άλγεβρα α΄ λυκείου γιώργος αποστόλου
178
) • ΑΛΓΕΒΡΑ Α’ ΛΥΚΕΙΟΥ ΑΠΟΣΤΟΛΟΥ ΓΙΩΡΓΟΣ ΜΑΘΗΜΑΤΙΚΟΣ • 113 ΕΡΩΤΗΣΕΙΣ ΜΕ ΑΠΑΝΤΗΣΕΙΣ ΘΕΩΡΙΑΣ • 152 ΛΥΜΕΝΑ ΘΕΜΑΤΑ ΜΕ ΜΕΘΟΔΟΛΟΓΙΕΣ • 217 ΑΣΚΗΣΕΙΣ ΠΡΟΣ ΛΥΣΗ • 27 ΘΕΜΑΤΑ ΕΞΕΤΑΣΕΩΝ
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Transcript of άλγεβρα α΄ λυκείου γιώργος αποστόλου
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)
113
152
217
27
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Copyright 2014
3 , 2014
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1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1 - 71.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.2 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2 141.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.2 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1 252.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.2 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2 382.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2.2 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.3 452.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.4 552.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
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3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.1 1 x+ = 0 693.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.2 x = 833.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.3 2 873.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.1 1 1014.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.2 1104.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.3 1144.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.4 1204.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.1 1255.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.2 1305.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.3 1375.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
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6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1436.1 1436.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.2 1506.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.3 f(x) = x+ 1626.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
8.1 177
177
8.2 177
177
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- -
-
1.
1.1 -
1.1.1
1.1 ;
, , (- ). , .
1.2 ;
, .
1.3 ;
.
1.4 ;
, - .
1.5 ;
, , - .
-
1.6 ;
, . .
1.7 ;
1. 2. 3. ( , ).4. ( ).5. .6. .7. .8. .
1. A2. AB = A B3. A B.4. A B.5. (AB) (B A).6. (A B) = A B.7. A.8. (A B) = A B.
1.8 ;
. A B =
- 8
-
1.1.2 - 1.1 1 5. , 2 2 .
i. .ii. 2 .iii. .
1.1 i.
= {ZZ,ZMZ,ZMM,MZZ,MZM,MM}
ii. A = {ZZ,MM}iii. B = {ZMM,MZM}
1.1:
1.2 4 1 4. , , .
i. .ii. 2 .
1.2 -, , .1 2 1 2 3 4
1 (1,1) (1,2) (1, 3) (1,4)2 (2,1) (2,2) (2, 3) (2,4)3 (3,1) (3,2) (3, 3) (3,4)4 (4,1) (4,2) (4, 3) (4,4)
- 9
-
i. = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2),(3, 3), (3, 4), (4, 1), (4, 2), (4, 3), (4, 4)}
ii. A = {(2, 2), (4, 4)}
1.3 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = {1, 3, 5, 7, 8}, B = {2, 4, 6, 8} . :
i. ii. iii. A B iv. A Bv. AB vi. B A
1.3 Venn
i. .A = {2, 4, 6, 9, 10}
ii. .B = {1, 3, 5, 7, 9, 10}
iii. A B .A B = {1, 2, 3, 4, 5, 6, 7, 8}
iv. A B .A B = {8}
v. - AB = {1, 3, 5, 7}
vi. - B A = {2, 4, 6}
1.2: Venn
1.4 A = {2, 3, 5}, B = {1, 3, 5, 6} = {1, 2, 3, 4, 5, 6} :
i. (A B) A B. ;ii. (A B) A B. ;iii. (AB) (A B) = A.
- 10
-
1.4 i. A B = {1, 2, 3, 5, 6} (A B) = {4}A = {1, 4, 6}, B = {2, 4} A B = {4} (A B) = A B
ii. A B = {3, 5} (A B) = {1, 2, 4, 6}A = {1, 4, 6}, B = {2, 4} A B = {1, 2, 4, 6} (A B) = A B
iii. ,AB = {2} A B = {3, 5} (AB) (A B) = {2, 3, 5} = A
- 11
-
1.1.3
1. 4 (, , , ) 4 (, , , ). - , .
2. {2, 5} . :
i. ii. : 2iii. : 2iv. A B,A B, (AB) (B A)
3. .
i. .ii. : .iii. : .
4. . .
i. .ii. : .
5. 2 () 3 ().i. .ii. :
6. - : : Venn :() () () () ()
7. , , Venn :
i. .ii. .
8. = {0, 1, 2, 3, 4, 5, 6} A = {1, 2, 3, 4, 6}, B = {0, 2, 3}. :
i. ii. A Biii. (A B) iv. A Bv. (A B) vi. A (B A)
9. = {1, 0, 1, 2, 3, 4, 5} A = {1, 2, 3, 4}, B = {1, 3, 5}.
i. (A B) A B (A B) = A B
ii.
- 12
-
A (B A) A (B A) = A B.
10. = {3,1, 0, 1, 2, 3, 4, 6} A = {1, 2, 3, 4}, B = {3,1, 0, 2}. :() B A = A B() (AB) = B A
11. = {0, 1, 2, 3, 4, 6} - A = {1, 2, 3, 4}, B = {0, 2, 3}. :
i. A = (AB) (A B) = (A B) (A B)ii. B = (B A) (B A) = (B A) (B A)
- 13
-
1.2
1.2.1 1.9 .
,
fA. fA =
.
1.10 .
, ( ). , , .
1.11 .
( ) , - :
P (A) =N(A)
N()
N(A) .N()
1.12 .
= 1, 2, ..., . i , P (i) :
1. 0 P (i) 12. P (1) + P (2) + ...+ P () = 1
P (i) i. P (A) A = {1, 2, ..., } P (1) +P (2) + ...+P (), P () = 0.
- 14
-
1.13 : P (A B) = P (A) + P (B)
P (A B) = N(A B)N()
=N(A) +N(B)
N()
=N(A)
N()+N(B)
N()
= P (A) + P (B)
1.14 : P (A) = 1 P (A)
A A = A A = .
P (A A) = P () P (A) + P (A) = 1 P (A) = 1 P (A)
1.15 : P (A B) = P (A) + P (B) P (A B)
P (A B) = N(A B)N()
=N(A) +N(B)N(A B)
N()
=N(A)
N()+N(B)
N() N(A B)
N()
= P (A) + P (B) P (A B)
1.16 , . A B, P (A) P (B).
A B N(A) N(B)
N(A)N()
N(B)N()
P (A) P (B)
1.17 : P (AB) = P (A) P (A B)
- 15
-
P (AB) = N(AB)N()
=N(A))N(A B)
N()
=N(A)
N() N(A B)
N()
= P (A) P (A B)
- 16
-
1.2.2 - 1.5 10 , 15 , 20 , 5 30 . , .
1.5 : N(K) = 10, N() = 15, N() = 20, N(M) = 5, N(A) = 30,N() = N(K) +N() +N() +N(M) +N(A) = 10 + 15 + 20 + 5 + 30 = 80:
P (K) =N(K)
N()=
10
80=
1
8
P () =N()
N()=
15
80=
3
16
P () =N()
N()=
20
80=
1
4
P (M) =N(M)
N()=
5
80=
1
16
P (A) =N(A)
N()=
30
80=
3
8
1.6 " .i. :
: 5: , 4: 6
2.ii. , .
1.6 i. = {1, 2, 3, 4, 5, 6} N() = 6 :A = {1, 2, 3, 4} N(A) = 4B = {4, 6} N(B) = 2 = {2, 3, 6} N() = 3 :
P (A) =N(A)
N()=
4
6=
2
3
P (B) =N(B)
N()=
2
6=
1
3
P () =N()
N()=
3
6=
1
2
ii. A B = {4} 6= A,B . A = {2, 3} 6= A, . B = {6} 6= , B .
1.7 , = {1, 2, 3, 4}
- 17
-
i. P (1) =1
4, P (2) =
1
5, P (3) =
3
8,
P (4).
ii. P (1) = P (3) =1
4 P (2) = 2P (4)
P (2), P (4)iii. A = {1, 2} B = {2, 3}
P (1) =1
6, P (A) =
1
4, P (B) =
1
3 P (4)
1.7 i. ,P () = 1 P (1) + P (2) + P (3) + P (4) = 1
14
+1
5+
3
8+ P (4) = 1
3340
+ P (4) = 1
P (4) = 133
40
P (4) =7
40ii. ,
P () = 1 P (1) + P (2) + P (3) + P (4) = 1
14
+ 2P (4) +1
4+ P (4) = 1
12
+ 3P (4) = 1
3P (4) = 11
2
3P (4) =1
2
P (4) =1
6
P (2) = 21
6=
1
3iii. ,
P (A) =1
4 P (1) + P (2) =
1
4
16
+ P (2) =1
4
P (2) =1
4 1
6
P (2) =1
12
- 18
-
,
P (B) =1
3 P (2) + P (3) =
1
3
112
+ P (3) =1
3
P (3) =1
3 1
12
P (3) =1
4,
P () = 1 P (1) + P (2) + P (3) + P (4) = 1
16
+1
12+
1
4+ P (4) = 1
12
+ P (4) = 1
P (4) = 11
2
P (4) =1
2
1.8 A,B :
P (B) =3
4, P (A B) = 3
5 P (A B) = 3
10 :
i. P (B)ii. P (A)iii. P (A B)iv. P [(AB) (B A)]
1.8 i. P (B) = 1 P (B) = 1 34
=1
4ii.P (A B) = P (A) + P (B) P (A B) 3
5= P (A) +
1
4 3
10
35 1
4+
3
10= P (A)
P (A) = 1320
iii. P (A B) = P (AB) = P (A) P (A B) = 1320 3
10=
7
10
iv. P [(AB) (B A)] = P (A) + P (B) 2P (A B) = 1320
+1
4 2 3
10=
3
10
1.9 A,B
P (A) =2
5 P (B) =
3
4i. A,B .
ii. ,3
20 P (A B) 2
5
iii. P (AB) 14
- 19
-
1.9 i. P (A B) = 0 P (A B = P (A) + P (B) = 2
5+
3
4=
23
20> 1 .
A,B .
1.1 , A B 6= P (A B) 6= 0. . P (AB) = 0 P (AB) = P (A)+P (B) .
ii. :3
20 P (A B) 3
20 P (A) + P (B) P (A B)
320 2
5+
3
4 P (A B)
P (A B) 25
+3
4 3
20 P (A B) 1
.
P (A B) 25 P (A B) P (A)
, A B A
1.2 , .
iii. :P (AB) 1
4 P (A) P (A B) 1
4
25 P (A B) 1
4
25 1
4 P (A B)
320 P (A B)
i.
1.10 A,B P (A) = 0, 4, P (B) = 0, 3 P (A B) = 0, 1, :
i. A B.ii. A B.iii. A.iv. A B.
- 20
-
1.3
P (A)
P (AB) = P (A B) P (A B)
( , ) P (A B)
( ) P ((AB) (B A))
P ((A B)) = P (A B) P (A)
P ((A B)) = P (A B) A B :P (A B) = P (A) + P (B) P (A B) = 0, 4 + 0, 3 0, 1 = 0, 6.
1.10 i.ii. A B :P (A B) = 1 P (A B) = 1 0, 6 = 0, 4.
iii. A :P (AB) = P (A) P (A B) = 0, 4 0, 1 = 0, 3.
iv. A B :P [(AB) (B A)] = P (A) + P (B) 2P (A B)= 0, 4 + 0, 3 2 0, 1 = 0, 5.
