Questions — AQA (3620 questions)

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AQA Paper 3 2024 June Q3
1 marks Easy -2.0
One of the graphs shown below **cannot** have an equation of the form $$y = a^x \quad \text{where } a > 0$$ Identify this graph. Tick (\(\checkmark\)) one box. [1 mark] \includegraphics{figure_3}
AQA Paper 3 2024 June Q4
2 marks Easy -1.2
A curve has equation \(y = x^4 + 2^x\) Find an expression for \(\frac{dy}{dx}\) [2 marks]
AQA Paper 3 2024 June Q5
3 marks Moderate -0.8
The diagram below shows a sector of a circle \(OAB\). The chord \(AB\) divides the sector into a triangle and a shaded segment. Angle \(AOB\) is \(\frac{\pi}{6}\) radians. The radius of the sector is 18 cm. \includegraphics{figure_5} Show that the area of the shaded segment is $$k(\pi - 3) \text{cm}^2$$ where \(k\) is an integer to be found. [3 marks]
AQA Paper 3 2024 June Q6
5 marks Easy -1.2
\begin{enumerate}[label=(\alph*)] \item Find \(\int \left(6x^2 - \frac{5}{\sqrt{x}}\right) dx\) [3 marks] \item The gradient of a curve is given by $$\frac{dy}{dx} = 6x^2 - \frac{5}{\sqrt{x}}$$ The curve passes through the point \((4, 90)\). Find the equation of the curve. [2 marks]
AQA Paper 3 2024 June Q7
5 marks Moderate -0.8
The graphs with equations $$y = 2 + 3x - 2x^2 \text{ and } x + y = 1$$ are shown in the diagram below. \includegraphics{figure_7} The graphs intersect at the points \(A\) and \(B\) \begin{enumerate}[label=(\alph*)] \item On the diagram above, shade and label the region, \(R\), that is satisfied by the inequalities $$0 \leq y \leq 2 + 3x - 2x^2$$ and $$x + y \geq 1$$ [2 marks] \item Find the exact coordinates of \(A\) [3 marks]
AQA Paper 3 2024 June Q8
8 marks Moderate -0.3
The temperature \(\theta\) °C of an oven \(t\) minutes after it is switched on can be modelled by the equation $$\theta = 20(11 - 10e^{-kt})$$ where \(k\) is a positive constant. Initially the oven is at room temperature. The maximum temperature of the oven is \(T\) °C The temperature predicted by the model is shown in the graph below. \includegraphics{figure_8} \begin{enumerate}[label=(\alph*)] \item Find the room temperature. [2 marks] \item Find the value of \(T\) [2 marks] \item The oven reaches a temperature of 86 °C one minute after it is switched on.
  1. Find the value of \(k\). [2 marks]
  2. Find the time it takes for the temperature of the oven to be within 1°C of its maximum. [2 marks]
AQA Paper 3 2024 June Q9
9 marks Moderate -0.3
Figure 1 below shows a circle. **Figure 1** \includegraphics{figure_9} 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 $$x^2 + y^2 = 12y - 8x - 27$$ \begin{enumerate}[label=(\alph*)] \item 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. [3 marks] \item State the coordinates of \(P\) [1 mark] \item Find the \(y\)-coordinate of \(Q\) [2 marks] \item The line segment \(QR\) is a tangent to the circle as shown in Figure 2 below. **Figure 2** \includegraphics{figure_9d} The point \(R\) has coordinates \((9, -3)\). Find the angle \(QPR\) Give your answer in radians to three significant figures. [3 marks]
AQA Paper 3 2024 June Q10
5 marks Challenging +1.2
It is given that $$f'(x) = 5x^3 + x$$ Use differentiation from first principles to prove that $$f''(x) = 15x^2 + 1$$ [5 marks]
AQA Paper 3 2024 June Q11
10 marks Challenging +1.2
The curve \(C\) with equation $$y = (x^2 - 8x) \ln x$$ is defined for \(x > 0\) and is shown in the diagram below. \includegraphics{figure_11} 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. [10 marks]
