AQA Further Paper 1 (Further Paper 1) 2024 June

The roots of the equation \(20x^3 - 16x^2 - 4x + 7 = 0\) are \(\alpha\), \(\beta\) and \(\gamma\) Find the value of \(\alpha\beta + \beta\gamma + \gamma\alpha\) Circle your answer. [1 mark] \(-\frac{4}{5}\) \(-\frac{1}{5}\) \(\frac{1}{5}\) \(\frac{4}{5}\)

The complex number \(z = e^{\frac{i\pi}{3}}\) Which one of the following is a real number? Circle your answer. [1 mark] \(z^4\) \(z^5\) \(z^6\) \(z^7\)

The function f is defined by $$f(x) = x^2 \quad (x \in \mathbb{R})$$ Find the mean value of \(f(x)\) between \(x = 0\) and \(x = 2\) Circle your answer. [1 mark] \(\frac{2}{3}\) \(\frac{4}{3}\) \(\frac{8}{3}\) \(\frac{16}{3}\)

Which one of the following statements is correct? Tick (\(\checkmark\)) one box. [1 mark] \(\lim_{x \to 0}(x^2 \ln x) = 0\) \(\square\) \(\lim_{x \to 0}(x^2 \ln x) = 1\) \(\square\) \(\lim_{x \to 0}(x^2 \ln x) = 2\) \(\square\) \(\lim_{x \to 0}(x^2 \ln x)\) is not defined. \(\square\)

The points \(A\), \(B\) and \(C\) have coordinates \(A(5, 3, 4)\), \(B(8, -1, 9)\) and \(C(12, 5, 10)\) The points \(A\), \(B\) and \(C\) lie in the plane \(\Pi\)

1. Find a vector that is normal to the plane \(\Pi\) [3 marks]
2. Find a Cartesian equation of the plane \(\Pi\) [2 marks]

The sequence \(u_1, u_2, u_3, \ldots\) is defined by $$u_1 = 1$$ $$u_{n+1} = u_n + 3n$$ Prove by induction that for all integers \(n \geq 1\) $$u_n = \frac{1}{2}n^2 - \frac{3}{2}n + 1$$ [4 marks]

The complex numbers \(z\) and \(w\) satisfy the simultaneous equations $$z + w^* = 5$$ $$3z^* - w = 6 + 4i$$ Find \(z\) and \(w\) [5 marks]

The ellipse \(E\) has equation $$x^2 + \frac{y^2}{9} = 1$$ The line with equation \(y = mx + 4\) is a tangent to \(E\) Without using differentiation show that \(m = \pm\sqrt{7}\) [4 marks]

1. It is given that $$p = \ln\left(r + \sqrt{r^2 + 1}\right)$$ Starting from the exponential definition of the sinh function, show that \(\sinh p = r\) [4 marks]
2. Solve the equation $$\cosh^2 x = 2\sinh x + 16$$ Give your answers in logarithmic form. [4 marks]

The complex numbers \(z\) and \(w\) are defined by $$z = \cos\frac{\pi}{4} + i\sin\frac{\pi}{4}$$ and $$w = \cos\frac{\pi}{6} + i\sin\frac{\pi}{6}$$ By evaluating the product \(zw\), show that $$\tan\frac{5\pi}{12} = 2 + \sqrt{3}$$ [6 marks]

1. Find \(\frac{d}{dx}(x^2\tan^{-1} x)\) [1 mark]
2. Hence find \(\int 2x \tan^{-1} x \, dx\) [4 marks]

The line \(L_1\) has equation $$\mathbf{r} = \begin{pmatrix} 4 \\ 2 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 3 \\ -1 \end{pmatrix}$$ The transformation T is represented by the matrix $$\begin{pmatrix} 2 & 1 & 0 \\ 3 & 4 & 6 \\ -5 & 2 & -3 \end{pmatrix}$$ The transformation T transforms the line \(L_1\) to the line \(L_2\)

1. Show that the angle between \(L_1\) and \(L_2\) is 0.701 radians, correct to three decimal places. [4 marks]
2. Find the shortest distance between \(L_1\) and \(L_2\) Give your answer in an exact form. [6 marks]

1. Use de Moivre's theorem to show that $$\cos 3\theta = 4\cos^3 \theta - 3\cos \theta$$ [3 marks]
2. Use de Moivre's theorem to express \(\sin 3\theta\) in terms of \(\sin \theta\) [2 marks]
3. Hence show that $$\cot 3\theta = \frac{\cot^3 \theta - 3\cot \theta}{3\cot^2 \theta - 1}$$ [4 marks]

Solve the differential equation $$\frac{dy}{dx} + y\tanh x = \sinh^3 x$$ given that \(y = 3\) when \(x = \ln 2\) Give your answer in an exact form. [7 marks]

A curve is defined parametrically by the equations $$x = \frac{3}{2}t^3 + 5$$ $$y = t^{\frac{3}{2}} \quad (t \geq 0)$$ Show that the arc length of the curve from \(t = 0\) to \(t = 2\) is equal to 26 units. [5 marks]

The curve \(C\) has polar equation \(r = 2 + \tan \theta\) The curve \(C\) meets the line \(\theta = \frac{\pi}{4}\) at the point \(A\) The point \(B\) has polar coordinates \((4, 0)\) The diagram shows part of the curve \(C\), and the points \(A\) and \(B\) \includegraphics{figure_16}

1. Show that the area of triangle \(OAB\) is \(3\sqrt{2}\) units. [2 marks]
2. Find the area of the shaded region. Give your answer in an exact form. [7 marks]

By making a suitable substitution, show that $$\int_{-2}^{1} \sqrt{x^2 + 6x + 8} \, dx = 2\sqrt{15} - \frac{1}{2}\cosh^{-1}(4)$$ [7 marks]

In this question use \(g = 9.8\) m s\(^{-2}\) Two light elastic strings each have one end attached to a small ball \(B\) of mass 0.5 kg The other ends of the strings are attached to the fixed points \(A\) and \(C\), which are 8 metres apart with \(A\) vertically above \(C\) The whole system is in a thin tube of oil, as shown in the diagram below. \includegraphics{figure_18} The string connecting \(A\) and \(B\) has natural length 2 metres, and the tension in this string is \(7e\) newtons when the extension is \(e\) metres. The string connecting \(B\) and \(C\) has natural length 3 metres, and the tension in this string is \(3e\) newtons when the extension is \(e\) metres.

1. Find the extension of each string when the system is in equilibrium. [3 marks]
2. It is known that in a large bath of oil, the oil causes a resistive force of magnitude \(4.5v\) newtons to act on the ball, where \(v\) m s\(^{-1}\) is the speed of the ball. Use this model to answer part (b)(i) and part (b)(ii).
   1. The ball is pulled a distance of 0.6 metres downwards from its equilibrium position towards \(C\), and released from rest. Show that during the subsequent motion the particle satisfies the differential equation $$\frac{d^2x}{dt^2} + 9\frac{dx}{dt} + 20x = 0$$ where \(x\) metres is the displacement of the particle below the equilibrium position at time \(t\) seconds after the particle is released. [3 marks]
   2. Find \(x\) in terms of \(t\) [5 marks]
3. State one limitation of the model used in part (b) [1 mark]