Questions — AQA AS Paper 1 (120 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
AQA AS Paper 1 2023 June Q15
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
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
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
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 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
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
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
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
1 marks
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
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
3 marks
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
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
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
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
1 marks
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
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
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
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
3 marks
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
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.