Questions — AQA Paper 2 (149 questions)

Browse by module
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
Browse by board
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
Sort by: Default | Easiest first | Hardest first
AQA Paper 2 2020 June Q9
10 marks Standard +0.8
A cylinder is to be cut out of the circular face of a solid hemisphere. The cylinder and the hemisphere have the same axis of symmetry. The cylinder has height \(h\) and the hemisphere has a radius of \(R\). \includegraphics{figure_9}
  1. Show that the volume, \(V\), of the cylinder is given by $$V = \pi R^2 h - \pi h^3$$ [3 marks]
  2. Find the maximum volume of the cylinder in terms of \(R\). Fully justify your answer. [7 marks]
AQA Paper 2 2020 June Q10
1 marks Easy -2.0
A vehicle is driven at a constant speed of \(12\text{ ms}^{-1}\) along a straight horizontal road. Only one of the statements below is correct. Identify the correct statement. Tick (\(\checkmark\)) one box. The vehicle is accelerating The vehicle's driving force exceeds the total force resisting its motion The resultant force acting on the vehicle is zero The resultant force acting on the vehicle is dependent on its mass [1 mark]
AQA Paper 2 2020 June Q11
1 marks Easy -1.8
A number of forces act on a particle such that the resultant force is \(\begin{pmatrix} 6 \\ -3 \end{pmatrix}\) N One of the forces acting on the particle is \(\begin{pmatrix} 8 \\ -5 \end{pmatrix}\) N Calculate the total of the other forces acting on the particle. Circle your answer. \(\begin{pmatrix} 2 \\ -2 \end{pmatrix}\) N \quad \(\begin{pmatrix} 14 \\ -8 \end{pmatrix}\) N \quad \(\begin{pmatrix} -2 \\ 2 \end{pmatrix}\) N \quad \(\begin{pmatrix} -14 \\ 8 \end{pmatrix}\) N [1 mark]
AQA Paper 2 2020 June Q12
1 marks Easy -1.8
A particle, \(P\), is moving with constant velocity \(8\mathbf{i} - 12\mathbf{j}\) A second particle, \(Q\), is moving with constant velocity \(a\mathbf{i} + 9\mathbf{j}\) \(Q\) travels in a direction which is parallel to the motion of \(P\). Find \(a\). Circle your answer. \(-6\) \quad \(-5\) \quad \(5\) \quad \(6\) [1 mark]
AQA Paper 2 2020 June Q13
3 marks Moderate -0.8
A uniform rod, \(AB\), has length \(7\) metres and mass \(4\) kilograms. The rod rests on a single fixed pivot point, \(C\), where \(AC = 2\) metres. A particle of weight \(W\) newtons is fixed at \(A\), as shown in the diagram. \includegraphics{figure_13} The system is in equilibrium with the rod resting horizontally.
  1. Find \(W\), giving your answer in terms of \(g\). [2 marks]
  2. Explain how you have used the fact that the rod is uniform in part (a). [1 mark]
AQA Paper 2 2020 June Q14
7 marks Standard +0.3
At time \(t\) seconds a particle, \(P\), has position vector \(\mathbf{r}\) metres, with respect to a fixed origin, such that $$\mathbf{r} = (t^3 - 5t^2)\mathbf{i} + (8t - t^2)\mathbf{j}$$
  1. Find the exact speed of \(P\) when \(t = 2\) [4 marks]
  2. Bella claims that the magnitude of acceleration of \(P\) will never be zero. Determine whether Bella's claim is correct. Fully justify your answer. [3 marks]
AQA Paper 2 2020 June Q15
5 marks Moderate -0.8
A particle is moving in a straight line with velocity \(v\text{ ms}^{-1}\) at time \(t\) seconds as shown by the graph below. \includegraphics{figure_15}
  1. Use the trapezium rule with four strips to estimate the distance travelled by the particle during the time period \(20 \leq t \leq 100\) [4 marks]
  2. Over the same time period, the curve can be very closely modelled by a particular quadratic. Explain how you could find an alternative estimate using this quadratic. [1 mark]
AQA Paper 2 2020 June Q16
5 marks Standard +0.3
Two particles \(A\) and \(B\) are released from rest from different starting points above a horizontal surface. \(A\) is released from a height of \(h\) metres. \(B\) is released at a time \(t\) seconds after \(A\) from a height of \(kh\) metres, where \(0 < k < 1\) Both particles land on the surface \(5\) seconds after \(A\) was released. Assuming any resistance forces may be ignored, prove that $$t = 5(1 - \sqrt{k})$$ Fully justify your answer. [5 marks]
AQA Paper 2 2020 June Q17
6 marks Standard +0.3
A ball is projected forward from a fixed point, \(P\), on a horizontal surface with an initial speed \(u\text{ ms}^{-1}\), at an acute angle \(\theta\) above the horizontal. The ball needs to first land at a point at least \(d\) metres away from \(P\). You may assume the ball may be modelled as a particle and that air resistance may be ignored. Show that $$\sin 2\theta \geq \frac{dg}{u^2}$$ [6 marks]
AQA Paper 2 2020 June Q18
13 marks Standard +0.3
Block \(A\), of mass \(0.2\) kg, lies at rest on a rough plane. The plane is inclined at an angle \(\theta\) to the horizontal, such that \(\tan \theta = \frac{7}{24}\) A light inextensible string is attached to \(A\) and runs parallel to the line of greatest slope until it passes over a smooth fixed pulley at the top of the slope. The other end of this string is attached to particle \(B\), of mass \(2\) kg, which is held at rest so that the string is taut, as shown in the diagram below. \includegraphics{figure_18}
  1. \(B\) is released from rest so that it begins to move vertically downwards with an acceleration of \(\frac{543}{625}\) g ms\(^{-2}\) Show that the coefficient of friction between \(A\) and the surface of the inclined plane is \(0.17\) [8 marks]
  2. In this question use \(g = 9.81\text{ ms}^{-2}\) When \(A\) reaches a speed of \(0.5\text{ ms}^{-1}\) the string breaks.
