Kosovo 15.09.2018 – Functions

This question comes form a 2008 exam paper from Kosovo:

Kosovo 2a

The graph of the function is given in the figure, accordingly, what is the value of f(-1) + f(0)?

Firstly, I like multiple choice questions when it is clearly thoughtful. I wonder to the extent that it is necessary in this context, but that is another question.

Even though functions are thought at GCSE mathematics, I would hazard a guess that even questions like the above would not be too familiar. Often transformations of functions are graphed, and substitution into functions are calculated algebraically speaking. But the combination of the two is not as frequent it feels.

Thoughts on this question:

Could we combine with transformations of graphs (e.g., asking to sketch, and then asking for values such as 2f(3) + -f(0))
Could we include the graph of another function, say g(x), and ask for combined or even composite values (e.g., f(1) + g(2))
Could we include questions about inverse functions, where defined?

Similar problems – me:

The graph of two function are shown in the picture below.

Kosovo2a2.png

What is the value of:
Kosovo2a3

Similar problems – world:

When I find similar problems I will link them here 🙂

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Mozambique 10.09.2018 – Inequalities

The following question is from a 2013 paper from Mozambique

Mozambique 1a

This question is unusual from an English school perspective for a couple of reasons:
1) It is not too often that the notation of braces is used, except occasionally for piece-wise functions.

2) It is not too frequent that inequality questions are so involved, without incorporating anything beyond linear equations.

Thoughts on this question:

Could we incorporate polynomials of higher degree?
Could we include inequalities with circles, ellipses, and so forth?
(How) can such inequalities be used for introducing Linear Programming?
Similar problems – me:

Mozambique 1b.png

Similar problems – world:

When I find similar problems I will link them here 🙂

Canada 03.09.2018 – Series and Proof

Here is what we might call a classic problem, coming from a textbook published in 1889.

Canada 1a.png

One of the many things I enjoy about historical texts of this kind is that they show us the expectations of our predecessors, as well as continuity in topics.

Here we have (at least) two topics – one of which taught at A level mathematics (series), the other Further Mathematics (proof).

I have made mention before of my affinity for such questions, and this holds true still. To see contexts colliding and to work through a more complex situations is more challenging and more fun.

Thoughts on this question:
What would change if instead of a geometric series we were working with an arithmetic series? Or indeed working with sequences given by polynomials?

If we are resolved to a geometric sequence:

Would if the common first term is no longer assumed? E.g., what if its differs for each series, or what if it is no longer 1 but 2 (etc.) for each series?
What if the ratios are fractions? Decimals? Surds?

In these cases is a (aesthetic) closed form feasible?
What methods of proof are most appropriate for such problems?

Similar problems – me:

Canada 1b

Similar problems – world:

(in combining topics) Papua New Guinea 25.08.2018 – Trigonometry Summation

(in combining topics) Peru 24.08.2018 – Logarithms

When I find more similar problems I will link them here 🙂

Kazakhstan 30.08.2018 – Indices

A rather typical problem when it comes to indices from Kazakhstan:

Kazakhstan 1a

Or, at least, by typical I mean typical for the region and its neighbours. I have rarely found anything of this nature in England. Though if there are any contemporary examples (in exams or textbooks) I’d be pleased to see them.

The skills checked here include fractional, negative, and mixed indices (the latter is surprisingly rare to see – as are decimal indices). The fact that we have numerical rather than algebraic bases means that we are expected to determine a value, and so the expectation of knowing square and cube numbers is clear (and clearly beyond the usual bound in English schools of 12 squared and – perhaps – 5 cubed).

Thoughts on this question:

Could we include not only fractional but also decimal indices. How about surd indices? In the latter case, we would have to resolve this in some way (noting that, for example Kazakhstan 1b )

Similar problems – me:

Kazakhstan 1c

Similar problems – world:

For this kind of problem there will be plentiful examples of similar questions. I will include them here when I see more:

Ireland 27.08.2018 – Inequalities

From a 2015 exam paper from Ireland:

Ireland 1a.png

Part (i) is a reasonably standard problem when it comes to inequalities. This is taught often in countries such as Russia, where it is used to show more complex results involving inequalities.

I like that this problem has been applied to a geometric situation.

Thoughts on this question:

Most problems I have seen of this nature go on to ask to prove more inequality facts; since this question considers a geometric situation I wonder if similar inequalities can be applied to: –

  1. Other geometric situations
  2. Other situations entirely

Similar problems – me:
Ireland 1b

Similar problems – world:

Here are some standard problems from a Russian textbook dealing with inequalities:

Ireland 1c.png

When I find more similar problems I will link them here 🙂

Ethiopia 26.08.2018 – Differentiation

Not entirely certain when this question was published, but it comes from Ethiopia:

Ethiopia 1a.png

One thing I enjoy particularly about this question is that it is not a standard differentiation problem. I also am quite the fan of multiple choice questions when common mistakes/misconceptions can be checked.

Thoughts on this question:

If we are to apply the chain rule, could we add numerous other functions? In particular, can we have multiple functions which ‘cancel’ one another out to lead to a ‘nice’ result, despite how complex the problem appears at first?

If we are to determine h(x) first, could we use what we have determined about h(x) to say anything about f(x) or g(x)? Perhaps at particular values or points of interest (say when h(x), h'(x), or h”(x) = 0?).

Similar problems – me:

I was inspired by the above question to make the following problem, which can be extended in many ways, I would be very excited to see how this is approached and extended!

Ethiopia 1b

Similar problems – world:

When I find similar problems I will link them here

Papua New Guinea 25.08.2018 – Trigonometry Summation

Here is a very interesting problem from a 2009 exam paper from Papua New Guinea:

PapuaNewGuinea1a

Series are taught at A level, so in theory nothing in this problem is unattainable. A very nice problem that looks more challenging than it is!

Thoughts on this question:

What other similar results can we derive – with trigonometry ratios or otherwise?
Can we know use this series to solve a problem which looks complicated?

In the finite case, what if we have a arithmetic rather than geometric progression?

Similar problems – me:

This in reality utilises the above result,

PapuaNewGuinea1b.png

Similar problems – world:

(in combining topics) Peru 24.08.2018 – Logarithms

When I find more similar problems I will link them here 🙂