Simultaneous equations and planesEdexcel A-Level Further Maths: Subtopic test

1. 1
   Three planes have equations $x+y+z=6$, $\ 2x-y+z=3$ and $x+2y-z=2$.

   (a)

   The equations are written as $\mathbf{M}\begin{pmatrix}x\\ y\\ z\end{pmatrix}=\begin{pmatrix}6\\3\\2\end{pmatrix}$. Find $\det\mathbf{M}$.

   [1 mark]

   • A$5$
   • B$-7$
   • C$7$
   • D$0$

   (b)

   What does the value of $\det\mathbf{M}$ tell you about the three planes?

   [1 mark]

   • AThey form a sheaf, meeting in a line
   • BThey form a triangular prism
   • CThey are three parallel planes
   • DThey meet at exactly one point

   (c)

   Use the inverse of $\mathbf{M}$, found using your calculator, to solve the equations.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Three planes have equations $x+2y-z=4$, $\ 2x+y+z=5$ and $3x+3y=k$, where $k$ is a constant.

   (a)

   For how many values of $k$ do the equations have a unique solution?

   [1 mark]

   • AEvery value of $k$
   • BNo value of $k$
   • COnly $k=9$
   • DEvery value except $k=9$

   (b)

   Which describes the three planes when $k=9$?

   [1 mark]

   • AA sheaf, meeting in a line
   • BThey meet at a single point
   • CA triangular prism
   • DThree parallel planes

   (c)

   Describe the geometrical configuration of the three planes when $k=6$, giving a reason.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Three planes have equations $\Pi_1:\ x+2y+3z=5$, $\ \Pi_2:\ 2x+4y+6z=7$ and $\Pi_3:\ x-y+z=1$.

   (a)

   Explain why the equations have no solution, and describe the geometrical configuration of the three planes.

   [3 marks]

   (b)

   The equation of $\Pi_2$ is changed to $2x+4y+6z=c$. Find the value of $c$ for which the equations have infinitely many solutions, and describe the configuration of the planes for this value of $c$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Three planes have equations $x+2y+z=1$, $\ 2x+5y+3z=4$ and $3x+7y+az=5$, where $a$ is a constant.

   (a)

   (i) Show that the determinant of the coefficient matrix is $a-4$, and state the values of $a$ for which the equations have a unique solution.
   (ii) Given that $a=3$, use your calculator to solve the equations.

   [6 marks]

   (b)

   Given that $a=4$, show that the equations are consistent, describe the geometrical configuration of the planes, and find the equations of the line they have in common.

   [6 marks]

   Total for question 4: 12 marks

End of questions