1.11 A,B
P (A) 1125, P (B) 13
25, :
i. P (A B) 2625 P (A B).
ii. A B 6= .
1.11 i. :
P (A) 1125
1 P (A) 1125
P (A) 1 1125
P (A) 1425
(1)
:
P (B) 1325
1 P (B) 1325
P (B) 1 1325
P (A) 1225
(2)
, (1), (2) P (A) + P (B) 2625
(3)
P (A B) 2625 P (A B), :
- 21
-
P (A B) 2625 P (A B) P (A) + P (B) P (A B) 26
25 P (A B)
P (A) + P (B) 2625
(3).ii. A B = P (A B) = 0
P (A B) = P (A) + P (B) 2625
> 1 . , A B 6=
- 22
-
1.2.3
1. 1 30. :i. 3.ii. 4.iii. 6.
:i. A B ii. A iii. B iv. A B
2. 2 () -(). .
i. 2 .ii. 2 .iii. 2 .iv. .
3. 3 . , 10.
4. 28 , , -
. , 3
14,
, , .
5. A,B , P (A) = 0, 25, P (AB) = 0, 5 P [(AB)] = 0, 2, :
i. A ii. A Biii. A B iv. B
6. A,B , P (A) = 0, 6, P (A B) = 0, 35 P (A B) = 0, 55, :
i. A ii. Biii. AB iv. A Bv. A B vi. A B
7. A,B , P (A) = 0, 35, P (B) = 0, 3 P (A B) = 0, 15, :
i. A ii. A Biii. AB iv. A Bv. A B vi. A B
8. A,B ,
P (A) =1
8, P (B) =
1
2
P (A B) = 58:
i. , .ii. :
i. A ii. (A B)iii. AB iv. A Bv. A B vi. A B
9. A,B , P (A)+P (B) = 1, 2, P (B) = 0, 3 P (AB) = 0, 2,
- 23
-
:i. B A ii. A B
10. A,B ,
10 P (A) P (B) = 3, 10 P (A) 10 P (B) = 1 P (A B) = 15,
:i. A ii. Biii. A B iv. A Bv. A B vi. A B
11. A,B , P (AB) = 0, 3, P (B) = 0, 3 P (B A) = 0, 3, :
i. .ii. .iii. .iv. .
12. A,B , P (A B) = 0, 8, P (B) = 0, 4 P (A) = 0, 5, :
i. .ii. .iii. .iv. .v. .
13. A,B , P (A) = 0, 8, P (B) = 0, 4 .
I. , .II. :
i. 0, 2 P (A B) 0, 4ii. 0, 8 P (A B) 1
14. A,B , P (A) 3 P (A), P (B) 0, 25.
i. P (A B) + P (A B) 1.ii. .
- 24
-
-
-
2.
2.1
2.1.1
2.1 ;
, , - .
2.2 ;
, :
,
, 6= 0. - , , , .
2.3 ;
, , , , .
2.4 ;
i. + = + ii. + ( + ) = (+ ) + iii. + 0 = 0 + = 0 .iv. + () = () + = 0 0.
-
2.5 ;
i. =
ii. ( ) = ( )
iii. 1 = 1 = 1 .
iv. 1
=1
= 1, 6= 0 1.
2.6 ;
+ () =
2.7 ;
( 1
) =
2.8 ;
i. ( = = ) + = + .
ii. ( = = ) = .
iii. = + = + .
iv. = = .
v. = 0
= 0
= 0
vi. 6= 0
6= 0
6= 0
- 26
-
2.9 ;
, 2,
= ,
:i. 1 = ii. 0 = 1, 6= 0iii. =
1
, 6= 0
2.10 ;
1. = + 4. = ( )
2.
= 5.
=(
)3. =
() 6. (
)=
(
) 2.1:
2.11 , = = ; ;
= , = . , : (2)2 = 22 = 4 2 6= 2.
2.12 ;
-.
- 27
-
2.13 .
:
i.(+ )2 = 2 + 2 + 2
ii.( )2 = 2 2 + 2iii.( )(+ ) = 2 2iv.3 + 3 = (+ )(2 + 2)v.3 3 = ( )(+ + 2)vi.(+ )3 = 3 + 32 + 32 + 3
vii.( )3 = 3 32 + 32 3
2.2:
i.(+ )2 = (+ )(+ )
= 2 + + + 2
= 2 + 2 + 2
ii.( )2 = ( )( )
= 2 + 2
= 2 2 + 2iii.
( )(+ ) = 2 + + 2
= 2 2iv.
(+ )(2 + 2) = 3 2 + 2 + 2 2 + 3
= 3 + 3
v.( )(+ + 2) = 3 + 2 + 2 2 2 3
= 3 3vi.
(+ )3 = (+ )2(+ )
= (2 + 2 + 2)(+ )
= 3 + 2 + 22 + 22 + 2+ 3
= 3 + 32 + 32 + 3
vii.( )3 = ( )2( )
= (2 2 + 2)( )= 3 2 22 + 22 + 2 3
= 3 32 + 32 3
- 28
-
2.14 .
i.2 + 2 = (+ )2 2ii.3 + 3 = (+ )3 3(+ )iii.( )2 = (+ )2 : ( )2 = (+ )2iv.( )2 = ( )2 : ( )2 = ( )2v.(+ + )2 = 2 + 2 + 2 + 2 + 2 + 2vi.( + )2 = 2 + 2 + 2 2 + 2 2vii. = ( )(1 + 2 + ...+ 2 + 1)viii. + + = 0 3 + 3 + 3 = 3
2.3:
:
i.(+ )2 2 = 2 + 2 + 2 2
= 2 + 2
ii.(+ )3 3(+ ) = 3 + 32 + 32 + 3 32 32
= 3 + 3
iii.( )2 = [(+ )]2
= (+ )2
iv.( )2 = [( )]2
= ( )2
v.(+ + )2 = (+ + )(+ + )
= 2 + + + + 2 + + + + 2
= 2 + 2 + 2 + 2 + 2 + 2
vi.( + )2 = ( + )( + )
= 2 + + 2 + + 2
= 2 + 2 + 2 2 + 2 2vii. .viii. + + = 0 =
- 29
-
3 + 3 + 3 = ( )3 + 3 + 3
= [( + )]3 + 3 + 3
= ( + )3 + 3 + 3
= (3 + 32 + 32 + 3) + 3 + 3
= 3 32 32 3 + 3 + 3
= 32 32
= 3( )= 3
2.15 ;
I. .i. ii. iii. .
II. .
2.16 .
i.
=
= 6= 0
ii.
=
=
6= 0
iii.
=
+
= +
6= 0
iv.
=
=
=+
+ ( + ) 6= 0
2.4:
:
i.
=
=
= ii.
=
=
=
=
- 30
-
iii.
=
+ 1 =
+ 1
+
= +
iv. :
=
= , = = , :
+ = +
= ( + )
=+
+
:
=
=
=+
+
2.17 ;
, . :
i. , . : + = ( + ) :() ... ()
. .
ii. () , , :() ()
iii.
2 2 = ( )(+ ) :(+ )2 = 2 + 2 + 2,( )2 = 2 2 + 2 .
iv. 3 + 3 = (+ )(2 + 2),3 3 = ( )(+ + 2) :(+ )3 = 3 + 32 + 32 + 3,( )3 = 3 32 + 32 3 .
v.
- 31
-
2 1 :(+ )2 = 2 + 2 + 2,( )2 = 2 2 + 2(+ )3 = 3 + 32 + 32 + 3,( )3 = 3 32 + 32 3
vi. x2 x+ = 24. :
() > 0, x2 x+ = (x x1)(x x2) x1,2 = +
2
() = 0, x2 x+ = (x x1)2 x1 =2
() < 0, . :
x2 (+ )x+ = (x )(x )
x2 + (+ )x+ = (x+ )(x+ )
2.18 ;
i. , :
=
ii. , : = 0 = 0 = 0
- 32
-
2.1.2 -
2.1 : (2 + 2)(x2 + y2) = (x+ y)2 + (y x)2 (1)
2.1
(1) 2x2 + 2y2 + 2x2 + 2y2 = 2x2 + 2xy + 2y2 + 2y2 2yx+ 2x2
0 = 0
.
2.2 = 5 : (+ 2) + ( 2) 2
2.2
(+ 2) + ( 2) 2 = 2 + 2+ 2 2 2= 2 + 2 2 + 2 2= ( )2 + 2( )
=5= 52 + 2 5= 35
2.3 : 2 > .
2.3 =1
2:
1
4>
1
2 , .
2 > .
2.4 , + .
2.4 , =
+ . + =
:
+ =
+
=
=
=
, . + .
- 33
-
2.5 ;
2.5 ,
3 + 1
3 + 7. :
3 + 1
3 + 7 = 8 R .
2.6 2 , .
2.6 , = 2+ 1, Z. :
= 2+ 1 2 = (2+ 1)2
2 = 42 + 2+ 1 2 = 2(22 + ) + 1 2 ,
. .
2.7
2 .
2.7
2 , :
2 =
, , N
. :
2 =
2 =
2
2
2 = 22
2 = 2
:
2 = 22 (2)2 = 22
42 = 22
2 = 22
2 = 2
=
2
2=
, .
2 .
- 34
-
2.8 :i. 2x5 + 4x4y + 6x2
ii. 3x+ x+ 3y + yiii. x2 6x+ 9iv. x2 4v. x3 8vi. x2 5x+ 6vii. x2 6x+ 9
2.8 i. 2x5 + 4x4y + 6x2 = 2x2(x3 + 2x2y + 3)ii. 3x+ x+ 3y + y
1 : 3x+ x+ 3y + y = 3(x+ y) + (x+ y) = (x+ y)(3 + )2 : 3x+ x+ 3y + y = x(3 + ) + y(3 + ) = (3 + )(x+ y)
iii. x2 6x+ 9 = x2 2 3 x+ 32 = (x 3)2iv. x2 4 = x2 22 = (x 2)(x+ 2)v. x3 8 = x3 23 = (x 2)(x2 + 2x+ 22) = (x 2)(x2 + 2x+ 4)vi. x2 5x+ 6
= 24 = (5)24 1 6 = 2524 = 1 > 0. :
x1 = +
2=(5) +
1
2 1=
5 + 1
2=
6
2= 3
x1 =
2=(5)
1
2 1=
5 12
=4
2= 2
x2 5x+ 6 = (x 3)(x 2).
vii. x2 6x+ 9 = 2 4 = (6)2 4 1 9 = 36 36 = 0,
x1 =2
=(6)2 1
=6
2= 3.
x2 6x+ 9 = (x 3)2.
- 35
-
2.1.3
1. :
A =[(x2y3
)2 (xy3)4] : ( x3y1
)3i. : A = x9y9
ii. x = 2012 y =1
2012.
2. A =[(xy1
)2:(x3y7
)1]2 x = 0, 4 y = 2, 5
3. i (+ )2 ( )2 = 4
ii :
(999
1000+
1000
999
)2(
999
1000 1000
999
)2.
4. :i. 8 4ii. 52 10 202iii. 252 25iv. ( )(x 2y) ( )(2x 3y)v. (x y) (y x)vi. (3 4)2 (4 3)3vii. 2 3+ 3viii. xy2 + x 1 y2ix. (x2 + y2) + xy(2 + 2)x. 16x2y4 44xi. 33 273xii. 9 (+ 3)2xiii. 25x2 + 10x+ 1xiv. 2 42 + 44xv. 2x2 3x+ 2xvi. x2 3x+ 2
5. :A = [x(xy + 2) x(2 y2)] : xy x = 4, 125 y = 3, 875
6. + =1
, A = (+ )3 3 3.
