A Level H2 Maths question spotting thoughts for AY 2018

The first edition of the 9758 A Level H2 Maths examination which happened last year was quite uneventful, though I wouldn't venture as far as to label it a walk in the park in the absolute sense. In hindsight students typically tripped up because of carelessness committed during the calculation process and failure to consult the help sheet MF26 appropriately, rather than having encountered outright speed bumps as far as problem design was concerned. Therefore, while I wouldn't expect this year's papers to be genuinely insurmountable, based on a personal review of actual block tests, preliminary examinations as well as revision packages furnished by top and middle tier junior colleges in recent times, it appears teachers are somewhat leaning towards the nostalgic - which by implication could see 2018 favoring a disproportionately large quantity of "archaic" methodologies being poked and prodded once more. From the bottom of my heart I pray I am dead wrong, then again under the circumstances ain't it much safer to "have a gun and not need it than to need a gun and not have it" be overprepared than underprepared? (yikes Tarantino creeping from within....) Without further ado here's a summarized inventory of my forecasts which takes significant heed of the good ol' days:

Complex Numbers

The main deal here centres on deriving more complicated trigonometric identities via complex numbers theorems. I would like to share about two possible instances arising by citing actual problems. The first happens as follows:

Knowledge of binomial series expansion is assumed* here, however you may wish to read about a primer on accomplishing the task in an efficient manner. That being said, with regards to solving the above problem posed, deriving the below generic association between a complex number and its conjugate would certainly be helpful:

The second type of problem involves obtaining an expression for tan(nθ):

Once again binomial series expansion is involved and this is an intermediate result one must strive to achieve:

Thereafter can you satisfactorily reconcile both real and imaginary components to tie up loose ends?

Functions

Are you aware of what a floor function is?

And here's embedding the said concept within a problem concerning periodic piece-wise continuous functions:

The solution for (i) is offered below, and it shouldn't be all that difficult to figure out (ii). (Note though the integral requiring evaluation is not exactly about computing the area under the existing graph)

Inequalities Hybridized With Integration Techniques

The problem typically tasks the student with solving a mildly difficult inequality at the onset, subsequently posing an integral which references the context of the particular said inequality. The integral itself would naturally involve modulus expressions - care must therefore be exercised to treat them before invoking the actual integration process. A proper example would best articulate affairs:

Assuming you can crack the inequality wide open (and you should be able to), here's getting you started off with regards to tackling the integral:

How would you proceed to dismantle the modulus signs present?

Statistics

While I do not expect folks to run into any stiff headwinds here since mostly standard fare graphic calculator commands are being deployed to establish answers, it would still be prudent to remain cognizant of some really old school question templates. One always comes to mind: enacting a recurrence relation and thus solving for the mode of a binomial distribution. An example is furnished below.

And that pretty much wraps things up; bear in mind the pointers offered herein are meant to supplement your revision efforts, not serve as an entirely standalone substitute. As I am drafting this you have slightly more than 2 weeks before Paper 1 (on 9 November) begins, so just hang in there for a wee bit longer........the A levels shall end soon enough and then pop goes the proverbial champagne.

Good luck kids. Peace.

*Addendum: R-formula (Trigonometry), Remainder/Factor Theorem, partial fractions, differentiation from first principles and various other concepts previously acquired at the O Levels also fall under the ambit of assumed knowledge.