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AQA AS Paper 1 2023 June Q9
3 marks Moderate -0.8
9 A continuous curve has equation \(y = \mathrm { f } ( x )\) The curve passes through the points \(A ( 2,1 ) , B ( 4,5 )\) and \(C ( 6,1 )\) It is given that \(f ^ { \prime } ( 4 ) = 0\) Jasmin made two statements about the nature of the curve \(y = \mathrm { f } ( x )\) at the point \(B\) :
Statement 1: There is a turning point at \(B\) Statement 2: There is a maximum point at \(B\) 9
  1. Draw a sketch of the curve \(y = \mathrm { f } ( x )\) such that Statement 1 is correct and Statement 2 is correct.
    [0pt] [1 mark] \includegraphics[max width=\textwidth, alt={}, center]{9cd7f38d-a2a1-4fd3-9ed9-cb389e8ee3b6-10_593_588_1043_817} 9
  2. Draw a sketch of the curve \(y = \mathrm { f } ( x )\) such that Statement 1 is correct and Statement 2 is not correct. \includegraphics[max width=\textwidth, alt={}, center]{9cd7f38d-a2a1-4fd3-9ed9-cb389e8ee3b6-11_607_597_497_813} 9
  3. Draw a sketch of the curve \(y = \mathrm { f } ( x )\) such that Statement 1 is not correct and Statement 2 is not correct.
    [0pt] [1 mark] \includegraphics[max width=\textwidth, alt={}, center]{9cd7f38d-a2a1-4fd3-9ed9-cb389e8ee3b6-11_605_597_1541_813} 1 Charlie buys a car for \(\pounds 18000\) on 1 January 2016.
    The value of the car decreases exponentially.
    The car has a value of \(\pounds 12000\) on 1 January 2018.
AQA AS Paper 1 2023 June Q10
8 marks Moderate -0.3
10
  1. Charlie says:
    • because the car has lost \(\pounds 6000\) after two years, after another two years it will be worth £6000.
    Charlie's friend Kaya says:
    • because the car has lost one third of its value after two years, after another two years it will be worth \(\pounds 8000\).
    Explain whose statement is correct, justifying the value they have stated.
    10
  2. The value of Charlie's car, \(\pounds V , t\) years after 1 January 2016 may be modelled by the equation \(V = A \mathrm { e } ^ { - k t }\)
    where \(A\) and \(k\) are positive constants.
    Find the value of \(t\) when the car has a value of \(\pounds 10000\), giving your answer to two significant figures.
    [5 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
    [1 mark] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
AQA AS Paper 1 2023 June Q11
7 marks Standard +0.3
11
  1. A circle has equation $$x ^ { 2 } + y ^ { 2 } - 10 x - 6 = 0$$ Find the centre and the radius of the circle.
    11
  2. An equilateral triangle has one vertex at the origin, and one side along the line \(x = 8\), as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{9cd7f38d-a2a1-4fd3-9ed9-cb389e8ee3b6-14_697_750_1311_735} 11
    1. Show that the vertex at the origin lies inside the circle \(x ^ { 2 } + y ^ { 2 } - 10 x - 6 = 0\) 11
  4. (ii) Prove that the triangle lies completely within the circle \(x ^ { 2 } + y ^ { 2 } - 10 x - 6 = 0\)
AQA AS Paper 1 2023 June Q12
1 marks Easy -1.8
12 A particle, initially at rest, starts to move forward in a straight line with constant acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) After 6 seconds the particle has a velocity of \(3 \mathrm {~ms} ^ { - 1 }\) Find the value of \(a\) Circle your answer. \(\begin{array} { l l l l } - 2 & - 0.5 & 0.5 & 2 \end{array}\)
AQA AS Paper 1 2023 June Q13
1 marks Easy -1.8
13 A resultant force of \(\left[ \begin{array} { c } - 2 \\ 6 \end{array} \right] \mathrm { N }\) acts on a particle.
The acceleration of the particle is \(\left[ \begin{array} { c } - 6 \\ y \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 2 }\) Find the value of \(y\) Circle your answer.