7. x(+ x) 6= 0,
A =x
+x(+ x)2
2(
1
+ x 1x
), , x.
8. i. : (2 + 2)(x2 + y2) (x+ y)2 = (y x)2ii. 25 26 .
9. 1
= 5, :
i. 2 +1
2
ii. 3 13
iii. +1
- 36
-
iv. 3 +1
310. :
i.3 22 +
2 ii.
(2 ) + 2 22 1
iii.( 1
)2
3 + 2
(+ 1)3
iv.2 + + 1
+ 1
2 13 1
v. (x+ y)2 (x1 + y1)2
vi.x+ y
x y x1 y1
x2 y2
11. :(x3 + y3
x2 y2
):
(x2
x y y)
= 1.
12. :i. , + .ii. , 6= 0, , .
- 37
-
2.2
2.2.1 2.19 ;
> , > 0. < . : > 0 < 0 , > :
> > 0
, .
2.20 ;
> =
2.21 ;
i. ( > 0 > 0) + > 0ii. ( < 0 < 0) + < 0iii. , > 0
> 0
iv. , < 0 < 0
v. 2 0 R = = 0
vi. 2 + 2 = 0 = = 0vii. 2 + 2 6= 0 6= 0 6= 0
2.22 .
i. ( > > ) > ii. > + > + iii. >
>0 > iv. >
>0 >
v. >
-
vii. ( > > ) + > + viii. , , , > 0 ( > > ) > ix. , > 0 :
> >
2.23 , : > >
>
;
, , 15 > 5 5 > 1, 15
5>
5
1.
( )
!!! .
2.24 , : > > > ;
, : 10 > 9 8 > 1, 10 8 > 9 1.( )
!!! .
2.25 , :2 > 2 > ;
, : (3)2 > 22, 3 > 2.( )
- 39
-
2.2.2 -
2.1 , 1 , ( ), , .
2.9 , , , ... > 1 0, :
>
>
1>
1
1 0, ...: +1
2
2.11
+1
2 + 1
2
2 + 1 2 2 + 1 2 0 (+ 1)2 0
.
2.12 < 0, ...: +1
2
2.12 .
2.13 ... 2 + + 2 0
- 40
-
2.13
2 + + 2 0 22 + 2 + 22 0 2 + 2 + 2 + 2 + 2 0 (+)2 + 2 + 2 0
.
2.14 ... 2 + 2 0
2.14 .
2.15 ... 2 + 2 0
2.15 : 2 0 2 0, , 2 + 2 0
2.16 ... 2 0 = 0
2.16 i. = 0 2 = 0 , 2 0ii. 2 0 = 0 < 0 . = 0.
, 2 0 = 0
2.17 x 0 y > 0 ... x+ y > 0
2.17 x 0 x+ y y > 0
2.18 x 0 y < 0 ... x+ y < 0
2.18 .
2.19 x y > ... x+ > y +
2.19 x y x y 0 > > 0, (x y) + ( ) > 0 (x+ ) (y + ) > 0 x+ > y +
2.20 x y < ... x+ < y +
2.20 .
2.21 ... 2 + 2 > 0 6= 0 6= 0
2.21 :i. = 0 6= 0 , 2 = 0 2 > 0 : 2 + 2 > 0ii. = 0 6= 0 , 2 = 0 2 > 0 : 2 + 2 > 0
- 41
-
iii. 6= 0 6= 0 , 2 > 0 2 > 0 : 2 + 2 > 0: 2 + 2 > 0 = = 0 2 + 2 = 0 . 6= 0 6= 0
2.22 : 0 < <
i. 1,
,
ii.
1,
.
2.22 i. , 0 < < ,
< 1
> 1
< 1 0
.
2.23 2 < x < 4 1 < y < 2, :
i. x+ yii. 2x 3y
iii.x2
y
iv.1
x+
1
y
2.23 i. : 2 < x < 4 1 < y < 2 , : 3 < x+ y < 6
ii. : 2 < x < 4 4 < 2x < 8 1 < y < 2 6 < 3y < 3 , : 2 < 2x 3y < 5
!!! .
iii. : 2 < x < 4 4 < x2 < 16 1 < y < 2 1
2 0, :(
1 +x
y
)2+(
1 +y
x
)2 8
14. 4 < x < 5 2 < y < 6, :
i. x+ yii. 2x+ 3yiii. 3x 2yiv. xyv. 2xy + 4vi.
x
y15. 3 < x < 2 4 < y < 1,
xy.16. x, y , -
:i. (x 2)2 + (y + 1)2 0ii. x2 + y2 2x+ 4y + 5 0
- 44
-
2.3
2.3.1 2.26 ;
A .
2.1:
A O, OA, ||. :
|| ={, 0, < 0
2.27 ;
i. || = | | 0ii. || 0, || , || iii. ||2 = 2iv. | | = || ||
v. | |= ||||
vi. ||| ||| |+ | ||+ ||vii. | | = ||
2.28 ;
i. > 0 :i. |x| = x = x = ii. |x| = || x = x =
ii. > 0 :i. |x| < < x < ii. |x| > x < x >
- 45
-
2.29 :i. | | = || ||
ii. | |= ||||
iii. |+ | ||+ ||
i.| | = || || | |2 = (|| ||)2
| |2 = ||2 ||2
( )2 = 2 2 .
ii.
| |= ||||
| |2=
(||||
)2
(
)2=||2
||2
(
)2=2
2
.iii. , , |+ | ||+ ||, , :
|+ | ||+ || |+ |2 (||+ ||)2
(+ )2 ||2 + 2||||+ ||2
2 + 2 + 2 2 + 2||||+ 2
|| . > 0, , .
2.30 [,] :+
2
[, ] , , ,. [,] x0, : = :
(MA) = (MB) d(x0, ) = d(x0, ) |x0 | = |x0 | x0 = x0, < x0 < 2x0 = +
x0 =+
2
x0 =+
2 ,
- 46
-
[,], =
2 [,].
2.31 |xx0| < .
x0 R > 0 :
|x x0| < x (x0 , x0 + ) x0 < x < x0 +
2.32 |xx0| > .
x0 R > 0 :
|x x0| > x (, x0 ) (x0 + ,+) x < x0 x > x0 +
- 47
-
2.3.2
2.24 :i) |2 2 3| = |2 2 + 3|
ii)
||=||
2.24 i) 2 2 3 22 + 3 , || = | |.
ii)
||=|| 2 = ||2 .
2.25 :i) ii) 2iii) 2+ 1iv) 2 + 2
2.25 i) || = | | ={ , 0 , < 0
ii) 2 0 = 2, | 2| = 2 2 < 0 = < 2, | 2| = + 2 - 2. 2 = 0 (1) . = 2 .
: | 2| ={ 2 , 2+ 2 , < 2
iii) 2+ 1 = 0 = = 12 (2), : |2+1| =
{2+ 1 , < 122 1 , 12
iv) 2 + 2 > 0 : |2 + 2| = 2 + 2
2.2: (1)
2.3: (2)
- 48
-
2.26 A = 2|x 1|+ 3
2.26 x 1 = 0 = x = 1 (3) :
i) x < 1 : A = 2|x 1|+ 3 = 2(x+ 1) + 3 = 2x+ 2 + 3 = 2x+ 5ii) x 1 : A = 2|x 1|+ 3 = 2(x 1) + 3 = 2x 2 + 3 = 2x+ 1
A ={2x+ 5 , x < 12x+ 1 , x 1
2.4: (3)
2.27
A = 3|1 x|+ 2|x 2|+ 5
2.27 (4) (5). (4) (5), , (6) x < 1, 1 x 2 x > 2
1 x < 1 :
A = 3(1 x) + 2(x+ 2) + 5 = 3 + 3x 2x+ 4 + 5 = x+ 6
2 1 x 2 :
A = 3(1 + x) + 2(x+ 2) + 5 = 3 3x 2x+ 4 + 5 = 5x+ 12
3 x > 2 :
A = 3(1 + x) + 2(x 2) + 5 = 3 3x+ 2x 4 + 5 = x+ 4
A =
x+ 6 , x < 15x+ 12 , 1 x 2x+ 4 , x > 2
2.5: (4)
- 49
-
2.6: (5)
2.7: (6)
2.28 x, y, w R i) |x|+ |y|+ |w| = 0;ii) |x|+ |y|+ |w| > 0;
2.28 R || 0 i) x = y = w = 0 ii) x, y, w 6= 0
2.29 |x+ y 3|+ |3x 2y + 1| = 0, x, y.
2.29 , x+ y 3 = 0
3x 2y + 1 = 0
x = 1
y = 2
2.30 i) |+ | = ||+ ||;ii) : ||| ||| |+ |.
=;
- 50
-
2.30 i)
|+ |2 = (||+ ||)2 (+ )2 = (||+ ||)2 2 + 2 + 2 = ||2 + ||2 + 2|||| 2 + 2 + 2 = 2 + 2 + 2|||| = |||| 0
, ii)
||| ||| |+ | ||| |||2 |+ |2 (|| ||)2 (+ )2 ||2 + ||2 2|||| 2 + 2 + 2 |||| |||| ||
, = 0 , .
2.31 | | 1 | | 2, | | 3
2.31 :
| | =| + | | |+ | | 1 + 2 =
3
2.32 |x| < 2, |3x2 5x+ 1| 23
2.32 :
|3x2 5x+ 1| =|3x2 + (5x) + 1| |3x2|+ | 5x|+ 1 =3|x|2 + 5|x|+ 1 3 22 + 5 2 + 1 =12 + 10 + 1 =
23
- 51
-
2.33 :
i) + 4+ 1
= 2, || = 2ii)
|x|y|+ y|x|||xy|
= 2, x, y x, y 6= 0.
2.33 i)+ 4+ 1 = 2
|+ 4||+ 1|
= 2
|+ 4| = 2|+ 1| |+ 4|2 = 4|+ 1|2 (+ 4)2 = 4(+ 1)2 2 + 8+ 16 = 4(2 + 2+ 1)...2 = 4||2 = 22 || = 2
ii)
|x|y|+ y|x|||xy|
= 2
|x|y|+ y|x|| = 2|xy| |x|y|+ y|x||2 = 4|xy|2 (x|y|+ y|x|)2 = 4(xy)2 x2y2 + 2xy|xy|+ x2y2 = 4x2y2 2xy|xy| = 2x2y2 (x, y 6= 0)|xy| = xy xy > 0
x, y .
- 52
-
2.3.3
1. () | 3|() | 4|() |3 |+ |4 |() |
2
3| |
3
2|2. 3 < x < 4
A = |x 3|+ |x 4|
3. A = |x 3| |4 x| :() x < 3() x > 4
4. < < :
= 2| |+ 3| | | |
5. 0 < < :
= 3| | 2| |+ |+ | 5|2+ |
6. 6= ,
7.
