Inequalities & Simultaneous Equations (CIE A Level Maths: Pure 1)

Exam Questions

4 hours48 questions
1a
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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.

1b
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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.

1c
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2 marks

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2a
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2 marks

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

2b
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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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3a
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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.
3b
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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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4a
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1 mark

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

4b
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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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5
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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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6
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4 marks

Substitute y equals space x space plus space 3 into the equation 2 x to the power of 2 space end exponent minus space y to the power of 2 space end exponent equals space 5 x space plus space 3 in order to solve the equations simultaneously.

Clearly state which values of x correspond to which values of y from your solutions.

 

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7
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4 marks

Solve the simultaneous equations

 y equals space 2 x space minus space 1

x to the power of 2 space end exponent plus space y to the power of 2 space end exponent minus space 2 space equals space 0

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8
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4 marks

Solve the simultaneous equations        

                      y space minus space x space minus space 1 space equals space 0

                     open parentheses 2 x space plus space 1 close parentheses to the power of 2 space end exponent space minus space 3 y to the power of 2 space end exponent space plus space 3 x space minus space 10 space equals space 0

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9
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4 marks

Solve the simultaneous equations

                        x space plus space 3 y space minus space equals space 0

                        x to the power of 2 space end exponent plus space 9 y space equals space 2 space minus space y squared

 

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1a
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2 marks

By eliminating from the equations
                3 x to the power of 2 space end exponent space plus space 4 y space equals space 83

                3 x space plus space 2 y space equals space minus 11

 show that x to the power of 2 space end exponent minus space 2 x space minus space 35 space equals space 0.

1b
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5 marks

Hence solve the simultaneous equations
                3 x to the power of 2 space end exponent plus space 4 y space equals 83
3 x space plus space 2 y space equals negative 11 subscript blank space

                 

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2a
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2 marks

By eliminating from the equations

                 x squared space plus space 10 x space plus space y to the power of 2 space end exponent space equals space minus 20

                 y space equals space 2 x space plus space 10

show that x to the power of 2 space end exponent plus space 10 x space plus space 24 space equals space 0.

2b
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5 marks

Hence solve the simultaneous equations
                x to the power of 2 space end exponent plus space 10 x space plus space y squared space equals space minus 20

                y space equals space 2 x space plus space 10

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3a
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2 marks

By eliminating y from the equations

                 x to the power of 2 space end exponent minus space 8 y space equals space minus 40

                 3 x space plus space 2 y space equals space 4
show that x to the power of 2 space end exponent plus space 12 x space plus space 24 space equals space 0.

3b
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5 marks

Hence solve the simultaneous equations
                x squared space minus space 8 y space equals space minus 40 subscript blank subscript blank

                3 x space plus space 2 y space equals space 4

giving x and y in the form a space plus-or-minus space b square root of 3, where a and b are integers.

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4a
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3 marks

      x squared minus 2 y equals 10

      2 x minus y equals k

are simultaneous equations, where k is a constant.

By eliminating  from the equations show that x squared minus 4 x plus 2 left parenthesis k minus 5 right parenthesis equals 0.

4b
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2 marks

By considering the discriminant of x squared minus 4 x plus 2 left parenthesis k minus 5 right parenthesis equals 0 find the value of k for which the simultaneous equations have only one solution.

4c
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3 marks

Find the solution to the simultaneous equations for the value of k that you found in part (b).

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5
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4 marks

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

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6
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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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7
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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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8a
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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

8b
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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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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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7 marks

Solve the simultaneous equations

                       x plus y equals 4

                       x squared minus 4 x minus 3 y equals 0

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2a
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2 marks

By eliminating  from the equations

                 4 x squared plus 5 x y plus 9 y squared equals 36

                 y equals 2 over 3 space x plus 2

show that x squared plus 3 x equals 0.

2b
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5 marks

Hence solve the simultaneous equations
                   4 x squared plus 5 x y plus 9 y squared equals 36

                    y equals 2 over 3 space x plus 2

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3a
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2 marks

By eliminating  from the equations

      15 x squared minus 4 y squared equals 12

      4 x plus 2 y equals 1

show that x squared minus 8 x plus 13 equals 0.

3b
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5 marks

Hence solve the simultaneous equations
                  15 x squared minus 4 y squared equals 12

                   4 x plus 2 y equals 1

giving x and y in the form a plus-or-minus space b space square root of 3 comma end root  where a and b are rational numbers.

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4a
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5 marks

x squared minus y plus 3 x equals k

20 x minus 4 y equals negative 5

are simultaneous equations, where k is a constant.

Given that the simultaneous equations have exactly one solution, find the value of the constant k.

4b
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3 marks

Find the solution to the simultaneous equations for the value of k that you found in part (a).

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5
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5 marks

A firework is launched inside a large shed with a sloping roof. In relation to the horizontal distance from the point it was launched, the height of the firework, h space m comma can be modelled by the quadratic equation

      h equals 0.84 x minus 0.07 x squared

The sloping roof of the shed can be modelled with the equation

      h equals 2 plus 0.1 x

screenshot-2023-07-25-at-12-30-26-pm

Determine whether, according to the model, the firework will hit the roof of the shed before escaping out the open end of the shed on the right of the diagram.

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6
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3 marks

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

 

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7
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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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8
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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 k is a constant, has two distinct real roots.  Find the possible values of k.

