David Cameron has urged the British public to report illegal immigrants and vowed to change the citizenship test so that it includes questions on British history.

The text of the prime minister's key immigration speech confirms that he wants to make forced marriage a criminal offence, and makes it clear that Downing Street backs Theresa May's desire to rewrite the immigration rules so "misinterpretations" of the right to family life do not block immigration deportations.

But his proposal to require families who want to sponsor overseas relatives to settle in Britain to post a financial bond worth thousands of pounds has already sparked a strong warning from Keith Vaz, the chairman of the Commons home affairs committee, that it will "antagonise settled communities in Britain and enrage our allies such as India".

Vaz said the speech marked the third time the financial bond had been proposed over the last decade, adding: "It was shelved before because it was unworkable. A more effective method must be found in order for immigrants to pay for public services."

The proposal to make forced marriage a criminal offence will initially be limited to introducing a criminal penalty for those who breach a forced marriage prevention order.

More than 257 such orders have been made since they were introduced in November 2008, but Cameron appears to have acknowledged Home Office concerns that a new offence of forced marriage may be difficult to enforce as it involves the victim having to testify against close relatives.

He said the creation of an offence of forced marriage must ensure that it does not prevent or hinder victims reporting what has happened to them.

In his speech to the Institute for Government in London, the prime minister said he wanted even tougher action against illegal immigrants in Britain.

"For our part in government, we are creating a new national crime agency with a dedicated border policing command which will have responsibility for safeguarding the security of our border," he said.

"But I want everyone in the country to help … including by reporting suspected illegal immigrants to our Border Agency through the Crimestoppers phoneline or through the Border Agency website. Together, we will reclaim our borders and send illegal immigrants home."

He said the drive to improve the UK Borders Agency's record of finding illegal migrants and deporting them already included using credit reference agencies to ensure they cannot get easy access to credit and a targeted campaign this summer that involved 600 operations and resulted in more than 550 arrests.

Cameron also unveiled changes to the citizenship test taken by all those who apply for a British passport. He said he had been to citizenship ceremonies and thought they were moving and they worked, "but here, too, changes are needed".

"So let me say one more thing abut the journey to becoming a British citizen," he added. "We're also going to change the citizenship test. There's a whole chapter in the citizenship handbook on British history but, incredibly, there is no question on British history in the actual test.

"Instead you'll find questions on the roles and powers of the main institutions of Europe and the benefits system within the UK. So we are going to revise the whole test … and put British history and culture at the heart of it."

The test is centred on how British society works, so typical current questions include questions about the Queen's ceremonial duties, how the House of Commons is elected and the role of the prime minister.

Questions on British customs and traditions include asking to name the dates of each patron saint's day for the four UK countries and the dates of the main Christian festivals.

But whether those who are already British citizens would be able to pass such a test remains open to doubt. A series of polls taken over the past decade show that a large proportion of those under 30 have difficulty in identifying major figures in British history.

One BBC poll showed that fewer than half could identify Sir Francis Drake as the key figure in the defeat of the Spanish Armada, and one-third did not know that William the Conquerer won the Battle of Hastings.