Questions — AQA (3620 questions)

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AQA Further Paper 2 Specimen Q10
8 marks Challenging +1.8
Evaluate the improper integral \(\int_0^{\infty} \frac{4x - 30}{(x^2 + 5)(3x + 2)} \, dx\), showing the limiting process used. Give your answer as a single term. [8 marks]
AQA Further Paper 2 Specimen Q11
8 marks Challenging +1.8
The diagram shows a sketch of a curve \(C\), the pole \(O\) and the initial line. \includegraphics{figure_11} The polar equation of \(C\) is \(r = 4 + 2\cos \theta\), \quad \(-\pi \leq \theta \leq \pi\)
  1. Show that the area of the region bounded by the curve \(C\) is \(18\pi\) [4 marks]
  2. Points \(A\) and \(B\) lie on the curve \(C\) such that \(-\frac{\pi}{2} < \theta < \frac{\pi}{2}\) and \(AOB\) is an equilateral triangle. Find the polar equation of the line segment \(AB\) [4 marks]
AQA Further Paper 2 Specimen Q12
11 marks Standard +0.8
\(\mathbf{M} = \begin{pmatrix} -1 & 2 & -1 \\ 2 & 2 & -2 \\ -1 & -2 & -1 \end{pmatrix}\)
  1. Given that 4 is an eigenvalue of M, find a corresponding eigenvector. [3 marks]
  2. Given that \(\mathbf{MU} = \mathbf{UD}\), where D is a diagonal matrix, find possible matrices for D and U. [8 marks]
AQA Further Paper 2 Specimen Q13
7 marks Challenging +1.8
S is a singular matrix such that \(\det \mathbf{S} = \begin{vmatrix} a & a & x \\ x-b & a-b & x+1 \\ x^2 & a^2 & ax \end{vmatrix}\) Express the possible values of \(x\) in terms of \(a\) and \(b\). [7 marks]
AQA Further Paper 2 Specimen Q14
9 marks Challenging +1.2
Given that the vectors a and b are perpendicular, prove that \(|(\mathbf{a} + 5\mathbf{b}) \times (\mathbf{a} - 4\mathbf{b})| = k|\mathbf{a}||\mathbf{b}|\), where \(k\) is an integer to be found. Explicitly state any properties of the vector product that you use within your proof. [9 marks]
AQA Further Paper 2 Specimen Q15
10 marks Challenging +1.3
  1. Show that \((1-\frac{1}{4}e^{2i\theta})(1-\frac{1}{4}e^{-2i\theta}) = \frac{1}{16}(17-8\cos 2\theta)\) [3 marks]
  2. Given that the series \(e^{2i\theta} + \frac{1}{4}e^{4i\theta} + \frac{1}{16}e^{6i\theta} + \frac{1}{64}e^{8i\theta} + \ldots\) has a sum to infinity, express this sum to infinity in terms of \(e^{2i\theta}\) [2 marks]
  3. Hence show that \(\sum_{n=1}^{\infty} \frac{1}{4^{n-1}} \cos 2n\theta = \frac{16\cos 2\theta - 4}{17 - 8\cos 2\theta}\) [4 marks]
  4. Deduce a similar expression for \(\sum_{n=1}^{\infty} \frac{1}{4^{n-1}} \sin 2n\theta\) [1 mark]
AQA Further Paper 2 Specimen Q16
9 marks Challenging +1.8
A designer is using a computer aided design system to design part of a building. He models part of a roof as a triangular prism \(ABCDEF\) with parallel triangular ends \(ABC\) and \(DEF\), and a rectangular base \(ACFD\). He uses the metre as the unit of length. \includegraphics{figure_16} The coordinates of \(B\), \(C\) and \(D\) are \((3, 1, 11)\), \((9, 3, 4)\) and \((-4, 12, 4)\) respectively. He uses the equation \(x - 3y = 0\) for the plane \(ABC\). He uses \(\mathbf{r} - \begin{pmatrix} -4 \\ 12 \\ 4 \end{pmatrix} \times \begin{pmatrix} 4 \\ -12 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}\) for the equation of the line \(AD\). Find the volume of the space enclosed inside this section of the roof. [9 marks]
AQA Further Paper 3 Mechanics 2021 June Q1
1 marks Easy -1.2
A spring of natural length 50 cm and modulus of elasticity \(\lambda\) newtons has an elastic potential energy of 4 J when compressed by 5 cm. Find the value of \(\lambda\) Circle your answer. [1 mark] 8 16 800 1600
