AQA Paper 2 (Paper 2) 2021 June

1 Four possible sketches of \(y = a x ^ { 2 } + b x + c\) are shown below.
Given \(b ^ { 2 } - 4 a c = 0\) and \(a , b\) and \(c\) are non-zero constants, which sketch is the only one that could possibly be correct? Tick ( \(\checkmark\) ) one box. A
\includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-02_383_303_995_550}
\includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-02_113_111_1128_1009} B
\includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-02_113_111_1562_1009} C
\includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-02_223_300_1868_548}
\includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-02_108_109_2001_1009} D
\includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-02_388_301_2305_549}
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2 A curve has equation \(y = \mathrm { f } ( x )\) The curve has a point of inflection at \(x = 7\)
It is given that \(\mathrm { f } ^ { \prime } ( 7 ) = a\) and \(\mathrm { f } ^ { \prime \prime } ( 7 ) = b\), where \(a\) and \(b\) are real numbers. Identify which one of the statements below must be true.
Circle your answer.
\(\mathrm { f } ^ { \prime } ( 7 ) \neq 0\)
\(\mathrm { f } ^ { \prime } ( 7 ) = 0\)
\(\mathrm { f } ^ { \prime \prime } ( 7 ) \neq 0\)
\(\mathrm { f } ^ { \prime \prime } ( 7 ) = 0\)

3 A sequence is defined by $$u _ { 1 } = a \text { and } u _ { n + 1 } = - 1 \times u _ { n }$$ Find \(\sum _ { n = 1 } ^ { 95 } u _ { n }\)
Circle your answer.
\(- a\)
0
\(a\)
95a

4

1. On Figure 1 add a sketch of the graph of $$y = | 3 x - 6 |$$ 4
2. Find the coordinates of the points of intersection of the two graphs.
   Fully justify your answer.
   \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-05_2488_1716_219_153}

5 Express $$\frac { 5 ( x - 3 ) } { ( 2 x - 11 ) ( 4 - 3 x ) }$$ in the form $$\frac { A } { ( 2 x - 11 ) } + \frac { B } { ( 4 - 3 x ) }$$ where \(A\) and \(B\) are integers.

6 Show that the solution of the equation $$5 ^ { x } = 3 ^ { x + 4 }$$ can be written as $$x = \frac { \ln 81 } { \ln 5 - \ln 3 }$$ Fully justify your answer.

7 A circle has equation $$x ^ { 2 } + y ^ { 2 } - 6 x - 8 y = p$$ 7

1. 1. State the coordinates of the centre of the circle.
      7
2. (ii) Find the radius of the circle in terms of \(p\).
   7
3. The circle intersects the coordinate axes at exactly three points. Find the two possible values of \(p\).

8 Kai is proving that \(n ^ { 3 } - n\) is a multiple of 3 for all positive integer values of \(n\). Kai begins a proof by exhaustion.
Step 1 $$n ^ { 3 } - n = n \left( n ^ { 2 } - 1 \right)$$ Step 2 When \(n = 3 m\), where \(m\) is a \(n ^ { 3 } - n = 3 m \left( 9 m ^ { 2 } - 1 \right)\) non-negative integer which is a multiple of 3 Step 3 When \(n = 3 m + 1\), $$\begin{aligned} & n ^ { 3 } - n = ( 3 m + 1 ) \left( ( 3 m + 1 ) ^ { 2 } - 1 \right)
& = ( 3 m + 1 ) \left( 9 m ^ { 2 } \right)
& = 3 ( 3 m + 1 ) \left( 3 m ^ { 2 } \right) \end{aligned}$$ Step 5 Therefore \(n ^ { 3 } - n\) is a multiple of 3 for all positive integer values of \(n\) 8

1. Explain the two mistakes that Kai has made after Step 3. Step 4 \section*{which is a multiple of 3
   which is a multiple of 3}
   \includegraphics[max width=\textwidth, alt={}]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-10_67_134_964_230}
   别
   all positive integer values of \(n\) \section*{a} \includegraphics[max width=\textwidth, alt={}]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-10_58_49_1037_370} 墐 pount \(\_\_\_\_\)
   \(\_\_\_\_\) " \(\_\_\_\_\)
   8
2. Correct Kai's argument from Step 4 onwards.

