Questions Paper 3 - A-Level Maths

AQA Paper 3 2024 June Q6

Find $\int \left( 6 x ^ { 2 } - \frac { 5 } { \sqrt { x } } \right) \mathrm { d } x$ 6
The gradient of a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 2 } - \frac { 5 } { \sqrt { x } }$$ The curve passes through the point $( 4,90 )$. Find the equation of the curve.

AQA Paper 3 2024 June Q7

7 The graphs with equations $$y = 2 + 3 x - 2 x ^ { 2 } \text { and } x + y = 1$$ are shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-10_791_721_550_719} The graphs intersect at the points $A$ and $B$
7

On the diagram above, shade and label the region, $R$, that is satisfied by the inequalities $$0 \leq y \leq 2 + 3 x - 2 x ^ { 2 }$$ and $$x + y \geq 1$$ 7
Find the exact coordinates of $A$

AQA Paper 3 2024 June Q8

2 marks

8 The temperature $\theta ^ { \circ } \mathrm { C }$ of an oven $t$ minutes after it is switched on can be modelled by the equation $$\theta = 20 \left( 11 - 10 \mathrm { e } ^ { - k t } \right)$$ where $k$ is a positive constant.
Initially the oven is at room temperature.
The maximum temperature of the oven is $T ^ { \circ } \mathrm { C }$
The temperature predicted by the model is shown in the graph below.
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-12_750_1319_870_424} 8

Find the room temperature.
8
Find the value of $T$
[0pt] [2 marks]
Question 8 continues on the next page 8
The oven reaches a temperature of $86 ^ { \circ } \mathrm { C }$ one minute after it is switched on. 8
1. Find the value of $k$.
  8
(ii) Find the time it takes for the temperature of the oven to be within $1 ^ { \circ } \mathrm { C }$ of its maximum.
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-15_2493_1759_173_119} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{deec0d32-b031-4227-bc80-7150a0acbc94-16_805_869_459_651}
\end{figure} The centre of the circle is $P$ and the circle intersects the $y$-axis at $Q$ as shown in Figure 1. The equation of the circle is $$x ^ { 2 } + y ^ { 2 } = 12 y - 8 x - 27$$

AQA Paper 3 2024 June Q9

Express the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$ where $a , b$ and $k$ are constants to be found.
9
State the coordinates of $P$ 9
Find the $y$-coordinate of $Q$
\section*{Question 9 continues on the next page} 9
The line segment $Q R$ is a tangent to the circle as shown in Figure 2 below. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{deec0d32-b031-4227-bc80-7150a0acbc94-18_885_1180_456_495}
\end{figure} The point $R$ has coordinates $( 9 , - 3 )$.
Find the angle QPR
Give your answer in radians to three significant figures.
It is given that $$f ( x ) = 5 x ^ { 3 } + x$$ Use differentiation from first principles to prove that $$f ^ { \prime } ( x ) = 15 x ^ { 2 } + 1$$

AQA Paper 3 2024 June Q11

10 marks

11 The curve $C$ with equation $$y = \left( x ^ { 2 } - 8 x \right) \ln x$$ is defined for $x > 0$ and is shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-20_862_632_502_767} The shaded region, $R$, lies below the $x$-axis and is bounded by $C$ and the $x$-axis.
Show that the area of $R$ can be written as $$p + q \ln 2$$ where $p$ and $q$ are rational numbers to be found.
[0pt] [10 marks]
\section*{END OF SECTION A}

AQA Paper 3 2024 June Q12

1 marks

12 A random sample of 84 students was asked how many revision websites they had visited in the past month. The data is summarised in the table below.

Number of websites	Frequency
0	1
1	4
2	18
3	16
4	5
5	37
6	2
7	1

AQA Paper 3 2024 June Q14

14 The annual cost of energy in 2021 for each of the 350 households in Village A can be modelled by a random variable $\pounds X$ It is given that $$\sum x = 945000 \quad \sum x ^ { 2 } = 2607500000$$ 14

Calculate the mean of $X$. 14
Calculate the standard deviation of $X$.
14
For households in Village B the annual cost of energy in 2021 has mean $\pounds 3100$ and standard deviation £325 Compare the annual cost of energy in 2021 for households in Village A and Village B.

AQA Paper 3 2024 June Q15

7 marks

15 It is given that $$X \sim \mathrm {~B} ( 48,0.175 )$$ 15

Find the mean of $X$
[0pt] [1 mark] 15
Show that the variance of $X$ is 6.93
[0pt] [1 mark] 15
Find $\mathrm { P } ( X < 10 )$
[0pt] [1 mark] 15
$\quad$ Find $\mathrm { P } ( X \geq 6 )$
[0pt] [2 marks]
15
$\quad$ Find $\mathrm { P } ( 9 \leq X \leq 15 )$
[0pt] [2 marks] L
15
The aeroplanes used on a particular route have 48 seats. The proportion of passengers who use this route to travel for business is known to be 17.5\% Make two comments on whether it would be appropriate to use $X$ to model the number of passengers on an aeroplane who are travelling for business using this route.