AQA Further Paper 3 Mechanics 2021 June Q2
1 marks Easy -1.8
A force of magnitude 7 N acts at each end of a rod of length 20 cm, forming a couple. The forces act at right angles to the rod, as shown in the diagram below. \includegraphics{figure_2} Find the magnitude of the resultant moment of the couple. Circle your answer. [1 mark] 1.4 N m 2.8 N m 140 N m 280 N m
AQA Further Paper 3 Mechanics 2021 June Q3
3 marks Moderate -0.5
A ball has mass 0.4 kg and is hit by a wooden bat. The speed of the ball just before it is hit by the bat is \(6 \text{ m s}^{-1}\) The velocity of the ball immediately after being hit by the bat is perpendicular to its initial velocity. The speed of the ball just after it is hit by the bat is \(8 \text{ m s}^{-1}\) Show that the impulse on the ball has magnitude 4 N s [3 marks]
AQA Further Paper 3 Mechanics 2021 June Q4
4 marks Standard +0.3
A spring has stiffness \(k\)
  1. Determine the dimensions of \(k\) [1 mark]
  2. One end of the spring is attached to a fixed point. A particle of mass \(m\) kg is attached to the other end of the spring. The particle is set into vertical motion and moves up and down, taking \(t\) seconds to complete one oscillation. A possible model for \(t\) is $$t = pm^a g^b k^c$$ where \(p\) is a dimensionless constant and \(g \text{ m s}^{-2}\) is the acceleration due to gravity. Find the values of \(a\), \(b\) and \(c\) for this model to be dimensionally consistent. [3 marks]
AQA Further Paper 3 Mechanics 2021 June Q5
7 marks Standard +0.8
A uniform lamina has the shape of the region enclosed by the curve \(y = x^2 + 1\) and the lines \(x = 0\), \(x = 4\) and \(y = 0\) The diagram below shows the lamina. \includegraphics{figure_5}
  1. Find the coordinates of the centre of mass of the lamina, giving your answer in exact form. [4 marks]
  2. The lamina is suspended from the point where the curve intersects the line \(x = 4\) and hangs in equilibrium. Find the angle between the vertical and the longest straight edge of the lamina, giving your answer correct to the nearest degree. [3 marks]
AQA Further Paper 3 Mechanics 2021 June Q6
4 marks Standard +0.3
A ball of mass \(m\) kg is held at rest at a height \(h\) metres above a horizontal surface. The ball is released and bounces on the surface. The coefficient of restitution between the ball and the surface is \(e\) Prove that the kinetic energy lost during the first bounce is given by $$mgh(1 - e^2)$$ [4 marks]
AQA Further Paper 3 Mechanics 2021 June Q7
9 marks Challenging +1.8
A light string has length 1.5 metres. A small sphere is attached to one end of the string. The other end of the string is attached to a fixed point O A thin horizontal bar is positioned 0.9 metres directly below O The bar is perpendicular to the plane in which the sphere moves. The sphere is released from rest with the string taut and at an angle \(\alpha\) to the downward vertical through O The string becomes slack when the angle between the two sections of the string is 60° Ben draws the diagram below to show the initial position of the sphere, the bar and the path of the sphere. \includegraphics{figure_7}
  1. State two reasons why Ben's diagram is not a good representation of the situation. [2 marks]
  2. Using your answer to part (a), sketch an improved diagram. [1 mark]
  3. Find \(\alpha\), giving your answer to the nearest degree. [6 marks]
