AQA Paper 3 (Paper 3) 2018 June

A circle has equation \((x - 4)^2 + (y + 4)^2 = 9\) What is the area of the circle? Circle your answer. [1 mark] \(3\pi\) \quad \(9\pi\) \quad \(16\pi\) \quad \(81\pi\)

A curve has equation \(y = x^5 + 4x^3 + 7x + q\) where \(q\) is a positive constant. Find the gradient of the curve at the point where \(x = 0\) Circle your answer. [1 mark] \(0\) \quad \(4\) \quad \(7\) \quad \(q\)

The line \(L\) has equation \(2x + 3y = 7\) Which one of the following is perpendicular to \(L\)? Tick one box. [1 mark] \(2x - 3y = 7\) \(3x + 2y = -7\) \(2x + 3y = -\frac{1}{7}\) \(3x - 2y = 7\)

Sketch the graph of \(y = |2x + a|\), where \(a\) is a positive constant. Show clearly where the graph intersects the axes. [3 marks] \includegraphics{figure_4}

Show that, for small values of \(x\), the graph of \(y = 5 + 4\sin\frac{x}{2} + 12\tan\frac{x}{3}\) can be approximated by a straight line. [3 marks]

A function \(f\) is defined by \(f(x) = \frac{x}{\sqrt{2x - 2}}\)

1. State the maximum possible domain of \(f\). [2 marks]
2. Use the quotient rule to show that \(f'(x) = \frac{x - 2}{(2x - 2)^{\frac{3}{2}}}\). [3 marks]
3. Show that the graph of \(y = f(x)\) has exactly one point of inflection. [7 marks]
4. Write down the values of \(x\) for which the graph of \(y = f(x)\) is convex. [1 mark]

1. Given that \(\log_a y = 2\log_a 7 + \log_a 4 + \frac{1}{2}\), find \(y\) in terms of \(a\). [4 marks]
2. When asked to solve the equation $$2\log_a x = \log_a 9 - \log_a 4$$ a student gives the following solution: \(2\log_a x = \log_a 9 - \log_a 4\) \(\Rightarrow 2\log_a x = \log_a \frac{9}{4}\) \(\Rightarrow \log_a x^2 = \log_a \frac{9}{4}\) \(\Rightarrow x^2 = \frac{9}{4}\) \(\therefore x = \frac{3}{2}\) or \(-\frac{3}{2}\) Explain what is wrong with the student's solution. [1 mark]

1. Prove the identity \(\frac{\sin 2x}{1 + \tan^2 x} = 2\sin x \cos^3 x\) [3 marks]
2. Hence find \(\int \frac{4\sin 4\theta}{1 + \tan^2 2\theta} d\theta\) [6 marks]

Helen is creating a mosaic pattern by placing square tiles next to each other along a straight line. \includegraphics{figure_9} The area of each tile is half the area of the previous tile, and the sides of the largest tile have length \(w\) centimetres.

1. Find, in terms of \(w\), the length of the sides of the second largest tile. [1 mark]
2. Assume the tiles are in contact with adjacent tiles, but do not overlap. Show that, no matter how many tiles are in the pattern, the total length of the series of tiles will be less than \(3.5w\). [4 marks]
3. Helen decides the pattern will look better if she leaves a 3 millimetre gap between adjacent tiles. Explain how you could refine the model used in part (b) to account for the 3 millimetre gap, and state how the total length of the series of tiles will be affected. [2 marks]

Prove by contradiction that \(\sqrt[3]{2}\) is an irrational number. [7 marks]

The table below shows the probability distribution for a discrete random variable \(X\).

