A Level H2 Maths question spotting thoughts for AY 2019

The 2018 A Level H2 Maths examination while largely manageable still succeeded in delivering a fair bit of pain, as students were up in arms over an integration techniques problem calling for the computation of a curve's arc length (relevant to the F Maths syllabus), that colossal 13 mark context-based irritant which exacted tedious algebraic manipulation coupled with suitable use of differentiation techniques alongside one triple Venn diagram conundrum posed among others. Let's of course not simply dismiss the functions snafu erroneously articulated in paper 1 which didn't escape numerous keen eyes. Should folks therefore batten down the hatches this time round in anticipation of something even more severe occurring? I'd reckon not, but it certainly wouldn't hurt to shore up preparation efforts by getting acquainted with less routine musings - hence the thrust of my debatably accurate "prognostications". Here goes:

Graphing Techniques

Vertical asymptotes remain unaffected when the graph of y=f'(x) is premised on y=f(x), but can previously absent ones arise as a consequence of undergoing this particular transformation? The answer is actually yes, as long as points exist on the original curve such that the tangents to them are parallel to the y-axis.

In the upper half of a hyperbola example illustrated above, the tangents to both x-intercepts at (-11,0) and (1,0) are infinitely steep, thus implying that x=-11 and x=1 shall end up as vertical asymptotes happening in the y=f'(x) sketch subsequently produced. Also, no x-intercepts are expected when pursuing the graphical derivative of y=f(x) on account not a single stationary point transpired initially.

On an unrelated note, I cannot emphasize enough the importance of properly comprehending the asymptotic behaviour of graphs; simply relying on the graphic calculator output and thereafter applying visual approximates to equations of existing asymptotes is strongly discouraged. Considering y=[2/(e^x-1)]+1 for a moment, I had students mistaking curves on both sides of the y-axis to be simultaneously restricted by the x-axis, ie y=0 "appeared" as a common horizontal asymptote on the GC display. Well sorry to burst your bubble because it ain't so.

Pop quiz: can you prove that f(x)=[2/(e^x-1)]+1 is an odd function for x∈ℝ, x≠0 ?

Integration Techniques

Integrals featuring purely fractional exponential functions can on occasion confound, then again surveying them from a different perspective could enable surprising breakthroughs. Check out these two instances to better appreciate what I am driving at:


Sometimes achieving the value of a scalar product via a deceptively simple triangular construct may require adept manipulation of information indirectly furnished - the following problem expounds this need unambiguously ( I have therefore included a pertinent hint to get you started off just in case).

Should you feel inclined to pursue the remaining parts of the question, they happen as such:


Discovering the domain and range of a typical composite function should be right up your alley, what then if you are confronted by three concatenated functions, such as hgf(x)? Should one be cognizant of how stuff are strung together as far as a two-tier process is concerned, then it is merely extrapolating that understanding once more to accommodate an additional plant. In any case I have detailed the required strategies accompanied by a box diagram below:


I am not expecting anything devastatingly insurmountable to creep into the latter part of paper 2, that being said you may wish to take special note of the negative binomial distribution (aka geometric distribution), where the nth success is assigned to a particular trial. Eg. Imagine tossing an unbiased die 10 times consecutively, what would be the probability alluding to the eighth throw displaying the third "6" ? (Food for thought: recognizing the first seven throws must yield exactly two "6"s, could you thus formulate a downsized binomial distribution to adequately address affairs?)

You may or may not wish to have a go at another problem:

End here I must, given a final round of consolidation with my charges still awaits. As cautioned in previous editions the pointers offered herein are merely meant to supplement your revision efforts, nothing more. Stay frosty in the meanwhile for the upcoming As, 'em holidays shall beckon sooner than you think - trust me on this.

Good luck kids. Peace.