A government-backed study has undermined claims by Ed Balls, the schools secretary, that GCSE standards have been maintained, by showing that some science papers include questions so simple that they require no knowledge of the subject.

The findings also demonstrated that examination boards were allowing scientifically wrong answers to be marked as correct and that maths was only being tested “in a very limited way”.

The report was passed to the government in July, only a few weeks before GCSE results were released, when Balls accused critics of exam standards of “rubbishing the achievements of young people”.

The report was not publicised by the Department for Children, Schools and Families, although it had partly funded the research with the Gatsby Foundation, a charity set up by Lord Sainsbury, the former science minister.

It appeared only on the website of Score, a science education group that did the research and whose member organisations include the Royal Society, Britain’s foremost scientific body.

Sir Martin Taylor, the society’s vice-president, said: “If we have science exams that do not test the quality of mathematics needed to do good science, or if we have questions that do not require scientific knowledge to answer them, then we do not have an examination system that is fit for purpose.”

The researchers’ job was to assess the level of science and mathematical knowledge required in science GCSEs.

They found that in some papers candidates could have their answers marked as correct even if they “did not reflect correct science”. In others no mathematical knowledge was required, while in a few questions no scientific knowledge was needed. “Of particular concern were questions which appeared to be general knowledge,” says the report.

Further concerns about GCSE standards have been raised by a maths paper set by the Edexcel exam board on November 5, mainly for pupils unable to take it in the summer. The paper, known as a “higher tier”, is intended for the country’s brighter young mathematicians, less than two years before they apply to university.

The question worth the most marks showed a cube with sides 5cm in length and asked: “Work out the total surface area of the cube. State the units of your answer.”

The paper continued: “The weight of the cube is 87 grams, correct to the nearest gram. What is the minimum the weight could be?”

Another asked candidates to add together Á and Ä and to work out the answer to É x Í.

Richard Pike, chief executive of the Royal Society of Chemistry, said the maths paper was easier than 11+ practice papers from 1960 with which he had compared it.

“That is an extraordinary indictment of the current UK education system,” said Pike. “We cannot continue to live the lie of ever-increasing standards while businesses struggle to recruit staff with numeracy skills, or who understand the quantitative basis of science.”

The failure of maths GCSEs to stretch the brightest has been demonstrated by results obtained at St Paul’s school, west London, where 159 out of 167 pupils scored A* grades. The other eight achieved As.

Michael Gove, the shadow schools secretary, said too many maths and science exams were “simply not stretching enough” and that ministers had debased standards “to flatter their own record”.

He added: “Young people are betrayed by an educational establishment that has failed to ensure our curriculum and our exams equip the next generation to the level of students in other countries.”

An Edexcel spokeswoman said: “We have confidence that GCSE maths papers are testing candidates at the right level,” adding that new exams were being developed by the board.

A spokesman for the schools department said that Ofqual, the new exams regulator, was already taking action on science papers using “tough powers to maintain standards”.

Vernon Coaker, the schools minister, said: “There have been huge improvements in school standards over the last decade.” He added that he wanted to see “a properly informed and mature debate over exams standards, instead of puerile arguments where critics handpick individual questions”.

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