    1. Find the distance travelled by \(A\) after the string breaks until first coming to rest. [4 marks]
    2. State an assumption that could affect the validity of your answer to part (b)(i). [1 mark]
AQA Paper 2 2020 June Q19
8 marks Standard +0.3
A particle moves so that its acceleration, \(a\text{ ms}^{-2}\), at time \(t\) seconds may be modelled in terms of its velocity, \(v\text{ ms}^{-1}\), as $$a = -0.1v^2$$ The initial velocity of the particle is \(4\text{ ms}^{-1}\)
  1. By first forming a suitable differential equation, show that $$v = \frac{20}{5 + 2t}$$ [6 marks]
  2. Find the acceleration of the particle when \(t = 5.5\) [2 marks]
AQA Paper 2 2024 June Q1
1 marks Easy -2.0
One of the equations below is the equation of a circle. Identify this equation. [1 mark] Tick \((\checkmark)\) one box. \((x + 1)^2 - (y + 2)^2 = -36\) \((x + 1)^2 - (y + 2)^2 = 36\) \((x + 1)^2 + (y + 2)^2 = -36\) \((x + 1)^2 + (y + 2)^2 = 36\)
AQA Paper 2 2024 June Q2
1 marks Easy -1.2
The graph of \(y = f(x)\) intersects the \(x\)-axis at \((-3, 0)\), \((0, 0)\) and \((2, 0)\) as shown in the diagram below. \includegraphics{figure_2} The shaded region \(A\) has an area of 189 The shaded region \(B\) has an area of 64 Find the value of \(\int_{-3}^{2} f(x) \, dx\) Circle your answer. [1 mark] \(-253\) \(\quad\) \(-125\) \(\quad\) \(125\) \(\quad\) \(253\)
AQA Paper 2 2024 June Q3
1 marks Easy -1.8
Solve the inequality $$(1 - x)(x - 4) < 0$$ [1 mark] Tick \((\checkmark)\) one box. \(\{x : x < 1\} \cup \{x : x > 4\}\) \(\{x : x < 1\} \cap \{x : x > 4\}\) \(\{x : x < 1\} \cup \{x : x \geq 4\}\) \(\{x : x < 1\} \cap \{x : x \geq 4\}\)
AQA Paper 2 2024 June Q4
3 marks Moderate -0.8
Use logarithms to solve the equation $$5^{x-2} = 7^{1570}$$ Give your answer to two decimal places. [3 marks]
AQA Paper 2 2024 June Q5
3 marks Moderate -0.3
Given that $$y = \frac{x^3}{\sin x}$$ find \(\frac{dy}{dx}\) [3 marks]
AQA Paper 2 2024 June Q6
6 marks Standard +0.8
It is given that $$(2 \sin \theta + 3 \cos \theta)^2 + (6 \sin \theta - \cos \theta)^2 = 30$$ and that \(\theta\) is obtuse. Find the exact value of \(\sin \theta\). Fully justify your answer. [6 marks]
AQA Paper 2 2024 June Q7
5 marks Moderate -0.3
On the first day of each month, Kate pays £50 into a savings account. Interest is paid on the total amount in the account on the last day of each month. The interest rate is 0.2% At the end of the \(n\)th month, the total amount of money in Kate's savings account is £\(T_n\) Kate correctly calculates \(T_1\) and \(T_2\) as shown below: \(T_1 = 50 \times 1.002 = 50.10\) \(T_2 = (T_1 + 50) \times 1.002\) \(= ((50 \times 1.002) + 50) \times 1.002\) \(= 50 \times 1.002^2 + 50 \times 1.002\) \(\approx 100.30\)
  1. Show that \(T_3\) is given by $$T_3 = 50 \times 1.002^3 + 50 \times 1.002^2 + 50 \times 1.002$$ [1 mark]
  2. Kate uses her method to correctly calculate how much money she can expect to have in her savings account at the end of 10 years.
