AQA Paper 2 (Paper 2) 2018 June

Which of these statements is correct? Tick one box. [1 mark] \(x = 2 \Rightarrow x^2 = 4\) \(x^2 = 4 \Rightarrow x = 2\) \(x^2 = 4 \Leftrightarrow x = 2\) \(x^2 = 4 \Rightarrow x = -2\)

Find the coefficient of \(x^2\) in the expansion of \((1 + 2x)^7\) Circle your answer. [1 mark] 42 4 21 84

The graph of \(y = x^3\) is shown. \includegraphics{figure_1} Find the total shaded area. Circle your answer. [1 mark] \(-68\) 60 68 128

A curve, C, has equation \(y = x^2 - 6x + k\), where \(k\) is a constant. The equation \(x^2 - 6x + k = 0\) has two distinct positive roots.

1. Sketch C on the axes below. [2 marks]
2. Find the range of possible values for \(k\). Fully justify your answer. [4 marks]

Prove that 23 is a prime number. [2 marks]

Find the coordinates of the stationary point of the curve with equation \((x + y - 2)^2 = e^y - 1\) [7 marks]

A function f has domain \(\mathbb{R}\) and range \(\{y \in \mathbb{R} : y \geq c\}\) The graph of \(y = f(x)\) is shown. \includegraphics{figure_2} The gradient of the curve at the point \((x, y)\) is given by \(\frac{dy}{dx} = (x - 1)e^x\) Find an expression for f(x). Fully justify your answer. [8 marks]

1. Determine a sequence of transformations which maps the graph of \(y = \sin x\) onto the graph of \(y = \sqrt{3} \sin x - 3 \cos x + 4\) Fully justify your answer. [7 marks]
2. 1. Show that the least value of \(\frac{1}{\sqrt{3} \sin x - 3 \cos x + 4}\) is \(\frac{2 - \sqrt{3}}{2}\) [2 marks]
   2. Find the greatest value of \(\frac{1}{\sqrt{3} \sin x - 3 \cos x + 4}\) [1 mark]

A market trader notices that daily sales are dependent on two variables: number of hours, \(t\), after the stall opens total sales, \(x\), in pounds since the stall opened. The trader models the rate of sales as directly proportional to \(\frac{8 - t}{x}\) After two hours the rate of sales is £72 per hour and total sales are £336

1. Show that $$x \frac{dx}{dt} = 4032(8 - t)$$ [3 marks]
2. Hence, show that $$x^2 = 4032t(16 - t)$$ [3 marks]
3. The stall opens at 09.30.
   1. The trader closes the stall when the rate of sales falls below £24 per hour. Using the results in parts (a) and (b), calculate the earliest time that the trader closes the stall. [6 marks]
   2. Explain why the model used by the trader is not valid at 09.30. [2 marks]

A garden snail moves in a straight line from rest to 1.28 cm s\(^{-1}\), with a constant acceleration in 1.8 seconds. Find the acceleration of the snail. Circle your answer. [1 mark] 2.30 m s\(^{-2}\) 0.71 m s\(^{-2}\) 0.0071 m s\(^{-2}\) 0.023 m s\(^{-2}\)

A uniform rod, AB, has length 4 metres. The rod is resting on a support at its midpoint C. A particle of mass 4 kg is placed 0.6 metres to the left of C. Another particle of mass 1.5 kg is placed \(x\) metres to the right of C, as shown. \includegraphics{figure_3} The rod is balanced in equilibrium at C. Find \(x\). Circle your answer. [1 mark] 1.8 m 1.5 m 1.75 m 1.6 m

The graph below shows the velocity of an object moving in a straight line over a 20 second journey. \includegraphics{figure_4}

1. Find the maximum magnitude of the acceleration of the object. [1 mark]
2. The object is at its starting position at times 0, \(t_1\) and \(t_2\) seconds. Find \(t_1\) and \(t_2\) [4 marks]

In this question use \(g = 9.8\) m s\(^{-2}\) A boy attempts to move a wooden crate of mass 20 kg along horizontal ground. The coefficient of friction between the crate and the ground is 0.85

1. The boy applies a horizontal force of 150 N. Show that the crate remains stationary. [3 marks]
2. Instead, the boy uses a handle to pull the crate forward. He exerts a force of 150 N, at an angle of 15° above the horizontal, as shown in the diagram. \includegraphics{figure_5} Determine whether the crate remains stationary. Fully justify your answer. [5 marks]

A quadrilateral has vertices A, B, C and D with position vectors given by $$\overrightarrow{OA} = \begin{pmatrix} 3 \\ 5 \\ 1 \end{pmatrix}, \overrightarrow{OB} = \begin{pmatrix} -1 \\ 2 \\ 7 \end{pmatrix}, \overrightarrow{OC} = \begin{pmatrix} 0 \\ 7 \\ 6 \end{pmatrix} \text{ and } \overrightarrow{OD} = \begin{pmatrix} 4 \\ 10 \\ 0 \end{pmatrix}$$

1. Write down the vector \(\overrightarrow{AB}\) [1 mark]
2. Show that ABCD is a parallelogram, but not a rhombus. [5 marks]

A driver is road-testing two minibuses, A and B, for a taxi company. The performance of each minibus along a straight track is compared. A flag is dropped to indicate the start of the test. Each minibus starts from rest. The acceleration in m s\(^{-2}\) of each minibus is modelled as a function of time, \(t\) seconds, after the flag is dropped: The acceleration of A = \(0.138 t^2\) The acceleration of B = \(0.024 t^3\)

1. Find the time taken for A to travel 100 metres. Give your answer to four significant figures. [4 marks]
2. The company decides to buy the minibus which travels 100 metres in the shortest time. Determine which minibus should be bought. [4 marks]
3. The models assume that both minibuses start moving immediately when \(t = 0\) In light of this, explain why the company may, in reality, make the wrong decision. [1 mark]

In this question use \(g = 9.81\) m s\(^{-2}\) A particle is projected with an initial speed \(u\), at an angle of 35° above the horizontal. It lands at a point 10 metres vertically below its starting position. The particle takes 1.5 seconds to reach the highest point of its trajectory.

1. Find \(u\). [3 marks]
2. Find the total time that the particle is in flight. [3 marks]

A buggy is pulling a roller-skater, in a straight line along a horizontal road, by means of a connecting rope as shown in the diagram. \includegraphics{figure_6} The combined mass of the buggy and driver is 410 kg A driving force of 300 N and a total resistance force of 140 N act on the buggy. The mass of the roller-skater is 72 kg A total resistance force of R newtons acts on the roller-skater. The buggy and the roller-skater have an acceleration of 0.2 m s\(^{-2}\)

1. 1. Find R. [3 marks]
   2. Find the tension in the rope. [3 marks]
2. State a necessary assumption that you have made. [1 mark]
3. The roller-skater releases the rope at a point A, when she reaches a speed of 6 m s\(^{-1}\) She continues to move forward, experiencing the same resistance force. The driver notices a change in motion of the buggy, and brings it to rest at a distance of 20 m from A.
   1. Determine whether the roller-skater will stop before reaching the stationary buggy. Fully justify your answer. [5 marks]
   2. Explain the change in motion that the driver noticed. [2 marks]