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 x 0 y - 3 - 5 1 5 The diagram shows the graph of the cubic function ( ) y x f = . The intercepts of the curve with the axes are all integers. (a) Find the set of values of x for which ( )x 0 f 1 . [1] (b) Find an expression for ( )x f . [3]

1(a) 3 1 − x B1 1(b) ( )( )( ) 1 3 1 5 3 − + − − x x x 3 B1 for a negative cubic function B1 for a cubic function multiplied by 1 3 B1 for ( )( )( ) 3 1 5 + − − x x x

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.

4 (a) Show that x x 2 5 3 2 + - can be written in the form a x b c 2 + + ` j , where a, b and c are constants. [3] (b) Hence write down the coordinates of the stationary point on the curve with equation y x x 2 5 3 2 = + - . [2] (c) On the axes below, sketch the graph of y x x 2 5 3 2 = + - , stating the coordinates of the intercepts with the axes. [3] y x O (d) Write down the value of k for which the equation x x k 2 5 3 2 + - = has exactly 3 distinct solutions. [1]

4(a) 2 5 49 2 4 8   + −     x 3 B1 for 2 5 4   = +     b x or ( ) 2 1.25 + x B1 for 49 8 = − c or 6.125 − 4(b) 5 49 , 4 8   − −     oe 2 B1 for 5 4 − as part of a set of coordinates or 5 4 = − x , FT on − their b B1 for 49 8 − as part of a set of coordinates or 49 8 = − y FT on their c Need to be using their answer to (a) and not using differentiation as ‘Hence’. B1 for 5 49 , 4 8 − − 4(c) 3 B1 for correct shape, with maximum in the second quadrant and cusps on the x-axes and reasonable curvature for 3 x B1 for ( ) 3, 0 − and ( ) 0.5, 0 either seen on the graph or stated, must have attempted a correct shape B1 for ( ) 0, 3 either seen on the graph or stated, must have attempted a correct shape 4(d) 49 8 oe B1 FT on their c from (a) Allow 49 8 from other methods To earn marks, you need to correctly identify the 'b' term (the coefficient of x inside the squared term) and the 'c' term (the constant outside the squared term) when rewriting the expression in the specified form. To earn marks, you must use your answer from part (a) to directly state the x and y coordinates of the stationary point. The x-coordinate comes from the term inside the squared bracket, and the y-coordinate comes from the constant term outside the bracket. To earn marks, you must draw a graph with the correct 'W' shape, showing a maximum in the second quadrant and cusps on the x-axis. You also need to correctly label the x-intercepts and the y-intercept, either on the graph or stated separately. To earn the mark, you need to state the value of k that corresponds to the y-coordinate of the maximum point of the graph, which is the 'c' value from part (a).

Cambridge IGCSE Additional Mathematics · 0606

IGCSE Additional Mathematics (0606) Equations, inequalities and graphs past paper questions

Questions on solving equations and inequalities, including modulus and graphical methods, where a clear sketch usually earns the method marks.

42 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 IGCSE Additional Mathematics 0606 exams, 2020–2025

Across 2020–2025, Equations, inequalities and graphs accounted for 55 questions worth 307 marks in Cambridge IGCSE Additional Mathematics (0606) papers.