Asked four times whether she had known of the in-flight malfunction she refused to answer.
Speaking on the BBC's Andrew Marr Show she said: "I have absolute faith in our Trident missiles. When I made that speech in the House of Commons, what we were talking about was whether or not we should renew our Trident, whether or not we should have Trident missiles and an independent nuclear deterrent in the future."
Pressed by Mr Marr, who said, "Did you know that it had happened?" Ms May said: "I think we should defend our country."
Mr Marr asked: "This is a very serious incident. Did you know about it?"
Ms May again skirted the issue and did not give a yes-or-no answer.
"Prime Minister, did you know?" Mr Marr asked.
Ms May said: "There are tests that take place all the time, regularly, for our nuclear deterrent."
Finally Mr Marr conceded: "I'm not going to get an answer."
The wayward Trident missile was reportedly unarmed at the time of the malfunction, which occurred shortly after it was launched by the submarine HMS Vengeance off the coast of Florida last June. The Commons voted to renew the nuclear deterrent in July.
It was intended to be fired 5,600 miles to a sea target off the west coast of Africa but may have veered off towards America instead.
A source told the Sunday Times: "There was a major panic at the highest level of government and the military after the first test of our nuclear deterrent in four years ended in disastrous failure."
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