Edexcel A Level Maths: Pure

Topic Questions

2.3 Simultaneous Equations

1
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2 marks

Solve the simultaneous equations

x plus y equals 8
x minus y equals 4

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2
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3 marks

Solve the simultaneous equations

2 x plus 5 y equals 3
6 x minus 10 y equals 34

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3a
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1 mark

Show that the equation 4 x plus 2 y minus 6 equals 0 can be written as y equals 3 minus 2 x.

3b
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3 marks

By substituting the result from part (a) into the equation 3 x plus 2 y minus 1 equals 1 spacesolve the equations

4 x plus 2 y minus 6 equals 0
3 x plus 2 y minus 1 equals 1

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4
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3 marks

Solve the simultaneous equations

5 x minus 2 y minus 16 equals 0
3 x plus 7 y plus 15 equals 0

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5
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4 marks

Substitute y equals x plus 3 spaceinto the equation 2 x squared minus y squared equals 5 x plus 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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6
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4 marks

Solve the simultaneous equations

y equals 2 x minus 1
x squared plus y squared minus 2 equals 0

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7
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3 marks

Solve the simultaneous equations

y equals 2 x plus 3
4 x minus 3 y plus 4 equals negative 1

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8
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4 marks

Solve the simultaneous equations

y minus x minus 1 equals 0
left parenthesis 2 x plus 1 right parenthesis squared minus 3 y squared plus 3 x minus 10 equals 0

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9
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4 marks

Solve the simultaneous equations

x plus 3 y minus 1 equals 0
x squared plus 9 y equals 2 minus y squared

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1
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4 marks

Use elimination to solve the simultaneous equations

3 x plus 4 y equals negative 13
2 x minus 3 y equals 14

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2
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4 marks

Use substitution to solve the simultaneous equations

4 y minus 5 x equals negative 2
3 x plus 2 y equals negative 12

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3
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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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4a
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2 marks

By eliminating y  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.

4b
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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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5a
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2 marks

By eliminating y  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.

5b
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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 b square root of 3, where a and b are rational numbers.

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6a
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4 marks

3 k x plus y equals negative 4
2 k x minus 4 y equals 2

are simultaneous equations, where k is a constant.

Solve the equations for x and space y, giving your answer for x in terms of the constant k.

6b
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1 mark

For what value of the constant k will the values of x and y in the solution be equal?

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7a
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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.

7b
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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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8
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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 m, 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

2-3-edexcel-alevel-maths-pure-q8hard

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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1
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4 marks

Use elimination to solve the simultaneous equations

7 x plus 4 y equals 17
3 x minus 2 y equals 11

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2
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4 marks

Use substitution to solve the simultaneous equations

2 x minus 5 y equals 4
x plus y equals negative 5

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3a
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2 marks

By eliminating y from the equations

3 x squared plus 4 y equals 83
3 x plus 2 y equals negative 11

show that x squared minus 2 x minus 35 equals 0.

3b
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5 marks

Hence solve the simultaneous equations

3 x squared plus 4 y equals 83
3 x plus 2 y equals negative 11

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4a
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2 marks

By eliminating y from the equations

x squared plus 10 x plus y squared equals negative 20
y equals 2 x plus 10

show that x squared plus 10 x plus 24 equals 0.

4b
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5 marks

Hence solve the simultaneous equations

x squared plus 10 x plus y squared equals negative 20
y equals 2 x plus 10

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5a
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2 marks

By eliminating y from the equations

x squared minus 8 y equals negative 40
3 x plus 2 y equals 4

show that x squared plus 12 x plus 24 equals 0.

5b
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5 marks

Hence solve the simultaneous equations

x squared minus 8 y equals negative 40
3 x plus 2 y equals 4

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

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6a
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3 marks

4 x minus k y equals 13
3 x plus 2 k y equals 7

are simultaneous equations, where k is a constant.

Show that x equals 3.

6b
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1 mark

Find an expression for y in terms of the constant k.

6c
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1 mark

Given that space y equals 3, find the value of k.

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7a
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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 y from the equations show that x squared minus 4 x plus 2 left parenthesis k minus 5 right parenthesis equals 0.

7b
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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.

7c
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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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8
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6 marks

You are asked to advise a client on which parcel delivery service to use to deliver parcels of differing sizes.  Linear Deliveries Inc. charges a flat rate of £2.25 per parcel, plus 40p times the mass of the parcel in kilograms.  Square Deal Delivery Solutions charges a flat rate of £4 per parcel, plus 1p times the square of the parcel’s mass in kilograms.  Under what circumstances would you advise your client to use each of the two delivery services?  Be sure to show clear mathematical justifications for your answer.

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1
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4 marks

Use elimination to solve the simultaneous equations

6 x minus 15 y equals negative 1
9 x plus 20 y equals 7

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2
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4 marks

Use substitution to solve the simultaneous equations

4 x plus 3 y equals 1
5 y minus 2 x equals negative 1

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3
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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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4
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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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5a
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2 marks

By eliminating space y 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.

5b
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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 space y in the form a plus-or-minus b square root of c, where a and b are rational numbers and c is a prime number.

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6a
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4 marks

5 x plus left parenthesis k plus 1 right parenthesis y equals negative 20
7 x minus 2 k y equals 2 y plus 6

are simultaneous equations, where k is a constant.

Solve the equations for x and space y, giving your answer forspace y in terms of the constant k.

6b
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1 mark

For what value of the constant k do the equations not have a solution?

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7a
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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.

7b
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3 marks

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

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8a
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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 equals negative 0.02 x left parenthesis x minus k right parenthesis, where k greater or equal than 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 plus 5 h equals 25

2-3-edexcel-alevel-maths-pure-q8vhard

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.

8b
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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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