[0pt] [1 mark] 231018
\includegraphics[max width=\textwidth, alt={}]{9cd7f38d-a2a1-4fd3-9ed9-cb389e8ee3b6-17_2496_1721_214_148}
AQA AS Paper 1 2023 June Q15
4 marks Easy -1.2
15 A particle is moving in a straight line such that its velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), changes with respect to time, \(t\) seconds, as shown in the graph below. \includegraphics[max width=\textwidth, alt={}, center]{9cd7f38d-a2a1-4fd3-9ed9-cb389e8ee3b6-18_501_889_406_667} 15
  1. Show that the acceleration of the particle over the first 4 seconds is \(3.5 \mathrm {~ms} ^ { - 2 }\) 15
  2. The particle is initially at a fixed point \(P\) Show that the displacement of the particle from \(P\), when \(t = 9\), is 62 metres.
AQA AS Paper 1 2023 June Q16
7 marks Easy -1.2
16 A toy remote control speed boat is launched from one edge of a small pond and moves in a straight line across the pond's surface. The boat's velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is modelled in terms of time, \(t\) seconds after the boat is launched, by the expression $$v = 0.9 + 0.16 t - 0.06 t ^ { 2 }$$ 16
  1. Find the acceleration of the boat when \(t = 2\) 16
  2. Find the displacement of the boat, from the point where it was launched, when \(t = 2\)
AQA AS Paper 1 2023 June Q17
4 marks Moderate -0.8
17 A particle, \(P\), is initially at rest on a smooth horizontal surface. A resultant force of \(\left[ \begin{array} { c } 12 \\ 9 \end{array} \right] \mathrm { N }\) is then applied to \(P\), so that it moves in a straight line.
17
  1. Find the magnitude of the resultant force. 17
  2. Two fixed points \(A\) and \(B\) have position vectors $$\overrightarrow { O A } = \left[ \begin{array} { l } 3 \\ 7 \end{array} \right] \text { metres } \quad \text { and } \quad \overrightarrow { O B } = \left[ \begin{array} { c } k \\ k - 1 \end{array} \right] \text { metres }$$ with respect to a fixed origin, \(O\) \(P\) moves in a straight line parallel to \(\overrightarrow { A B }\) 17
    1. Find \(\overrightarrow { A B }\) in terms of \(k\) 17
  4. (ii) Find the value of \(k\)
AQA AS Paper 1 2023 June Q18
6 marks Standard +0.3
18 A rescue van is towing a broken-down car by using a tow bar. The van and the car are moving with a constant acceleration of \(0.6 \mathrm {~ms} ^ { - 2 }\) along a straight horizontal road as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{9cd7f38d-a2a1-4fd3-9ed9-cb389e8ee3b6-22_77_163_511_991} The van has a total mass of 2780 kg
The car has a total mass of 1620 kg
The van experiences a driving force of \(D\) newtons. The van experiences a total resistance force of \(R\) newtons.
The car experiences a total resistance force of \(0.6 R\) newtons. 18
  1. The tension in the tow bar, \(T\) newtons, may be modelled by $$T = k D - 18$$ where \(k\) is a constant. Find \(k\) 18
  2. State one assumption that must be made in answering part (a).
AQA AS Paper 1 2024 June Q1
1 marks Easy -1.8
1 It is given that \(\tan \theta ^ { \circ } = k\), where \(k\) is a constant.
Find \(\tan ( \theta + 180 ) ^ { \circ }\) Circle your answer. \(- k\) \(- \frac { 1 } { k }\) \(\frac { 1 } { k }\) \(k\)
AQA AS Paper 1 2024 June Q2
1 marks Easy -1.8
2 Curve \(C\) has equation \(y = \frac { 1 } { ( x - 1 ) ^ { 2 } }\) State the equations of the asymptotes to curve \(C\) Tick ( ✓ ) one box.
[0pt] [1 mark] \(x = 0\) and \(y = 0\) \includegraphics[max width=\textwidth, alt={}, center]{f4f303a2-f029-42be-93a0-046b0c81e3c0-02_111_113_1975_735} \(x = 0\) and \(y = 1\) \includegraphics[max width=\textwidth, alt={}, center]{f4f303a2-f029-42be-93a0-046b0c81e3c0-02_113_113_2126_735} \(x = 1\) and \(y = 0\) \includegraphics[max width=\textwidth, alt={}, center]{f4f303a2-f029-42be-93a0-046b0c81e3c0-02_111_113_2279_735} \(x = 1\) and \(y = 1\) \includegraphics[max width=\textwidth, alt={}, center]{f4f303a2-f029-42be-93a0-046b0c81e3c0-02_117_113_2426_735}
AQA AS Paper 1 2024 June Q3
4 marks Moderate -0.8
3 Express \(\frac { \sqrt { 3 } + 3 \sqrt { 5 } } { \sqrt { 5 } - \sqrt { 3 } }\) in the form \(a + \sqrt { b }\), where \(a\) and \(b\) are integers. Fully justify your answer.