() |x+ 2|() |3x|() |4x 3|() |x2 + 1|() |2 3x|
8. () |x|+ |x+ 1|() 2|x 2|+ 3|x+ 1|
()|3 x|x 3
() 2|x| 3|x 1| 7|x 5|() 7| x|+ 6|2 x| |x+ 1|
9. x 6= 0 y 6= 0 :
|x|x
+|y|y
10. : |x y| |x w|+ |w y|.11. > ,
() =+ + | |
2
() =+ | |
212. x y
() |x|+ |y| = 0;() |x|+ |y| > 0;
13. , R, :
() |+ 1|+ |2 + 3| = 0
- 53
-
() (+ 2)2 + |3 5 + 2| = 014. , :
() |2 2+ 1|+ |2 1| = 0() | 1|+ |+ 1| = 0
15. 1,
,
, 0 < <
()
()
1 ,
16. : |x| 2 |y| 5,, :
() |3x+ y| 11() |2x 3y + 11| < 32
17. || 1, : |33 + 22 8| < 1318. || 2 || 1
2, : | 4 3| 7
19. .
|x 4| 2 d(x, 4) 2 2 x 6 [2, 6]|x+ 3| < 4|x 4| > 2|x+ 3| 4
d(x, 5) < 1
d(x,1) > 2d(x, 5) 1d(x,1) 2
x < 5 x > 15 x 1
(2, 2)(,2]
[2,)
- 54
-
2.4
2.4.1 2.33 ;
, , . : 0,
x2 = .
2.34 ;
i.2 = ||, R
ii.2
= , 0iii. =, , 0
iv.
=
, 0, > 0
2.35 - ;
-
, , , N. : 0,
x = .
!!! 1 = 2
=
2.36 - ;
i. = || R
ii.
= = , 0
iii. =
, , 0
iv.
=
, 0, > 0
v.
=
, 0
vi. =
, 0
vii. = (
)
viii. =
ix. , 0 < <
2.37 ;
> 0, Z, N,
=
- 55
-
, N 0 = 0
2.38 :i. =
, , 0
ii.
=
, 0, > 0
iii.
=
, 0
iv. =
, 0
i. =
(
) = (
)
( )(
) =
= .
ii.
=
(
)=
(
)
()
() =
=
.
iii.
=
(
)=()
[(
)]=
()
= .
iv. =
=
() =
- 56
-
2.4.2
2.2 , , - ( 0). ( 5)
, :
2 = ||, R. ( 1)
, . ( 2)
, - :
A
A
A
A3 3
32 3
3 + 3
2
( 8, 9) .
, . > 0, Z, N, :
=
2.34
i. A =
(
2 2)2
ii. B =x4 2x2 + 1
iii. =42
24
2.34 i.
A =
(
2 2)2
= |
2 2|=
2 + 2
2 2 < 0,
2 < 2.
ii.B =
x4 2x2 + 1
=
(x2 1)2
= |x2 1| x21 , .
iii. =
42
24
=
(2)2
(2)2
= |2| |2|= 2|| ||2
- 57
-
2.35 : 3
8 5
18 + 2
72 5
50
2.35 :8 24 22 21
8 = 23
18 29 33 31
18 = 2 32
72 236 218 29 33 31
72 = 23 32
50 225 55 51
50 = 2 52
, :
3
8 5
18 + 2
72 5
50 = 3
23 5
232 + 2
2332 5
2 52
= 3
22
2 5
2
32 + 2
23
32
5
2
52
= 2
2 3 5
2 3 + 2 2
2 3 5
2 5= 6
2 15
2 + 12
2 25
2
= 22
2
- 58
-
2.36 :i.
3 2
2ii.
8 + 2
15
2.36 i. 2
2 ,
( )2 = 2 2 + 2
:3 2
2 =
2 + 1 2
2
=
22
+ 12 2
2
=
(
2 1)2
= |
2 1|=
2 1
2 > 1
ii. 2
15 ,
(+ )2 = 2 + 2 + 2
:8 + 2
15 =
5 + 3 + 2
5 3
=
52
+
32
+ 2
5
3
=
(
5 +
3)2
= |
5 +
3|=
5 +
3
5 +
3 > 0
2.37 :
i. A =4
3
3
3
ii. B =
23
3
4
- 59
-
2.37 i.
A =4
3
3
3
=12
3
3
=12
32 3
=12
33
=24
33
=8
3ii.
B =
2
3
3
4
=
3
23 3
4
=6
23 3
4
=6
26 32 4
=12
26 32 22
=12
28 32
2.38 i.
11 xii. 3|x| 1
iii.
2 5xiv. 52 + 8
2.38 , 0.
i.11 x 0 x 11
x [, 11]ii.
|x| 1 0 |x| 1 x 1, x 1 x (,1] [1,+)
iii.2 5x 0 5x 2
x 25
x (, 25
]iv.
2 + 8 0 R
- 60
-
2.39 :i. A =
3
3
3
3 +
3ii. B =
7 4
3
7 + 4
3
2.39 i.
A =
3
3
3
3 +
3
=
3
(3
3)(3 +
3)
=
3
32
32
=
3
9 3=
3
6
=
3
3
2
=
32
2
= 3
2ii.
B =
7 4
3
7 + 4
3
=
(7 4
3)(7 + 4
3)
=
72 42
32
=
49 16 3=
1
= 1
2.40 : 3
3
2
2.40 , ...(2,3)=6 :
=
3
3 =23
32 = 6
9
2 =3.2
23 = 6
8 6
9 > 6
8 3
3 >
2
2.41 = 5 +
3 = 3 +
5
2.41 , , - . , 0 2 2 :
- 61
-
2 = (5 +
3)2
= 25 + 3 + 2 5
3
= 28 + 10
3
2 = (3 +
5)2
= 9 + 5 + 2 3
5
= 14 + 6
5
2, 2
2 2 = 28 + 10
3 (14 + 6
5)
= 28 + 10
3 14 6
5
= 14 + 2(5
3 3
5)
(5
3)2 = 25 3 = 75 (3
5)2 = 9 5 = 455
3 > 3
5 5
3 3
5 > 0 2 2 > 0 2 > 2
>
2.42 -
i.23
ii.12
iii.3
4
53
iv.2
5 1v.
47 +
3
2.42 i.
3 :
23
=2
33
3
=2
3
32
=2
3
3ii.
2 :
- 62
-
12
=1
22
2
=
2
22
=
2
2iii. 4
543 = 4
5 :
34
53=
3 4
54
5 4
53
=3 4
54
54
=3 4
5
5iv.
5 + 1 :
25 1
=2 (
5 + 1)
(
5 1) (
5 + 1)
=2 (
5 + 1)
52 12
=2 (
5 + 1)
5 1
=2 (
5 + 1)
4v.
7
3 :
47 +
3=
4 (
7
3)
(
7 +
3) (
7
3)
=4 (
7
3)
72
32
=4 (
7
3)
7 3
=4 (
7
3)
4
=
7
3
- 63
-
2.43 .
23
3 3
2
2.43 , - .
3 3 = ( )(2 + 2)
= 3
3 = 3
2 ,
3
32 3
33
2 +3
22
:
23
3 3
2=
2( 3
32 3
3 3
2 + 3
22)
( 3
3 3
2)( 3
32 3
3 3
2 + 3
22)
=2( 3
32 3
3 3
2 + 3
22)
3
33 3
23
=2( 3
32 3
3 3
2 + 3
22)
3 2= 2(
3
32 3
33
2 +3
22)
= 2(3
9 3
6 +3
4)
- 64
-
2.4.3
1. :()
( 4)2
()
(20)2()
(x 1)2
()
x2
42. : A = 22 + 3 52 =
3 +
3 =
3
5.3. :
i. A =
2
ii. B =
x2
x
(x+ 1)2
x+ 1
iii. =x
|x|x2
x
iv.
(
3 1)2 +
(
2 + 1)2
4. :
1
(
3 2)2+
1
(
3 + 2)2= 4
5. :i. A =
4 2 +
2 4
ii. B =
(2
2)2 +
(
5 1)2
iii.
21 +
13 +
7 +
4
6. :i. A =
3 2
2
ii. B =
13 4
3iii. =
17 12
2
iv. =
4 + 2
3v. E =
5 +
24 +
5
24vi. Z =
13 +
48
7. :i. 13
8 5
32 +
128 7
18
ii. (
16 3
5 +
20
9) (1 +
5)
iii.1
2
27 5
3
48 +
3
9
108
iv. (
8
18) (
50 +
72
32)v. 8
24 + 2
150 3
6 +
54vi. 7
45 3
125 + 4
5
500 + 5
12 2
3 + 6
27
8. :i.
2
2
2
2 +
2 = 2ii. 3
2 3
3 +
5 3
3
5 = 2
9. :
i.25
1275
= 10
ii.
216
7550
= 18
- 65
-
10. =
2 +
2 +
3, =
2
2 +
3 =
2 +
3 : = 1
11. x i.x 1
ii.
1 3xiii.x2 + 2
x2 2x+ 1
iv.
3 |x|
v.x+
1
xvi.
5 + |x|vii.
|x| 4
viii.|x+ 1| 1
12. A = (x 5
x+ 3) (
x 5 +
x+ 3)
i. x .ii. =-8.
13. A =x+ 2
x 1 +
x 2
x 1
i. x .ii. .
14. :i.
3 2
5
3ii.
2 + 3 2 +
7iii.
7
5 +
2iv. 1
2
3
2v.
3 +
2
6 + 1
15. i. 3
3
ii. 3 3
iii.
9x2
2y
3x
iv. 33612
v.
3
64x
vi.
x
y
3
y2
x2
vii.4
3
2
3
viii.4
23
322 3
3
ix. 43+2
4
x.
23
3
4
xi. 6 3
16. :
i.
2 3
2 = 3
2
ii. 9
28 6
25 = 2 18
213
iii.
52 3
5 6
54 = 25
5
iv.5
2
2 3
2 = 3
2
- 66
-
v. 4
33 3
3 = 3 12
3
17.
i.32
ii.13
iii.55
iv.17
v.13
2
vi.3
34
27
vii.5
65
viii.x
4x3
ix.2
39x2
x.
37
52
xi.75
318.
i.1
3
2
ii.4
5
3
iii.2
23 1
iv.2
2
5 2
2
v.1
7 5
vi.6 +
25
2
vii.8
7
5
viii.
7 +
67
6
19. :
i.
35
3+
5
5 +
3= 4
ii.1
(2
3)2 1
(2 +
3)2= 8
3
20. :
i3
3 2
23
2= 5 +
6
- 67
-
ii , > 0 :
= (+ ) +
21. i. (3 + 2
7)2 (3 2
7)2
ii. :
37 + 12
7
37 12
7 = 622. =
3
2 =
5 + 123.
i.7
2 +
3
5
ii.8
2
1
2 +
524.
i.2
3
4 3
2
ii.4
3
5 125.
i.
5 3
10
ii. 3
5
6 4
3
4iii. 4
6 1 + 5
2
26. :
A =[
23 ( 2)
12 (1)
32
]3 =
2
2 =
13
2
- 68
-
1 x+ = 0
x =
2
3.
3.1 1 x+ = 0
3.1.1 3.1 ;
, f(x) = g(x), f, g R. , .
3.2 x+ = 0 , R;
:x+ = 0 x = :
1. 6= 0 : x = x = , .
2. = 0 : x = 0x = :
i. 6= 0 , 0x = 6= 0 , , .
ii. = 0 , 0x = 0 x R, , .
-
3.3 x = :i. ii. iii.
i. , : 6= 0ii. , : = = 0iii. , : = 0 6= 0
- 70
-
3.1.2 3.1
i. 2(x+ 1) = x+ 3ii. x+ 3 = 2(x 3) xiii. 5(x 3) x = 4x 15
3.1 1 , , - 6= 0 , .
