Which of the following is the correct solution to the inequality x2 - 25 ≤ 0 ?
A) x ≤ -5
B) x ≥ 5
C) -5 ≤ x ≤ 5
D) x ≤ -5 or x ≥ 5
(Total for Question 1 is 1 mark)
2
Solve the inequality x2 - 16 > 0.
(Total for Question 2 is 1 mark)
3
Solve the inequality x2 - 7x + 10 ≤ 0.
(Total for Question 3 is 1 mark)
4
Solve the inequality 3x2 - 5x - 2 ≥ 0.
(Total for Question 4 is 1 mark)
5
Solve the inequality x2 + 3 < 4x.
(Total for Question 5 is 1 mark)
6
Solve the inequality x2 ≥ 121. Give your answer using set notation with the union symbol.
(Total for Question 6 is 1 mark)
7
Solve the inequality 9 - x2 < 0.
(Total for Question 7 is 1 mark)
8
Solve the inequality x2 - 8x + 12 ≤ 0.
(Total for Question 8 is 2 marks)
9
Solve the inequality x2 - x - 20 < 0. Give your answer using set notation.
(Total for Question 9 is 2 marks)
10
Solve the inequality 20 - x - x2 ≤ 0.
(Total for Question 10 is 2 marks)
11
Find the set of values of x that satisfy both x2 - 16 < 0 and x - 2 > 0.
(Total for Question 11 is 2 marks)
12
Solve the inequality x2 + 2x - 1 > 0, giving your answer in surd form.
(Total for Question 12 is 2 marks)
13
Solve the inequality x2 - 4x - 3 < 0, giving your answer in surd form.
(Total for Question 13 is 2 marks)
14
Solve the inequality 2x2 + 3x - 9 ≥ 0, giving your answer in set notation.
(Total for Question 14 is 3 marks)
15
Find the set of values of x that satisfy both x2 - 9 ≤ 0 and 2x + 3 > 0.
(Total for Question 15 is 3 marks)
16
A rectangular lawn has length (x + 7) metres and width (x - 2) metres, where x > 2. The area of the lawn is greater than 70 m2. Form and solve an inequality to find the range of possible values of x.
(Total for Question 16 is 3 marks)
17
The function f(x) = √15 - 2x - x2 is defined for real values of x. Find the set of values of x for which f(x) is defined.
(Total for Question 17 is 3 marks)
18
The height, h metres, of a ball t seconds after being thrown is given by h = 2 + 18t - 6t2. Find the values of t for which the ball is more than 14 metres above the ground.
(Total for Question 18 is 4 marks)
19
Solve the inequality x4 - 29x2 + 100 < 0.
(Total for Question 19 is 5 marks)
Mark scheme · A4D Quadratic Inequalities: Fluency and Exam Drill
Question 1
B1 correct option identified (C)
Answer: C) -5 ≤ x ≤ 5
Question 2
B1 x < -4 or x > 4 (oe) cao
Answer: x < -4 or x > 4
Question 3
B1 2 ≤ x ≤ 5 (oe) cao
Answer: 2 ≤ x ≤ 5
Question 4
B1 x ≤ -1/3 or x ≥ 2 (oe) cao
Answer: x ≤ -1/3 or x ≥ 2
Question 5
B1 1 < x < 3 (oe) cao
Answer: 1 < x < 3
Question 6
B1 {x : x ≤ -11} U {x : x ≥ 11} (oe) cao
Answer: {x : x ≤ -11} U {x : x ≥ 11}
Question 7
B1 x < -3 or x > 3 (oe) cao
Answer: x < -3 or x > 3
Question 8
M1 factorise to (x-2)(x-6)
A1 2 ≤ x ≤ 6 (oe) cao
Answer: 2 ≤ x ≤ 6
Question 9
M1 factorise to (x-5)(x+4) and identify critical values x = -4 and x = 5
A1 correct set notation {x : -4 < x < 5} (oe)
Answer: {x : -4 < x < 5}
Question 10
M1 rearrange (e.g. multiply by -1, reversing the inequality) to x2+x-20≥0 and factorise to (x+5)(x-4)
A1 x ≤ -5 or x ≥ 4 (oe)
Answer: x ≤ -5 or x ≥ 4
Question 11
M1 solve x2-16<0 to obtain -4<x<4, and x-2>0 to obtain x>2
A1 correct combined interval 2 < x < 4 (ft)
Answer: 2 < x < 4
Question 12
M1 apply the quadratic formula with a=1, b=2, c=-1 to find roots x = -1 - √2 and x = -1 + √2
A1 x < -1 - √2 or x > -1 + √2 (oe)
Answer: x < -1 - √2 or x > -1 + √2
Question 13
M1 apply the quadratic formula with a=1, b=-4, c=-3 to find roots x = 2 - √7 and x = 2 + √7
A1 2 - √7 < x < 2 + √7 (oe)
Answer: 2 - √7 < x < 2 + √7
Question 14
M1 factorise to (2x-3)(x+3)
A1 correct critical values x = -3 and x = 3/2
A1 correct set notation {x : x ≤ -3} U {x : x ≥ 3/2} (oe)
Answer: {x : x ≤ -3} U {x : x ≥ 3/2}
Question 15
M1 solve x2-9≤0 to obtain -3≤x≤3
M1 solve 2x+3>0 to obtain x>-1.5
A1 correct combined interval -1.5 < x ≤ 3 (ft)
Answer: -1.5 < x ≤ 3
Question 16
M1 form inequality (x+7)(x-2)>70, i.e. x2+5x-84>0
M1 factorise to (x+12)(x-7)
A1 correct final answer x > 7, rejecting x < -12 as it contradicts x > 2 (cao)
Answer: x > 7
Question 17
M1 set up condition 15-2x-x2 ≥ 0 for the square root to be defined
M1 rearrange (e.g. multiply by -1) and factorise to (x+5)(x-3) ≤ 0
A1 correct final domain -5 ≤ x ≤ 3 (oe)
Answer: -5 ≤ x ≤ 3
Question 18
M1 form inequality 2+18t-6t2>14
M1 rearrange and simplify (e.g. divide by -6, reversing) to t2-3t+2<0
M1 factorise to (t-1)(t-2)
A1 correct final answer 1 < t < 2 (seconds) (oe)
Answer: 1 < t < 2 (seconds)
Question 19
M1 substitute u=x2 to obtain u2-29u+100<0
M1 factorise to (u-25)(u-4)<0
A1 correct critical values u = 4 and u = 25, giving 4 < u < 25
M1 convert to 4 < x2 < 25, splitting into x2>4 and x2<25
A1 fully correct final answer -5 < x < -2 or 2 < x < 5 (oe)