Find the parameter values giving a unique solution.

01 / Standard#Your turn

kx+2y-z=p,\quad ky+z=q,\quad kx+3y=6

Official paper · jm02-2025 · 5(b)(i) · PDF 7

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the determinant of the coefficient matrix.

Hint 2

Subtract the first row from the third.

Worked solution

The row operation preserves the determinant.
$\begin{vmatrix}k&2&-1\\0&k&1\\k&3&0\end{vmatrix}=\begin{vmatrix}k&2&-1\\0&k&1\\0&1&1\end{vmatrix}=k(k-1)$
A nonsingular coefficient matrix gives exactly one solution for every p,q.
$k\ne0,1$

Unique solution exactly when k≠0 and k≠1.

Checks and common pitfalls: A zero determinant needs a separate consistency analysis, not an automatic no-solution conclusion.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The row operation preserves the determinant.
$\begin{vmatrix}k&2&-1\\0&k&1\\k&3&0\end{vmatrix}=\begin{vmatrix}k&2&-1\\0&k&1\\0&1&1\end{vmatrix}=k(k-1)$
A nonsingular coefficient matrix gives exactly one solution for every p,q.
$k\ne0,1$

Think first. Reveal a hint when the class is ready.

2025 JM02