(a) Find Values of \( k \) for No Solution:

Form the augmented matrix:

\[ \begin{bmatrix} 1 & 2 & -3 & k \\ 3 & 1 & 2 & 4 \\ 5 & 0 & 7 & 5 \end{bmatrix} \quad (M1) \]

Perform row operation \( R_1 – 2R_2 \):

\[ \begin{bmatrix} -5 & 0 & -7 & k – 8 \\ 3 & 1 & 2 & 4 \\ 5 & 0 & 7 & 5 \end{bmatrix} \quad (A1) \]

Perform row operation \( R_1 + R_3 \):

\[ \begin{bmatrix} 0 & 0 & 0 & k – 3 \\ 3 & 1 & 2 & 4 \\ 5 & 0 & 7 & 5 \end{bmatrix} \quad (A1) \]

For no solution, the system is inconsistent when \( k – 3 \neq 0 \), i.e., \( k \neq 3 \quad (A1) \).

[4 marks]

(b) Geometrical Relationship:

Two planes intersect in a line, and the third plane is parallel to that line of intersection (A1).

[1 mark]