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ΜΑΘΗΜΑΤΙΚΑA΄ Γυμνασίου A΄ Tεύχος
ΠΑΙ∆ΑΓΩΓΙΚΟ ΙΝΣΤΙΤΟΥΤΟΥΠΗΡΕΣΙΑ ΑΝΑΠΤΥΞΗΣ ΠΡΟΓΡΑΜΜΑΤΩΝ
YΠΟΥΡΓΕΙΟ ΠΑΙ∆ΕΙΑΣ ΚΑΙ ΠΟΛΙΤΙΣΜΟΥ
ΜΑΘΗΜΑΤΙΚΑ
A΄ Γ
YΜΝΑΣΙΟΥ A
΄ TΕΥΧΟΣ
Αναθεωρημένη Έκδοση
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M
Y
CM
MY
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CMY
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MATH COVERS.A-A&B.ai 1 15/04/2014 12:56 μ.μ.MATH COVERS.A-A&B.ai 1 15/04/2014 12:56 μ.μ.
Μαθηματικά
A΄ Γυμνασίου, Α΄ Τεύχος
Συγγραφή:
Αθανασίου Ανδρέας
Αντωνιάδης Μάριος
Γιασουμής Νικόλας
Έλληνα Αγγέλα
Ιωάννου Ιωάννης
Ματθαίου Κυριάκος
Μουσουλίδου – Νικολαΐδου Μαριλένα
Παπαγιάννης Κωνσταντίνος
Τιμοθέου Σάββας
Φιλίππου Ανδρέας
Συντονιστής: Χρίστου Κωνσταντίνος, Καθηγητής Πανεπιστημίου Κύπρου
Εποπτεία:
Θεοφίλου Στέλιος, Επιθεωρητής Μέσης Εκπαίδευσης
Κωστή Αντώνιος, Επιθεωρητής Μέσης Εκπαίδευσης
Παντελή Παντελής, Επιθεωρητής Μέσης Εκπαίδευσης
Παπαγιάννη Όλγα, Επιθεωρήτρια Μέσης Εκπαίδευσης
Γλωσσική επιμέλεια: Κατσουρά Ευφροσύνη, Λειτουργός Υπηρεσίας Ανάπτυξης Προγραμμάτων
Παλάτου – Χριστόφια Μαριάννα, Λειτουργός Υπηρεσίας Ανάπτυξης
Προγραμμάτων
Σχεδιασμός εξωφύλλου: Σιαμμάς Χρύσης, Λειτουργός Υπηρεσίας Ανάπτυξης Προγραμμάτων
Συντονισμός έκδοσης: Παρπούνας Χρίστος, Συντονιστής Υπηρεσίας Ανάπτυξης Προγραμμάτων
Α΄ έκδοση 2012
Β΄ έκδοση 2014
Εκτύπωση: Ariagraf & ΣΙΑ ΕΕ
© ΥΠΟΥΡΓΕΙΟ ΠΑΙΔΕΙΑΣ ΚΑΙ ΠΟΛΙΤΙΣΜΟΥ
ΠΑΙΔΑΓΩΓΙΚΟ ΙΝΣΤΙΤΟΥΤΟ ΚΥΠΡΟΥ
ΥΠΗΡΕΣΙΑ ΑΝΑΠΤΥΞΗΣ ΠΡΟΓΡΑΜΜΑΤΩΝ
ISBN: 978-9963-0-4726-0
Στο εξώφυλλο χρησιμοποιήθηκε ανακυκλωμένο χαρτί σε ποσοστό τουλάχιστον 50%, προερχόμενο από
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( ) ( 1,3) + (+5) + (+3,2) + + (+1,3) + + ( ) ( 3) + (+1) + (+3,2) + ( 1,1) + (+1,3) + ( 5)
( ) , , 0.
, . ( 1,3) + (+5) + (+3,2) + + (+1,3) + + = ( 1,3) + (+1,3) + + + + (+3,2) + (+5) = 0 + 0 + 3,2 + 5 = +8,2
( )
. ,
. ( 3) + (+1) + (+3,2) + ( 1,1) + (+1,3) + ( 5) = 3 + 1 + 3,2 1,1 + 1,3 5 = 3 11 5 + 3,2 + 1,3 + 1 = 19 + 5,5 = 13,5
- - -
141
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( ) + ( ) + ( ) 0,3 + 0,15 ( ) 2,35 + 4,2
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( ) (+8) + (+14) ( ) 3 + 25( ) ( 9) + (+15) ( ) +3 25( ) ( 13) + ( 15) ( ) 28 2( ) (+6) + ( 19) ( ) 24 + ( 30)
( ) (+16) + ( 16) ( ) 17 + 17
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( ) ( 2,95) + ( 1,2) ( ) 1,20 + 2( ) + + ( ) 2 3 ( ) + ( 2,15) ( ) 1 + 4 ( ) 1 + ( ) 3 + 1
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:
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142
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( ) (+8) + ( 3,5) + (+3,5)
( ) ( 7,2) + (+3) + (+6,1) + ( 9) + ( 3)
( ) +61 + 29 41 29
( ) (+20) + (+14) + ( 1013) + (+6) + (+1013)
( ) ( 5) + + + (+12) +
( ) + + + + ( 3)
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7. = 5 + 12 9 = +7 3 + 1,
+ .
