Further Mathematics Paper 2, May/June. 2014  
Questions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Main
General Comments

Question 2

A binary operation  is defined on the set of rational numbers by
            mn = , n0 and m0

  1. Find -3  2.
  2. Show whether or not  is associative

This question was also reported to be very popular among the candidates and they performed well in it.
In part (a) majority of the candidates were reported to have correctly substituted -3 for m and 2 for n in the expression  to obtain  = .
In part (b) candidates’ performance was reported not to be as good as it was in part (a). It was also observed that majority of them were able to show that the operation was not associative using specific figures instead of variables. Candidates needed to be led to know that when proving a theorem or principle, you do not rely on constants but on variables so that you can generalize your results. Candidates’ expected response was as follow:

If m,n and p are rational numbers such that m ≠ n ≠ p, then for the operation
to be associative, m  (n  p) = (m n)  p. Taking the left hand side of the sign of equality, m(np) = m () = .  Taking the right hand side, (m * n) * P = () * p . Since the left hand side is not equal to the right hand side, it implied that the operation was not associative.

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