 3 } }$
5
Use the result from part (a) to obtain an approximation to $\sqrt [ 3 ] { 1.18 }$ giving your answer to 4 decimal places.
5
Explain why substituting $x = \frac { 1 } { 2 }$ into your answer to part (a) does not lead to a valid approximation for $\sqrt [ 3 ] { 4 }$.

AQA Paper 3 Specimen Q6

8 marks

6 Find the value of $\int _ { 1 } ^ { 2 } \frac { 6 x + 1 } { 6 x ^ { 2 } - 7 x + 2 } \mathrm {~d} x$, expressing your answer in the form
$m \ln 2 + n \ln 3$, where $m$ and $n$ are integers.
[0pt] [8 marks]

AQA Paper 3 Specimen Q7

8 marks

7 The diagram shows part of the graph of $y = \mathrm { e } ^ { - x ^ { 2 } }$
\includegraphics[max width=\textwidth, alt={}, center]{be9ad375-d3cd-488c-9e3b-a4cd035d0d1d-10_376_940_607_392} The graph is formed from two convex sections, where the gradient is increasing, and one concave section, where the gradient is decreasing. 7

Find the values of $x$ for which the graph is concave. 7
The finite region bounded by the $x$-axis and the lines $x = 0.1$ and $x = 0.5$ is shaded.
\includegraphics[max width=\textwidth, alt={}, center]{be9ad375-d3cd-488c-9e3b-a4cd035d0d1d-11_372_937_584_355} Use the trapezium rule, with 4 strips, to find an estimate for $\int _ { 0.1 } ^ { 0.5 } e ^ { - x ^ { 2 } } d x$ Give your estimate to four decimal places.
[0pt] [3 marks]
7
Explain with reference to your answer in part (a), why the answer you found in part (b) is an underestimate.
[0pt] [2 marks] 7
By considering the area of a rectangle, and using your answer to part (b), prove that the shaded area is 0.4 correct to 1 decimal place.
[0pt] [3 marks]
\section*{END OF SECTION A
TURN OVER FOR SECTION B}

AQA Paper 3 Specimen Q8

2 marks

8 Edna wishes to investigate the energy intake from eating out at restaurants for the households in her village. She wants a sample of 100 households.
She has a list of all 2065 households in the village.
Ralph suggests this selection method.
"Number the households 0000 to 2064. Obtain 100 different four-digit random numbers between 0000 and 2064 and select the corresponding households for inclusion in the investigation." 8

What is the population for this investigation?
Circle your answer.
[0pt] [1 mark] Edna and Ralph
The 2065
The energy households intake for the The 100 in the village village from households eating out selected 8
What is the sampling method suggested by Ralph?
Circle your answer.
[0pt] [1 mark] Opportunity Random number Continuous random variable Simple random

AQA Paper 3 Specimen Q9

2 marks

9 A survey has found that, of the 2400 households in Growmore, 1680 eat home-grown fruit and vegetables. 9

Using the binomial distribution, find the probability that, out of a random sample of 25 households in Growmore, exactly 22 eat home-grown fruit and vegetables.
[0pt] [2 marks]
9
Give a reason why you would not expect your calculation in part (a) to be valid for the 25 households in Gifford Terrace, a residential road in Growmore.
Shona calculated four correlation coefficients using data from the Large Data Set.
In each case she calculated the correlation coefficient between the masses of the cars and the CO2 emissions for varying sample sizes. A summary of these calculations, labelled A to D, are listed in the table below.
\cline { 2 - 3 } \multicolumn{1}{c|}{} Sample size
Correlation
coefficient
A 3827 0.088
B 3735 0.246
C 24 0.400
D 1250 - 1.183
Shona would like to use calculation A to test whether there is evidence of positive correlation between mass and CO2 emissions. She finds the critical value for a one-tailed test at the 5\% level for a sample of size 3827 is 0.027

AQA Paper 3 Specimen Q10

3 marks

1. State appropriate hypotheses for Shona to use in her test. 10
(ii) Determine if there is sufficient evidence to reject the null hypothesis.
Fully justify your answer.
[0pt] [1 mark] 10
Shona's teacher tells her to remove calculation $D$ from the table as it is incorrect.
Explain how the teacher knew it was incorrect.
[0pt] [1 mark] 10
Before performing calculation B, Shona cleaned the data. She removed all cars from the Large Data Set that had incorrect masses. Using your knowledge of the large data set, explain what was incorrect about the masses which were removed from the calculation.
[0pt] [1 mark] 10
Apart from CO 2 and CO emissions, state one other type of emission that Shona could investigate using the Large Data Set. 10
Wesley claims that calculation C shows that a heavier car causes higher CO 2 emissions. Give two reasons why Wesley's claim may be incorrect.