8. , = 10, y = 7 z = 8.
( ) + | 1| ( ) 6 + |z| ( ) z + ( 5) ( ) y + z ( ) + + ( ) | + y|
9. : +3 +1 1 +1
0 2 1 3
10.
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143
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/
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+ + = + + = + +
15. :
( ) 5 14 ( ) +2 3 5 + 12 15
( ) 4 + 3 15 5 ( ) 3 5 6 12 3
( ) 5 + 3 + 4 ( ) + 2 3 + 2 + 5
16. , : ( ) + = 0 ( ) + = ( ) + = 0 + =
144
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24
.
, :
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5 .
145
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. . .,
. +2, 5.
o ,
(+2) ( 5) .
, (+2) ( 5) = (+2) + (+5) = +7
.
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-
. -
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2 .
.
.
. .
146
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«+ : +( 4 + 2 1) = 4 + 2 1 = 3 « -
: ( 4 + 2 1) = +4 2 + 1 = +3
1. : ( ) (+9) (+12) ( ) 7,3 ( 15,14) ( ) (+3 4) ( 2 + 3 1) ( ) (+9) (+12) = (+9) + ( 12) = 3
+12,
, 12
( ) 7,3 ( 15,14) = +7,3 + (+15,14) = 22,44 15,4,
, +15,4.
( ) (+3 4) ( 2 + 3 1)
(+3 4) ( 2 + 3 1)
. = +3 4 + 2 3 + 1 = +3 4 = 1
. = ( 1) ( 3 + 3) = ( 1) 0 = 1
147
4: –
2. 38,9 660,2 .
;
: 660,2 ( 38,9) = 660,2 + (+38,9) = 660,2 + 38,9
= 699,1
1. :
( ) ( 17) (+13) ( ) (+27) ( 8)
( ) (+5) (+7) ( ) 4 ( 19)
( ) ( 20) (+5) ( ) 19 ( 19) ( ) 0 ( 25) ( ) (+8) ( 10)
2. :
( ) (+1,2) (+2,6) ( ) 4,9 ( 1,9) ( ) ( 11) (+4,2) ( ) 1,5
( ) 15 14 ( ) 2
3. 3 :
( ) 10, 8, 6, 4, , , ( ) 10, 7, 4, 1, , ,
( ) 19, 14, 9, 4, , , ( ) 1 , 3, 4 , 6 , , ,
4. >, = , < ,
:
( ) 2000 . . . . . ( 2000) ( ) (+5) . . . . . (+5) ( 10)
( ) 2 + 3 . . . . . 2 4 ( ) 4 . . . . . 4 ( ) | 1| | 2| . . . . . ( 1) ( 2) ( ) 3 ( 2) . . . . . | 1 + 2|
148
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5. – N . 1962
235 , 2002 182 .
2002 1962.
6. , :
( ) ( 5) = +1 ( ) (+11) = +4
( ) 5 + = 13 ( ) +12 = 12
( ) | 11| + = +20 ( ) | 3 | = +10 ( ) + = ( ) 1 + = 2
( ) 2 + > 0 ( ) 7 > 0
7. :
( ) (+10) + ( 15) + ( 3) ( 1) (+7)
( ) ( 7) + ( 5) (+8) + ( 7) ( 15)
( ) 12 (11 3) + (5 7) (8 + 6)
( ) (13,7 2,6) + 14,8 ( 8,7 + 5)
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8. = 3 = + , :
= 3 + = | |
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149
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.
= ( 2,3 + 7,2) ( 5,9) = ( 104,99 4,8) ( 100 + 2,99) ( 100,3)
11. + ,
:
12. . . .. ,
, .
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08: 00. ( )
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17: 00;
150
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829,84 160 57
504 18
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«A_En_4_Ginomeno_1
.
151
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.
. 0 .
.
. .
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6. .
152
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+ : (+3) (+7) = +(7 3) = +21 ( 8)( 1) = +(8 1) = +8
: (+3)( 7) = (7 3) = 21 , , = ( ) = ( ) ( + ) = + ( ) = .
: ( 2)(+3) = (+3)( 2) [( 2)(+7)]( 1) = ( 2)[(+7)( 1)] ( 2)[( 4) + (+6)] = ( 2)( 4) + ( 2)( 6) = =
: ( 3)(+1) = ( 3) : ( 3)( 1)( 2) = 6 ( 1)( 5)( 4)( 2) = +40
(+3) (+7) (+3)(+7) ---
153
4: –
0
=
:
4 .
