Questions — AQA AS Paper 1 (133 questions)

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All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
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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 2024 June Q4
7 marks Moderate -0.3
    1. By using a suitable trigonometric identity, show that the equation $$\sin \theta \tan \theta = 4 \cos \theta$$ can be written as $$\tan^2 \theta = 4$$ [1 mark]
    2. Hence solve the equation $$\sin \theta \tan \theta = 4 \cos \theta$$ where \(0^\circ < \theta < 360^\circ\) Give your answers to the nearest degree. [3 marks]
  2. 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. [3 marks]
AQA AS Paper 1 2024 June Q5
3 marks Easy -1.2
A student is looking for factors of the polynomial \(f(x)\) They suggest that \((x - 2)\) is a factor of \(f(x)\) The method they use to check this suggestion is to calculate \(f(-2)\) They correctly calculate that \(f(-2) = 0\) They conclude that their suggestion is correct.
  1. Make one comment about the student's method. [1 mark]
  2. Make two comments about the student's conclusion. [2 marks] 1 2
AQA AS Paper 1 2024 June Q6
4 marks Moderate -0.3
Determine the set of values of \(x\) which satisfy the inequality $$3x^2 + 3x > x + 6$$ Give your answer in exact form using set notation. [4 marks]
AQA AS Paper 1 2024 June Q7
5 marks Moderate -0.8
A triangular field of grass, \(ABC\), has boundaries with lengths as follows: $$AB = 234 \text{ m} \qquad BC = 225 \text{ m} \qquad AC = 310 \text{ m}$$ The field is shown in the diagram below. \includegraphics{figure_7}
  1. Find angle \(A\) [2 marks]
  2. Farmers calculate the number of sheep they can keep in a field, by allowing one sheep for every \(1200 \text{ m}^2\) of grass. Find the maximum number of sheep which can be kept in the field \(ABC\) [3 marks]
AQA AS Paper 1 2024 June Q8
6 marks Moderate -0.3
It is given that $$\ln x - \ln y = 3$$
  1. Express \(x\) in terms of \(y\) in a form not involving logarithms. [3 marks]
  2. Given also that $$x + y = 10$$ find the exact value of \(y\) and the exact value of \(x\) [3 marks]
AQA AS Paper 1 2024 June Q9
5 marks Moderate -0.8
A curve has equation \(y = f(x)\) where $$f(x) = x(6 - x)$$
  1. Find \(f'(x)\) [2 marks]
  2. The diagram below shows the graph of \(y = f(x)\) On the same diagram sketch the gradient function for this curve, stating the coordinates of any points where the gradient function cuts the axes. [3 marks] \includegraphics{figure_9}
AQA AS Paper 1 2024 June Q10
6 marks Standard +0.3
It is given that $$\frac{\mathrm{d}y}{\mathrm{d}x} = (x + 2)(2x - 1)^2$$ and when \(x = 6\), \(y = 900\) Find \(y\) in terms of \(x\) [6 marks]
AQA AS Paper 1 2024 June Q11
5 marks Standard +0.3
It is given that for the continuous function \(g\) • \(g'(1) = 0\) • \(g'(4) = 0\) • \(g''(x) = 2x - 5\)
  1. Determine the nature of each of the turning points of \(g\) Fully justify your answer. [3 marks]
  2. Find the set of values of \(x\) for which \(g\) is an increasing function. [2 marks]
AQA AS Paper 1 2024 June Q12
6 marks Moderate -0.8
The monthly mean temperature of a city, \(T\) degrees Celsius, may be modelled by the equation $$T = 15 + 8 \sin (30m - 120)^\circ$$ where \(m\) is the month number, counting January = 1, February = 2, through to December = 12