AQA Paper 3 Specimen Q11

3 marks

11 Terence owns a local shop. His shop has three checkouts, at least one of which is always staffed. A regular customer observed that the probability distribution for $N$, the number of checkouts that are staffed at any given time during the spring, is $$\mathrm { P } ( N = n ) = \left\{ \begin{array} { c c } \frac { 3 } { 4 } \left( \frac { 1 } { 4 } \right) ^ { n - 1 } & \text { for } n = 1,2
k & \text { for } n = 3 \end{array} \right.$$ 11

Find the value of $k$.
[0pt] [1 mark]
11
Find the probability that a customer, visiting Terence's shop during the spring, will find at least 2 checkouts staffed.
[0pt] [2 marks]

AQA Paper 3 Specimen Q12

10 marks

12 During the 2006 Christmas holiday, John, a maths teacher, realised that he had fallen ill during 65\% of the Christmas holidays since he had started teaching. In January 2007, he increased his weekly exercise to try to improve his health.
For the next 7 years, he only fell ill during 2 Christmas holidays. 12

Using a binomial distribution, investigate, at the $5 \%$ level of significance, whether there is evidence that John's rate of illness during the Christmas holidays had decreased since increasing his weekly exercise.
[0pt] [6 marks] 12
State two assumptions, regarding illness during the Christmas holidays, that are necessary for the distribution you have used in part (a) to be valid. For each assumption, comment, in context, on whether it is likely to be correct.
[0pt] [4 marks]

AQA Paper 3 Specimen Q13

7 marks

13 In the South West region of England, 100 households were randomly selected and, for each household, the weekly expenditure, $\pounds X$, per person on food and drink was recorded. The maximum amount recorded was $\pounds 40.48$ and the minimum amount recorded was £22.00 The results are summarised below, where $\bar { X }$ denotes the sample mean. $$\sum x = 3046.14 \quad \sum ( x - \bar { x } ) ^ { 2 } = 1746.29$$ 13

1. Find the mean of $X$
  Find the standard deviation of $X$
  [0pt] [2 marks] 13
(ii) Using your results from part (a)(i) and other information given, explain why the normal distribution can be used to model $X$.
[0pt] [2 marks] 13
(iii) Find the probability that a household in the South West spends less than $\pounds 25.00$ on food and drink per person per week.
13
For households in the North West of England, the weekly expenditure, $\pounds Y$, per person on food and drink can be modelled by a normal distribution with mean $\pounds 29.55$ It is known that $\mathrm { P } ( Y < 30 ) = 0.55$
Find the standard deviation of $Y$, giving your answer to one decimal place.
[0pt] [3 marks]

AQA Paper 3 Specimen Q14

7 marks

14 A survey during 2013 investigated mean expenditure on bread and on alcohol.
The 2013 survey obtained information from 12144 adults.
The survey revealed that the mean expenditure per adult per week on bread was 127p.
14

For 2012, it is known that the expenditure per adult per week on bread had mean 123p, and a standard deviation of 70p. 14
1. Carry out a hypothesis test, at the $5 \%$ significance level, to investigate whether the mean expenditure per adult per week on bread changed from 2012 to 2013. Assume that the survey data is a random sample taken from a normal distribution.
  [0pt] [5 marks] 14
(ii) Calculate the greatest and least values for the sample mean expenditure on bread per adult per week for 2013 that would have resulted in acceptance of the null hypothesis for the test you carried out in part (a)(i). Give your answers to two decimal places.
[0pt] [2 marks] 14
The 2013 survey revealed that the mean expenditure per adult, per week on alcohol was 324p. The mean expenditure per adult per week on alcohol for 2009 was 307p.
A test was carried out on the following hypotheses relating to mean expenditure per adult per week on alcohol in 2013.
$\mathrm { H } _ { 0 } : \mu = 307$
$\mathrm { H } _ { 1 } : \mu \neq 307$
This test resulted in the null hypothesis, $\mathrm { H } _ { 0 }$, being rejected.
State, with a reason, whether the test result supports the following statements:
14
1. the mean UK expenditure on alcohol per adult per week increased by 17 p from 2009 to 2013; 14
(ii) the mean UK consumption of alcohol per adult per week changed from 2009 to 2013.