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= (3 4) = 12
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( ) 2 (+2)
= (2 2) = 4
. , .
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= +6 . , .
( ) = =
. , .
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( ) = + = +1
. , +1.
154
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2. 1000 .
30 000 , : ( ) 3 ; ( ) 3 ;
( ) 1000 , 3 : ( 3)( 1000) = +3000
, 30000 + 3000 = 33000 .
( ) 1000 , 3 : (+3)( 1000) = 3000. , 30000 3000 = 27000 .
3. , = 0,5. = +
= + = 0,5 0,5 + 0,5 = 0,5 (0,5 + 0,5) (0,5 0,5) = 0,5 1 0 = 0
-.
- -
.
4. : = ( 2)(+3)( 4)(+5) + ( 2)( 3)( 4)(+5)
. 2
. 3 . = ( 2)(+3)( 4)(+5) + ( 2)( 3)( 4)(+5)
= +(2 3 4 5) (2 3 4 5) = 0
, . , .
5. :
( ) 2( + 5) ( ) (4 6)
155
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- , -
- .
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.
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2. : ( ) ( 7) (+2) ( ) ( 2) ( 5)
( ) ( 3) 0 ( ) 2 (+3) 3 ( 12) ( ) 5 ( 11)
( ) ( 6,2) ( 3) ( ) ( 1) 0
( ) 8 ( ) ( 2,1) ( 3,2)( ) ( ) 1
3. :
( ) (+3) ( 1) (+2) ( ) (+3) ( 1) ( 2)
( ) (+3) ( 1) ( 2) ( 2) ( ) (+3) (+1) (+2) ( 1)
( ) (+3) 0 (+2,5) (+1,2) ( ) ( 1) ( 1) … ( 1)
( ) + ( ) ( 1) (+1) ( 1) (+1) … (+1)
( ) 1 0 ( ) (+1) ( 1) (+1) ( 1) … (+1)
4. , :
+3 2,5 3 +7 1 +
156
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.
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( ) 6 2 (5 + 4) ( ) 6 ( 2) ( 8)
( ) 2 ( 4) ( 1) ( 1) ( ) 3 ( 6) + 3 ( 8 + 2)
( ) 5 ( 4 + 3) 2 21 ( ) 2 ( 6 + 10) ( 15)
( ) ( 3 + 3) ( 3) ( 3) ( ) ( 1) ( 1) … ( 1) ( 1)
8. , :
( ) ( 3) = ( ) + + = ( ) = ( ) 2 = +1 ( ) ( 5) = 0 ( ) +2 = 2 (+1)( ) ( 3) = 1 ( ) ( 2012) = ( 1) ( 2)
( ) 1, 2, 4, 8, , , ( ) 2, 2, 2, 2, , ,
( ) 1, 10, 100, 1000, , , ( ) 1, 3, 9, 27, , ,
157
4: –
9. 8 6 .
10. 2,5 , +9 , 0 , 1 , +10 , 2100. :
( ) , ( ) .
11. , = 1. = 2 + ( + 1) = (3 + 2)(1 )( + 0,5) = ( 2010)( 2011)( 2012) + ( 2012)( 2011)(+2010) = ( 2)(+3)(+5) + ( 2)( 3)( 5)
12. :
( ) 3( + 2) ( ) 3( 1) + 5( ) (6 + 32) ( ) 6 ( 5) + 7
( ) 5(3 2) + 7 ( ) (4 + ) ( + 2)
( ) 4( + 7) 6(4 ) ( ) 5( 11) + 2
13. 350 €6 €9.
( ) €6, :
i. €9 ,
ii. €6, iii. €9, iv. .
( ) 120 €6, .
14. := 2( 3 ) + 4(2 ) 2(3 2 ).
, + = 4.
158
4: –
2 . ;
1 2 3 4 (+3) 3 ( 3) (+3) 9 ( 3) (+3) 27 ( 3) (+3) 81 ( 3) (+3) 243 ( 3) (+3) 729 ( 3) (+3) 2187 ( 3) (+3) 6561 ( 3) (+3) 19683 ( 3) (+3) 59049 ( 3) (+3) 14348907 ( 3) (+3) 43046721 ( 3) (+3) 3486784401 ( 3)
; , .
;
3 ;
159
4: –
A > , > .
: (+7) = +343
, . < >
: ( 3) = +9 , . < < .
: ( 2) = 8
1. : ( ) 5
( ) ( 5) ( ) (+4) ( ) ( 1256)
( ) 5 , (5) .
( ) ( 5) , ( 5) (2)
.
( ) (+4) , (+4) .
( ) ( 1256) , ( 1256) (27) .
