Questions - AQA Paper 3 - A-Level Maths

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.

AQA Paper 3 2020 June Q18

Find the probability that:
18
1. a box contains exactly 5 shirts with a colour defect
  18
(ii) a box contains fewer than 15 shirts with a sewing defect
18
(iii) a box contains at least 20 shirts which do not have a fabric defect.
Question 18 continues on the next page 18
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
1. Using a $5 \%$ level of significance, find the critical region for $x$.
  18
(ii) In her sample she finds 13 shirts with a fabric defect. Complete the test stating her conclusion in context.

AQA Paper 3 2021 June Q1

1 marks

1 The graph of $y = \arccos x$ is shown.
\includegraphics[max width=\textwidth, alt={}, center]{0abb6343-017d-4629-8a2d-cfb405dc2d14-02_599_856_758_591} State the coordinates of the end point $P$.
Circle your answer.
[0pt] [1 mark] $$( - \pi , 1 ) \quad ( - 1 , \pi ) \quad \left( - \frac { \pi } { 2 } , 1 \right) \quad \left( - 1 , \frac { \pi } { 2 } \right)$$

AQA Paper 3 2021 June Q2

2 Simplify fully $$\frac { ( x + 3 ) ( 6 - 2 x ) } { ( x - 3 ) ( 3 + x ) } \quad \text { for } x \neq \pm 3$$ Circle your answer.
-2
2
$\frac { ( 6 - 2 x ) } { ( x - 3 ) }$
$\frac { ( 2 x - 6 ) } { ( x - 3 ) }$
$3 \mathrm { f } ( x ) = 3 x ^ { 2 }$
Obtain $\lim _ { h \rightarrow 0 } \frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }$
Circle your answer. $$\frac { 3 h ^ { 2 } } { h } \quad x ^ { 3 } \quad \frac { 3 ( x + h ) ^ { 2 } - 3 x ^ { 2 } } { h } \quad 6 x$$

AQA Paper 3 2021 June Q4

Show that the first three terms, in descending powers of $x$, of the expansion of $$( 2 x - 3 ) ^ { 10 }$$ are given by $$1024 x ^ { 10 } + p x ^ { 9 } + q x ^ { 8 }$$ where $p$ and $q$ are integers to be found.
4
Find the constant term in the expansion of $$\left( 2 x - \frac { 3 } { x } \right) ^ { 10 }$$

AQA Paper 3 2021 June Q5

5 marks

5 A gardener is creating flowerbeds in the shape of sectors of circles. The gardener uses an edging strip around the perimeter of each of the flowerbeds.
The cost of the edging strip is $\pounds 1.80$ per metre and can be purchased for any length.
One of the flowerbeds has a radius of 5 metres and an angle at the centre of 0.7 radians as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{0abb6343-017d-4629-8a2d-cfb405dc2d14-05_544_579_676_730} 5

1. Find the area of this flowerbed.
  \section*{Question 5 continues on the next page} 5
(ii) Find the cost of the edging strip required for this flowerbed.
5
A flowerbed is to be made with an area of $20 \mathrm {~m} ^ { 2 }$
5
1. Show that the cost, $\pounds C$, of the edging strip required for this flowerbed is given by $$C = \frac { 18 } { 5 } \left( \frac { 20 } { r } + r \right)$$ where $r$ is the radius measured in metres.
  5
(ii) Hence, show that the minimum cost of the edging strip for this flowerbed occurs when $r \approx 4.5$ Fully justify your answer.
[0pt] [5 marks]

AQA Paper 3 2021 June Q6

4 marks

6 Given that $x > 0$ and $x \neq 25$, fully simplify $$\frac { 10 + 5 x - 2 x ^ { \frac { 1 } { 2 } } - x ^ { \frac { 3 } { 2 } } } { 5 - \sqrt { x } }$$ Fully justify your answer.
[0pt] [4 marks]
\includegraphics[max width=\textwidth, alt={}, center]{0abb6343-017d-4629-8a2d-cfb405dc2d14-09_2488_1716_219_153}

AQA Paper 3 2021 June Q7

7 A building has a leaking roof and, while it is raining, water drips into a 12 litre bucket. When the rain stops, the bucket is one third full.
Water continues to drip into the bucket from a puddle on the roof.
In the first minute after the rain stops, 30 millilitres of water drips into the bucket.
In each subsequent minute, the amount of water that drips into the bucket reduces by $2 \%$. During the $n$th minute after the rain stops, the volume of water that drips into the bucket is $W _ { n }$ millilitres. 7

$\quad$ Find $W _ { 2 }$
7
Explain why $$W _ { n } = A \times 0.98 ^ { n - 1 }$$ and state the value of $A$.
7
Find the increase in the water in the bucket 15 minutes after the rain stops. Give your answer to the nearest millilitre. 7
Assuming it does not start to rain again, find the maximum amount of water in the bucket.
7
After several hours the water has stopped dripping.
Give two reasons why the amount of water in the bucket is not as much as the answer found in part (d).

