IGCSEMaths: ExtendedCIEExam QuestionsAlgebra & SequencesForming & Solving Equations

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First teaching 2023

First exams 2025

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Forming & Solving Equations (CIE IGCSE Maths: Extended)

Exam Questions

5 hours63 questions

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Forming & Solving Equations

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Forming & Solving Equations

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1
Sme Calculator3 marks

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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2
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5 n is the mean of the three numbers 391, n and n minus 1.

Find the value of n.


n = ................................................... 

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3
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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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4
Sme Calculator5 marks

Julie and Liam write down the same number.

Julie multiplies the number by 5 and then adds 4 to the result.
She writes down her answer.

Liam subtracts the number from IO
He writes down his answer.

Julie's answer is two thirds of Liam's answer.

Work out the number that Julie and Liam started with.
You must show your working.

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5
Sme Calculator4 marks

A pattern is made using identical rectangular tiles.

q2-meduim-2-13-forming-and-solving-equations-edexcel-gcse-maths

Find the total area of the pattern.

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6
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Gemma has the same number of sweets as Betty.

Gemma gives 24 of her sweets to Betty.
Betty now has 5 times as many sweets as Gemma.

Work out the total number of sweets that Gemma and Betty have.

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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
Sme Calculator4 marks

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

A B C spaceis a triangle.

q10-meduim-2-13-forming-and-solving-equations-edexcel-gcse-maths

Angle A B C space=  angle B C A.

The length of side space A B spaceis left parenthesis 3 x space long dash space 5 right parenthesis cm.
The length of side space A C spaceis left parenthesis 19 space long dash x right parenthesis cm.
The length of side space B C spaceis 2 x cm.

Work out the perimeter of the triangle.
Give your answer as a number of centimetres.

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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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11
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 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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The diagram shows a trapezium.

cie-igcse-2020-may-june-p2-tz1-q6

Work out the value of x.


x equals .................................................. 

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3
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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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4
Sme Calculator5 marks

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 Calculator4 marks

The diagram shows a triangle.

q1-hard-2-13-forming-and-solving-equations-edexcel-gcse-maths

In the diagram, all the measurements are in metres.

The perimeter of the triangle is 56 m.
The area of the triangle is A m2.

Work out the value of A.

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6
Sme Calculator3 marks

Here is a right-angled triangle.

q2-1-hard-2-13-forming-and-solving-equations-edexcel-gcse-maths

Four of these triangles are joined to enclose the square space A B C D space as shown below.

q2-2-hard-2-13-forming-and-solving-equations-edexcel-gcse-mathsShow that the area of the square space A B C D spaceis x squared space plus space y squared

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7
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Becky has some marbles.
Chris has two times as many marbles as Becky.
Dan has seven more marbles than Chris.

They have a total of 57 marbles.

Dan says,
"If I give some marbles to Becky, each of us will have the same number of marbles."

Is Dan correct?
You must show how you get your answer.

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8
Sme Calculator4 marks

The diagram shows the plan of a floor.

q10-hard-2-13-forming-and-solving-equations-edexcel-gcse-maths

The area of the floor is 138 m2.

Work out the value of x.

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1a
Sme Calculator1 mark

In this question, all measurements are in centimetres. 

q21-paper2-spec2025-cie-igcse-further-maths

The height of the triangle is h and the height of the rectangle is (h space plus space 2).
The length of the base of the triangle is xand the length of the rectangle is (x space plus space 1).
The area of the triangle is 11cm2 and the area of the rectangle is 39cm2.

Write down an expression, in terms of x, for the height of the rectangle. 

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1b
Sme Calculator3 marks

Show that space 2 x squared space – space 15 x space plus space 22 space equals space 0 space.

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1c
Sme Calculator5 marks

By factorising and solving 2 x squared space – space 15 x space plus space 22 space equals space 0 , find the two possible heights of the triangle.

h space equals space.............. space or space h space equals space................

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2
Sme Calculator3 marks

m is inversely proportional to the square of open parentheses p minus 1 close parentheses.
When p equals 4 comma space m equals 5.

Find m when p equals 6.


m equals ..................................................

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3
Sme Calculator7 marks

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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4
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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5a
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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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5b
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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6
Sme Calculator2 marks

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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7
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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8
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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9
Sme Calculator4 marks

q1-veryhard-2-13-forming-and-solving-equations-edexcel-gcse-maths

A B C D is a square with a side length of 4 x
M is the midpoint of D C.
N is the point on A D where N D space equals space x

B M N is a right-angled triangle.

Find an expression, in terms of x, for the area of triangle B M N.
Give your expression in its simplest form.

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10
Sme Calculator5 marks

q3-veryhard-2-13-forming-and-solving-equations-edexcel-gcse-maths

The lengths of the sides are in centimetres.

The area of triangle bold T subscript bold 1 is equal to the area of triangle bold T subscript bold 2.
Work out the value of x, giving your answer in the form a space plus space square root of b where a and b are integers.

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11
Sme Calculator4 marks

The diagram shows a rectangle, A B D E , and two congruent triangles, space A F E spaceand B C D.

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

area of rectangle A B D E = area of triangle A F E + area of triangle B C D

A B space colon thin space A E space equals space 1 space colon space 3

Work out the length of A E.

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

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