AQA Paper 3 (Paper 3) 2021 June

The graph of \(y = \arccos x\) is shown. \includegraphics{figure_1} State the coordinates of the end point \(P\). Circle your answer. [1 mark] \((-\pi, 1)\) \quad \((-1, \pi)\) \quad \(\left(-\frac{\pi}{2}, 1\right)\) \quad \(\left(-1, \frac{\pi}{2}\right)\)

Simplify fully $$\frac{(x + 3)(6 - 2x)}{(x - 3)(3 + x)} \quad \text{for } x \neq \pm 3$$ Circle your answer. [1 mark] \(-2\) \quad \(2\) \quad \(\frac{(6 - 2x)}{(x - 3)}\) \quad \(\frac{(2x - 6)}{(x - 3)}\)

\(f(x) = 3x^2\) Obtain \(\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\) Circle your answer. [1 mark] \(\frac{3h^2}{h}\) \quad \(x^3\) \quad \(\frac{3(x + h)^2 - 3x^2}{h}\) \quad \(6x\)

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

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 £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{figure_5}

1. 1. Find the area of this flowerbed. [2 marks]
   2. Find the cost of the edging strip required for this flowerbed. [3 marks]
2. A flowerbed is to be made with an area of 20 m²
   1. Show that the cost, £\(C\), 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. [3 marks]
   2. Hence, show that the minimum cost of the edging strip for this flowerbed occurs when \(r \approx 4.5\) Fully justify your answer. [5 marks]

Given that \(x > 0\) and \(x \neq 25\), fully simplify $$\frac{10 + 5x - 2x^{\frac{1}{2}} - x^{\frac{3}{2}}}{5 - \sqrt{x}}$$ Fully justify your answer. [4 marks]

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]

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]

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. 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]

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

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

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]

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

1. 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]

\(A\) 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]

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]

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]

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]

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]