Matrix representations of standard transformationsEdexcel International A Level Further Maths: Subtopic test

1. 1
   Single transformations of the plane are represented by $2\times2$ matrices acting on position vectors, with the origin $O$ as the fixed point. The point $P$ has coordinates $(3,-2)$.

   (a)

   Which matrix represents reflection in the line $y=-x$?

   [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$\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$

   (b)

   Find the image of $P$ under reflection in the line $y=x$.

   [1 mark]

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

   (c)

   The stretch parallel to the $x$-axis with scale factor $3$ maps $P$ to $P'$. Write down the matrix of the stretch and find the coordinates of $P'$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The matrix $\mathbf{M}=\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ represents a single transformation $T$ of the plane.

   (a)

   Which of these describes $T$?

   [1 mark]

   • AA rotation through $90^\circ$ anticlockwise about $O$
   • BReflection in the line $y=x$
   • CA rotation through $90^\circ$ clockwise about $O$
   • DReflection in the line $y=-x$

   (b)

   Find the image of the point $(4,1)$ under $T$.

   [1 mark]

   • A$(1,-4)$
   • B$(-1,4)$
   • C$(-4,-1)$
   • D$(4,-1)$

   (c)

   Show that $\mathbf{M}^2$ represents an enlargement, and state its centre and scale factor.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Triangle $T$ has vertices $A(1,1)$, $B(3,1)$ and $C(3,4)$.

   (a)

   A rotation through $60^\circ$ anticlockwise about $O$ has matrix $\mathbf{R}$. Write down $\mathbf{R}$ with exact entries and find the exact coordinates of the image of $B$.

   [3 marks]

   (b)

   The stretch parallel to the $y$-axis with scale factor $3$ maps $T$ onto triangle $T'$. Write down the matrix of the stretch, find the coordinates of the vertices of $T'$ and find the area of $T'$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In this question every transformation is a single geometrical transformation of the plane, represented by a $2\times2$ matrix that acts on position vectors.

   (a)

   The matrix $\mathbf{M}$ maps $(1,0)$ to $(0,-1)$ and $(0,1)$ to $(-1,0)$.
   (i) Write down $\mathbf{M}$.
   (ii) Describe fully the transformation represented by $\mathbf{M}$.
   (iii) Find the equation of the image of the line $y=3x$ under this transformation.

   [6 marks]

   (b)

   A stretch $\mathsf{S}$ parallel to the $x$-axis maps the point $(4,-3)$ to $(10,-3)$.
   (i) Find the matrix of $\mathsf{S}$.
   (ii) Find the equation of the image of the line $y=2x+3$ under $\mathsf{S}$.
   (iii) Write down the equation of the line of invariant points of $\mathsf{S}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions