Questions — AQA Paper 3 (141 questions)

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AQA Paper 3 2023 June Q11
1 marks Easy -1.8
A and B are mutually exclusive events. Which one of the following statements must be correct? Tick (\(\checkmark\)) one box. [1 mark] \(P(A \cup B) = P(A) \times P(B)\) \(P(A \cup B) = P(A) - P(B)\) \(P(A \cap B) = 0\) \(P(A \cap B) = 1\)
AQA Paper 3 2023 June Q12
8 marks Easy -1.3
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 \((32, 0.4)\)
  1. State one assumption, in context, required for this distribution to be used. [1 mark]
  2. Find the probability that exactly 7 of the drivers on a particular day pass their test. [1 mark]
  3. Find the probability that, at most, 16 of the drivers on a particular day pass their test. [1 mark]
  4. Find the probability that more than 12 of the drivers on a particular day pass their test. [2 marks]
  5. Find the mean number of drivers per day who pass their test. [1 mark]
  6. Find the standard deviation of the number of drivers per day who pass their test. [2 marks]
AQA Paper 3 2023 June Q13
4 marks Moderate -0.8
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.
  1. Find the probability that both coins were of the same type. [2 marks]
  2. Find the probability that both coins are bronze, given that at least one of the coins is bronze. [2 marks]
AQA Paper 3 2023 June Q14
10 marks Standard +0.3
The mass of aluminium cans recycled each day in a city may be modelled by a normal distribution with mean 24 500 kg and standard deviation 5 200 kg.
  1. State the probability that the mass of aluminium cans recycled on any given day is not equal to 24 500 kg. [1 mark]
  2. To reduce costs, the city's council decides to collect aluminium cans for recycling less frequently. Following the decision, it was found that over a 24-day period a total mass of 641 520 kg of aluminium cans was recycled. It can be assumed that the distribution of the mass of aluminium cans recycled is still normal with standard deviation 5 200 kg, and that the 24-day period can be regarded as a random sample. Investigate, at the 5% level of significance, whether the mean daily mass of aluminium cans recycled has changed. [7 marks]
  3. 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. [2 marks]
AQA Paper 3 2023 June Q15
11 marks Moderate -0.8
  1. A random sample of eight cars was selected from the Large Data Set. The masses of these cars, in kilograms, were as follows. 950 989 1247 1415 1506 1680 1833 2040 It is given that, for the population of cars in the Large Data Set: lower quartile = 1167 median = 1393 upper quartile = 1570
    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. [3 marks]
    2. Write down the statistical name for the mass that should be removed in part (a)(i). [1 mark]
  2. The table shows the probability distribution of the number of previous owners, \(N\), for a sample of cars taken from the Large Data Set.
    \(n\)0123456 or more
    \(P(N = n)\)0.140.370.9k0.250.4k1.7k0
    Find the value of \(P(1 \leq N < 5)\) [4 marks]
  3. An expert team is investigating whether there have been any changes in CO₂ 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.
    1. Using your knowledge of the Large Data Set, explain how the team can collect this sample. [2 marks]
    2. Describe one disadvantage of quota sampling. [1 mark]
AQA Paper 3 2023 June Q16
9 marks Standard +0.3
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
    1. Find \(P(D < 5.2)\) [1 mark]
    2. Find \(P(D > 7)\) [1 mark]
    3. The supermarket only accepts apples with diameters between 5 cm and 8 cm. Find the proportion of apples that the supermarket accepts. [1 mark]
  2. 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. [6 marks]
AQA Paper 3 2023 June Q17
6 marks Standard +0.3
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. [6 marks]
AQA Paper 3 2024 June Q1
1 marks Easy -1.8
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. [1 mark] Tick (\(\checkmark\)) one box. \(0.1 + 0.2 + 0.4 + 0.8 + \ldots\) \(1 - 1 + 1 - 1 + \ldots\) \(128 - 64 + 32 - 16 + \ldots\) \(1 + 2 + 4 + 8 + \ldots\)
AQA Paper 3 2024 June Q2
1 marks Easy -1.8
The quadratic equation $$4x^2 + bx + 9 = 0$$ has one repeated real root. Find \(b\) Circle your answer. [1 mark] \(b = 0\) \quad \(b = \pm 12\) \quad \(b = \pm 13\) \quad \(b = \pm 36\)
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]