AQA AS Paper 2 (AS Paper 2) 2020 June

Identify the expression below that is equivalent to \(e^{-\frac{2}{5}}\) Circle your answer. [1 mark] \(\frac{1}{\sqrt[5]{e^2}}\) \quad \(-\sqrt{e^5}\) \quad \(-\sqrt[5]{e^2}\) \quad \(\frac{1}{\sqrt{e^5}}\)

It is given that \(y = \frac{1}{x}\) and \(x < -1\) Determine which statement below fully describes the possible values of \(y\). Tick (\(\checkmark\)) one box. [1 mark] \(-\infty < y < -1\) \(y > -1\) \(-1 < y < 0\) \(y < 0\)

It is given that $$y = 3x^4 + \frac{2}{x} - \frac{x}{4} + 1$$ Find an expression for \(\frac{d^2y}{dx^2}\) [3 marks]

Find all the solutions of $$9 \sin^2 x - 6 \sin x + \cos^2 x = 0$$ where \(0° \leq x \leq 180°\) Give your solutions to the nearest degree. Fully justify your answer. [4 marks]

Joseph is expanding \((2 - 3x)^7\) in ascending powers of \(x\). He states that the coefficient of the fourth term is 15120 Joseph's teacher comments that his answer is almost correct. Using a suitable calculation, explain the teacher's comment. [4 marks]

A circle has equation $$x^2 + y^2 + 10x - 4y - 71 = 0$$

1. Find the centre of the circle. [2 marks]
2. Hence, find the equation of the tangent to the circle at the point \((1, 10)\), giving your answer in the form \(ax + by + c = 0\) where \(a\), \(b\) and \(c\) are integers. [4 marks]

The population of a country was 3.6 million in 1989. It grew exponentially to reach 6 million in 2019. Estimate the population of the country in 2049 if the exponential growth continues unchanged. [2 marks]

1. Using \(y = 2^{2x}\) as a substitution, show that $$16^x - 2^{(2x+3)} - 9 = 0$$ can be written as $$y^2 - 8y - 9 = 0$$ [2 marks]
2. Hence, show that the equation $$16^x - 2^{(2x+3)} - 9 = 0$$ has \(x = \log_2 3\) as its only solution. Fully justify your answer. [4 marks]

1. 1. Find $$\int (4x - x^3) dx$$ [2 marks]
   2. Evaluate $$\int_{-2}^{2} (4x - x^3) dx$$ [1 mark]
2. Using a sketch, explain why the integral in part (a)(ii) does not give the area enclosed between the curve \(y = 4x - x^3\) and the \(x\)-axis. [2 marks]
3. Find the area enclosed between the curve \(y = 4x - x^3\) and the \(x\)-axis. [2 marks]

A curve has gradient function $$\frac{dy}{dx} = 3x^2 - 12x + c$$ The curve has a turning point at \((-1, 1)\)

1. Find the coordinates of the other turning point of the curve. Fully justify your answer. [6 marks]
2. Find the set of values of \(x\) for which \(y\) is increasing. [2 marks]

A fire crew is tackling a grass fire on horizontal ground. The crew directs a single jet of water which flows continuously from point \(A\). \includegraphics{figure_11} The path of the jet can be modelled by the equation $$y = -0.0125x^2 + 0.5x - 2.55$$ where \(x\) metres is the horizontal distance of the jet from the fire truck at \(O\) and \(y\) metres is the height of the jet above the ground. The coordinates of point \(A\) are \((a, 0)\)

1. 1. Find the value of \(a\). [3 marks]
   2. Find the horizontal distance from \(A\) to the point where the jet hits the ground. [1 mark]
2. Calculate the maximum vertical height reached by the jet. [4 marks]
3. A vertical wall is located 11 metres horizontally from \(A\) in the direction of the jet. The height of the wall is 2.3 metres. Using the model, determine whether the jet passes over the wall, stating any necessary modelling assumption. [3 marks]

