Moderate -0.8 This is a straightforward multi-part question on 2D transformations requiring standard matrix operations (reflection, shear, composition) and interpretation of determinants. All parts involve routine recall and application of well-practiced techniques with no novel problem-solving or geometric insight required.
\(P\), \(Q\) and \(T\) are three transformations in 2-D.
\(P\) is a reflection in the \(x\)-axis. \(\mathbf{A}\) is the matrix that represents \(P\).
Write down the matrix \(\mathbf{A}\).
[1]
\(Q\) is a shear in which the \(y\)-axis is invariant and the point \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\) is transformed to the point \(\begin{pmatrix} 1 \\ 2 \end{pmatrix}\). \(\mathbf{B}\) is the matrix that represents \(Q\).
Find the matrix \(\mathbf{B}\).
[2]
\(T\) is \(P\) followed by \(Q\). \(\mathbf{C}\) is the matrix that represents \(T\).
Determine the matrix \(\mathbf{C}\).
[2]
\(L\) is the line whose equation is \(y = x\).
Explain whether or not \(L\) is a line of invariant points under \(T\).
[2]
An object parallelogram, \(M\), is transformed under \(T\) to an image parallelogram, \(N\).
Explain what the value of the determinant of \(\mathbf{C}\) means about
• the area of \(N\) compared to the area of \(M\).
• the orientation of \(N\) compared to the orientation of \(M\).
[3]
\begin{enumerate}[label=(\alph*)]
\item $P$, $Q$ and $T$ are three transformations in 2-D.
$P$ is a reflection in the $x$-axis. $\mathbf{A}$ is the matrix that represents $P$.
Write down the matrix $\mathbf{A}$.
[1]
\item $Q$ is a shear in which the $y$-axis is invariant and the point $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ is transformed to the point $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$. $\mathbf{B}$ is the matrix that represents $Q$.
Find the matrix $\mathbf{B}$.
[2]
\item $T$ is $P$ followed by $Q$. $\mathbf{C}$ is the matrix that represents $T$.
Determine the matrix $\mathbf{C}$.
[2]
\item $L$ is the line whose equation is $y = x$.
Explain whether or not $L$ is a line of invariant points under $T$.
[2]
\item An object parallelogram, $M$, is transformed under $T$ to an image parallelogram, $N$.
Explain what the value of the determinant of $\mathbf{C}$ means about
• the area of $N$ compared to the area of $M$.
• the orientation of $N$ compared to the orientation of $M$.
[3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2025 Q5 [10]}}