For every integer (a) there is an additive inverse (-a) ..... 12- The sum of edges of cuboids 18 cm if its length triple
Cairo Governorate Nozha Directorate of Education Nozha Language Schools “Ismailia Road” Branch
Department : Maths Form : 6th Primary Revision Sheet 2014 2nd term The set of integers
The set of counting numbers ={1,2,3,4,…} N={0,1,2,3,…} + 0 Z Z={…,-3,-2,-1,0,1,2,3,…} Z = Z- ∪ {0} ∪ Z+ ............................................................................................................................................. Important remarks: 1) 0 ∉Z- and 0∉ Z+ so zero is neither positive nor negative number + 2) N⊂ Z , Z ⊂ Z , Z- ⊂ Z and {0} ⊂ Z 3) Z = N∪ Z4) Z+∩ Z- =∅ 5) Z – N = Z- , N – Z = ∅ ............................................................................................................................................ Property of addition in Z 1)closure property : The sum of any two integers is an integer a+b=c
“Zis closed under addition”
a,b,c ∈ Z
2)Commutative property: If a,b are two integers then a + b = b + a 3)Associative property: If a , b ,c are integers then : a+b+c= (a+b)+c = a+(b+c) 4)Additive identity (neutral) element in Z: For any integer a we have :a +0 = 0+a = a Zero is the additive Identity element in Z 5)The additive inverse (opposite) property: For every integer (a) there is an additive inverse (-a) Where a +(-a)=0 1
Note that: 1)Z is closed under subtraction operation 2)subtraction operation in Z is not “commutative , associative” 3)the additive inverse of zero is zero . 4)the additive inverse of (–a) is a . ………………………………………………………………………………………. Properties of multiplication in Z: + × + = + and - × - = + 1) Z is closed under multiplication. + × - = - and - × + = 2) commutative property is satisfy in multiplication. 3) Associative property is satisfy in multiplication. 4) The multiplicative identity is one a× ×1=1× ×a=a 5) multiplicative is distributed over addition and subtraction a× ×(b+c)=a× ×b +a× ×c Note that: 1)Division is not always possible in Z. 2)Division operation in Z is not commutative . 3)Division operation in Z is not associative . ……………………………………………………………………………………… Repeated multiplication
Base →
m
n
3
4 ←power
=3×3×3×3=81
m+n
×a =a * am ÷ a n= a m-n
*a
: a≠0
2
Unit 1 1- Put ( √ ) if the number is an integer ( × ) if the is not an integer : a) 10 e)
10 2
b)
3 8
f) – 15
2 3
c) 1111
d)
g) 2.1
h) – 101
2- Complete the following sets of numbers : a) { 2 , 3 , 4 , …… , ………. , 7 , ……….. } b) { 0 , 1 , 2 , …………. , 4 , …….. , 6 , ………. } c) { – 5 , – 4 , – 3 , ……… , – 1 , 0 , ……….. } d) { – 10 , – 9 , – 8 , …….. , – 6 , ……….. – 4 , – 3 , ……. } e) { – 20 , ……….. , – 18 , – 17 , ………. , ……… , …….. , – 13 , – 12 } 3- Write the following sets of numbers using the listing method : a) The set of integers greater than – 2 b) The set of integers smaller than – 2 c) The set of integers greater than – 8 but less than – 4 . d) The set of integers between – 5 and 1 e) The set of negative integers .
4- Arrange the following integers in ascerding order : a) 2 , – 2 , – 1 , 0 , 1 b) 0 , 3 , 1 , 2 , – 1
c) – 1 , – 2 , – 3 , – 4 , 5 d) – 98 , – 99 , – 102 , – 11 , – 100 , – 101
e) 7 , – 7 , 8 , – 8 , 1 , – 1
3
5- Write the smallest integer that makes sentence true :a) 5 < ……………….. b) – 6 < ………………. c) 0 < ……………… d) – 5 < …………….. e) – 11 < …………….. f) – 8 < ……………. 6- Write the integer that will make the sentence true : a) 5 > ………………. b) – 3
>
……………….
c) – 4
>
………………
d) 10
>
………………
e) – 11
>
………………
f) – 101
>
………………
7- Find the result : a) – 7 b) 9 c) 5 – 4 d) – 2 + – 13 e) – 4 + 3 f) – 9 + – 3 h) – 12 – 12 g) – 100 – – 50 8- Insert the correct sign ( < , = or > ) :a) 0
…………….
– 11
b) – 2
……………
2
c) – 1
…………...
