Find invariant lines through origin

A question is this type if and only if it asks to find lines of the form y = mx that are mapped onto themselves (possibly with reversed direction) under a transformation.

2 questions · Standard +0.0

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Edexcel CP AS 2019 June Q1
6 marks Moderate -0.3
1. $$\mathbf { M } = \left( \begin{array} { l l } 4 & - 5 \\ 2 & - 7 \end{array} \right)$$
  1. Show that the matrix \(\mathbf { M }\) is non-singular. The transformation \(T\) of the plane is represented by the matrix \(\mathbf { M }\).
    The triangle \(R\) is transformed to the triangle \(S\) by the transformation \(T\).
    Given that the area of \(S\) is 63 square units,
  2. find the area of \(R\).
  3. Show that the line \(y = 2 x\) is invariant under the transformation \(T\).
OCR Further Pure Core 1 2018 March Q2
10 marks Standard +0.3
2 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 1 & a \\ 3 & 0 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 4 & 2 \\ 3 & 3 \end{array} \right)\).
  1. Find the value of \(a\) such that \(\mathbf { A B } = \mathbf { B A }\).
  2. Prove by counter example that matrix multiplication for \(2 \times 2\) matrices is not commutative.
  3. A triangle of area 4 square units is transformed by the matrix B. Find the area of the image of the triangle following this transformation.
  4. Find the equations of the invariant lines of the form \(y = m x\) for the transformation represented by matrix \(\mathbf { B }\).