\(x\)	1	2	3	4	5
P(\(X = x\))	\(k\)	\(2k\)	\(4k\)	\(2k\)	\(k\)

Find the value of \(k\). Circle your answer. [1 mark] \(\frac{1}{2}\) \quad \(\frac{1}{4}\) \quad \(\frac{1}{10}\) \quad \(1\)

The histogram below shows the heights, in cm, of male A-level students at a particular school. \includegraphics{figure_12} Which class interval contains the median height? Circle your answer. [1 mark] \([155, 160)\) \quad \([160, 170)\) \quad \([170, 180)\) \quad \([180, 190]\)

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

Year	2011	2012	2013	2014	\% change since 2011
Other takeaway food brought home	0	0	0	0	\(-29\)

Sarah claims that the \(-29\%\) change since 2011 is incorrect, as there is no change between 2011 and 2014. Using your knowledge of the Large Data Set to justify your answer, explain whether Sarah's claim is correct. [3 marks]

A teacher in a college asks her mathematics students what other subjects they are studying. She finds that, of her 24 students: 12 study physics 8 study geography 4 study geography and physics

1. A student is chosen at random from the class. Determine whether the event 'the student studies physics' and the event 'the student studies geography' are independent. [2 marks]
2. It is known that for the whole college: the probability of a student studying mathematics is \(\frac{1}{5}\) the probability of a student studying biology is \(\frac{1}{6}\) the probability of a student studying biology given that they study mathematics is \(\frac{3}{8}\) Calculate the probability that a student studies mathematics or biology or both. [4 marks]

Abu visits his local hardware store to buy six light bulbs. He knows that 15% of all bulbs at this store are faulty.

1. State a distribution which can be used to model the number of faulty bulbs he buys. [1 mark]
2. Find the probability that all of the bulbs he buys are faulty. [1 mark]
3. Find the probability that at least two of the bulbs he buys are faulty. [2 marks]
4. Find the mean of the distribution stated in part (a). [1 mark]
5. State two necessary assumptions in context so that the distribution stated in part (a) is valid. [2 marks]

A survey of 120 adults found that the volume, \(X\) litres per person, of carbonated drinks they consumed in a week had the following results: $$\sum x = 165.6 \quad \sum x^2 = 261.8$$

1. 1. Calculate the mean of \(X\). [1 mark]
   2. Calculate the standard deviation of \(X\). [2 marks]
2. Assuming that \(X\) can be modelled by a normal distribution find
   1. P\((0.5 < X < 1.5)\) [2 marks]
   2. P\((X = 1)\) [1 mark]
3. Determine with a reason, whether a normal distribution is suitable to model this data. [2 marks]
4. It is known that the volume, \(Y\) litres per person, of energy drinks consumed in a week may be modelled by a normal distribution with standard deviation 0.21 Given that P\((Y > 0.75) = 0.10\), find the value of \(\mu\), correct to three significant figures. [4 marks]

Suzanne is a member of a sports club. For each sport she competes in, she wins half of the matches.

1. After buying a new tennis racket Suzanne plays 10 matches and wins 7 of them. Investigate, at the 10% level of significance, whether Suzanne's new racket has made a difference to the probability of her winning a match. [7 marks]
2. After buying a new squash racket, Suzanne plays 20 matches. Find the minimum number of matches she must win for her to conclude, at the 10% level of significance, that the new racket has improved her performance. [5 marks]

In a region of England, the government decides to use an advertising campaign to encourage people to eat more healthily. Before the campaign, the mean consumption of chocolate per person per week was known to be 66.5g, with a standard deviation of 21.2g

1. After the campaign, the first 750 available people from this region were surveyed to find out their average consumption of chocolate.
   1. State the sampling method used to collect the survey. [1 mark]
   2. Explain why this sample should not be used to conduct a hypothesis test. [1 mark]
2. A second sample of 750 people revealed that the mean consumption of chocolate per person per week was 65.4g Investigate, at the 10% level of significance, whether the advertising campaign has decreased the mean consumption of chocolate per person per week. Assume that an appropriate sampling method was used and that the consumption of chocolate is normally distributed with an unchanged standard deviation. [6 marks]