3.1 i.2(x+ 1) = x+ 3 2x+ 2 = x+ 3
2x+ x = 3 2 3x = 1 x = 1
3
.
ii.x+ 3 = 2(x 3) x x+ 3 = 2x 6 x
x 2x+ x = 6 3 0x = 8
, .
iii.5(x 3) x = 4x 15 5x 15 x = 4x 15
5x x 4x = 15 15 0x = 0
x R, , .
3.2 :i. (x 2)2 (x 1)2 = (1 3x)2 (2 3x)2 (1)ii. x2 7x+ 6 = 0 (2)iii.
2x+ 7
4 2x+ 5
3=
2 x4 2 + x
2(3)
iv.2x+ 1
x2 x 1x
=x
(x 1)2(4)
v.6
x 3 2x+ 3
=12
x2 9(5)
- 71
-
3.2 i.(1) x2 4x+ 4 (x2 2x+ 1) = 1 6x+ 9x2 (4 12x+ 9x2)
x2 4x+ 4 x2 + 2x 1 = 1 6x+ 9x2 4 + 12x 9x2
4x+ 4 + 2x 1 = 1 6x 4 + 12x 2x+ 3 = 6x 3 2x 6x = 3 3 8x = 6
x = 68
=3
4
ii.(2) x2 (6 + 1)x+ 6 1 = 0
(x 6) (x 1) = 0 x 6 = 0 x 1 = 0 x = 6 x = 1
iii.(3) 122x+ 7
4 122x+ 5
3= 12
2 x4 122 + x
2 3(2x+ 7) 4(2x+ 5) = 3(2 x) 6(2 + x) 6x+ 21 8x 20 = 6 3x 12 6x 6x+ 21 8x 20 = 6 3x 12 6x 6x 8x+ 3x+ 6x = 6 12 21 + 20 7x = 7 x = 1
iv.(4) 2x+ 1
x(x 1) 1x
=x
(x 1)2, x 6= 0, x 6= 1
x(x 1)2 2x+ 1x(x 1)
x(x 1)2 1x
= x(x 1)2 x(x 1)2
(x 1)(2x+ 1) (x 1)2 = x2
2x2 + x 2x 1 x2 + 2x 1 = x2
x = 2
v.(5) 6
x 3 2x+ 3
=12
(x 3)(x+ 3), x 6= 3, x 6= 3
(x 3)(x+ 3) 6x 3
(x 3)(x+ 3) 2x+ 3
= (x 3)(x+ 3) 12(x 3)(x+ 3)
6(x+ 3) 2(x 3) = 12 6x+ 18 2x+ 6 = 12 6x 2x = 12 18 6 4x = 12 x = 3
- 72
-
(5)
3.3 , (+ 1)x 2(+ 2x) = ( 2) 9 .
3.3 x = = = 0. .
(+ 1)x 2(+ 2x) = ( 2) 9 x+ x 2 4x = 2 2 ( 3)x = 2 9
:
3 = 0
2 9
= 3
= 3 = 3
3.4 , (2 + 3)x 4(x ) = 2 + 3 .
3.4 x = = 0 6= 0. .
(2 + 3)x 4(x ) = 2 + 3 x2 + 3x 4x+ 4 = 2 + 3 x2 + 3x 4x = 2 + 3 4 x(2 1) = 2 4+ 3
:2 1 = 0
2 4+ 3 6= 0
= 1 = 1
6= 3 6= 1 = 1
=1.
3.5 ,
2x 2
= 4x+ 1 .
3.5 x = = = 0. . 6= 0. :
- 73
-
2x 2
= 4x+ 1 2x 2
= 4x+ 1
3x 2 = 4x+ 3x 4x = + 2 x(3 4) = + 2
:
3 4 6= 0 (2 4) 6= 0
6= 0
2 4 6= 0 6= 0,2, 2
3.6 2x 1 = x+ (1) R.
3.6 , : x =
2x 1 = x+ 2x x = 1 + (2 1)x = 1 +
:I. 2 1 6= 0 6= 1 R {1, 1}
:
x =1 +
2 1=
1 +
( 1)(+ 1)=
1
1II. =-1, (1) : 0x = 0 ,
.III. =1, (1) : 0x = 2 ,
.
3.7 2x + (x 2) = 1 (1) , R.
3.7 , : x =
2x+ (x 2) = 1 2x+ x 2 = 1 2x+ x = 2+ 1 x(2 + ) = 2+ 1
:I. 2 + 6= 0 (+ 1) 6= 0 6= 0 6= 1 :
x =2+ 12 +
.
- 74
-
II. =0, : 0x = 1.i. 1 6= 0 6= 1 .ii. 1 = 0 = 1 .
III. =-1, : 0x = 3.i. 3 6= 0 6= 3 .ii. 3 = 0 = 3 .
3.8 , x = 1
(2 + 1)(x 1) + x = 2 5+ 5 (1)
3.8 1 , (1) x = 1 :
1 = 2 5+ 5 2 5+ 4 = 0
= 1
= 4
3.9 , x = 0
(+ 3)x 2 3 = 4(x ) + ( 2) (1)
3.9 x = 6= 0
(1) x+ 3x 2 3 = 4x 4+ 2 2 x+ 3x 4x = 4+ 2 2+ 2+ 3 x( 1) = 2 4+ 3
1 6= 0 6= 1 :
x =2 4+ 3
1
x = 0, :
0 =2 4+ 3
1 2 4+ 3 = 0
:
6= 1
2 4+ 3 = 0
6= 1
= 1 = 3 = 3
- 75
-
!!! x = 0, 0, 0 .
3.10 :
2(x 1) 5(x ) = 11x+ 4 (1)
, :
(x 7) + 4(x+ 1) = 22 (2)
.
3.10 (1) x = , = 0 6= 0
(1) 2x 2 5x+ 5) = 11x+ 4 2x 5x 11x = 2 5+ 4 x(2 16) = 2 5+ 4
:
2 16 = 0
2 5+ 4 6= 0
= 4 = 4
6= 4 6= 1 = 4
= 4 (2), :
4(x 7) + 4(x+ 1) = 2(4)2 4x+ 28 + 4x+ 4 = 32 0x = 0
(2) .
3.11 :i. 2|x+ 1| 6 = 0ii. |2x+ 7|+ 9 = 0iii. |x 1| 3|x+ 5| = 0iv. |x+ 1| = x+ 1v. |2x 4| = 2x+ 4vi. |3x 6| = 2x 2vii. |x2 4|+ |3x 6| = 0
3.11 i. |f(x)| = > 0 x = . , :
- 76
-
2|x+ 1| 6 = 0 2|x+ 1| = 6 |x+ 1| = 3
x+ 1 = 3
x+ 1 = 3
x = 2
x = 4ii. |f(x)| = < 0
: |2x+ 7|+ 9 = 0 |2x+ 7| = 9 .
iii. |f(x)| = |g(x)|
f(x) = g(x)
f(x) = g(x)
|x 1| 3|x+ 5| = 0 |x 1| = 3|x+ 5|
x 1 = 3(x+ 5)
x 1 = 3(x+ 5)
x 1 = 3x+ 15
x 1 = 3x 15
2x = 16
4x = 14
x = 8
x = 72
iv. : |f(x)| = f(x) f(x) > 0 |x+ 1| = x+ 1 x+ 1 > 0 x > 1
v. : |f(x)| = f(x) f(x) < 0 |2x 4| = 2x+ 4 2x 4 < 0 2x < 4 x < 2
vi. |f(x)| = g(x), |f(x)| 0 g(x) 0
|f(x)| = g(x)
g(x) 0
f(x) = g(x)
f(x) = g(x)
- 77
-
|3x 6| = 2x 2
2x 2 0
3x 6 = 2x 2
3x 6 = 2x+ 2
x 1
x = 4
x =8
5
x = 4
x =8
5vii. |f(x)|+ |g(x)| = 0 |f(x)| 0 |g(x)| 0 : f(x) = g(x) = 0
:
|x2 4|+ |3x 6| = 0
x2 4 = 0
3x 6 = 0
x = 2
x = 2
x = 2
3.12 : (x 3)3 8x3 + (x+ 3)3 = 0 ( )
3.12 Euler :
+ + = 0 3 + 3 + 3 = 3
, : (x 3)3 (2x)3 + (x+ 3)3 = 0 : x 3 2x+ x+ 3 = 0 :
- 78
-
(x 3)3 8x3 + (x+ 3)3 = 0 (x 3)3 (2x)3 + (x+ 3)3 = 0 (x 3)3 (2x)3 + (x+ 3)3 = 0 3(x 3)(2x)(x+ 3) = 0
x 3 = 0
2x = 0
x+ 3 = 0
x = 3
x+ 0
x = 3
- 79
-
3.1.3
1. :i. 19x = 7x 24ii. 7x 15 = 27iii. 4x 3(2x 1) = 7x 42iv. 2(3x 1) 3(2x 1) = 4v. 6(x+ 4) = 4(x 1) + 8vi. 3(x+ 1) 4(2x 1) + 5x+ 2 = 0vii. 7x+ 16 = 15x 4(2x 4)viii. x(x 2)2 = x2 4x+ 4
2. :
i.x
2=x+ 1
3
ii.x+ 1
2=
1
5
iii.2x 4
2= 5x
iv.1 4x
5 x+ 1
4=x 4
20+
5
4
v. 2x 5 x3
= 53
+7x
3
vi.5(4x 3)
7 (3x+ 2) = 2 x
7 20
vii.1
3x 1
2(x+ 3) = 1
6x 3
23. :
i.5(x 2)x+ 2
2x 6x+ 3
= 3
ii.2x 32x 4
6 = x 53x 6
112
iii.x+ 1
x 2+x+ 1
x 2=
2x2 + 4
x2 4iv.
x
x 1=
1
x2 xv.
x+ 1
x2 1+
2
x2 2x+ 1= 0
vi.1
x 1+
1
x+ 1=
2
x2 1vii.
3
x+ 2 2x
=x 4x2 + 2x
viii.x2 xx2 1
=x
x+ 1
ix.x3 8x 2
= x2 + 4
x.
