SURDS
[ESTIMATED TIME: 75 minutes] r = .................... Q18 GCSE
(+ IGCSE) EXAM QUESTION PRACTICE
(Total 3 marks)
1. [2 marks]
19. Express 98 in the form a*b where a and b are integers and a > 1.
5.! Natasha!invested!$2600!!for!4!years!at!3%!per!annum!compound!interest.!
!
! (a)! Calculate!the!value!of!her!investment!at!the!end!of!4!years.!
! .......................... Q19
!
! 2. (Total 2 marks) [3 marks]
!
! 23 Express 48 + 108 in the form k 6 where k is a surd.
!
!
$ .........................................
(3)
! (b)! Work!out!the!amount!of!interest!that!Natasha!earned!in!these!4!years.!
!
!
!
!
!
N18957A 19 Turn over
$ .........................................
(2)
! ..............................................................
!
! 3. (Total for Question 23 is 3 marks) [2 marks]
!
6.! Show!that! 27 + 147!can!be!expressed!in!the!form!! !,!where!a!and!b!are!integers.!
!
!
!
!
Do NOT write in this space
!
!
............................................
(2)
!
Contains questions which have been reproduced with the kind Questions compiled by:
permission of Pearson Education Limited UK @Maths4Everyone
4. [3 marks]
19 Simplify (7 + 2 50 2)
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Give your answer in the form a + b 18 where a and b are integers.
Show your working clearly.
DO NOT WRITE IN THIS AREA
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5. (Total for Question 19 is 3 marks) [3 marks]
20 Show that (6 − 8 ) = 44 − 24 2
2
Show each stage of your working clearly.
DO NOT WRITE IN THIS AREA
(Total for Question 20 is 3 marks)
5 9
21 Solve + =2
( x + 2) ( x − 2)
20
*P45863A02024*
Show clear algebraic working.
6. [4 marks]
x. (a) Show that 48 + 108 can be expressed in the form ! !, where a and b are integers.
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(2)
(b) Show that 5 − 12 6 − 3 = 36 − 17 3
Show each stage of your working.
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(2)
7. [3 marks]
3 + 27
19 Show that can be expressed in the form k where k is an integer.
2
State the value of k.
k = ..............................................................
(Total for Question 19 is 3 marks)
4 3
20 Simplify fully +
x 2− x
8. [4 marks]
17 (a) Show that (3 + 2 2 )(4 − 2 ) = 8 + 5 2
Show your working clearly.
Leave
blank
18.
Diagram NOT
a cm 6.8 cm accurately drawn
(2)
10 + 3 2
(b) Rationalise the denominator and simplify fully
64° 35° 2
Show your working clearly.
Calculate the value of a.
Give your value correct to 3 significant figures.
...............................
a = ..................... (2)Q18
9. (Total for Question
(Total17
3 is 4 marks)
marks) [2 marks]
12
19. Show that =3 2
8
Do NOT write in this space.
Q19
21
*P44620A02124*(Total 2 marks) Turn over
10. [5 marks]
16 (a) Expand (5 + 3 2 ) 2
Give your answer in the form (a + b 2 ), where a and b are integers.
Show your working clearly.
.......................................................
(2)
2 q
(b) (5 + 3 2 ) = p + , where p and q are integers.
8
Find the value of q.
q = .........................................
(3)
(Total for Question 16 is 5 marks)
18
*P43074A01824*
y = ........................................................
11. [3 marks]
18 Solve 5x2 + 2x!"!#!$!% (Total for Question 16 is 5 marks)
Give your solutions correct
2
to 3 significant figures.
17 Show
Givenyour (5 − x )clearly.
that working = y − 20 2 where x and y are positive integers, find the value of
x and the value of y.
x = ............................
y = ............................
.. . .. . . .. . . .. . .. . . .. . .. . . .. . . .. . .. . . .. . . .. . .. . . .. . . .. . .. . . ..... ...............
12. (Total for Question 17 is 3 marks) [3 marks]
(Total for Question 18 is 3 marks)
19 (3 + a )(4 + a ) = 17 + k a where a and k are positive integers.
15
*P42940A01520*
Find the value of a and the value of k.
Turn over
a = . .. . . .. . .. . . .. . . ...... ...............
k = . .. . . .. . . .. . .. . . .. . .............. .....
(Total for Question 19 is 3 marks)
13
*P41036A01320* Turn over
13. [3 marks]
18 A trapezium ABCD has an area of 5 6 cm2.
Diagram NOT
A 4 cm B
accurately drawn
5 6 cm2 3 cm
D k cm C
AB = 4 cm.
BC = 3 cm.
DC = k cm.
Calculate the value of k, giving your answer in the form a b – c
where a, b and c are positive integers.
Show each step in your working.
k = ......................................
(Total for Question 18 is 3 marks)
Do NOT write in this space.
17
*P43028A01724* Turn over
14. [5 marks]
( )(
19 (a) Show that 5 − 8 7 + )
2 = 31 − 9 2
Show each stage of your working.
(3)
Given that c is a prime number,
3c − c
(b) rationalise the denominator of
c
Simplify your answer.
......................................................
(2)
(Total for Question 19 is 5 marks)
17
*P44389A01724* Turn over
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(3)
15. (Total for Question 21 is 5 marks) [3 marks]
( )
2
22 a + 8a = 54 + b 2
a and b are positive integers.
Find the value of a and the value of b.
Show your working clearly.
a = ......................................................
b = ......................................................
16. (Total for Question 22 is 3 marks) [3 marks]
22 (a + b )2 = 49 + 12 b where a and b are integers, and b is prime. 17
*P42933A01720*
Find the value of a and the value of b Turn over
a = .....................................................
b = .....................................................
17. (Total for Question 22 is 3 marks) [3 marks]
(6 − 5 )(6 + 5 )
23 ABC is a triangle.
19 Simplify fully
AB = 12 cm 31
DO NOT WRITE IN THIS AREA
DO NOT WRITE IN THIS AREA
You must show your working.
AC = 14 cm
The area of triangle ABC is 72 cm2
Find, in degrees, the two possible sizes of angle BAC.
Give your answers correct to the nearest degree.
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(Total for Question 19 is 3 marks)
20 Prove algebraically that the difference between the squares of any two consecutive
integers is equal to the sum of these two integers.
REA
DO NO
18. [3 marks]
18 + 10
Express 18 + 10 in the form p + q 2 , where p and q are integers.
Express 2 in the form p + q 2 , where p and q are integers.
2
Show clear working out.
Show clear working out.
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........................................................
(3)
(3)
19. [4 marks]
33
Rationalise the denominator and simplify fully 33
Rationalise the denominator and simplify fully 4 + 5
Show clear working out. 4 + 5
Show clear working out.
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(4)
(4)
20. [4 marks]
39
Express in the form a + b 3 , where a and b are integers
4− 3
Show clear 39
working out.form a + b 3 , where a and b are integers
Express in the
4− 3
Show clear working out.
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(4)
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21. (4) [4 marks]
7− 5
Simplify , giving your answer in the form a + b 5 , where a and b are integers.
2+ 5
Show 7 − 5
clear working out. your answer in the form a + b 5 , where a and b are integers.
Simplify , giving
2+ 5
Show clear working out.
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(4)
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(4)
22. [4 marks]
3
Show that can be written in the form m + n , where m and n are integers.
27 − 18
3
Show that can be written in the form m + n , where m and n are integers.
27 − 18
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(4)
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23. (4) [4 marks]
16
Show that − 8=6 2
162
Show that − 8=6 2
2
(4)
(4)