1
d) 2 – – 1
…………..
0
e) – 3 – – 3 ………….
0
f) 9 + – 12 ………….
0 4
9- Complete : a) Z ∩ N = ………… b) Z
+
∩ Z =………..
c) Z ∩ N = ………… d) Z – Z– = …………. e) – – 45 = ……….. f) Z = ……….. ∪ ………….∪ ……. h) Z ∩ Z+ = …………
10- State which of the following statements are true and which are false : a) 52
∈
N
b) { 0 }
⊂
N
c) – 10
⊂
N
d) – 7
∈
Z
e) – 7
∈
N
∈
N
f)
1 3
11- Write the numbers between the two integers numbers : a) – 4 , 2 b) – 1 , 0 c) – 7 , 7 d) – 7 , – 5 12-Using number line to represent the operations : a) – 3 ×( – 3 ) b) 7× ( – 5 ) c) 2 – ( – 3 ) d) 10 – ( 5 ) – ( 3 )
5
13- Write the set of integers represents the statements : a) A > – 1 b) A < 7 c) A < – 1 + 5 d) 3 < A < 7 e) 7 < B < – 7 + 101 f) – 4 < B < – 10 + 10 g) 4 ≤ Z ≤ 10 14- Put the suitable sign ∈ , ∉ , ⊂ , ⊄ :a) – 9 + 3 …………… Z b) { 9 } …….………… Z c)
3 5
……….……. Z 6−6 8
….………. Z
e) { – 3 ,
7 } ……… Z 11
d)
f)
9 7+7
…………… Z
15- Using the addition properties to find the answer : a) -5 +(-6) +5
b) 15+(-8)+25
c) (-200) + 170 + ( – 200 )
16- Make sure of closure properties are In Z . In sets of numbers : X={–2,0,1,1}
, Y = { – 3 , zero , 1 , 3 }
6
17- Find the result : a) ( – 131 ) × ( – 3 ) b) – 4 × 5 c) – 8 × 1 d) – 9 × ( 7 ) e) – 9 ( 7 ) × 3 f) Zero × – (11) g) – ( – 6 ) × ( – 2 ) h) – 80 × ( – 80) × – 1 18- Find the result : a) ( – 32 ) ÷ 8 b) ( 420 ) ÷ (– 151 ) c) 65 × ( – 13 ) d) – 5 × ( – 6) × ( – 1) e) ( – 1300 × 26 f) – 6 × ( – 3) × ( – 3) × ( – 4) 19- Find the result using distributive property: a) ( – 4 ) × [ 4 + ( – 1 ) ] b) – 5 × [ – 4 + ( – 5 ] c) 6 × [ ( – 6 ) + 6 ] d) 15 × [ ( – 15 ) – zero ]
e) 64 × 99 7
f) 26 × 101
g) 3 ×(-2) +3× 5
h) 147 ×69 - 47×69
20- If X = 3 , Y = -2 , 7 = -7 find the result : a) 3 X + Y – 27 b) 3 × Y Z
c) 3 X + 2 Y + 3 Z d) ( 3 X ÷ Y ) × ( 3 Z ÷ X )
21- Find the value of X : a) 8 X = – 32
b) 3X = 45
c) 3 X + 3 = 48
d) 3 X + 5 = 2 X + 1
8
22- Find the value of following : a) ( – 8 ) 3 b) ( – 6 ) 2 × 2 2 c) ( – 2 ) 3 + ( 3 ) 3 d) ( – 1) 100 + ( – 1) 101 e) ( – 4) 3 × ( – 1 ) 5 f) ( – 5 ) 5 ÷ ( – 1 ) 5 g) ( 3 ) 3 ÷ ( 3 ) 3 23- Find the result : a)
35 × 36 = ……… 33 × 3
b)
(−4) 3 × (−4) 8 (−4) 5
c)
(8) 4 × (−8) 3 (−8) 7
d)
(19)6 × (9)5 (9)3 × (19)5
e)
(1) 7 × (7) 8 (7) 8 × (1) 5
f)
(−6) 3 × (−6) 5 ( − 6) 7
g)
(2)5 × (−5)3 (−5)3 × (2)5
24- Arrange the following Ascending : ( – 2 ) 5 , ( 2 ) 5 , ( – 2 ) 7 , ( – 2 ) 8 , ( – 2) 10 , ( – 3 ) 3
9
25- Find the value if X = -3 , Y = 2 a) 3 X 3 b) 3 X 3 + 4 Y 4 c) 3 X + 5 d) 3 X 3 + 5 Y 5 + 3 26- Put suitable sign { < , > , = } a) 3 3 ……………….. 27 b) ( – 5 ) 2 ………….. 25 c) ( – 3) 3 ………….. ( 3 ) 3 d) e)
1 × 7 5 …….…….. 1 49 1 1 × 7 5 × 3 ……… 73 5 7 7
2- Complete : a) 4 , 7 , ……… , 13 , 16 , ………… , ………. b) 128 , 64 , ……… , 16 , 8 , ……… , ……… c) ……… , 15 , 12 , 9 , ……… , ……… d) 1 , 4 , 9 , 16 , 25 , ……… , ………. , …….. e)
1 , 2
1 1 , 4 8
,
1 , …… , ……… , ……… 16
10
Unit 2 1- Determine the degree of the following: a) X + 5 = 7 b) X + 3 < 5 c) 2 X 2 – 2 = 14 d) 3 N – 2 < – 2 e) X 3 – 4 X 2 = zero 2- Determine the solution set : a) X + 4 = 12
b) 2 X + 3 = 13
