(a)     Show that the following system of equations has an infinite number of solutions.

     \(x + y + 2z =  - 2\)

     \(3x - y + 14z = 6\)

     \(x + 2y =  - 5\)

The system of equations represents three planes in space.

(b)     Find the parametric equations of the line of intersection of the three planes.

(a)     EITHER

\(\left( {\begin{array}{*{20}{c}}1&1&2\\3&{ - 1}&{14}\\1&2&0\end{array}\left| \begin{array}{c} - 2\\6\\ - 5\end{array} \right.} \right) \to \left( {\begin{array}{*{20}{c}}1&1&2\\0&1&{ - 2}\\0&0&0\end{array}\left| \begin{array}{c} - 2\\ - 3\\0\end{array} \right.} \right)\)     M1

 

row of zeroes implies infinite solutions, (or equivalent).     R1

 

Note:     Award M1 for any attempt at row reduction.

 

OR

\(\left| {\begin{array}{*{20}{c}}1&1&2\\3&{ - 1}&{14}\\1&2&0\end{array}} \right| = 0\)     M1

\(\left| {\begin{array}{*{20}{c}}1&1&2\\3&{ - 1}&{14}\\1&2&0\end{array}} \right| = 0\) with one valid point     R1

OR

\(x + y + 2z =  - 2\)

\(3x - y + 14z = 6\)

\(x + 2y =  - 5\)   \( \Rightarrow x =  - 5 - 2y\)

substitute \(x =  - 5 - 2y\) into the first two equations:

\( - 5 - 2y + y + 2z =  - 2\)

\(3( - 5 - 2y) - y + 14z = 6\)     M1

\( - y + 2z = 3\)

\( - 7y + 14z = 21\)

the latter two equations are equivalent (by multiplying by 7) therefore an infinite number of solutions.     R1

OR

for example, \(7 \times {{\text{R}}_1} - {{\text{R}}_2}\) gives \(4x + 8y =  - 20\)     M1

this equation is a multiple of the third equation, therefore an infinite

number of solutions.     R1

[2 marks]

 

(b)     let \(y = t\)     M1

then \(x =  - 5 - 2t\)     A1

\(z = \frac{{t + 3}}{2}\)     A1

OR

let \(x = t\)     M1

then \(y = \frac{{ - 5 - t}}{2}\)     A1

\(z = \frac{{1 - t}}{4}\)     A1

OR

let \(z = t\)     M1

then \(x = 1 - 4t\)     A1

\(y =  - 3 + 2t\)     A1

OR

attempt to find cross product of two normal vectors:

eg: \(\left| {\begin{array}{*{20}{c}}i&j&k\\1&1&2\\1&2&0\end{array}} \right| =  - 4i + 2j + k\)     M1A1

\(x = 1 - 4t\)

\(y =  - 3 + 2t\)

\(z = t\)     A1

(or equivalent)

[3 marks]

 

Total [5 marks]

[N/A]