Questions — AQA Paper 3 (123 questions)

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AQA Paper 3 2022 June Q2
2 The shaded region, shown in the diagram below, is defined by $$x ^ { 2 } - 7 x + 7 \leq y \leq 7 - 2 x$$
\includegraphics[max width=\textwidth, alt={}]{6ad3bac9-bf08-443d-8be2-b0c26209ffe8-03_801_830_456_607}
Identify which of the following gives the area of the shaded region.
Tick \(( \checkmark )\) one box. $$\begin{array} { l c } \int ( 7 - 2 x ) \mathrm { d } x - \int \left( x ^ { 2 } - 7 x + 7 \right) \mathrm { d } x & \square
\int _ { 0 } ^ { 5 } \left( x ^ { 2 } - 5 x \right) \mathrm { d } x & \square
\int _ { 0 } ^ { 5 } \left( 5 x - x ^ { 2 } \right) \mathrm { d } x & \square
\int _ { 0 } ^ { 5 } \left( x ^ { 2 } - 9 x + 14 \right) \mathrm { d } x & \square \end{array}$$
AQA Paper 3 2022 June Q3
3 The function f is defined by $$f ( x ) = 2 x + 1$$ Solve the equation $$\mathrm { f } ( x ) = \mathrm { f } ^ { - 1 } ( x )$$ Circle your answer. $$x = - 1 \quad x = 0 \quad x = 1 \quad x = 2$$
AQA Paper 3 2022 June Q4
4 Find $$\int \left( x ^ { 2 } + x ^ { \frac { 1 } { 2 } } \right) \mathrm { d } x$$
AQA Paper 3 2022 June Q5
5
  1. Sketch the graph of $$y = \sin 2 x$$ for \(0 ^ { \circ } \leq x \leq 360 ^ { \circ }\)
    \includegraphics[max width=\textwidth, alt={}, center]{6ad3bac9-bf08-443d-8be2-b0c26209ffe8-05_1095_1246_534_402} 5
  2. The equation $$\sin 2 x = A$$ has exactly two solutions for \(0 ^ { \circ } \leq x \leq 360 ^ { \circ }\)
    State the possible values of \(A\).
AQA Paper 3 2022 June Q6
6 A design for a surfboard is shown in Figure 1. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{6ad3bac9-bf08-443d-8be2-b0c26209ffe8-06_415_1403_447_322}
\end{figure} The curve of the top half of the surfboard can be modelled by the parametric equations $$\begin{aligned} & x = - 2 t ^ { 2 }
& y = 9 t - 0.7 t ^ { 2 } \end{aligned}$$ for \(0 \leq t \leq 9.5\) as shown in Figure 2, where \(x\) and \(y\) are measured in centimetres. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{6ad3bac9-bf08-443d-8be2-b0c26209ffe8-06_383_1342_1379_351}
\end{figure} 6
  1. Find the length of the surfboard.
    6
    1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
      6
  3. (ii) Hence, show that the width of the surfboard is approximately one third of its length.
    \(7 \quad\) A planet takes \(T\) days to complete one orbit of the Sun.
    \(T\) is known to be related to the planet's average distance \(d\), in millions of kilometres, from the Sun. A graph of \(\log _ { 10 } T\) against \(\log _ { 10 } d\) is shown with data for Mercury and Uranus labelled.
    \includegraphics[max width=\textwidth, alt={}, center]{6ad3bac9-bf08-443d-8be2-b0c26209ffe8-08_752_1447_580_296}
AQA Paper 3 2022 June Q7
7
    1. Find the equation of the straight line in the form $$\log _ { 10 } T = a + b \log _ { 10 } d$$ where \(a\) and \(b\) are constants to be found.
      7
  2. (ii) Show that $$T = \mathrm { K } d ^ { \mathrm { n } }$$ where K and n are constants to be found.
    7
  3. Neptune takes approximately 60000 days to complete one orbit of the Sun.
    Use your answer to 7(a)(ii) to find an estimate for the average distance of Neptune from the Sun.
AQA Paper 3 2022 June Q8
8 Water is poured into an empty cone at a constant rate of \(8 \mathrm {~cm} ^ { 3 } / \mathrm { s }\) After \(t\) seconds the depth of the water in the inverted cone is \(h \mathrm {~cm}\), as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{6ad3bac9-bf08-443d-8be2-b0c26209ffe8-10_468_481_497_778} When the depth of the water in the inverted cone is \(h \mathrm {~cm}\), the volume, \(V \mathrm {~cm} ^ { 3 }\), is given by $$V = \frac { \pi h ^ { 3 } } { 12 }$$ 8
  1. Show that when \(t = 3\) $$\frac { \mathrm { d } V } { \mathrm {~d} h } = 6 \sqrt [ 3 ] { 6 \pi }$$ 8
  2. Hence, find the rate at which the depth is increasing when \(t = 3\)
    Give your answer to three significant figures.
