AQA Paper 2 (Paper 2) 2018 June

1 Which of these statements is correct? Tick one box. $$\begin{aligned} & x = 2 \Rightarrow x ^ { 2 } = 4
& x ^ { 2 } = 4 \Rightarrow x = 2
& x ^ { 2 } = 4 \Leftrightarrow x = 2
& x ^ { 2 } = 4 \Rightarrow x = - 2 \end{aligned}$$

2 Find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 1 + 2 x ) ^ { 7 }\)
Circle your answer. 4242184
\includegraphics[max width=\textwidth, alt={}, center]{5e475c96-6bbb-44fb-a411-706dac20e2d6-03_1442_1168_219_365} Find the total shaded area. Circle your answer.
-68 60686128

4 A curve, \(C\), has equation \(y = x ^ { 2 } - 6 x + k\), where \(k\) is a constant. The equation \(x ^ { 2 } - 6 x + k = 0\) has two distinct positive roots. 4

1. Sketch \(C\) on the axes below.
   \includegraphics[max width=\textwidth, alt={}, center]{5e475c96-6bbb-44fb-a411-706dac20e2d6-04_1013_1016_534_513} 4
2. Find the range of possible values for \(k\). Fully justify your answer.
   \begin{center} \begin{tabular}{ | l | } \hline

5 Prove that 23 is a prime number.
[0pt] [2 marks]
\end{tabular} \end{center}

6 Find the coordinates of the stationary point of the curve with equation $$( x + y - 2 ) ^ { 2 } = \mathrm { e } ^ { y } - 1$$ \(7 \quad\) A function f has domain \(\mathbb { R }\) and range \(\{ y \in \mathbb { R } : y \geq \mathrm { e } \}\) The graph of \(y = \mathrm { f } ( x )\) is shown.
\includegraphics[max width=\textwidth, alt={}, center]{5e475c96-6bbb-44fb-a411-706dac20e2d6-08_922_1108_447_466} The gradient of the curve at the point \(( x , y )\) is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = ( x - 1 ) \mathrm { e } ^ { x }\)
Find an expression for \(\mathrm { f } ( x )\).
Fully justify your answer.

8

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.
   8
2. (ii) Find the greatest value of \(\frac { 1 } { \sqrt { 3 } \sin x - 3 \cos x + 4 }\)

9 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 \(\pounds 72\) per hour and total sales are \(\pounds 336\)
9

1. Show that $$x \frac { \mathrm {~d} x } { \mathrm {~d} t } = 4032 ( 8 - t )$$ 9
2. Hence, show that $$x ^ { 2 } = 4032 t ( 16 - t )$$ \(\mathbf { 9 }\) (c) The stall opens at 09.30. 9
3. 1. The trader closes the stall when the rate of sales falls below \(\pounds 24\) per hour.
      Using the results in parts (a) and (b), calculate the earliest time that the trader closes the stall.
      9
4. (ii) Explain why the model used by the trader is not valid at 09.30.

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

11 A uniform rod, \(A B\), 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[max width=\textwidth, alt={}, center]{5e475c96-6bbb-44fb-a411-706dac20e2d6-16_277_908_1521_568} The rod is balanced in equilibrium at \(C\).
Find \(x\). Circle your answer.
[0pt] [1 mark]
\(1.8 \mathrm {~m} \quad 1.5 \mathrm {~m} \quad 1.75 \mathrm {~m} \quad 1.6 \mathrm {~m}\)

12 The graph below shows the velocity of an object moving in a straight line over a 20 second journey.
\includegraphics[max width=\textwidth, alt={}, center]{5e475c96-6bbb-44fb-a411-706dac20e2d6-17_579_1682_406_169} 12

1. Find the maximum magnitude of the acceleration of the object. 12
2. The object is at its starting position at times \(0 , t _ { 1 }\) and \(t _ { 2 }\) seconds.
   Find \(t _ { 1 }\) and \(t _ { 2 }\)

13 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) 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 13

1. The boy applies a horizontal force of 150 N . Show that the crate remains stationary.
   13
2. Instead, the boy uses a handle to pull the crate forward. He exerts a force of 150 N , at an angle of \(15 ^ { \circ }\) above the horizontal, as shown in the diagram.
   \includegraphics[max width=\textwidth, alt={}, center]{5e475c96-6bbb-44fb-a411-706dac20e2d6-19_244_915_408_561} Determine whether the crate remains stationary.
   Fully justify your answer.

14 A quadrilateral has vertices \(A , B , C\) and \(D\) with position vectors given by $$\overrightarrow { O A } = \left[ \begin{array} { l } 3
5
1 \end{array} \right] , \overrightarrow { O B } = \left[ \begin{array} { r } - 1
2
7 \end{array} \right] , \overrightarrow { O C } = \left[ \begin{array} { l } 0
7
6 \end{array} \right] \text { and } \overrightarrow { O D } = \left[ \begin{array} { r } 4
10
0 \end{array} \right]$$ 14

1. Write down the vector \(\overrightarrow { A B }\) 14
2. Show that \(A B C D\) is a parallelogram, but not a rhombus.

15 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 \(\mathrm { ms } ^ { - 2 }\) of each minibus is modelled as a function of time, \(t\) seconds, after the flag is dropped: The acceleration of \(\mathrm { A } = 0.138 t ^ { 2 }\)
The acceleration of \(\mathrm { B } = 0.024 t ^ { 3 }\)
15

1. Find the time taken for A to travel 100 metres.
   Give your answer to four significant figures.
   15
2. The company decides to buy the minibus which travels 100 metres in the shortest time. Determine which minibus should be bought.
   15
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.
   A particle is projected with an initial speed \(u\), at an angle of \(35 ^ { \circ }\) 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.

16

1. \(\quad\) Find \(u\). 16 In this question use \(g = 9.81 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) In this question use \(\boldsymbol { g } = 9.81 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) 16
2. Find the total time that the particle is in flight.

17 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[max width=\textwidth, alt={}, center]{5e475c96-6bbb-44fb-a411-706dac20e2d6-24_240_1006_402_516} 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 \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
17

1. 1. Find \(R\).
      

      17	17 (ii) Find the tension in the rope.

      17
   2. The roller-skater releases the rope at a point \(A\), when she reaches a speed of \(6 \mathrm {~ms} ^ { - 1 }\) 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 \(A\). 17
   3. 1. Determine whether the roller-skater will stop before reaching the stationary buggy.
         Fully justify your answer.
         17
   4. (ii) Explain the change in motion that the driver noticed.