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AQA Further Paper 2 Specimen Q2
3 marks Easy -1.2
Given that \(z\) is a complex number and that \(z^*\) is the complex conjugate of \(z\) prove that \(zz^* - |z|^2 = 0\) [3 marks]
AQA Further Paper 2 Specimen Q3
3 marks Standard +0.8
The transformation T is defined by the matrix M. The transformation S is defined by the matrix \(\mathbf{M}^{-1}\). Given that the point \((x, y)\) is invariant under transformation T, prove that \((x, y)\) is also an invariant point under transformation S. [3 marks]
AQA Further Paper 2 Specimen Q4
4 marks Standard +0.3
Solve the equation \(z^3 = i\), giving your answers in the form \(e^{i\theta}\), where \(-\pi < \theta \leq \pi\) [4 marks]
AQA Further Paper 2 Specimen Q5
4 marks Standard +0.3
Find the smallest value \(\theta\) of for which \((\cos \theta + i \sin \theta)^5 = \frac{1}{\sqrt{2}}(1 - i)\) \(\{\theta \in \mathbb{R} : \theta > 0\}\) [4 marks]
AQA Further Paper 2 Specimen Q6
5 marks Standard +0.3
Prove that \(8^n - 7n + 6\) is divisible by 7 for all integers \(n \geq 0\) [5 marks]
AQA Further Paper 2 Specimen Q7
5 marks Challenging +1.2
A small, hollow, plastic ball, of mass \(m\) kg is at rest at a point \(O\) on a polished horizontal surface. The ball is attached to two identical springs. The other ends of the springs are attached to the points \(P\) and \(Q\) which are 1.8 metres apart on a straight line through \(O\). The ball is struck so that it moves away from \(O\), towards \(P\) with a speed of 0.75 m s\(^{-1}\). As the ball moves, its displacement from \(O\) is \(x\) metres at time \(t\) seconds after the motion starts. The force that each of the springs applies to the ball is \(12.5mx\) newtons towards \(O\). The ball is to be modelled as a particle. The surface is assumed to be smooth and it is assumed that the forces applied to the ball by the springs are the only horizontal forces acting on the ball.
  1. Find the minimum distance of the ball from \(P\), in the subsequent motion. [5 marks]
AQA Further Paper 2 Specimen Q7
2 marks Moderate -0.5
  1. In practice the minimum distance predicted by the model is incorrect. Is the minimum distance predicted by the model likely to be too big or too small? Explain your answer with reference to the model. [2 marks]
AQA Further Paper 2 Specimen Q8
5 marks Standard +0.8
Given that \(I_n = \int_0^{\frac{\pi}{2}} \sin^n x \, dx\) \quad \(n \geq 0\) show that \(n I_n = (n-1)I_{n-2}\) \quad \(n \geq 2\) [5 marks]
AQA Further Paper 2 Specimen Q9
6 marks Challenging +1.2
A student claims: "Given any two non-zero square matrices, A and B, then \((\mathbf{AB})^{-1} = \mathbf{B}^{-1}\mathbf{A}^{-1}\)"
  1. Explain why the student's claim is incorrect giving a counter example. [2 marks]
  2. Refine the student's claim to make it fully correct. [1 mark]
  3. Prove that your answer to part (b) is correct. [3 marks]
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]