IGCSEMaths: ExtendedCIETopic Questions2. Algebra & Graphs2.9 Forming & Solving Equations

Syllabus Edition

First teaching 2021

Last exams 2024

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Forming & Solving Equations (CIE IGCSE Maths: Extended)

Topic Questions

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1
Sme Calculator3 marks

Ahmed sells different types of cake in his shop.
The cost of each cake depends on its type and its size.

Every small cake costs $ x and every large cake costs $ left parenthesis 2 x space plus space 1 right parenthesis.

The number of small cherry cakes that can be bought for $4 is the same as the number of large cherry cakes that can be bought for $13.

Find the cost of a small cherry cake.

$ ................................................

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2
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Naga has n marbles.
Panav has three times as many marbles as Naga.
Naga loses 5 marbles and Panav buys 10 marbles.
Together they now have more than 105 marbles.

Write down and solve an inequality in n.

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3
Sme Calculator10 marks

cie-igcse-spec-p4-q5a-diagram

The perimeter of the rectangle is 80 cm.
The area of the rectangle is Acm2.

i)
Show that  x squared space space – space 40 x space plus space A space equals space 0.

[3]
ii)
When A space equals space 300, solve the equation space x squared space space – space 40 x space plus space A space equals space 0 space by factorising.
  
x = .................. or x = .................  [3]
iii)
When A space equals space 200, solve the equation space space x squared space space – space 40 x space plus space A space equals space 0 space using the quadratic formula.
Show all your working and give your answers correct to 2 decimal places.
  
x= .................. or x = .................  [4]

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4
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The cost of one ruler is r cents.
The cost of one protractor is p cents.

The total cost of 5 rulers and 1 protractor is 245 cents.
The total cost of 2 rulers and 3 protractors is 215 cents.

Write down two equations in terms of r and p and solve these equations to find the cost of one protractor.

........................................... cents 

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5
Sme Calculator9 marks

cie-igcse-2019-oct-nov-p4-tz2-q6b

The area of triangle DEF is 70 m2.

i)
Show that  4 x squared plus 11 x minus 300 equals 0.

 

[4]

ii)
Use the quadratic formula to solve 4 x squared plus 11 x minus 300 equals 0.
Show all your working and give your answers correct to 2 decimal places.

 

x = .......................... or x = .......................... [4]

iii)
Find the length of DE.

 

DE = ............................................... m [1]

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6a
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Paulo and Jim each buy sacks of rice but from different shops.
Paulo pays $72 for sacks costing $m each.
Jim pays $72 for sacks costing $left parenthesis m plus 0.9 right parenthesis each.

i)
Find an expression, in terms of m, for the number of sacks Paulo buys.
 
 

[1]

ii)
Find an expression, in terms of m, for the number of sacks Jim buys.
 
 

[1]

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6b
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Paulo buys 4 more sacks than Jim.
 
Write down an equation, in terms of m, and show that it simplifies to  10 m squared plus 9 m minus 162 equals 0.

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6c
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i)
Solve  10 m squared plus 9 m minus 162 equals 0. 
 
 

m = .................... or m = .................... [3]

ii)
Find the number of sacks of rice that Paulo buys.
 
 

[1]

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7a
Sme Calculator3 marks

The diagram shows a right-angled triangle ABC.

cie-igcse-2018-oct-nov-p4-tz2-q4

The area of this triangle is 30 cm2.

Show that  2 x squared plus 3 x minus 20 equals 0.

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7b
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Use factorisation to solve the equation  2 x squared plus 3 x minus 20 equals 0.
  

x = ...................... or x = ...................... 

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7c
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Calculate BC.
 

BC = .......................................... cm 

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8a
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In this question, all measurements are in metres.

cie-igcse-2018-may-jun-2-p4-q7
The diagram shows a right-angled triangle.

Show that 5 x squared minus 12 x minus 27 equals 0.

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8b
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Solve 5 x squared minus 12 x minus 27 equals 0.
Show all your working and give your answers correct to 2 decimal places.

x equals......................... or x equals......................... 

