(a)     If \(a = 4\) find the coordinates of the point of intersection of the three planes.

(b)     (i)     Find the value of \(a\) for which the planes do not meet at a unique point.

  (ii)     For this value of \(a\) show that the three planes do not have any common point.

(a)     let \({\boldsymbol{A}} = \left( {\begin{array}{*{20}{c}}
  1&1&2 \\
  2&{ - 1}&3 \\
  5&{ - 1}&4
\end{array}} \right)\) , \({\boldsymbol{X}} = \left( {\begin{array}{*{20}{c}}
  x \\
  y \\
  z
\end{array}} \right)\) , and \({\boldsymbol{B}} = \left( {\begin{array}{*{20}{c}}
  2 \\
  2 \\
  5
\end{array}} \right)\)     (M1)

point of intersection is \(\left( {\frac{{11}}{{12}},\frac{7}{{12}},\frac{1}{4}} \right)\)   or \(\left( {{\text{or }}\left( {{\text{0}}{\text{.917, 0}}{\text{.583, 0}}{\text{.25}}} \right)} \right)\)     A1

(b)     METHOD 1

(i)     \(\det \left( {\begin{array}{*{20}{c}}
  1&1&2 \\
  2&{ - 1}&3 \\
  5&{ - 1}&4
\end{array}} \right) = 0\)     M1

\( - 3a + 24 = 0\)     (A1)

\(a = 8\)     A1     N1

(ii)     consider the augmented matrix \(\left( {\begin{array}{*{20}{ccc|c}}
  1&1&2&2 \\
  2&{ - 1}&3&2 \\
  5&{ - 1}&4&5
\end{array}} \right)\)     M1

use row reduction to obtain \(\left( {\begin{array}{*{20}{ccc|c}}
  1&1&2&2 \\
  0&{ - 3}&{ - 1}&{ - 2} \\
  0&0&0&{ - 1}
\end{array}} \right)\) or \(\left( {\begin{array}{*{20}{ccc|c}}
  1&0&{\frac{5}{3}}&0 \\
  0&1&{\frac{1}{3}}&0 \\
  0&0&0&1
\end{array}} \right)\) (or equivalent)     A1

any valid reason     R1

(e.g. as the last row is not all zeros, the planes do not meet)     N0

METHOD 2

use of row reduction (or equivalent manipulation of equations)     M1

e.g. \(\left( {\begin{array}{*{20}{c}}
  1&1&2&2 \\
  2&{ - 1}&3&2 \\
  5&{ - 1}&a&5
\end{array}} \right) \Rightarrow \left( {\begin{array}{*{20}{c}}
  1&1&2&2 \\
  0&{ - 3}&{ - 1}&{ - 2} \\
  0&{ - 6}&{a - 10}&{ - 5}
\end{array}} \right)\)     A1A1

Note: Award an A1 for each correctly reduced row.

(i)     \(a -10 = -2 \Rightarrow a = 8\)     M1A1     N1

(ii)     when \(a = 8\) , row 3 \( \ne \) 2 \( \times \) row 2     R1     N0

[8 marks]

Few students were able to do this question efficiently. Many students were able to do part (a) by manipulating equations, whereas calculator methods would yield the solution quickly and easily. Part (b) was poorly attempted and it was apparent that many students used a lot of time manipulating equations without real understanding of what they were looking for.