2. : ( ) ( 4) ( ) 4 ( ) ( 3)
( ) (+4,2) ( ) (+1) ( ) ( 1)
160
4: –
( ) ( 4) = +(4 ) = 16
( ) 4 = (4) = 16
( ) ( 3) = 27( ) (+4,2) = 1( ) (+1) = +1
( ) ( 1) = +1
1. : ( 2)
6,3 2
2. :
3. :
( ) ( 1) = 1 ( ) ( 2) = 16 ( ) (+3) = 27 ( ) ( 1) = 1 ( ) 2 = 16 ( ) 6 = 36 ( ) = ( ) ( 2,7) = 2,7 ( ) (0,5) = 0,125
4. :
( ) ( 5) ( ) 2 ( ) ( )
( ) ( 1) ( ) ( 18)
( ) ( 3) ( ) ( 10)
5.
: ( ) 25 ( ) 8 ( ) 1
( ) ( 3) ( ) ( ) 2012 ( ) (+2,15)
161
4: –
6. < , =, > . .
( ) ( 2) = 8 ( ) ( 5) = 5
( ) 3 = +9 ( ) ( 1) = 1
( ) ( 5) = +25 ( ) ( 1) = 1
( ) 3 = 1 ( ) =
( ) (3 5) > 0 ( ) ( 5 ) < 0
7. < , , > ,
:
( ) (+2) … … ( 2) ( ) ( 7) … … (+7)
( ) (+12) … … ( 12) ( ) (+113) … … ( 113)
( ) ( 2) … … (+2) ( ) (+4) … … ( 4)
8. ( 6) , 6 , ( 6) , 6
, .
9.
.
.
.
. .
10. = 1, : = 2 4 + 1 5
11. :
= ( 1) + ( 1) + ( 1) + + ( 1) + ( 1)
162
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(1)
, .
, 2 .
12 , ;
08: 00 2 ,
8 ;
03: 00, 08: 00 2 ;
(2)
12 15.
.
.
163
4: –
.
+ : (+12) (+2) = +(12 2) = +6 ( 30) ( 5) = +(30 5) = +6
: ( 45) (+5) = (45 5) = 9 (+2,1) ( 7) = (2,1 7) = 0,3 = , 0
: ( 5) (+2) = ( 5) + = = = =
: = = = = = 5 – = = 0 , 0, 0
: = = =
--
164
4: –
. , , 0
( + ) = + ( ) =
: [( 4) + (+6)] ( 2) = ( 4) ( 2) + (+6) ( 2)
1. : ( ) 12 3 ( ) 24 ( 3) ( )
( ) ( )
( ) 12 3 = (12 3) = 4 . , .
( ) 24 ( 3) = +(24 3) = 8 . ,
.
( ) = = 5 . ,
.
( ) = = +6 -
. . , .
( ) = · = = 2
, -
.
2. N 85 .
, = ( ) C ,
.
= , 0 = = , 0
165
4: –
= ( ) = ( ) 85. = ( ) .
= .
= 65 .
3. : ( ) ( 3 + 1 + 2) ( 2)
( ) [ 52 + ( 6) ] ( 1) (8 9)
( ) .
( 3 + 1 + 2) ( 2) . = 0 ( 2) -
. = (+1) 0 = +1
.
( ) ( )
. [ 52 + ( 6) ] ( 1) (8 9) = [ 52 + 36] ( 1) ( 1)= ( 16) ( 1) ( 1) = (+16) ( 1) = +16 + 1 = +17
. .
1. := = 20 5 = ( 4) ( 3) = ( 2) (+5) = 8 ( 10) = ( 12) ( 16)
166
4: –
2. : ( ) ( 10) ( 2) ( ) ( 15) 5 ( ) 18 ( 3)
( ) ( 70) ( 7) ( ) ( 2,1) ( 3) ( ) ( 4,5) 9
( ) 50 ( 0,5) ( ) ( 12,24) ( 4) ( ) 80 ( 2)
( ) ( ) 1 : 5 ( ) ( 7) 3. 20 15.
4. +2 +4.
5. <, =, >,
: ( ) (+6) ( 2) . . . . . +5 2 ( ) ( 12) ( 3) . . . . . 12 ( 3)
( ) (+6) . . . . . 0 ( ) ( 1) . . . . . ( 1) ( 1)
( ) (+2) . . . . . 2 2 ( ) (+6) . . . . . + ( 10) 6. ,
: ( ) (+32) = (+8) ( ) ( 24) ( 6) = ( ) ( 25) = +5 ( ) (+24) = +2
( ) ( 3) = 4 ( ) ( 5 ) = 5
( ) ( 4) = ( ) (+3) = 9
( ) = ( ) = 2
( ) = +2 ( ) = 2
7.
.
= ( )
.
