AQA Further Paper 2 (Further Paper 2) 2023 June

Given that \(y = \sin x + \sinh x\), find \(\frac{d^2y}{dx^2} + y\) Circle your answer. [1 mark] \(2\sin x\) \quad \(-2\sin x\) \quad \(2\sinh x\) \quad \(-2\sinh x\)

Which one of the expressions below is not equal to zero? Circle your answer. [1 mark] \(\lim_{x \to \infty} (x^2e^{-x})\) \quad \(\lim_{x \to 0} (x^5 \ln x)\) \quad \(\lim_{x \to \infty} \left(\frac{e^x}{x^5}\right)\) \quad \(\lim_{x \to 0^+} (x^3e^x)\)

The determinant \(A = \begin{vmatrix} 1 & 1 & 1 \\ 2 & 0 & 2 \\ 3 & 2 & 1 \end{vmatrix}\) Which one of the determinants below has a value which is not equal to the value of \(A\)? Tick (\(\checkmark\)) one box. [1 mark] \(\begin{vmatrix} 3 & 1 & 3 \\ 2 & 0 & 2 \\ 3 & 2 & 1 \end{vmatrix}\) \quad \(\square\) \(\begin{vmatrix} 1 & 2 & 3 \\ 1 & 0 & 2 \\ 1 & 2 & 1 \end{vmatrix}\) \quad \(\square\) \(\begin{vmatrix} 2 & 2 & 2 \\ 1 & 0 & 1 \\ 3 & 2 & 1 \end{vmatrix}\) \quad \(\square\) \(\begin{vmatrix} 1 & 1 & 1 \\ 3 & 2 & 1 \\ 2 & 0 & 2 \end{vmatrix}\) \quad \(\square\)

It is given that \(f(x) = \cosh^{-1}(x - 3)\) Which of the sets listed below is the greatest possible domain of the function \(f\)? Circle your answer. [1 mark] \(\{x : x \geq 4\}\) \quad \(\{x : x \geq 3\}\) \quad \(\{x : x \geq 1\}\) \quad \(\{x : x \geq 0\}\)

Josh and Zoe are solving the following mathematics problem: The curve \(C_1\) has equation $$\frac{x^2}{16} - \frac{y^2}{9} = 1$$ The matrix \(\mathbf{M} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\) maps \(C_1\) onto \(C_2\) Find the equations of the asymptotes of \(C_2\) Josh says that to solve this problem you must first carry out the transformation on \(C_1\) to find \(C_2\), and then find the asymptotes of \(C_2\) Zoe says that you will get the same answer if you first find the asymptotes of \(C_1\), and then carry out the transformation on these asymptotes to obtain the asymptotes of \(C_2\) Show that Zoe is correct. [5 marks]

1. Express \(-5 - 5\text{i}\) in the form \(re^{i\theta}\), where \(-\pi < \theta \leq \pi\) [2 marks]
2. The point on an Argand diagram that represents \(-5 - 5\text{i}\) is one of the vertices of an equilateral triangle whose centre is at the origin. Find the complex numbers represented by the other two vertices of the triangle. Give your answers in the form \(re^{i\theta}\), where \(-\pi < \theta \leq \pi\) [3 marks]

Show that $$\sum_{r=11}^{n+1} r^3 = \frac{1}{4}(n^2 + an + b)(n^2 + an + c)$$ where \(a\), \(b\) and \(c\) are integers to be found. [3 marks]

\(\mathbf{A}\) is a non-singular \(2 \times 2\) matrix and \(\mathbf{A}^T\) is the transpose of \(\mathbf{A}\)

1. Using the result $$(\mathbf{AB})^T = \mathbf{B}^T\mathbf{A}^T$$ show that $$(\mathbf{A}^{-1})^T = (\mathbf{A}^T)^{-1}$$ [3 marks]
2. It is given that \(\mathbf{A} = \begin{pmatrix} 4 & 5 \\ -1 & k \end{pmatrix}\), where \(k\) is a real constant.
   1. Find \((\mathbf{A}^{-1})^T\), giving your answer in terms of \(k\) [2 marks]
   2. State the restriction on the possible values of \(k\) [1 mark]

The complex number \(z\) is such that $$z = \frac{1 + \text{i}}{1 - k\text{i}}$$ where \(k\) is a real number.

1. Find the real part of \(z\) and the imaginary part of \(z\), giving your answers in terms of \(k\) [2 marks]
2. In the case where \(k = \sqrt{3}\), use part (a) to show that $$\cos \frac{7\pi}{12} = \frac{\sqrt{2} - \sqrt{6}}{4}$$ [5 marks]

The region \(R\) on an Argand diagram satisfies both \(|z + 2\text{i}| \leq 3\) and \(-\frac{\pi}{6} \leq \arg(z) \leq \frac{\pi}{2}\)

1. Sketch \(R\) on the Argand diagram below. [3 marks] \includegraphics{figure_10a}
2. Find the maximum value of \(|z|\) in the region \(R\), giving your answer in exact form. [5 marks]

The line \(l_1\) passes through the points \(A(6, 2, 7)\) and \(B(4, -3, 7)\)

1. Find a Cartesian equation of \(l_1\) [2 marks]
2. The line \(l_2\) has vector equation \(\mathbf{r} = \begin{pmatrix} 8 \\ 9 \\ c \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}\) where \(c\) is a constant.
   1. Explain how you know that the lines \(l_1\) and \(l_2\) are not perpendicular. [2 marks]
   2. The lines \(l_1\) and \(l_2\) both lie in the same plane. Find the value of \(c\) [5 marks]

The function \(f\) is defined by $$f(n) = 3^{3n+1} + 2^{3n+4} \quad (n \in \mathbb{Z}^+)$$ Prove by induction that \(f(n)\) is divisible by 19 for \(n \geq 1\) [6 marks]

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]

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]

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]

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]