Inequalities (Edexcel A Level Maths: Pure): Exam Questions

3 hours40 questions
12 marks

Solve

6 x minus 7 less or equal than 35

writing your answer in set notation.

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2
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3 marks

Solve

(i) 2 x greater or equal than 8

(ii) 3 plus 2 x less than 11

(iii) 5 plus x greater than 4 x minus 1

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34 marks

Solve

(i) 2 x minus 9 greater or equal than 5 left parenthesis x minus 3 right parenthesis

(ii) 3 left parenthesis 5 minus x right parenthesis less than 2 left parenthesis 9 minus 2 x right parenthesis

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4a1 mark

Write down the solutions to

left parenthesis x minus 3 right parenthesis left parenthesis x minus 8 right parenthesis equals 0

4b2 marks

Sketch the curve

space y equals left parenthesis x minus 3 right parenthesis left parenthesis x minus 8 right parenthesis

Label clearly the points at which the curve meets the x-axis.

4c2 marks

Hence solve the inequality

left parenthesis x minus 3 right parenthesis left parenthesis x minus 8 right parenthesis less than 0

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52 marks

On the axes below, sketch the region bounded by the following inequalities:

x greater or equal than 0
y less or equal than 4
x less or equal than 5
y greater or equal than 1

Label your region R.

2-4-edexcel-alevel-maths-pure-q5easy

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63 marks

Sketch the curve with equation

y equals 9 minus x squared

and hence solve the inequality

9 minus x squared space greater or equal than 0

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7a2 marks

Find, in terms of k, the discriminant of

x squared plus 8 x plus 4 k

7b2 marks

Hence find the values of for k which the equation 

x squared plus 8 x plus 4 k equals 0

has two real and distinct solutions.

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83 marks

Write down the three inequalities that define the region R shown in the diagram below.

2-4-edexcel-alevel-maths-pure-q8easy

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92 marks

Solve

6 less or equal than 8 x minus 2 less or equal than 22

writing your answer in the form

a less or equal than x less or equal than b

where a and b are integers to be found.

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102 marks

Solve the inequality

3 x plus 4 less or equal than 5 left parenthesis x minus 1 right parenthesis

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113 marks

Solve the inequality

negative 2 less or equal than 3 x minus 4 less or equal than 5

writing your answer in set notation.

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123 marks

Write down the three inequalities that define the region R shown in the diagram below.

2-4-edexcel-alevel-maths-pure-q8medium

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13 marks

In this question you should show all stages of your working.

Solutions relying on calculator technology are not acceptable.

Using algebra, solve the inequality

x squared minus x greater than 20

writing your answer in set notation.

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23 marks

Using algebra, solve the inequality

x squared minus 5 x greater than 6

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34 marks

On the axes below sketch the region defined by the inequalities

x plus 2 y greater than 3
y less or equal than x plus 4
y plus 3 x less than 8

Label your region R.

2-4-edexcel-alevel-maths-pure-q4medium

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44 marks

Given that

x squared plus 3 x greater than 4

and that

4 x plus 1 greater than 4

find the possible values of x.

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5a3 marks

The cross section of a tunnel is in the shape of the region defined by the inequalities

y less or equal than 5 minus x squared over 5

y greater or equal than 0

where x andspace y are measured in metres.

Sketch the region on the axes below and label it R.

2-4-edexcel-alevel-maths-pure-q7medium
5b2 marks

Write down the maximum height and maximum width of the tunnel.

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63 marks

The equation

x squared plus k x plus 4 equals 0

where k is a constant, has no real solutions.

Find the possible value(s) of k.

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4 marks

A stone is fired vertically upwards from ground level.

The vertical distance above the ground, d metres at time t seconds after launch, is given by

d left parenthesis t right parenthesis equals 12 t minus 4.9 t squared

Find the length of time that the stone spends at a height greater than 2 metres above the ground.

Give your answer to 3 significant figures.

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85 marks

On the axes below sketch the region defined by the inequalities

space y plus x greater than x squared
5 y less than 20 minus 4 x
space y minus 1 greater or equal than 0

Label your region R.

2-4-edexcel-alevel-maths-pure-q4hard

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94 marks

Find the three inequalities that define the region R shown in the diagram below.

2-4-edexcel-alevel-maths-pure-q8hard

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12 marks

Given that x and y are integers such that

  • x less than 0

  • left parenthesis x plus y right parenthesis squared less than 9 x squared plus y squared

show that y greater than 4 x

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25 marks
Graph showing a shaded region R below line l, and above a quadratic curve C which intersects the x-axis at O and 6. Line l intersects the y-axis at 60, and the curve C at two points, on of which has coordinates (10, 80).
Figure 3

Figure 3 shows a sketch of a curve C and a straight line l.

Given that

  • C has equation y equals straight f open parentheses x close parentheses where straight f open parentheses x close parentheses is a quadratic expression in x

  • C cuts the x-axis at 0 and 6

  • l cuts the y-axis at 60 and intersects C at the point open parentheses 10 comma space 80 close parentheses

use inequalities to define the region R shown shaded in Figure 3.

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34 marks

The equation

k x squared plus 2 k x plus 4 equals 0

where k is a constant has two distinct real roots.

Find the possible values of k.

