Moderate -0.8 This question tests standard recognition of transformation matrices (stretch and rotation) and basic matrix multiplication. Part (i) requires identifying a diagonal matrix as a stretch and recognizing rotation matrix entries, then performing routine matrix multiplication. Part (ii) is a straightforward discriminant calculation to show a quadratic has no real roots. All techniques are direct applications of bookwork with no problem-solving insight required.
Describe fully the single transformation represented by the matrix \(\mathbf { A }\).
Describe fully the single transformation represented by the matrix \(\mathbf { B }\).
The transformation represented by \(\mathbf { A }\) followed by the transformation represented by \(\mathbf { B }\) is equivalent to the transformation represented by the matrix \(\mathbf { C }\).
Find \(\mathbf { C }\).
\(\mathbf { M } = \left( \begin{array} { c c } 2 k + 5 & - 4 \\ 1 & k \end{array} \right)\), where \(k\) is a real number.
Show that \(\operatorname { det } \mathbf { M } \neq 0\) for all values of \(k\).
6.\\
(i)
$$\mathbf { A } = \left( \begin{array} { l l }
3 & 0 \\
0 & 1
\end{array} \right) \quad \mathbf { B } = \left( \begin{array} { r r }
- \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \\
- \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 }
\end{array} \right)$$
\begin{enumerate}[label=(\alph*)]
\item Describe fully the single transformation represented by the matrix $\mathbf { A }$.
\item Describe fully the single transformation represented by the matrix $\mathbf { B }$.
The transformation represented by $\mathbf { A }$ followed by the transformation represented by $\mathbf { B }$ is equivalent to the transformation represented by the matrix $\mathbf { C }$.
\item Find $\mathbf { C }$.\\
(ii) $\mathbf { M } = \left( \begin{array} { c c } 2 k + 5 & - 4 \\ 1 & k \end{array} \right)$, where $k$ is a real number.
Show that $\operatorname { det } \mathbf { M } \neq 0$ for all values of $k$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2015 Q6 [10]}}