Linear transformationsAQA A-Level Further Maths: Subtopic test

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

   (a)

   Which of the following describes $T$?

   [1 mark]

   • ARotation through $90^\circ$ clockwise about the origin
   • BReflection in the line $y=x$
   • CRotation through $90^\circ$ anticlockwise about the origin
   • DReflection in the line $y=-x$

   (b)

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

   [1 mark]

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

   (c)

   The point $P$ is mapped by $T$ to $(-4,5)$. Find the coordinates of $P$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The unit square $OABC$ has vertices $O(0,0)$, $A(1,0)$, $B(1,1)$ and $C(0,1)$. It is transformed by the matrix $\mathbf{N}=\begin{pmatrix}1&2\\0&1\end{pmatrix}$.

   (a)

   Find the image of the vertex $B$.

   [1 mark]

   • A$(3,1)$
   • B$(1,3)$
   • C$(2,1)$
   • D$(1,2)$

   (b)

   Which of the following describes the transformation represented by $\mathbf{N}$?

   [1 mark]

   • AA stretch of scale factor $2$ parallel to the $x$-axis
   • BA shear in which the $y$-axis is invariant and $(1,0)$ maps to $(1,2)$
   • CAn enlargement of scale factor $2$ about the origin
   • DA shear in which the $x$-axis is invariant and $(0,1)$ maps to $(2,1)$

   (c)

   Describe fully the single transformation represented by $\mathbf{N}^2$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Transformation $P$ is a rotation through $90^\circ$ clockwise about the origin, and transformation $Q$ is a reflection in the line $y=x$.

   (a)

   Find the single matrix that represents $P$ followed by $Q$.

   [3 marks]

   (b)

   Find the single matrix that represents $Q$ followed by $P$ and describe the single transformation it represents. State, with a reason, whether the order of the two transformations matters.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In three dimensions, $R$ is the rotation through $90^\circ$ anticlockwise about the $x$-axis (as seen looking from the positive $x$-axis towards the origin), $T$ is the reflection in the plane $z=0$, and $U$ is the transformation with matrix $\mathbf{U}=\begin{pmatrix}0&0&1\\0&1&0\\-1&0&0\end{pmatrix}$.

   (a)

   (i) Write down the matrices that represent $R$ and $T$.
   (ii) Find the matrix that represents $R$ followed by $T$.
   (iii) Find the image of the point $(2,3,4)$ under $R$ followed by $T$.

   [6 marks]

   (b)

   (i) Describe fully the transformation represented by $\mathbf{U}$.
   (ii) Find $\mathbf{U}^2$.
   (iii) Hence describe the transformation represented by $\mathbf{U}^2$ and find the image of $(1,2,3)$ under it.

   [6 marks]

   Total for question 4: 12 marks

End of questions