Question 3
- A binary operation * is defined on the set of real numbers, R,by
- Find the range of values of n for which 3 + 14n – 5n2 ≤ 0.
x * y = x + y – 3xy. Where x, y ∈ R. find the identity element in R under the
operation *.
Observation
The Chief Examiner reported that majority of the Candidates did not complete part (b).
The general performance of Candidates in the question was described as fair.
In part (a), if we let e be the identity element, then x * e = e * x = x + e – 3xe = x and
this yielded e(1 – 3x) = 0 so that e = 0.
In part (b) they were able to factorize 3 + 14n – 5n2 ≤ 0 to get (3 – n)(1 + 5n) ≤ 0. However,
majority of them could not investigate before concluding. To investigate, see below
| 𝑛≤−15 | −15≤𝑛≤3 | 𝑛≥3 | |
| (3 – n)
(1 + 5n) |
+ – |
+ + |
- + |
| (3 – n)(1 + 5n) | – | + | – |
Therefore, the range of values of n is 𝑛≤−1⁄5 or 𝑛≥ 3.