Questions — AQA Paper 2 (149 questions)

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AQA Paper 2 2019 June Q1
1 marks Easy -1.8
Identify the graph of \(y = 1 - |x + 2|\) from the options below. Tick (\(\checkmark\)) one box. [1 mark] \includegraphics{figure_1}
AQA Paper 2 2019 June Q2
1 marks Easy -1.8
Simplify \(\sqrt{a^{\frac{2}{3}} \times a^{\frac{2}{5}}}\) Circle your answer. [1 mark] \(a^{\frac{2}{15}}\) \quad \(a^{\frac{4}{15}}\) \quad \(a^{\frac{8}{15}}\) \quad \(a^{\frac{16}{15}}\)
AQA Paper 2 2019 June Q3
1 marks Easy -1.8
Each of these functions has domain \(x \in \mathbb{R}\) Which function does not have an inverse? Circle your answer. [1 mark] \(f(x) = x^3\) \quad \(f(x) = 2x + 1\) \quad \(f(x) = x^2\) \quad \(f(x) = e^x\)
AQA Paper 2 2019 June Q4
4 marks Standard +0.3
\(x^2 + bx + c\) and \(x^2 + dx + e\) have a common factor \((x + 2)\) Show that \(2(d - b) = e - c\) Fully justify your answer. [4 marks]
AQA Paper 2 2019 June Q5
7 marks Standard +0.3
Solve the differential equation $$\frac{dt}{dx} = \frac{\ln x}{x^2 t} \quad \text{for } x > 0$$ given \(x = 1\) when \(t = 2\) Write your answer in the form \(t^2 = f(x)\) [7 marks]
AQA Paper 2 2019 June Q6
6 marks Standard +0.8
A curve has equation $$y = a \sin x + b \cos x$$ where \(a\) and \(b\) are constants. The maximum value of \(y\) is 4 and the curve passes through the point \(\left(\frac{\pi}{3}, 2\sqrt{3}\right)\) as shown in the diagram. \includegraphics{figure_6} Find the exact values of \(a\) and \(b\). [6 marks]
AQA Paper 2 2019 June Q7
10 marks Challenging +1.2
  1. Sketch the graph of any cubic function that has both three distinct real roots and a positive coefficient of \(x^3\) [2 marks]
  2. The function f(x) is defined by $$f(x) = x^3 + 3px^2 + q$$ where \(p\) and \(q\) are constants and \(p > 0\)
    1. Show that there is a turning point where the curve crosses the \(y\)-axis. [3 marks]
    2. The equation \(f(x) = 0\) has three distinct real roots. By considering the positions of the turning points find, in terms of \(p\), the range of possible values of \(q\). [5 marks]
AQA Paper 2 2019 June Q8
11 marks Moderate -0.3
Theresa bought a house on 2 January 1970 for £8000. The house was valued by a local estate agent on the same date every 10 years up to 2010. The valuations are shown in the following table.
Year19701980199020002010
Valuation price£8000£19000£36000£82000£205000
The valuation price of the house can be modelled by the equation $$V = pq^t$$ where \(V\) pounds is the valuation price \(t\) years after 2 January 1970 and \(p\) and \(q\) are constants.
  1. Show that \(V = pq^t\) can be written as \(\log_{10} V = \log_{10} p + t \log_{10} q\) [2 marks]
  2. The values in the table of \(\log_{10} V\) against \(t\) have been plotted and a line of best fit has been drawn on the graph below. \includegraphics{figure_8b} Using the given line of best fit, find estimates for the values of \(p\) and \(q\). Give your answers correct to three significant figures. [4 marks]
  3. Determine the year in which Theresa's house will first be worth half a million pounds. [3 marks]
  4. Explain whether your answer to part (c) is likely to be reliable. [2 marks]
AQA Paper 2 2019 June Q9
9 marks Standard +0.3
  1. Show that the first two terms of the binomial expansion of \(\sqrt{4 - 2x^2}\) are $$2 - \frac{x^2}{2}$$ [2 marks]
  2. State the range of values of \(x\) for which the expansion found in part (a) is valid. [2 marks]
  3. Hence, find an approximation for $$\int_0^{0.4} \sqrt{\cos x} \, dx$$ giving your answer to five decimal places. Fully justify your answer. [4 marks]
  4. A student decides to use this method to find an approximation for $$\int_0^{1.4} \sqrt{\cos x} \, dx$$ Explain why this may not be a suitable method. [1 mark]
AQA Paper 2 2019 June Q10
1 marks Easy -2.0
The diagram below shows a velocity-time graph for a particle moving with velocity \(v \text{ m s}^{-1}\) at time \(t\) seconds. \includegraphics{figure_10} Which statement is correct? Tick (\(\checkmark\)) one box. [1 mark] The particle was stationary for \(9 \leq t \leq 12\) The particle was decelerating for \(12 \leq t \leq 20\) The particle had a displacement of zero when \(t = 6\) The particle's speed when \(t = 4\) was \(-12 \text{ m s}^{-1}\)
AQA Paper 2 2019 June Q11
1 marks Easy -1.8
A wooden crate rests on a rough horizontal surface. The coefficient of friction between the crate and the surface is 0.6 A forward force acts on the crate, parallel to the surface. When this force is 600 N, the crate is on the point of moving. Find the weight of the crate. Circle your answer. [1 mark] 1000 N \quad 100 kg \quad 360 N \quad 36 kg