1 + x
1 x 1 x
1 + x1 + x
1 x 1
=3
14 x
4. :(2x 1)77 + 3(x2 1) = 1 2(x 1) 3 = 2 (2x 1)
5. Ri. ( 1)x = 1ii. ( 2)x = iii. ( 1)x = 1
- 80
-
iv. ( 1)x = 2 + v. x(2 + 1) + 3 = 4( x) 2x
vi.x+
x+
3= 1
vii.x+ 1
x x 1x+
=2(+ 1)2
x2 2
viii.x+
x =
x2
x2 26. , R
i. x+ = x+ ii. (3x+ ) + 7 2 = 2 + 3(1 + x)iii. ( )x = 2 (+ )xiv. (x+ )2 (x )2 = 2(+ ) + 6= 0v. (x 1)(x+ 1) + (x 1) (x2 + 1) = 0
vi.x
=x
vii.x
x 1
=
1
7. R ( 2)x = 2 2:i ii iii
8. R (2 + 3)(x 10) = 2 11+ 18 x = 10.
9. R (x+ 2) 15 = 3(x ) + ( 3) x = 1.
10. (3 1)x+ 9x2 = 1 , .11. R (2+5+6)x = +3
2 .12. ( 3)2 + 3 = 2( 1) 52,
(+ 2)x = (+ x) + 2(x 1), .13. , R, (2 9)x = + 3
(4 2)x = + 2 .14. , R, (2 4)x = 2 9
( 2)x = + + 1 .15. (+ 2)(x 1) = 3(x+ 1) 2(2+ 1), , ,
(x 1) 3(x+ 1) = 2 + 4(+ 1 x) 3 .16. (1 x) = 4x + 1 ,
(x+ 1) + ( 7)x = (+ ) .17. (x 2)3 + (x+ 3)3 (2x+ 1)3 = 0,
(x+ 22) 3 = 2(x+ ) .18. , R
x
x
= 1 ;
19. :i. |2x 3| = 5ii. |2x 4| = |x 1|iii. |x 2| = 2x 1iv. |2x 1| = x 2
v.|x|+ 4
3 |x|+ 4
5=
2
3
- 81
-
vi.2|x|+ 1
3 |x| 1
2=
1
2
vii.3 x3 + x
= 4viii. |x 1||x 2| = |x 1|ix. |2|x| 1| = 3x.x2 2x+ 1 = |3x 5|
20. :
i.|x+ 1|+ 4
3 3|x+ 1| 1
15=|x+ 1| 4
5+ 2
ii. x2 2|x| 15 = 0iii. |x+ 4|+ |x 2| 2|x| = 0
21. :i. 2|x+ 3| = + 4ii. |x2 4| = 2 + 2 1
22. :. |x2 9|+ |6 2x| = 0. |x2 x|+ |x2 3x+ 2| 0. |x y + 1|+ |4 y2| = 0
- 82
-
3.2 x =
3.2.1 3.4 x = , R, N
I. > 0 , x =
II. > 0 , x =
III. < 0 , x = ||
IV. < 0 ,
- 83
-
3.2.2 3.13
i. x2 = 4ii. x4 = 34iii. x3 = 8iv. x3 = 125v. x20 = 0
3.13 i. x2 = 4 x =
4 = 2ii. x4 = 34 .iii. x3 = 8 x = 3
8 = 2
iv. x3 = 125 x = 3| 125| = 5
v. x20 = 0 x = 0
3.14 i. x4 + 8x = 0ii. x5 = 16xiii. x6 = 32xiv. (2x+ 6)3 = 8
3.14 i.x4 + 8x = 0 x(x3 + 8 = 0)
x = 0
x3 = 8
x = 0
x = 3| 8|
x = 0
x = 2
ii.x5 = 16x x5 16x = 0
x(x4 16) = 0
x = 0
x4 = 16
x = 0
x = 4
16
x = 0
x = 2
- 84
-
iii.x6 = 32x x6 + 32x = 0
x(x5 + 32) = 0
x = 0
x5 = 32
x = 0
x = 5| 32|
x = 0
x = 2
iv.(2x+ 6)3 = 8 2x+ 6 = 3
8
2x+ 6 = 2 2x = 4 x = 2
- 85
-
3.2.3
1. :i. x3 125 = 0ii. x5 243 = 0iii. x7 1 = 0iv. x3 + 125 = 0v. x5 + 243 = 0vi. x7 + 1 = 0vii. x2 64 = 0viii. x4 81 = 0ix. x6 64 = 0
x. x5 8x2 = 0xi. x4 + x = 0xii. x5 + 16x = 0
2. i. (x+ 1)3 = 64ii. 1 + 125x3 = 0iii. (x 1)4 27(x 1) = 0iv. (x2 1)5 = 81(x2 1)v. |x|9 = |x|6vi. (x3 29)3 + 8 = 0
- 86
-
3.3 2
3.3.1 3.5 2 ;
x2 + x+ = 0, 6= 0.
3.6 2 ;
= 2 4 .
> 0, x1,2 =
2 = 0, x =
2
= 0, ().
3.7 Vietta .
x2 + x+ = 0, 6= 0 > 0, x1, x2 :
S = x1 + x2 =
P = x1 x2 =
:
: x1 = +
2, x2 =
2
S = x1 + x2
= +
2+
2
= +
2
=22
=
P = x1 x2
= +
2
2
=()2
2
42
=2 (2 4)
42
=2 2 + 4
42
=4
42
=
- 87
-
3.8 x2 + x+ = 0, 6= 0 > 0, Vietta;
:
x2 + x+ = 0 x2 + x+
= 0
x2 (x1 + x2)x+ x1x2 = 0 x2 Sx+ P = 0
3.9 ;
x2 + x + = 0 , = x
- 88
-
3.3.2
3.15 :i. x2 + 2x = 0ii. 2x2 + 6x = 0iii. x2 4 = 0iv. 3x2 + 16 = 0v. 2x2 5x+ 3 = 0vi. x2 6x+ 9 = 0vii. 3x2 + 4x+ 2 = 0
3.15 i.x2 + 2x = 0 x(x+ 2) = 0
x = 0
x+ 2 = 0
x = 0
x = 2ii.
2x2 + 6x = 0 2x(x+ 3) = 0
x = 0
x+ 3 = 0
x = 0
x = 3iii.
x2 4 = 0 x2 = 4x = 2
iv.3x2 + 16 = 0 3x2 = 16
v. 2x2 5x+ 3 = 0 x2 + x+ = 0
= 2, = 5, = 3, = 2 4 = (5)2 4 2 3 = 1 > 0
x1,2 =
2
x1,2 =(5)
1
2 2=
5 14
=
x1 =
5 + 1
4
x2 =5 1
4
=
x1 =
6
4
x2 =4
4
=
x1 =
3
2
x2 = 1
vi. x2 6x+ 9 = 0 = 2 4 = (6)2 4 1 9 = 0 x =
6
2= 3
vii. 3x2 + 4x+ 2 = 0 = 2 4 = (42 4 3 2 = 8 < 0 .
- 89
-
3.16 :i. x2 7|x|+ 12 = 0ii. (x 1)2 + 4|x 1| 5 = 0
iii.(x+
1
x
)2 5
(x+
1
x
)+ 6 = 0
iv. 4x4 + 11x2 3 = 0v.
2
x+
2x 3x 2
+2 x2
x2 2x= 0
3.16 i.x2 7|x|+ 12 = 0 |x|2 7|x|+ 12 = 0 |x| =
2 7 + 12 = 0
= 1
1 = 4
2 = 31 = 4
2 = 3
|x|===
|x| = 4
|x| = 3
x = 4
x = 3ii. (x 1)2 + 4|x 1| 5 = 0 |x 1|2 + 4|x 1| 5 = 0 |x 1| =
: 2 + 4 5 = 0, = 36, :1 = 5
2 = 1
|x1|=====
|x 1| = 5
|x 1| = 1
, |x 1| = 1 x 1 = 1
x = 0
x = 2
iii.(x+
1
x
)2 5
(x+
1
x
)+ 6 = 0, x 6= 0
(x+
1
x
)=
- 90
-
:
2 5 + 6 = 0
1 = 2
2 = 3
x+1
x=
====
x+
1
x= 2
x+1
x= 3
x2 2x+ 1 = 0
x2 3x+ 1 = 0
x = 1
x =3 +
5
2
x =3
5
2iv.4x4 + 11x2 3 = 0, x2 = 42 + 11 3 = 0
1 = 3
2 =1
4
x2 = 3
x2 =1
4
x = 1
2
x =1
2
v. :2
x+
2x 3x 2
+2 x2
x2 2x= 0 x 6= 0 x 6= 2
2
x+
2x 3x 2
+2 x2
x2 2x= 0 2
x+
2x 3x 2
+2 x2
x(x 2)= 0
x(x 2) 2x
+ x(x 2)2x 3x 2
+ x(x 2) 2 x2
x(x 2)= 0
2(x 2) + x(2x 3) + 2 x2 = 0 2x 4 + 2x2 3x+ 2 x2 = 0 x2 x 2 = 0, = 9
x1 = 2
x2 = 1
- 91
-
3.17 R 2 (2 3+ 2)x2 x+ 1 = 0
3.17 2 : 2 3+ 2 6= 0
1,2 =3
9 82
6= 0 1,2 6= 2, 1
3.18 R :(2 3+ 2)x2 + x ( 1)( 2) = 0 .
3.18 :1. 2 , , x2
2 3+ 2 6= 0 6= 1 6= 22. ,
> 0 1 + 4(2 3+ 2)( 1)( 2) > 0 1 + 4(2 3+ 2)2 > 0 R.
R {1, 2}
3.19 : x2 + 2 = 2 2x, . R
3.19 x2 + 2 = 2 2x x2 + 2x+ 2 2 = 0 = (2)2 4(2 2) = 42 42 + 42 = 42
1. = 42 = 0 = 0 x = 22. 6= 0
x1,2 =2
42
2
=
2+ 2
2
+2+ 2
2
=
+
- 92
-
3.20 2x2 23x+ 4 1 = 0, 6= 0.i. .ii. .
3.20 i. = 2 4
= (23)2 42(4 1)= 46 46 + 42
= 42
ii. .
x1,2 =
2=
23
42
22=
23 222
x1 =2 + 1
x2 =
2 1
3.2 x2 + x+ = 0, .
6= 0, 0 6= 0, < 0
6= 0, = 0 2 6= 0, > 0
6= 0, 0, S = 0 6= 0, 0, P = 1 6= 0, > 0, P > 0 6= 0, > 0, P < 0
6= 0, > 0, P > 0, S > 0 6= 0, > 0, P > 0, S < 0
3.21 :i. x2 + 2x ( 2) = 0, 6= 0ii. x2 + (+ )x+ = 0, 6= 0
3.21 i. x2 + 2x ( 2) = 0 2 6= 0, 0 : 6= 0
0 22 4[( 2)] 0 4 + 42 + 8 0 (2 + 2)2 0
. 2 .ii. x2 + (+ )x+ = 0, 6= 0
2 , 6= 0, 0 : 6= 0
- 93
-
0 (+ )2 4 0 2 + 2 + 2 4 0 2 2 + 2 0 ( )2 0
. 2 .
3.22 R, x2 + 2x+ = 0, 6= 0 .
3.22 x2 + 2x+ = 0 , 6= 0, = 0 : 6= 0
= 0 22 42 = 0 2 = 1 = 1
3.23 6= , (2 + 2)x2 + 2(+ )x+ 2 = 0 . = .
3.23 (2 + 2)x2 + 2( + )x + 2 = 0 , < 0
< 0 4(+ )2 8(2 + 2) < 0 42 + 42 + 8 82 82 < 0 (42 + 42 8) < 0 (2 2)2 < 0
. = = 0 .
3.24 R, 2x2 + ( 9)x+ 2 + 3+ 4 = 0 .
3.24 :
= 0 ( 9)2 4 2 (2 + +3+ 4) = 0 2 18+ 81 82 24 32 = 0 72 42+ 49 = 0 2 + 6 7 = 0
- 94
-
1,2 =6
64
2=6 8
2 = 1 = 7
3.25 x2 8x+ 2 = 0 , R 18.
3.25 :
0 (8)2 4 1 ( 2) 0 64 4+ 8 0 72 4 18
18.
3.26 x2 2x + ( + )2 = 0, 6= (1), , x2x+ = 0 (2) .
3.26 (1) :
= 0 (2)2 4 1 (+ )2 = 0 42 = 4(+ )2
2 = (+ )2 (3)
(2) : = 2 4 = (+ )2 4 = ( )2 > 0 6=
3.27 5 2
6 5 + 2
6
3.27 : x2 Sx+ P = 0 S = x1 + x2, P = x1x2.S = x1 + x2 = 5 2
6 + 5 + 2
6 = 10
P = x1x2 = (5 2
6)(5 + 2
6) = 52 (2
6)2 = 1 : x2 10x+ 1 = 0
3.28 : x2 (
5 +
3)x+
15 = 0
3.28 : x2 Sx+ P = 0 S = x1 + x2 =
5 +
3, P = x1x2 =
15 =
3
5. x1 =
5, x2 =
3
- 95
-
3.29 x1, x2 x2 + 9x 5 = 0(1), 2 ,1 = x1 + 2x2 2 = x2 + 2x1.