AQA AS Paper 1 2024 June Q4
7 marks Standard +0.3
4
    1. By using a suitable trigonometric identity, show that the equation $$\sin \theta \tan \theta = 4 \cos \theta$$ can be written as $$\tan ^ { 2 } \theta = 4$$ 4
  2. (ii) Hence solve the equation $$\sin \theta \tan \theta = 4 \cos \theta$$ where \(0 ^ { \circ } < \theta < 360 ^ { \circ }\) Give your answers to the nearest degree.
    4
  3. Deduce all solutions of the equation $$\sin 3 \alpha \tan 3 \alpha = 4 \cos 3 \alpha$$ where \(0 ^ { \circ } < \alpha < 180 ^ { \circ }\) Give your answers to the nearest degree.
AQA AS Paper 1 2024 June Q5
3 marks Easy -1.2
5 A student is looking for factors of the polynomial \(\mathrm { f } ( x )\) They suggest that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\) The method they use to check this suggestion is to calculate \(\mathrm { f } ( - 2 )\) They correctly calculate that \(\mathrm { f } ( - 2 ) = 0\) They conclude that their suggestion is correct. 5
  1. Make one comment about the student's method.
    [0pt] [1 mark] 5
  2. Make two comments about the student's conclusion. 1 \(\_\_\_\_\) 2 \(\_\_\_\_\)
AQA AS Paper 1 2024 June Q6
4 marks Moderate -0.8
6 Determine the set of values of \(x\) which satisfy the inequality $$3 x ^ { 2 } + 3 x > x + 6$$ Give your answer in exact form using set notation.
[0pt] [4 marks]
AQA AS Paper 1 2024 June Q7
5 marks Moderate -0.3
7 A triangular field of grass, \(A B C\), has boundaries with lengths as follows: $$A B = 234 \mathrm {~m} \quad B C = 225 \mathrm {~m} \quad A C = 310 \mathrm {~m}$$ The field is shown in the diagram below. 7
  1. Find angle \(A\) 7
  2. Farmers calculate the number of sheep they can keep in a field, by allowing one sheep for every \(1200 \mathrm {~m} ^ { 2 }\) of grass. Find the maximum number of sheep which can be kept in the field \(A B C\) [0pt] [3 marks]
AQA AS Paper 1 2024 June Q8
6 marks Moderate -0.8
8 It is given that $$\ln x - \ln y = 3$$ 8
  1. Express \(x\) in terms of \(y\) in a form not involving logarithms.
    8
  2. Given also that $$x + y = 10$$ find the exact value of \(y\) and the exact value of \(x\)
AQA AS Paper 1 2024 June Q9
5 marks Easy -1.2
9 A curve has equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = x ( 6 - x )$$ 9
  1. \(\quad\) Find \(\mathrm { f } ^ { \prime } ( x )\) 9
  2. The diagram below shows the graph of \(y = \mathrm { f } ( x )\) On the same diagram sketch the gradient function for this curve, stating the coordinates of any points where the gradient function cuts the axes. \includegraphics[max width=\textwidth, alt={}, center]{f4f303a2-f029-42be-93a0-046b0c81e3c0-11_922_1198_1475_406} It is given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = ( x + 2 ) ( 2 x - 1 ) ^ { 2 }$$ and when \(x = 6 , y = 900\) Find \(y\) in terms of \(x\)
AQA AS Paper 1 2024 June Q11
5 marks Moderate -0.3
11 It is given that for the continuous function \(g\)
  • \(g ^ { \prime } ( 1 ) = 0\)
  • \(\mathrm { g } ^ { \prime } ( 4 ) = 0\)
  • \(\mathrm { g } ^ { \prime \prime } ( x ) = 2 x - 5\)
11
  1. Determine the nature of each of the turning points of \(g\) Fully justify your answer.
    11
  2. Find the set of values of \(x\) for which \(g\) is an increasing function.
AQA AS Paper 1 2024 June Q12
6 marks Moderate -0.8
12 The monthly mean temperature of a city, \(T\) degrees Celsius, may be modelled by the equation $$T = 15 + 8 \sin ( 30 m - 120 ) ^ { \circ }$$ where \(m\) is the month number, counting January \(= 1\), February \(= 2\), through to December = 12 12