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8c
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Calculate the perimeter of the triangle.

............................................ m 

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8d
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Calculate the smallest angle of the triangle.

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9
Sme Calculator4 marks

At a football match, the price of an adult ticket is $x and the price of a child ticket is $left parenthesis x minus 2.50 right parenthesis.
There are 18 500 adults and 2400 children attending the football match.
The total amount paid for the tickets is $320 040.

Find the price of an adult ticket.

$ ................................................ 

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1a
Sme Calculator3 marks

Ahmed sells different types of cake in his shop.
The cost of each cake depends on its type and its size.

Every small cake costs $ x and every large cake costs $ left parenthesis 2 x space plus space 1 right parenthesis.

The total cost of 3 small lemon cakes and 2 large lemon cakes is $12.36.

Find the cost of a small lemon cake.

$ ................................................ 

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1b
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The cost of 18 small chocolate cakes is the same as the cost of 7 large chocolate cakes.

Find the cost of a small chocolate cake.

$ ................................................ [3]

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2
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cie-igcse-2020-oct-nov-p4-tz2-q8b

In the diagram, K, L and M lie on the circle, centre O.
Angle KML = space 2 x degree spaceand reflex angle KOL = 11 x degree.

Find the value of x.
  

x space equals ................................................ 

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3
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The angles of a quadrilateral are x degree, left parenthesis x plus 5 right parenthesis degree, left parenthesis 2 x minus 25 right parenthesis degree and left parenthesis x plus 10 right parenthesis degree.

Find the value of x.

x = ................................................... 

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4
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Oranges cost 21 cents each.
Alex buys x oranges and Bobbie buys left parenthesis x plus 2 right parenthesis oranges.
The total cost of these oranges is $4.20.

Find the value of x.

x = ................................................... 

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5
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In this part, all measurements are in metres.

cie-igcse-2018-may-jun-2-p4-q4b
The diagram shows a rectangle.
The area of the rectangle is 310 m2.

Work out the value of w.

w equals................................................ 

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6
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i)
Factorise  y squared plus 5 y minus 84.

[2]
ii)
cie-igcse-2018-may-jun-3-p4-q5bii
The area of the rectangle is 84 cm2.

Find the perimeter.

.......................................... cm [3]

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7
Sme Calculator3 marks

Here is a cuboid.

q5-meduim-2-13-forming-and-solving-equations-edexcel-gcse-maths

All measurements are in centimetres.
x is an integer.
The total volume of the cuboid is less than 900 cm3

Show that xless-than or slanted equal to 5

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8
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The diagram shows a garden in the shape of a rectangle.

q8-meduim-2-13-forming-and-solving-equations-edexcel-gcse-maths

All measurements are in metres.
The perimeter of the garden is 32 metres.

Work out the value of x.

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9
Sme Calculator5 marks

Redlands School sent x students to a revision day.

   St Samuel's School sent twice as many students as Redlands School.
   Francis Long School sent 7 fewer students than Redlands School.

   Each student paid £ 15 for the revision day.
   The students paid a total of £1155

   Work out how many students were sent by each school to the revision day.
   You must show all your working.

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10
Sme Calculator4 marks

Here is a rectangle.

q12-meduim-2-13-forming-and-solving-equations-edexcel-gcse-maths

All measurements are in centimetres.
The area of the rectangle is 48 cm2.
Show that  y space equals space 3

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1
Sme Calculator6 marks

Alan invests $200 at a rate of r percent sign per year compound interest.
After 2 years the value of his investment is $206.46.

i)
Show that     r to the power of 2 space end exponent plus space 200 r space minus space 323 space equals space 0.
[3]
ii)
Solve the equation   r to the power of 2 space end exponent plus space 200 r space minus space 323 space equals space 0  to find the rate of interest.
Show all your working and give your answer correct to 2 decimal places.
 
r space equals space................................................ [3]

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2
Sme Calculator7 marks

Ahmed sells different types of cake in his shop.
The cost of each cake depends on its type and its size.

Every small cake costs $ x and every large cake costs $ left parenthesis 2 x space plus space 1 right parenthesis.