  1. Using this model, calculate the monthly mean temperature of the city for May, the fifth month. [2 marks]
  2. Using this model, find the month with the highest mean temperature. [2 marks]
  3. Climate change may affect the parameters, 8, 30, 120 and 15, used in this model.
    1. State, with a reason, which parameter would be increased because of an overall rise in temperatures. [1 mark]
    2. State, with a reason, which parameter would be increased because of the occurrence of more extreme temperatures. [1 mark]
AQA AS Paper 1 2024 June Q13
1 marks Easy -1.8
A particle is moving in a straight line with constant acceleration \(a\) m s\(^{-2}\) The particle's velocity, \(v\) m s\(^{-1}\), varies with time, \(t\) seconds, so that $$v = 3 - 4t$$ Deduce the value of \(a\) Circle your answer. [1 mark] \(-4\) \qquad \(-1\) \qquad \(3\) \qquad \(4\)
AQA AS Paper 1 2024 June Q14
1 marks Easy -1.8
Two forces, \(\mathbf{F}_1 = 3\mathbf{i} + 2\mathbf{j}\) newtons and \(\mathbf{F}_2 = \mathbf{i} - 3\mathbf{j}\) 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 (\(\checkmark\)) one box. [1 mark] \includegraphics{figure_14}
AQA AS Paper 1 2024 June Q15
4 marks Moderate -0.8
A graph indicating how the velocity, \(v\) m s\(^{-1}\), of a particle changes with respect to time, \(t\) seconds, is shown below. \includegraphics{figure_15}
  1. Find the total distance travelled by the particle over the 8 second period shown. [3 marks]
  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. [1 mark]
AQA AS Paper 1 2024 June Q16
3 marks Easy -1.2
In this question use \(g = 9.8\) m s\(^{-2}\) A ball is launched vertically upwards from the Earth's surface with velocity \(u\) m s\(^{-1}\) The ball reaches a maximum height of 15 metres. You may assume that air resistance can be ignored. Find the value of \(u\) [3 marks]
AQA AS Paper 1 2024 June Q17
4 marks Moderate -0.8
A particle moves in a straight line with acceleration \(a\) m s\(^{-2}\), at time \(t\) seconds, where $$a = 10 - 6t$$ The particle's velocity, \(v\) m s\(^{-1}\), and displacement, \(r\) metres, are both initially zero. Show that $$r = t^2(5 - t)$$ Fully justify your answer. [4 marks]
AQA AS Paper 1 2024 June Q18
6 marks Moderate -0.3
It is given that two points \(A\) and \(B\) have position vectors $$\overrightarrow{OA} = \begin{bmatrix} 5 \\ -1 \end{bmatrix} \text{ metres} \quad \text{and} \quad \overrightarrow{OB} = \begin{bmatrix} 13 \\ 5 \end{bmatrix} \text{ metres.}$$
  1. Show that the distance from \(A\) to \(B\) is 10 metres. [3 marks]
  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 m 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 \(R\) [3 marks]
AQA AS Paper 1 2024 June Q19
8 marks Moderate -0.3
Two objects, \(M\) and \(N\), are connected by a light inextensible string that passes over a smooth peg. \(M\) has a mass of 0.6 kilograms. \(N\) has a mass of 0.5 kilograms. \(M\) and \(N\) are initially held at rest, with the string taut, as shown in the diagram below. \includegraphics{figure_19} \(M\) and \(N\) are released at the same instant and begin to move vertically. You may assume that air resistance can be ignored.