9 A robotic arm which is attached to a flat surface at the origin \(O\), is used to draw a graphic design. The arm is made from two rods \(O P\) and \(P Q\), each of length \(d\), which are joined at \(P\).
A pen is attached to the arm at \(Q\).
The coordinates of the pen are controlled by adjusting the angle \(O P Q\) and the angle \(\theta\) between \(O P\) and the \(x\)-axis. For this particular design the pen is made to move so that the two angles are always equal to each other with \(0 \leq \theta \leq \frac { \pi } { 2 }\) as shown in Figure 2. \begin{figure}[h] \end{figure} 9

1. Show that the \(x\)-coordinate of the pen can be modelled by the equation $$x = d \left( \cos \theta + \sin \left( 2 \theta - \frac { \pi } { 2 } \right) \right)$$ 9
2. Hence, show that $$x = d \left( 1 + \cos \theta - 2 \cos ^ { 2 } \theta \right)$$ 9
3. It can be shown that $$x = \frac { 9 d } { 8 } - d \left( \cos \theta - \frac { 1 } { 4 } \right) ^ { 2 }$$ State the greatest possible value of \(x\) and the corresponding value of \(\cos \theta\)
   9
4. Figure 3 below shows the arm when the \(x\)-coordinate is at its greatest possible value. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-14_570_773_456_630}
   \end{figure} Find, in terms of \(d\), the exact distance \(O Q\).
   \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-15_2488_1716_219_153}

10 The function h is defined by $$\mathrm { h } ( x ) = \frac { \sqrt { x } } { x - 3 }$$ where \(h\) has its maximum possible domain.
10

1. Find the domain of h .
   Give your answer using set notation. 10
2. Alice correctly calculates $$h ( 1 ) = - 0.5 \text { and } h ( 4 ) = 2$$ She then argues that since there is a change of sign there must be a value of \(x\) in the interval \(1 < x < 4\) that gives \(\mathrm { h } ( x ) = 0\) Explain the error in Alice's argument.
   [0pt] [2 marks]
   10
3. By considering any turning points of h , determine whether h has an inverse function. Fully justify your answer.
   [0pt] [6 marks]

11 A particle's displacement, \(r\) metres, with respect to time, \(t\) seconds, is defined by the equation $$r = 3 \mathrm { e } ^ { 0.5 t }$$ Find an expression for the velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of the particle at time \(t\) seconds.
Circle your answer.
\(v = 1.5 \mathrm { e } ^ { 0.5 t }\)
\(v = 6 \mathrm { e } ^ { 0.5 t }\)
\(v = 1.5 t \mathrm { e } ^ { 0.5 t }\)
\(v = 6 t e ^ { 0.5 t }\)

12 A particle has a speed of \(6 \mathrm {~ms} ^ { - 1 }\) in a direction relative to unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-18_307_542_1528_749} The velocity of this particle can be expressed as a vector \(\left[ \begin{array} { l } v _ { 1 }
v _ { 2 } \end{array} \right] \mathrm { ms } ^ { - 1 }\)
Find the correct expression for \(v _ { 2 }\)
Circle your answer.
[0pt] [1 mark]
\(v _ { 2 } = 6 \cos 30 ^ { \circ }\)
\(v _ { 2 } = 6 \sin 30 ^ { \circ }\)
\(v _ { 2 } = - 6 \sin 30 ^ { \circ }\)
\(v _ { 2 } = - 6 \cos 30 ^ { \circ }\)

13 A vehicle, of total mass 1200 kg , is travelling along a straight, horizontal road at a constant speed of \(13 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) This vehicle begins to accelerate at a constant rate.
After 40 metres it reaches a speed of \(17 \mathrm {~ms} ^ { - 1 }\)
Find the resultant force acting on the vehicle during the period of acceleration.

