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 integers.
2 −1
(3 marks)
1
3 Show that can be written as 2− 2
1
1+
2
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(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.
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(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)
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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
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(4 marks)
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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
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(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)
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