  1. It is given that \(M\) and \(N\) move with acceleration \(a\) m s\(^{-2}\) By forming two equations of motion show that $$a = \frac{1}{11}g$$ [5 marks]
  2. The speed of \(N\), 0.5 seconds after its release, is \(\frac{g}{k}\) m s\(^{-1}\) where \(k\) is a constant. Find the value of \(k\) [2 marks]
  3. State one assumption that must be made for the answer in part (b) to be valid. [1 mark]
AQA AS Paper 1 Specimen Q1
1 marks Easy -1.8
The curve \(y = \sqrt{x}\) is translated onto the curve \(y = \sqrt{x + 4}\) The translation is described by a vector. Find this vector. Circle your answer. [1 mark] \(\begin{bmatrix} 4 \\ 0 \end{bmatrix}\) \(\begin{bmatrix} -4 \\ 0 \end{bmatrix}\) \(\begin{bmatrix} 0 \\ 4 \end{bmatrix}\) \(\begin{bmatrix} 0 \\ -4 \end{bmatrix}\)
AQA AS Paper 1 Specimen Q2
1 marks Moderate -0.8
Consider the two statements, A and B, below. A: \(x^2 - 6x + 8 > 0\) B: \(x > 4\) Choose the most appropriate option below. Circle your answer. [1 mark] \(A \Rightarrow B\) \(A \Leftarrow B\) \(A \Leftrightarrow B\) There is no connection between A and B
AQA AS Paper 1 Specimen Q3
4 marks Easy -1.3
  1. Write down the value of \(p\) and the value of \(q\) given that:
    1. \(\sqrt{3} = 3^p\) [1 mark]
    2. \(\frac{1}{9} = 3^q\) [1 mark]
  2. Find the value of \(x\) for which \(\sqrt{3} \times 3^x = \frac{1}{9}\) [2 marks]
AQA AS Paper 1 Specimen Q4
3 marks Easy -1.2
Show that \(\frac{5\sqrt{2} + 2}{3\sqrt{2} + 4}\) can be expressed in the form \(m + n\sqrt{2}\), where \(m\) and \(n\) are integers. [3 marks]
AQA AS Paper 1 Specimen Q5
2 marks Moderate -0.5
Jessica, a maths student, is asked by her teacher to solve the equation \(\tan x = \sin x\), giving all solutions in the range \(0° \leq x \leq 360°\) The steps of Jessica's working are shown below. \(\tan x = \sin x\) Step 1 \(\Rightarrow\) \(\frac{\sin x}{\cos x} = \sin x\) Write \(\tan x\) as \(\frac{\sin x}{\cos x}\) Step 2 \(\Rightarrow\) \(\sin x = \sin x \cos x\) Multiply by \(\cos x\) Step 3 \(\Rightarrow\) \(1 = \cos x\) Cancel \(\sin x\) \(\Rightarrow\) \(x = 0°\) or \(360°\) The teacher tells Jessica that she has not found all the solutions because of a mistake. Explain why Jessica's method is not correct. [2 marks]
AQA AS Paper 1 Specimen Q6
4 marks Standard +0.3
A parallelogram has sides of length 6 cm and 4.5 cm. The larger interior angles of the parallelogram have size \(\alpha\) Given that the area of the parallelogram is 24 cm², find the exact value of \(\tan \alpha\) [4 marks]
AQA AS Paper 1 Specimen Q7
4 marks Moderate -0.8
Determine whether the line with equation \(2x + 3y + 4 = 0\) is parallel to the line through the points with coordinates \((9, 4)\) and \((3, 8)\). [4 marks]
AQA AS Paper 1 Specimen Q8
6 marks Moderate -0.8
  1. Find the first three terms, in ascending powers of \(x\), of the expansion of \((1 - 2x)^{10}\) [3 marks]
  2. Carly has lost her calculator. She uses the first three terms, in ascending powers of \(x\), of the expansion of \((1 - 2x)^{10}\) to evaluate \(0.998^{10}\) Find Carly's value for \(0.998^{10}\) and show that it is correct to five decimal places. [3 marks]
AQA AS Paper 1 Specimen Q9
5 marks Moderate -0.3
  1. Given that \(f(x) = x^2 - 4x + 2\), find \(f(3 + h)\) Express your answer in the form \(h^2 + bh + c\), where \(b\) and \(c \in \mathbb{Z}\). [2 marks]
  2. The curve with equation \(y = x^2 - 4x + 2\) passes through the point \(P(3, -1)\) and the point \(Q\) where \(x = 3 + h\). Using differentiation from first principles, find the gradient of the tangent to the curve at the point \(P\). [3 marks]