AQA Further Paper 3 Mechanics 2021 June Q8
11 marks Challenging +1.2
In this question use \(g = 9.8 \text{ m s}^{-2}\) A lift is used to raise a crate of mass 250 kg The lift exerts an upward force of magnitude \(P\) newtons on the crate. When the crate is at a height of \(x\) metres above its initial position $$P = k(x + 1)(12 - x) + 2450$$ where \(k\) is a constant. The crate is initially at rest, at the point where \(x = 0\)
  1. Show that the work done by the upward force as the crate rises to a height of 12 metres is given by $$29400 + 360k$$ [3 marks]
  2. The speed of the crate is \(3 \text{ m s}^{-1}\) when it has risen to a height of 12 metres. Find the speed of the crate when it has risen to a height of 15 metres. [5 marks]
  3. Find the height of the crate when its speed becomes zero. [2 marks]
  4. Air resistance has been ignored. Explain why this is reasonable in this context. [1 mark]
AQA Further Paper 3 Mechanics 2021 June Q9
10 marks Challenging +1.8
In this question use \(g = 9.81 \text{ m s}^{-2}\) A conical pendulum is made from an elastic string and a sphere of mass 0.2 kg The string has natural length 1.6 metres and modulus of elasticity 200 N The sphere describes a horizontal circle of radius 0.5 metres at a speed of \(v \text{ m s}^{-1}\) The angle between the elastic string and the vertical is \(\alpha\)
  1. Show that $$62.5 - 200 \sin \alpha = 1.962 \tan \alpha$$ [5 marks]
  2. Use your calculator to find \(\alpha\) [1 mark]
  3. Find the value of \(v\) [4 marks]
AQA Further Paper 3 Mechanics 2024 June Q1
1 marks Easy -1.8
A particle moves in a circular path so that at time \(t\) seconds its position vector, \(\mathbf{r}\) metres, is given by $$\mathbf{r} = 4\sin(2t)\mathbf{i} + 4\cos(2t)\mathbf{j}$$ Find the velocity of the particle, in m s\(^{-1}\), when \(t = 0\) Circle your answer. [1 mark] \(8\mathbf{i}\) \quad \(-8\mathbf{j}\) \quad \(8\mathbf{j}\) \quad \(8\mathbf{i} - 8\mathbf{j}\)
AQA Further Paper 3 Mechanics 2024 June Q2
1 marks Easy -1.2
As a particle moves along a straight horizontal line, it is subjected to a force \(F\) newtons that acts in the direction of motion of the particle. At time \(t\) seconds, \(F = \frac{t}{5}\) Calculate the magnitude of the impulse on the particle between \(t = 0\) and \(t = 3\) Circle your answer. [1 mark] 0.3 N s \quad 0.6 N s \quad 0.9 N s \quad 1.8 N s
AQA Further Paper 3 Mechanics 2024 June Q3
1 marks Moderate -0.8
A conical pendulum consists of a light string and a particle of mass \(m\) kg The conical pendulum completes horizontal circles with radius \(r\) metres and angular speed \(\omega\) radians per second. The string makes an angle \(\theta\) with the downward vertical. The tension in the string is \(T\) newtons. The conical pendulum and the forces acting on the particle are shown in the diagram. \includegraphics{figure_3} Which one of the following statements is correct? Tick (\(\checkmark\)) one box. [1 mark] \(T \cos \theta = mr\omega^2\) \quad \(\square\) \(T \sin \theta = mr\omega^2\) \quad \(\square\) \(T \cos \theta = \frac{m\omega^2}{r}\) \quad \(\square\) \(T \sin \theta = \frac{m\omega^2}{r}\) \quad \(\square\)
AQA Further Paper 3 Mechanics 2024 June Q4
5 marks Moderate -0.8
A particle of mass 3 kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point on a smooth horizontal surface. The particle is set into motion so that it moves with a constant speed 4 m s\(^{-1}\) in a circular path with radius 0.8 metres on the horizontal surface.