AQA Paper 3 2021 June Q8

6 marks

8 Given that $$\int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 3 } } x \cos x d x = a \pi + b$$ find the exact value of $a$ and the exact value of $b$. Fully justify your answer.
[0pt] [6 marks]

AQA Paper 3 2021 June Q9

9 A function f is defined for all real values of $x$ as $$f ( x ) = x ^ { 4 } + 5 x ^ { 3 }$$ The function has exactly two stationary points when $x = 0$ and $x = - \frac { 15 } { 4 }$
9

1. Find $\mathrm { f } ^ { \prime \prime } ( x )$
  9
(ii) Determine the nature of the stationary points.
Fully justify your answer.
9
State the range of values of $x$ for which $$f ( x ) = x ^ { 4 } + 5 x ^ { 3 }$$ is an increasing function.
9
A second function g is defined for all real values of $x$ as $$\mathrm { g } ( x ) = x ^ { 4 } - 5 x ^ { 3 }$$ 9
1. State the single transformation which maps f onto g .
  9
(ii) State the range of values of $x$ for which g is an increasing function.

AQA Paper 3 2021 June Q10

10 Anke has collected data from 30 similar-sized cars to investigate any correlation between the age of the car and the current market value. She calculates the correlation coefficient. Which of the following statements best describes her answer of - 1.2 ?
Tick ( $\checkmark$ ) one box. Definitely incorrect □ Probably incorrect
\includegraphics[max width=\textwidth, alt={}, center]{0abb6343-017d-4629-8a2d-cfb405dc2d14-16_108_113_1265_959} Probably correct
\includegraphics[max width=\textwidth, alt={}, center]{0abb6343-017d-4629-8a2d-cfb405dc2d14-16_106_113_1400_959} Definitely correct □

AQA Paper 3 2021 June Q11

11 The random variable $X$ is such that $X \sim \mathrm {~B} ( n , p )$
The mean value of $X$ is 225 The variance of $X$ is 144 Find $p$.
Circle your answer.
0.36
0.6
0.64
0.8

AQA Paper 3 2021 June Q12

3 marks

12 An electoral register contains 8000 names. A researcher decides to select a systematic sample of 100 names from the register.
Explain how the researcher should select such a sample.
[0pt] [3 marks]

AQA Paper 3 2021 June Q13

2 marks

13 The table below is an extract from the Large Data Set.

Propulsion Type	Region	Engine Size	Mass	$\mathrm { CO } _ { 2 }$	Particulate Emissions
2	London	1896	1533	154	0.04
2	North West	1896	1423	146	0.029
2	North West	1896	1353	138	0.025
2	South West	1998	1547	159	0.026
2	London	1896	1388	138	0.025
2	South West	1896	1214	130	0.011
2	South West	1896	1480	146	0.029
2	South West	1896	1413	146	0.024
2	South West	2496	1695	192	0.034
2	South West	1422	1251	122	0.025
2	South West	1995	2075	175	0.034
2	London	1896	1285	140	0.036
2	North West	1896	0	146

1. Calculate the mean and standard deviation of $\mathrm { CO } _ { 2 }$ emissions in the table.
  [0pt] [2 marks]
  13
(ii) Any value more than 2 standard deviations from the mean can be identified as an outlier. Determine, using this definition of an outlier, if there are any outliers in this sample of $\mathrm { CO } _ { 2 }$ emissions. Fully justify your answer.
13
Maria claims that the last line in the table must contain two errors. Use your knowledge of the Large Data Set to comment on Maria's claim.
$14 \quad A$ and $B$ are two events such that $$\begin{aligned} & \mathrm { P } ( A \cap B ) = 0.1
& \mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \right) = 0.2
& \mathrm { P } ( B ) = 2 \mathrm { P } ( A ) \end{aligned}$$

AQA Paper 3 2021 June Q14

$\quad$ Find $\mathrm { P } ( A )$
14
$\quad$ Find $\operatorname { P } ( B \mid A )$
14
Determine if $A$ and $B$ are independent events.

