Combined and inverse transformationsEdexcel International A Level Further Maths: Subtopic test

1. 1
   The matrix $\mathbf{P}=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ represents a rotation through $90^\circ$ anticlockwise about $O$, and the matrix $\mathbf{Q}=\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}$ represents a stretch parallel to the $x$-axis with scale factor $2$.

   (a)

   Which matrix represents the transformation $\mathbf{P}$ followed by $\mathbf{Q}$?

   [1 mark]

   • A$\begin{pmatrix} 0 & -2 \\ 1 & 0 \end{pmatrix}$
   • B$\begin{pmatrix} 0 & -1 \\ 2 & 0 \end{pmatrix}$
   • C$\begin{pmatrix} 2 & -1 \\ 1 & 1 \end{pmatrix}$
   • D$\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$

   (b)

   Find the image of the point $(1,1)$ under $\mathbf{P}$ followed by $\mathbf{Q}$.

   [1 mark]

   • A$(-1,2)$
   • B$(-1,1)$
   • C$(-2,1)$
   • D$(2,1)$

   (c)

   Find the single matrix that represents $\mathbf{Q}$ followed by $\mathbf{P}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The matrix $\mathbf{M}=\begin{pmatrix} 4 & 3 \\ 2 & 2 \end{pmatrix}$ represents a transformation of the plane.

   (a)

   Find the determinant of $\mathbf{M}$.

   [1 mark]

   • A$14$
   • B$-2$
   • C$8$
   • D$2$

   (b)

   Find $\mathbf{M}^{-1}$.

   [1 mark]

   • A$\begin{pmatrix} 2 & -3 \\ -2 & 4 \end{pmatrix}$
   • B$\begin{pmatrix} 1 & -\frac32 \\ -1 & 2 \end{pmatrix}$
   • C$\begin{pmatrix} 1 & \frac32 \\ 1 & 2 \end{pmatrix}$
   • D$\begin{pmatrix} \frac14 & \frac13 \\ \frac12 & \frac12 \end{pmatrix}$

   (c)

   A region of area $7\text{ cm}^2$ is transformed by $\mathbf{M}$. Find the area of its image.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A triangle $T$ has vertices $O(0,0)$, $A(2,0)$ and $B(0,3)$. The matrix $\mathbf{N}=\begin{pmatrix} 1 & 2 \\ -1 & 3 \end{pmatrix}$ represents a transformation of the plane, and $T$ is mapped onto triangle $T'$.

   (a)

   Find the coordinates of $A'$ and $B'$, the images of $A$ and $B$.

   [3 marks]

   (b)

   Find $\mathbf{N}^{-1}$ and hence find the coordinates of the point that is mapped by $\mathbf{N}$ onto $(7,4)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The matrix $\mathbf{A}=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ represents reflection in the line $y=x$, and the matrix $\mathbf{B}=\begin{pmatrix} 1 & 0 \\ 0 & 3 \end{pmatrix}$ represents a stretch parallel to the $y$-axis with scale factor $3$.

   (a)

   The transformation $\mathbf{A}$ followed by $\mathbf{B}$ is represented by $\mathbf{C}$.
   (i) Find $\mathbf{C}$.
   (ii) Find $\det\mathbf{C}$ and state the area scale factor of the transformation and whether it reverses orientation.
   (iii) Find $\mathbf{C}^{-1}$.

   [6 marks]

   (b)

   (i) Find the matrix that represents $\mathbf{B}$ followed by $\mathbf{A}$.
   (ii) Show that this combined transformation maps the line $y=x$ onto the line $y=\frac13x$.
   (iii) Without multiplying the matrices again, state with a reason whether $\mathbf{A}$ followed by $\mathbf{B}$ and $\mathbf{B}$ followed by $\mathbf{A}$ have the same area scale factor.

   [6 marks]

   Total for question 4: 12 marks

End of questions