c) 4 X – 2 = 13 d) X + 2 < 5
i) – X + 1 < 5
3- Find the solution set in Z a) X + 8 = 19
b) 5 X + 1 = 2 1
c) 3 X + 7 = 25
d) 2 X = 3 X + 21
11
4- Find the solution set in N a) X – 15 = 40
b) 3 X – 4 = – 21
c) 3 X – 2 = 6 X + 12
d) 4 ( X – 2 ) = 3 ( X – 5 )
e) ( 3 X – 5 ) + 4 = X – 13
5- Find the solution set in N and Z a) 3 X = 7
b) 4 X + 12 = 5
c) 3 X – 12 = 8
6- Find the sum of 3 consecutive natural numbers if their sum is 27 .Find these numbers
7- Find the sum of 2 even consecutive numbers if their sum is 30
8- Rectangle its width half its length and its perimeter is 36 cm . Find length and width
12
9- If the age of Omer more than age of Ahmed by 3 years . In the next year the sum of ages is 41 years . Find the age of each one of them
10- Find the two consecutive odd numbers their sum is 16.
11- If the age of man is , triple his son and the sum of their ages is 72 find their ages .
12- The sum of edges of cuboids 18 cm if its length triple its width and its height double its width. Find the three dimensions
13- Find the solution set : a) 2 X – 5 ≤ - 8
X ∈Z
b) 3 X + 2 ≤ 13
X ∈ N
c) 3 X – 7 ≤ 5 + 2
X ∈Z
d) 3 < x – 1 ≤ 9
X ∈Z
13
Unit ( 3 ) 1- In the lattice : draw the following points : a) ( 0 , 10 ) b) ( 0 , 2 ) c) ( – 3 , 6 ) d) ( 3 , 6 ) then calculate the perimeter and the area of the figure. 2- In the lattice : determine the position of the following points X(–2,2),Y(2,2) Z ( 2,9), L(–2,9) Then find the area and the perimeter of the figure XYZL
3- In the lattice : determine the position of the following points A ( 2, 2 ) , B (– 2 , 2 ) and C ( 0 , 5 ) , then find the area and the perimeter of the figure ABC and discuss the symmilarity of the figure?
4- In the lattice : draw the following figure by using the following points A ( 3 , 2 ) , B ( 0 , 5 ) C ( – 3 , 2 ) and D ( 0 , – 1 ) , then find the perimeter and the area of the Figure?
5- In the figure : find the image of these two points A ( 3 , 2 ) And B ( – 3 , 1 ) by translation ( X + 3 , Y + 2 ) .
14
6- In the lattice: find the image of the line segment XY where : X ( 2 , 3 ) , Y ( – 2 , 0 ) By translation ( X + 3 , y + y ) 7- In the lattice : ABC where A ( 0 , 1 ) , B ( 2 , 3 ) and C ( – 1 , 4 ) find the image of ABC by translation ( X + 2 , Y + 3 )
15
16
17
8- In the previous figure : complete: 1) A` B` = ………..……………….
A
A`
2) AC = ……..…………………… 3) A`C` \\ ……………………… 4) M ( ∠ B` ) = M ( ∠ …………)
C
B
B`
C`
5) M ( ∠ C ) = M ( ∠ ………) 6) BC \\ …………………………
9- Determine in the cartesian plane the following: a) Image of XY where X ( 3 , 0 ) Y ( 0 , 1 ) by translation ( X + 3 , Y + 2 ) , what is name the figure XX`YY` ?, why? b) Image of quadrilateral ABCD by translation ( 3 , - 4 )
10- Find the area and the circumference of a circle whose diameter 14 cm., π
22 7
11- In the opposite figure rectangle ABCD it’s length 12 cm and its width 7 cm Find the area of the shaded
7 cm
12 cm
12- Find the circumference of a circle whose area 616 cm2
13- In the opposite figure the surface area of half circle is 100.48 cm2 , find the perimeter.
