AQA AS Paper 2 (AS Paper 2) 2018 June

Given that \(\frac{dy}{dx} = \frac{1}{6x^2}\), find \(y\). Circle your answer. \(\frac{-1}{3x^3} + c\) \quad \(\frac{1}{2x^3} + c\) \quad \(\frac{-1}{6x} + c\) \quad \(\frac{-1}{3x} + c\) [1 mark]

Figure 1 shows \(y = f(x)\). \includegraphics{figure_1} Which figure below shows \(y = f(2x)\)? Tick one box. \includegraphics{figure_2} \quad \includegraphics{figure_3} \quad \includegraphics{figure_4} \quad \includegraphics{figure_5} [1 mark]

Express as a single logarithm \(2\log_a 6 - \log_a 3\) [2 marks]

Solve the equation \(\tan^2 2\theta - 3 = 0\) giving all the solutions for \(0° \leq \theta \leq 360°\) [4 marks]

\(f'(x) = \left(2x - \frac{3}{x}\right)^2\) and \(f(3) = 2\) Find \(f(x)\). [4 marks]

Points \(A(-7, -7)\), \(B(8, -1)\), \(C(4, 9)\) and \(D(-11, 3)\) are the vertices of a quadrilateral \(ABCD\).

1. Prove that \(ABCD\) is a rectangle. [4 marks]
2. Find the area of \(ABCD\). [2 marks]

1. Express \(2x^2 - 5x + k\) in the form \(a(x - b)^2 + c\) [3 marks]
2. Find the values of \(k\) for which the curve \(y = 2x^2 - 5x + k\) does not intersect the line \(y = 3\) [3 marks]

A circle of radius 6 passes through the points \((0, 0)\) and \((0, 10)\).

1. Sketch the two possible positions of the circle. [1 mark]
2. Find the equations of the two circles. [3 marks]

It is given that \(\cos 15° = \frac{1}{2}\sqrt{2 + \sqrt{3}}\) and \(\sin 15° = \frac{1}{2}\sqrt{2 - \sqrt{3}}\) Show that \(\tan^2 15°\) can be written in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are integers. Fully justify your answer. [3 marks]

In the binomial expansion of \((1 + x)^n\), where \(n \geq 4\), the coefficient of \(x^4\) is \(\frac{1}{2}\) times the sum of the coefficients of \(x^2\) and \(x^3\) Find the value of \(n\). [5 marks]

Rakti makes open-topped cylindrical planters out of thin sheets of galvanised steel. She bends a rectangle of steel to make an open cylinder and welds the joint. She then welds this cylinder to the circumference of a circular base. \includegraphics{figure_11} The planter must have a capacity of \(8000\text{cm}^3\) Welding is time consuming, so Rakti wants the total length of weld to be a minimum. Calculate the radius, \(r\), and height, \(h\), of a planter which requires the minimum total length of weld. Fully justify your answers, giving them to an appropriate degree of accuracy. [9 marks]

Trees in a forest may be affected by one of two types of fungal disease, but not by both. The number of trees affected by disease A, \(n_A\), can be modelled by the formula $$n_A = ae^{0.1t}$$ where \(t\) is the time in years after 1 January 2017. The number of trees affected by disease B, \(n_B\), can be modelled by the formula $$n_B = be^{0.2t}$$ On 1 January 2017 a total of 290 trees were affected by a fungal disease. On 1 January 2018 a total of 331 trees were affected by a fungal disease.

1. Show that \(b = 90\), to the nearest integer, and find the value of \(a\). [3 marks]
2. Estimate the total number of trees that will be affected by a fungal disease on 1 January 2020. [1 mark]
3. Find the year in which the number of trees affected by disease B will first exceed the number affected by disease A. [3 marks]
4. Comment on the long-term accuracy of the model. [1 mark]

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

\(x\)	0	1	2	3	4 or more
P(X = x)	0.35	0.25	\(k\)	0.14	0.1

Find the value of \(k\). Circle your answer. 0.14 \quad 0.16 \quad 0.18 \quad 1 [1 mark]

Given that \(\sum x = 364\), \(\sum x^2 = 19412\), \(n = 10\), find \(\sigma\), the standard deviation of \(X\). Circle your answer. 24.8 \quad 44.1 \quad 616.2 \quad 1941.2 [1 mark]

Nicola, a darts player, is practising hitting the bullseye. She knows from previous experience that she has a probability of 0.3 of hitting the bullseye with each dart. Nicola throws eight practice darts.

1. Using a binomial distribution, calculate the probability that she will hit the bullseye three or more times. [2 marks]
2. Nicola throws eight practice darts on three different occasions. Calculate the probability that she will hit the bullseye three or more times on all three occasions. [2 marks]
3. State two assumptions that are necessary for the distribution you have used in part (a) to be valid. [2 marks]

Kevin is the Principal of a college. He wishes to investigate types of transport used by students to travel to college. There are 3200 students in the college and Kevin decides to survey 60 of them. Describe how he could obtain a simple random sample of size 60 from the 3200 students. [4 marks]

The table below is an extract from the Large Data Set, showing the purchased quantities of fats and oils for the South East of England in 2014.

Description	Purchased quantity
Butter	42
Soft margarine	16
Olive oil	17
Other vegetable and salad oils	28

Kim claims that more olive oil was purchased in the South East than soft margarine. Explain why Kim may be incorrect. [2 marks]

Jennie is a piano teacher who teaches nine pupils. She records how many hours per week they practice the piano along with their most recent practical exam score.

Student	Practice (hours per week)	Practical exam score (out of 100)
Donovan	50	64
Vazquez	6	71
Higgins	3	55
Begum	2.5	47
Collins	1	80
Coldbridge	4	61
Nedbalek	4.5	65
Carter	8	83
White	11	92

1. Identify two possible outliers by name, giving a possible explanation for the position on the scatter diagram of each outlier. [4 marks]
2. Jennie discards the two outliers.
   1. Describe the correlation shown by the scatter diagram for the remaining points. [1 mark]
   2. Interpret this correlation in the context of the question. [1 mark]

Martin grows cucumbers from seed. In the past, he has found that 70% of all seeds successfully germinate and grow into cucumber plants. He decides to try out a new brand of seed. The producer of this brand claims that these seeds are more likely to successfully germinate than other brands of seeds. Martin sows 20 of this new brand of seed and 18 successfully germinate. Carry out a hypothesis test at the 5% level of significance to investigate the producer's claim. [7 marks]