Sketch the graph of the function. You are not required to find the coordinates of the maximum.

Find the value of k .

     A1

Note: Award A1 for intercepts of 0 and 2 and a concave down curve in the given domain .

Note: Award A0 if the cubic graph is extended outside the domain [0, 2] .

[1 mark]

$\int_0^2 {kx(x + 1)(2 - x){\text{d}}x = 1}$     (M1)

Note: The correct limits and =1 must be seen but may be seen later.

$k\int_0^2 {( - {x^3} + {x^2} + 2x){\text{d}}x = 1}$     A1

$k\left[ { - \frac{1}{4}{x^4} + \frac{1}{3}{x^3} + {x^2}} \right]_0^2 = 1$     M1

$k\left( { - 4 + \frac{8}{3} + 4} \right) = 1$     (A1)

$k = \frac{3}{8}$     A1

[5 marks]

Most candidates completed this question well. A number extended the graph beyond the given domain.

Most candidates completed this question well. A number extended the graph beyond the given domain.