AQA Paper 3 (Paper 3) 2022 June

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 \(|x| < 4\)

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] \(\int (7 - 2x) \, dx - \int (x^2 - 7x + 7) \, dx\) \(\int_0^5 (x^2 - 5x) \, dx\) \(\int_0^5 (5x - x^2) \, dx\) \(\int_0^5 (x^2 - 9x + 14) \, dx\)

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 \(x = 2\)

Find $$\int \left(x^2 + x^{\frac{1}{2}}\right) dx$$ [2 marks]

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]

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]
2. 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]

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. 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]

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]

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]

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]
3. 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]

\(X \sim \text{N}(14, 0.35)\) 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

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

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]

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 \(X\).

1. State two assumptions necessary for \(X\) to be modelled by a binomial distribution. [2 marks]
2. Assume that \(X\) can be modelled by a binomial distribution.
   1. Find P(\(X = 1\)) [1 mark]
   2. Find P(\(X < 4\)) [2 marks]
   3. Find P(\(X \geq 10\)) [2 marks]
3. 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 $$Y \sim \text{B}(10, p)$$ The standard deviation of \(Y\) is 1.5 Calculate the possible values of \(p\). [3 marks]

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.

1. Identify the population in their investigation. [1 mark]
2. 1. State the name of this method of sampling. [1 mark]
   2. Describe one advantage of this sampling method. [1 mark]

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.

1. Represent this information by fully completing the Venn diagram below. [3 marks] \includegraphics{figure_16}
2. A household is chosen at random from the sample.
   1. Find the probability that the household owns a cat only. [1 mark]
   2. Find the probability that the household owns at least two of the three types of pet. [2 marks]
   3. Find the probability that the household owns a cat or a dog or both, given that the household does not own a tortoise. [2 marks]
3. Determine whether a household owning a cat and a household owning a tortoise are independent of each other. Fully justify your answer. [2 marks]

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. [6 marks]

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.

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. [3 marks]
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.
   1. Find the probability that the height of a randomly selected male athlete is 1.82 metres. [1 mark]
   2. Find the probability that the height of a randomly selected male athlete is between 1.70 metres and 1.90 metres. [1 mark]
   3. Two male athletes are chosen at random. Calculate the probability that both of their heights are between 1.70 metres and 1.90 metres. [1 mark]
3. The summarised data for the heights, \(h\) metres, of a random sample of 40 male athletes at the Winter Olympics is given below. $$\sum h = 69.2 \quad\quad \sum (h - \bar{h})^2 = 2.81$$ Use this data to calculate estimates of the mean and standard deviation of the heights of male athletes at the Winter Olympics. [3 marks]
4. Using your answers from part (c), compare the heights of male athletes at the Summer Olympics and male athletes at the Winter Olympics. [2 marks]

A bank runs a campaign to promote Internet banking accounts to their customers. Before the campaign, 42% of their customers had an Internet banking account. One week after the campaign started, 35 customers were surveyed at random and 18 of them were found to have registered for an Internet banking account. Using a binomial distribution, carry out a hypothesis test at the 10% significance level to investigate the claim that, since the campaign, there has been an increase in the proportion of customers registered for an Internet banking account. [6 marks]