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43 marks

Using algebra, solve the inequality

left parenthesis x plus 2 right parenthesis squared greater than 5

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53 marks

Use algebra to solve the inequality

fraction numerator 5 over denominator 3 x squared plus 2 end fraction less or equal than 2

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65 marks

The equation

left parenthesis k x right parenthesis squared plus left parenthesis k minus 2 right parenthesis x plus 1 equals 0

where k is a constant, has two distinct real roots. 

Find the possible values of k.

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75 marks

Find the values of x that satisfy both of the inequalities below.

x squared plus 4 x minus 3 less or equal than 2 minus x squared minus 5 x

8 minus 2 x squared less or equal than 2 x left parenthesis 2 x plus 1 right parenthesis

Give your answer in set notation.

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8a1 mark

An electronics company produces cables.

The company sells cables individually for left parenthesis 40 minus c right parenthesis pence each.

Write down an expression, in terms of c, for the total income (in pence) made from selling c cables.

8b4 marks

The cost to the company of making c cables is left parenthesis 200 plus 10 c right parenthesis pence.

By forming and solving a suitable inequality, find the minimum number of cables the company must sell in order to make a profit.

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94 marks

Find the values of t that satisfy both of the inequalities below.

t squared minus 2 t minus 15 less than 0

t squared plus 14 less or equal than 9 t

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104 marks

Use algebra to solve the inequality

fraction numerator 4 x squared minus 11 over denominator left parenthesis x plus 1 right parenthesis squared end fraction greater or equal than 4

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113 marks

On the axes below sketch the region defined by the inequalities

x squared minus 9 less or equal than y

y less or equal than left parenthesis 2 plus x right parenthesis left parenthesis 2 minus x right parenthesis

Label this region R.

2-4-edexcel-alevel-maths-pure-q4vhard

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126 marks

Use algebra to solve the inequality

negative 6 less or equal than x squared plus 3 x minus 4 less or equal than 6

writing your answer in set notation.

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14 marks
A graph with a parabola opening upwards, vertex at the origin, labelled x-axis and y-axis. Origin is marked O, axes have arrows indicating positive direction.
Figure 4

Figure 4 shows a sketch of the graph of y equals straight g open parentheses x close parentheses, where

straight g open parentheses x close parentheses equals open curly brackets table row cell open parentheses x minus 2 close parentheses squared plus 1 end cell cell x less or equal than 2 end cell row cell 4 x minus 7 end cell cell x greater than 2 end cell end table close

Find all the values of x for which

straight g open parentheses x close parentheses greater than 28

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25 marks
Graph depicting curve C and line I, intersecting at (-2, 13) and (0, 25). Shaded region R is bounded by the curve, line, and y-axis.
Figure 1

Figure 1 shows a sketch of a curve C with equation y equals straight f open parentheses x close parentheses and a straight line l.

The curve C meets l at the points (-2, 13) and (0, 25) as shown.

The shaded region R is bounded by C and l as shown in Figure 1.

Given that

  • straight f open parentheses x close parentheses is a quadratic function in x

  • (-2, 13) is the minimum turning point of y equals straight f open parentheses x close parentheses

use inequalities to define R.

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5 marks

The equation 

left parenthesis k plus 1 right parenthesis t squared plus 2 left parenthesis k plus 2 right parenthesis t equals 3 left parenthesis k plus 3 right parenthesis

has real roots.

Find the possible values of k.

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45 marks

Use algebra to solve the inequality

2 x squared plus 1 less or equal than x squared plus 10 x minus 8 less than 2 x squared minus 7 x plus 52

writing your answer in interval notation.

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53 marks

Find the three inequalities that define the region R shown in the diagram below.

2-4-q8-inequalities-a-level-maths

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6
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5 marks

A stone is projected vertically upwards from an initial height of 2 metres above the ground.

  • The vertical height of the stone above its initial position, d subscript 1 metres, at time t seconds after launch, is given by

 d subscript 1 left parenthesis t right parenthesis equals 13.2 t minus 4.9 t squared

At the same time, a second stone is projected vertically upwards from an height of 2.3 metres above the ground.

  • The vertical height of the second stone above its initial position, d subscript 2 metres, at time t seconds after launch, is given by

 d subscript 2 left parenthesis t right parenthesis equals 13 t minus 4.9 t squared

Find the length of time during which both stones are greater than 4 metres above the ground.

Give your answer to 3 significant figures.

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7a1 mark

A company produces x chairs andspace y tables in a day. 

They sell every chair and every table they produce. 

Due to the manufacturing processes involved, the number of chairs and tables they can make in a day are limited by the following five inequalities:

y less or equal than x plus 20
y greater or equal than 3 x minus 45

y less or equal than negative 2 x plus 80
x greater or equal than 0

y greater or equal than 0

Explain the significance of the inequalities x greater or equal than 0 space and y greater or equal than 0 in the context.

7b4 marks

On the axes below, sketch the region within which the company can produce x chairs andspace y tables per day.

2-4-edexcel-alevel-maths-pure-q11vhard
7c3 marks

The company’s profit, £ P per day, is given by the formula

P equals 3 x plus 2 y

A financial advisor tells the company that their maximum profit is found when x and y lie on a vertex of the region found in part (b), but did not say which vertex.

Use this information to find the number of chairs and tables the company should make in order to maximise its daily profit.

Show your working clearly.

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