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

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

   (a)

   Find the image of the point $(2,5)$ under the transformation.

   [1 mark]

   • A$(16,2)$
   • B$(-4,2)$
   • C$(2,-4)$
   • D$(6,2)$

   (b)

   Which of these points is an invariant point of the transformation?

   [1 mark]

   • A$(1,0)$
   • B$(2,-1)$
   • C$(3,3)$
   • D$(0,2)$

   (c)

   Find the coordinates of all the invariant points of the transformation.

   [2 marks]

   Total for question 1: 4 marks

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

   (a)

   How many invariant points does the transformation have?

   [1 mark]

   • AOnly the origin
   • BNone
   • CA line of them
   • DEvery point

   (b)

   Which of the following lines is invariant under the transformation?

   [1 mark]

   • A$y=x$
   • B$y=1$
   • C$y=2x+1$
   • D$y=0$

   (c)

   Find the equations of all the invariant lines through the origin.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The matrix $\mathbf{B}=\begin{pmatrix}4&2\\ 1&3\end{pmatrix}$ represents a linear transformation of the plane.

   (a)

   Show that the invariant lines through the origin of the form $y=mx$ satisfy $2m^2+m-1=0$.

   [3 marks]

   (b)

   Hence find the equations of the two invariant lines through the origin. Explain whether any point on either line, other than the origin, is an invariant point.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The matrix $\mathbf{C}=\begin{pmatrix}1&2\\ 0&3\end{pmatrix}$ represents a linear transformation of the plane.

   (a)

   (i) Find all the invariant points of the transformation.
   (ii) Show that the line $y=x+c$ is invariant for every real value of $c$.

   [6 marks]

   (b)

   The point $P(1,4)$ is mapped to $P'$. Find $P'$, and hence find the equation of the invariant line through $P$ and $P'$. Verify, by considering the image of a general point of this line, that it is invariant.

   [6 marks]

   Total for question 4: 12 marks

End of questions