AQA Paper 2 (Paper 2) 2022 June

1 A circle has centre \(( 4 , - 5 )\) and radius 6
Find the equation of the circle.
Tick ( \(\checkmark\) ) one box. $$\begin{aligned} & ( x - 4 ) ^ { 2 } + ( y + 5 ) ^ { 2 } = 6
& ( x + 4 ) ^ { 2 } + ( y - 5 ) ^ { 2 } = 6
& ( x - 4 ) ^ { 2 } + ( y + 5 ) ^ { 2 } = 36
& ( x + 4 ) ^ { 2 } + ( y - 5 ) ^ { 2 } = 36 \end{aligned}$$ □
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2 State the value of $$\lim _ { h \rightarrow 0 } \frac { \sin ( \pi + h ) - \sin \pi } { h }$$ Circle your answer.
\(\cos h\)
-1
0
1

3 The function f is concave and is represented by one of the graphs below. Identify the graph which represents f . Tick ( \(\checkmark\) ) one box.
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-03_709_561_632_191}
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-03_117_111_927_826}
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-03_716_570_630_1082} □
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-03_711_563_1503_191}
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-03_711_565_1503_1085}
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-03_117_113_1800_1717}

4 The diagram shows a triangle \(A B C\).
\(A B\) is the shortest side. The lengths of \(A C\) and \(B C\) are 6.1 cm and 8.7 cm respectively. The size of angle \(A B C\) is \(38 ^ { \circ }\)
Find the size of the largest angle.
Give your answer to the nearest degree.
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-05_2488_1716_219_153}

5 The binomial expansion of \(( 2 + 5 x ) ^ { 4 }\) is given by $$( 2 + 5 x ) ^ { 4 } = A + 160 x + B x ^ { 2 } + 1000 x ^ { 3 } + 625 x ^ { 4 }$$ 5

1. Find the value of \(A\) and the value of \(B\).
   [0pt] [2 marks]
   L
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2. Show that $$( 2 + 5 x ) ^ { 4 } - ( 2 - 5 x ) ^ { 4 } = C x + D x ^ { 3 }$$ where \(C\) and \(D\) are constants to be found.
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3. Hence, or otherwise, find $$\int \left( ( 2 + 5 x ) ^ { 4 } - ( 2 - 5 x ) ^ { 4 } \right) \mathrm { d } x$$

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1. Asif notices that \(24 ^ { 2 } = 576\) and \(2 + 4 = 6\) gives the last digit of 576 He checks two more examples: $$\begin{array} { l c } 27 ^ { 2 } = 729 & 29 ^ { 2 } = 841
   2 + 7 = 9 & 2 + 9 = 11
   \text { Last digit } 9 & \text { Last digit } 1 \end{array}$$ Asif concludes that he can find the last digit of any square number greater than 100 by adding the digits of the number being squared. Give a counter example to show that Asif's conclusion is not correct. 6
2. Claire tells Asif that he should look only at the last digit of the number being squared. $$\begin{array} { c c } 27 ^ { 2 } = 729 & 24 ^ { 2 } = 576
   7 ^ { 2 } = 49 & 4 ^ { 2 } = 16
   \text { Last digit } 9 & \text { Last digit } 6 \end{array}$$ Using Claire's method determine the last digit of \(23456789 { } ^ { 2 }\)
   [0pt] [1 mark] 6
3. Given Claire's method is correct, use proof by exhaustion to show that no square number has a last digit of 8

7 The curve \(y = 15 - x ^ { 2 }\) and the isosceles triangle \(O P Q\) are shown on the diagram The curve \(y = 15 - x ^ { 2 }\) and the isosceles triangle \(O P Q\) are shown on the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-10_759_810_388_614} Vertices \(P\) and \(Q\) lie on the curve such that \(Q\) lies vertically above some point ( \(q , 0\) ) The line \(P Q\) is parallel to the \(x\)-axis. 7

1. Show that the area, \(A\), of the triangle \(O P Q\) is given by $$A = 15 q - q ^ { 3 } \quad \text { for } 0 < q < c$$ where \(c\) is a constant to be found.
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2. Find the exact maximum area of triangle \(O P Q\). Fully justify your answer.

8

1. Sketch the graph of \(y = \frac { 1 } { x ^ { 2 } }\)
   \includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-12_1273_1083_404_482} 8
2. The graph of \(y = \frac { 1 } { x ^ { 2 } }\) can be transformed onto the graph of \(y = \frac { 9 } { x ^ { 2 } }\) using a stretch in one direction. Beth thinks the stretch should be in the \(y\)-direction.
   Paul thinks the stretch should be in the \(x\)-direction.
   State, giving reasons for your answer, whether Beth is correct, Paul is correct, both are correct or neither is correct.
   [0pt] [3 marks]

9 Given that $$\log _ { 2 } x ^ { 3 } - \log _ { 2 } y ^ { 2 } = 9$$ show that $$x = A y ^ { p }$$ where \(A\) is an integer and \(p\) is a rational number.
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-15_2488_1716_219_153}

10 A gardener has a greenhouse containing 900 tomato plants. The gardener notices that some of the tomato plants are damaged by insects.
Initially there are 25 damaged tomato plants.
The number of tomato plants damaged by insects is increasing by \(32 \%\) each day.
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1. The total number of plants damaged by insects, \(x\), is modelled by $$x = A \times B ^ { t }$$ where \(A\) and \(B\) are constants and \(t\) is the number of days after the gardener first noticed the damaged plants. 10
2. 1. Use this model to find the total number of plants damaged by insects 5 days after the gardener noticed the damaged plants.
      10
3. (ii) Explain why this model is not realistic in the long term.
   10
4. A refined model assumes the rate of increase of the number of plants damaged by insects is given by $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { x ( 900 - x ) } { 2700 }$$ 10
5. 1. Show that $$\int \left( \frac { A } { x } + \frac { B } { 900 - x } \right) \mathrm { d } x = \int \mathrm { d } t$$ where \(A\) and \(B\) are positive integers to be found.
      

