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AQA Further Paper 2 2023 June Q13
11 marks Challenging +1.8
The quadratic equation \(z^2 - 5z + 8 = 0\) has roots \(\alpha\) and \(\beta\)
  1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha\beta\) [2 marks]
  2. Without finding the value of \(\alpha\) or the value of \(\beta\), show that \(\alpha^4 + \beta^4 = -47\) [4 marks]
  3. Find a quadratic equation, with integer coefficients, which has roots \(\alpha^3 + \beta\) and \(\beta^3 + \alpha\) [5 marks]
AQA Further Paper 2 2023 June Q14
10 marks Challenging +1.3
The function \(f\) is defined by $$f(x) = \frac{1}{4x^2 + 16x + 19} \quad (x \in \mathbb{R})$$
  1. Show, without using calculus, that the graph of \(y = f(x)\) has a stationary point at \(\left(-2, \frac{1}{3}\right)\) [3 marks]
  2. Show that \(\int_{-2}^{-\frac{1}{2}} f(x) \, dx = \frac{\pi\sqrt{3}}{18}\) [5 marks]
  3. Find the value of \(\int_{-2}^{\infty} f(x) \, dx\) Fully justify your answer. [2 marks]
AQA Further Paper 2 2023 June Q15
10 marks Challenging +1.2
  1. Given that \(z = \cos \theta + \text{i} \sin \theta\), use de Moivre's theorem to show that $$z^n - z^{-n} = 2\text{i} \sin n\theta$$ [2 marks]
  2. The series \(S\) is defined as $$S = \sin \theta + \sin 3\theta + \ldots + \sin(2n - 1)\theta$$ Use part (a) to express \(S\) in the form $$S = \frac{1}{2\text{i}}(G_1) - \frac{1}{2\text{i}}(G_2)$$ where each of \(G_1\) and \(G_2\) is a geometric series. [3 marks]
  3. Hence, show that $$S = \frac{\sin^2(n\theta)}{\sin \theta}$$ [5 marks]
AQA Further Paper 2 2023 June Q16
16 marks Hard +2.3
A bungee jumper of mass \(m\) kg is attached to an elastic rope. The other end of the rope is attached to a fixed point. The bungee jumper falls vertically from the fixed point. At time \(t\) seconds after the rope first becomes taut, the extension of the rope is \(x\) metres and the speed of the bungee jumper is \(v\) m s\(^{-1}\)
  1. A model for the motion while the rope remains taut assumes that the forces acting on the bungee jumper are • the weight of the bungee jumper • a tension in the rope of magnitude \(kx\) newtons • an air resistance force of magnitude \(Rv\) newtons where \(k\) and \(R\) are constants such that \(4km > R^2\)
    1. Show that this model gives the result $$x = e^{-\frac{Rt}{2m}} \left( A \cos \frac{\sqrt{4km - R^2}}{2m} t + B \sin \frac{\sqrt{4km - R^2}}{2m} t \right) + \frac{mg}{k}$$ where \(A\) and \(B\) are constants, and \(g\) m s\(^{-2}\) is the acceleration due to gravity. You do not need to find the value of \(A\) or the value of \(B\) [6 marks]
    2. It is also given that: \(k = 16\) \(R = 20\) \(m = 62.5\) \(g = 9.8\) m s\(^{-2}\) and that the speed of the bungee jumper when the rope becomes taut is 14 m s\(^{-1}\) Show that, to the nearest integer, \(A = -38\) and \(B = 16\) [6 marks]
  2. A second, simpler model assumes that the air resistance is zero. The values of \(k\), \(m\) and \(g\) remain the same. Find an expression for \(x\) in terms of \(t\) according to this simpler model, giving the values of all constants to two significant figures. [4 marks]
AQA Further Paper 2 2024 June Q1
1 marks Easy -1.8
It is given that $$\begin{bmatrix} 2 \\ 1 \\ 9 \end{bmatrix} \times \begin{bmatrix} 5 \\ \lambda \\ -6 \end{bmatrix} = 0$$ where \(\lambda\) is a constant. Find the value of \(\lambda\) Circle your answer. [1 mark] \(-28\) \quad\quad \(-8\) \quad\quad \(8\) \quad\quad \(28\)
AQA Further Paper 2 2024 June Q2
1 marks Moderate -0.8
The movement of a particle is described by the simple harmonic equation $$\ddot{x} = -25x$$ where \(x\) metres is the displacement of the particle at time \(t\) seconds, and \(\ddot{x}\) m s\(^{-2}\) is the acceleration of the particle. The maximum displacement of the particle is 9 metres. Find the maximum speed of the particle. Circle your answer. [1 mark] \(15\) m s\(^{-1}\) \quad\quad \(45\) m s\(^{-1}\) \quad\quad \(75\) m s\(^{-1}\) \quad\quad \(135\) m s\(^{-1}\)
AQA Further Paper 2 2024 June Q3
1 marks Easy -1.2
