INSTRUCTOR PLT WITH A FOREIGN PVT PLT GETTING NIGHT CURRENCY; ON A RADAR VECTOR FROM TPA APCH CTL; FLEW INTO THE CLASS B AIRSPACE WITHOUT A CLRNC AND HAD A NMAC.

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ON NIGHT FLT WITH PVT PLT EARNING HIS NIGHT FLYING PRIVILEGES THE FOLLOWING OCCURRED. OUR ACFT DEPARTED BOW (BARTOW) ON A HEADING TOWARDS THE SW. PLT CONTACTED LAL (LAKELAND) AND REQUESTED PERMISSION TO TRANSITION THEIR AIRSPACE. LAL ATC ADVISED US TO FLY OVER LAL ARPT AND RPT (HEADING ALTERED TO R WNW). PLT COMPLIED AND SHORTLY AFTER WAS ADVISED TO CONTACT TPA APCH CTL. UPON ESTABLISHING CONTACT WITH TPA APCH PLT WAS ADVISED OF SQUAWK CODE AND TO FLY A HDG OF W (270 DEGS) AND MAINTAIN 2500 FT. PLT COMPLIED WITH THESE INSTRUCTIONS. THE INSTRUCTIONS GIVEN BY TPA APCH LEAD ACFT DIRECTLY TO TPA INTL. THESE INSTRUCTIONS WERE GIVEN APPROX 1 MI E OF PLANT CITY ARPT. THIS ACFT EXPERIENCED A NEAR MISS APPROX 1-2 MI S OF TPA INTL. CTLR WAS CONTACTED AND HE ADVISED THAT HE WAS NOT PICKING UP OUR XPONDER AND TO REMAIN CLR OF CLASS B AIRSPACE. PLT WAS ALREADY WELL INTO CLASS B AIRSPACE DUE TO VECTORS GIVEN BY THIS CTLR AND UNDERSTOOD HE WAS CLRED INTO TPA INTL AIRSPACE. SHORTLY AFTER NEAR MISS; ATC CTLR ADVISED PLT TO CONTACT TPA APCH ON NEW SECTOR FREQ. XPONDER SIGNAL WAS PICKED UP FINE AND THIS CTLR ADVISED STUDENT PLT TO CALL TPA APCH TWR UPON LNDG AT DEST. STUDENT PLT COMPLIED; WAS ADVISED OF NEAR MISS AND VIOLATION OF CLASS B AIRSPACE. THIS PROB AROSE DUE TO PLT COMPLYING WITH CTLR'S ASSIGNED HDG (270 DEGS). CONTRIBUTING FACTORS INCLUDE LAL ATC ALTERING ACFT HEADING TO R OF COURSE TO OVERFLY LAL ARPT; PLT UNDERSTANDING HE WAS CLRED INTO CLASS B AIRSPACE DUE TO ASSIGNED HDG AND XPONDER CODE WHILE RPTING OVER PLANT CITY; (PROX ISSUE) FAILURE TO IDENT/LOCATE TPA INTL PRIOR TO A NEAR MISS. CORRECTIVE ACTION: PLT SHOULD HAVE QUERIED ATC TO VERIFY RADAR CONTACT PRIOR TO ENTERING CLASS B; INITIAL TPA APCH CTLR SHOULD HAVE ADVISED ACFT TO REMAIN OUTSIDE OF CLASS B AIRSPACE IF RADAR (SECONDARY) WAS NOT PICKING UP XPONDER. BECAUSE THE CTLR HE PASSED OUR ACFT OFF TO; PICKED UP XPONDER CODE; I QUESTION THIS.