1 (a) On the axes, sketch the graph of ( )( )( ) y x x x 5 1 3 2 2 = + - - , stating the intercepts with the coordinate axes. [3] y O x (b) Hence find the values of x for which ( )( )( ) x x x 5 1 3 2 2 0 2 + - - . [2] e o

1(a) 3 B1 for a well-drawn cubic graph in the correct orientation with both arms extending beyond x-axis B1 for 1, 2 = − = x x and 2 3 = x either on the graph or stated with a cubic graph B1 for 20 = y either on the graph or stated with a cubic graph 1(b) 2 1 3 −< < x B1 Must be found from a cubic graph

1 On the axes below, sketch the graph of ( )( )( ) y x x x 4 1 2 1 3 4 =- + - + stating the intercepts with the coordinate axes. [3] 0 y x

1 3 B1 for a cubic shape with a maximum in the first quadrant, a minimum in the third quadrant, extending into the second and 4th quadrants. The extensions must not curve incorrectly and not lead to a complete stationary point. B1 for x-intercepts 1 4, ,3 2 − − either on diagram or stated but must be with a cubic graph. B1 for y-intercept 3 either on diagram or stated but must be with a cubic graph.

2 (a) On the axes, sketch the graph of y x 4 3 = - , stating the intercepts with the coordinate axes. [2] y O x (b) Solve the inequality x 4 3 7 H - . [3]

2(a) 2 B1 for symmetrical V shape in the correct quadrant, touching the x-axis. Must have straight lines. B1 for 4 3 = x and 4 = y only, either seen or stated on a modulus graph. 2(b) 1 − x ≤ , 11 3 x≥ or 3.67 or better 3 B1 for –1 from a correct method. B1 for 11 3 or 3.67 or better, from a correct method.

1 - 6 - 5 - 4 - 3 - 2 - 1 0 y x 1 2 3 4 5 6 5 10 15 20 (a) On the axes, draw the graphs of y x 5 3 2 = + - and y x 11 = - . [4] (b) Using the graphs, or otherwise, solve the inequality x x 11 5 3 2 1 - + - . [2]

1(a) 4 M1 for ∨ shape of 5 3 2 y x = + − with vertex at 2 , 5 3       A1 for correct graph with y-intercept ( ) 0, 7 M1 for correct straight line for 11 y x = − A1 for correct straight line with y-intercept ( ) 0,1 1 1(b) 2 x > or 2 x < − B2 Mark final answer for B2 B1 FT for exactly two correct critical values or correct FT critical values soi, FT dependent on at least M1 in (a)

1 x 0 - 2 - 1 1 2 3 4 5 6 7 8 - 2 2 4 6 8 10 y (a) On the axes draw the graphs of y x 5 = - and y x 6 2 7 = - - . [4] (b) Use your graphs to solve the inequality x x 5 6 2 7 2 - - - . [2]

1(a) 4 M1 for 5 = − y x : ∨ shape with vertex at (5, 0) A1 Correct graph with y-intercept at (0, 5) M1 for 6 2 7 = − − y x : ∧ shape with vertex at (3.5, 6) A1 Correct graph with y-intercept at (0, −1) 1(b) x 6 final answer B2 B1 for exactly two correct critical values or B1 FT for exactly two correct FT critical values soi, FT dependent on at least M1 in (a) If the CVs are decimal allow BOD for reasonable values

Cambridge O Level Additional Mathematics · 4037

O-Level Additional Mathematics (4037) Equations, inequalities and graphs past paper questions

Revise O-Level Additional Mathematics (4037) — Equations, inequalities and graphs with real Cambridge past-paper questions grouped by topic, each shown with its marking scheme. Working one topic at a time targets exactly the skill each question tests.

21 free questions from 2016–2021, each with its marking scheme. Open the full topical bank to filter by subtopic, paper, and year.

Equations, inequalities and graphs in O-Level Additional Mathematics 4037 exams, 2020–2025

Across 2020–2025, Equations, inequalities and graphs accounted for 24 questions worth 124 marks in Cambridge O Level Additional Mathematics (4037) papers.