AQA Paper 3 2021 June Q15

15 A team game involves solving puzzles to escape from a room. Using data from the past, the mean time to solve the puzzles and escape from one of these rooms is 65 minutes with a standard deviation of 11.3 minutes. After recent changes to the puzzles in the room, it is claimed that the mean time to solve the puzzles and escape has changed. To test this claim, a random sample of 100 teams is selected.
The total time to solve the puzzles and escape for the 100 teams is 6780 minutes.
Assuming that the times are normally distributed, test at the $2 \%$ level the claim that the mean time has changed.

AQA Paper 3 2021 June Q16

16 The discrete random variable $X$ has the probability function $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c c } c ( 7 - 2 x ) & x = 0,1,2,3
k & x = 4
0 & \text { otherwise } \end{array} \right.$$ where $c$ and $k$ are constants.
16

Show that $16 c + k = 1$
16
Given that $\mathrm { P } ( X \geq 3 ) = \frac { 5 } { 8 }$ find the value of $c$ and the value of $k$.

AQA Paper 3 2021 June Q17

17 James is playing a mathematical game on his computer.
The probability that he wins is 0.6
As part of an online tournament, James plays the game 10 times.
Let $Y$ be the number of games that James wins.
17

State two assumptions, in context, for $Y$ to be modelled as $B ( 10,0.6 )$
17
$\quad$ Find $\mathrm { P } ( Y = 4 )$
17
$\quad$ Find $\mathrm { P } ( Y \geq 4 )$
17
After practising the game, James claims that he has increased his probability of winning the game. In a random sample of 15 subsequent games, he wins 12 of them.
Test a $5 \%$ significance level whether James's claim is correct.
\begin{center} \begin{tabular}{|l|l|l|l|} \hline \multicolumn{4}{|c|}{\begin{tabular}{l}

AQA Paper 3 2021 June Q18

7 marks

18 (b)
The weight, $Y$ grams, of marmalade in a jar can be modelled as a normal variable with mean $\mu$ and standard deviation $\sigma$
18 (b) (i)
18 (b) (i) $\_\_\_\_$ $\_\_\_\_$ $346 - \mu = 1.96 \sigma$
Fully justify your answer. $\_\_\_\_$
[0pt] [3 marks] $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$
\end{tabular}}
\hline \end{tabular} \end{center} 18 (b) (ii) Given further that $$\mathrm { P } ( Y < 336 ) = 0.14$$ find $\mu$ and $\sigma$
[0pt] [4 marks]
\includegraphics[max width=\textwidth, alt={}, center]{0abb6343-017d-4629-8a2d-cfb405dc2d14-28_2492_1721_217_150}

AQA Paper 3 2022 June Q1

1 marks

1 State the range of values of $x$ for which the binomial expansion of $$\sqrt { 1 - \frac { x } { 4 } }$$ is valid. Circle your answer.
[0pt] [1 mark] $$| x | < \frac { 1 } { 4 } \quad | x | < 1 \quad | x | < 2 \quad | x | < 4$$

Questions — AQA Paper 3 (123 questions)

Browse by module

Browse by board

AQA Paper 3 2020 June Q12

AQA Paper 3 2020 June Q13

AQA Paper 3 2020 June Q14

AQA Paper 3 2020 June Q15

AQA Paper 3 2020 June Q16

AQA Paper 3 2020 June Q17

AQA Paper 3 2020 June Q18

AQA Paper 3 2021 June Q1

AQA Paper 3 2021 June Q2

AQA Paper 3 2021 June Q4

AQA Paper 3 2021 June Q5

AQA Paper 3 2021 June Q6

AQA Paper 3 2021 June Q7

AQA Paper 3 2021 June Q8

AQA Paper 3 2021 June Q9

AQA Paper 3 2021 June Q10

AQA Paper 3 2021 June Q11

AQA Paper 3 2021 June Q12

AQA Paper 3 2021 June Q13

AQA Paper 3 2021 June Q14

AQA Paper 3 2021 June Q15

AQA Paper 3 2021 June Q16

AQA Paper 3 2021 June Q17

AQA Paper 3 2021 June Q18

AQA Paper 3 2022 June Q1