# Curious case of an error in an exam paper

\(\)Here is a question from A level Edexcel Core 1 January 2013 paper:

The equation \( (k+3) x^2+6x+k=5 \)

where \( k\) is a constant, has two distinct real solutions for \(x\).

Show that \( k\) satisfies \( k^2-2k-24<0 \)

Hence find the set of possible values of \( k\).

This is a question of the type that often comes up, and students dutifully follow the familiar routine of forming an expression for the discriminant and then solving the inequality. Except that in this particular case, the question has a twist, as the coefficient at \( x^2\) depends on the parameter \( k\) and equals zero when \(k=-3 \enspace \) turning the seemingly quadratic equation into a linear one. Apart from that, \(k=-3 \enspace \) does belong to the set of solutions for the inequality, so the final answer should exclude \(k=-3 \enspace \) as the linear equation has a single solution rather than two distinct ones.

I wouldn’t be surprised if this peculiarity had been left unnoticed by students. What is worrying is that the mark scheme also ignores this, producing the wrong answer that includes \(k=-3\). It is even more disturbing given that a similar twisted question had come up earlier, in January 2009 paper (Question 7), also with the wrong answer in the mark scheme.

If you ask me, I would not include a question with a twist like that in the paper. It presents an unnecessary complication, and knowledge of discriminants can be tested without it. This kind of question would be good for a class discussion, but not for an exam that is stressful enough.

Could it be that the twist was unintentional, and the numerous people involved in producing papers and mark schemes simply didn’t notice? Or they did but still decided to leave the wrong answers in the mark schemes? Either way, I feel genuinely sorry for those students who might have spotted the anomaly at \(k=-3 \enspace \) and produced the correct answer, but lost marks because of the faulty mark scheme.