AQA Further AS Paper 1 (Further AS Paper 1) 2018 June

\(z = 3 - i\) Determine the value of \(zz*\) Circle your answer. [1 mark] \(10\) \(\qquad\) \(\sqrt{10}\) \(\qquad\) \(10 - 2i\) \(\qquad\) \(10 + 2i\)

Three matrices \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\) are given by $$\mathbf{A} = \begin{pmatrix} 5 & 2 & -3 \\ 0 & 7 & 6 \\ 4 & 1 & 0 \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 1 & 0 \\ 3 & -5 \\ -2 & 6 \end{pmatrix} \quad \text{and } \mathbf{C} = \begin{pmatrix} 6 & 4 & 3 \\ 1 & 2 & 0 \end{pmatrix}$$ Which of the following **cannot** be calculated? Circle your answer. [1 mark] \(\mathbf{AB}\) \(\qquad\) \(\mathbf{AC}\) \(\qquad\) \(\mathbf{BC}\) \(\qquad\) \(\mathbf{A}^2\)

Which of the following functions has the fourth term \(-\frac{1}{720}x^6\) in its Maclaurin series expansion? Circle your answer. [1 mark] \(\sin x\) \(\qquad\) \(\cos x\) \(\qquad\) \(e^x\) \(\qquad\) \(\ln(1 + x)\)

Sketch the graph given by the polar equation $$r = \frac{a}{\cos \theta}$$ where \(a\) is a positive constant. [2 marks] \includegraphics{figure_4}

Describe fully the transformation given by the matrix \(\begin{pmatrix} -\frac{1}{2} & -\frac{\sqrt{3}}{2} & 0 \\ \frac{\sqrt{3}}{2} & -\frac{1}{2} & 0 \\ 0 & 0 & 1 \end{pmatrix}\) [3 marks]

1. Matthew is finding a formula for the inverse function \(\text{arsinh } x\). He writes his steps as follows: Let \(y = \sinh x\) \(y = \frac{1}{2}(e^x - e^{-x})\) \(2y = e^x - e^{-x}\) \(0 = e^x - 2y - e^{-x}\) \(0 = (e^x)^2 - 2ye^x - 1\) \(0 = (e^x - y)^2 - y^2 - 1\) \(y^2 + 1 = (e^x - y)^2\) \(\pm \sqrt{y^2 + 1} = e^x - y\) \(y + \sqrt{y^2 + 1} = e^x\) To find the inverse function, swap \(x\) and \(y\): \(x + \sqrt{x^2 + 1} = e^y\) \(\ln\left(x + \sqrt{x^2 + 1}\right) = y\) \(\text{arsinh } x = \ln\left(x + \sqrt{x^2 + 1}\right)\) Identify, and explain, the error in Matthew's proof. [2 marks]
2. Solve \(\ln\left(x + \sqrt{x^2 + 1}\right) = 3\) [1 mark]

Find two invariant points under the transformation given by \(\begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}\) [2 marks]

\(2 - 3i\) is one root of the equation $$z^3 + mz + 52 = 0$$ where \(m\) is real.

1. Find the other roots. [3 marks]
2. Determine the value of \(m\). [2 marks]

1. Sketch the graph of \(y^2 = 4x\) [1 mark] \includegraphics{figure_9a}
2. Ben is using a 3D printer to make a plastic bowl which holds exactly \(1000\text{cm}^3\) of water. Ben models the bowl as a region which is rotated through \(2\pi\) radians about the \(x\)-axis. He uses the finite region enclosed by the lines \(x = d\) and \(y = 0\) and the curve with equation \(y^2 = 4x\) for \(y \geq 0\)
   1. Find the depth of the bowl to the nearest millimetre. [4 marks]
   2. What assumption has Ben made about the bowl? [1 mark]

1. Prove by induction that, for all integers \(n \geq 1\), $$\sum_{r=1}^{n} r^3 = \frac{1}{4}n^2(n + 1)^2$$ [4 marks]
2. Hence show that $$\sum_{r=1}^{2n} r(r - 1)(r + 1) = n(n + 1)(2n - 1)(2n + 1)$$ [4 marks]

Four finite regions \(A\), \(B\), \(C\) and \(D\) are enclosed by the curve with equation $$y = x^3 - 7x^2 + 11x + 6$$ and the lines \(y = k\), \(x = 1\) and \(x = 4\), as shown in the diagram below. \includegraphics{figure_11} The areas of \(B\) and \(C\) are equal. Find the value of \(k\). [3 marks]

1. Show that the matrix \(\begin{pmatrix} 5 - k & 2 \\ k^3 + 1 & k \end{pmatrix}\) is singular when \(k = 1\). [1 mark]
2. Find the values of \(k\) for which the matrix \(\begin{pmatrix} 5 - k & 2 \\ k^3 + 1 & k \end{pmatrix}\) has a negative determinant. Fully justify your answer. [5 marks]

The graph of the rational function \(y = f(x)\) intersects the \(x\)-axis exactly once at \((-3, 0)\) The graph has exactly two asymptotes, \(y = 2\) and \(x = -1\)

1. Find \(f(x)\) [2 marks]
2. Sketch the graph of the function. [3 marks] \includegraphics{figure_13b}
3. Find the range of values of \(x\) for which \(f(x) \leq 5\) [4 marks]

1. Sketch, on the Argand diagram below, the locus of points satisfying the equation $$|z - 3| = 2$$ [1 mark] \includegraphics{figure_14a}
2. There is a unique complex number \(w\) that satisfies both $$|w - 3| = 2 \quad \text{and} \quad \arg(w + 1) = \alpha$$ where \(\alpha\) is a constant such that \(0 < \alpha < \pi\)
   1. Find the value of \(\alpha\). [2 marks]
   2. Express \(w\) in the form \(r(\cos \theta + i \sin \theta)\). Give each of \(r\) and \(\theta\) to two significant figures. [4 marks]

1. Show that $$\frac{1}{r + 2} - \frac{1}{r + 3} = \frac{1}{(r + 2)(r + 3)}$$ [1 mark]
2. Use the method of differences to show that $$\sum_{r=1}^{n} \frac{1}{(r + 2)(r + 3)} = \frac{n}{3(n + 3)}$$ [3 marks]

Two matrices \(\mathbf{A}\) and \(\mathbf{B}\) satisfy the equation $$\mathbf{AB} = \mathbf{I} + 2\mathbf{A}$$ where \(\mathbf{I}\) is the identity matrix and \(\mathbf{B} = \begin{pmatrix} 3 & -2 \\ -4 & 8 \end{pmatrix}\) Find \(\mathbf{A}\). [3 marks]

Find the exact solution to the equation $$\sinh \theta(\sinh \theta + \cosh \theta) = 1$$ [4 marks]

\(\alpha\), \(\beta\) and \(\gamma\) are the real roots of the cubic equation $$x^3 + mx^2 + nx + 2 = 0$$ By considering \((\alpha - \beta)^2 + (\gamma - \alpha)^2 + (\beta - \gamma)^2\), prove that $$m^2 \geq 3n$$ [4 marks]

A theme park has two zip wires. Sarah models the two zip wires as straight lines using coordinates in metres. The ends of one wire are located at \((0, 0, 0)\) and \((0, 100, -20)\) The ends of the other wire are located at \((10, 0, 20)\) and \((-10, 100, -5)\)

1. Use Sarah's model to find the shortest distance between the zip wires. [7 marks]
2. State one way in which Sarah's model could be refined. [1 mark]