Here's an alternative solution to the 2007 exam question 3b. I bet you thought the answer was x^{3}-20x+24. That's too boring.

Of course, a better question would be that if two distinct points on E: y^{2}=x^{3}-5x+3 are A(x_{1}, y_{1}), B(x_{2}, y_{2}) with x_{1}≠x_{2} show that if λ=(y_{2}-y_{1})/(x_{2}-x_{1}) then there is a third point C on E through AB given by (λ^{2}-x_{1}-x_{2}, λ(λ^{2}-2x_{1}-x_{2})+y_{1}).

Such a shame they don't ask better questions in HSC exams thesedays!

**Induction (again):**

The concluding statements for inductions provided by many candidates show that they incorrectly think that a proof by induction is actually an iterative proof, in which you imagine that the recipe should be repeated as many times as necessary in order to verify the statement for whichever positive integer is of interest. In fact, the Principle of Mathematical Induction is that every set with the property that, for each integer n in the set, n+1 is also in the set and which also contains 1 contains all positive integers. So, having established that the statement is true for 1 and, if true for some integer, is also true for the next integer, the correct conclusion is to simply state that, by induction, the statement is true for all positive integers. - 2007 HSC Examiners' Report. I was told by a member of the exam committee that at least 75% of teachers disagree with the exam committee. This is because most maths teachers blindly teach from textbooks written by amateurs without thinking about what they are teaching. Not only is this a very boring way to teach, it also will lead to errors such as the one indicated by the 2007 HSC Examiners' report.

Although this has been corrected in recent years in published solutions to 4 Unit papers, the published solutions to 3 unit papers are still bedevilled by the dreaded mantra, true for n=1, so true for n=2, so true for n=3, etc, therefore true for all positive integers - whereas in 4 unit solutions, we now see by induction, the statement is true for all positive integers.

So we now have another problem of confused 4 unit students (who have to sit both papers) with conflicting conclusions to published induction proofs. The only thing teachers can do is to tell them that the published 3 unit solutions are wrong.