Which of the following is the correct solution to the inequality x2 - 16 ≤ 0 ?
A) x ≤ -4
B) x ≥ 4
C) -4 ≤ x ≤ 4
D) x ≤ -4 or x ≥ 4
(Total for Question 1 is 1 mark)
2
Solve the inequality x2 - 9 > 0.
(Total for Question 2 is 2 marks)
3
Solve the inequality x2 - 6x + 8 ≤ 0.
(Total for Question 3 is 3 marks)
4
Solve the inequality 2x2 - 7x - 4 ≥ 0.
(Total for Question 4 is 3 marks)
5
Solve the inequality x2 + 6 < 5x.
(Total for Question 5 is 3 marks)
6
The graph of y = x2 + 2x - 8 is sketched below. Use the sketch to write down the solution to x2 + 2x - 8 ≥ 0.
(Total for Question 6 is 2 marks)
7
Solve the inequality x2 - 5x - 14 < 0. Give your answer using set notation.
(Total for Question 7 is 3 marks)
8
Solve the inequality x2 - x - 6 ≤ 0. Hence write down all the integer values of x that satisfy the inequality.
(Total for Question 8 is 4 marks)
9
Solve the inequality 16 - x2 < 0.
(Total for Question 9 is 3 marks)
10
Solve the inequality x2 ≥ 49. Give your answer using set notation with the union symbol.
(Total for Question 10 is 3 marks)
11
Find the set of values of x that satisfy both x2 - 9 < 0 and x + 1 > 0.
(Total for Question 11 is 4 marks)
12
A rectangular lawn has length (x + 3) metres and width (x - 1) metres, where x > 1. The area of the lawn is greater than 45 m2. Form and solve an inequality to find the range of possible values of x.
(Total for Question 12 is 5 marks)
13
Solve the inequality x2 + 4x - 3 > 0, giving your answer in surd form.
(Total for Question 13 is 4 marks)
14
Show that x2 - 6x + 10 > 0 for all real values of x.
(Total for Question 14 is 3 marks)
15
Solve the inequality x2 + 2x + 5 < 0. Explain what your answer tells you about the graph of y = x2 + 2x + 5.
(Total for Question 15 is 3 marks)
16
Solve the inequality 12 - 4x - x2 < 0.
(Total for Question 16 is 4 marks)
17
Solve the inequality x4 - 5x2 + 4 < 0.
(Total for Question 17 is 6 marks)
18
The function f(x) = √20 - x - x2 is defined for real values of x. Find the set of values of x for which f(x) is defined.
(Total for Question 18 is 4 marks)
19
The height, h metres, of a ball t seconds after being thrown is given by h = 1 + 15t - 5t2. Find the values of t for which the ball is more than 11 metres above the ground.
(Total for Question 19 is 5 marks)
20
The equation x2 + kx + (k + 3) = 0 has no real roots. Find the set of values of k.