The function g is defined by $$g(x) = \text{sech } x \quad\quad (x \in \mathbb{R})$$ Which one of the following is the range of g? Tick (\(\checkmark\)) one box. [1 mark] \(-\infty < g(x) \leq -1\) \quad \(\square\) \(-1 \leq g(x) < 0\) \quad \(\square\) \(0 < g(x) \leq 1\) \quad \(\square\) \(1 \leq g(x) \leq \infty\) \quad \(\square\)
AQA Further Paper 2 2024 June Q4
1 marks Moderate -0.8
The function f is a quartic function with real coefficients. The complex number \(5i\) is a root of the equation \(f(x) = 0\) Which one of the following must be a factor of \(f(x)\)? Circle your answer. [1 mark] \((x^2 - 25)\) \quad\quad \((x^2 - 5)\) \quad\quad \((x^2 + 5)\) \quad\quad \((x^2 + 25)\)
AQA Further Paper 2 2024 June Q5
3 marks Standard +0.3
The first four terms of the series \(S\) can be written as $$S = (1 \times 2) + (2 \times 3) + (3 \times 4) + (4 \times 5) + ...$$
  1. Write an expression, using \(\sum\) notation, for the sum of the first \(n\) terms of \(S\) [1 mark]
  2. Show that the sum of the first \(n\) terms of \(S\) is equal to $$\frac{1}{3}n(n + 1)(n + 2)$$ [2 marks]
AQA Further Paper 2 2024 June Q6
3 marks Moderate -0.3
The cubic equation $$x^3 + 5x^2 - 4x + 2 = 0$$ has roots \(\alpha\), \(\beta\) and \(\gamma\) Find a cubic equation, with integer coefficients, whose roots are \(3\alpha\), \(3\beta\) and \(3\gamma\) [3 marks]
AQA Further Paper 2 2024 June Q7
4 marks Standard +0.8
The matrices \(\mathbf{A}\) and \(\mathbf{B}\) are defined as follows. $$\mathbf{A} = \begin{bmatrix} p - 2 & p - 1 \\ 0 & 1 \end{bmatrix} \quad\quad \mathbf{B} = \begin{bmatrix} 1 & 2p - 1 \\ 0 & 4 - p \end{bmatrix}$$ Find the values of \(p\) such that \(\mathbf{A}\) and \(\mathbf{B}\) are commutative under matrix multiplication. Fully justify your answer. [4 marks]
AQA Further Paper 2 2024 June Q8
4 marks Standard +0.8
The vectors \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\) are such that \(\mathbf{a} \times \mathbf{b} = \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}\) and \(\mathbf{a} \times \mathbf{c} = \begin{bmatrix} 0 \\ 0 \\ 3 \end{bmatrix}\) Work out \((\mathbf{a} - 4\mathbf{b} + 3\mathbf{c}) \times (2\mathbf{a})\) [4 marks]
AQA Further Paper 2 2024 June Q9
4 marks Standard +0.8
A curve passes through the point \((-2, 4.73)\) and satisfies the differential equation $$\frac{dy}{dx} = \frac{y^2 - x^2}{2x + 3y}$$ Use Euler's step by step method once, and then the midpoint formula $$y_{r+1} = y_{r-1} + 2hf(x_r, y_r), \quad x_{r+1} = x_r + h$$ once, each with a step length of \(0.02\), to estimate the value of \(y\) when \(x = -1.96\) Give your answer to five significant figures. [4 marks]
AQA Further Paper 2 2024 June Q10
4 marks Standard +0.8
The matrix \(\mathbf{C}\) is defined by $$\mathbf{C} = \begin{bmatrix} 3 & 2 \\ -4 & 5 \end{bmatrix}$$ Prove that the transformation represented by \(\mathbf{C}\) has no invariant lines of the form \(y = kx\) [4 marks]
AQA Further Paper 2 2024 June Q11
3 marks Standard +0.8
Latifa and Sam are studying polynomial equations of degree greater than 2, with real coefficients and no repeated roots. Latifa says that if such an equation has exactly one real root, it must be of degree 3 Sam says that this is not correct. State, giving reasons, whether Latifa or Sam is right. [3 marks]
AQA Further Paper 2 2024 June Q12
5 marks Challenging +1.2
The transformation \(S\) is represented by the matrix \(\mathbf{M} = \begin{bmatrix} 1 & -6 \\ 2 & 7 \end{bmatrix}\) The transformation \(T\) is a reflection in the line \(y = x\sqrt{3}\) and is represented by the matrix \(\mathbf{N}\) The point \(P(x, y)\) is transformed first by \(S\), then by \(T\) The result of these transformations is the point \(Q(3, 8)\) Find the coordinates of \(P\) Give your answers to three decimal places. [5 marks]
AQA Further Paper 2 2024 June Q13
8 marks Standard +0.8
  1. Use the method of differences to show that $$\sum_{r=2}^{n} \frac{1}{(r - 1)r(r + 1)} = \frac{1}{4} - \frac{1}{2n} + \frac{1}{2(n + 1)}$$ [5 marks]
  2. Find the smallest integer \(n\) such that $$\sum_{r=2}^{n} \frac{1}{(r - 1)r(r + 1)} > 0.24999$$ [3 marks]
AQA Further Paper 2 2024 June Q14
10 marks Standard +0.8
The matrix \(\mathbf{M}\) is defined as $$\mathbf{M} = \begin{bmatrix} 5 & 2 & 1 \\ 6 & 3 & 2k + 3 \\ 2 & 1 & 5 \end{bmatrix}$$ where \(k\) is a constant.
