| Exam Board | OCR |
|---|---|
| Module | FP1 AS (Further Pure 1 AS) |
| Year | 2021 |
| Session | June |
| Marks | 7 |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Find line of invariant points |
| Difficulty | Standard +0.3 This is a multi-part question covering standard FP1 topics (determinants, invariant points/lines) with mostly routine calculations. Part (a) requires finding a matrix from given conditions (image point and area scale factor), part (b) involves solving (B-λI)x=0 for λ=1, and part (c) verifies an invariant line property by substitution. While it requires understanding of transformations and eigenvalue concepts, the techniques are straightforward applications of standard methods with no novel problem-solving required. Slightly easier than average A-level due to being heavily scaffolded. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03h Determinant 2x2: calculation4.03m det(AB) = det(A)*det(B) |
3 The matrices $\mathbf { A }$ and $\mathbf { B }$ are given by $\mathbf { A } = \left( \begin{array} { r r } t & 6 \\ t & - 2 \end{array} \right)$ and $\mathbf { B } = \left( \begin{array} { r r } 2 t & 4 \\ t & - 2 \end{array} \right)$ where $t$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Show that $| \mathrm { A } | = | \mathrm { B } |$.
\item Verify that $| \mathrm { AB } | = | \mathrm { A } | | \mathrm { B } |$.
\item Given that $| \mathbf { A B } | = - 1$ explain what this means about the constant $t$.
The $2 \times 2$ matrix $A$ represents a transformation $T$ which has the following properties.
\begin{itemize}
\item The image of the point $( 0,1 )$ is the point $( 3,4 )$.
\item An object shape whose area is 7 is transformed to an image shape whose area is 35 .
\item T has a line of invariant points.\\
(a) Find a possible matrix for $\mathbf { A }$.
\end{itemize}
The transformation $S$ is represented by the matrix $B$ where $B = \left( \begin{array} { l l } 3 & 1 \\ 2 & 2 \end{array} \right)$.\\
(b) Find the equation of the line of invariant points of S .\\
(c) Show that any line of the form $y = x + c$ is an invariant line of S .
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 AS 2021 Q3 [7]}}