49 58 50 41 40 20
167
4: –
8. : ( ) +1 ( ) 1 ( ) 0
9. :
( ) ( 12 4) ( 4) ( ) ( 9 + 3) ( 2 4) ( ) ( 2 + 3 1) ( 3 + 2) ( ) ( 2 5 1) ( 3 + 1)
( ) + + ( ) : ( 4 2)
( ) ( )
10. :
( ) ( 20) 2 ( 2) ( ) ( 21) + 6 3 ( ) 10 + 3 2 7 ( ) / ,
5.
11. , :
( ) ( 10). . . . . ( 2). . . . . (+1) = 21
( ) ( 5). . . . . ( 2). . . . . (+4) = 1
( ) 6. . . . . ( 7). . . . . (+2) = 44
( ) ( 8). . . . . ( 2). . . . . ( 2 ) = 9
12. :
( ) 45 35 ( 5)
( ) 15 ( 2) ( 1 + 3)
( ) ( 2) ( 3) ( 12) ( 1)
( ) 127 25 ( 1 + 2)
( ) 2 [ 3 2 ( 1 1)] ( ) 2 + 2 + ( 1)
( ) ( )( ) ( ) ( )( )( )( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )
( ) ( ) ( )
168
4: –
13. , = 12, = 4 = 6:
( ) ( )
( ) ( + ) ( + 2) ( ) ( )( + )
( ) ( ) ( )
14. , 11: ( 5) + (+4) ( 2) ( 1) ( 3)
15. , .
.
16. = 2 = 3, :
( ) 3 + 2 + ( ) ( ) ( 1) +
17. : ( ) 3( 3) ( ) 6 ( 2 + 1) ( ) ( 4) + 3 1 (+2) ( ) (2 3) ( 5) + (3 5) ( ) ( 7) ( 7) + ( 3) + 5 1
( ) 7,25 ( 1 + 2) + ( 3) ( 4,2)
( ) ( 2) ( 2) ( 5) +2 ( 3)
( ) ( ) ( )( )
169
4: –
1, , , … , .
1 1 0,111111111111 …. 0,5 0,1
0,333333333333 … 0,090909090909 …
0,25 0,833333333333 …
0,2 0,076923076923 … 0,1666666666666 … 0,0714285714285714 …
0,142857142857 … 0,066666666666666 … 0,125 0,625
.
,
2 / 5, .
.
170
4: –
: 0,333 … = 0, 3 , 3 1. 34,23181818 … = 34,2318 , 18 2.
: 0, 3 34,2318
: = 0, 3125 , = 0, 416 ( 10 = ( , , 0).
= 2, 171
10 3, 172
-
171
4: –
1. : ( ) 2 ( ) ( )
( ) 2 2 = 2 = 2,6.
( ) 5 8
, = 0,625
( ) ,
5 3 .
5,000 3
, = = 1,666 = 1, 6 3 1,666 20 18 20 18 20
2. 0, 3 .
= 0,333 … . 10 = 3,333 … 10. 10 = 3,333 … 0,333 … 9 = 3 =
-
. 0, 3 =
5,000 8 48 0,625 20 16 40 40 0
172
4: –
3. 0,586 . = 0,58666 … . 100 = 58,666 … 100 -
. 1000 = 586,666 … 1000.
1000 100 = 586,666 … 58,666 … 900 = 528 = =
-.
0,586 =
1. :
( ) 4, 7 ( ) 2, 3
( ) 3,417 ( ) 0,28
2. 1 , . 3. :
( ) 1, 1 1,11111111
( ) 3,68 3,68
( ) 4,912 4,9102
4. :
0,69 0, 69 0,069 0, 6 0,69
173
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;
13 10
174
4: –
.
175
4: –
: 2 6 = 3 2 . 6 3 .
:
+ 3 = 5 = 2. 2 1 = 3 = 2. + 3 = 5 2 1 = 3 .
, .
.
+ = + = =
+ =
2 8 = +15 3
= 2
: 2 + 3 = 8 + 15 =
-
, ( ).
.
176
4: –
1. 4 + 3 = 5 5 3 4 + 3 = 5 5 3
.
4 + 3 = 5 5 3 4 5 + 3 = 5 3 2 = 8 . = . = = 4 .
2. : + 1 =
+ 1 =+ =
[3,2,1] = 6 - . = ( )
-. 4 + 6 = 3 2( + 3) .4 + 6 = 3 2 6
.4 3 + 2 = 6 6
. 3 = 12 33 = 123 . = 4 .
177
4: –
3. 30 5 10 .
( ) .
( ) €2,40, .
( ) 5 ,
10 30 .
5 : 10 : 30
, 5 : 5
10 : 10(30 ) = 5 + 10(30 ) = 5 + 300 10 = 300 5
( ) = 240 300 5 = 240
5 = 240 300 5 = 60 = = 12
, 5 12, 10 18. 4. ,
= . 3
, 7
. , 38 .