    \(9 \quad\) Assume that \(a\) and \(b\) are integers such that $$a ^ { 2 } - 4 b - 2 = 0$$
AQA Paper 3 2022 June Q9
9
  1. Prove that \(a\) is even. 9
  2. Hence, prove that \(2 b + 1\) is even and explain why this is a contradiction.
    9
  3. Explain what can be deduced about the solutions of the equation $$a ^ { 2 } - 4 b - 2 = 0$$
AQA Paper 3 2022 June Q10
2 marks
10 The function f is defined by $$f ( x ) = \frac { x ^ { 2 } + 10 } { 2 x + 5 }$$ where \(f\) has its maximum possible domain. The curve \(y = \mathrm { f } ( x )\) intersects the line \(y = x\) at the points \(P\) and \(Q\) as shown below.
\includegraphics[max width=\textwidth, alt={}, center]{6ad3bac9-bf08-443d-8be2-b0c26209ffe8-14_908_1125_685_459} 10
  1. State the value of \(x\) which is not in the domain of f . 10
  2. Explain how you know that the function f is many-to-one.
    10
    1. Show that the \(x\)-coordinates of \(P\) and \(Q\) satisfy the equation $$x ^ { 2 } + 5 x - 10 = 0$$ [2 marks]
      10
  4. (ii) Hence, find the exact \(x\)-coordinate of \(P\) and the exact \(x\)-coordinate of \(Q\).
    10
  5. Show that \(P\) and \(Q\) are stationary points of the curve. Fully justify your answer.
    10
  6. Using set notation, state the range of f .
AQA Paper 3 2022 June Q12
1 marks
12 The box plot below shows summary data for the number of minutes late that buses arrived at a rural bus stop.
\includegraphics[max width=\textwidth, alt={}, center]{6ad3bac9-bf08-443d-8be2-b0c26209ffe8-17_515_1614_1384_212} Identify which term best describes the distribution of this data.
Circle your answer.
[0pt] [1 mark]
negatively skewed
normal
positively skewed
symmetrical
AQA Paper 3 2022 June Q13
4 marks
13 A reporter is writing an article on the \(\mathrm { CO } _ { 2 }\) emissions from vehicles using the Large Data Set. The reporter claims that the Large Data Set shows that the CO2 emissions from all vehicles in the UK have declined every year from 2002 to 2016. Using your knowledge of the Large Data Set, give two reasons why this claim is invalid.
[0pt] [2 marks]
\includegraphics[max width=\textwidth, alt={}, center]{6ad3bac9-bf08-443d-8be2-b0c26209ffe8-19_2488_1716_219_153} \begin{center} \begin{tabular}{|l|l|l|} \hline 14 & 14 (b) (ii) Find \(\mathrm { P } ( X < 4 )\) \(\text { Find } \mathrm { P } ( X < 4 )\) [2 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) & \begin{tabular}{l} A customer service centre records every call they receive.
It is found that \(30 \%\) of all calls made to this centre are complaints.
A sample of 20 calls is selected.
The number of calls in the sample which are complaints is denoted by the random variable \(X\).
State two assumptions necessary for \(X\) to be modelled by a binomial distribution.
AQA Paper 3 2022 June Q14
3 marks
14
  1. State two assumptions necessary for \(X\) to be modelled by a binomial distribution. \(\_\_\_\_\) [2 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
    14
  2. Assume that \(X\) can be modelled by a binomial distribution. \(\_\_\_\_\)
    [0pt] [1 mark]
    \end{tabular}
    \hline & &
    \hline \end{tabular} \end{center} 14
  3. (iii) Find \(\mathrm { P } ( X \geq 10 )\)
    14
  4. In a random sample of 10 calls to a school, the number of calls which are complaints, \(Y\), may be modelled by a binomial distribution $$Y \sim \mathrm {~B} ( 10 , p )$$ The standard deviation of \(Y\) is 1.5 Calculate the possible values of \(p\).
AQA Paper 3 2022 June Q15
15 Researchers are investigating the average time spent on social media by adults on the electoral register of a town. They select every 100th adult from the electoral register for their investigation.
15
  1. Identify the population in their investigation.
    15
    1. State the name of this method of sampling.
      15
  3. (ii) Describe one advantage of this sampling method.
    \includegraphics[max width=\textwidth, alt={}, center]{6ad3bac9-bf08-443d-8be2-b0c26209ffe8-23_2488_1716_219_153}
AQA Paper 3 2022 June Q16
16 A sample of 240 households were asked which, if any, of the following animals they own as pets:
  • cats (C)
  • dogs (D)
  • tortoises ( \(T\) )
The results are shown in the table below.