  1. Find the acceleration of the particle. [2 marks]
  2. Find the tension in the string. [1 mark]
  3. Show that the angular speed of the particle is 48 revolutions per minute correct to two significant figures. [2 marks]
AQA Further Paper 3 Mechanics 2024 June Q5
4 marks Standard +0.3
When a sphere of radius \(r\) metres is falling at \(v\) m s\(^{-1}\) it experiences an air resistance force \(F\) newtons. The force is to be modelled as $$F = kr^\alpha v^\beta$$ where \(k\) is a constant with units kg m\(^{-2}\)
  1. State the dimensions of \(F\) [1 mark]
  2. Use dimensional analysis to find the value of \(\alpha\) and the value of \(\beta\) [3 marks]
AQA Further Paper 3 Mechanics 2024 June Q6
10 marks Standard +0.3
In this question use \(g = 9.8\) m s\(^{-2}\) A light elastic string has natural length 3 metres and modulus of elasticity 18 newtons. One end of the elastic string is attached to a particle of mass 0.25 kg The other end of the elastic string is attached to a fixed point \(O\) The particle is released from rest at a point \(A\), which is 4.5 metres vertically below \(O\)
  1. Calculate the elastic potential energy of the string when the particle is at \(A\) [2 marks]
  2. The point \(B\) is 3 metres vertically below \(O\) Calculate the gravitational potential energy gained by the particle as it moves from \(A\) to \(B\) [2 marks]
  3. Find the speed of the particle at \(B\) [3 marks]
  4. The point \(C\) is 3.6 metres vertically below \(O\) Explain, showing any calculations that you make, why the speed of the particle is increasing the first time that the particle is at \(C\) [3 marks]
AQA Further Paper 3 Mechanics 2024 June Q7
10 marks Standard +0.3
A sphere, of mass 0.2 kg, moving on a smooth horizontal surface, collides with a fixed wall. Before the collision the sphere moves with speed 5 m s\(^{-1}\) at an angle of 60° to the wall. After the collision the sphere moves with speed \(v\) m s\(^{-1}\) at an angle of \(\theta\)° to the wall. The velocities are shown in the diagram below. \includegraphics{figure_7} The coefficient of restitution between the wall and the sphere is 0.7
  1. Assume that the wall is smooth.
    1. Find the value of \(v\) Give your answer to two significant figures. [4 marks]
    2. Find the value of \(\theta\) Give your answer to the nearest whole number. [2 marks]
    3. Find the magnitude of the impulse exerted on the sphere by the wall. Give your answer to two significant figures. [2 marks]
  2. In reality the wall is not smooth. Explain how this would cause a change in the magnitude of the impulse calculated in part (a)(iii). [2 marks]
AQA Further Paper 3 Mechanics 2024 June Q8
10 marks Challenging +1.2
The finite region enclosed by the line \(y = kx\), the \(x\)-axis and the line \(x = 5\) is rotated through 360° around the \(x\) axis to form a solid cone.
    1. Use integration to show that the position of the centre of mass of the cone is independent of \(k\) [4 marks]
    2. State the distance between the base of the cone and its centre of mass. [1 mark]
  2. State one assumption that you have made about the cone. [1 mark]
  3. The plane face of the cone is placed on a rough inclined plane. The coefficient of friction between the cone and the plane is 0.8 The angle between the plane and the horizontal is gradually increased from 0° Find the range of values of \(k\) for which the cone slides before it topples. [4 marks]
AQA Further Paper 3 Mechanics 2024 June Q9
8 marks Challenging +1.8
A small sphere, of mass \(m\), is attached to one end of a light inextensible string of length \(a\) The other end of the string is attached to a fixed point \(O\) The sphere is at rest in equilibrium directly below \(O\) when it is struck, giving it a horizontal impulse of magnitude \(mU\) After the impulse, the sphere follows a circular path in a vertical plane containing the point \(O\) until the string becomes slack at the point \(C\) At \(C\) the string makes an angle of 30° with the upward vertical through \(O\), as shown in the diagram below. \includegraphics{figure_9}
  1. Show that $$U^2 = \frac{ag}{2}\left(4 + 3\sqrt{3}\right)$$ where \(g\) is the acceleration due to gravity. [6 marks]
  2. With reference to any modelling assumptions that you have made, explain why giving your answer as an inequality would be more appropriate, and state this inequality. [2 marks]