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

3 hours43 questions
1
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3 marks

Solve the inequalities:

(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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2
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4 marks

Solve the inequalities:

(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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3a
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2 marks

Write down the solutions to left parenthesis x minus 3 right parenthesis left parenthesis x minus 8 right parenthesis equals 0.

3b
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2 marks

Sketch the graph of space y equals left parenthesis x minus 3 right parenthesis left parenthesis x minus 8 right parenthesis, clearly showing the coordinates of the points where the graph intercepts the x-axis.

3c
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2 marks

Hence, or otherwise, solve the inequality left parenthesis x minus 3 right parenthesis left parenthesis x minus 8 right parenthesis less than 0.

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4a
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2 marks

Find the discriminant for the quadratic function x squared plus 8 x plus 15.

4b
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2 marks

Write down the number of real solutions to the equation x squared plus 8 x plus 15 space equals 0.

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5
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4 marks

On the axes below, show the region bounded by the 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

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

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6a
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3 marks
(i)
Solve the equation 9 minus x squared equals 0.
(ii)
Use symmetry to write down the coordinates of the turning point on the graph of y equals 9 minus x squared.
6b
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3 marks

Sketch the graph of 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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7a
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1 mark

Write down, in terms of k, the discriminant of x squared plus 8 x plus 4 k.

7b
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2 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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8
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3 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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9
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4 marks

The total cost to a company manufacturing c cables is left parenthesis 500 plus 3 c right parenthesis pence.

The total income from selling all c cables is left parenthesis 5 c minus 3500 right parenthesis pence.

What is the minimum number of cables the company needs to sell in order to recover their costs?

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10
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4 marks

The equation x squared plus k x plus 4 equals 0, where k is a constant, has no real roots.

Find the possible value(s) of k.

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11
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4 marks

Solve the inequality 6 x minus 7 less or equal than 35, giving your answer in set notation.

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12
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3 marks

Solve the inequality 6 less or equal than 8 x minus 2 less or equal than 22.

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1
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3 marks

Solve the inequality 3 x plus 4 less or equal than 5 left parenthesis x minus 1 right parenthesis.

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2
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4 marks

Solve the inequality x squared minus 5 x greater than 6.

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3
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4 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 value(s) of k.

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4
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5 marks

On the axes below show the region satisfied 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 this region R.

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

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5
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5 marks

Find the values of x that satisfy the inequalities

x squared plus 3 x greater than 4
4 x plus 1 greater than 4

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6
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4 marks

Solve the inequality negative 2 less or equal than 3 x minus 4 less or equal than 5, giving your answer in set notation.

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7a
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3 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

On the axes below show the region satisfying the inequalities

2-4-edexcel-alevel-maths-pure-q7medium

7b
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2 marks

Given that x andspace y are in metres write down the height and the maximum width of the tunnel.

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8
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4 marks

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

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

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9
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4 marks

The total cost to a company manufacturing c cables is left parenthesis 100 plus 5 c right parenthesis pence.

The total income from selling all c cables is left parenthesis 30 c minus c squared right parenthesis pence.

What is the minimum number of cables the company needs to sell in order to recover their costs?

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10
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4 marks

A stone is projected vertically upwards from ground level.

The distance above the ground, d m at t seconds after launch, is given by

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

How long does the stone remain 2 m above the ground?

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1
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3 marks

Solve the inequality left parenthesis x plus 2 right parenthesis squared greater than 5.

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2
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4 marks

Solve the inequality fraction numerator 5 over denominator 3 x squared plus 2 end fraction less or equal than 2.

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3
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4 marks

The equation left parenthesis k x right parenthesis squared plus left parenthesis k minus 2 right parenthesis x plus 1 equals 0, where