| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 1 (Further Pure Core 1) |
| Year | 2018 |
| Session | March |
| Marks | 10 |
| Topic | Linear transformations |
| Type | Find invariant lines through origin |
| Difficulty | Standard +0.3 This is a multi-part question covering standard Further Maths topics (matrix multiplication, commutativity, determinants, and invariant lines via eigenvalues). Part (iv) requires finding eigenvalues and relating them to lines y=mx, which is a routine Further Maths technique. While more advanced than basic A-level, it's a straightforward application of learned methods without requiring novel insight. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03c Matrix multiplication: properties (associative, not commutative)4.03g Invariant points and lines4.03h Determinant 2x2: calculation4.03i Determinant: area scale factor and orientation |
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 $\mathbf { B } = \left( \begin{array} { l l } 4 & 2 \\ 3 & 3 \end{array} \right)$.\\
(i) Find the value of $a$ such that $\mathbf { A B } = \mathbf { B A }$.\\
(ii) Prove by counter example that matrix multiplication for $2 \times 2$ matrices is not commutative.\\
(iii) 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.\\
(iv) Find the equations of the invariant lines of the form $y = m x$ for the transformation represented by matrix $\mathbf { B }$.
\hfill \mbox{\textit{OCR Further Pure Core 1 2018 Q2 [10]}}