2004 HSC Mathematics Extension 2 Paper

[frog0101]

Hi,
For Q7(a)(iii), how did students know that they needed to sub the left hand side of the inequality into the inequality proved in the part prior, what thought process lead them to that?

Thanks

2004 HSC Mathematics Extension 2:
http://www.k6.boardofstudies.nsw.edu.au/wps/wcm/connect/c03be093-eab2-405b-9de4-b2f9986e3f96/maths-ext2-hsc-exam-2004.pdf?MOD=AJPERES&CACHEID=ROOTWORKSPACE-c03be093-eab2-405b-9de4-b2f9986e3f96-lGd7Z3z

[RuiAce]

The link between the two parts was essentially in that
1) There were 3 terms in the LHS
2) Note that a 9 appears on the RHS. Coincidentally, if \(n = 3\), then \(n^2 = 9\). Which matches out perfectly.
3) The ≥ inequality appeared in both parts ii and iii

That's enough reason to suggest that taking \(n=3\), and attempting to find three terms to sub in, was a good idea.

(As for what to sub in, the intuitive 'guess' would be to just jump straight to \( a_1 = \operatorname{cosec}^2\theta\), \(a_2 = \sec^2\theta\) and \(a_3 = \cot^2\theta\). It's not immediately obvious that this is heading in the right direction, but it's the only thing that makes sense.)