FP1: Transformations using matricesEdexcel International A Level Further Maths: Topic test

1. 1
   A linear transformation $U$ of the plane maps $(1,0)$ to $(0,-2)$ and $(0,1)$ to $(3,1)$.

   (a)

   Which matrix represents $U$?

   [1 mark]

   • A$\begin{pmatrix}0&-2\\ 3&1\end{pmatrix}$
   • B$\begin{pmatrix}3&0\\ 1&-2\end{pmatrix}$
   • C$\begin{pmatrix}0&3\\ -2&1\end{pmatrix}$
   • D$\begin{pmatrix}-2&1\\ 0&3\end{pmatrix}$

   (b)

   Find the image of the point $(2,-1)$ under $U$.

   [1 mark]

   • A$(-3,-5)$
   • B$(2,5)$
   • C$(-5,-3)$
   • D$(-3,-3)$

   (c)

   Find the coordinates of the point that $U$ maps to $(6,-1)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Transformation $F$ is a reflection in the line $y=-x$ and transformation $G$ is an enlargement, centre $(0,0)$, with scale factor $3$.

   (a)

   Which matrix represents $F$?

   [1 mark]

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

   (b)

   Which matrix represents $G$ followed by $F$?

   [1 mark]

   • A$\begin{pmatrix}0&3\\ 3&0\end{pmatrix}$
   • B$\begin{pmatrix}0&-\frac13\\ -\frac13&0\end{pmatrix}$
   • C$\begin{pmatrix}-3&0\\ 0&-3\end{pmatrix}$
   • D$\begin{pmatrix}0&-3\\ -3&0\end{pmatrix}$

   (c)

   The point $P$ has coordinates $(2,5)$. Find the coordinates of the image of $P$ under $G$ followed by $F$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The matrix $\mathbf{M}=\begin{pmatrix}\frac{\sqrt3}{2}&-\frac12\\ \frac12&\frac{\sqrt3}{2}\end{pmatrix}$ represents a single transformation of the plane.

   (a)

   Describe fully the transformation represented by $\mathbf{M}$.

   [3 marks]

   (b)

   Find $\mathbf{M}^{-1}$ and use it to find the coordinates of the point that is mapped by $\mathbf{M}$ to $(\sqrt3,1)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The transformation $S$ is a stretch parallel to the $x$-axis with scale factor $2$, and the transformation $R$ is a rotation through $90^\circ$ clockwise about the origin.

   (a)

   (i) Write down the matrices $\mathbf{S}$ and $\mathbf{R}$. (ii) Find the matrix representing $S$ followed by $R$, and the matrix representing $R$ followed by $S$. (iii) Hence state whether the order of the two transformations affects the result, giving a reason.

   [6 marks]

   (b)

   A triangle of area $7$ is transformed by $S$ followed by $R$. (i) Find the area of its image. (ii) Find the matrix of the transformation that maps the image back to the original triangle, and describe this transformation as a rotation followed by a stretch.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Triangle $T$ has vertices $(0,0)$, $(6,0)$ and $(0,4)$ and is transformed by the matrix $\mathbf{W}=\begin{pmatrix}3&1\\ -1&2\end{pmatrix}$.

   (a)

   Find the image of the vertex $(6,0)$.

   [1 mark]

   • A$(18,-6)$
   • B$(18,6)$
   • C$(3,-1)$
   • D$(-6,18)$

   (b)

   Find the area of the image of $T$.

   [1 mark]

   • A$60$
   • B$72$
   • C$7$
   • D$84$

   (c)

   Show that $\mathbf{W}^{-1}=\frac17\begin{pmatrix}2&-1\\ 1&3\end{pmatrix}$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Transformation $A$ is a stretch parallel to the $y$-axis with scale factor $3$ and transformation $B$ is a reflection in the $x$-axis.

   (a)

   Which matrix represents $A$?

   [1 mark]

   • A$\begin{pmatrix}3&0\\ 0&1\end{pmatrix}$
   • B$\begin{pmatrix}3&0\\ 0&3\end{pmatrix}$
   • C$\begin{pmatrix}1&0\\ 0&3\end{pmatrix}$
   • D$\begin{pmatrix}1&0\\ 0&\frac13\end{pmatrix}$

   (b)

   Which matrix represents $A$ followed by $B$?

   [1 mark]

   • A$\begin{pmatrix}-1&0\\ 0&3\end{pmatrix}$
   • B$\begin{pmatrix}1&0\\ 0&-3\end{pmatrix}$
   • C$\begin{pmatrix}3&0\\ 0&-1\end{pmatrix}$
   • D$\begin{pmatrix}1&0\\ 0&3\end{pmatrix}$

   (c)

   The point $P$ has coordinates $(4,-2)$. Find the image of $P$ under $B$ followed by $A$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The matrix $\mathbf{U}=\begin{pmatrix}2&k\\ 1&3\end{pmatrix}$, where $k$ is a constant, represents a linear transformation under which the area of any region is multiplied by $4$.

   (a)

   Find the two possible values of $k$.

   [3 marks]

   (b)

   For $k=2$, the transformation maps a point $P$ to $(6,5)$. Find $\mathbf{U}^{-1}$ and hence the coordinates of $P$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   $\mathbf{Z}=\begin{pmatrix}0&-2\\ 2&0\end{pmatrix}$ represents an enlargement with centre $(0,0)$ and scale factor $k$ $(k>0)$, followed by a rotation through angle $\theta$ anticlockwise about $(0,0)$, where $0^\circ<\theta<360^\circ$.

   (a)

   (i) Find the value of $k$ and the value of $\theta$. (ii) Find $\det\mathbf{Z}$ and state its geometrical meaning.

   [6 marks]

   (b)

   The transformation $X$ is a reflection in the line $y=x$. (i) Write down the matrix $\mathbf{X}$. (ii) Find the matrix representing $X$ followed by $Z$. (iii) Show that this matrix represents a reflection in the $y$-axis followed by an enlargement with centre $(0,0)$, and find its inverse.

   [6 marks]

   Total for question 8: 12 marks

End of questions