  1. Given that \(\mathbf{M}\) is a non-singular matrix, find \(\mathbf{M}^{-1}\) in terms of \(k\) [5 marks]
  2. State any restrictions on the value of \(k\) [1 mark]
  3. Using your answer to part (a), show that the solution to the set of simultaneous equations below is independent of the value of \(k\) \(5x + 2y + z = 1\) \(6x + 3y + (2k + 3)z = 4k + 3\) \(2x + y + 5z = 9\) [4 marks]
AQA Further Paper 2 2024 June Q15
7 marks Standard +0.8
The diagram shows the line \(y = 5 - x\) \includegraphics{figure_15}
  1. On the diagram above, sketch the graph of \(y = |x^2 - 4x|\), including all parts of the graph where it intersects the line \(y = 5 - x\) (You do not need to show the coordinates of the points of intersection.) [3 marks]
  2. Find the solution of the inequality $$|x^2 - 4x| > 5 - x$$ Give your answer in an exact form. [4 marks]
AQA Further Paper 2 2024 June Q16
9 marks Challenging +1.2
The function f is defined by $$f(x) = \frac{ax + 5}{x + b}$$ where \(a\) and \(b\) are constants. The graph of \(y = f(x)\) has asymptotes \(x = -2\) and \(y = 3\)
  1. Write down the value of \(a\) and the value of \(b\) [2 marks]
  2. The diagram shows the graph of \(y = f(x)\) and its asymptotes. The shaded region \(R\) is enclosed by the graph of \(y = f(x)\), the \(x\)-axis and the \(y\)-axis. \includegraphics{figure_16}
    1. The shaded region \(R\) is rotated through \(360°\) about the \(x\)-axis to form a solid. Find the volume of this solid. Give your answer to three significant figures. [3 marks]
    2. The shaded region \(R\) is rotated through \(360°\) about the \(y\)-axis to form a solid. Find the volume of this solid. Give your answer to three significant figures. [4 marks]
AQA Further Paper 2 2024 June Q17
9 marks Standard +0.8
The Argand diagram below shows a circle \(C\) \includegraphics{figure_17}
  1. Write down the equation of the locus of \(C\) in the form $$|z - w| = a$$ where \(w\) is a complex number whose real and imaginary parts are integers, and \(a\) is an integer. [2 marks]
  2. It is given that \(z_1\) is a complex number representing a point on \(C\). Of all the complex numbers which represent points on \(C\), \(z_1\) has the least argument.
    1. Find \(|z_1|\) Give your answer in an exact form. [3 marks]
    2. Show that \(\arg z_1 = \arcsin\left(\frac{6\sqrt{3} - 2}{13}\right)\) [4 marks]
AQA Further Paper 2 2024 June Q18
4 marks Standard +0.8
In this question you may use results from the formulae booklet without proof. Use the binomial series for \((1 + x)^n\) and the Maclaurin's series for \(\sin x\) to find the series expansion for \(\frac{1}{(1 + \sin \theta)^4}\) up to and including the term in \(\theta^3\) [4 marks]
AQA Further Paper 2 2024 June Q19
10 marks Challenging +1.2
Solve the differential equation $$\frac{d^2y}{dx^2} + 4\frac{dy}{dx} - 45y = 21e^{5x} - 0.3x + 27x^2$$ given that \(y = \frac{37}{225}\) and \(\frac{dy}{dx} = 0\) when \(x = 0\) [10 marks]
AQA Further Paper 2 2024 June Q20
9 marks Challenging +1.3
The integral \(I_n\) is defined by $$I_n = \int_0^{\frac{\pi}{4}} \cos^n x \, dx \quad\quad (n \geq 0)$$
  1. Show that $$I_n = \left(\frac{n-1}{n}\right)I_{n-2} + \frac{1}{n\left(2^{\frac{n}{2}}\right)} \quad\quad (n \geq 2)$$ [6 marks]
  2. Use the result from part (a) to show that $$\int_0^{\frac{\pi}{4}} \cos^6 x \, dx = \frac{a\pi + b}{192}$$ where \(a\) and \(b\) are integers to be found. [3 marks]
AQA Further Paper 2 Specimen Q1
1 marks Easy -1.8
Given that \(z_1 = 4e^{i\frac{\pi}{3}}\) and \(z_2 = 2e^{i\frac{\pi}{4}}\) state the value of \(\arg\left(\frac{z_1}{z_2}\right)\) Circle your answer. [1 mark] \(\frac{\pi}{12}\) \quad \(\frac{4}{3}\) \quad \(\frac{7\pi}{12}\) \quad \(2\)