18
14- A circle is divided into four equal parts if the surface area of one part is 346.5 cm2 find the circumference of the circle. Where π =
22 7
15- Find the lateral area and the total surface area of each of the following. a) A cube of edge 1.8 cm long
b) A cuboid with square base of side length 10 cm and height 12 cm.
16- A cube of length 6 cm find the ratio between its lateral area and its total surface area.
17- A cuboid- shaped box without a lid, has a base with inner dimensions 1.2 and 1.7 m and height 70 cm it is wanted to cover it from inside with iron sheets that cost L.E 10 per square metre find the cost of the iron sheets . 18- If the total surface area of a cube is 216 cm2 find
a) The side length of the cube
b) The volume of the cube
19- The inner dimensions of swimming pool are 25 , 12 and 2.25 meter it is needed to cover its inner ground and sides with square ceramic tiles of side length 15 cm How many tiles are needed? 19
20- The sum of length of the edge of a cube is 132 cm. find:a) The area of one of its faces
b) Its later al area and its total surface area.
21- If we double the dimensions of a cube what is the ratio between the lateral and total surface area .
20
Unit ( 4 ) Sheet 1 1- The following table shows a class about his favorite TV programmes we obtained the following results:Kind of programme
Culture
Sport
Music
News
No. of pupils
27
15
12
6
Represent these Data by pie charts.
2- The following table shows the number of hours of weekly study of Sami in different subjects. Subjects
Arabic
English
Math
Science
Social
No. of hours
10
7
9
8
6
Represent these Data by pie charts.
3- The following table shows the number of maths which Asher of played with his team in three sport periods:Period
First
Second
Third
No. of matches
7
10
13
Represent these Data as pie charts
21
4- In the following table shows the number of tourists who visited Egypt in 2006 Nationality
American
Arab
European
Other national
No. of tourists
340
1922
6260
561
Represent these Data as pie charts 5-The following table shows the TV. Programmers, these pupils watch pt In 6th Subject
Enteriment
Culcutre
News
Sports
Derma
Games
No. of hours
10
7
11
4
5
9
Represent these Data as pie charts
6- The following table shows the percentage of needed in gradients to make a kind of cake. In gradient
Milk
Sugar
Flour
Butter
Percentage
10 %
25 %
50 %
………
Complete table, then represent these Data as pie charts. 7- The monthly income for family is L.E 800 spend 30 % of his money for housing 35 % for other requirement and save the rest of the money. Represent these Data as pie charts, then find the sum of saved money.
22
1- Complete:a- Random experiment:- ………………………….. b- Sam:-…………………………………. c- Event:-………………………………………….. d- Probability:- …………………………………… e- ple space2- If a die is rolled once, then write the sample space and find:a- Probability of getting a number more than 3. b- Probability of getting the number 3. 3- 40 cards are numbered from 1 to 40 if the card is chosen at random then find:the probability than chosen card has:a-an even number b- a number divisible by 7 c-the number divisible by 3 d- a number less than 3 e-a prefect square number f- the number divisible by 2,3 g- the number divisible by 2,5
4- If a die is rolled twice then write sample space and find:a- Probability of getting a number more than 3 b- Probability of getting the number 5
23
5- A box contains: 7 red apples, 3 yellow apples, 5 green apples if an apple is chosen at random, then find probability that apple choosen is:a- Red. b- Yellow. c- Not yellow. d- Blue. e- Gree & and red. f- Not red. 7- Ahmed in 1st prep class contains 70 pupils where 18 of them are girls if a pupils is chosen at random, then find the probability that the chosen pupil is:a- A boy. b- Girl. c- Ahmed. 8- These are 40 pupils in a class 30 of then succeeded in manth, 24 succeeded in science 20 succeeded in both if the student is chosen at radom find that the chosen pupil:a-Succeeded I math. b- Failed in science. c-Succeeded in science. d- Failed in math and science.
9- If a die rolled once then write sample space and find:a- Probability of a is less than 5 b- Probability of B ≥ 3 c- Probability of f < 6 d- Probability of 1 < C < 6
24