A student plots the scatter diagram below showing the mass in kilograms against the CO₂ emissions in grams per kilogram for a sample of cars in the Large Data Set. \includegraphics{figure_12} Their teacher tells them to remove an error to clean the data. Identify the data point which should be removed. Circle your answer below. [1 mark] \(A\) \quad \(B\) \quad \(C\) \quad \(D\)

The random variable \(X\) is such that \(X \sim B\left(n, \frac{1}{3}\right)\) The standard deviation of \(X\) is 4 Find the value of \(n\). Circle your answer. [1 mark] 9 \quad 12 \quad 18 \quad 72

A retail company has 5200 employees in 100 stores throughout the United Kingdom. The company recently introduced a new reward scheme for its staff. The management team wanted to sample the staff to find out their opinions of the new scheme. Three possible sampling methods were suggested: Method A \quad Choose 100 people who work at the largest store Method B \quad Choose one person at random from each of the 100 stores Method C \quad List all employees in alphabetical order and assign each a number from 1 to 5200 Choose a random number between 1 and 52 Choose this person and every 52nd person on the list thereafter.

1. Give one disadvantage of using Method A compared with using Method B. [1 mark]
2. Give one advantage of using Method B compared with using Method C. [1 mark]
3. 1. Identify the method of sampling used in Method C. [1 mark]
   2. Give a reason why Method C does not provide a random sample. [1 mark]

A random sample of ten CO₂ emissions was selected from the Large Data Set. The emissions in grams per kilogram were: 13 \quad 45 \quad 45 \quad 0 \quad 49 \quad 77 \quad 49 \quad 49 \quad 49 \quad 78

1. Find the standard deviation of the sample. [1 mark]
2. An environmentalist calculated the average CO₂ emissions for cars in the Large Data Set registered in 2002 and in 2016. The averages are listed below.

   Year of registration	2002	2016
   Average CO₂ emission	171.2	120.4

   The environmentalist claims that the average CO₂ emissions for 2002 and 2016 combined is 145.8 Determine whether this claim is correct. Fully justify your answer. [2 marks]

A mathematical puzzle is published every day in a newspaper. Over a long period of time Paula is able to solve the puzzle correctly 60% of the time.

1. For a randomly chosen 14-day period find the probability that:
   1. Paula correctly solves exactly 8 puzzles [1 mark]
   2. Paula correctly solves at least 7 but not more than 11 puzzles. [2 marks]
2. State one assumption that is necessary for the distribution used in part (a) to be valid. [1 mark]

A game consists of spinning a circular wheel divided into numbered sectors as shown below. \includegraphics{figure_17} On each spin the score, \(X\), is the value shown in the sector that the arrow points to when the spinner stops. The probability of the arrow pointing at a sector is proportional to the angle subtended at the centre by that sector.

1. Show that \(P(X = 1) = \frac{5}{18}\) [1 mark]
2. Complete the probability distribution for \(X\) in the table below.

   \(x\)	1
   \(P(X = x)\)	\(\frac{5}{18}\)

   [2 marks]

1. Bag A contains 7 blue discs, 4 red discs and 1 yellow disc. Two discs are drawn at random from bag A without replacement. Find the probability that exactly one of the discs is blue. [2 marks]
2. Bag A contains 7 blue discs, 4 red discs and 1 yellow disc. Bag B contains 3 blue discs and 6 red discs. A disc is drawn at random from Bag A and placed in Bag B. A disc is then drawn at random from Bag B. Find the probability that the disc drawn from Bag B is red. [3 marks]

It is known from historical data that 15% of the residents of a town buy the local weekly newspaper, 'Local News'. A new free weekly paper is introduced into the town. The owners of 'Local News' are interested to know whether the introduction of the free newspaper has changed the proportion of residents who buy their paper. In a random sample of 50 residents of the town taken after the free newspaper was introduced, it was found that 3 of them purchased 'Local News' regularly. Investigate, at the 5% significance level, whether this sample provides evidence that the proportion of local residents who buy 'Local News' has changed. [6 marks]