Moderate -0.3 This is a straightforward Further Maths FP1 question requiring students to find invariant points by solving (M-I)x=0, which is a standard technique. While it's a Further Maths topic (making it harder than typical A-level), it's a routine application of a well-practiced method with minimal steps, placing it slightly below average difficulty overall.
3 Find the equation of the line of invariant points under the transformation given by the matrix \(\mathbf { M } = \left( \begin{array} { r r } 3 & - 1 \\ 2 & 0 \end{array} \right)\).
\(\begin{pmatrix} 3 & -1 \\ 2 & 0 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix} \Rightarrow x = 3x - y,\ y = 2x\)
M1, A1
M1 for \(\begin{pmatrix} 3 & -1 \\ 2 & 0 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix}\) (allow if implied). \(\begin{pmatrix} 3 & -1 \\ 2 & 0 \end{pmatrix}\begin{pmatrix} k \\ mk \end{pmatrix} = \begin{pmatrix} K \\ mK \end{pmatrix}\) can lead to full marks if correctly used
\(\Rightarrow y = 2x\)
A1 [3]
Lose second A1 if answer includes two lines
## Question 3:
$\begin{pmatrix} 3 & -1 \\ 2 & 0 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix} \Rightarrow x = 3x - y,\ y = 2x$ | M1, A1 | M1 for $\begin{pmatrix} 3 & -1 \\ 2 & 0 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix}$ (allow if implied). $\begin{pmatrix} 3 & -1 \\ 2 & 0 \end{pmatrix}\begin{pmatrix} k \\ mk \end{pmatrix} = \begin{pmatrix} K \\ mK \end{pmatrix}$ can lead to full marks if correctly used
$\Rightarrow y = 2x$ | A1 **[3]** | Lose second A1 if answer includes two lines
---
3 Find the equation of the line of invariant points under the transformation given by the matrix $\mathbf { M } = \left( \begin{array} { r r } 3 & - 1 \\ 2 & 0 \end{array} \right)$.
\hfill \mbox{\textit{OCR MEI FP1 2005 Q3 [3]}}