3.9 Matrix transformationsIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   Transformation $T$ is represented by the matrix $\mathbf{M}=\begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}$, so that a point $(x,y)$ maps to the point with position vector $\mathbf{M}\begin{pmatrix}x \\ y\end{pmatrix}$.

   (a)

   Find the image of the point $(3,2)$ under $T$.

   [1 mark]

   • A$\begin{pmatrix}2 \\ -3\end{pmatrix}$
   • B$\begin{pmatrix}-3 \\ -2\end{pmatrix}$
   • C$\begin{pmatrix}-2 \\ 3\end{pmatrix}$
   • D$\begin{pmatrix}3 \\ -2\end{pmatrix}$

   (b)

   Which of the following describes $T$?

   [1 mark]

   • AA reflection in the line $y=x$
   • BA rotation of $90^\circ$ anticlockwise about the origin
   • CA rotation of $90^\circ$ clockwise about the origin
   • DA reflection in the $y$-axis

   (c)

   $T$ is followed by a reflection in the $x$-axis, which has matrix $\mathbf{R}=\begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}$. Find the single matrix that represents this combined transformation.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Triangle $S$ has area $8$ cm$^2$. It is transformed by the matrix $\mathbf{N}=\begin{pmatrix}3 & 1 \\ 2 & 2\end{pmatrix}$ to give triangle $S'$.

   (a)

   Find the area of $S'$.

   [1 mark]

   • A$32$ cm$^2$
   • B$12$ cm$^2$
   • C$2$ cm$^2$
   • D$4$ cm$^2$

   (b)

   A second transformation has matrix $\begin{pmatrix}1 & 0 \\ 0 & 3\end{pmatrix}$. This transformation is

   [1 mark]

   • Aa horizontal stretch with scale factor $3$
   • Ban enlargement with scale factor $3$, centre the origin
   • Ca translation of $3$ units in the positive $y$ direction
   • Da vertical stretch with scale factor $3$

   (c)

   $S'$ is then enlarged by scale factor $2$, centre the origin, to give triangle $S''$. Find the area of $S''$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A transformation $U$ maps a point with position vector $\begin{pmatrix}x \\ y\end{pmatrix}$ to $\begin{pmatrix}2 & 0 \\ 0 & 2\end{pmatrix}\begin{pmatrix}x \\ y\end{pmatrix}+\begin{pmatrix}1 \\ -3\end{pmatrix}$. Triangle $ABC$ has vertices $A(1,2)$, $B(4,2)$ and $C(1,6)$, with lengths in cm.

   (a)

   Find the coordinates of the images $A'$, $B'$ and $C'$ of the vertices of triangle $ABC$ under $U$.

   [3 marks]

   (b)

   (i) Find the area of triangle $ABC$.
   (ii) Write down the determinant of $\begin{pmatrix}2 & 0 \\ 0 & 2\end{pmatrix}$.
   (iii) Hence find the area of triangle $A'B'C'$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A designer plots a spiral of points. The first point is $P_0=(8,0)$ and each point, written as a column vector, is found from the previous one using $P_{n+1}=\mathbf{R}P_n$, where $\mathbf{R}=\begin{pmatrix}0 & -0.5 \\ 0.5 & 0\end{pmatrix}$. Coordinates are in cm.

   (a)

   (i) Find the coordinates of $P_1$ and $P_2$.
   (ii) $\mathbf{R}$ represents a rotation followed by an enlargement, both with centre the origin. Describe each fully.
   (iii) A triangle of area $12$ cm$^2$ is transformed by $\mathbf{R}$. Find the area of its image.

   [6 marks]

   (b)

   (i) Find the distance $P_0P_1$.
   (ii) Explain why the distance $P_{n+1}P_{n+2}$ is half the distance $P_nP_{n+1}$ for every $n$.
   (iii) Hence find the total length of the infinite path $P_0P_1P_2P_3\ldots$

   [6 marks]

   Total for question 4: 12 marks

End of questions