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AQA Paper 3 2022 June Q18

12 marks Moderate -0.8

18 In a particular year, the height of a male athlete at the Summer Olympics has a mean 1.78 metres and standard deviation 0.23 metres. The heights of $95 \%$ of male athletes are between 1.33 metres and 2.22 metres.
18

Comment on whether a normal distribution may be suitable to model the height of a male athlete at the Summer Olympics in this particular year.
18
You may assume that the height of a male athlete at the Summer Olympics may be modelled by a normal distribution with mean 1.78 metres and standard deviation 0.23 metres. 18
1. Find the probability that the height of a randomly selected male athlete is 1.82 metres.
  18
(ii) Find the probability that the height of a randomly selected male athlete is between 1.70 metres and 1.90 metres.
18
(iii) Two male athletes are chosen at random. Calculate the probability that both of their heights are between 1.70 metres and 1.90 metres. 18
The summarised data for the heights, $h$ metres, of a random sample of 40 male athletes at the Winter Olympics is given below. $$\sum h = 69.2 \quad \sum ( h - \bar { h } ) ^ { 2 } = 2.81$$ Use this data to calculate estimates of the mean and standard deviation of the heights of male athletes at the Winter Olympics.
[0pt] [3 marks]
18
Using your answers from part (c), compare the heights of male athletes at the Summer Olympics and male athletes at the Winter Olympics.
[0pt] [2 marks] \includegraphics[max width=\textwidth, alt={}, center]{6ad3bac9-bf08-443d-8be2-b0c26209ffe8-30_2488_1721_219_153} \includegraphics[max width=\textwidth, alt={}, center]{6ad3bac9-bf08-443d-8be2-b0c26209ffe8-31_2492_1722_217_150}
\includegraphics[max width=\textwidth, alt={}]{6ad3bac9-bf08-443d-8be2-b0c26209ffe8-36_2496_1721_214_148}

Sketch the curve with equation $$y = x ^ { 2 } ( 2 x + a )$$ where $a > 0$ \includegraphics[max width=\textwidth, alt={}, center]{6fba7e53-de46-460b-9bef-f1a6962f2e7d-06_1070_922_541_648} 6
The polynomial $\mathrm { p } ( x )$ is given by $$\mathrm { p } ( x ) = x ^ { 2 } ( 2 x + a ) + 36$$ 6
1. It is given that $x + 3$ is a factor of $\mathrm { p } ( x )$ Use the factor theorem to show $a = 2$ 6
(ii) State the transformation which maps the curve with equation $$y = x ^ { 2 } ( 2 x + 2 )$$ onto the curve with equation $$y = x ^ { 2 } ( 2 x + 2 ) + 36$$ 6
(iii) The polynomial $x ^ { 2 } ( 2 x + 2 ) + 36$ can be written as $( x + 3 ) \left( 2 x ^ { 2 } + b x + c \right)$ Without finding the values of $b$ and $c$, use your answers to parts (a) and (b)(ii) to explain why $$b ^ { 2 } < 8 c$$

AQA Paper 3 2023 June Q7

14 marks Standard +0.3

7 A new design for a company logo is to be made from two sectors of a circle, ORP and OQS, and a rhombus OSTR, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{6fba7e53-de46-460b-9bef-f1a6962f2e7d-08_509_584_408_817} The points $P , O$ and $Q$ lie on a straight line and the angle $R O S$ is $\theta$ radians.
A large copy of the logo, with $P Q = 5$ metres, is to be put on a wall.
7

Show that the area of the logo, $A$ square metres, is given by $$A = \frac { 25 } { 8 } ( \pi - \theta + 2 \sin \theta )$$ \section*{-
7
1. Show that the maximum value of $A$ occurs when $\theta = \frac { \pi } { 3 }$ Fully justify your answer.} 7
(ii) Find the exact maximum value of $A$ 7
Without further calculation, state how your answers to parts (b)(i) and (b)(ii) would change if $P Q$ were increased to 10 metres. \includegraphics[max width=\textwidth, alt={}, center]{6fba7e53-de46-460b-9bef-f1a6962f2e7d-11_2488_1716_219_153} Use the substitution $u = x ^ { 5 } + 2$ to show that $$\int _ { 0 } ^ { 1 } \frac { x ^ { 9 } } { \left( x ^ { 5 } + 2 \right) ^ { 3 } } \mathrm {~d} x = \frac { 1 } { 180 }$$

AQA Paper 3 2023 June Q9

17 marks Standard +0.3

9 A water slide is the shape of a curve $P Q$ as shown in Figure 1 below. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{6fba7e53-de46-460b-9bef-f1a6962f2e7d-14_592_1278_427_470}