3.29 x1, x2 (1) x1 + x2 = 9 x1x2 = 5 1 = x1 + 2x2 2 = x2 + 2x1 x2 Sx+ P = 0:
S = 1 + 2
= x1 + 2x2 + x2 + 2x1
= 3(x1 + x2)
= 3(9)= 27
P = 12
= (x1 + 2x2)(x2 + 2x1)
= 5x1x2 + 2(x21 + x
22)
= 5x1x2 + 2[(x1 + x2)2 2x1x2
= 157
: x2 + 27x+ 157 = 0
3.30 x1, x2 x2 + x 12 = 0, - :
i. x1 + x2ii. x1x2iii. x21 + x
22
iv. x31 + x32
v. (x1 x2)2vi. |x1 x2|
vii.x21x2
+x22x1
3.30 x2 + x 10 = 0, = 1, = 1, = 12 :i. S = x1 + x2 =
= 1
ii. P = x1x2 =
= 12
iii. x21 + x22 = (x1 + x2)
2 2x1x2 = (1)2 2 (12) = 1 + 24 = 25iv. x31 + x
32 = (x1 + x2)(x
21 x1x2 + x22) = 1(25 + 12) = 37
v. (x1 x2)2 = x21 2x1x2 + x22 = 25 2 (12) = 49vi. |x1 x2| =
(x1 x2)2 =
49 = 7
vii.x21x2
+x22x1
=x31 + x
32
x1x2=3712
=37
12
- 96
-
3.31 x2 + 2x 8 = 0i. .ii. ,
.
3.31 x2 + 2x 8 = 0 = 1, = 2, = 8
i. = 2 4 = 42 + 32 > 0 1, 2.
ii. , 1 = 2 =
2
Vietta :1 + 2 = 2 + 2 = 2(1)1 2 = 8 2 = 8 3 = 8 = 2:1 = 22 = 4(1) 2 + 4 = 2 = 1
3.32 R 3x2 + (x+ 1) x2 = 2(1 10x) .
3.32 x2 + x+ = 0. :3x2 + (2 + + 20)x+ 2 = 0 = 3, = 2 + + 20, = 2 : > 0 (1) S = x1 + x2 = 0 (2) (1) , (2)
S = 0 2 + + 20
3= 0 2 + + 20 = 0=
1 = 4
2 = 5 (1).
i. = 4 = (2 + + 20)2 4 3 ( 2) =[(4)2 + (4) + 20]2 4 3 [(4) 2] = 72 > 0 = 4 .
ii. = 5 = (2 ++ 20)2 4 3 ( 2) = (52 + 5 + 20)2 4 3 (5 2) = 36 < 0 = 5 .
- 97
-
3.3.3
1. :i. 2x2 18 = 0ii. x2 2x 80 = 0iii. 2x2 3x+ 6 = 0iv. (x+ 1)2 (x 1)(x+ 2 = 2x(3 x))v. (2x 1)2 3(2x 1) = 0vi. 2x(x 3) = (x+ 1)2 (x 1)(x+ 2)vii. (1
2)x2 2(1 +
2)x = 3
2 + 1
viii. 3x2 (1 +
3)x+ 1 = 0ix. x2 + (
2 1)x
2 = 0
2. R, x2 2x+ 1 = 0 :i .ii .iii .iv .
3. R x2 2x+ 2 = 0 2. , , .
4. x2 + x+ = 0 1, :3 + 3 + 3 = 3
5. , :i. x2 + (2 3)x = 0, 6= 0ii. x2 2(+ 2)x 2 = 0, Riii. x2 2x+ 2 + 2 + 2 = 0, 6= 0iv. x2 x 2 = 0
6. :i. x2 9x+ 142 = 0ii. x2 (2 )x 2 = 0iii. x2 2x+ 2 2 = 0iv. (x+ )(x ) = 2+ 1v. 4x2 4x+ 2 2 = 0vi. (x+ )(x ) = 2(x )2 +
7. x2 + 6x + = 0 , .
8. x2 ( + 1)x + 2 = 0 x2 + x = 0, 6= 2, , .
9. f(x) = x2 + x+ , 6= 0. f(0) = 1, f(1) = 7, f(2) = 21, f(x) = 0 .
10. x2 (5
2)x+ 6 3
2 = 0
i. .ii. .
11. x2 + x+ = 0 6= 0, ,
1
x2 + x+ = 0
12. :i. 1, 0
ii.1
4, 8
iii. 4, 12
- 98
-
iv. 4, 4v. 1 +
3, 1
3
vi. 4, 313. :
i. , 1ii.
+
,+
iii. 2 3, 2+ 3iv.
,
14. x1, x2 2x2x8 = 0, :
i. x1 + x2ii. x1x2iii. x21 + x
22
iv.x1x2
+x2x1
v.1
x21+
1
x22vi. x1x22 + x2x
21
15. x2 + x 1 = 0, Ri. .ii. x1, x2 , , -
i. x1 + x2ii. x1x2iii. x21 + x
22
iv.x1x2
+x2x1
v.1
x21+
1
x22vi. x1x22 + x2x
21
16. x2 + x+ = 0 , 2 4 = 1
17. x1, x2 x2 x+ 3 = 0, :|(x1 + x2)x ( 1)x| < x1x2
18. ( 1)x2 2x+ 1 = 0, , , :
i. ii. 0iii. iv. .
19. f(x) = x2 + x+ 1, 2 f(x) = 0. :
f
(S
2+
)= f
(S
2 )
,R20. R 1, 2 x2+x+1 = 0
21 + 31 + 2 + 212 = 221. x2 + x+ = 0, 6= 0, x1, x2 ,
:
- 99
-
|x1 x2| =
||
- 100
-
1
4.
4.1 1
4.1.1
4.1 , R x+ > 0
x+ > 0 x >
i. > 0x > x >
ii. < 0
x > x <
iii. = 0 0x > x R , > 0 , 0.
4.2 x+ > 0 , R x;
i. , = 0 0ii. , = 0 < 0
4.3 - ;
i. |x| < < x < ii. |x| > x < x >
-
4.4 x+ , R
x+ = 0 x =
x
+x+ 0
- 102
-
4.1.2
4.1
i.5x
3 1
9> (x 3)5
9ii. (x 1)2 + 3 < x2 3(x+ 1)
4.1 i.5x
3 1
9> (x 3)5
9 15x 1 > 5(x 3) 10x > 14
x > 1410
x > 75ii.
(x 1)2 + 3 < x2 3(x+ 1) x2 2x+ 1 + 3 < x2 3x 3 x < 7
4.2 2(4x 1) 3 > 10 5(x 1)
4.2
2(4x 1) 2 > 10 4(x 1) 8x+ 2 2 > 10 4x+ 4 8x+ 4x > 2 + 2 + 10 + 4 4x > 14
x < 72
4.3 x+ 3 < 2(x 3) x
4.3
x+ 3 < 2(x 3) x x+ 3 < 2x 6 x x 2x+ x < 6 3 0x < 8
, .
4.4 5(x 3) x > 4x 16
4.4
5(x 3) x = 4x 15 5x 15 x > 4x 16 5x x 4x > 15 16 0x > 1
, x R, , .
- 103
-
4.5
i.4x
3 5
9>x 3
3ii. (x 1)2 + 3 < x2 + 2(x+ 1)
4.5 i.4x
3 5
9>x 9
3 12x 5 > 3x 45 8x > 40 x < 5
ii.(x 1)2 + 3 < x2 + 2(x+ 1) x2 2x+ 1 + 3 < x2 + 2x+ 2
4x < 4 x > 1
1 < x < 5
4.6 (x 3) > 1 + 2x
4.6
(x 3) > 1 + 2x x 3 > 1 + 2x x 2x > 1 + 3 ( 2)x > 1 + 3 (1)
2 > 0 > 2 (1) x > 1 + 3
2i.ii. 2 < 0 < 2
(1) x < 1 + 3 2
iii. 2 = 0 = 2, (1) : 0x > 7 .
4.7 (x 3) 6 (x 3)
4.7
(x 3) 6 (x 3) x 32 6 x 32
( )x 6 3(2 2) (1)
> 0 >
(1) x 6 2 2
x 6 3(+ )
i.ii. < 0 <
(1) x > 2 2
x > 3(+ )
iii. = 0 = , (1) : 0x 6 0 .
- 104
-
4.8 (x 1) + x > (x+ 2) 3
4.8
(x 1) + x > (x+ 2) 3 x + x > x+ 2 3 x+ x x > + 2 3 (+ 1 ) > + 2 3 (1)
+ 1 > 0 (1) x > + 2 3
+ 1 i.ii. + 1 < 0
(1) x < + 2 3+ 1
iii. + 1 = 0 = + 1, (1) : 0x > 3 1 (2) :
() 3 1 < 0 < 13
(2) : 0x > 3 1 > 0 x.() 3 1 > 0 > 1
3 (2) : 0x > 3 1 < 0 .
() 3 1 = 0 = 13
(2) : 0x = 3 1 > 0 :
4.9 :i. |x 3| < 2ii. |2x 3| > 5iii. |x 7| < 4iv. |2x 6| 6 0v. |x+ 7| 0
4.9 i.|x 3| < 2 2 < x 3 < 2
2 + 3 < x < 3 + 2 1 < x < 5 x (1, 5)
ii.|2x 3| > 5 2x 3 < 5 2x 3 > 5
2x < 5 + 3 2x > 5 + 3 2x < 2 2x > 8 x < 1 x > 4 x (,1) (4,+)
iii. |x 7| < 4
- 105
-
iv.|2x 6| 6 0 2x 6 = 0
2x = 6 x = 3
v. |x+ 7| 0 x R
4.10 : |x 1| < |x 2|
4.1 |A(x)| < |B(x)| .
4.10
|x 1| < |x 2| |x 1|2 < |x 2|2
(x 1)2 < (x 2)2
x2 2x+ 1 < x2 4x+ 4 4x 2x < 4 1
x < 32
4.11 : 2|x 1| < |x 2|+ 3
4.2 |A(x)|+ |B(x)|+ |(x)|+ < 0 .
4.11 .
x 1 2 +x 1 0 +
+x 2
0 + :
i. x (, 1),
2(x+ 1) < (x+ 2) + 3 2x+ 2 < x+ 2 + 3 x < 3 x > 3
x (, 1) x (3, 1)
ii. x (1, 2),
2(x 1) < (x+ 2) + 3 2x 2 < x+ 2 + 3 3x < 7
x < 73
x (1, 2) x (1, 2)
- 106
-
iii. x (2,+) ,2(x 1) < (x 2) + 3 2x 2 < x 2 + 3
x < 3 x (2,+) x (2, 3)
- 107
-
4.1.3
1. :
i.3x 5
2+ x >
7x+ 4
5
ii.2x 1
3
5x+ 2
12>x 3
4+ 1
iii.2x 4
5 3x
4 7x+ 16
20
iv.x 2
2+
1 2x5
x 2
3 x 3
4
x+ 5
6+ 2 >
5(x 1)12
4. ( ):
3x
2 1 > x+ 1
4 2
x(2x+ 1)
4 1 > x
2 + 1
2
5. :
x+x 1
2>x 3
4 x 2
3
x+1
2 x+ 3
2
6. :
2x x 18
> x
x 4 + x+ 12 < 0
7. (2 + )2x > 3 + 4x R8. :
i. (x 1) < x+ 3
ii.x
2 1
(2
3+
3
4
)x
iii. 22 + 2x > 2x+ 2
- 108
-
iv. (x )2 ( x)2 < + 9. ( 2)x = 2 4, R.
