Find the solution set for quadratic inequalities — Paper 2 · IGCSE Mathematics - Additional 0606

1 Solve the inequality (2 - x)(x + 9) 1 10 . [4]

4 marks

Mark scheme: Question Answer Marks Partial Marks 1 x 2 + 7 x − 8 ( > 0) 2 M1for expanding and collecting terms x < –8 or x > 1 2 M1 for factorising ( x + 8 )( x − 1) > 0

This question in 0606/21 Oct/Nov 2018

1 Solve the inequality (x - 3)(x + 4) 2 x + 13 . [3]

3 marks

Mark scheme: Question Answer Marks Partial Marks 1 x 2 + x − 12 > x + 13 M1 expand and simplify → x 2 … 25 A1 x > 5 or x < − 5 A1 or x > 5 , x < −5 or x > 5 and x < − 5

This question in 0606/22 Oct/Nov 2018

1 Find the values of x for which x(6x + 7) H 20. [3]

3 marks

Mark scheme: Question Answer Marks Partial Marks 1 6 x 2 + 7 x − 20[*0] M1 where * may be any inequality sign or = 4 5 A1 Critical values , − 3 2 5 4 A1 FT their critical values using outside x ≤ − or x ≥ final answer 2 3 regions

This question in 0606/21 May/June 2019

1 Find the values of x for which 9x2 + 18x - 1 1 x + 1. [3]

3 marks

Mark scheme: Question Answer Marks Partial Marks 1 For attempting to solve M1 where * may be any inequality sign or = 9 x 2 + 17 x − 2[*0] 1 A1 Critical values , − 2 9 1 A1 FT their critical values from ax 2 + bx + c < 0 with a −<2 x < final answer 9 > 0

This question in 0606/23 May/June 2019

1 Find the values of x for which 12x 2 - 20x + 5 1 ( 2 x + 1)( x - 1). [4]

4 marks

Mark scheme: Question Answer Marks Partial Marks 1 Expands right hand side and attempts to M1 collect terms Factorises or solves their 3-term quadratic M1 2 3 A1 correct CVs , 5 2 2 3 A1 FT their CVs, provided both M marks < x < mark final answer awarded 5 2

This question in 0606/22 Feb/March 2020

1 Solve the inequality ( x - 8)( x - 10) 2 35 . [4] + 1

4 marks

Mark scheme: Question Answer Marks Partial Marks 1 x2 – 18x + 45 (= 0) B1 Expand and simplify to three terms. (x – 15)(x – 3)(= 0) M1 Factorise or use formula on their 3 term 2 quadratic or complete the square 18 ± 18 −×4 45 or x = 2 or (x – 9)2 = –45 + 81 x = 15 and x = 3 A1 x < 3 or x > 15 A1 oe Do not accept ‘and’. or (–∞, 3) ∪ (15, ∞) Do not accept 3 > x > 15. Mark final answer.

This question in 0606/22 Oct/Nov 2020

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

3 marks

Mark scheme: Question Answer Marks Partial Marks 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

This question in 0606/21 Oct/Nov 2021

4 (a) Find the range of values of x satisfying the inequality ( 5x - 1)( 6 - x) 1 0 . [2] 2 1 (b) Show that the equation ( 2k + 1) x - 4kx + 2k - 1 = 0 , where k !- , has distinct, real roots. 2 [3]

5 marks

Mark scheme: 4(a) 1 M1 CVs , 6 5 1 A1 mark final answer x < , x > 6 5 4(b) (4k)2 – 4(2k + 1)(2k – 1) M1 16k2 – 4(4k2 – 1) A1 or 16 k 2  16 k 2  8k  8k  4 or better 4 > 0 A1

This question in 0606/23 May/June 2022

1 Solve the following inequality. ( 2x + 3)( x - 4) 2 ( 3x + 4)( x - 1) [5]

5 marks

Mark scheme: Question Answer Marks Guidance 1 2 x 2 − 8 x + 3 x − 12 * 3 x 2 − 3 x + 4 x − 4 B1 Correctly expands all brackets * is any inequality or equals sign  0* x 2 + 6 x + 8 B1 Collects terms to correct 3-term quadratic in solvable form  0*( x + 2 )( x + 4 ) M1 Factorises or solves their 3-term quadratic −4 and −2 A1 Correct critical values −4 < x < −2 mark final answer A1

This question in 0606/23 Oct/Nov 2022

1 (a) Solve the inequality 3x 2 - 12x + 16 2 3x + 4 . [3] (b) (i) Write 3x 2 - 12x + 16 in the form a ( x + b) 2 + c where a, b and c are integers. [3] (ii) Hence, write down the equation of the tangent to the curve y = 3x 2 - 12x + 16 at the minimum point of the curve. [1]

7 marks

Mark scheme: Question Answer Marks Partial Marks 1(a) 3 x 2  15 x  12 [* 0] oe where * is any B1 inequality sign or = Factorises or solves their 3-term quadratic M1 FT their 3-term quadratic x < 1 or x > 4 mark final answer A1 1(b)(i) 2 3 2 3  x  2   4 B2 for 3  x  2  2 or B1 for  x  2  or a = 3, b = –2 and 2 B1 for a  x  b   4 with numerical values of a and b or c = 4 1(b)(ii) y = their 4 B1 STRICT FT their 4 from part (i)

This question in 0606/22 May/June 2023

1 (a) Solve the equation 2 8 - 4x + 5 = 25 . [3] 2 57 - 9x (b) Solve the inequality 16 x - 5x - 3 1 . [4] 6

7 marks

Mark scheme: Question Answer Marks Partial Marks 1(a) 8 – 4x = 10 oe soi M1 and 8 – 4x = −10 oe soi OR 16 x 2 − 64 x − 36  = 0 oe 1 9 A2 mark final answer x = − , x = 2 2 1 9 A1 for x = − or x = 2 2 1(b) −30 x 2 + 105 x − 75 *0  oe M1 condone one sign or arithmetic error where * is any inequality sign or = Critical values 2.5 and 1 2 M1 for factorises or solves a 3-term quadratic to find critical values x < 1 x > 2.5 A1 mark final answer

This question in 0606/22 Feb/March 2024