(Total for Question 20 is 5 marks)
Mark scheme · A4 Quadratic Inequalities
Question 1
B1 correct option identified (C)
Answer: C) -4 ≤ x ≤ 4
Question 2
M1 factorise to (x-3)(x+3) and identify critical values x = -3 and x = 3
A1 correct inequality x < -3 or x > 3 (oe)
Answer: x < -3 or x > 3
Question 3
M1 factorise to (x-2)(x-4)
A1 correct critical values x = 2 and x = 4
A1 correct inequality 2 ≤ x ≤ 4 (oe)
Answer: 2 ≤ x ≤ 4
Question 4
M1 factorise to (2x+1)(x-4)
A1 correct critical values x = -1/2 and x = 4
A1 correct inequality x ≤ -1/2 or x ≥ 4 (oe)
Answer: x ≤ -1/2 or x ≥ 4
Question 5
M1 rearrange to x2 - 5x + 6 < 0
M1 factorise to (x-2)(x-3)
A1 correct inequality 2 < x < 3 (oe)
Answer: 2 < x < 3
Question 6
B1 correctly reads roots x = -4 and x = 2 from the sketch
B1 correct inequality x ≤ -4 or x ≥ 2 (oe)
Answer: x ≤ -4 or x ≥ 2
Question 7
M1 factorise to (x-7)(x+2)
A1 correct critical values x = -2 and x = 7
A1 correct set notation {x : -2 < x < 7} (oe)
Answer: {x : -2 < x < 7}
Question 8
M1 factorise to (x-3)(x+2)
A1 correct critical values x = -2 and x = 3
A1 correct inequality -2 ≤ x ≤ 3 (oe)
B1 correct list of integers -2, -1, 0, 1, 2, 3 (ft from their inequality)
Answer: -2 ≤ x ≤ 3; integers -2, -1, 0, 1, 2, 3
Question 9
M1 rearrange to x2 > 16, correctly reversing the inequality
A1 correct critical values x = -4 and x = 4
A1 correct inequality x < -4 or x > 4 (oe)
Answer: x < -4 or x > 4
Question 10
M1 identify critical values x = -7 and x = 7
A1 correct inequality x ≤ -7 or x ≥ 7 (oe)
B1 correct set notation {x : x ≤ -7} U {x : x ≥ 7} (oe)
Answer: {x : x ≤ -7} U {x : x ≥ 7}
Question 11
M1 solve x2-9<0 to obtain -3<x<3
A1 correct -3 < x < 3
M1 solve x+1>0 to obtain x>-1
A1 correct combined interval -1 < x < 3 (ft)
Answer: -1 < x < 3
Question 12
M1 form area expression (x+3)(x-1)
M1 form inequality x2+2x-3>45 or equivalent x2+2x-48>0
M1 factorise to (x+8)(x-6)
A1 correct critical values x = -8 and x = 6
A1 correct final answer x > 6, rejecting x < -8 as it contradicts x > 1 (cao)
Answer: x > 6
Question 13
M1 apply the quadratic formula with a=1, b=4, c=-3
A1 correct roots x = -2 - √7 and x = -2 + √7 simplified
M1 identify correct inequality structure (outside the roots) using sign reasoning
A1 fully correct answer x < -2 - √7 or x > -2 + √7 (oe)
Answer: x < -2 - √7 or x > -2 + √7
Question 14
M1 attempt to complete the square
A1 correct form (x-3)2 + 1
A1 cso: since (x-3)2 ≥ 0 for all real x, (x-3)2+1 ≥ 1 > 0 for all real x
Answer: x2 - 6x + 10 = (x-3)2 + 1 > 0 for all real x
Question 15
M1 attempt discriminant b2-4ac = 4-20 = -16, or complete the square to (x+1)2+4
A1 correct conclusion: no real solutions (discriminant negative / minimum value 4 > 0)
B1 correct interpretation: the graph of y=x2+2x+5 never crosses or touches the x-axis (lies entirely above it)
Answer: No real solutions; the graph lies entirely above the x-axis
Question 16
M1 rearrange (e.g. multiply by -1, reversing the inequality) to x2+4x-12>0
M1 factorise to (x+6)(x-2)
A1 correct critical values x = -6 and x = 2
A1 correct inequality x < -6 or x > 2 (oe)
Answer: x < -6 or x > 2
Question 17
M1 substitute u=x2 to obtain u2-5u+4<0
M1 factorise to (u-1)(u-4)<0
A1 correct critical values u = 1 and u = 4, giving 1 < u < 4
M1 convert to 1 < x2 < 4, splitting into x2>1 and x2<4
A1 correct critical x values -2, -1, 1, 2
A1 fully correct final answer -2 < x < -1 or 1 < x < 2 (oe)
Answer: -2 < x < -1 or 1 < x < 2
Question 18
M1 set up condition 20-x-x2 ≥ 0 for the square root to be defined
M1 rearrange (e.g. multiply by -1) and factorise to (x+5)(x-4) ≤ 0
A1 correct critical values x = -5 and x = 4
A1 correct final domain -5 ≤ x ≤ 4 (oe)
Answer: -5 ≤ x ≤ 4
Question 19
M1 form inequality 1+15t-5t2>11
M1 rearrange and simplify (e.g. divide by -5, reversing) to t2-3t+2<0