= . , 2 3,
3(2 3) 7. = = = = 2 3 = = 3(2 3) 7 = 6 9 7 = 6 16
178
4: –
= + + + + = 38 + 6 16 + + 2 3 + 2 3 = 38 + 6 + + 2 + 2 = 38 + 16 + 3 + 3 12 = 60 1212 = 6012 = 5
, = = 5 = = 6 16 = 14
1. . ,
( ).
( )
( )
179
4: –
2. : ( ) 7 + = 5 ( ) 4 = 16
( ) 6 = 3 ( ) 3 4 = 19
( ) 11 = 4 1 ( ) 5 + 2 = 9 2
( ) 8 4 + 3 + 26 = 3 4 + 2 ( ) 2 + 4 = 2 + 4
3. : ( ) 5( + 1) = 3(3 1)
( ) + (5 7) = 5
( ) 5( 2) 2( 1) = 11 ( ) 4( 3) 5 = 0 ( ) 7( 3) 12 = 13
( ) 3( + 2) ( 2) = 4( + 1)
4. :
( ) = ( ) =
( ) = 1 ( ) =
( ) = ( ) =
( ) 1 = + ( ) = 2
5.
:
( ) 3 9 = 18 ( ) 2 = 6 ( ) 5 = 9 3 = 9 = 6 + 2 = 9 5 = 3 = 8 = 4
6. , 10.
7. , , = 5 + 9, = 7 + 5. .
180
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9. 280 . 20
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181
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( ) ( 2) ( 3) ( ) 2 3
( ) ( 3) ( 5) + (+4) ( ) 3 + 4 | 5| ( ) ( 3) | 5 + 4| ( ) ( 3) ( 5) (+4) ( ) ( 3 5) ( 4) ( ) ( 5) ( 4 + 3)
2. :
( ) (+1) = 1 ( ) (+1) = 1 ( ) ( 2) = 64
( ) = ( ) ( 25) = 25 ( ) ( 4) = 64
3. ,
:
( ) 10 + /
( ) 10 10 /
( ) 10 < 0 /
( ) | 10| < 0 /
( ) | 10| = ( 10) /
( ) 10 < | 10| /
( ) ( 10) > ( 10) /
4. :
( ) ( 3) ( ) (+1) ( ) 2 ( ) ( 2) ( ) ( ) ( 2) ( ) ( 3 + 2) ( ) ( 6 + 1) ( ) ( ) +
5. :
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182
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6. <, =, >, :
( ) 0 . . . . . 1 ( ) 5 . . . . . 6
( ) 5 . . . . . | 5| ( ) 5 . . . . . 12
( ) +5 . . . . . 5 ( ) 3 . . . . . 3
( ) 2010 . . . . . 2011 ( ) 3 2 1 . . . . . (+3)
( ) 9 + 2 . . . . . ( 1) ( ) ( 3 + 1) . . . . . ( 16) ( 4)
( ) 3 ( 2). . . . . 3 ( 2) ( ) 0 ( 1) . . . . . | 1| 1
7.
:
8. ,
: ( ) (+14) + = +5 ( ) ( 2,4) = 9,2
( ) ( 20) = +100 ( ) ( 8) = 1
( ) ( 2 1) = 1 ( ) (+2) = 4
( ) 12 + = 7 ( ) +2 = +7
( ) (+3 1) = 6 ( ) (+2 4) = | 2| ( ) = 2 + 3 ( ) = 2014 + 2014
9. , .
.( )
. ( ) > , | | > | |. ( ) , | | = | | | |. ( ) , | | = . ( ) .
1 5
5 1 0 3
183
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11. 48
28 .
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i. 2 , +4 ii. 2 , 4 iii. +2 , 4 iv. +2 , +4
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( ) ( 1 3) (4 7) ( ) (12 24) ( 3)
( ) ( 6) ( 3) + ( 4) ( ) ( 5) 4 ( 3)
( ) 8 ( 7) 8 7 ( ) 11 4 + ( 8) ( 3)
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
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184
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( ) : 1( ) ( 2) (+1) 7
( ) ( 1) 2 (+5 1)( ) ( 2) ( 2) ( 3) ( 3 + 4)
( ) ( 4 + 2) 4 [ (+5) ( 3) ] + ( 2)
16. , .
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22. + = 5 = 3,
: ( ) 3 + ( 3 ) ( ) 5 ( ) + 4
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186
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199
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8. ( ) : : = 100 + + 2 ( ) = 357 €3,57
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4. ( )
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H 78
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2. ( ) = 7 ( ) = 30
( ) = 15 ( ) = 64
( ) = 3125 ( ) = 510
( ) = 57 ( ) = 5
( ) = 5 ( ) = 15
3. ( ) = 11 ( ) = 7
( ) = 3 ( ) = 6
( ) = 3 ( ) = 25
( ) = 15 ( ) = 2
4.