AQA Paper 3 2024 June Q16

16 A medical student believes that, in adults, there is a negative correlation between the amount of nicotine in their blood stream and their energy level. The student collected data from a random sample of 50 adults. The correlation coefficient between the amount of nicotine in their blood stream and their energy level was - 0.45 Carry out a hypothesis test at the $2.5 \%$ significance level to determine if this sample provides evidence to support the student's belief. For $n = 50$, the critical value for a one-tailed test at the $2.5 \%$ level for the population correlation coefficient is 0.2787

AQA Paper 3 2024 June Q17

17 In 2019, the lengths of new-born babies at a clinic can be modelled by a normal distribution with mean 50 cm and standard deviation 4 cm . 17

This normal distribution is represented in the diagram below. Label the values 50 and 54 on the horizontal axis. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{deec0d32-b031-4227-bc80-7150a0acbc94-29_375_531_644_817} \captionsetup{labelformat=empty} \caption{Length (cm)}
\end{figure} 17
State the probability that the length of a new-born baby is less than 50 cm .
17
Find the probability that the length of a new-born baby is more than 56 cm .
17
Find the probability that the length of a new-born baby is more than 40 cm but less than 60 cm .
17
Determine the length exceeded by 95\% of all new-born babies at the clinic.
17
In 2020, the lengths of 40 new-born babies at the clinic were selected at random.
The total length of the 40 new-born babies was 2060 cm .
Carry out a hypothesis test at the $10 \%$ significance level to investigate whether the mean length of a new-born baby at the clinic in 2020 has increased compared to 2019. You may assume that the length of a new-born baby is still normally distributed with standard deviation 4 cm .

AQA Paper 3 2024 June Q18

4 marks

(ii)
[0pt] [2 marks]
\end{tabular}}
\hline \end{tabular} \end{center}

AQA Paper 3 2024 June Q19

2 marks

19 It is known that 80\% of all diesel cars registered in 2017 had carbon monoxide (CO) emissions less than $0.3 \mathrm {~g} / \mathrm { km }$. Talat decides to investigate whether the proportion of diesel cars registered in 2022 with CO emissions less than $0.3 \mathrm {~g} / \mathrm { km }$ has changed. Talat will carry out a hypothesis test at the 10\% significance level on a random sample of 25 diesel cars registered in 2022. 19

1. State suitable null and alternative hypotheses for Talat's test. 19
(ii) Using a 10\% level of significance, find the critical region for Talat's test.
19
(iii) In his random sample, Talat finds 18 cars with CO emissions less than $0.3 \mathrm {~g} / \mathrm { km }$. State Talat's conclusion in context. 19
Talat now wants to use his random sample of 25 diesel cars, registered in 2022, to investigate whether the proportion of diesel cars in England with CO emissions more than $0.5 \mathrm {~g} / \mathrm { km }$ has changed from the proportion given by the Large Data Set. Using your knowledge of the Large Data Set, give two reasons why it is not possible for Talat to do this.
[0pt] [2 marks]

Edexcel Paper 3 2018 June Q1

Helen believes that the random variable $C$, representing cloud cover from the large data set, can be modelled by a discrete uniform distribution.
1. Write down the probability distribution for $C$.
2. Using this model, find the probability that cloud cover is less than 50\%
Helen used all the data from the large data set for Hurn in 2015 and found that the proportion of days with cloud cover of less than $50 \%$ was 0.315
Comment on the suitability of Helen's model in the light of this information.
Suggest an appropriate refinement to Helen’s model.

Edexcel Paper 3 2018 June Q2

Tessa owns a small clothes shop in a seaside town. She records the weekly sales figures, $\pounds w$, and the average weekly temperature, $t ^ { \circ } \mathrm { C }$, for 8 weeks during the summer.
The product moment correlation coefficient for these data is - 0.915
1. Stating your hypotheses clearly and using a $5 \%$ level of significance, test whether or not the correlation between sales figures and average weekly temperature is negative.
2. Suggest a possible reason for this correlation.
Tessa suggests that a linear regression model could be used to model these data.
State, giving a reason, whether or not the correlation coefficient is consistent with Tessa’s suggestion.
State, giving a reason, which variable would be the explanatory variable. Tessa calculated the linear regression equation as $w = 10755 - 171 t$
Give an interpretation of the gradient of this regression equation.

Edexcel Paper 3 2018 June Q3

In an experiment a group of children each repeatedly throw a dart at a target. For each child, the random variable $H$ represents the number of times the dart hits the target in the first 10 throws.

Peta models $H$ as $\mathrm { B } ( 10,0.1 )$

State two assumptions Peta needs to make to use her model.
Using Peta's model, find $\mathrm { P } ( H \geqslant 4 )$ For each child the random variable $F$ represents the number of the throw on which the dart first hits the target. Using Peta's assumptions about this experiment,
find $\mathrm { P } ( F = 5 )$ Thomas assumes that in this experiment no child will need more than 10 throws for the dart to hit the target for the first time. He models $\mathrm { P } ( F = n )$ as $$\mathrm { P } ( F = n ) = 0.01 + ( n - 1 ) \times \alpha$$ where $\alpha$ is a constant.
Find the value of $\alpha$
Using Thomas' model, find $\mathrm { P } ( F = 5 )$
Explain how Peta's and Thomas' models differ in describing the probability that a dart hits the target in this experiment.

