Set k=2 and suppose the system has more than one solution. Find the relation among p,q,r; then solve when p=5,q=1,r=−4.

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Set k=2 and suppose the system has more than one solution. Find the relation among p,q,r; then solve when p=5,q=1,r=−4.

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

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

Official original and suggested answers ↗ · Suggested answer PDF page 12

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

At k=2 the second coefficient row equals the sum of the first and third.

Hint 2

Take z as a free parameter after checking the right-side relation.

Worked solution

The coefficient dependence forces the same relation on the right side.
$R_2=R_1+R_3\implies q=p+r\iff p-q+r=0$
The first two rows are independent, so this condition is also sufficient and leaves one free variable.
$\begin{vmatrix}2&1\\1&2\end{vmatrix}=3\ne0$
The specified right side is consistent; set z=t and solve the first two equations.
$5-1-4=0,\quad2x+y-t=5,\quad x+2y+t=1$
Substitution also satisfies the third equation.
$(x,y,z)=(3+t,-1-t,t),\quad t\in\mathbb R$

p−q+r=0; for the specified values, (x,y,z)=(3+t,−1−t,t), t∈ℝ.

Checks and common pitfalls: State the free parameter domain and retain the compatibility condition; one particular solution is not the full answer.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The coefficient dependence forces the same relation on the right side.
$R_2=R_1+R_3\implies q=p+r\iff p-q+r=0$
The first two rows are independent, so this condition is also sufficient and leaves one free variable.
$\begin{vmatrix}2&1\\1&2\end{vmatrix}=3\ne0$
The specified right side is consistent; set z=t and solve the first two equations.
$5-1-4=0,\quad2x+y-t=5,\quad x+2y+t=1$
Substitution also satisfies the third equation.
$(x,y,z)=(3+t,-1-t,t),\quad t\in\mathbb R$

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

Set k=2 and suppose the system has more than one solution. Find the relation among p,q,r; then solve when p=5,q=1,r=−4.

2021 JM02

Try it. Leave your reasoning visible.

Set k=2 and suppose the system has more than one solution. Find the relation among p,q,r; then solve when p=5,q=1,r=−4.