[S1_20160117AFNS] Much Ado About Nothing?

Problem

 

Introduction

     Assuming no typing errors, this is a tricky Secondary 2 question involving an equation with algebraic fractions.  How to solve it?  How to present the solution?

Solution

Remarks

[1]   Once the LHS expression has no meaning, it would not even make sense to continue.

[2]   This is a proof by contradiction type of argument.

It turns out that not all algebraic equations are soluble (or solvable).  This problem is one case in point.  The “unknown”  m  cannot be ⁵/₂ because that would make the expression undefined.  But if you substitute any other value, you always end up with nonsense like “15 = 0”.  So no matter what, there is no solution.  In other words, there is no value of  m  that you can substitute into the equation that makes it a true statement.

Heuristics Used

H05. Work backwards

H08. Make suppositions

Suitable Levels

* Lower Secondary Mathematics (Sec 2 ~ grade 8)

* GCE ‘O’ Level “Elementary” Mathematics

* other syllabuses that involve algebra

* any learner who is interested in algebra

[IB-HL_FTPR20150402] Periodicity cos a fortiori

     In this article, I give a rigorous proof of the period of cosine functions, using an a fortiori argument, signaled by the use of the words “in particular”.  When something is generally true for, say, all real numbers, it is “all the more” true for a particular case of a real number like zero.  I use this logic to nail down the answer.

Question

Solution

Suitable Levels

* revision for IB Mathematics HL / SL

* other syllabuses that teach trigonometry