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100 = 10, = 9 , …
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11. €12
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13. 14 16
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16. = 12 , = 5 , = 13
17. ( ) 24
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( ) = 28 ( ) = 110
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6. = 17, = 27, = 7
201
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8. ( ) 111 ( ) 10010
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9. ( ) 6 ( ) 2 + 4
10. ( ) ( )
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( ) = + 1080
12. ( ) , 6 = 11( ) , 4 = 20
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( ) , 7 = 12
( ) , 6 = 54
( ) , 4 + 12 = 26
13. 3, 9, 27, 81, … 14. : 36 : 12
15. 194, 195
17. 242, 243, 244
18. = 16 , : 4 , = 16
19. = 26, = 40
20. €4
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202
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5. ( ) 11110110( ) ( ) 10011111( ) ( ) 1100100( )
8. ( ) 580
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3
93
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8. = 12, 24, 36, ….
1
1. ( ) 1520, 4404, 3850 ( ) 765, 4404
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3. ( ) 2 3 4 6 7 8 9 ( ) 0 2 3 6 8
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( ) 0 ( ) 4 5
4. ( ) 100 ( ) 100
( ) 102 ( ) 108
( ) 100 ( ) 105
5.
6. 0 3 9
7. 90
8. 102
9. 6
10. : 64 : 90
14. ( ) 37 , 111, , 3 , 37
204
:
11
1. 56 = 2 7 , 108 = 2 3 , 73 = 73 , 84 = 2 3 7 , 65 = 5 13
: 73 2. 42
3. 30
4. . . 2 + 3 = 5
5. ( ) 32 = 3 + 29
( ) 20 , 23, 29, 31, 37, 40
6. 211
7.
11. 28
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15 = 3 5 3|(2 3 5 7 )5|(2 3 5 7 ) 15|(2 3 5 7 )
( )
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13.
116
1.
(32, 36) = 4, [32, 36] = 288 (24, 45) = 3, [24, 45] = 360 (32,48,80) = 16, [32, 48, 80] = 480 (252,294, 378) = 42, [252,294, 378] = 5292 2. , 4
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7. 150 , : 3, : 5
8. 24 , 1
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10. . . 5, 9
11. = 24
12. = 13 = 26 = 52
13. 14 35
15. , 3, 6
16. 4
119
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7. 12 13 15 17 21 31 51 71
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10. 1080
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208
:
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6. ( ) +8 ( ) 10,1 ( ) +20 ( ) +40
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8. ( ) 9 ( ) 14
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9. +3 4 +1 2 0 +2 1 +4 3
1 6 +1 0 2 4 5 +2 3
10. 38,92
11. = 0,9 = 5,3 = 6,2
12. ( ) ( )
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15. ( ) 9 ( ) 9 ( ) 11 2 ( ) 9 14
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16. ( ) = = 0
( ) = 0
( ) = = 0
1 7
1. ( ) 30 ( ) +35
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2. ( ) 1,4 ( ) +6,8
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6. ( ) 4 ( ) +15
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9. ( )
( ) : 2 ( 1) = 1
( ) : +2 (+11) = 9
( )
10. A = 10,8 11, B = 87,52 88
11. ( ) ( ) +,
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155
1. ( ) 1 ( )
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2. ( ) 14 ( ) +10
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210
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( ) ( 2100) (+10)
11. = 2, = +1, = 0, = +10
12. ( ) 3 6 ( ) 3 + 8
211
:
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( ) + 4 ( ) 3y + 55
13. ( ) i
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iii. 9(350 ) iv. 3 + 3150
( ) €2790
14. 2
16
1.