AQA Paper 3 Specimen Q15

6 marks

15 A sample of 200 households was obtained from a small town.
Each household was asked to complete a questionnaire about their purchases of takeaway food.
$A$ is the event that a household regularly purchases Indian takeaway food.
$B$ is the event that a household regularly purchases Chinese takeaway food.
It was observed that $\mathrm { P } ( B \mid A ) = 0.25$ and $\mathrm { P } ( A \mid B ) = 0.1$
Of these households, 122 indicated that they did not regularly purchase Indian or Chinese takeaway food.
A household is selected at random from those in the sample.
Find the probability that the household regularly purchases both Indian and Chinese takeaway food.
[0pt] [6 marks]

Edexcel Paper 3 2019 June Q1

\hspace{0pt} [In this question position vectors are given relative to a fixed origin $O$ ]

At time $t$ seconds, where $t \geqslant 0$, a particle, $P$, moves so that its velocity $\mathbf { v ~ m ~ s } ^ { - 1 }$ is given by $$\mathbf { v } = 6 t \mathbf { i } - 5 t ^ { \frac { 3 } { 2 } } \mathbf { j }$$ When $t = 0$, the position vector of $P$ is $( - 20 \mathbf { i } + 20 \mathbf { j } ) \mathrm { m }$.

Find the acceleration of $P$ when $t = 4$
Find the position vector of $P$ when $t = 4$

Edexcel Paper 3 2019 June Q2

A particle, $P$, moves with constant acceleration $( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }$

At time $t = 0$, the particle is at the point $A$ and is moving with velocity ( $- \mathbf { i } + 4 \mathbf { j }$ ) $\mathrm { m } \mathrm { s } ^ { - 1 }$
At time $t = T$ seconds, $P$ is moving in the direction of vector ( $3 \mathbf { i } - 4 \mathbf { j }$ )

Find the value of $T$. At time $t = 4$ seconds, $P$ is at the point $B$.
Find the distance $A B$.

Edexcel Paper 3 2019 June Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8399dae8-1b9d-4564-a95b-7ab857368b86-06_339_812_242_628} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Two blocks, $A$ and $B$, of masses $2 m$ and $3 m$ respectively, are attached to the ends of a light string. Initially $A$ is held at rest on a fixed rough plane.
The plane is inclined at angle $\alpha$ to the horizontal ground, where $\tan \alpha = \frac { 5 } { 12 }$
The string passes over a small smooth pulley, $P$, fixed at the top of the plane.
The part of the string from $A$ to $P$ is parallel to a line of greatest slope of the plane. Block $B$ hangs freely below $P$, as shown in Figure 1. The coefficient of friction between $A$ and the plane is $\frac { 2 } { 3 }$
The blocks are released from rest with the string taut and $A$ moves up the plane.
The tension in the string immediately after the blocks are released is $T$.
The blocks are modelled as particles and the string is modelled as being inextensible.

Show that $T = \frac { 12 m g } { 5 }$ After $B$ reaches the ground, $A$ continues to move up the plane until it comes to rest before reaching $P$.
Determine whether $A$ will remain at rest, carefully justifying your answer.
Suggest two refinements to the model that would make it more realistic.

Edexcel Paper 3 2019 June Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8399dae8-1b9d-4564-a95b-7ab857368b86-10_417_844_244_612} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A ramp, $A B$, of length 8 m and mass 20 kg , rests in equilibrium with the end $A$ on rough horizontal ground. The ramp rests on a smooth solid cylindrical drum which is partly under the ground. The drum is fixed with its axis at the same horizontal level as $A$. The point of contact between the ramp and the drum is $C$, where $A C = 5 \mathrm {~m}$, as shown in Figure 2. The ramp is resting in a vertical plane which is perpendicular to the axis of the drum, at an angle $\theta$ to the horizontal, where $\tan \theta = \frac { 7 } { 24 }$ The ramp is modelled as a uniform rod.