Petra spends $20 on small coffee cakes and $10 on large coffee cakes.
The total number of cakes is 45.

Write an equation in terms of x.
Solve this equation to find the cost of a small coffee cake.
Show all your working.
  

$ ................................................ 

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3a
Sme Calculator5 marks

All the lengths in this question are in centimetres.

cie-igcse-2020-may-jun-p4-tz3-q5

The diagram shows a shape ABCDEF made from two rectangles.
The total area of the shape is 342 cm2.

Show that x squared plus x minus 72 equals 0.

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3b
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Solve  space x squared plus x minus 72 equals 0 space by factorisation.
 

x equals .................... or  x equals.................... 

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3c
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Work out the perimeter of the shape ABCDEF.
 

 ............................................ cm 

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3d
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Calculate angle DBC.
 

Angle DBC = ................................................ 

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4
Sme Calculator4 marks

cie-igcse-2019-oct-nov-1-q1e

A, B and C lie on the circle, centre O.
Angle AOC = left parenthesis 3 x plus 22 right parenthesis degree and angle ABC = 5 x degree.

Find the value of x.

x = ................................................... 

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5
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Carol walks 12 km at x km/h and then a further 6 km at left parenthesis x minus 1 right parenthesis km/h.
The total time taken is 5 hours.

i)
Write an equation, in terms of x, and show that it simplifies to  5 x squared minus 23 x plus 12 equals 0.

[3]

ii)
Factorise  5 x squared minus 23 x plus 12.

[2]

iii)
Solve the equation  5 x squared minus 23 x plus 12 equals 0.

x = .................... or x = ................... [1]

iv)
Write down Carol’s walking speed during the final 6 km.

........................................... km/h [1]

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6
Sme Calculator3 marks

Petra records the score in each test she takes.
 
The mean of the first n scores is x.
The mean of the first left parenthesis n minus 1 right parenthesis scores is left parenthesis x plus 1 right parenthesis.

Find the n text th end text score in terms of n and x.
Give your answer in its simplest form.

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7a
Sme Calculator5 marks

cie-igcse-2019-may-jun-p4-tz1-q7b 
The difference between the areas of the two rectangles is 62 cm2.

i)
Show that x squared plus 2 x minus 63 equals 0.
 
[3]
ii)
Factorise x squared plus 2 x minus 63.
 
[2]

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7b
Sme Calculator2 marks

Solve the equation x squared plus 2 x minus 63 equals 0 to find the difference between the perimeters of the two rectangles.
 

.............................................. cm

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8
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In a shop, the price of a monthly magazine is $m and the price of a weekly magazine is $left parenthesis m minus 0.75 right parenthesis.
One day, the shop receives
•  $168 from selling monthly magazines
•  $207 from selling weekly magazines.
The total number of these magazines sold during this day is 100.

i)
Show that  50 m squared minus 225 m plus 63 equals 0.

[3]
ii)
Find the price of a monthly magazine.
Show all your working.

$ ............................................... [3]

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9
Sme Calculator4 marks

Klaus buys x silver balloons and y gold balloons for a party.
 
He buys

  • more gold balloons than silver balloons
  • at least 15 silver balloons
  • less than 50 gold balloons
  • a total of no more than 70 balloons.

Write down four inequalities, in terms of x and/or y, to show this information.

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10
Sme Calculator2 marks

straight f left parenthesis x right parenthesis equals x squared over 4 minus 4 over x, x not equal to 0      straight g left parenthesis x right parenthesis equals 9 over x,  x not equal to 0.

The exact answer to straight f open parentheses x close parentheses space equals space straight g open parentheses x close parentheses is  cube root of k .
 
Use algebra to find the value of k.
 

k = ................................................. 

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11
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cie-igcse-2019-may-jun-p4-tz2-q11a

The sequence of diagrams above is made up of small lines and dots.

The number of dots in Diagram n is a n squared plus b n plus 1.
 
Find the value of a and the value of b.
 

a = .............................................. 
b = .............................................. 

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