Browsing as Guest. Progress, bookmarks and attempts are disabled.
Log in to track your work.
FM June 2023 p11 q04
4202
The matrix M is given by \(\mathbf{M} = \begin{pmatrix} a & b^2 \\ c^2 & a \end{pmatrix}\), where \(a, b, c\) are real constants and \(b \neq 0\).
Show that M does not represent a rotation about the origin.
Find the equations of the invariant lines, through the origin, of the transformation in the x–y plane represented by M.
It is given that M represents the sequence of two transformations in the x–y plane given by an enlargement, centre the origin, scale factor 5 followed by a shear, x-axis fixed, with (0, 1) mapped to (5, 1).
Find M.
The triangle DEF in the x–y plane is transformed by M onto triangle PQR. Given that the area of triangle DEF is 12 cm2, find the area of triangle PQR.
Solution
(a) For a matrix to represent a rotation, it must be of the form \(\begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}\). Comparing with \(\mathbf{M} = \begin{pmatrix} a & b^2 \\ c^2 & a \end{pmatrix}\), we equate \(b^2 = -c^2\), which is impossible for real \(b\) and \(c\) with \(b \neq 0\).
(b) Consider the transformation \(\begin{pmatrix} a & b^2 \\ c^2 & a \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax + b^2y \\ c^2x + ay \end{pmatrix}\). For invariant lines, \(y = mx\) and \(Y = mX\). Solving \(c^2x + amx = m(ax + b^2mx)\) gives \(c^2 + am = ma + b^2m^2 \Rightarrow c^2 = b^2m^2\). Thus, \(y = \frac{c}{b}x\) and \(y = -\frac{c}{b}x\).
(c) The enlargement matrix is \(\begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix}\) and the shear matrix is \(\begin{pmatrix} 1 & 5 \\ 0 & 1 \end{pmatrix}\). The combined transformation is \(\begin{pmatrix} 1 & 5 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix} = \begin{pmatrix} 5 & 25 \\ 0 & 5 \end{pmatrix}\).
(d) The area of triangle PQR is \(12 \times \det(\mathbf{M}) = 12 \times 25 = 300 \text{ cm}^2\).