\end{figure} The curve can be modelled by the parametric equations $$\begin{aligned} & x = t - \frac { 1 } { t } + 4.8 \\ & y = t + \frac { 2 } { t } \end{aligned}$$ where $0.2 \leq t \leq 3$ The horizontal distance from $O$ is $x$ metres. The vertical distance above the point $O$ at ground level is $y$ metres. $P$ is the point where $t = 0.2$ and $Q$ is the point where $t = 3$ 9

To make sure speeds are safe at $Q$, the difference in height between $P$ and $Q$ must be less than 7 metres. Show that the slide meets this safety requirement.
9
1. Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$ 9
(ii) A vertical support, $R S$, is to be added between the ground and the lowest point on the slide as shown in Figure 2 below. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{6fba7e53-de46-460b-9bef-f1a6962f2e7d-16_590_1278_475_470}
\end{figure} Find the length of $R S$ 9
(iii) Find the acute angle the slide makes with the horizontal at $Q$ Give your answer to the nearest degree.
\section*{END OF SECTION A}

AQA Paper 3 2023 June Q10

1 marks Easy -1.8

10 Which of the following is not a possible value for a product moment correlation coefficient? Circle your answer. $$- \frac { 6 } { 5 } \quad - \frac { 3 } { 5 } \quad 0$$

AQA Paper 3 2023 June Q11

1 marks Easy -1.8

11 A and B are mutually exclusive events.
Which one of the following statements must be correct?
Tick ( $\checkmark$ ) one box. $$\begin{aligned} & P ( A \cup B ) = P ( A ) \times P ( B ) \\ & P ( A \cup B ) = P ( A ) - P ( B ) \\ & P ( A \cap B ) = 0 \\ & P ( A \cap B ) = 1 \end{aligned}$$ □
□
□
□ \includegraphics[max width=\textwidth, alt={}, center]{6fba7e53-de46-460b-9bef-f1a6962f2e7d-19_2488_1716_219_153} \begin{center} \begin{tabular}{|l|l|} \hline \begin{tabular}{l}

AQA Paper 3 2023 June Q12

8 marks Moderate -0.8

12
12

\end{tabular} &

It is known that, on average, $40 \%$ of the drivers who take their driving test at a local test centre pass their driving test.

Each day 32 drivers take their driving test at this centre.

The number of drivers who pass their test on a particular day can be modelled by the distribution B (32, 0.4)

State one assumption, in context, required for this distribution to be used.

[1 mark] $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$

Find the probability that exactly 7 of the drivers on a particular day pass their test.

[1 mark] $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$

Find the probability that, at most, 16 of the drivers on a particular day pass their test.

[1 mark] $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$

Find the probability that more than 12 of the drivers on a particular day pass their test.

[2 marks] $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$

\hline \end{tabular} \end{center}

Find the mean number of drivers per day who pass their test.

[1 mark]

Find the standard deviation of the number of drivers per day who pass their test.

AQA Paper 3 2023 June Q13

4 marks Moderate -0.8

13 There are two types of coins in a money box:

20\% are bronze coins
80\% are silver coins

Craig takes out a coin at random and places it back in the money box.
Craig then takes out a second coin at random.
13

Find the probability that both coins were of the same type.
13
Find the probability that both coins are bronze, given that at least one of the coins is bronze. \includegraphics[max width=\textwidth, alt={}, center]{6fba7e53-de46-460b-9bef-f1a6962f2e7d-23_2488_1716_219_153} \begin{center} \begin{tabular}{|l|l|} \hline \begin{tabular}{l}

AQA Paper 3 2023 June Q14

10 marks Easy -1.8

14
14

\end{tabular} &

The mass of aluminium cans recycled each day in a city may be modelled by a normal distribution with mean 24500 kg and standard deviation 5200 kg .

State the probability that the mass of aluminium cans recycled on any given day is not equal to 24500 kg .

[1 mark]

\hline \end{tabular} \end{center} 14

A member of the council claims that if a different sample of 24 days had been used the hypothesis test in part (b) would have given the same result. Comment on the validity of this claim.

AQA Paper 3 2023 June Q15

11 marks Easy -1.2

A random sample of eight cars was selected from the Large Data Set. The masses of these cars, in kilograms, were as follows. $\begin{array} { l l l l l l l l } 950 & 989 & 1247 & 1415 & 1506 & 1680 & 1833 & 2040 \end{array}$ It is given that, for the population of cars in the Large Data Set: $$\begin{aligned} \text { lower quartile } & = 1167 \\ \text { median } & = 1393 \\ \text { upper quartile } & = 1570 \end{aligned}$$ 15
1. It was decided to remove any of the masses which fall outside the following interval. median $- 1.5 \times$ interquartile range $\leq$ mass $\leq$ median $+ 1.5 \times$ interquartile range Show that only one of the eight masses in the sample should be removed.
  15
(ii) Write down the statistical name for the mass that should be removed in part (a)(i).
15

The table shows the probability distribution of the number of previous owners, $N$, for a sample of cars taken from the Large Data Set.

