Mathematics · May / June 2023 · Paper 42 · Question 8

Correct expansion of a pair of brackets x2 – 4x + [1]x – 4 or x2 – 4x – 2x + 8 or x2 + [1]x – 2x – 2 x3 – 4x2 + x2 – 4x – 2x2 + 8x – 2x + 8 leading to and stating [y = ] x3 – 5x2 + 2x + 8 [2]

M1 accept x2 – 3x – 4 or x2 – 6x + 8 or x2 – [1]x – 2 A1 Accept x3 – 3x2 – 4x – 2x2 + 6x + 8 or x3 – 6x2 +[1] x2 + 8x – 6x + 8 or x3 –[1] x2 – 2x – 4x2 + 4x + 8 leading to and stating [y = ] x3 – 5x2 + 2x + 8

Correct labelled sketch positive cubic Crossing x-axis at –1, 2 and 4 only Crossing y – axis at 8 only [4]

B1 for positive cubic

B2 for three intercepts only with x -axis labelled at – 1, 2 and 4

or B1 for 1 or 2 correctly labelled x – intercepts

B1 for a single intercept on y-axis labelled at 8 but not if line y = 8

3x2 – 10x – 8 [= 0] 2 − x = 4 and x = 3 2 112 − (4, 0) and , oe 3 27 [y =] 10x – 40 and 292 + [y =] 10 x 27 [13]

M3 B2 for derivative = 3x2 – 10x + 2 isw OR

B1 for derivative with 3x2 or –10x given in expression isw M1dep on

B1 for their first derivative = 10

B1 B1 B2 B1 for each or for two different equations of the form [y = ] 10x + c (c must be numeric) or for c = –40 and 292 27