A student pilot reported that they conducted an IFR check ride flight in an aircraft with a known malfunctioning transponder.

Narrative

During cruise; during an IFR flight (checkride with DPE) to ZZZ; Tower called out and asked if we had the current altimeter. I reported that wehad switched over to ZZZ altimeter - which was .1 off (or 10 feet).Tower (ZZZ departure) indicated that we were on the edge; showing 2600 ft (we'd beengiven maintain 3000 but were struggling to maintain altitude given turbulence and highdensity altitude due to the heat). I responded that we were at 2900 feet and the transponder was off.The examiner asked me something along the lines of; 'Is it always like that?' and; 'What'sit supposed to be?'I responded candidly that the transponder had been consistently reporting 200-300 feet off;that it was a known issue that I'd brought up multiple times with my instructor. To theexaminer's second question; I guessed 200-300 feet; flustered and not knowing the exacttransponder reg off the top of my head (I did know the mandatory check was in the last 24months and that it had been done).The examiner scrolled on his phone for the next 5-10 minutes; intermittently checking fortraffic as I was under the foggles. After we completed our first approach at ZZZ (theLPV for RNAV-XX going missed with the published missed); he noted that we were havingtrouble climbing due to the heat and that the regulation stated the transponder would haveto be within 175 feet.He let me know that the checkride would result in disapproval and that we would discuss onthe ground.He cited concerns with my Aeronautical Decision Making (ADM) on 'not calling' off the flight due to weather and turbulence;citing the serious challenge we would have performing the three approaches and unusualattitude recovery given the turbulence.I responded that I'd continued the flight as my spouse and I have our first baby due any daynow and I'd been flying in this awful; hot and turbulent afternoon weather consistently.Additionally; I knew that the instructor was about to be traveling and it would be unlikelyfor me to get a retest or continuation soon after.The key issue I'm reporting here; however; is the normalization of deviance with regard tothe transponder issue.I reported and discussed this component of the flight with my instructor; in addition toother details of the failed checkride.I told the instructor that I didn't blame him; and that I was PIC and responsible. However;I will note that I brought up this issue with the instructor multiple times in the weekspreceding on this aircraft; and that ATC had also called it out consistently (several times)when we were in the air.The instructor simply said something like; 'Oh; these planes are old - I don't know why ATCkeeps calling it out.' The instructor is also young; and while a bright;well-connected and mentored young man; does lack longevity of experience.When I raised the issue with the instructor post-checkride; he checked to see if the flightschool was indeed empty and then candidly shared how 'it's a constant battle' with theseflight schools on maintaining aircraft properly; citing the misaligned incentives therein.He cited other flight schools where the planes were so dangerous that DPE's often wouldn'ttake students for checkrides from there.And he did re-affirm that; while this was unfortunate; that he believed in the fundamentalsafety of the aircraft from the flight school.Interestingly; in our discussion literally the day before the checkride about this issue; Ihad noted that 200-300 feet was a large deviation but that the buffer given between planes*should* be 500 feet or more. So; I was a part of the normalization as well. Additionally; Ihad become friends with the instructor and trusted him on all matters of aviation; so thatlikely clouded my judgment.Also; the pressure to get the checkride done before the baby arrives certainly affected myjudgment. Lastly; I was quite stressed the day of the checkride as I'd been working on IFRfor years and was very hopeful.I'll admit that the incentive in the moment (which didn't even occur to me) was to actsurprised at the deviation on the transponder in front of the examiner. Instead; I was openand honest. And the examiner was quite upset with me on the ground stating that I had; 'put[him] in an illegal IFR aircraft.' Note: this was absolutely never my intention.Had I known the transponder deviation was outside of regs; I never would have flown.Additionally; there was a TAA aircraft the school has with a better autopilot (the turbulence was so bad our autopilot failed to hold heading and even shut off a few timesduring the flight); but I chose not to switch aircraft so close to the checkride (and theradio had a very quiet sound whenever I was on mic in that TAA aircraft).It's hard to say with this transponder who exactly is to blame. Is it me? Certainly; I wasPIC and responsible for the aircraft. The A&P Mechanic? He'd signed off on the transponder.The owner of the flight school? He owned the planes. Or the instructor? Who failed to squawkthe issue and raise it to attention after I called it out multiple times.I think we likely all share the blame. I'm just glad that the recurring issue was caught by the examiner before it resulted in any near misses.