The FDNY for the first time in its history will allow someone who failed its crucial physical fitness test to join the Bravest, The Post has learned.

Rebecca Wax, 33, is set to graduate Tuesday from the Fire Academy without passing the Functional Skills Training test, a grueling obstacle course of job-related tasks performed in full gear with a limited air supply, an insider has revealed.

“They’re going to allow the first person to graduate without passing because this administration has lowered the standard,” said the insider, who is familiar with the training.

Upon graduation, Wax would be assigned to a firehouse and tasked with the full duties of a firefighter.

Some FDNY members are angry.

“We’re being asked to go into a fire with someone who isn’t 100 percent qualified,” the source said. “Our job is a team effort. If there’s a weak link in the chain, either civilians or our members can die.”

Wax’s graduation comes as the city celebrates the FDNY’s 150th anniversary and as the department is under pressure by the de Blasio administration to hire more women.

Only 44 of the FDNY’s 10,500 firefighters are female.

While Wax fell short, two other female probies in the graduating class passed the FST with flying colors.

“They’re kicking butt. They’re doing better than 50 percent of the class,” the insider said. “When they get assigned to a firehouse, they’ll be welcomed with open arms because they’ve done what everyone else has gone through.”

Other female firefighters aren’t pleased about Wax’s treatment, either.

“A lot of the girls in the field are pissed because they feel like they’re getting lumped into the same category of a female getting special treatment and not meeting the same standards as the males,” the insider added. “It devalues what the women in the field have accomplished.”

“We’re being asked to go into a fire with someone who isn’t 100 percent qualified. Our job is a team effort. If there’s a weak link in the chain, either civilians or our members can die.”

In the FST exam, probies must breathe through a mask attached to an air tank while carrying up to 50 pounds of gear.

They must climb six flights of stairs, stretch hose lines, raise ladders, perform tasks that simulate breaking doors and pulling down ceilings, and drag dummies through tunnels with no visibility.

They must complete the course in 17 minutes, 50 seconds or less.

Despite many attempts over the Fire Academy’s 18-week training course, Wax completed the test just once — but it took her more than 22 minutes, the source said.

In numerous tries, Wax struggled and was too slow. While fit probies finish with air left in their tanks, she had to stop when hers ran out, the source added.

“She’s in the best shape of her life, and it’s still not good enough,” he said.

Under the FDNY’s new hiring policy, probies must earn at least 75 percent on the combined requirements of academics, hands-on skills and physical fitness.

Wax had a high grade-point average on her academics, which officials determined offset her FST deficiency, the insider said.

Last December, Fire Commissioner Daniel Nigro told a City Council hearing on the FDNY’s efforts to recruit women that he had changed FST requirements to lower obstacles.

“We still grade the people. You can still fail it if you go beyond the time, but you’re not automatically failed from the program,” he said.

He also indicated he wanted to act before a possible sex discrimination lawsuit after the city paid $98 million to settle a lawsuit that accused the FDNY of discriminating against minorities.

“We must no longer wait for a judge’s ruling to tell us what fairness means,” Nigro said.

United Women Firefighters, an organization of FDNY women, objects to the FST test, contending it unfairly bars females.

The FST requires a much higher level of fitness than the Candidate Physical Ability Test, which applicants must pass to enter the academy.

Experts say the FST mirrors the demands of the job, training would-be firefighters to work in stressful environments while conserving their air supply.

Wax had previously received another break to join the FDNY after pleading in June 2011 for a City Council bill that raised the age limit from 29 to 35 for applicants like her who first took the entrance exam in 2007. A hiring freeze imposed by federal Judge Nicholas Garaufis in the discrimination suit put them in limbo.

“I want nothing more than to be a New York City firefighter,” Wax, then 29, had testified at a hearing.

Wax has worked as a civilian FDNY employee since 2009, lately as a community coordinator, making $52,338 last year.

Reached outside her Upper East Side apartment Friday, she would not speak to a reporter.

Nigro, questioned by The Post during an FDNY anniversary celebration at a Midtown firehouse Saturday, refused to comment on Wax’s fitness failure.

“Every one of our 305 probies have passed and will be graduating Tuesday,” he said. “She has met the requirements.”

The probationary class started with 320 recruits, meaning 15 dropped out or must retake the course.