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9
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5 marks

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

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

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10
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5 marks

Find the values of x that satisfy the inequalities

x squared plus x less than 2
x squared less than 4

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11a
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4 marks

Solve the inequality negative 2 less or equal than x squared minus 4 less or equal than 5.

11b
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4 marks

Find the values of x that satisfy the inequalities

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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12a
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3 marks

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

x squared plus y squared less or equal than 25 y greater or equal than 0

On the axes below show the region satisfying the inequalities

2-4-edexcel-alevel-maths-pure-q7hard

12b
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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.

12c
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2 marks

Find the area of the cross-section of the tunnel.

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13a
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2 marks

An electronics company can produce c cables at a total cost of left parenthesis 200 plus 10 c right parenthesis pence.
The cables can be sold for left parenthesis 40 minus c right parenthesis pence each.

Show that the total income from selling c cables is left parenthesis 40 c minus c squared right parenthesis pence

13b
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4 marks

What is the minimum number of cables the company needs to sell in order to make a profit?

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14
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4 marks

A stone is projected vertically upwards from a height of 1.5 m.
It’s height, above its starting position, d m at time t seconds after launch, is given by

d left parenthesis t right parenthesis equals 16 t minus 4.9 t squared

How long does the stone remain 3 m above the ground?

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1
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7 marks

Solve the simultaneous equations

                        4 x squared plus 2 x minus 6 y equals 4

                        2 x minus 3 y equals negative 1

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2
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7 marks

Solve the simultaneous equations

                        9 x squared minus 7 x y plus 4 y squared equals 36

                        3 x plus 2 y equals negative 6

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3a
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2 marks

By eliminating  from the equations
                  8 y squared minus 3 x squared minus 4 x equals negative 11 over 2

                  3 x plus 4 y equals 1

show that  3 x squared minus 14 x plus 12 equals 0.

3b
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5 marks

Hence solve the simultaneous equations
8 y squared minus 3 x squared minus 4 x equals negative 11 over 2

3 x plus 4 y equals 1

giving x and y in the form a space plus-or-minus space b square root of c comma end rootwhere a and b are rational numbers and c is a prime number.

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4a
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6 marks

x squared plus 2 y squared equals 25

x minus y equals k

are simultaneous equations, where k is a constant.

Find the respective sets of values for k for which the simultaneous equations have one, two, and no solutions.

 

4b
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3 marks

Given that the simultaneous equations have exactly one solution, find all possible pairs open parentheses x comma y close parentheses that might correspond to that solution. Give all your values for x and y in the form a square root of 6 comma where a is a rational number.

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5a
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2 marks

The goal in a video game is to have a unicorn leap as far as possible in a horizontal direction without being destroyed by the death ray that is being fired overhead.  You hack into the game code and find that the height of the unicorn, h, is being modelled in relation to the horizontal distance from the point it jumps by the quadratic equation h space equals space minus 0.02 x open parentheses x minus k close parentheses comma where k space greater or equal than space 0  is a parameter that can be controlled by the player’s actions, and x is the horizontal distance in metres.  You also find that the path of the death ray is being modelled by the equation x space plus space 5 h space equals space 25.

screenshot-2023-07-25-at-2-47-13-pm

The value of h can never be less than zero, and if the path of the unicorn crosses or touches the path of the death ray, the unicorn is considered to have been destroyed.

Ignoring the problem of the death ray, explain why the parameter k represents the horizontal distance leapt by the unicorn.

5b
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6 marks

Your friend’s personal best in the game is a leap of 21.5 m without the unicorn being destroyed. He is determined to keep playing until his unicorn has leapt 22 m safely.  Determine whether or not your friend has a chance of reaching this goal.

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6
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4 marks

Solve the simultaneous inequalities

t squared minus 2 t minus 15 less than 0 and
t squared plus 14 less or equal than 9 t.

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7
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4 marks

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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8
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3 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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9a
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3 marks

On the axes below show the region satisfied 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

9b
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1 mark

Write down the equation(s) of any line(s) of symmetry of the region R.

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10
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6 marks

Solve the inequality negative 6 less or equal than x squared plus 3 x minus 4 less or equal than 6, giving your answer in set notation.

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11
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5 marks

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, giving your answer in interval notation.

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12a
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2 marks

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

y less or equal than 6 minus x squared over 6 y greater or equal than 0

On the axes below show the region satisfying the inequalities

2-4-edexcel-alevel-maths-pure-q7vhard

12b
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2 marks

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

12c
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3 marks

12d
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2 marks

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13
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3 marks

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

2-4-edexcel-alevel-maths-pure-q8vhard

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14a
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5 marks

An electronics company can produce c cables at a total cost of left parenthesis 160 plus 12 c right parenthesis pence.
The cables can then be sold for left parenthesis 38 minus c right parenthesis pence each.

Find the minimum and maximum number of cables the company needs to sell in order to make a profit?

14b
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1 mark

How many cables does the company need to sell to make the maximum profit?

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15
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5 marks

A stone is projected vertically upwards from a height of 2 m.
It’s height, above it’s starting position, d subscript 1 m, 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 upwards from a height of 2.3 m.
It’s height, above its starting position, is given by

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

For how long are both stones simultaneously at least 4 m above the ground?

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