Questions — AQA AS Paper 1 (133 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 PURE 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 PURE S1 S2 S3 S4 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 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 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
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AQA AS Paper 1 2022 June Q14
3 marks Moderate -0.8
A ball is released from rest from a height \(h\) metres above horizontal ground and falls freely downwards. When the ball reaches the ground, its speed is \(v\) m s\(^{-1}\), where \(v \leq 10\) Show that $$h \leq \frac{50}{g}$$ [3 marks]
AQA AS Paper 1 2022 June Q15
5 marks Moderate -0.3
Two particles, \(P\) and \(Q\), are initially at rest at the same point on a horizontal plane. A force of \(\begin{bmatrix} 4 \\ 0 \end{bmatrix}\) N is applied to \(P\). A force of \(\begin{bmatrix} 8 \\ 15 \end{bmatrix}\) N is applied to \(Q\).
  1. Calculate, to the nearest degree, the acute angle between the two forces. [2 marks]
  2. The particles begin to move under the action of the respective forces. \(P\) and \(Q\) have the same mass. \(P\) has an acceleration of magnitude 5 m s\(^{-2}\) Find the magnitude of the acceleration of \(Q\). [3 marks]
AQA AS Paper 1 2022 June Q16
6 marks Moderate -0.3
Jermaine and his friend Meena are walking in the same direction along a straight path. Meena is walking at a constant speed of \(u\) m s\(^{-1}\) Jermaine is walking 0.2 m s\(^{-1}\) more slowly than Meena. When Jermaine is \(d\) metres behind Meena he starts to run with a constant acceleration of 2 m s\(^{-2}\), for a time of \(t\) seconds, until he reaches her.
  1. Show that $$d = t^2 - 0.2t$$ [4 marks]
  2. When Jermaine's speed is 7.8 m s\(^{-1}\), he reaches Meena. Given that \(u = 1.4\) find the value of \(d\). [2 marks]
AQA AS Paper 1 2022 June Q17
8 marks Moderate -0.8
\includegraphics{figure_17} A car and caravan, connected by a tow bar, move forward together along a horizontal road. Their velocity \(v\) m s\(^{-1}\) at time \(t\) seconds, for \(0 \leq t < 20\), is given by $$v = 0.5t + 0.01t^2$$
  1. Show that when \(t = 15\) their acceleration is 0.8 m s\(^{-2}\) [2 marks]
  2. The car has a mass of 1500 kg The caravan has a mass of 850 kg When \(t = 15\) the tension in the tow bar is 800 N and the car experiences a resistance force of 100 N
    1. Find the total resistance force experienced by the caravan when \(t = 15\) [2 marks]
    2. Find the driving force being applied by the car when \(t = 15\) [3 marks]
  3. State one assumption you have made about the tow bar. [1 mark]
AQA AS Paper 1 2023 June Q1
1 marks Easy -1.8
At a point \(P\) on a curve, the gradient of the tangent to the curve is 10 State the gradient of the normal to the curve at \(P\) Circle your answer. [1 mark] \(-10\) \quad \(-0.1\) \quad \(0.1\) \quad \(10\)
AQA AS Paper 1 2023 June Q2
1 marks Easy -1.8
Identify the expression below which is equivalent to \(\left(\frac{2x}{5}\right)^{-3}\) Circle your answer. [1 mark] \(\frac{8x^3}{125}\) \quad \(\frac{125x^3}{8}\) \quad \(\frac{125}{8x^3}\) \quad \(\frac{8}{125x^3}\)
AQA AS Paper 1 2023 June Q3
3 marks Moderate -0.3
The coefficient of \(x^2\) in the binomial expansion of \((1 + ax)^6\) is \(\frac{20}{3}\) Find the two possible values of \(a\) [3 marks]
AQA AS Paper 1 2023 June Q4
5 marks Moderate -0.3
It is given that \(5\cos^2 \theta - 4\sin^2 \theta = 0\)
  1. Find the possible values of \(\tan \theta\), giving your answers in exact form. [3 marks]
  2. Hence, or otherwise, solve the equation $$5\cos^2 \theta - 4\sin^2 \theta = 0$$ giving all solutions of \(\theta\) to the nearest \(0.1°\) in the interval \(0° \leq \theta \leq 360°\) [2 marks]
AQA AS Paper 1 2023 June Q5
7 marks Moderate -0.3
  1. Given that \(y = x\sqrt{x}\), find \(\frac{dy}{dx}\) [2 marks]
  2. The line, \(L\), has equation \(6x - 2y + 5 = 0\) \(L\) is a tangent to the curve with equation \(y = x\sqrt{x} + k\) Find the value of \(k\) [5 marks]
AQA AS Paper 1 2023 June Q6
6 marks Moderate -0.8
  1. The curve \(C_1\) has equation \(y = 2x^2 - 20x + 42\) Express the equation of \(C_1\) in the form $$y = a(x - h)^2 + c$$ where \(a\), \(b\) and \(c\) are integers. [3 marks]
