| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2018 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Matrices |
| Type | Matrix satisfying given equation |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question testing standard matrix operations. Part (i) involves routine inverse and squaring of 2×2 matrices, then solving a simple equation for k. Part (ii) requires decomposing a transformation matrix into stretch and rotation components using standard formulas. All techniques are direct applications of F1 syllabus content with no novel problem-solving required, making it slightly easier than average even for Further Maths. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products4.03n Inverse 2x2 matrix4.03o Inverse 3x3 matrix |
7. (i)
$$\mathbf { A } = \left( \begin{array} { r r }
6 & k \\
- 3 & - 4
\end{array} \right) , \text { where } k \text { is a real constant, } k \neq 8$$
Find, in terms of $k$,
\begin{enumerate}[label=(\alph*)]
\item $\mathbf { A } ^ { - 1 }$
\item $\mathbf { A } ^ { 2 }$
Given that $\mathbf { A } ^ { 2 } + 3 \mathbf { A } ^ { - 1 } = \left( \begin{array} { r r } 5 & 9 \\ - 3 & - 5 \end{array} \right)$
\item find the value of $k$.\\
(ii)
$$\mathbf { M } = \left( \begin{array} { c c }
- \frac { 1 } { 2 } & - \sqrt { 3 } \\
\frac { \sqrt { 3 } } { 2 } & - 1
\end{array} \right)$$
The matrix $\mathbf { M }$ represents a one way stretch, parallel to the $y$-axis, scale factor $p$, where $p > 0$, followed by a rotation anticlockwise through an angle $\theta$ about $( 0,0 )$.\\
(a) Find the value of $p$.\\
(b) Find the value of $\theta$.
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\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2018 Q7 [11]}}