( 2) 2 3 ( 2)( 2)( 2) ( 6,3) 6,3 2 ( 6,3)( 6,3)
13 4
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( ) + ( )
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( ) + ( )
( ) +
4. ( ) 25 ( ) 16
( ) ( ) ( ) 1 ( ) 1
( ) 27 ( ) 100
5. ( ) 25 = ( 5) = (+5) = (+25)
( ) 8 = ( 2) = ( 8)
( ) . . 1 = ( 1) = (+1) = ( 2012)
6. ( ) ( )
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( ) ( )
( ) ( )
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212
:
7. ( ) = ( ) =
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8. ( 6) < 6 < ( 6) < 6
9. : 0 , 1 : 0 , 10
: 0 1 10. = 16
11. = 0
165
1. = 5, = 4 , = +1 , = , = , = + < < < < <
2. ( ) +5 ( ) 3
( ) 6 ( ) +10 ( ) +0,7 ( ) 0,5
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( ) 40 ( )
( ) ( ) +98
3. ±1 , ±5
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( ) < ( ) =
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6. ( ) +4 ( ) +4
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( ) +12 ( ) +25
( ) + ( ) 3
( ) 8 ( ) +2
( ) 24 ( ) . . …
7. : 45 : 50
8. ( )
( )
( )
213
:
9. ( ) +4 ( ) +1
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( ) ( ) +
10. ( ) 12 ( ) 19
( ) +9
( ) ( 20) [2 ( 2)], [( 21) + 6] 3, 10 + 3 (2 7)
11. ( ) ( 10) ( 2) + (+1) = 21 ( ) ( 5) ( 2) + (+4) = 1
( ) 6 ( 7) (+2) = 44 ( ) ( 8) ( 2) ( 2) = 9 12. ( ) 38 ( ) +19
( ) 6 ( ) +102
( ) 0 ( ) +4
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( ) ( )
13. ( ) 3 ( ) 2
( ) 3 ( ) 160
( ) 30 ( ) 7
14. ( 5) + [(+4) ( 2) ( 1)] ( 3)
15. ( ) ( )
( ) ( )
( )
16. ( ) +9 ( ) 19
17. ( ) 27 ( ) 7
( ) 16 ( ) 3
( ) 2 ( ) 4
( ) 31 ( )
172
1. ( ) ( )
( ) ( )
2. 1, 27 , 0,375 , 0,83
3. ( ) 1, 1 > 1,11111111 ( ) 3,68 > 3,68
214
:
( ) 4,912 > 4, 102
4. 0,069 0, 6 0,69 0, 69 0,69
178
1. ( ) 2 ( ) 6
2. ( ) = 12 ( ) = 4
( ) = 3 ( ) = 5
( ) = 3 ( ) = 1
3. ( ) = 2 ( ) =
( ) = 19 ( ) = 2
( ) = 10 ( ) = 2
4. ( ) = 3 ( ) = 6
( ) = 2 ( ) = 3
( ) = 9 ( ) =
( ) = 8 ( ) = 2
6. : 5
7. : 57
8. : 7 cm , : 34 cm , : 10 cm , : 35 cm
9. : 140 , : 80, : 60
10. = 16 cm = 15 cm
11. : €19 : €7
13. : 10
14. 43
181
1. ( ) +6 ( ) 5
( ) +1 ( ) 5
( ) +6 ( ) 4
( ) +2 ( ) +60 ( ) +2 ( ) +5
215
:
2. ( ) + ( ) +
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3. ( ) ( )
( ) ( )
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4. ( ) 9 ( ) 1
( ) 16 ( ) 1
( ) ( ) 16
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( ) 1 ( ) 0
5. ( ) 1,5 ( ) 4, 3
( ) 1, 81 ( ) 0,875
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7.
8. ( ) 9 ( ) +6,8
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10.
11.
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12. ( ) +12 ( ) +4
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14. ( ) = 5 ( ) = ±12 ( ) = 0 , = ±1 , = ±2
15. ( ) 0 ( )
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16. ( ) ( )
( ) ( )
( ) ( )
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17. A , i , ii , iii
18. ( ) > ( ) <
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19. ( ) ( )
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21. = 26 = +7
22. ( ) +9 ( ) 25
23. ( ) = 2 ( ) = +3
( ) = 6 ( ) = 1
( ) = +1 ( ) = 2
( ) = 10 ( ) = +7
( ) = 6 ( ) = +2
24. 121
25. : 33 : 6
26. : 80 : 20 : 15
27. : €60
: €20 : €20
: €60
217
:
28. : 10 : 40
29. : 30
30. : 36 : 12
16
1. 17
2. < 0
3.
4. : 30 10
5. 0
6. ( ) : 16 : 18
( ) 6
7. ,
8.
9. = 3
10. 4
11. 32
12. 19683
{ } =
{1,2,3,4, . . . } {0,1,2,3,4, . . . } {0, ±1, ±2, ±3, ±4, . . . } {1,2,3,4, . . . } { 1, 2, 3, 4, . . . } { / : ,
/
o
=
·
Ans
^
SHIFT 2nd F
AC
DEL
Abs
(-)
a b/c 23
S D a b/c
CLR
M+
M-
M
3 + 4 3+4 7 2,34 – 1,1 2.34 – 1.1 1.243 2 3 2 6 2 2 2 32 5 10 5 3 5 103 5000
3 4 2
2
2 3 4 4 4 2 5 5 25 2 (7 3) 2 (7 3) 8 2 (7 3) 2 8 2 ( 3) 2 ( 3) 5 | 3| | 3| 3 0,25
9 6 2
9 – 6 2 + 6
3 (9 6: 2) 6: (9 6: 2) 3 – 6 17
+ = 3 4
· – 2 3 4 1 · 1 =
3 x 2 =
2 5 =
5 P 3 =
3 a b/c 4 =
3 a b/c 4 =
SHIFT 23 2 3 4 =
2 a b/c 3 a b/c 4 =
4 =
5 X² =
= 3 – 7 ( x 2 )
x 2 = 3 – = 7 Ans
2 ( (-) 3 ) =–
= 3 (-) Abs
S D
a b/c SHIFT
= 2 ÷ 6 – 9
3 – 6 ÷ =