The NSW woman stabbed her children to death one night in their bedrooms years ago, before attempting to take her own life.
The woman's husband made the horrific discovery after being notified his children were absent from school and went to check on them.
Upon finding her wounded in the house, he asked "what have you done? ... where are (the kids)?" according to the agreed facts.
"I killed them," she replied.
The family cannot be identified for legal reasons.
The mother appeared in NSW Supreme Court on Tuesday for a hearing over whether she should be found not criminally responsible for the deaths due to mental impairment.
The course of action was endorsed by both the woman's lawyer and the prosecutor, but Justice Richard Cavanagh questioned the strength of the evidence about her mental health impairment.
The mother's internet search history, suicide notes and seemingly contradictory psychological reports cast doubt on the extent of her impairment, he said.
"This is not a case involving schizophrenia, for example, there's no suggestion of deluded beliefs, there's no suggestion of psychotic episodes ... that so affected the defendant's thought processes that she wasn't able to reason", he said.
"It needs to be more than a mental health impairment."
The suicide note, which police found in the house the day after the killings, said she couldn't "leave my boys alone in such a bad world without my support" and "I hope you can forgive me".
Justice Cavanagh did not dispute the mother had major depression, but questioned whether that met the threshold for impairment.
"There's obviously a difference between her belief about her own attempted suicide and whether she believed what she was planning to do ... was wrong," he said.
He also highlighted the woman's internet searches prior to the alleged murder, which appeared to show she was "clearly planning the event".
Those searches included terms such as "bodies (sic) main arteries", "suicide parents and kids", scientific articles, news and true crime stories on filicide - the term for murdering one's children - "murder suicide mother kids" and tens of other similar queries.
The woman also read forum posts about what stabbing someone is like, investigated lethal doses of medication and researched the locations of vital blood vessels including how long it takes them to bleed out.
The searches, which began a month before the killings, were interspersed with dropping her kids off at school, going to their athletics training and taking them to McDonald's for dinner, court documents seen by AAP show.
But the woman's lawyer Madeleine Avenell SC argued her client's actions might not have reflected an understanding that killing her children was wrong.
"She knows that the action is wrong in the sense that she can say 'yes it's wrong to kill my child', but she didn't have the ability to exercise ... that wrongness," she said.
The hearing over the mother's mental health will resume on June 24, giving lawyers more time to consult with the psychiatrists.
Justice Cavanagh's concerns over the expert material were valid, defence solicitor Paul McGirr told reporters outside court.
"This is a very sensitive matter and it has to be done properly, particularly when you're dealing with extreme charges," he said.
The accused murderer, who dialled into court from custody, hung her head low during the proceedings, leaving only a tangle of hair visible.
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