14 A motorised scooter is travelling along a straight path with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over time \(t\) seconds as shown by the following graph.
\includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-20_1120_1134_420_452} Noosha says that, in the period \(\mathbf { 1 2 } \leq \boldsymbol { t } \leq \mathbf { 3 6 }\), the scooter travels approximately 130 metres. Determine if Noosha is correct, showing clearly any calculations you have used.

15 A cyclist is towing a trailer behind her bicycle. She is riding along a straight, horizontal path at a constant speed.
\includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-22_371_723_447_657} A tension of \(T\) newtons acts on the connecting rod between the bicycle and the trailer.
The cyclist is causing a constant driving force of 40 N to be applied whilst pedalling forwards on her bicycle. The constant resistance force acting on the trailer is 12 N
15

1. State the value of \(T\) giving a clear reason for your answer.
   15
2. State one assumption you have made in reaching your answer to part (a).

16 A straight uniform rod, \(A B\), has length 6 m and mass 0.2 kg A particle of weight \(w\) newtons is fixed at \(A\).
A second particle of weight \(3 w\) newtons is fixed at \(B\).
The rod is suspended by a string from a point \(x\) metres from \(B\).
The rod rests in equilibrium with \(A B\) horizontal and the string hanging vertically as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-24_410_1148_767_445} Show that $$x = \frac { 3 w + 0.3 g } { 2 w + 0.1 g }$$ \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-25_2488_1716_219_153}

17 A ball is released from a great height so that it falls vertically downwards towards the surface of the Earth. 17

1. Using a simple model, Andy predicts that the velocity of the ball, exactly 2 seconds after being released from rest, is \(2 g \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Show how Andy has obtained his prediction.
   17
2. Using a refined model, Amy predicts that the ball's acceleration, \(a \mathrm {~ms} ^ { - 2 }\), at time \(t\) seconds after being released from rest is $$a = g - 0.1 v$$ where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of the ball at time \(t\) seconds. Find an expression for \(v\) in terms of \(t\).
   17
3. Comment on the value of \(v\) for the two models as \(t\) becomes large.

18 Two particles, \(P\) and \(Q\), are projected at the same time from a fixed point \(X\), on the ground, so that they travel in the same vertical plane.
\(P\) is projected at an acute angle \(\theta ^ { \circ }\) to the horizontal, with speed \(u \mathrm {~ms} ^ { - 1 }\)
\(Q\) is projected at an acute angle \(2 \theta ^ { \circ }\) to the horizontal, with speed \(2 u \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
Both particles land back on the ground at exactly the same point, \(Y\).
Resistance forces to motion may be ignored.
18

1. Show that $$\cos 2 \theta = \frac { 1 } { 8 }$$ 18
2. \(\quad P\) takes a total of 0.4 seconds to travel from \(X\) to \(Y\).
   Find the time taken by \(Q\) to travel from \(X\) to \(Y\).
   18
3. State one modelling assumption you have chosen to make in this question.
   [0pt] [1 mark]
   \begin{center} \begin{tabular}{|l|l|} \hline 19 & \begin{tabular}{l} Two skaters, Jo and Amba, are separately skating across a smooth, horizontal surface of ice.
   Both are moving in the same direction, so that their paths are straight and are parallel to each other.
   Jo is moving with constant velocity \(( 2.8 \mathbf { i } + 9.6 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)
   At time \(t = 0\) seconds Amba is at position ( \(2 \mathbf { i } - 7 \mathbf { j }\) ) metres and is moving with a constant speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
   Explain why Amba's velocity must be in the form \(k ( 2.8 \mathbf { i } + 9.6 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(k\) is a constant.
   [0pt] [1 mark] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

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1. (ii) Verify that \(k = 0.8\)
   [0pt] [1 mark] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
   19
2. Find the position vector of Amba when \(t = 4\)
   [0pt] [3 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
   \end{tabular}
   \hline \end{tabular} \end{center} 19
3. At both \(t = 0\) and \(t = 4\) there is a distance of 5 metres between Jo and Amba's positions. Determine the shortest distance between their two parallel lines of motion.
   Fully justify your answer.
   \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-32_2492_1721_217_150}