AQA Further AS Paper 1 (Further AS Paper 1) 2020 June

Express the complex number \(1 - i\sqrt{3}\) in modulus-argument form. Tick \((\checkmark)\) one box. [1 mark] \(2\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)\) \(2\left(\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3}\right)\) \(2\left(\cos\left(-\frac{\pi}{3}\right) + i\sin\left(-\frac{\pi}{3}\right)\right)\) \(2\left(\cos\left(-\frac{2\pi}{3}\right) + i\sin\left(-\frac{2\pi}{3}\right)\right)\)

Given that \(1 - i\) is a root of the equation \(z^3 - 3z^2 + 4z - 2 = 0\), find the other two roots. Tick \((\checkmark)\) one box. [1 mark] \(-1 + i\) and \(-1\) \(1 + i\) and \(1\) \(-1 + i\) and \(1\) \(1 + i\) and \(-1\)

Given \((x - 1)(x - 2)(x - a) < 0\) and \(a > 2\) Find the set of possible values of \(x\). Tick \((\checkmark)\) one box. [1 mark] \(\{x : x < 1\} \cup \{x : 2 < x < a\}\) \(\{x : 1 < x < 2\} \cup \{x : x > a\}\) \(\{x : x < -a\} \cup \{x : -2 < x < -1\}\) \(\{x : -a < x < -2\} \cup \{x : x > -1\}\)

The matrices \(\mathbf{A}\) and \(\mathbf{B}\) are such that $$\mathbf{A} = \begin{bmatrix} 2 & a & 3 \\ 0 & -2 & 1 \end{bmatrix} \quad \text{and} \quad \mathbf{B} = \begin{bmatrix} 1 & -3 \\ -2 & 4a \\ 0 & 5 \end{bmatrix}$$

1. Find the product \(\mathbf{AB}\) in terms of \(a\). [2 marks]
2. Find the determinant of \(\mathbf{AB}\) in terms of \(a\). [1 mark]
3. Show that \(\mathbf{AB}\) is singular when \(a = -1\) [2 marks]

1. Show that $$r^2(r + 1)^2 - (r - 1)^2r^2 = pr^3$$ where \(p\) is an integer to be found. [1 mark]
2. Hence use the method of differences to show that $$\sum_{r=1}^{n} r^3 = \frac{1}{4}n^2(n + 1)^2$$ [3 marks]

Anna has been asked to describe the transformation given by the matrix $$\begin{bmatrix} 1 & 0 & 0 \\ 0 & -\frac{\sqrt{3}}{2} & -\frac{1}{2} \\ 0 & \frac{1}{2} & -\frac{\sqrt{3}}{2} \end{bmatrix}$$ She writes her answer as follows: The transformation is a rotation about the \(x\)-axis through an angle of \(\theta\), where $$\sin \theta = \frac{1}{2} \quad \text{and} \quad -\sin \theta = -\frac{1}{2}$$ $$\theta = 30°$$ Identify and correct the error in Anna's work. [2 marks]

Prove by induction that, for all integers \(n \geq 1\), the expression \(7^n - 3^n\) is divisible by 4 [4 marks]

1. Prove that $$\tanh^{-1} x = \frac{1}{2}\ln\left(\frac{1 + x}{1 - x}\right)$$ [5 marks]
2. Prove that the graphs of $$y = \sinh x \quad \text{and} \quad y = \cosh x$$ do not intersect. [3 marks]

The quadratic equation \(2x^2 + px + 3 = 0\) has two roots, \(\alpha\) and \(\beta\), where \(\alpha > \beta\).

