Find the values of $k$ such that the equation ${x^3} + {x^2} - x + 2 = k$ has three distinct real solutions.

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]

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.