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1. NUMBER, SET NOTATION
AND LANGUAGE
1.
Consider the nine numbers 0.4, 2 ,
3
5 ,  , 7, 9, 23, 40, 121.
Write down
(a) the two prime numbers,
(b) the two square numbers,
(c) the irrational numbers.
2.
(a) Consider the numbers 2 ,  , 47 , 3 11, 7 , 3 17 , e, 1.277
5
32
20
Write down
(i)
The four rational numbers, and
(ii) The four irrational numbers.
(b) Write down the prime numbers between 30 and 50.
(c) Write down the tens digit and thousands digit of the 2568349
(d) Express 180 as the product of its prime factors.
(e) Given two numbers 60 and 126, find
(i)
the largest integer which is a factor of both 60 and 126,
(ii) the smallest integer which is an exact multiple of both 60 and 126.
Number, Set Notation and Language
1
3.
(a) Express 63 as the product of its prime factors.
(b) What is the smallest positive integer value of n for which 63n is a multiple of 35.
4.
(a)
Find the HCF of 12 and 15.
(b) Find the HCF of 28, 42 and 70.
5.
Find the LCM of
(a) 12 and 15
(b) 3, 4 and 8
6.
What is the minimum number of tiles of dimensions 21 cm x 21 cm required to form a square?
7.
The LCM of 6 and a certain number is 24. Find the number (If there is more than one answer,
give the smallest number).
8.
Find the HCF of 36, 24, 144 and 96.
Number, Set Notation and Language
2
9.
The highest point of an iceberg is +62 m from sea level. Its lowest point is -103 m from sea level.
What is the height of the iceberg?
10. At noon, the temperature of a certain place registered 35  C. At 01 00 the next day, the
temperature was  5 C. What was the difference in the temperature?
It was found that the temperature after 01 00 further decreased by 0.5 C every hour. What was
the temperature at 04 00?
11. Find the value of 1 5  ( 3  2 ) .
9
4
6
12. Find the value of 2(45  13)  4  2 .
13. Find the value of 16  32  (2)3 .
14. Find the value of
3  6 1 1 .
4 7
7
15. Giving your answer in two decimal places, find the value of 4.426 7.201
20 .34
Number, Set Notation and Language
3
16. Evaluate the following giving your answer correct to 2 significant figures
(4.75 2  2.14 2 ) 1.08
12 .93  2.33 2
.
17. Use your calculator to evaluate the following giving your answer correct to 3 decimal places
7.14  2.56 2
15 .21
 3 8.19 6.35 .
9.73
18. Evaluate the following by taking a  2, b  6, c  4 and d=5
(a) (b  c )  a
a
(b) ( ab  c)d
d
2
c
19. Evaluate the following when p  3, q  1 , x  4 and y   2
2
(a)
3
y
q
4( x  )
p
(b) ( 12  3 p)( x  y)
q
20. Write down the next two terms in the following sequence:
2 , 3 , 4 , 5 , ..., ...
3 5 7 9
Number, Set Notation and Language
4
21. Write down the general term of the following sequence 1 1 ,2 1 ,3 1 ,4 1 , ...
2
3
4
5
22. Write down the next two terms and the nth term in each of the following sequences:
(a)
(b)
(c)
23. (a)
1 , 1 , 1 , 1 , 1 , ....
2 3 4 5 6
5 , 10 , 17 , 26 , 37 , ....
2 3 4 5 6
7 , 14 , 23 , 34 , 47 , ....
3 4 5 6 7
The nth term of a sequence of numbers is n ( n 2  1) . Find the sum of the first three terms.
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(b) A manufacturing company uses automated guided vehicles (AGVs) on the factory floor.
Three such vehicles leave a station every 20 minutes, 40 minutes and 50 minutes
respectively. If they leave the station at 09 00, find when they next leave the station at the
same time.
24. Look at the following pictures:
How many rectangles (including squares) of all sizes are there in
(a) the 5th pattern,
(b) the 10th pattern.
Number, Set Notation and Language
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25. The figures shows an equilateral triangle X1Y1Z1 of length 10 cm. X 2, Y2 and Z2 divide
X1Y1, Y1Z1 and Z1X1 respectively in the ratio 1 : 1 and form the equilateral triangle X 2Y2Z2 .
Following the same pattern, X 3, Y3 and Z3 divide X 2Y2 , Y2Z 2 and Z2 X 2 respectively to give
triangle X 3Y3Z3 . This process is repeated indefinitely to give
X 4Y4Z4 , … , X nYn Z n .
(a) Find X 2Y2 .
(b) Find X 2 X 3 : X1 X 2 .
(c) If an insect crawls along the path X1, X 2 , ..., X n , show that the
total distance traveled can’t exceed 10 cm.
X1
Z3
X2
X3
Y1
Z2
Y3
Y2
Z1
EXTENDED
26. In a group of 150 pupils, 100 study Biology and 70 study Chemistry. It is given that:
 ={Pupils in the group}
B = {Pupils who study Biology}
C = {Pupils who study Chemistry}
(a) Find the greatest possible value of n( B  C ' ) .
(b) Describe in words the set notation B  C '  Ø.
(c) Express in set notation the statement “Some pupils study neither Biology nor Chemistry.”
27. The sets P, Q and R satisfy all the following condition:
(1) P  Q  Q and
(2) P  R  Ø and
(3) n(Q  R )  0 .
Represent these sets clearly on a labeled Venn diagram.
Number, Set Notation and Language
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28. Given that:
  {x : x  Z and 1  x  50 }
A = {x : x is prime}
B = { x : 21x  x 2  54}
C = {x : x is even }
(a) Explain the meaning of “ x  Z ”.
(b) List the elements of A, B and A  B .
(c) Find n( B  C ' ) .
(d) Draw a clearly labeled Venn diagram to illustrate the relationship between the sets A, C
and  .
29. A youth club has facilities for members to play pool, darts and table-tennis. Every member plays
at least one of the three games. P, D and T represent the sets of members who play pool, darts
and table-tennis respectively. Express each of the following in set language and illustrate each by
means of a Venn diagram.
(a) The set of members who only play pool.
(b) The set of members who play exactly 2 games, neither of which is darts.
Number, Set Notation and Language
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30. (i)
(ii)

A
B

A
B
For each of the Venn diagrams above, express the shaded region in set notation.
31. Express each of the following statements in appropriate set notation
(i)
x is not an element of set A
(ii) The number of elements not in set B is 16.
(iii) Sets C and D have no common elements.
32. Given that
  {x : 5  x  5}
A  {x : 8  2 x  1}
B  {x : x 2  x  2}
Find the values of x which define the set A  B .
Number, Set Notation and Language
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33. In a class of 25 members, 15 take History, 17 take Geography and 3 take neither subject. How
many members take both subjects?
34. In a class of 36 students, each student learning at least one foreign language, chosen from French,
German and Japanese. 13 students learn only French, 5 learn only German and 7 learn both
French and German. All those who learn Japanese do not learn French or German. How many
students learn Japanese?
35. A class of 38 students took two tests in local History. 32 students passed Test I and 34 passed
Test II. What is the maximum number of students who did not pass both tests?
36. A survey was conducted on 80 people who watched the news on TV last Saturday. The results
are as follows:
34 people watched the news in English,
36 people watched the news in Chinese.
The people who watched the news in Malay did not watch the news in other languages. Nobody
watched the news in Tamil.
What is the maximum number of people who watched the news in Malay?
Number, Set Notation and Language
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