AQA Further Paper 1 (Further Paper 1) 2019 June

Which one of these functions has the set \(\{x : |x| < 1\}\) as its greatest possible domain? Circle your answer. [1 mark] \(\cosh x\) \quad \(\cosh^{-1} x\) \quad \(\tanh x\) \quad \(\tanh^{-1} x\)

The first two non-zero terms of the Maclaurin series expansion of \(f(x)\) are \(x\) and \(-\frac{1}{2}x^3\) Which one of the following could be \(f(x)\)? Circle your answer. [1 mark] \(xe^{\frac{1}{2}x^2}\) \quad \(\frac{1}{2}\sin 2x\) \quad \(x \cos x\) \quad \((1 + x^3)^{-\frac{1}{2}}\)

The function \(f(x) = x^2 - 1\) Find the mean value of \(f(x)\) from \(x = -0.5\) to \(x = 1.7\) Give your answer to three significant figures. Circle your answer. [1 mark] \(-0.521\) \quad \(-0.434\) \quad \(-0.237\) \quad \(0.786\)

Solve the equation \(2z - 5iz^* = 12\) [4 marks]

A plane has equation \(\mathbf{r} \cdot \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} = 7\) A line has equation \(\mathbf{r} = \begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}\) Calculate the acute angle between the line and the plane. Give your answer to the nearest \(0.1°\) [3 marks]

1. Show that $$\cosh^3 x + \sinh^3 x = \frac{1}{4}e^{mx} + \frac{3}{4}e^{nx}$$ where \(m\) and \(n\) are integers. [3 marks]
2. Hence find \(\cosh^6 x - \sinh^6 x\) in the form $$\frac{a \cosh(kx) + b}{8}$$ where \(a\), \(b\) and \(k\) are integers. [5 marks]

Three non-singular square matrices, \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{R}\) are such that $$\mathbf{AR} = \mathbf{B}$$ The matrix \(\mathbf{R}\) represents a rotation about the \(z\)-axis through an angle \(\theta\) and $$\mathbf{B} = \begin{pmatrix} -\cos \theta & \sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

1. Show that \(\mathbf{A}\) is independent of the value of \(\theta\). [3 marks]
2. Give a full description of the single transformation represented by the matrix \(\mathbf{A}\). [1 mark]

1. If \(z = \cos \theta + i \sin \theta\), use de Moivre's theorem to prove that $$z^n - \frac{1}{z^n} = 2i \sin n\theta$$ [3 marks]
2. Express \(\sin^5 \theta\) in terms of \(\sin 5\theta\), \(\sin 3\theta\) and \(\sin \theta\) [4 marks]
3. Hence show that $$\int_0^{\frac{\pi}{3}} \sin^5 \theta \, d\theta = \frac{53}{480}$$ [3 marks]

1. Solve the equation \(z^3 = \sqrt{2} - \sqrt{6}i\), giving your answers in the form \(re^{i\theta}\) where \(r > 0\) and \(0 \leq \theta < 2\pi\) [5 marks]
2. The transformation represented by the matrix \(\mathbf{M} = \begin{pmatrix} 5 & 1 \\ 1 & 3 \end{pmatrix}\) acts on the points on an Argand Diagram which represent the roots of the equation in part (a). Find the exact area of the shape formed by joining the transformed points. [4 marks]

The points \(A(5, -4, 6)\) and \(B(6, -6, 8)\) lie on the line \(L\). The point \(C\) is \((15, -5, 9)\).

1. \(D\) is the point on \(L\) that is closest to \(C\). Find the coordinates of \(D\). [6 marks]
2. Hence find, in exact form, the shortest distance from \(C\) to \(L\). [2 marks]

Find the general solution of the differential equation $$x \frac{dy}{dx} - 2y = \frac{x^3}{\sqrt{4 - 2x - x^2}}$$ where \(0 < x < \sqrt{5} - 1\) [7 marks]

Three planes have equations \begin{align} 4x - 5y + z &= 8
3x + 2y - kz &= 6
(k - 2)x + ky - 8z &= 6 \end{align} where \(k\) is a real constant. The planes do not meet at a unique point.

1. Find the possible values of \(k\). [3 marks]
2. For each value of \(k\) found in part (a), identify the configuration of the given planes. Fully justify your answer, stating in each case whether or not the equations of the planes form a consistent system. [5 marks]

The equation \(z^3 + kz^2 + 9 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\).

1. 1. Show that $$\alpha^2 + \beta^2 + \gamma^2 = k^2$$ [3 marks]
   2. Show that $$\alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2 = -18k$$ [4 marks]
2. The equation \(9z^3 - 40z^2 + rz + s = 0\) has roots \(\alpha\beta + \gamma\), \(\beta\gamma + \alpha\) and \(\gamma\alpha + \beta\).
   1. Show that $$k = -\frac{40}{9}$$ [1 mark]
   2. Without calculating the values of \(\alpha\), \(\beta\) and \(\gamma\), find the value of \(s\). Show working to justify your answer. [6 marks]

In this question use \(g = 10 \text{ m s}^{-2}\) A light spring is attached to the base of a long tube and has a mass \(m\) attached to the other end, as shown in the diagram. The tube is filled with oil. When the compression of the spring is \(c\) metres, the thrust in the spring is \(9mc\) newtons. \includegraphics{figure_14} The mass is held at rest in a position where the compression of the spring is \(\frac{20}{9}\) metres. The mass is then released from rest. During the subsequent motion the oil causes a resistive force of \(6mv\) newtons to act on the mass, where \(v \text{ m s}^{-1}\) is the speed of the mass. At time \(t\) seconds after the mass is released, the displacement of the mass above its starting position is \(x\) metres.

1. Find \(x\) in terms of \(t\). [10 marks]
2. State, giving a reason, the type of damping which occurs. [1 mark]

The diagram shows part of a spiral curve. The point \(P\) has polar coordinates \((r, \theta)\) where \(0 \leq \theta \leq \frac{\pi}{2}\) The points \(T\) and \(S\) lie on the initial line and \(O\) is the pole. \(TPQ\) is the tangent to the curve at \(P\). \includegraphics{figure_15}

1. Show that the gradient of \(TPQ\) is equal to $$\frac{\frac{dr}{d\theta} \sin \theta + r \cos \theta}{\frac{dr}{d\theta} \cos \theta - r \sin \theta}$$ [4 marks]
2. The curve has polar equation $$r = e^{(\cot b)\theta}$$ where \(b\) is a constant such that \(0 < b < \frac{\pi}{2}\) Use the result of part (a) to show that the angle between the line \(OP\) and the tangent \(TPQ\) does not depend on \(\theta\). [7 marks]