  2. Write down the coordinates of the minimum point of \(C_1\) [1 mark]
  3. The curve \(C_1\) is mapped onto the curve \(C_2\) by a stretch in the \(y\)-direction. The minimum point of \(C_2\) is at \((5, -4)\) Find the equation of \(C_2\) [2 marks]
AQA AS Paper 1 2023 June Q7
5 marks Moderate -0.8
Points \(P\) and \(Q\) lie on the curve with equation \(y = x^4\) The \(x\)-coordinate of \(P\) is \(x\) The \(x\)-coordinate of \(Q\) is \(x + h\)
  1. Expand \((x + h)^4\) [2 marks]
  2. Hence, find an expression, in terms of \(x\) and \(h\), for the gradient of the line \(PQ\) [1 mark]
  3. Explain how to use the answer from part (b) to obtain the gradient function of \(y = x^4\) [2 marks]
AQA AS Paper 1 2023 June Q8
7 marks Standard +0.3
  1. Show that $$\int_1^a \left(6 - \frac{12}{\sqrt{x}}\right) dx = 6a - 24\sqrt{a} + 18$$ [3 marks]
  2. The curve \(y = 6 - \frac{12}{\sqrt{x}}\), the line \(x = 1\) and the line \(x = a\) are shown in the diagram below. The shaded region \(R_1\) is bounded by the curve, the line \(x = 1\) and the \(x\)-axis. The shaded region \(R_2\) is bounded by the curve, the line \(x = a\) and the \(x\)-axis. \includegraphics{figure_8} It is given that the areas of \(R_1\) and \(R_2\) are equal. Find the value of \(a\) Fully justify your answer. [4 marks]
AQA AS Paper 1 2023 June Q9
3 marks Moderate -0.8
A continuous curve has equation \(y = f(x)\) The curve passes through the points \(A(2, 1)\), \(B(4, 5)\) and \(C(6, 1)\) It is given that \(f'(4) = 0\) Jasmin made two statements about the nature of the curve \(y = f(x)\) at the point \(B\): Statement 1: There is a turning point at \(B\) Statement 2: There is a maximum point at \(B\)
  1. Draw a sketch of the curve \(y = f(x)\) such that Statement 1 is correct and Statement 2 is correct. [1 mark]
  2. Draw a sketch of the curve \(y = f(x)\) such that Statement 1 is correct and Statement 2 is not correct. [1 mark]
  3. Draw a sketch of the curve \(y = f(x)\) such that Statement 1 is not correct and Statement 2 is not correct. [1 mark]
AQA AS Paper 1 2023 June Q10
8 marks Moderate -0.8
Charlie buys a car for £18000 on 1 January 2016. The value of the car decreases exponentially. The car has a value of £12000 on 1 January 2018.
  1. Charlie says: • because the car has lost £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 £8000. Explain whose statement is correct, justifying the value they have stated. [2 marks]
  2. The value of Charlie's car, £\(V\), \(t\) years after 1 January 2016 may be modelled by the equation $$V = Ae^{-kt}$$ where \(A\) and \(k\) are positive constants. Find the value of \(t\) when the car has a value of £10000, giving your answer to two significant figures. [5 marks]
  3. Give a reason why the model, in this context, will not be suitable to calculate the value of the car when \(t = 30\) [1 mark]
AQA AS Paper 1 2023 June Q11
7 marks Standard +0.3
  1. A circle has equation $$x^2 + y^2 - 10x - 6 = 0$$ Find the centre and the radius of the circle. [2 marks]
  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{figure_11}
    1. Show that the vertex at the origin lies inside the circle \(x^2 + y^2 - 10x - 6 = 0\) [1 mark]
    2. Prove that the triangle lies completely within the circle \(x^2 + y^2 - 10x - 6 = 0\) [4 marks]
AQA AS Paper 1 2023 June Q12
1 marks Easy -1.8
A particle, initially at rest, starts to move forward in a straight line with constant acceleration, \(a \text{ m s}^{-2}\) After 6 seconds the particle has a velocity of \(3 \text{ m s}^{-1}\) Find the value of \(a\) Circle your answer. [1 mark] \(-2\) \quad \(-0.5\) \quad \(0.5\) \quad \(2\)
AQA AS Paper 1 2023 June Q13
1 marks Easy -1.8
A resultant force of \(\begin{bmatrix} -2 \\ 6 \end{bmatrix}\) N acts on a particle. The acceleration of the particle is \(\begin{bmatrix} -6 \\ y \end{bmatrix} \text{ m s}^{-2}\) Find the value of \(y\) Circle your answer. [1 mark] \(2\) \quad \(3\) \quad \(10\) \quad \(18\)
AQA AS Paper 1 2023 June Q14
4 marks Moderate -0.3
A ball, initially at rest, is dropped from a vertical height of \(h\) metres above the Earth's surface. After 4 seconds the ball's height above the Earth's surface is \(0.2h\) metres.
