(a) Represent the system as an augmented matrix: \[ \begin{pmatrix} 1 & 2 & -3 & k \\ 3 & 1 & 2 & 4 \\ 5 & 0 & 7 & 5 \end{pmatrix} \] (M1).

Perform row operation \( R_1 – 2R_2 \): \[ \begin{pmatrix} -5 & 0 & -7 & k – 8 \\ 3 & 1 & 2 & 4 \\ 5 & 0 & 7 & 5 \end{pmatrix} \] (A1).

Perform row operation \( R_1 + R_3 \): \[ \begin{pmatrix} 0 & 0 & 0 & k – 3 \\ 3 & 1 & 2 & 4 \\ 5 & 0 & 7 & 5 \end{pmatrix} \] (A1).

For no solutions, the system must be inconsistent, which occurs when the first row is \( [0 \ 0 \ 0 \ | \ k – 3] \) with \( k – 3 \neq 0 \). Thus, the system has no solutions if \( k \neq 3 \), so the set of values is \( k \in \mathbb{R}, k \neq 3 \) (A1).

[4 marks]

(b) The system represents three planes. When \( k \neq 3 \), the system is inconsistent (no common point). Geometrically, two planes intersect in a line, and the third plane is parallel to this line of intersection but does not contain it (A1).

[1 mark]

Total [5 marks]