O Level Cambridge Maths 2 hours 53 questions
All Questions
Surds
Simplifying Surds / Rationalising Denominators
Medium (21 questions) /49 Scan here to return to the course
or visit savemyexams.com
Hard (17 questions) /50
Very Hard (15 questions) /50
Total Marks /149
© 2024 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 1
Medium Questions
1 Simplify.
32 + 98
(2 marks)
© 2024 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 2
12
2 (a) Rationalise the denominator of .
3
(2 marks)
(b) Work out the value of ( 2 + 8 )2.
(2 marks)
3 Work out the value of ( 12 − 3 ) 2 .
(2 marks)
10
4 Rationalise the denominator of .
5
Give your answer in its simplest form.
(2 marks)
12
5 Show that
3+2
can be written in the form a b where a is a simplified fraction and b is an integer.
© 2024 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 3
(2 marks)
6 Show that (6 + 2 12 ) 2 = 12(7 + 4 3 )
Show each stage of your working.
(3 marks)
20 + 80
7 Show that can be expressed in the form a where a is an integer.
3
Show your working clearly.
(3 marks)
8 Show that 45 + 20 = 5 5
Show your working clearly.
(2 marks)
10
9 Rationalise the denominator of
6
Give your answer in its simplest form
© 2024 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 4
(2 marks)
10 Show that 20 = 2 5 .
(2 marks)
11 Write (1 + 3 ) 2 in the form a + b 3 .
(3 marks)
12 Simplify 6× 3.
(2 marks)
13 Simplify fully by rationalising the denominator.
20
5
(3 marks)
© 2024 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 5
14 Simplify fully.
200
(2 marks)
15 Write 12 + 75 in the form k 3 .
(3 marks)
16 Simplify fully.
50 + 2
(2 marks)
17 Work out 2 10 × 80 × 18
Give your answer as an integer.
(3 marks)
© 2024 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 6
10
18 Rationalise the denominator and simplify
3 5
(2 marks)
19 8300 = 100 × 83
Circle the number that is closest in value to 8300
19 90 830 900
(1 mark)
14
20 Show that can be written in the form a b where a and b are integers.
7
(2 marks)
21 Simplify.
75 + 27
(2 marks)
© 2024 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 7
Hard Questions
1 Rationalise the denominator.
1
2 +1
(2 marks)
© 2024 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 8
5
2 (a) Rationalise the denominator of .
2
(2 marks)
(b) Expand and simplify (2 + 3 ) 2 − (2 − 3 ) 2 .
(2 marks)
3 Expand (1 + 2 ) (3 − 2 )
Give your answer in the form a + b 2 where a and b are integers.
(2 marks)
© 2024 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 9
4 (a) Martin did this question.
14
Rationalise the denominator of
2+ 3
Here is how he answered the question.
14 14 × (2 − 3)
=
2+ 3 (2 + 3 ) (2 − 3)
28 − 14 3
=
4 + 2 3 − 2 3 +3
28 − 14 3
=
7
= 4− 2 3
Martin's answer is wrong.
Find Martin's mistake.
(1 mark)
(b) Sian did this question.
5
Rationalise the denominator of
12
Here is how she answered the question.
5 5 12
=
12 12 × 12
5 ×3 2
=
12
5 2
=
4
© 2024 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 10
Sian's answer is wrong.
Find Sian's mistake.
(1 mark)
(6 − 5 ) (6 + 5 )
5 Simplify fully
31
You must show your working.
(3 marks)
6 5( 8 + 18 ) can be written in the form a 10 where a is an integer.
Find the value of a .
(3 marks)
(4 − 3 ) (4 + 3 )
7 Show that simplifies to 13 .
13
(2 marks)
© 2024 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 11
8 (a) Express 3+ 12 in the form a 3 where a is an integer.
(2 marks)
⎛⎜ 1 ⎞⎟7 b
(b) Express ⎜⎜ ⎟⎟ in the form where b and c are integers.
⎝ 3 ⎠
c
(3 marks)
4
9 Rationalise the denominator of
7− 5
Show each stage of your working.
Give your answer in the form a + b 5 where a and b are fractions in their simplest
forms.
(3 marks)
10 Given that y is a prime number,
3 a +b y
express in the form where a , b and c are integers.
2− y c−y
© 2024 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 12
(2 marks)
6
11 Without using a calculator, rationalise the denominator of
3− 7
Simplify your answer.
You must show each stage of your working.
(3 marks)
8
12 Show that can be written in the form n + n , where n is an integer.
8 −2
Show your working clearly.
(3 marks)
2
13 Express in the form p + q
3 −1
where p and q are integers.
Show your working clearly.
(2 marks)
© 2024 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 13
14
14 Show that can be written in the form a + b 2.
3− 2
(5 marks)
150 − 6
15 Show that simplifies to an integer.
2× 3
(3 marks)
16 Show that (5 3 − 12 ) 2 simplifies to an integer.
(3 marks)
17 Rationalise the denominator.
3
2+ 5
© 2024 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 14
Give your answer in its simplest form.
(3 marks)
© 2024 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 15
Very Hard Questions
a= 8 +2
1 b = 8 −2
T = a2 − b 2
Work out the value of T .
Give your answer in the form c 2 where c is an integer.
(4 marks)
6− 8
2 Show that can be written in the form a + b 2 where a and b are
2 −1
integers.
(3 marks)
1
3 Show that can be written as 2− 2
1
1+
2
© 2024 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 16
(3 marks)
4 a= 8 +4
b = 8 −4
( a – b ) ( a + b ) can be written in the form y 4y
Find the value of y
Show your working clearly.
(3 marks)
a+ 4b
5 Rationalise the denominator of where a is an integer and b is a prime
a− 4b
number.
Simplify your answer.
(3 marks)
2 a+ a
6 Show that can be written in the form
6−3 2 b
where a and b are integers.
Show your working clearly.
© 2024 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 17
(3 marks)
8
7 Express in the form a +b where a and b are integers.
5 −1
Show each stage of your working clearly.
(3 marks)
8 The area of a rectangle is 18 cm2
The length of the rectangle is ( 7 + 1) cm.
Without using a calculator and showing each stage of your working, find the width of the
rectangle.
Give your answer in the form a b +c where a , b and c are integers.
..................... cm
(3 marks)
© 2024 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 18
4+ 8
9 Show that can be written in the form a + b 2 , where a and b are integers.
2 −1
Show each stage of your working clearly and give the value of a and the value of b .
(3 marks)
10 In the following equation, n is an integer greater than 1.
( 2 )n = k 2
i) Find k when n = 7.
k = ...................... [2]
ii) Find n when k = 64.
n = ..................... [2]
(4 marks)
(4 + 2 5 ) 3 5 +7
11 Show that can be simplified to
5 −1 2
© 2024 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 19
(4 marks)
© 2024 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 20
12 (a) Here is a sequence.
5 5 3 15 15 3
Work out the next term.
(1 mark)
(b) Find the n th term.
(3 marks)
2
13 Simplify 80 + 2
9
a 5
Give your answer in the form where a and b are integers.
b
(3 marks)
28
14 Work out 18 −
50
a
Give your answer in the form where a and b are integers.
b
© 2024 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 21
(4 marks)
2 6 3 c d
15 Show that − can be written in the form
5 10 10
where c and d are integers.
(3 marks)
© 2024 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 22
