Invariant points and linesAQA A-Level Further Maths: Subtopic test

1. 1
   The transformation $T$ of the plane has matrix $\mathbf{M}=\begin{pmatrix}1&2\\0&3\end{pmatrix}$.

   (a)

   Which of the following points is invariant under $T$?

   [1 mark]

   • A$(0,2)$
   • B$(1,1)$
   • C$(2,-1)$
   • D$(4,0)$

   (b)

   Which of the following lines is invariant under $T$?

   [1 mark]

   • A$y=x$
   • B$y=2x$
   • C$y=3x$
   • D$x=0$

   (c)

   Show that the line $y=2x$ is not invariant under $T$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The matrix $\mathbf{N}=\begin{pmatrix}-1&0\\0&1\end{pmatrix}$ represents a reflection in the $y$-axis.

   (a)

   Which line is a line of invariant points under the reflection?

   [1 mark]

   • AThe $x$-axis
   • BThe line $y=x$
   • CThe $y$-axis
   • DThe line $y=-x$

   (b)

   Which line is invariant under the reflection, but is not a line of invariant points?

   [1 mark]

   • A$y=x$
   • B$y=5$
   • C$x=3$
   • D$y=2x$

   (c)

   Find the equation of the image of the line $y=2x+3$ under the reflection, and hence state whether this line is invariant.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The matrix $\mathbf{S}=\begin{pmatrix}2&-1\\1&0\end{pmatrix}$ represents a transformation $S$ of the plane.

   (a)

   Find the invariant points of $S$.

   [3 marks]

   (b)

   Show that, for every constant $c$, the line $y=x+c$ is invariant under $S$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The matrix $\mathbf{T}=\begin{pmatrix}4&-2\\1&1\end{pmatrix}$ represents a transformation $T$ of the plane.

   (a)

   (i) Show that the origin is the only invariant point of $T$.
   (ii) Find the equations of the invariant lines of $T$ of the form $y=mx$.

   [6 marks]

   (b)

   Show that $T$ has no invariant line of the form $y=mx+k$ with $k\neq0$.

   [6 marks]

   Total for question 4: 12 marks

End of questions