OCR MEI Further Pure Core AS (Further Pure Core AS) 2018 June

The matrices \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\) are defined as follows: $$\mathbf{A} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 2 & 0 & 3 \\ 1 & -1 & 3 \end{pmatrix}, \quad \mathbf{C} = \begin{pmatrix} 1 & 3 \end{pmatrix}.$$ Calculate all possible products formed from two of these three matrices. [4]

Find, to the nearest degree, the angle between the vectors \(\begin{pmatrix} 1 \\ 0 \\ -2 \end{pmatrix}\) and \(\begin{pmatrix} -2 \\ 3 \\ -3 \end{pmatrix}\). [3]

Find real numbers \(a\) and \(b\) such that \((a - 3i)(5 - i) = b - 17i\). [5]

Find a cubic equation with real coefficients, two of whose roots are \(2 - i\) and \(3\). [5]

A transformation of the \(x\)-\(y\) plane is represented by the matrix \(\begin{pmatrix} \cos \theta & 2 \sin \theta \\ 2 \sin \theta & -\cos \theta \end{pmatrix}\), where \(\theta\) is a positive acute angle.

1. Write down the image of the point \((2, 3)\) under this transformation. [2]
2. You are given that this image is the point \((a, 0)\). Find the value of \(a\). [5]

Find the invariant line of the transformation of the \(x\)-\(y\) plane represented by the matrix \(\begin{pmatrix} 2 & 0 \\ 4 & -1 \end{pmatrix}\). [4]

1. Express \(\frac{1}{2r-1} - \frac{1}{2r+1}\) as a single fraction. [2]
2. Find how many terms of the series $$\frac{2}{1 \times 3} + \frac{2}{3 \times 5} + \frac{2}{5 \times 7} + \ldots + \frac{2}{(2r-1)(2r+1)} + \ldots$$ are needed for the sum to exceed \(0.999999\). [7]

Prove by induction that \(\begin{pmatrix} 1 & 1 \\ 0 & 2 \end{pmatrix}^n = \begin{pmatrix} 1 & 2^n - 1 \\ 0 & 2^n \end{pmatrix}\) for all positive integers \(n\). [6]

Fig. 9 shows a sketch of the region OPQ of the Argand diagram defined by $$\left\{z : |z| \leq 4\sqrt{2}\right\} \cap \left\{z : -\frac{1}{4}\pi \leq \arg z \leq \frac{1}{4}\pi\right\}.$$ \includegraphics{figure_9}

1. Find, in modulus-argument form, the complex number represented by the point P. [2]
2. Find, in the form \(a + ib\), where \(a\) and \(b\) are exact real numbers, the complex number represented by the point Q. [3]
3. In this question you must show detailed reasoning. Determine whether the points representing the complex numbers
   lie within this region. [4]

Three planes have equations \begin{align} -x + 2y + z &= 0
2x - y - z &= 0
x + y &= a \end{align} where \(a\) is a constant.

1. Investigate the arrangement of the planes:
   [6]
2. Chris claims that the position vectors \(-\mathbf{i} + 2\mathbf{j} + \mathbf{k}\), \(2\mathbf{i} - \mathbf{j} - \mathbf{k}\) and \(\mathbf{i} + \mathbf{j}\) lie in a plane. Determine whether or not Chris is correct. [2]