GCSE
SURDS
r = ....................
Q18
(+ IGCSE) EXAM QUESTION PRACTICE
(Total 3 marks)
[2 marks]
[ESTIMATED TIME: 75 minutes]
1.
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.!
..........................
!
!
2.
(Total 2 marks)
!
!
! 23 Express 48 + 108 in the form k 6 where k is a surd.
!
!
!
!
!
!
!
!
Q19
[3 marks]
$ .........................................
(3)
(b)!
N18957A
Work!out!the!amount!of!interest!that!Natasha!earned!in!these!4!years.!
19
Turn over
$ .........................................
(2)
..............................................................
!
!
(Total for Question 23 is 3 marks)
3.
[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
permission of Pearson Education Limited UK
Questions compiled by:
@Maths4Everyone
4.
[3 marks]
19 Simplify (7 + 2 50
2)
DO NOT WRITE IN THIS AREA
Give your answer in the form a + b 18 where a and b are integers.
Show your working clearly.
(Total for Question 19 is 3 marks)
5.
DO NOT WRITE IN THIS AREA
........................................................
[3 marks]
20 Show that (6 − 8 ) 2 = 44 − 24 2
Show each stage of your working clearly.
DO NOT WRITE IN THIS AREA
(Total for Question 20 is 3 marks)
21 Solve
20
5
9
+
=2
( x + 2) ( x − 2)
*P45863A02024*
Show clear algebraic working.
6.
x.
[4 marks]
(a)
Show that 48 + 108 can be expressed in the form ! !, where a and b are integers.
............................................
(2)
(b)
Show that 5 − 12 6 − 3 = 36 − 17 3
Show each stage of your working.
............................................
(2)
7.
[3 marks]
3 + 27
can be expressed in the form
2
State the value of k.
19 Show that
k where k is an integer.
k = ..............................................................
(Total for Question 19 is 3 marks)
20 Simplify fully
4
3
+
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.
a cm
6.8 cm
(b) Rationalise the denominator and simplify fully
64°
35°
Show your working clearly.
Calculate the value of a.
Give your value correct to 3 significant figures.
Diagram NOT
accurately drawn
(2)
10 + 3 2
2
...............................
a = ..................... (2)Q18
(Total for Question
4 marks)
(Total17
3 is
marks)
9.
19. Show that
[2 marks]
12
=3 2
8
Do NOT write in this space.
*P44620A02124*(Total 2 marks)
Q19
21
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)
(b) (5 + 3 2 ) 2 = p +
Find the value of q.
q
, where p and q are integers.
8
q = .........................................
(3)
(Total for Question 16 is 5 marks)
18
*P43074A01824*
y = ........................................................
11.
(Total for Question 16 is 5 marks)
18 Solve 5x2 + 2x!"!#!$!%
Give your solutions correct
to
3
significant
figures.
2
(5 − x )clearly.
= y − 20 2 where x and y are positive integers, find the value of
17 Show
Givenyour
that working
x and the value of y.
[3 marks]
x = ............................
y = ............................
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... .... .... .... ..
(Total for Question 17 is 3 marks)
(Total for Question 18 is 3 marks)
12.
19 (3 + a )(4 + a ) = 17 + k a where a and k are positive integers.
*P42940A01520*
Find the value of a and the value of k.
[3 marks]
15
Turn over
a = . . . . . . . . . . . . . . . . . ... .... .... .... .... .
k = . . . . . . . . . . . . . . . . . . . .. ... .... .... .... ..
(Total for Question 19 is 3 marks)
*P41036A01320*
13
Turn over
13.
[3 marks]
18 A trapezium ABCD has an area of 5 6 cm2.
A
4 cm
3 cm
5 6 cm2
D
k cm
Diagram NOT
accurately drawn
B
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.
*P43028A01724*
17
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,
(b) rationalise the denominator of
3c − c
c
Simplify your answer.
......................................................
(2)
(Total for Question 19 is 5 marks)
*P44389A01724*
17
Turn over
(3)
(Total for Question 21 is 5 marks)
15.
22
(
a + 8a
)
2
[3 marks]
= 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 = ......................................................
(Total for Question 22 is 3 marks)
16.
22 (a +
b )2 = 49 + 12 b
where a and b are integers, and b is prime.
[3 marks]
17
*P42933A01720*
Turn over
Find the value of a and the value of b
a = .....................................................
b = .....................................................
(Total for Question 22 is 3 marks)
[3 marks]
Find, in degrees, the two possible sizes of angle BAC.
Give your answers correct to the nearest degree.
.........................................
(Total for Question 19 is 3 marks)
20 Prove algebraically that the difference between the squares of any two consecutive
DO NOT WRITE IN THIS AREA
23 ABC is a triangle.
(6 − 5 )(6 + 5 )
19 Simplify fully
31
AB = 12 cm
You must show your working.
AC = 14 cm
The area of triangle ABC is 72 cm2
DO NOT WRITE IN THIS AREA
17.
18.
[3 marks]
18 + 10
Express 18 + 10 in the form p + q 2 , where p and q are integers.
Express
in the form p + q 2 , where p and q are integers.
2
2
Show clear working out.
Show clear working out.
........................................................
........................................................
(3)
(3)
19.
[4 marks]
33
Rationalise the denominator and simplify fully
33
Rationalise the denominator and simplify fully 4 + 5
4+ 5
Show clear working out.
Show clear working out.
........................................................
........................................................
(4)
(4)
20.
[4 marks]
39
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.
Express
........................................................
(4)
........................................................
(4)
21.
[4 marks]
7− 5
Simplify
, giving your answer in the form a + b 5 , where a and b are integers.
2+ 5
7
− 5
Show 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.
........................................................
(4)
........................................................
(4)
22.
Show that
Show that
[4 marks]
3
27 − 18
3
27 − 18
can be written in the form
m + n , where m and n are integers.
can be written in the form
m + n , where m and n are integers.
23.
16
Show that
− 8=6 2
2
16
Show that
− 8=6 2
2
........................................................
(4)
........................................................
(4)
[4 marks]
(4)
(4)