David Cameron has made a strong defence of his decision to employ a Nepalese nanny, while at the same time refusing to say that his government will meet its target to cut the number of net migrants to the UK to fewer than 100,000 by next year.

James Brokenshire, the immigration minister, caused consternation within the government last week when he accused a metropolitan elite of employing home helps from abroad, making it more difficult for the government to meet its immigration targets.

Downing Street has insisted Cameron followed all the "proper processes" when Gita Lima made her application for British citizenship in 2010. The prime minister's wife, Samantha, was named as her employer on the application but neither she nor her husband wrote letters in support, Downing Street said.

Speaking during his visit to Israel, Cameron said Lima had passed a British citizenship test and was a woman who worked hard and wanted to get on. He also revealed that one of the questions she was asked in the test was about the role of the British cabinet. He said he refused to give her any advice.

Lima was "excellent and originally came from Nepal", he said. He added: "She carried out her exams to become a British citizen and does a fantastic job. She certainly fit the description of someone who works hard and wants to get on. I can confirm I had given her no assistance in the exam she took."

He said Lima once asked him what the role was of the cabinet. "I won't tell you my answer. I will leave you to speculate," he told the press.

He said immigration levels needed to come down, and had done so by almost 30%, with immigration from outside the EU having fallen by a third since 1998.

"We have put in place a cap on economic migration from outside the EU because we should be training British young people to do more of the jobs that are available. We have closed down around 700 bogus colleges, because people were abusing the student route into Britain, and taken a range of steps to make sure family reunion is family reunion and not another way of breaking the rules."

Cameron has employed Lima for several years since recruiting her through Kalayaan, a charity that helps immigrants to escape abusive bosses. She took temporary leave of absence in 2012, during which an Australian nanny filled in.