Solving simultaneous equations with matricesAQA A-Level Further Maths: Subtopic test

1. 1
   Three planes have equations $x+y+z=6$, $2x-y+z=6$ and $x+3y-z=2$. A calculator may be used.

   (a)

   Which of the following is the matrix form of the three equations?

   [1 mark]

   • A$\begin{pmatrix}1&1&1\\2&-1&1\\1&3&-1\end{pmatrix}\begin{pmatrix}x\\ y\\ z\end{pmatrix}=\begin{pmatrix}6\\2\\6\end{pmatrix}$
   • B$\begin{pmatrix}1&1&1\\2&1&1\\1&3&-1\end{pmatrix}\begin{pmatrix}x\\ y\\ z\end{pmatrix}=\begin{pmatrix}6\\6\\2\end{pmatrix}$
   • C$\begin{pmatrix}1&1&1\\2&-1&1\\1&3&-1\end{pmatrix}\begin{pmatrix}x\\ y\\ z\end{pmatrix}=\begin{pmatrix}6\\6\\2\end{pmatrix}$
   • D$\begin{pmatrix}1&1&1\\2&-1&1\\1&-3&1\end{pmatrix}\begin{pmatrix}x\\ y\\ z\end{pmatrix}=\begin{pmatrix}6\\6\\2\end{pmatrix}$

   (b)

   The determinant of the coefficient matrix is $8$. What does this tell you about the three planes?

   [1 mark]

   • AThey meet at exactly one point
   • BThey meet in a line
   • CThey form a triangular prism
   • DTwo of the planes are parallel

   (c)

   The inverse of the coefficient matrix is $\frac18\begin{pmatrix}-2&4&2\\3&-2&1\\7&-2&-3\end{pmatrix}$. Use it to solve the equations.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Three planes have equations $x+y-z=2$, $2x+2y-2z=4$ and $x-y+z=0$.

   (a)

   What is the relationship between the first two planes?

   [1 mark]

   • AThey are parallel and distinct
   • BThey are the same plane
   • CThey are perpendicular
   • DThey meet in a line

   (b)

   Which statement about the solutions of the three equations is correct?

   [1 mark]

   • AThere is no solution
   • BThere is exactly one solution
   • CThere are exactly two solutions
   • DThere are infinitely many solutions, lying on a line

   (c)

   Find the general solution of the three equations.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Three planes have equations $x+2y+z=4$, $2x+y-z=1$ and $3x+3y=k$, where $k$ is a constant.

   (a)

   Show that the three equations have no solution when $k=7$, and state the geometrical arrangement of the planes.

   [3 marks]

   (b)

   Find the value of $k$ for which the equations have infinitely many solutions, find these solutions and describe the geometrical arrangement of the planes.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Three planes have equations $x+y+z=1$, $x+2y+3z=3$ and $2x+3y+az=b$, where $a$ and $b$ are constants. The coefficient matrix has determinant $a-4$.

   (a)

   (i) Show that the determinant of the coefficient matrix is $a-4$.
   (ii) Explain why the equations have a unique solution when $a\neq4$, and describe the planes geometrically in this case.
   (iii) When $a=5$ and $b=7$, solve the equations.

   [6 marks]

   (b)

   Let $a=4$.
   (i) Show that the equations are consistent only when $b=4$.
   (ii) When $b=4$, find the general solution and describe the planes geometrically.
   (iii) Describe the arrangement of the planes when $b\neq4$.

   [6 marks]

   Total for question 4: 12 marks

End of questions