  1. Using this model, calculate the monthly mean temperature of the city for May, the fifth month.
    12
  2. Using this model, find the month with the highest mean temperature.
    12
  3. Climate change may affect the parameters, 8, 30, 120 and 15, used in this model. 12
    1. State, with a reason, which parameter would be increased because of an overall rise in temperatures.
      [0pt] [1 mark]
      12
  5. (ii) State, with a reason, which parameter would be increased because of the occurrence of more extreme temperatures. \section*{END OF SECTION A}
AQA AS Paper 1 2024 June Q13
1 marks Easy -1.8
13 A particle is moving in a straight line with constant acceleration a \(\mathrm { m } \mathrm { s } ^ { - 2 }\) The particle's velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), varies with time, \(t\) seconds, so that $$v = 3 - 4 t$$ Deduce the value of \(a\) Circle your answer.
[0pt] [1 mark]
-4
-1
3
4
AQA AS Paper 1 2024 June Q14
1 marks Easy -1.3
14
Two forces, \(\mathbf { F } _ { \mathbf { 1 } } = 3 \mathbf { i } + 2 \mathbf { j }\) newtons and \(\mathbf { F } _ { \mathbf { 2 } } = \mathbf { i } - 3 \mathbf { j }\) newtons, are added together to find a resultant force, \(\mathbf { R }\) newtons. This vector addition can be represented using a diagram.
Identify the diagram below which correctly represents this vector addition.
Tick ( ✓ ) one box. \includegraphics[max width=\textwidth, alt={}, center]{f4f303a2-f029-42be-93a0-046b0c81e3c0-17_497_645_762_153} \includegraphics[max width=\textwidth, alt={}, center]{f4f303a2-f029-42be-93a0-046b0c81e3c0-17_147_118_1110_817} \includegraphics[max width=\textwidth, alt={}, center]{f4f303a2-f029-42be-93a0-046b0c81e3c0-17_502_636_762_1080} \includegraphics[max width=\textwidth, alt={}, center]{f4f303a2-f029-42be-93a0-046b0c81e3c0-17_113_111_1110_1749} \includegraphics[max width=\textwidth, alt={}, center]{f4f303a2-f029-42be-93a0-046b0c81e3c0-17_508_796_1400_146} \includegraphics[max width=\textwidth, alt={}, center]{f4f303a2-f029-42be-93a0-046b0c81e3c0-17_499_643_1400_1078} \includegraphics[max width=\textwidth, alt={}, center]{f4f303a2-f029-42be-93a0-046b0c81e3c0-17_111_109_1758_1749}
AQA AS Paper 1 2024 June Q15
4 marks Moderate -0.3
15 A graph indicating how the velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of a particle changes with respect to time, \(t\) seconds, is shown below. \includegraphics[max width=\textwidth, alt={}, center]{f4f303a2-f029-42be-93a0-046b0c81e3c0-18_565_1004_411_502} 15
  1. Find the total distance travelled by the particle over the 8 second period shown.
    15
  2. A student claims that
    "The displacement of the particle is less than the distance travelled."
    State the range of values of \(t\) for which this claim is true.
AQA AS Paper 1 2024 June Q18
6 marks Standard +0.3
18 It is given that two points \(A\) and \(B\) have position vectors $$\overrightarrow { O A } = \left[ \begin{array} { c } 5 \\ - 1 \end{array} \right] \text { metres } \quad \text { and } \quad \overrightarrow { O B } = \left[ \begin{array} { c } 13 \\ 5 \end{array} \right] \text { metres. }$$ 18
  1. Show that the distance from \(A\) to \(B\) is 10 metres.
    [0pt] [3 marks]
    18
  2. A constant resultant force, of magnitude \(R\) newtons, acts on a particle so that it moves in a straight line passing through the same two points \(A\) and \(B\) At \(A\), the speed of the particle is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction from \(A\) to \(B\) The particle takes 2 seconds to travel from \(A\) to \(B\) The mass of the particle is 150 grams. Find the value of \(R\)
AQA AS Paper 1 2024 June Q19
8 marks Moderate -0.3
19
  1. It is given that \(M\) and \(N\) move with acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) By forming two equations of motion show that $$a = \frac { 1 } { 11 } g$$ 19
  2. The speed of \(N , 0.5\) seconds after its release, is \(\frac { g } { k } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) where \(k\) is a constant. Find the value of \(k\) 19
  3. State one assumption that must be made for the answer in part (b) to be valid.