The directive for the apology came after CBSE Chairman RK Chaturvedi met Union Human Resource Development Minister Prakash Javadekar in New Delhi today.
The CBSE also suspended four teachers for the "overzealous" act.
Examination officials at a school in Ernakulam in Kerala were booked for allegedly trimming the shirt sleeves of a student as part of the CBSE's steps to curb cheating in the National Eligibility cum Entrance Test (NEET), held on May 7.
It was a measure to "secure the sanctity" of the high- stake examination in which 11 lakh students vie for 95000 seats, it said.
"The incident reported in Kannur is unfortunate and a consequence of the overzealousness of a few persons involved in the process. However, the inconvenience caused to the students inadvertently in the process is regretted by the Board," CBSE spokesperson Rama Sharma said in New Delhi.
Sharma said the principal had been asked to apologise to the candidate who was made to remove her bra and added that the Board would "sensitise" the frisking staff at exam centres to prevent such incidents.
But the student in Kerala being asked to remove her bra triggered widespread outrage, leading to a political outcry in the state assembly and the suspension of four Kannur teachers.
Jamaluddin K, principal of TISK English Medium school, one of the NEET centres, told PTI that the teachers had been suspended pending an inquiry though he was yet to receive any complaint.
Kerala Chief Minister Pinarayi Vijayan said he would bring the matter to the Centre's notice and that the police had been asked to examine the incidents.
A furious Leader of the Opposition, Ramesh Chennithala, said the CBSE norms were "an insult to humanity".
The stripping of students and using metal detectors for physical examination were "cruel, inhuman and insulting" and created mental pressures on the students, he added.
The Kerala Child Rights Commission has sought a detailed report from the CBSE within 10 days, while the Kerala Human Rights Commission registered a suo motu case against the education board and demanded a high-level probe.
The don'ts included big buttons, brooches and heeled shoes for female candidates and kurta-pyjama, full-sleeve shirts and shoes for male candidates. The candidates were also not allowed to carry electronic devices inside the exam centre.
"Candidates were repeatedly instructed through the website, information bulletins, printed instructions on admit cards and individual communication through emails and SMSs about the steps to be taken before entering the exam centres in the high-value and high-stake examination," the CBSE spokesperson said.
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