Questions Paper 2 (413 questions)

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AQA Paper 2 2020 June Q2
1 marks Easy -2.0
Which one of the following equations has no real solutions? Tick (\(\checkmark\)) one box. \(\cot x = 0\) \(\ln x = 0\) \(|x + 1| = 0\) \(\sec x = 0\) [1 mark]
AQA Paper 2 2020 June Q3
3 marks Moderate -0.3
Find the coefficient of \(x^2\) in the binomial expansion of \(\left(2x - \frac{3}{x}\right)^8\) [3 marks]
AQA Paper 2 2020 June Q4
4 marks Standard +0.3
Using small angle approximations, show that for small, non-zero, values of \(x\) $$\frac{x \tan 5x}{\cos 4x - 1} \approx A$$ where \(A\) is a constant to be determined. [4 marks]
AQA Paper 2 2020 June Q5
6 marks Standard +0.3
Use integration by substitution to show that $$\int_{-\frac{3}{4}}^6 x\sqrt{4x + 1} \, dx = \frac{875}{12}$$ Fully justify your answer. [6 marks]
AQA Paper 2 2020 June Q6
8 marks Standard +0.3
The line \(L\) has equation $$5y + 12x = 298$$ A circle, \(C\), has centre \((7, 9)\) \(L\) is a tangent to \(C\).
  1. Find the coordinates of the point of intersection of \(L\) and \(C\). Fully justify your answer. [5 marks]
  2. Find the equation of \(C\). [3 marks]
AQA Paper 2 2020 June Q7
7 marks Standard +0.3
\(a\) and \(b\) are two positive irrational numbers. The sum of \(a\) and \(b\) is rational. The product of \(a\) and \(b\) is rational. Caroline is trying to prove \(\frac{1}{a} + \frac{1}{b}\) is rational. Here is her proof: Step 1 \quad \(\frac{1}{a} + \frac{1}{b} = \frac{2}{a + b}\) Step 2 \quad \(2\) is rational and \(a + b\) is non-zero and rational. Step 3 \quad Therefore \(\frac{2}{a + b}\) is rational. Step 4 \quad Hence \(\frac{1}{a} + \frac{1}{b}\) is rational.
    1. Identify Caroline's mistake. [1 mark]
    2. Write down a correct version of the proof. [2 marks]
  2. Prove by contradiction that the difference of any rational number and any irrational number is irrational. [4 marks]
AQA Paper 2 2020 June Q8
10 marks Standard +0.3
The curve defined by the parametric equations $$x = t^2 \text{ and } y = 2t \quad -\sqrt{2} \leq t \leq \sqrt{2}$$ is shown in Figure 1 below. \includegraphics{figure_1}
  1. Find a Cartesian equation of the curve in the form \(y^2 = f(x)\) [2 marks]
  2. The point \(A\) lies on the curve where \(t = a\) The tangent to the curve at \(A\) is at an angle \(\theta\) to a line through \(A\) parallel to the \(x\)-axis. The point \(B\) has coordinates \((1, 0)\) The line \(AB\) is at an angle \(\phi\) to the \(x\)-axis. \includegraphics{figure_1_extended}
    1. By considering the gradient of the curve, show that $$\tan \theta = \frac{1}{a}$$ [3 marks]
    2. Find \(\tan \phi\) in terms of \(a\). [2 marks]
    3. Show that \(\tan 2\theta = \tan \phi\) [3 marks]
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]