For the displayed system with constants k,p,q,r, find all k for which the solution is unique.

01 / Standard#Your turn

(E):\begin{cases}kx+y-z=p\\x+ky+z=q\\-x+y+kz=r\end{cases}

Official paper · jm02-2021 · 5(b)(i) · PDF 7

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the coefficient determinant.

Hint 2

Factor the cubic determinant in k.

Worked solution

Expand along the first row.
$\det A=k(k^2-1)-(k+1)-(k+1)=k^3-3k-2$
A square system is uniquely solvable for any right side exactly when this determinant is nonzero.
$\det A=(k+1)^2(k-2)\ne0\iff k\ne-1,2$

All real k except −1 and 2.

Checks and common pitfalls: A zero determinant does not distinguish no solution from infinitely many solutions without checking consistency.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Expand along the first row.
$\det A=k(k^2-1)-(k+1)-(k+1)=k^3-3k-2$
A square system is uniquely solvable for any right side exactly when this determinant is nonzero.
$\det A=(k+1)^2(k-2)\ne0\iff k\ne-1,2$

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

2021 JM02