Explain why the reaction from the drum on the ramp at point $C$ acts in a direction which is perpendicular to the ramp.
Find the magnitude of the resultant force acting on the ramp at $A$. The ramp is still in equilibrium in the position shown in Figure 2 but the ramp is not now modelled as being uniform. Given that the centre of mass of the ramp is assumed to be closer to $A$ than to $B$,
state how this would affect the magnitude of the normal reaction between the ramp and the drum at $C$.

Edexcel Paper 3 2019 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8399dae8-1b9d-4564-a95b-7ab857368b86-14_223_855_239_605} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} The points $A$ and $B$ lie 50 m apart on horizontal ground.
At time $t = 0$ two small balls, $P$ and $Q$, are projected in the vertical plane containing $A B$.
Ball $P$ is projected from $A$ with speed $20 \mathrm {~ms} ^ { - 1 }$ at $30 ^ { \circ }$ to $A B$.
Ball $Q$ is projected from $B$ with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at angle $\theta$ to $B A$, as shown in Figure 3.
At time $t = 2$ seconds, $P$ and $Q$ collide.
Until they collide, the balls are modelled as particles moving freely under gravity.

Find the velocity of $P$ at the instant before it collides with $Q$.
Find
1. the size of angle $\theta$,
2. the value of $u$.
State one limitation of the model, other than air resistance, that could affect the accuracy of your answers.

Edexcel Paper 3 2019 June Q1

Three bags, $A , B$ and $C$, each contain 1 red marble and some green marbles.

Bag $A$ contains 1 red marble and 9 green marbles only
Bag $B$ contains 1 red marble and 4 green marbles only
Bag $C$ contains 1 red marble and 2 green marbles only
Sasha selects at random one marble from bag $A$.
If he selects a red marble, he stops selecting.
If the marble is green, he continues by selecting at random one marble from bag $B$.
If he selects a red marble, he stops selecting.
If the marble is green, he continues by selecting at random one marble from bag $C$.

Draw a tree diagram to represent this information.
Find the probability that Sasha selects 3 green marbles.
Find the probability that Sasha selects at least 1 marble of each colour.
Given that Sasha selects a red marble, find the probability that he selects it from bag $B$.

Edexcel Paper 3 2019 June Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d1eaaae7-c1dc-4aee-ab54-59f35519a7a4-06_321_1822_294_127} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The partially completed box plot in Figure 1 shows the distribution of daily mean air temperatures using the data from the large data set for Beijing in 2015 An outlier is defined as a value
more than $1.5 \times$ IQR below $Q _ { 1 }$ or
more than $1.5 \times$ IQR above $Q _ { 3 }$
The three lowest air temperatures in the data set are $7.6 ^ { \circ } \mathrm { C } , 8.1 ^ { \circ } \mathrm { C }$ and $9.1 ^ { \circ } \mathrm { C }$
The highest air temperature in the data set is $32.5 ^ { \circ } \mathrm { C }$

Complete the box plot in Figure 1 showing clearly any outliers.
Using your knowledge of the large data set, suggest from which month the two outliers are likely to have come. Using the data from the large data set, Simon produced the following summary statistics for the daily mean air temperature, $x ^ { \circ } \mathrm { C }$, for Beijing in 2015 $$n = 184 \quad \sum x = 4153.6 \quad \mathrm {~S} _ { x x } = 4952.906$$
Show that, to 3 significant figures, the standard deviation is $5.19 ^ { \circ } \mathrm { C }$ Simon decides to model the air temperatures with the random variable $$T \sim \mathrm {~N} \left( 22.6,5.19 ^ { 2 } \right)$$
Using Simon's model, calculate the 10th to 90th interpercentile range. Simon wants to model another variable from the large data set for Beijing using a normal distribution.
State two variables from the large data set for Beijing that are not suitable to be modelled by a normal distribution. Give a reason for each answer.
\includegraphics[max width=\textwidth, alt={}, center]{d1eaaae7-c1dc-4aee-ab54-59f35519a7a4-09_473_1813_2161_127}
(Total for Question 2 is 11 marks)

