O-LevelAdditional MathematicsEquations, inequalities and graphs4037/11 · May/June 2022 · Q410 Marks

[Figure 4.a] The diagram shows the graph of y =|f(x)|, where f(x) is a cubic. Find the possible expressions for f(x).

(a)Find the possible expressions for f(x).[3]
(b)(i)On the axes below, sketch the graph of y = |2x+1| and the graph of y = |4(x-1)], stating the coordinates of the points where the graphs meet the coordinate axes.[3]
(ii)Find the exact solutions of the equation |2x+1|=|4(x-1)|.[4]

This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

[3] B1 for ±
B1 for 4
B1 for (x + 2)(x – 1)(x – 3)
[3] B1 for 2 V shapes which intersect twice in the first quadrant, with vertices on the x-axis, must be straight lines, not curves.
B1 for -0.5 and 1 on the x-axis
B1 for 1 and 4 on the y-axis
[4] M1 For attempt to solve to get x = (for 2x + 1 = 4(x – 1))
A1 (for x = 2.5)
M1 For attempt to solve to get x = (for 2x + 1 = -4(x - 1) oe)
A1 (for x = 0.5)
Alternative:
(M1) For attempt to square each equation and equate (for 4x² + 4x + 1 = 16x² – 32x + 16)
(M1) Dep on previous M mark for attempt to simplify to a 3-term quadratic equation, equated to zero and attempt to solve (for 12x² - 36x + 15 = 0 oe)
(A2) A1 for each (for x=2.5 x = 0.5)

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.