1 Find the possible values of the constant k such that the equation kx kx k 4 3 1 0 2 + + + = has two different real roots. [4]

1 ( ) ( ) 2 4 4 3 1 − + k k k M1 For use of the discriminant to obtain a two term quadratic expression. 2 4 4 0 − = k k M1 Dep to find critical values, allow if only one is found 0, 1 = = k k A1 For both critical values 0 k A1

1 (a) Write the expression x x 6 1 2 - + in the form ( ) x a b 2 + + , where a and b are constants. [2] (b) Hence write down the coordinates of the minimum point on the curve y x x 6 1 2 = - + . [1]

1(a) (x – 3)2 – 8 B2 B1 for (x – 3)2 + k where k ≠ −8 or a = −3 or (x + m)2 − 8 where m ≠ −3 or b = −8 1(b) (3, −8) B1 strict FT their a and b

1 Solve the inequality ( )( ) x x x 5 2 3 6 2 + - + . [3]

1 x2 + 3x – 10 − 3x − 6 * 0 oe M1 Condone one sign or arithmetic error * can be = or any inequality sign Critical Values: 4 and −4 A1 x > 4 or x < −4 A1 Mark final answer

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

2 Find the values of the constant k for which the equation ( ) kx k x 3 1 25 0 2 - + + = has equal roots. [4]

2 Uses 2 4 − b ac with at most one error in substitution: 2 ( 3( 1)) 4( )(25)*0 − + − k k M1 2 9 82 9*0 − + k k A1 Factorises or solves their 3-term quadratic M1 k = 1 9 or 9; mark final answer A1

Cambridge IGCSE Additional Mathematics · 0606

IGCSE Additional Mathematics (0606) Quadratic functions past paper questions

Quadratic-function questions test completing the square, the discriminant, and roots and turning points — the toolkit for curve sketching and later calculus.

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