Types of pet\(C\)\(D\)\(T\)\(C\) and \(D\)\(C\) and \(T\)\(D\) and \(T\)\(C , D\) and \(T\)
Number of
households
153704548213217
16
  1. Represent this information by fully completing the Venn diagram below.
    \includegraphics[max width=\textwidth, alt={}, center]{6ad3bac9-bf08-443d-8be2-b0c26209ffe8-24_858_1191_1032_427} 16
  2. A household is chosen at random from the sample.
    16
    1. Find the probability that the household owns a cat only. 16
  4. (ii) Find the probability that the household owns at least two of the three types of pet.
    16
  5. (iii) Find the probability that the household owns a cat or a dog or both, given that the household does not own a tortoise.
    16
  6. Determine whether a household owning a cat and a household owning a tortoise are independent of each other. Fully justify your answer.
AQA Paper 3 2022 June Q17
17 The number of working hours per week of employees in a company is modelled by a normal distribution with mean of 34 hours and a standard deviation of 4.5 hours. The manager claims that the mean working hours per week of the company's employees has increased. A random sample of 30 employees in the company was found to have mean working hours per week of 36.2 hours. Carry out a hypothesis test at the \(2.5 \%\) significance level to investigate the manager's claim.
\includegraphics[max width=\textwidth, alt={}, center]{6ad3bac9-bf08-443d-8be2-b0c26209ffe8-27_2490_1730_217_141}
AQA Paper 3 2022 June Q18
5 marks
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
  1. 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
  2. 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
  4. (ii) Find the probability that the height of a randomly selected male athlete is between 1.70 metres and 1.90 metres.
    18
  5. (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
  6. 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
  7. 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}
AQA Paper 3 2023 June Q1
1 The graph of \(y = \mathrm { f } ( x )\) is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{6fba7e53-de46-460b-9bef-f1a6962f2e7d-02_771_1324_676_447} One of the four equations listed below is the equation of the graph \(y = \mathrm { f } ( x )\)
Identify which one is the correct equation of the graph.
Tick ( \(\checkmark\) ) one box. $$\begin{aligned} & y = | x + 2 | + 3
& y = | x + 2 | - 3
& y = | x - 2 | + 3
& y = | x - 2 | - 3 \end{aligned}$$ □
□
□
AQA Paper 3 2023 June Q2
2 The trapezium rule is used to estimate the area of the shaded region in each of the graphs below. Identify the graph for which the trapezium rule produces an overestimate. Tick ( \(\checkmark\) ) one box.
\includegraphics[max width=\textwidth, alt={}, center]{6fba7e53-de46-460b-9bef-f1a6962f2e7d-03_524_424_539_497}
\includegraphics[max width=\textwidth, alt={}, center]{6fba7e53-de46-460b-9bef-f1a6962f2e7d-03_506_424_1096_497}
\includegraphics[max width=\textwidth, alt={}, center]{6fba7e53-de46-460b-9bef-f1a6962f2e7d-03_512_426_1640_497}
□
□
\includegraphics[max width=\textwidth, alt={}]{6fba7e53-de46-460b-9bef-f1a6962f2e7d-03_147_124_1256_1121} \(\square\)
\includegraphics[max width=\textwidth, alt={}, center]{6fba7e53-de46-460b-9bef-f1a6962f2e7d-03_515_431_2188_495}
AQA Paper 3 2023 June Q3
3 A curve with equation \(y = \mathrm { f } ( x )\) passes through the point (3, 7) Given that \(\mathrm { f } ^ { \prime } ( 3 ) = 0\) find the equation of the normal to the curve at ( 3,7 ) Circle your answer. $$y = \frac { 7 } { 3 } x \quad y = 0 \quad x = 3 \quad x = 7$$
AQA Paper 3 2023 June Q4
4 Express $$\frac { 5 - \sqrt [ 3 ] { x } } { x ^ { 2 } }$$ in the form $$5 x ^ { p } - x ^ { q }$$ where \(p\) and \(q\) are constants.
AQA Paper 3 2023 June Q5
5 A curve has equation \(y = 3 \mathrm { e } ^ { 2 x }\) Find the gradient of the curve at the point where \(y = 10\)
AQA Paper 3 2023 June Q6
6
  1. 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
  2. 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
  4. (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
  5. (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
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
  1. 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
  3. (ii) Find the exact maximum value of \(A\)
    7
  4. 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
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
  1. 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
  3. (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
  4. (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
10 Which of the following is not a possible value for a product moment correlation coefficient? Circle your answer. $$- \frac { 6 } { 5 } \quad - \frac { 3 } { 5 } \quad 0$$