    1. Find the amount of money Kate expects to have in her savings account at the end of 10 years. [3 marks]
    2. The amount of money in Kate's savings account at the end of 10 years may not be the amount she has correctly calculated. Explain why. [1 mark]
AQA Paper 2 2024 June Q8
7 marks Moderate -0.3
A zookeeper models the median mass of infant monkeys born at their zoo, up to the age of 2 years, by the formula $$y = a + b \log_{10} x$$ where \(y\) is the median mass in kilograms, \(x\) is age in months and \(a\) and \(b\) are constants. The zookeeper uses the data shown below to determine the values of \(a\) and \(b\).
Age in months (\(x\))324
Median mass (\(y\))6.412
  1. The zookeeper uses the data for monkeys aged 3 months to write the correct equation $$6.4 = a + b \log_{10} 3$$
    1. Use the data for monkeys aged 24 months to write a second equation. [1 mark]
    2. Show that $$b = \frac{5.6}{\log_{10} 8}$$ [3 marks]
    3. Find the value of \(a\). Give your answer to two decimal places. [1 mark]
  2. Use a suitable value for \(x\) to determine whether the model can be used to predict the median mass of monkeys less than one week old. [2 marks]
AQA Paper 2 2024 June Q9
13 marks Standard +0.3
    1. Find the binomial expansion of \((1 + 3x)^{-1}\) up to and including the term in \(x^2\) [2 marks]
    2. Show that the first three terms in the binomial expansion of $$\frac{1}{2 - 3x}$$ form a geometric sequence and state the common ratio. [5 marks]
  2. It is given that $$\frac{36x}{(1 + 3x)(2 - 3x)} = \frac{P}{(2 - 3x)} + \frac{Q}{(1 + 3x)}$$ where \(P\) and \(Q\) are integers. Find the value of \(P\) and the value of \(Q\) [3 marks]
    1. Using your answers to parts (a) and (b), find the binomial expansion of $$\frac{12x}{(1 + 3x)(2 - 3x)}$$ up to and including the term in \(x^2\) [2 marks]
    2. Find the range of values of \(x\) for which the binomial expansion of $$\frac{12x}{(1 + 3x)(2 - 3x)}$$ is valid. [1 mark]
AQA Paper 2 2024 June Q10
4 marks Standard +0.3
The function f is defined by $$f(x) = x^2 + 2 \cos x \text{ for } -\pi \leq x \leq \pi$$ Determine whether the curve with equation \(y = f(x)\) has a point of inflection at the point where \(x = 0\) Fully justify your answer. [4 marks]
AQA Paper 2 2024 June Q11
6 marks Moderate -0.8
  1. A student states that 3 is the smallest value of \(k\) in the interval \(3 < k < 4\) Explain the error in the student's statement. [1 mark]
  2. The student's teacher says there is no smallest value of \(k\) in the interval \(3 < k < 4\) The teacher gives the following correct proof: Step 1: Assume there is a smallest number in the interval \(3 < k < 4\) and let this smallest number be \(x\) Step 2: let \(y = \frac{3 + x}{2}\) Step 3: \(3 < y < x\) which is a contradiction. Step 4: Therefore, there is no smallest number in interval \(3 < k < 4\)
    1. Explain the contradiction stated in Step 3 [1 mark]
    2. Prove that there is no largest value of \(k\) in the interval \(3 < k < 4\) [4 marks]
AQA Paper 2 2024 June Q12
1 marks Easy -2.0
Two constant forces act on a particle, of mass 2 kilograms, so that it moves forward in a straight line. The two forces are: • a forward driving force of 10 newtons • a resistance force of 4 newtons. Find the acceleration of the particle. Circle your answer. [1 mark] \(2 \text{ m s}^{-2}\) \(\quad\) \(3 \text{ m s}^{-2}\) \(\quad\) \(5 \text{ m s}^{-2}\) \(\quad\) \(12 \text{ m s}^{-2}\)
AQA Paper 2 2024 June Q13
1 marks Easy -1.8
A car starting from rest moves forward in a straight line. The motion of the car is modelled by the velocity–time graph below: \includegraphics{figure_13} One of the following assumptions about the motion of the car is implied by the graph. Identify this assumption. [1 mark] Tick \((\checkmark)\) one box. The car never accelerates. The acceleration of the car is always positive. The acceleration of the car can change instantaneously. The acceleration of the car is never constant.
AQA Paper 2 2024 June Q14
3 marks Moderate -0.8
The displacement, \(r\) metres, of a particle at time \(t\) seconds is $$r = 6t - 2t^2$$
  1. Find the value of \(r\) when \(t = 4\) [1 mark]
  2. Determine the range of values of \(t\) for which the displacement is positive. [2 marks]