$\boldsymbol { n }$	0	1	2	3	4	5	6 or more
$\mathbf { P } ( \boldsymbol { N } = \boldsymbol { n } )$	0.14	0.37	0.9 k	0.25	0.4 k	1.7 k	0

Find the value of $\mathrm { P } ( 1 \leq N < 5 )$

An expert team is investigating whether there have been any changes in $\mathrm { CO } _ { 2 }$ emissions from all cars taken from the Large Data Set.

The team decided to collect a quota sample of 200 cars to reflect the different years and the different makes of cars in the Large Data Set.

Using your knowledge of the Large Data Set, explain how the team can collect this sample.

\includegraphics[max width=\textwidth, alt={}]{6fba7e53-de46-460b-9bef-f1a6962f2e7d-29_2488_1716_219_153}

\begin{center} \begin{tabular}{|l|} \hline \begin{tabular}{l}

AQA Paper 3 2023 June Q16

9 marks Moderate -0.3

16 A farm supplies apples to a supermarket.
The diameters of the apples, $D$ centimetres, are normally distributed with mean 6.5 and standard deviation 0.73
\end{tabular}
\hline 16

1. \hline [1 mark]
  \hline 16
(ii) Find $\mathrm { P } ( D > 7 )$ \hline [1 mark]
\hline
16
(iii) The supermarket only accepts apples with diameters between 5 cm and 8 cm . Find the proportion of apples that the supermarket accepts.
[1 mark] $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$

\hline \end{tabular} \end{center} 16
The farm also supplies plums to the supermarket. These plums have diameters that are normally distributed.
It is found that $60 \%$ of these plums have a diameter less than 5.9 cm .
It is found that $20 \%$ of these plums have a diameter greater than 6.1 cm .
Find the mean and standard deviation of the diameter, in centimetres, of the plums supplied by the farm.

AQA Paper 3 2023 June Q17

6 marks Standard +0.3

17 A council found that $70 \%$ of its new local businesses made a profit in their first year. The council introduced an incentive scheme for its residents to encourage the use of new local businesses. At the end of the scheme, a random sample of 25 new local businesses was selected and it was found that 21 of them had made a profit in their first year. Using a binomial distribution, investigate, at the $2.5 \%$ level of significance, whether there is evidence of an increase in the proportion of new local businesses making a profit in their first year. \includegraphics[max width=\textwidth, alt={}, center]{6fba7e53-de46-460b-9bef-f1a6962f2e7d-33_2488_1719_219_150} Question number Additional page, if required.
Write the question numbers in the left-hand margin.

AQA Paper 3 2024 June Q1

1 marks Easy -1.8

1 Each of the series below shows the first four terms of a geometric series. Identify the only one of these geometric series that is convergent.
[0pt] [1 mark] Tick $( \checkmark )$ one box. $0.1 + 0.2 + 0.4 + 0.8 + \ldots$ \includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-02_113_113_858_927} $1 - 1 + 1 - 1 + \ldots$ \includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-02_117_117_1014_927} $128 - 64 + 32 - 16 + \ldots$ \includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-02_118_117_1169_927} $1 + 2 + 4 + 8 + \ldots$ □

AQA Paper 3 2024 June Q2

1 marks Easy -1.2

2 The quadratic equation $$4 x ^ { 2 } + b x + 9 = 0$$ has one repeated real root. Find $b$ Circle your answer. $b = 0$ $b = \pm 12$ $b = \pm 13$ $b = \pm 36$

AQA Paper 3 2024 June Q4

2 marks Easy -1.2

4 A curve has equation $y = x ^ { 4 } + 2 ^ { x }$ Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$

AQA Paper 3 2024 June Q5

3 marks Moderate -0.5

5 The diagram below shows a sector of a circle $O A B$. The chord $A B$ divides the sector into a triangle and a shaded segment. Angle $A O B$ is $\frac { \pi } { 6 }$ radians.
The radius of the sector is 18 cm . \includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-06_467_428_614_790} Show that the area of the shaded segment is $$k ( \pi - 3 ) \mathrm { cm } ^ { 2 }$$ where $k$ is an integer to be found. \includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-07_2491_1753_173_123}

AQA Paper 3 2024 June Q6

5 marks Easy -1.2

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

8 marks Moderate -0.8

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

8 marks Moderate -0.3

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

9 marks Standard +0.3

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 $$f ^ { \prime } ( x ) = 15 x ^ { 2 } + 1$$

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\(\boldsymbol { n }\)	0	1	2	3	4	5	6 or more
\(\mathbf { P } ( \boldsymbol { N } = \boldsymbol { n } )\)	0.14	0.37	0.9 k	0.25	0.4 k	1.7 k	0

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AQA Paper 3 2022 June Q18

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