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8.6D Quadratic Inequalities: Fluency and Exam Drill

EDEXCEL Edexcel 1MA1 · Calculator allowed · about 60 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Solve the inequality x² - 5x + 6 < 0. Give your answer in inequality form.

(Total for Question 1 is 2 marks)

Solve the inequality x² - 9 ≤ 0. Give your answer in inequality form.

(Total for Question 2 is 2 marks)

Solve the inequality 2x² - 8x ≥ 0. Give your answer in inequality form.

(Total for Question 3 is 2 marks)

Solve the inequality -x² + 4x < 0. Give your answer in inequality form.

(Total for Question 4 is 2 marks)

Solve the inequality x² + 2x - 15 > 0. Give your answer in inequality form.

(Total for Question 5 is 2 marks)

Solve the inequality (x-1)(x+2) ≥ 0. Give your answer both in inequality form and in set notation.

(Total for Question 6 is 3 marks)

Solve the inequality x² - 4x + 4 ≤ 0. Give your answer in inequality form.

(Total for Question 7 is 2 marks)

Solve the inequality (1/2)x² - (3/2)x < 0. Give your answer in inequality form.

(Total for Question 8 is 2 marks)

Solve the inequality 3x² + x - 4 ≤ 0. Give your answer in inequality form.

(Total for Question 9 is 3 marks)

Solve the inequality -(2x² - 5x - 3) > 0. Give your answer in inequality form.

(Total for Question 10 is 3 marks)

Solve the inequality x² - x - 6 ≥ 0. Give your answer both in inequality form and in set notation.

(Total for Question 11 is 3 marks)

a) Solve the inequality x² - 6x + 5 < 0.
b) Express the solution from part (a) in set notation.

(a)Solve the inequality x² - 6x + 5 < 0.(2)

(b)Express the solution from part (a) in set notation.(2)

(Total for Question 12 is 4 marks)

Solve the inequality (x-2)(2x+3) > 0. Give your answer in inequality form.

(Total for Question 13 is 2 marks)

Solve the inequality x² + 4x + 3 < 0. Give your answer in inequality form.

(Total for Question 14 is 2 marks)

Solve the inequality x² - 2x - 1 > 0. Give your answer in exact form.

(Total for Question 15 is 3 marks)

Solve the inequality -x² + 6x - 8 ≥ 0. Give your answer in inequality form.

(Total for Question 16 is 3 marks)

Solve the inequality x² - 5x + 6 < x - 1. Give your answer in inequality form.

(Total for Question 17 is 3 marks)

For which integer values of n is n² - 7n + 10 < 0? Give your answer as a list of integers.

(Total for Question 18 is 2 marks)

Mark scheme · 8.6D Quadratic Inequalities: Fluency and Exam Drill

Question 1

M1 factorise to (x-2)(x-3) or identify roots x=2,3
A1 2 < x < 3 cao
Answer: 2 < x < 3 (inequality form: 2 < x < 3)

Question 2

M1 factorise to (x-3)(x+3) or identify roots x=-3,3
A1 -3 ≤ x ≤ 3 cao
Answer: -3 ≤ x ≤ 3

Question 3

M1 factorise to 2x(x-4) and identify roots x=0 and x=4
A1 x ≤ 0 or x ≥ 4 cao
Answer: x ≤ 0 or x ≥ 4

Question 4

M1 rearrange or factorise, e.g. -x(x-4) < 0 or multiply by -1 giving x² -4x > 0
A1 x < 0 or x > 4 cao
Answer: x < 0 or x > 4

Question 5

M1 factorise to (x+5)(x-3) or identify roots x=-5,3
A1 x < -5 or x > 3 cao
Answer: x < -5 or x > 3

Question 6

M1 identify roots x = 1 and x = -2 and sign on intervals
M1 state inequality form x ≤ -2 or x ≥ 1
A1 set notation: (-infty, -2] U [1, +infty) cao
Answer: Inequality form: x ≤ -2 or x ≥ 1. Set notation: (-infty, -2] U [1, +infty)

Question 7

M1 recognise (x-2)² ≤ 0 so repeated root x=2
A1 x = 2 cao
Answer: x = 2

Question 8

M1 multiply by 2 to get x² - 3x < 0 and factor to x(x-3)<0
A1 0 < x < 3 cao
Answer: 0 < x < 3

Question 9

M1 use quadratic formula or factorisation to find roots x = 1 and x = -4/3
M1 order the roots and decide region where quadratic ≤ 0 (parabola up)
A1 -4/3 ≤ x ≤ 1 cao
Answer: -4/3 ≤ x ≤ 1

Question 10

M1 multiply by -1 (flip inequality) to get 2x² - 5x - 3 < 0 and find roots
M1 calculate roots x = -1/2 and x = 3
A1 -1/2 < x < 3 cao
Answer: -1/2 < x < 3

Question 11

M1 factorise to (x-3)(x+2) and identify roots -2 and 3
M1 decide sign: x ≤ -2 or x ≥ 3
A1 set notation: (-infty, -2] U [3, +infty) cao
Answer: Inequality form: x ≤ -2 or x ≥ 3. Set notation: (-infty, -2] U [3, +infty)

Question 12

(a) M1 factorise to (x-1)(x-5) or identify roots 1 and 5
(a) A1 1 < x < 5 cao
(a) Answer: 1 < x < 5
(b) M1 convert interval to set notation
(b) A1 (1, 5) cao
(b) Answer: (1, 5)

Question 13

M1 find roots x = 2 and x = -3/2 and test sign regions
A1 x < -3/2 or x > 2 cao
Answer: x < -3/2 or x > 2

Question 14

M1 factorise to (x+1)(x+3) or complete the square to find roots -3 and -1
A1 -3 < x < -1 cao
Answer: -3 < x < -1

Question 15

M1 use quadratic formula to find roots x = 1 ± √2
M1 state sign regions for parabola opening up
A1 x < 1 - √2 or x > 1 + √2 cao
Answer: x < 1 - √2 or x > 1 + √2

Question 16

M1 multiply by -1 (flip inequality) to get x² -6x +8 ≤ 0 and factor to (x-2)(x-4)
M1 identify interval between roots
A1 2 ≤ x ≤ 4 cao
Answer: 2 ≤ x ≤ 4

Question 17

M1 bring all terms to LHS to form x² -6x +7 < 0 and use quadratic formula to find roots
M1 compute roots x = 3 ± √2 and identify interval between roots
A1 3 - √2 < x < 3 + √2 cao
Answer: 3 - √2 < x < 3 + √2

Question 18

M1 factorise to (n-5)(n-2) and identify range 2 < n < 5
A1 n = 3, 4 cao
Answer: n = 3, 4

Quadratic Inequalities: Fluency and Exam Drill - Worksheets, Questions and Revision