OCR Further Pure Core 2 (Further Pure Core 2) 2021 June

In this question you must show detailed reasoning. S is the 2-D transformation which is a stretch of scale factor 3 parallel to the x-axis. A is the matrix which represents S.

1. Write down A. [1]
2. By considering the transformation represented by \(\mathbf{A}^{-1}\), determine the matrix \(\mathbf{A}^{-1}\). [2]

Matrix B is given by \(\mathbf{B} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\). T is the transformation represented by B.

1. Describe T. [1]
2. Determine the matrix which represents the transformation S followed by T. [2]
3. Demonstrate, by direct calculation, that \((\mathbf{BA})^{-1} = \mathbf{A}^{-1}\mathbf{B}^{-1}\). [2]

1. Find the shortest distance between the point \((-6, 4)\) and the line \(y = -0.75x + 7\). [2]

Two lines, \(l_1\) and \(l_2\), are given by $$l_1: \mathbf{r} = \begin{pmatrix} 4 \\ 3 \\ -2 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 1 \\ -4 \end{pmatrix} \text{ and } l_2: \mathbf{r} = \begin{pmatrix} 11 \\ -1 \\ 5 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ -1 \\ 1 \end{pmatrix}$$

1. Find the shortest distance between \(l_1\) and \(l_2\). [3]
2. Hence determine the geometrical arrangement of \(l_1\) and \(l_2\). [2]

Three matrices, A, B and C, are given by \(\mathbf{A} = \begin{pmatrix} 1 & 2 \\ a & -1 \end{pmatrix}\), \(\mathbf{B} = \begin{pmatrix} 2 & -1 \\ 4 & 1 \end{pmatrix}\) and \(\mathbf{C} = \begin{pmatrix} 5 & 0 \\ -2 & 2 \end{pmatrix}\) where \(a\) is a constant.

1. Using A, B and C in that order demonstrate explicitly the associativity property of matrix multiplication. [4]
2. Use A and C to disprove by counterexample the proposition 'Matrix multiplication is commutative'. [2]

For a certain value of \(a\), \(\mathbf{A}\begin{pmatrix} x \\ y \end{pmatrix} = 3\begin{pmatrix} x \\ y \end{pmatrix}\)

1. Find

\includegraphics{figure_4} The figure shows part of the graph of \(y = (x - 3)\sqrt{\ln x}\). The portion of the graph below the x-axis is rotated by \(2\pi\) radians around the x-axis to form a solid of revolution, S. Determine the exact volume of S. [7]

\(C\) is the locus of numbers, \(z\), for which \(\ln\left(\frac{z + 7i}{z - 24}\right) = \frac{1}{4}\). By writing \(z = x + iy\) give a complete description of the shape of \(C\) on an Argand diagram. [7]