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AQA Paper 3 2021 June Q7
10 marks Moderate -0.8
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.
  1. Find \(W_2\) [1 mark]
  2. Explain why $$W_n = A \times 0.98^{n-1}$$ and state the value of \(A\). [2 marks]
  3. Find the increase in the water in the bucket 15 minutes after the rain stops. Give your answer to the nearest millilitre. [2 marks]
  4. Assuming it does not start to rain again, find the maximum amount of water in the bucket. [3 marks]
  5. 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). [2 marks]
AQA Paper 3 2021 June Q8
6 marks Standard +0.3
Given that $$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} x \cos x \, dx = a\pi + b$$ find the exact value of \(a\) and the exact value of \(b\). Fully justify your answer. [6 marks]
AQA Paper 3 2021 June Q9
9 marks Standard +0.3
A function f is defined for all real values of \(x\) as $$f(x) = x^4 + 5x^3$$ The function has exactly two stationary points when \(x = 0\) and \(x = -\frac{15}{4}\)
    1. Find \(f''(x)\) [2 marks]
    2. Determine the nature of the stationary points. Fully justify your answer. [4 marks]
  2. State the range of values of \(x\) for which $$f(x) = x^4 + 5x^3$$ is an increasing function. [1 mark]
  3. A second function g is defined for all real values of \(x\) as $$g(x) = x^4 - 5x^3$$
    1. State the single transformation which maps f onto g. [1 mark]
    2. State the range of values of \(x\) for which g is an increasing function. [1 mark]
AQA Paper 3 2021 June Q10
1 marks Easy -2.5
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. [1 mark] Definitely incorrect Probably incorrect Probably correct Definitely correct
AQA Paper 3 2021 June Q11
1 marks Easy -1.2
The random variable \(X\) is such that \(X \sim B(n, p)\) The mean value of \(X\) is 225 The variance of \(X\) is 144 Find \(p\). Circle your answer. [1 mark] 0.36 \quad 0.6 \quad 0.64 \quad 0.8
AQA Paper 3 2021 June Q12
3 marks Easy -1.8
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. [3 marks]
AQA Paper 3 2021 June Q13
6 marks Moderate -0.8
The table below is an extract from the Large Data Set.
Propulsion TypeRegionEngine SizeMassCO₂Particulate Emissions
2London189615331540.04
2North West189614231460.029
2North West189613531380.025
2South West199815471590.026
2London189613881380.025
2South West189612141300.011
2South West189614801460.029
2South West189614131460.024
2South West249616951920.034
2South West142212511220.025
2South West199520751750.034
2London189612851400.036
2North West18960146
    1. Calculate the mean and standard deviation of CO₂ emissions in the table. [2 marks]
    2. 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 CO₂ emissions. Fully justify your answer. [2 marks]
  2. 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. [2 marks]
AQA Paper 3 2021 June Q14
7 marks Standard +0.3
\(A\) and \(B\) are two events such that $$P(A \cap B) = 0.1$$ $$P(A' \cap B') = 0.2$$ $$P(B) = 2P(A)$$
  1. Find \(P(A)\) [4 marks]
  2. Find \(P(B|A)\) [2 marks]
  3. Determine if \(A\) and \(B\) are independent events. [1 mark]
AQA Paper 3 2021 June Q15
7 marks Standard +0.3
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. [7 marks]
AQA Paper 3 2021 June Q16
4 marks Moderate -0.3
The discrete random variable \(X\) has the probability function $$P(X = x) = \begin{cases} c(7 - 2x) & x = 0, 1, 2, 3 \\ k & x = 4 \\ 0 & \text{otherwise} \end{cases}$$ where \(c\) and \(k\) are constants.
  1. Show that \(16c + k = 1\) [2 marks]
  2. Given that \(P(X \geq 3) = \frac{5}{8}\) find the value of \(c\) and the value of \(k\). [2 marks]
AQA Paper 3 2021 June Q17
11 marks Standard +0.3
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.
  1. State two assumptions, in context, for \(Y\) to be modelled as \(B(10, 0.6)\) [2 marks]
  2. Find \(P(Y = 4)\) [1 mark]
  3. Find \(P(Y \geq 4)\) [2 marks]
  4. 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 at a 5% significance level whether James's claim is correct. [6 marks]
AQA Paper 3 2021 June Q18
10 marks Moderate -0.3
A factory produces jars of jam and jars of marmalade.