AQA Paper 3 2024 June Q12
1 marks Easy -1.8
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 websitesFrequency
01
14
218
316
45
537
62
71
Find the interquartile range of the number of websites visited by these 84 students. Circle your answer. [1 mark] 3 \quad 4 \quad 19 \quad 42
AQA Paper 3 2024 June Q13
1 marks Easy -2.5
The shaded region on one of the Venn diagrams below represents \((A \cup C) \cap B\) Identify this Venn diagram. Tick (\(\checkmark\)) one box. [1 mark] \includegraphics{figure_13}
AQA Paper 3 2024 June Q14
5 marks Moderate -0.8
The annual cost of energy in 2021 for each of the 350 households in Village A can be modelled by a random variable \(X\) It is given that $$\sum x = 945\,000 \quad \sum x^2 = 2\,607\,500\,000$$ \begin{enumerate}[label=(\alph*)] \item Calculate the mean of \(X\). [1 mark] \item Calculate the standard deviation of \(X\). [2 marks] \item For households in Village B the annual cost of energy in 2021 has mean £3100 and standard deviation £325 Compare the annual cost of energy in 2021 for households in Village A and Village B. [2 marks]
AQA Paper 3 2024 June Q15
9 marks Moderate -0.8
It is given that $$X \sim \text{B}(48, 0.175)$$ \begin{enumerate}[label=(\alph*)] \item Find the mean of \(X\) [1 mark] \item Show that the variance of \(X\) is 6.93 [1 mark] \item Find P(\(X < 10\)) [1 mark] \item Find P(\(X \geq 6\)) [2 marks] \item Find P(\(9 \leq X \leq 15\)) [2 marks] \item 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. [2 marks]
AQA Paper 3 2024 June Q16
4 marks Moderate -0.8
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 [4 marks]
AQA Paper 3 2024 June Q17
14 marks Moderate -0.8
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.
  1. This normal distribution is represented in the diagram below. Label the values 50 and 54 on the horizontal axis. [2 marks] \includegraphics{figure_17a}
  2. State the probability that the length of a new-born baby is less than 50 cm. [1 mark]
  3. Find the probability that the length of a new-born baby is more than 56 cm. [1 mark]
  4. Find the probability that the length of a new-born baby is more than 40 cm but less than 60 cm. [1 mark]
  5. Determine the length exceeded by 95% of all new-born babies at the clinic. [2 marks]
  6. 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. [7 marks]
AQA Paper 3 2024 June Q18
7 marks Easy -1.3
The Human Resources director in a company is investigating the graduate status and salaries of its employees. Event \(G\) is defined as the employee is a graduate. Event \(H\) is defined as the employee earns at least £40 000 a year. The director summarised the findings in the table of probabilities below.
\(H\)\(H'\)
\(G\)0.210.18
\(G'\)0.070.54
\begin{enumerate}[label=(\alph*)] \item An employee is selected at random.
  1. Find P(\(G\)) [1 mark]
  2. Find P[\((G \cap H)'\)] [2 marks]
  3. Find P(\(H | G'\)) [2 marks]
\item Determine whether the events \(G\) and \(H\) are independent. Fully justify your answer. [2 marks]
AQA Paper 3 2024 June Q19
9 marks Standard +0.3
It is known that 80% of all diesel cars registered in 2017 had carbon monoxide (CO) emissions less than 0.3 g/km. Talat decides to investigate whether the proportion of diesel cars registered in 2022 with CO emissions less than 0.3 g/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.
