Find all p for which the system has a unique solution.

01 / Standard#Your turn

\begin{cases}x+y+pz=1\\px+y+z=p\\x+py+z=q\end{cases}

Official paper · jm02-2022 · 5(b)(i) · PDF 7

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Compute the coefficient determinant.

Hint 2

Factor the resulting cubic in p.

Worked solution

Expand along the first row.
$\Delta=\begin{vmatrix}1&1&p\\p&1&1\\1&p&1\end{vmatrix}=1-p-(p-1)+p(p^2-1)=p^3-3p+2$
Require a nonzero determinant.
$\Delta=(p-1)^2(p+2)\ne0\iff p\ne1,-2$

p∈ℝ∖{1,−2}, for any q.

Checks and common pitfalls: The right-hand parameter q does not affect invertibility.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Expand along the first row.
$\Delta=\begin{vmatrix}1&1&p\\p&1&1\\1&p&1\end{vmatrix}=1-p-(p-1)+p(p^2-1)=p^3-3p+2$
Require a nonzero determinant.
$\Delta=(p-1)^2(p+2)\ne0\iff p\ne1,-2$

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

2022 JM02