AQA Further AS Paper 1 (Further AS Paper 1) 2019 June

Which of the following matrices is an identity matrix? Circle your answer. [1 mark] \(\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}\) \quad \(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\) \quad \(\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\) \quad \(\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}\)

Which of the following expressions is the determinant of the matrix \(\begin{bmatrix} a & 2 \\ b & 5 \end{bmatrix}\)? Circle your answer. [1 mark] \(5a - 2b\) \quad \(2a - 5b\) \quad \(5b - 2a\) \quad \(2b - 5a\)

Point \(P\) has polar coordinates \(\left(2, \frac{2\pi}{3}\right)\). Which of the following are the Cartesian coordinates of \(P\)? Circle your answer. [1 mark] \((1, -\sqrt{3})\) \quad \((-\sqrt{3}, 1)\) \quad \((\sqrt{3}, -1)\) \quad \((-1, \sqrt{3})\)

The line \(L\) has polar equation $$r = \frac{k}{\sin \theta}$$ where \(k\) is a positive constant.

1. Sketch \(L\). [1 mark]
2. State the minimum distance between \(L\) and the point \(O\). [1 mark]

A hyperbola \(H\) has the equation $$\frac{x^2}{a^2} - \frac{y^2}{4a^2} = 1$$ where \(a\) is a positive constant.

1. Write down the equations of the asymptotes of \(H\). [1 mark]
2. Sketch the hyperbola \(H\) on the axes below, indicating the coordinates of any points of intersection with the coordinate axes. The asymptotes have already been drawn. [2 marks]
3. The finite region bounded by \(H\), the positive \(x\)-axis, the positive \(y\)-axis and the line \(y = a\) is rotated through \(360°\) about the \(y\)-axis. Show that the volume of the solid generated is \(ma^3\), where \(m = 3.40\) correct to three significant figures. [5 marks]

1. On the axes provided, sketch the graph of $$x = \cosh(y + b)$$ where \(b\) is a positive constant. [4 marks]
2. Determine the minimum distance between the graph of \(x = \cosh(y + b)\) and the \(y\)-axis. [1 mark]

1. Show that $$\frac{1}{r-1} - \frac{1}{r+1} = \frac{A}{r^2-1}$$ where \(A\) is a constant to be found. [1 mark]
2. Hence use the method of differences to show that $$\sum_{r=2}^n \frac{1}{r^2-1} = \frac{an^2 + bn + c}{4n(n+1)}$$ where \(a\), \(b\) and \(c\) are integers to be found. [4 marks]

Given that \(z_1 = 2\left(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6}\right)\) and \(z_2 = 2\left(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4}\right)\)

1. Find the value of \(|z_1z_2|\) [1 mark]
2. Find the value of \(\arg\left(\frac{z_1}{z_2}\right)\) [1 mark]
3. Sketch \(z_1\) and \(z_2\) on the Argand diagram below, labelling the points as \(P\) and \(Q\) respectively. [2 marks]
4. A third complex number \(w\) satisfies both \(|w| = 2\) and \(-\pi < \arg w < 0\) Given that \(w\) is represented on the Argand diagram as the point \(R\), find the angle \(PRQ\). Fully justify your answer. [3 marks]

1. Saul is solving the equation $$2\cosh x + \sinh^2 x = 1$$ He writes his steps as follows: $$2\cosh x + \sinh^2 x = 1$$ $$2\cosh x + 1 - \cosh^2 x = 1$$ $$2\cosh x - \cosh^2 x = 0$$ $$\cosh x \neq 0 \therefore 2 - \cosh x = 0$$ $$\cosh x = 2$$ $$x = \pm \cosh^{-1}(2)$$ Identify and explain the error in Saul's method. [2 marks]
2. Anna is solving the different equation $$\sinh^2(2x) - 2\cosh(2x) = 1$$ and finds the correct answers in the form \(x = \frac{1}{p}\cosh^{-1}(q + \sqrt{r})\), where \(p\), \(q\) and \(r\) are integers. Find the possible values of \(p\), \(q\) and \(r\). Fully justify your answer. [5 marks]

1. Using the definition of \(\cosh x\) and the Maclaurin series expansion of \(e^x\), find the first three non-zero terms in the Maclaurin series expansion of \(\cosh x\). [3 marks]
2. Hence find a trigonometric function for which the first three terms of its Maclaurin series are the same as the first three terms of the Maclaurin series for \(\cosh(ix)\). [3 marks]

1. Curve \(C\) has equation $$y = \frac{x^2 + px - q}{x^2 - r}$$ where \(p\), \(q\) and \(r\) are positive constants. Write down the equations of its asymptotes. [2 marks]
2. Find the set of possible \(y\)-coordinates for the graph of $$y = \frac{x^2 + x - 6}{x^2 - 1}, \quad x \neq \pm 1$$ giving your answer in exact form. No credit will be given for solutions based on differentiation. [6 marks]

The matrix \(\mathbf{A}\) is given by $$\mathbf{A} = \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}$$

1. Prove by induction that, for all integers \(n \geq 1\), $$\mathbf{A}^n = \begin{bmatrix} 1 & 3^n - 1 \\ 0 & 3^n \end{bmatrix}$$ [4 marks]
2. Find all invariant lines under the transformation matrix \(\mathbf{A}\). Fully justify your answer. [6 marks]
3. Find a line of invariant points under the transformation matrix \(\mathbf{A}\). [2 marks]

Line \(l_1\) has Cartesian equation $$x - 3 = \frac{2y + 2}{3} = 2 - z$$

1. Write the equation of line \(l_1\) in the form $$\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}$$ where \(\lambda\) is a parameter and \(\mathbf{a}\) and \(\mathbf{b}\) are vectors to be found. [2 marks]
2. Line \(l_2\) passes through the points \(P(3, 2, 0)\) and \(Q(n, 5, n)\), where \(n\) is a constant.
   1. Show that the lines \(l_1\) and \(l_2\) are not perpendicular. [3 marks]
   2. Explain briefly why lines \(l_1\) and \(l_2\) cannot be parallel. [2 marks]
   3. Given that \(\theta\) is the acute angle between lines \(l_1\) and \(l_2\), show that $$\cos \theta = \frac{p}{\sqrt{34n^2 + qn + 306}}$$ where \(p\) and \(q\) are constants to be found. [3 marks]

The graph of \(y = x^3 - 3x\) is shown below. \includegraphics{figure_14} The two stationary points have \(x\)-coordinates of \(-1\) and \(1\) The cubic equation $$x^3 - 3x + p = 0$$ where \(p\) is a real constant, has the roots \(\alpha\), \(\beta\) and \(\gamma\). The roots \(\alpha\) and \(\beta\) are not real.

1. Explain why \(\alpha + \beta = -\gamma\) [1 mark]
2. Find the set of possible values for the real constant \(p\). [2 marks]
3. \(f(x) = 0\) is a cubic equation with roots \(\alpha + 1\), \(\beta + 1\) and \(\gamma + 1\)
   1. Show that the constant term of \(f(x)\) is \(p + 2\) [3 marks]
   2. Write down the \(x\)-coordinates of the stationary points of \(y = f(x)\) [1 mark]