Find all real triples for which the determinant vanishes.

01 / Standard#Your turn

(a+b+c)\big[(a-b)^2+(b-c)^2+(c-a)^2\big]=0

Official paper · jm02-2025 · 5(a)(ii) · PDF 7

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Either factor is zero.

Hint 2

A sum of real squares vanishes only when every square vanishes.

Worked solution

The linear factor describes one plane of solutions.
$a+b+c=0\implies(a,b,c)=(r,s,-r-s)$
The squared-difference factor describes the diagonal line.
$a=b=c\implies(a,b,c)=(r,r,r),\quad r,s\in\mathbb R$

All triples (r,s,−r−s), together with all triples (r,r,r).

Checks and common pitfalls: The two families form a union, not simultaneous conditions.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The linear factor describes one plane of solutions.
$a+b+c=0\implies(a,b,c)=(r,s,-r-s)$
The squared-difference factor describes the diagonal line.
$a=b=c\implies(a,b,c)=(r,r,r),\quad r,s\in\mathbb R$

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

2025 JM02