Core Pure: MatricesEdexcel A-Level Further Maths: Topic test

1. 1
   Transformation $\mathrm{U}$ is a stretch with scale factor $2$ parallel to the $y$-axis. Transformation $\mathrm{V}$ is a reflection in the line $y=x$.

   (a)

   Which matrix represents $\mathrm{U}$?

   [1 mark]

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

   (b)

   Which matrix represents $\mathrm{U}$ followed by $\mathrm{V}$?

   [1 mark]

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

   (c)

   Find the image of the point $(3,1)$ when it is transformed by $\mathrm{U}$ followed by $\mathrm{V}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The matrix $\mathbf{N}=\begin{pmatrix}2&-3\\4&5\end{pmatrix}$ represents a linear transformation of the plane.

   (a)

   What is the determinant of $\mathbf{N}$?

   [1 mark]

   • A$-2$
   • B$-22$
   • C$10$
   • D$22$

   (b)

   A triangle of area $5\text{ cm}^2$ is transformed by $\mathbf{N}$. What is the area of the image?

   [1 mark]

   • A$110\text{ cm}^2$
   • B$-110\text{ cm}^2$
   • C$22\text{ cm}^2$
   • D$\frac{5}{22}\text{ cm}^2$

   (c)

   Find $\mathbf{N}^{-1}$.

   [2 marks]

   Total for question 2: 4 marks

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

   (a)

   Find the invariant points of the transformation.

   [3 marks]

   (b)

   Find the equations of the invariant lines of the form $y=mx$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The matrix $\mathbf{M}=\begin{pmatrix}1&1&1\\1&2&3\\1&4&a\end{pmatrix}$, where $a$ is a constant, and three planes $\Pi_1:\ x+y+z=6$, $\Pi_2:\ x+2y+3z=14$ and $\Pi_3:\ x+4y+az=c$, where $c$ is a constant.

   (a)

   Show that $\det\mathbf{M}=a-7$. Given that $a=7$, find the value of $c$ for which the three planes meet in a line, and find the equations of that line.

   [6 marks]

   (b)

   Given that $a=8$, find $\mathbf{M}^{-1}$ and hence find the point where the three planes meet when $c=33$. Explain what the value of $\det\mathbf{M}$ tells you about the planes when $a=8$, whatever the value of $c$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The matrix $\mathbf{R}=\begin{pmatrix}0&-1&0\\1&0&0\\0&0&1\end{pmatrix}$ represents a rotation through $90^\circ$ anticlockwise about the $z$-axis, viewed from the positive $z$ direction.

   (a)

   What is the image of the point $(2,3,5)$ under this rotation?

   [1 mark]

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

   (b)

   Which matrix represents a reflection in the plane $z=0$?

   [1 mark]

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

   (c)

   The point $(1,2,4)$ is reflected in the plane $z=0$ and then transformed by the rotation. Find the coordinates of its image.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Three planes have equations $x+y=1$, $y+z=1$ and $x-z=2$.

   (a)

   What is the determinant of the coefficient matrix of the three equations?

   [1 mark]

   • A$-2$
   • B$0$
   • C$1$
   • D$2$

   (b)

   Which statement describes the arrangement of the three planes?

   [1 mark]

   • AThey meet at a single point.
   • BThey meet in a line, forming a sheaf.
   • CThey form a triangular prism.
   • DThey are three parallel planes.

   (c)

   Show that the three planes have no point in common.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Transformation $\mathrm{T}$ is a rotation through $45^\circ$ anticlockwise about the origin, followed by an enlargement with scale factor $2\sqrt2$ and centre the origin.

   (a)

   Show that the matrix representing $\mathrm{T}$ is $\begin{pmatrix}2&-2\\2&2\end{pmatrix}$.

   [3 marks]

   (b)

   A triangle of area $7$ is transformed by $\mathrm{T}$. Find the area of the image, and find the matrix that represents the inverse of $\mathrm{T}$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The matrix $\mathbf{H}=\begin{pmatrix}2&1\\3&4\end{pmatrix}$ represents a linear transformation of the plane.

   (a)

   Find the invariant points of the transformation and the equations of the two invariant lines through the origin.

   [6 marks]

   (b)

   The unit square with vertices $(0,0)$, $(1,0)$, $(1,1)$ and $(0,1)$ is transformed by $\mathbf{H}$ to a parallelogram. Find the coordinates of the images of the vertices other than the origin, find the area of the parallelogram, and find $\mathbf{H}^{-1}$. State whether the transformation preserves orientation.

   [6 marks]

   Total for question 8: 12 marks

End of questions