20 The graph of \(y = x^3 - 7x + 9\) and the line with equation \(y = k\), where \(k\) is an integer, have 3 points of intersection.
(a) Find the greatest possible value of the integer \(k\). (1)
Part of the graph of \(y = x^3 - 7x + 9\) is shown on the grid.
(b) By drawing a suitable straight line on the grid, find estimates of the solutions of the equation \(x^3 - 6x - 2 = 0\). Give your answers correct to 1 decimal place. (3)
Mark scheme (a)
Scheme
Marks
16
B1
(1)
Notes
B1: cao
Mark scheme (b)
Scheme
Marks
\(x^3 - 7x + 9 = 11 - x\) or \(-x + 11\) oe seen line \(x + y = 11\) drawn
M1 M1
\(\approx -2.3,\ -0.3,\ 2.6\)
A1
(3)
(4 marks)
Notes
M1: May be implied by line \(x + y = 11\)
A1: Accept coordinates ft from candidate’s line if first M1 scored, line has negative gradient and there are 3 points of intersection