  1. Assuming air resistance can be ignored, show that $$h = 10g$$ [3 marks]
  2. Assuming air resistance cannot be ignored, explain the effect that this would have on the value of \(h\) in part (a). [1 mark]
AQA AS Paper 1 2023 June Q15
4 marks Easy -1.2
A particle is moving in a straight line such that its velocity, \(v \text{ m s}^{-1}\), changes with respect to time, \(t\) seconds, as shown in the graph below. \includegraphics{figure_15}
  1. Show that the acceleration of the particle over the first 4 seconds is \(3.5 \text{ m s}^{-2}\) [1 mark]
  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. [3 marks]
AQA AS Paper 1 2023 June Q16
7 marks Moderate -0.8
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, \(v \text{ m s}^{-1}\), is modelled in terms of time, \(t\) seconds after the boat is launched, by the expression $$v = 0.9 + 0.16t - 0.06t^2$$
  1. Find the acceleration of the boat when \(t = 2\) [3 marks]
  2. Find the displacement of the boat, from the point where it was launched, when \(t = 2\) [4 marks]
AQA AS Paper 1 2023 June Q17
4 marks Moderate -0.8
A particle, \(P\), is initially at rest on a smooth horizontal surface. A resultant force of \(\begin{bmatrix} 12 \\ 9 \end{bmatrix}\) N is then applied to \(P\), so that it moves in a straight line.
  1. Find the magnitude of the resultant force. [1 mark]
  2. Two fixed points \(A\) and \(B\) have position vectors $$\overrightarrow{OA} = \begin{bmatrix} 3 \\ 7 \end{bmatrix} \text{ metres} \quad \text{and} \quad \overrightarrow{OB} = \begin{bmatrix} k \\ k-1 \end{bmatrix} \text{ metres}$$ with respect to a fixed origin, \(O\) \(P\) moves in a straight line parallel to \(\overrightarrow{AB}\)
    1. Find \(\overrightarrow{AB}\) in terms of \(k\) [1 mark]
    2. Find the value of \(k\) [2 marks]
AQA AS Paper 1 2023 June Q18
6 marks Standard +0.3
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 \(0.6 \text{ m s}^{-2}\) along a straight horizontal road as shown in the diagram below. \includegraphics{figure_18} 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 \(0.6R\) newtons.
  1. The tension in the tow bar, \(T\) newtons, may be modelled by $$T = kD - 18$$ where \(k\) is a constant. Find \(k\) [5 marks]
  2. State one assumption that must be made in answering part (a). [1 mark]
AQA AS Paper 1 2024 June Q1
1 marks Easy -1.8
It is given that \(\tan \theta^\circ = k\), where \(k\) is a constant. Find \(\tan (\theta + 180)^\circ\) Circle your answer. [1 mark] \(-k\) \qquad \(-\frac{1}{k}\) \qquad \(\frac{1}{k}\) \qquad \(k\)
AQA AS Paper 1 2024 June Q2
1 marks Easy -1.8
Curve \(C\) has equation \(y = \frac{1}{(x-1)^2}\) State the equations of the asymptotes to curve \(C\) Tick (\(\checkmark\)) one box. [1 mark] \(x = 0\) and \(y = 0\) \qquad \(\square\) \(x = 0\) and \(y = 1\) \qquad \(\square\) \(x = 1\) and \(y = 0\) \qquad \(\square\) \(x = 1\) and \(y = 1\) \qquad \(\square\)
AQA AS Paper 1 2024 June Q3
4 marks Moderate -0.8
Express \(\frac{\sqrt{3} + 3\sqrt{5}}{\sqrt{5} - \sqrt{3}}\) in the form \(a + b\sqrt{c}\), where \(a\) and \(b\) are integers. Fully justify your answer. [4 marks]