Edexcel Paper 3 2018 June Q4

Charlie is studying the time it takes members of his company to travel to the office. He stands by the door to the office from 0840 to 0850 one morning and asks workers, as they arrive, how long their journey was.
1. State the sampling method Charlie used.
2. State and briefly describe an alternative method of non-random sampling Charlie could have used to obtain a sample of 40 workers.
Taruni decided to ask every member of the company the time, $x$ minutes, it takes them to travel to the office.
State the data selection process Taruni used. Taruni's results are summarised by the box plot and summary statistics below.
\includegraphics[max width=\textwidth, alt={}, center]{65e4b254-fb7b-45c2-9702-32f034018193-10_378_1349_1050_367} $$n = 95 \quad \sum x = 4133 \quad \sum x ^ { 2 } = 202294$$
Write down the interquartile range for these data.
Calculate the mean and the standard deviation for these data.
State, giving a reason, whether you would recommend using the mean and standard deviation or the median and interquartile range to describe these data. Rana and David both work for the company and have both moved house since Taruni collected her data. Rana's journey to work has changed from 75 minutes to 35 minutes and David's journey to work has changed from 60 minutes to 33 minutes. Taruni drew her box plot again and only had to change two values.
Explain which two values Taruni must have changed and whether each of these values has increased or decreased.

Edexcel Paper 3 2018 June Q5

The lifetime, $L$ hours, of a battery has a normal distribution with mean 18 hours and standard deviation 4 hours.

Alice's calculator requires 4 batteries and will stop working when any one battery reaches the end of its lifetime.

Find the probability that a randomly selected battery will last for longer than 16 hours. At the start of her exams Alice put 4 new batteries in her calculator. She has used her calculator for 16 hours, but has another 4 hours of exams to sit.
Find the probability that her calculator will not stop working for Alice's remaining exams. Alice only has 2 new batteries so, after the first 16 hours of her exams, although her calculator is still working, she randomly selects 2 of the batteries from her calculator and replaces these with the 2 new batteries.
Show that the probability that her calculator will not stop working for the remainder of her exams is 0.199 to 3 significant figures. After her exams, Alice believed that the lifetime of the batteries was more than 18 hours. She took a random sample of 20 of these batteries and found that their mean lifetime was 19.2 hours.
Stating your hypotheses clearly and using a $5 \%$ level of significance, test Alice's belief.

Edexcel Paper 3 2018 June Q6

6. At time $t$ seconds, where $t \geqslant 0$, a particle $P$ moves in the $x - y$ plane in such a way that its velocity $\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }$ is given by $$\mathbf { v } = t ^ { - \frac { 1 } { 2 } } \mathbf { i } - 4 \mathbf { j }$$ When $t = 1 , P$ is at the point $A$ and when $t = 4 , P$ is at the point $B$.
Find the exact distance $A B$.

Edexcel Paper 3 2018 June Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{65e4b254-fb7b-45c2-9702-32f034018193-20_264_698_246_685} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A wooden crate of mass 20 kg is pulled in a straight line along a rough horizontal floor using a handle attached to the crate.
The handle is inclined at an angle $\alpha$ to the floor, as shown in Figure 1, where $\tan \alpha = \frac { 3 } { 4 }$
The tension in the handle is 40 N .
The coefficient of friction between the crate and the floor is 0.14
The crate is modelled as a particle and the handle is modelled as a light rod.
Using the model,

find the acceleration of the crate. The crate is now pushed along the same floor using the handle. The handle is again inclined at the same angle $\alpha$ to the floor, and the thrust in the handle is 40 N as shown in Figure 2 below. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{65e4b254-fb7b-45c2-9702-32f034018193-20_220_923_1457_571} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure}
Explain briefly why the acceleration of the crate would now be less than the acceleration of the crate found in part (a).

Edexcel Paper 3 2018 June Q8

\hspace{0pt} [In this question $\mathbf { i }$ and $\mathbf { j }$ are horizontal unit vectors due east and due north respectively and position vectors are given relative to the fixed point $O$.]

A particle $P$ moves with constant acceleration.
At time $t = 0$, the particle is at $O$ and is moving with velocity ( $2 \mathbf { i } - 3 \mathbf { j }$ ) $\mathrm { ms } ^ { - 1 }$
At time $t = 2$ seconds, $P$ is at the point $A$ with position vector ( $7 \mathbf { i } - 10 \mathbf { j }$ ) m.

Show that the magnitude of the acceleration of $P$ is $2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ At the instant when $P$ leaves the point $A$, the acceleration of $P$ changes so that $P$ now moves with constant acceleration ( $4 \mathbf { i } + 8.8 \mathbf { j }$ ) $\mathrm { m } \mathrm { s } ^ { - 2 }$ At the instant when $P$ reaches the point $B$, the direction of motion of $P$ is north east.
Find the time it takes for $P$ to travel from $A$ to $B$.

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