Edexcel Paper 3 2019 June Q3

3. Barbara is investigating the relationship between average income (GDP per capita), $x$ US dollars, and average annual carbon dioxide ( $\mathrm { CO } _ { 2 }$ ) emissions, $y$ tonnes, for different countries. She takes a random sample of 24 countries and finds the product moment correlation coefficient between average annual $\mathrm { CO } _ { 2 }$ emissions and average income to be 0.446

Stating your hypotheses clearly, test, at the $5 \%$ level of significance, whether or not the product moment correlation coefficient for all countries is greater than zero. Barbara believes that a non-linear model would be a better fit to the data.
She codes the data using the coding $m = \log _ { 10 } x$ and $c = \log _ { 10 } y$ and obtains the model $c = - 1.82 + 0.89 m$ The product moment correlation coefficient between $c$ and $m$ is found to be 0.882
Explain how this value supports Barbara's belief.
Show that the relationship between $y$ and $x$ can be written in the form $y = a x ^ { n }$ where $a$ and $n$ are constants to be found.

Edexcel Paper 3 2019 June Q4

Magali is studying the mean total cloud cover, in oktas, for Leuchars in 1987 using data from the large data set. The daily mean total cloud cover for all 184 days from the large data set is summarised in the table below.

Daily mean total cloud cover (oktas)	0	1	2	3	4	5	6	7	8
Frequency (number of days)	0	1	4	7	10	30	52	52	28

One of the 184 days is selected at random.

Find the probability that it has a daily mean total cloud cover of 6 or greater. Magali is investigating whether the daily mean total cloud cover can be modelled using a binomial distribution. She uses the random variable $X$ to denote the daily mean total cloud cover and believes that $X \sim \mathrm {~B} ( 8,0.76 )$ Using Magali's model,
1. find $\mathrm { P } ( X \geqslant 6 )$
2. find, to 1 decimal place, the expected number of days in a sample of 184 days with a daily mean total cloud cover of 7
Explain whether or not your answers to part (b) support the use of Magali's model. There were 28 days that had a daily mean total cloud cover of 8 For these 28 days the daily mean total cloud cover for the following day is shown in the table below.
Daily mean total cloud cover (oktas) 0 1 2 3 4 5 6 7 8
Frequency (number of days) 0 0 1 1 2 1 5 9 9
Find the proportion of these days when the daily mean total cloud cover was 6 or greater.
Comment on Magali's model in light of your answer to part (d).

Edexcel Paper 3 2019 June Q5

A machine puts liquid into bottles of perfume. The amount of liquid put into each bottle, $D \mathrm { ml }$, follows a normal distribution with mean 25 ml

Given that 15\% of bottles contain less than 24.63 ml

find, to 2 decimal places, the value of $k$ such that $\mathrm { P } ( 24.63 < D < k ) = 0.45$ A random sample of 200 bottles is taken.
Using a normal approximation, find the probability that fewer than half of these bottles contain between 24.63 ml and $k \mathrm { ml }$ The machine is adjusted so that the standard deviation of the liquid put in the bottles is now 0.16 ml Following the adjustments, Hannah believes that the mean amount of liquid put in each bottle is less than 25 ml She takes a random sample of 20 bottles and finds the mean amount of liquid to be 24.94 ml
Test Hannah's belief at the $5 \%$ level of significance. You should state your hypotheses clearly.

Edexcel Paper 3 2022 June Q1

\hspace{0pt} [In this question, position vectors are given relative to a fixed origin.] At time $t$ seconds, where $t > 0$, a particle $P$ has velocity $\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }$ where

$$\mathbf { v } = 3 t ^ { 2 } \mathbf { i } - 6 t ^ { \frac { 1 } { 2 } } \mathbf { j }$$

Find the speed of $P$ at time $t = 2$ seconds.
Find an expression, in terms of $t , \mathbf { i }$ and $\mathbf { j }$, for the acceleration of $P$ at time $t$ seconds, where $t > 0$ At time $t = 4$ seconds, the position vector of $P$ is ( $\mathbf { i } - 4 \mathbf { j }$ ) m.
Find the position vector of $P$ at time $t = 1$ second.

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Edexcel Paper 3 2019 June Q1

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