      10 (iii) Hence, find the number of days it takes from when the damage is first noticed until half of the plants are damaged by the insects. [2 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

11 A moon vehicle has a mass of 212 kg and a length of 3 metres.
On the moon the vehicle has a weight of 345 N
Calculate a value for acceleration due to gravity on the moon.
Circle your answer.
[0pt] [1 mark] $$0.614 \mathrm {~m} \mathrm {~s} ^ { - 2 } \quad 1.63 \mathrm {~m} \mathrm {~s} ^ { - 2 } \quad 1.84 \mathrm {~m} \mathrm {~s} ^ { - 2 } \quad 4.89 \mathrm {~m} \mathrm {~s} ^ { - 2 }$$

12 A car is travelling along a straight horizontal road with initial velocity \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
The car begins to accelerate at a constant rate \(a \mathrm {~ms} ^ { - 2 }\) for 5 seconds, to reach a final velocity of \(4 u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Express \(a\) in terms of \(u\).
Circle your answer.
[0pt] [1 mark]
\(a = 0.2 u\)
\(a = 0.4 u\)
\(a = 0.6 u\)
\(a = 0.8 u\)

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1. Show that $$h = 2.5 \sin ^ { 2 } \theta$$ 13 In this question use \(g = 9.8 \mathrm {~ms} ^ { - 2 }\) 13
2. Hence, given that \(0 ^ { \circ } \leq \theta \leq 60 ^ { \circ }\), find the maximum value of \(h\).
   13
3. Nisha claims that the larger the size of the ball, the greater the maximum vertical height will be. State whether Nisha is correct, giving a reason for your answer.

14 A \(\pounds 2\) coin has a diameter of 28 mm and a mass of 12 grams. A uniform rod \(A B\) of length 160 mm and a fixed load of mass \(m\) grams are used to check that a \(\pounds 2\) coin has the correct mass. The rod rests with its midpoint on a support.
A \(\pounds 2\) coin is placed face down on the rod with part of its curved edge directly above \(A\). The fixed load is hung by a light inextensible string from a point directly below the other end of the rod at \(B\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-22_195_766_854_639} 14

1. Given that the rod is horizontal and rests in equilibrium, find \(m\).
   14
2. State an assumption you have made about the \(\pounds 2\) coin to answer part (a).

15 A car is moving in a straight line along a horizontal road. The graph below shows how the car's velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) changes with time, \(t\) seconds.
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-23_509_746_456_648} Over the period \(0 \leq t \leq 15\) the car has a total displacement of - 7 metres.
Initially the car has velocity \(0 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
Find the next time when the velocity of the car is \(0 \mathrm {~ms} ^ { - 1 }\)
[0pt] [4 marks]

16 Two particles, \(P\) and \(Q\), move in the same horizontal plane. Particle \(P\) is initially at rest at the point with position vector \(( - 4 \mathbf { i } + 5 \mathbf { j } )\) metres and moves with constant acceleration \(( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\) Particle \(Q\) moves in a straight line, passing through the points with position vectors \(( \mathbf { i } - \mathbf { j } )\) metres and \(( 10 \mathbf { i } + c \mathbf { j } )\) metres.
\(P\) and \(Q\) are moving along parallel paths.
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1. Show that \(c = - 13\)
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2. 1. Find an expression for the position vector of \(P\) at time \(t\) seconds.
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3. (ii) Hence, prove that the paths of \(P\) and \(Q\) are not collinear.

17 A particle is moving such that its position vector, \(\mathbf { r }\) metres, at time \(t\) seconds, is given by $$\mathbf { r } = \mathrm { e } ^ { t } \cos t \mathbf { i } + \mathrm { e } ^ { t } \sin t \mathbf { j }$$ Show that the magnitude of the acceleration of the particle, \(a \mathrm {~ms} ^ { - 2 }\), is given by $$a = 2 \mathrm { e } ^ { t }$$ Fully justify your answer.
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-27_2490_1728_217_141}

18 An object, \(O\), of mass \(m\) kilograms is hanging from a ceiling by two light, inelastic strings of different lengths. The shorter string, of length 0.8 metres, is fixed to the ceiling at \(A\).
The longer string, of length 1.2 metres, is fixed to the ceiling at \(B\).
This object hangs 0.6 metres directly below the ceiling as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-28_252_940_667_552} 18

1. Show that the tension in the shorter string is over \(30 \%\) more than the tension in the longer string.
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2. The tension in the longer string is known to be \(2 g\) newtons. Find the value of \(m\).
   A rough wooden ramp is 10 metres long and is inclined at an angle of \(25 ^ { \circ }\) above the horizontal. The bottom of the ramp is at the point \(O\). A crate of mass 20 kg is at rest at the point \(A\) on the ramp.
   The crate is pulled up the ramp using a rope attached to the crate.
   Once in motion, the rope remains taut and parallel to the line of greatest slope of the ramp.
   \includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-30_252_842_804_598}

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1. The tension in the rope is 230 N
   The crate accelerates up the ramp at \(1.2 \mathrm {~ms} ^ { - 2 }\)
   Find the coefficient of friction between the crate and the ramp.
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2. 1. The crate takes 3.8 seconds to reach the top of the ramp.
      Find the distance \(O A\).
      [0pt] [3 marks]
      19
3. (ii) Other than air resistance, state one assumption you have made about the crate in answering part (b)(i).
   \includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-32_2492_1721_217_150}