In A Dictionary of Australian Colloquialisms (Sydney University Press in association with Oxford University Press Australia, 1990), Gerald Alfred Wilkes (1927-2020) explained that, in Australian politics, the expression Dorothy Dix, also Dorothy Dixer, denotes:

A parliamentary question asked by a member of the Government party so that the Minister may make a prepared answer, to his own advantage.

This expression refers to Dorothy Dix, the pseudonym of Elizabeth Meriwether Gilmer (1861-1951), U.S. journalist and writer of a popular question-and-answer column.

The expression Dorothy Dix occurs, for example, in a report by the Procedure Committee of the Australian Senate, published in Canberra, Australian Capital Territory, in September 2008:

Restructuring question time
[…]
Introduction
Question time is undoubtedly one of the most important accountability mechanisms in the Australian Parliament. The opportunity to put questions in parliament to ministers relating to matters for which they have responsibility plays a key role in the successful functioning of our parliamentary democracy. It is also one of the functions of the parliament that has the highest public profile.
However, there may be areas where this important opportunity for scrutiny of the executive government could be made more effective and efficient.
Issues
[…]
Because there are a significant number of ministers relative to the number of questions available each day, most ministers will not be subject to more than a few questions each day. This means that ministers can usually provide a reasonable answer from the briefing material available to them, without necessarily being familiar with the full detail of the subject matter.
The use of Dorothy Dix questions, made by government senators to ministers from their own party, is one of the areas of question time subject to particular criticism. This is often an opportunity to make ministerial statements or simply to criticise one’s opponents. This does not provide for an accountability check on ministers.

The earliest occurrences that I have found of the expression Dorothy Dix(er) are as follows, in chronological order:

1 & 2-: From the column Over the Speaker’s Chair, published in The Canberra Times (Canberra, Australian Capital Territory):

1-: Of Friday 27^th July 1934:

“DOROTHY DIX”
There were many questions on trade and finance matters. One of these came from Mr. Hutchin, and there were cries of “Dorothy Dix” when he asked it. Everybody knows Dorothy Dix in Parliament. She comes tripping along in the most innocent fashion, and young, susceptible members are easily taken in by her; but as they grow older and more experienced their interest wanes. When a Minister is anxious to make some information available, or to answer some outside criticism, he will often get a private member to ask a question on the subject. Whether this is what happened in the case of Mr. Hutchin’s question one does not know, but to members of the Opposition it looked as if Dorothy was in it.

2-: Of Wednesday 3^rd April 1935:

We all know Dorothy Dix, that pretty little creature who replies to all the young men and women who write in to her newspaper for helpful suggestions in their love affairs. But there is an idea among some members of Parliament that many of these letters are written “in the office,” and that Dorothy herself writes a lot of the delightful questions that she answers so glibly.
AND HERE SHE COMES—
When Sir Littleton Groom this afternoon asked the Minister for Health (Mr. Hughes) a question “without notice” relating to Sister Kenny’s method of treating infantile paralysis, and suggested full investigation at one of the Children’s hospitals, the Minister said the hon. member for Darling Downs had been good enough to let him know he was going to ask this question, and he had prepared a reply. This reply was in warm agreement with the suggestion, and gave some good advice to the doctors.
Members of the Opposition looked at one another and smiled. “Dorothy Dix,” they said.
A little later Colonel Harrison, the member for Bendigo, drew attention to a statement by Dr. Courtney, who recently retired from the Repatriation department, to the effect that war service, on the whole, had left no lasting ill effects upon the returned men. And Mr. Hughes (this time in his capacity as Minister for Repatriation) advanced to the table with his reply—all nicely typed out.
“Dorothy Dix again,” commented Mr. Makin. But Colonel Harrison shook his head, deprecatingly.
AVOIDING PITFALLS.
Unfolding his papers, Mr. Hughes said: “The hon. member was good enough——” Then he looked round, surprised at the laughter. “Oh, yes—oh, well—yes, in this way I am enabled to avoid the pitfalls which so many of my colleagues, I am sorry to say, are apt to fall into” said Mr. Hughes; and, quite unabashed, he proceeded to read the reply, in which he stated that he entirely dissented from the view expressed by Dr. Courtney, and had called for a special report on the subject.
There was more laughter when, in reply to a question “without notice” by Mr. Fairbairn, relating to our trade with Germany, Italy and France, Mr. White, the Minister for Customs, said the hon. member had been “good enough to wire him.” The figures given by Mr. White were in sterling, and the Opposition leader (Mr. Scullin) suggested that it would have been just as easy to put them in terms of Australian currency.

3-: From The Canberra Times (Canberra, Australian Capital Territory) of Thursday 16^th June 1938:

GOVERNMENT SOLVES B.M.A. DEADLOCK
ROYAL COMMISSION TO DECIDE
BILL TO PASS NEXT WEEK
IF COUNTRY PARTY IS SATISFIED

The deadlock between the Commonwealth Government and the British Medical Association on questions of the capitation fee and treatment under the national health insurance scheme will be referred to a Royal Commission by the Government.
This announcement by the Treasurer (Mr. Casey) in the House of Representatives yesterday took the House by surprise.
[…]
The decision was conveyed to Government supporters, it is believed, at the joint party meeting held before the House assembled.
[…]
There were Labour jeers and cries of “Dorothy Dix,” when during question time Mr. J. Lawson (U.A.P.) asked if the Government had reached any agreement with the B.M.A. to overcome the deadlock.