  1. The weight, \(X\) grams, of jam in a jar can be modelled as a normal variable with mean 372 and a standard deviation of 3.5
    1. Find the probability that the weight of jam in a jar is equal to 372 grams. [1 mark]
    2. Find the probability that the weight of jam in a jar is greater than 368 grams. [2 marks]
  2. The weight, \(Y\) grams, of marmalade in a jar can be modelled as a normal variable with mean \(\mu\) and standard deviation \(\sigma\)
    1. Given that \(P(Y < 346) = 0.975\), show that $$346 - \mu = 1.96\sigma$$ Fully justify your answer. [3 marks]
    2. Given further that $$P(Y < 336) = 0.14$$ find \(\mu\) and \(\sigma\) [4 marks]
AQA Paper 3 2022 June Q1
1 marks Easy -1.8
State the range of values of \(x\) for which the binomial expansion of $$\sqrt{1 - \frac{x}{4}}$$ is valid. Circle your answer. [1 mark] \(|x| < \frac{1}{4}\) \quad\quad \(|x| < 1\) \quad\quad \(|x| < 2\) \quad\quad \(|x| < 4\)
AQA Paper 3 2022 June Q2
1 marks Easy -1.8
The shaded region, shown in the diagram below, is defined by $$x^2 - 7x + 7 \leq y \leq 7 - 2x$$ \includegraphics{figure_2} Identify which of the following gives the area of the shaded region. Tick (\(\checkmark\)) one box. [1 mark] \(\int (7 - 2x) \, dx - \int (x^2 - 7x + 7) \, dx\) \(\int_0^5 (x^2 - 5x) \, dx\) \(\int_0^5 (5x - x^2) \, dx\) \(\int_0^5 (x^2 - 9x + 14) \, dx\)
AQA Paper 3 2022 June Q3
1 marks Easy -1.8
The function f is defined by $$f(x) = 2x + 1$$ Solve the equation $$f(x) = f^{-1}(x)$$ Circle your answer. [1 mark] \(x = -1\) \quad\quad \(x = 0\) \quad\quad \(x = 1\) \quad\quad \(x = 2\)
AQA Paper 3 2022 June Q4
2 marks Easy -1.8
Find $$\int \left(x^2 + x^{\frac{1}{2}}\right) dx$$ [2 marks]
AQA Paper 3 2022 June Q5
3 marks Easy -1.2
  1. Sketch the graph of $$y = \sin 2x$$ for \(0° \leq x \leq 360°\) \includegraphics{figure_5a} [2 marks]
  2. The equation $$\sin 2x = A$$ has exactly two solutions for \(0° \leq x \leq 360°\) State the possible values of \(A\). [1 mark]
AQA Paper 3 2022 June Q6
9 marks Standard +0.8
A design for a surfboard is shown in Figure 1. Figure 1 \includegraphics{figure_6_1} The curve of the top half of the surfboard can be modelled by the parametric equations $$x = -2t^2$$ $$y = 9t - 0.7t^2$$ for \(0 \leq t \leq 9.5\) as shown in Figure 2, where \(x\) and \(y\) are measured in centimetres. Figure 2 \includegraphics{figure_6_2}
  1. Find the length of the surfboard. [2 marks]
    1. Find an expression for \(\frac{dy}{dx}\) in terms of \(t\). [3 marks]
    2. Hence, show that the width of the surfboard is approximately one third of its length. [4 marks]
AQA Paper 3 2022 June Q7
7 marks Standard +0.3
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{figure_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. [3 marks]
    2. Show that $$T = K d^n$$ where K and n are constants to be found. [2 marks]
  2. Neptune takes approximately 60 000 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. [2 marks]
AQA Paper 3 2022 June Q8
10 marks Standard +0.3
Water is poured into an empty cone at a constant rate of 8 cm³/s After \(t\) seconds the depth of the water in the inverted cone is \(h\) cm, as shown in the diagram below. \includegraphics{figure_8} When the depth of the water in the inverted cone is \(h\) cm, the volume, \(V\) cm³, is given by $$V = \frac{\pi h^3}{12}$$
  1. Show that when \(t = 3\) $$\frac{dV}{dh} = 6 \sqrt[3]{6\pi}$$ [4 marks]
  2. Hence, find the rate at which the depth is increasing when \(t = 3\) Give your answer to three significant figures. [3 marks]
AQA Paper 3 2022 June Q9
6 marks Standard +0.8
Assume that \(a\) and \(b\) are integers such that $$a^2 - 4b - 2 = 0$$
  1. Prove that \(a\) is even. [2 marks]
  2. Hence, prove that \(2b + 1\) is even and explain why this is a contradiction. [3 marks]
  3. Explain what can be deduced about the solutions of the equation $$a^2 - 4b - 2 = 0$$ [1 mark]
AQA Paper 3 2022 June Q10
13 marks Standard +0.3
The function f is defined by $$f(x) = \frac{x^2 + 10}{2x + 5}$$ where f has its maximum possible domain. The curve \(y = f(x)\) intersects the line \(y = x\) at the points P and Q as shown below. \includegraphics{figure_10}
  1. State the value of \(x\) which is not in the domain of f. [1 mark]
  2. Explain how you know that the function f is many-to-one. [2 marks]
    1. Show that the \(x\)-coordinates of P and Q satisfy the equation $$x^2 + 5x - 10 = 0$$ [2 marks]
    2. Hence, find the exact \(x\)-coordinate of P and the exact \(x\)-coordinate of Q. [1 mark]
  4. Show that P and Q are stationary points of the curve. Fully justify your answer. [5 marks]
  5. Using set notation, state the range of f. [2 marks]
AQA Paper 3 2022 June Q11
1 marks Easy -1.8
\(X \sim \text{N}(14, 0.35)\) Find the standard deviation of \(X\), correct to two decimal places. Circle your answer. [1 mark] 0.12 \quad\quad 0.35 \quad\quad 0.59 \quad\quad 1.78
AQA Paper 3 2022 June Q12
1 marks Easy -2.0
The box plot below shows summary data for the number of minutes late that buses arrived at a rural bus stop. \includegraphics{figure_12} Identify which term best describes the distribution of this data. Circle your answer. [1 mark] negatively skewed \quad\quad normal \quad\quad positively skewed \quad\quad symmetrical
AQA Paper 3 2022 June Q13
2 marks Easy -1.8
A reporter is writing an article on the CO₂ 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. [2 marks]