Standard +0.3 This is a straightforward Further Maths question testing standard recall of transformation matrices and basic determinant properties. Part (i) requires writing down standard matrices (rotation and stretch) and multiplying them—routine matrix multiplication. Part (ii) involves computing a 2×2 determinant algebraically and applying the area scale factor property (area scales by |det M|). All techniques are direct applications of learned formulas with no novel problem-solving required, making it slightly easier than average even for Further Maths.
In part (i), the elements of each matrix should be expressed in exact numerical form.
(a) Write down the \(2 \times 2\) matrix that represents a rotation of \(210 ^ { \circ }\) anticlockwise about the origin.
(b) Write down the \(2 \times 2\) matrix that represents a stretch parallel to the \(y\)-axis with scale factor 5
The transformation \(T\) is a rotation of \(210 ^ { \circ }\) anticlockwise about the origin followed by a stretch parallel to the \(y\)-axis with scale factor 5
(c) Determine the \(2 \times 2\) matrix that represents \(T\)
$$\mathbf { M } = \left( \begin{array} { r r }
k & k + 3 \\
- 5 & 1 - k
\end{array} \right) \quad \text { where } k \text { is a constant }$$
(a) Find det \(\mathbf { M }\), giving your answer in simplest form in terms of \(k\).
A closed shape \(R\) is transformed to a closed shape \(R ^ { \prime }\) by the transformation represented by the matrix \(\mathbf { M }\).
Given that the area of \(R\) is 2 square units and that the area of \(R ^ { \prime }\) is \(16 k\) square units,
(b) determine the possible values of \(k\).
\begin{enumerate}
\item In part (i), the elements of each matrix should be expressed in exact numerical form.\\
(i) (a) Write down the $2 \times 2$ matrix that represents a rotation of $210 ^ { \circ }$ anticlockwise about the origin.\\
(b) Write down the $2 \times 2$ matrix that represents a stretch parallel to the $y$-axis with scale factor 5
\end{enumerate}
The transformation $T$ is a rotation of $210 ^ { \circ }$ anticlockwise about the origin followed by a stretch parallel to the $y$-axis with scale factor 5\\
(c) Determine the $2 \times 2$ matrix that represents $T$\\
(ii)
$$\mathbf { M } = \left( \begin{array} { r r }
k & k + 3 \\
- 5 & 1 - k
\end{array} \right) \quad \text { where } k \text { is a constant }$$
(a) Find det $\mathbf { M }$, giving your answer in simplest form in terms of $k$.
A closed shape $R$ is transformed to a closed shape $R ^ { \prime }$ by the transformation represented by the matrix $\mathbf { M }$.
Given that the area of $R$ is 2 square units and that the area of $R ^ { \prime }$ is $16 k$ square units,\\
(b) determine the possible values of $k$.\\
\hfill \mbox{\textit{Edexcel F1 2021 Q7 [9]}}