GCSE MATHEMATICS
Aiming for Grade 9
REVISION BOOKLET
Exam Dates:
________________
________________
________________
Name: ______________________________
1
,Contents
Page:
Number:
Surds 3
Algebraic proofs 7
Algebra:
Transformations of graphs 12
Equations of circles 16
Quadratic and other sequences 18
Completing the square 21
Inverse and composite functions 24
Expanding more than two binomials 27
Nonlinear simultaneous equations 29
Solving quadratic inequalities 32
Shape, Space and Measure:
Circle theorems 34
Vectors 38
Sine and cosine rules 46
Area under graphs 50
Data Handling:
Histograms 55
Capture-Recapture 63
Probability:
Set theory 66
Ratio and Proportion:
Proportion 69
Percentages – reverse 73
2
,Surds
Things to remember:
√ means square root;
To simplify surds, find all its factors;
To rationalise the denominator, find an equivalent fraction where the denominator is
rational.
Questions:
1. Work out
(5 + √3)(5− √3)
√22
Give your answer in its simplest form.
……………………………………
(Total 3 marks)
1
2. (a) Rationalise the denominator of
√3
……………………………………
(1)
(b) Expand (2 + √3)(1 + √3)
Give your answer in the form 𝑎 + 𝑏√3 where a and b are integers.
……………………………………
(2)
(Total 3 marks)
3
, 1
3. (a) Rationalise the denominator of
√7
……………………………………
(2)
(b) (i) Expand and simplify (√3 + √15)2
Give your answer in the form 𝑎 + 𝑏√3 where a and b are integers.
……………………………………
(ii) All measurements on the triangle are in centimetres.
ABC is a right-angled triangle.
k is a positive integer.
A C
Diagram NOT
accurately drawn
k 3 +
B
Find the value of k.
k = ……………………………………
(5)
(Total 7 marks)
4
Aiming for Grade 9
REVISION BOOKLET
Exam Dates:
________________
________________
________________
Name: ______________________________
1
,Contents
Page:
Number:
Surds 3
Algebraic proofs 7
Algebra:
Transformations of graphs 12
Equations of circles 16
Quadratic and other sequences 18
Completing the square 21
Inverse and composite functions 24
Expanding more than two binomials 27
Nonlinear simultaneous equations 29
Solving quadratic inequalities 32
Shape, Space and Measure:
Circle theorems 34
Vectors 38
Sine and cosine rules 46
Area under graphs 50
Data Handling:
Histograms 55
Capture-Recapture 63
Probability:
Set theory 66
Ratio and Proportion:
Proportion 69
Percentages – reverse 73
2
,Surds
Things to remember:
√ means square root;
To simplify surds, find all its factors;
To rationalise the denominator, find an equivalent fraction where the denominator is
rational.
Questions:
1. Work out
(5 + √3)(5− √3)
√22
Give your answer in its simplest form.
……………………………………
(Total 3 marks)
1
2. (a) Rationalise the denominator of
√3
……………………………………
(1)
(b) Expand (2 + √3)(1 + √3)
Give your answer in the form 𝑎 + 𝑏√3 where a and b are integers.
……………………………………
(2)
(Total 3 marks)
3
, 1
3. (a) Rationalise the denominator of
√7
……………………………………
(2)
(b) (i) Expand and simplify (√3 + √15)2
Give your answer in the form 𝑎 + 𝑏√3 where a and b are integers.
……………………………………
(ii) All measurements on the triangle are in centimetres.
ABC is a right-angled triangle.
k is a positive integer.
A C
Diagram NOT
accurately drawn
k 3 +
B
Find the value of k.
k = ……………………………………
(5)
(Total 7 marks)
4
