| Date | November 2009 | Marks available | 5 | Reference code | 09N.2.hl.TZ0.1 |
| Level | HL only | Paper | 2 | Time zone | TZ0 |
| Command term | Find | Question number | 1 | Adapted from | N/A |
Question
Find the values of \(k\) such that the equation \({x^3} + {x^2} - x + 2 = k\) has three distinct real solutions.
Markscheme
from GDC, sketch a relevant graph A1
maximum: \(y = 3\) or (–1, 3) A1
minimum: \(y = 1.81\) or (0.333, 1.81) \(\left( {{\text{or }}y = \frac{{49}}{{27}}{\text{ or }}\left( {\frac{1}{3},\frac{{49}}{{27}}} \right)} \right)\) A1
hence, \(1.81 < k < 3\) A1A1 N3
Note: Award A1 for \(1.81 \leqslant k \leqslant 3\) .
[5 marks]
Examiners report
Responses to this question were surprisingly poor. Few candidates recognised that the easier way to answer the question was to use a graph on the GDC. Many candidates embarked on fruitless algebraic manipulation which led nowhere.