1. 1. Write down the value of \(\alpha\beta\). [1 mark]
   2. Express \(\alpha + \beta\) in terms of \(p\). [1 mark]
2. Hence find \((\alpha - \beta)^2\) in terms of \(p\). [2 marks]
3. Hence find, in terms of \(p\), a quadratic equation with roots \(\alpha - 1\) and \(\beta + 1\) [4 marks]

1. Show that the equation $$y = \frac{3x - 5}{2x + 4}$$ can be written in the form $$(x + a)(y + b) = c$$ where \(a\), \(b\) and \(c\) are integers to be found. [3 marks]
2. Write down the equations of the asymptotes of the graph of $$y = \frac{3x - 5}{2x + 4}$$ [2 marks]
3. Sketch, on the axes provided, the graph of $$y = \frac{3x - 5}{2x + 4}$$ \includegraphics{figure_10} [3 marks]

Sketch the polar graph of $$r = \sinh \theta + \cosh \theta$$ for \(0 \leq \theta \leq 2\pi\) \includegraphics{figure_11} [3 marks]

The mean value of the function \(\mathbf{f}\) over the interval \(1 \leq x \leq 5\) is \(m\). The graph of \(y = \mathbf{g}(x)\) is a reflection in the \(x\)-axis of \(y = \mathbf{f}(x)\). The graph of \(y = \mathbf{h}(x)\) is a translation of \(y = \mathbf{g}(x)\) by \(\begin{bmatrix} 3 \\ 7 \end{bmatrix}\) Determine, in terms of \(m\), the mean value of the function \(\mathbf{h}\) over the interval \(4 \leq x \leq 8\) [2 marks]

Line \(l_1\) has equation $$\frac{x - 2}{3} = \frac{1 - 2y}{4} = -z$$ and line \(l_2\) has equation $$\mathbf{r} = \begin{bmatrix} -7 \\ 4 \\ -2 \end{bmatrix} + \mu \begin{bmatrix} 12 \\ a + 3 \\ 2b \end{bmatrix}$$

1. In the case when \(l_1\) and \(l_2\) are parallel, show that \(a = -11\) and find the value of \(b\). [4 marks]
2. In a different case, the lines \(l_1\) and \(l_2\) intersect at exactly one point, and the value of \(b\) is 3 Find the value of \(a\). [5 marks]

1. Given $$\frac{x + 7}{x + 1} \leq x + 1$$ show that $$\frac{(x + a)(x + b)}{x + c} \geq 0$$ where \(a\), \(b\), and \(c\) are integers to be found. [4 marks]
2. Briefly explain why this statement is incorrect. $$\frac{(x + p)(x + q)}{x + r} \geq 0 \Leftrightarrow (x + p)(x + q)(x + r) \geq 0$$ [1 mark]
3. Solve $$\frac{x + 7}{x + 1} \leq x + 1$$ [2 marks]

A segment of the line \(y = kx\) is rotated about the \(x\)-axis to generate a cone with vertex \(O\). The distance of \(O\) from the centre of the base of the cone is \(h\). The radius of the base of the cone is \(r\). \includegraphics{figure_15}

1. Find \(k\) in terms of \(r\) and \(h\). [1 mark]
2. Use calculus to prove that the volume of the cone is $$\frac{1}{3}\pi r^2 h$$ [3 marks]

\(\mathbf{A}\) and \(\mathbf{B}\) are non-singular square matrices.

1. Write down the product \(\mathbf{AA}^{-1}\) as a single matrix. [1 mark]
2. \(\mathbf{M}\) is a matrix such that \(\mathbf{M} = \mathbf{AB}\). Prove that \(\mathbf{M}^{-1} = \mathbf{B}^{-1}\mathbf{A}^{-1}\) [3 marks]

The polar equation of the circle \(C\) is $$r = a(\cos \theta + \sin \theta)$$ Find, in terms of \(a\), the radius of \(C\). Fully justify your answer. [4 marks]

The locus of points \(L_1\) satisfies the equation \(|z| = 2\) The locus of points \(L_2\) satisfies the equation \(\arg(z + 4) = \frac{\pi}{4}\)

1. Sketch \(L_1\) on the Argand diagram below. \includegraphics{figure_18} [1 mark]
2. Sketch \(L_2\) on the Argand diagram above. [1 mark]
3. The complex number \(a + ib\), where \(a\) and \(b\) are real, lies on \(L_1\) The complex number \(c + id\), where \(c\) and \(d\) are real, lies on \(L_2\) Calculate the least possible value of the expression $$(c - a)^2 + (d - b)^2$$ [3 marks]