AQA Paper 2 2019 June Q12
1 marks Easy -1.8
A particle, under the action of two constant forces, is moving across a perfectly smooth horizontal surface at a constant speed of \(10 \text{ m s}^{-1}\) The first force acting on the particle is \((400\mathbf{i} + 180\mathbf{j})\) N. The second force acting on the particle is \((p\mathbf{i} - 180\mathbf{j})\) N. Find the value of \(p\). Circle your answer. [1 mark] \(-400\) \quad \(-390\) \quad \(390\) \quad \(400\)
AQA Paper 2 2019 June Q13
5 marks Moderate -0.8
In a school experiment, a particle, of mass \(m\) kilograms, is released from rest at a point \(h\) metres above the ground. At the instant it reaches the ground, the particle has velocity \(v \text{ m s}^{-1}\)
  1. Show that $$v = \sqrt{2gh}$$ [2 marks]
  2. A student correctly used \(h = 18\) and measured \(v\) as 20 The student's teacher claims that the machine measuring the velocity must have been faulty. Determine if the teacher's claim is correct. Fully justify your answer. [3 marks]
AQA Paper 2 2019 June Q14
8 marks Moderate -0.3
A metal rod, of mass \(m\) kilograms and length 20 cm, lies at rest on a horizontal shelf. The end of the rod, \(B\), extends 6 cm beyond the edge of the shelf, \(A\), as shown in the diagram below. \includegraphics{figure_14}
  1. The rod is in equilibrium when an object of mass 0.28 kilograms hangs from the midpoint of \(AB\). Show that \(m = 0.21\) [3 marks]
  2. The object of mass 0.28 kilograms is removed. A number, \(n\), of identical objects, each of mass 0.048 kg, are hung from the rod all at a distance of 1 cm from \(B\). Find the maximum value of \(n\) such that the rod remains horizontal. [4 marks]
  3. State one assumption you have made about the rod. [1 mark]
AQA Paper 2 2019 June Q15
9 marks Moderate -0.3
Four buoys on the surface of a large, calm lake are located at \(A\), \(B\), \(C\) and \(D\) with position vectors given by $$\overrightarrow{OA} = \begin{bmatrix} 410 \\ 710 \end{bmatrix}, \overrightarrow{OB} = \begin{bmatrix} -210 \\ 530 \end{bmatrix}, \overrightarrow{OC} = \begin{bmatrix} -340 \\ -310 \end{bmatrix} \text{ and } \overrightarrow{OD} = \begin{bmatrix} 590 \\ -40 \end{bmatrix}$$ All values are in metres.
  1. Prove that the quadrilateral \(ABCD\) is a trapezium but not a parallelogram. [5 marks]
  2. A speed boat travels directly from \(B\) to \(C\) at a constant speed in 50 seconds. Find the speed of the boat between \(B\) and \(C\). [4 marks]
AQA Paper 2 2019 June Q16
16 marks Standard +0.8
An elite athlete runs in a straight line to complete a 100-metre race. During the race, the athlete's velocity, \(v \text{ m s}^{-1}\), may be modelled by $$v = 11.71 - 11.68e^{-0.9t} - 0.03e^{0.3t}$$ where \(t\) is the time, in seconds, after the starting pistol is fired.
  1. Find the maximum value of \(v\), giving your answer to one decimal place. Fully justify your answer. [8 marks]
  2. Find an expression for the distance run in terms of \(t\). [6 marks]
  3. The athlete's actual time for this race is 9.8 seconds. Comment on the accuracy of the model. [2 marks]
AQA Paper 2 2019 June Q17
9 marks Standard +0.3
Lizzie is sat securely on a wooden sledge. The combined mass of Lizzie and the sledge is \(M\) kilograms. The sledge is being pulled forward in a straight line along a horizontal surface by means of a light inextensible rope, which is attached to the front of the sledge. This rope stays inclined at an acute angle \(\theta\) above the horizontal and remains taut as the sledge moves forward. \includegraphics{figure_17} The sledge remains in contact with the surface throughout. The coefficient of friction between the sledge and the surface is \(\mu\) and there are no other resistance forces. Lizzie and the sledge move forward with constant acceleration, \(a \text{ m s}^{-2}\) The tension in the rope is a constant \(T\) Newtons.
  1. Show that $$T = \frac{M(a + \mu g)}{\cos \theta + \mu \sin \theta}$$ [7 marks]
  2. It is known that when \(M = 30\), \(\theta = 30°\), and \(T = 40\), the sledge remains at rest. Lizzie uses these values with the relationship formed in part (a) to find the value for \(\mu\) Explain why her value for \(\mu\) may be incorrect. [2 marks]
AQA Paper 2 2020 June Q1
1 marks Easy -2.0
Which one of these functions is decreasing for all real values of \(x\)? Circle your answer. \(f(x) = e^x\) \quad \(f(x) = -e^{1-x}\) \quad \(f(x) = -e^{x-1}\) \quad \(f(x) = -e^{-x}\) [1 mark]
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]