    1. State suitable null and alternative hypotheses for Talat's test. [1 mark]
    2. Using a 10% level of significance, find the critical region for Talat's test. [5 marks]
    3. In his random sample, Talat finds 18 cars with CO emissions less than 0.3 g/km. State Talat's conclusion in context. [1 mark]
  2. 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 g/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. [2 marks]
AQA Paper 3 Specimen Q1
1 marks Easy -1.8
The graph of \(y = x^2 - 9\) is shown below. \includegraphics{figure_1} Find the area of the shaded region. Circle your answer. [1 mark] \(-18\) \quad\quad \(-6\) \quad\quad \(6\) \quad\quad \(18\)
AQA Paper 3 Specimen Q2
6 marks Moderate -0.3
A wooden frame is to be made to support some garden decking. The frame is to be in the shape of a sector of a circle. The sector \(OAB\) is shown in the diagram, with a wooden plank \(AC\) added to the frame for strength. \(OA\) makes an angle of \(\theta\) with \(OB\). \includegraphics{figure_2}
  1. Show that the exact value of \(\sin\theta\) is \(\frac{4\sqrt{14}}{15}\) [3 marks]
  2. Write down the value of \(\theta\) in radians to 3 significant figures. [1 mark]
  3. Find the area of the garden that will be covered by the decking. [2 marks]
AQA Paper 3 Specimen Q3
13 marks Moderate -0.3
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 \(A\).
  1. Show that the area covered by the weed can be modelled by $$A = Be^{kt}$$ where \(B\) and \(k\) are constants and \(t\) is time in days since the weed was first noticed. [4 marks]
  2. When it was first noticed, the weed covered an area of 0.25 m². Twenty days later the weed covered an area of 0.5 m²
    1. State the value of \(B\). [1 mark]
    2. Show that the model for the area covered by the weed can be written as $$A = 2^{\frac{t}{20} - 2}$$ [4 marks]
    3. How many days does it take for the weed to cover half of the surface of the pond? [2 marks]
  3. State one limitation of the model. [1 mark]
  4. Suggest one refinement that could be made to improve the model. [1 mark]
AQA Paper 3 Specimen Q4
5 marks Standard +0.3
\(\int_1^2 x^3 \ln(2x) dx\) can be written in the form \(p\ln 2 + q\), where \(p\) and \(q\) are rational numbers. Find \(p\) and \(q\). [5 marks]
AQA Paper 3 Specimen Q5
11 marks Moderate -0.3
  1. Find the first three terms, in ascending powers of \(x\), in the binomial expansion of \((1 + 6x)^{\frac{1}{3}}\) [2 marks]
  2. Use the result from part (a) to obtain an approximation to \(\sqrt[3]{1.18}\) giving your answer to 4 decimal places. [2 marks]
  3. Explain why substituting \(x = \frac{1}{2}\) into your answer to part (a) does not lead to a valid approximation for \(\sqrt[3]{4}\). [1 mark]
AQA Paper 3 Specimen Q6
8 marks Challenging +1.2
Find the value of \(\int_1^2 \frac{6x + 1}{6x^2 - 7x + 2} dx\), expressing your answer in the form \(m\ln 2 + n\ln 3\), where \(m\) and \(n\) are integers. [8 marks]
AQA Paper 3 Specimen Q7
12 marks Standard +0.8
The diagram shows part of the graph of \(y = e^{-x^2}\) \includegraphics{figure_7} The graph is formed from two convex sections, where the gradient is increasing, and one concave section, where the gradient is decreasing.
  1. Find the values of \(x\) for which the graph is concave. [4 marks]
  2. The finite region bounded by the \(x\)-axis and the lines \(x = 0.1\) and \(x = 0.5\) is shaded. \includegraphics{figure_7b} Use the trapezium rule, with 4 strips, to find an estimate for \(\int_{0.1}^{0.5} e^{-x^2} dx\) Give your estimate to four decimal places. [3 marks]
  3. Explain with reference to your answer in part (a), why the answer you found in part (b) is an underestimate. [2 marks]
  4. 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. [3 marks]
AQA Paper 3 Specimen Q8
2 marks Easy -2.0
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."
  1. What is the population for this investigation? Circle your answer. [1 mark]
    Edna and RalphThe 2065 households in the villageThe energy intake for the village from eating outThe 100 households selected
  2. What is the sampling method suggested by Ralph? Circle your answer. [1 mark]
    OpportunityRandom numberContinuous random variableSimple random