Questions — AQA (3620 questions)

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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]
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]