Verify invariant line property

A question is this type if and only if it asks to show or verify that a given line (e.g., y = kx) is invariant under a transformation, or to find k such that a line is invariant.

3 questions · Standard +0.3

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OCR Further Pure Core 1 2021 November Q9
5 marks Standard +0.3
9 You are given that the matrix \(\left( \begin{array} { c c } 2 & 1 \\ - 1 & 0 \end{array} \right)\) represents a transformation T .
  1. You are given that the line with equation \(\mathrm { y } = \mathrm { kx }\) is invariant under T . Determine the value of \(k\).
  2. Determine whether the line with equation \(\mathrm { y } = \mathrm { kx }\) in part (a) is a line of invariant points under T .
OCR MEI Further Pure Core 2021 November Q6
4 marks Standard +0.3
6 Given that \(y = m x\) is an invariant line of the transformation with matrix \(\left( \begin{array} { r r } 1 & 2 \\ 2 & - 2 \end{array} \right)\), determine the possible values of \(m\). Section B (113 marks)
Answer all the questions.
OCR MEI Further Pure Core Specimen Q3
6 marks Standard +0.3
3 Transformation M is represented by matrix \(\mathbf { M } = \left( \begin{array} { l l } 2 & 3 \\ 1 & 4 \end{array} \right)\).
  1. On the diagram in the Printed Answer Booklet draw the image of the unit square under M .
  2. (A) Show that there is a constant \(k\) such that \(\mathbf { M } \binom { x } { k x } = 5 \binom { x } { k x }\) for all \(x\).
    (B) Hence find the equation of an invariant line under M .
    (C) Draw the invariant line from part (ii) (B) on your diagram for part (i).