R, , :i. ii. iii. 10iv. [4, 12)
10. (x 2) = x 2 , ( 2)x < 2 4
11. :i. |x+ 1| < 4ii. |2x 3| < 5iii. | x+ 5| < 2iv. |2x+ 12| < 1v. |2x 5| > 1vi. |3x+ 4| > 6vii. | x+ 21| > 9viii. |x2 25| 0
12. :i. 2 < |x+ 1| < 4ii. 3 < |2x 4| < 5iii. |x2 1| < 5iv. || x+ 5|+ 1| < 2v. |2x+ 1| < |x|vi. |2x 5| > |x+ 1|vii. |x 1| > x+ 2
13. :
i.|x 2|+ 4
3 3|x 2| 1
15 |x 1|+ |3 3x|
4
- 109
-
4.2
4.2.1
4.5 x2 + x+ 6= 0
x2 + x+ = (x2 +
x+
)
=
[x2 + 2x
2+
(2
42
)(2
42
)+
]=
[(x+
2
)2
2 442
]
= 2 4 :
x2 + x+ =
[(x+
2
)2
42
](1)
i. > 0
x2 + x+ =
[(x+
2
)2
42
]
=
(x+ 2
)2
(
2
)2=
(x+
2+
2
)(x+
2
2
)
=
(x
2
)(x +
2
)= (x 1)(x 2)
ii. = 0
x2 + x+ =
[(x+
2
)2]
=
[(x
2
)2]= (x )2
iii. < 0 (1) .
- 110
-
4.2.2 4.12 :
i. x2 3x+ 2ii. 2x2 8x+ 8iii. x2 x+ 2
4.12 i. x2 3x + 2 = 1 > 0. 1 = 2 2 = 1 x2 3x+ 2 = (x 2)(x 1)
ii. 2x2 8x+ 8 = 0. = 2 2x2 8x+ 8 = 2(x 2)2
iii. x2 x + 2 = 7 < 0. .
4.13 :x2 3x+ 22x2 8x+ 8
4.13 , - :x2 3x+ 22x2 8x+ 8
=(x 2)(x 1)
2(x 2)2=
x 12(x 2)
4.14 :
2x2 + (2 )x
4.14 2x2 + (2 )x x : = 2, = (2 ), =
= (2 )2 + 4 2 = 42 + 2 4 + 8= 42 + 2 + 4
= (2 + )2 > 0
:
- 111
-
1,2 =(2 )
(2 + )2
4
=(2 ) (2 + )
4
=
2
2x2 + (2 )x = 2(x 2
)(x+ )
4.15 :
4 + 4 + 22
4.15 , -, .
4 + 4 + 22 = 4 + 4 + 22 + 22 22
= 4 + 4 + 222 22
= (2 + 2)2 ()2
= (2 + 2 + )(2 + 2 )
- 112
-
4.2.3
1. :
i.2x2 + 8x 42
x2 49ii.
4x2 12x+ 92x2 5x+ 3
2. :
i.2 + 22
2 62
ii.x2 x+ x x2 3x+ 22
- 113
-
4.3
4.3.1
4.6 x2 + x+ , 6= 0 .
i.x 1 2 +
x2 + x+ 0 0 > 0
1, 2 x2 + x+ = 0
ii.x +
x2 + x+ 0 = 0
x2 + x+ = 0
iii.x +
x2 + x+ < 0
- 114
-
4.3.2
4.3 2 , !!!
4.16 :i. x2 5x+ 6ii. x2 + 2x+ 1iii. 3x2 + 2x+ 4
4.16 i. x2 5x+ 6 = 0 2 3, :
x 2 3 +x2 5x+ 6 + 0 0 +
:x2 5x+ 6 > 0 x (, 2) (3.+)x2 5x+ 6 < 0 x (2, 3)x2 5x+ 6 = 0 x = 2, x = 3
ii. x2 2x + 1 = 0 1, :
x 1 +x2 + 2x+ 1 + 0 +
:x2 + 2x+ 1 > 0 x R {1}x2 + 2x+ 1 = 0 x = 1
iii. 3x2+2x+4 = 0 , :
x +3x2 + 2x+ 4 +
:3x2 + 2x+ 4 > 0 x R
4.17 :i. x2 + 3x 4ii. x2 + 9 6xiii. 1
4(x2 4x+ 3) > 0
4.17 i. x2 + 3x 4 x2 + 3x 4 0 : x2 + 3x 4 = 0 1 -4 :
- 115
-
x -4 1 +x2 + 3x 4 + 0 0 +
x2 + 3x 4 x (4, 1)ii.
x2 + 9 6x x2 + 9 6x 0 (x 3)2 0 (x 3)2 = 0 x = 3
iii. 14
(x2 4x+ 3) > 0
: 14
(x2 4x+ 3) = 0 1 3 :
x 1 3 +
14
(x2 4x+ 3) 0 + 0
14
(x2 4x+ 3) > 0 x (1, 3)
4.18 : 2x 1 < x2 4 < 12
4.18 2x 1 < x2 4 < 12
2x 1 < x2 4
x2 4 < 12 . 2x 1 < x2 4 x2 2x 3 > 0 2x 1 < x2 4 = 0 -1 3, :
x -1 3 +x2 2x 3 + 0 0 +
x2 2x 3 > 0 x (,1) (3,+)
x2 4 < 12 x2 16 < 0 x2 16 = 0 -4 4, :
x -4 4 +x2 16 + 0 0 +
x2 16 < 0 x (4, 4) :x (4, 1) (3, 4)
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-
4.19 f(x) =x2 6x 16
4.19 x2 6x 16 0, 2 . x2 6x 16 = 0 x1 = 2 x2 = 6. :
x -2 6 +x2 6x 16 + 0 0 +
Df = (,2) (6,+)
4.20 : x2 + 3x+ + 5 = 0, R. :
i. ii. iii.
4.20 2 , .
= (3)2 4(+ 5)= 92 42 20= 52 20= 5(2 4)
= 0 : 1 = 0, 2 = 4. :
x 0 4 +2 4 + 0 0 +
:
i. , = 0 1 = 0, 2 = 4ii. , > 0 (, 0) (4,+)iii. , < 0 (0, 4)
4.21 R :x2 + ( 1)x+ ( 1) < 0 x R
4.21 x2 + ( 1)x+ ( 1) < 0 : < 0
< 0
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-
< 0 ... 32 + 2+ 1 < 0 1 =
1
3 2 = 1, :
x 13
1 +x2 + 3x 4 0 + 0
< 0
< 13
> 1
< 13
4.22 : 2 + 2 > , R
4.22 2 + 2 > 2 + 2 > 0 1 2 + 2 = ()2 4 1 2 = 32 0 0 2 > 0 : 2 + 2 0
4.23 2x2 + x+ , :
i. > 0ii. 2 + + > 0
4.23 , < 0 x2 2 > 0 : 2x2 + x+ > 0 x.:
i. x = 0 > 0ii. x = 1 2 + + > 0
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-
4.3.3
1. :i. 2x2 3x 2 6 0ii. x2 4x > 4iii. x2 4x+ 4 < 0iv. x2 + 4x > 10v. x2 4x+ 3 > 0vi. 2x2 + 10x 12 > 0vii. 3x2 < 27xviii. 4x2 + 16 < 0
2. x R :i. x2 4x 5 < 0 x2 x 6 > 0ii. x2 + 3x+ 5 < 0 x2 5x+ 6 > 0iii. x < 2x2 < 8x
3. R x2 + 3x+ > 0 x R.
4. (+ 2)x2 2x+ 3 < 0. 6= 25. i. , 2 + 2 > 0 , R
ii. A =
+
1
, 6= 0.6. R , , :
( 3)x2 (+ 2)x+ 2+ 1 = 0, 6= 37. x R : 8x2 + (1 x) > 7 x, : 3 0 A(x)B(x) > 0
.ii. :
A(x)
B(x)> (x) A(x)
B(x) (x) > 0
A(x) B(x)(x)B(x)
> 0
[A(x) B(x)(x)]B(x) > 0 , B(x) 6= 0.
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-
4.4.2 4.4 , .
4.24 :
(x2 5x+ 6)(x2 + 2x+ 1)
4.24 : x2 5x+ 6 = 0 2 3, :
x 2 3 +x2 5x+ 6 + 0 0 +
x2 2x+ 1 = 0 1, :
x 1 +x2 + 2x+ 1 + 0 +
:x 1 2 3 +
x2 5x+ 6 + | + 0 0 +x2 + 2x+ 1 + 0 + | + | +
(x2 5x+ 6)(x2 + 2x+ 1) + 0 + 0 0 + :
(x2 5x+ 6)(x2 + 2x+ 1) > 0 x (,1) (1, 2) (3,+)(x2 5x+ 6)(x2 + 2x+ 1) < 0 x (2, 3)(x2 5x+ 6)(x2 + 2x+ 1) = 0 x = 1, 2, 3
4.25 :
A =x2 4x+ 3x2 + 6x+ 9
4.5 , - .
> 0
, , > 0
4.25 x2 + 6x+ 9 6= 0 x 6= 3
A =x2 4x+ 3x2 + 6x+ 9
:
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-
(x2 4x+ 3)(x2 + 6x+ 9) x2 4x+ 3 = 0 1 3, :
x 1 3 +x2 4x+ 3 + 0 0 +
x2 + 6x+ 9 = 0 -3, :
x 3 +x2 + 6x+ 9 + 0 +
:x 3 1 3 +
x2 4x+ 3 + | + 0 0 +x2 + 6x+ 9 + 0 + | + | +
A =x2 4x+ 3x2 + 6x+ 9
+ 0 + 0 0 +
:(x2 4x+ 3)(x2 + 6x+ 9) > 0 x (,3) (3, 1) (3,+)(x2 4x+ 3)(x2 + 6x+ 9) < 0 x (1, 3)x2 4x+ 3x2 + 6x+ 9
= 0 x = 1, 3
4.26 x
x 1 2x+ 1
0
xi.x
3x 5 2x 1
xii.5
x+ 2+
1
x+ 3>
7
6
xiii. 3 0
x2 4 0
x >x2 4
|x| 2
x > 0
x2 > x2 4
x 2 x 2
x 2
A = [2,+]
6.3 f(x) =x
x2 4x+ , R
6.3 A = {x R|x2 4x+ 6= 0} x2 4x+ = 0 = 16 4 :
2 + + 0
:i. < 4 > 0, x2 4x + = 0 ,
x1,2 =4
16 42
= 2
4 f A = R {2
4 , 2 +
4 }
ii. = 4 = 0, x2 4x+ = 0 , x = 2 f A = R {2}
iii. > 4 < 0, x2 4x+ = 0 f A = R
6.4 f(x) =x2 3x+ 1
x 3
6.2 f , y R, f(x) = y ( x) .
6.4 A = R {3}
f(x) = y x2 3x+ 1x 3
= y
y(x 3) = x2 3x+ 1 x2 + x(y 3) + 3y + 1 = 0
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-
0
x 6= 3
(y + 3)2 4(3y + 1) 0
32 + 3(y 3) + 3y + 1 6= 0
y2 6y + 5 0
1 6= 0 ( )
y 1 y 5
f(A) = (, 1] [5,+)
6.5 f(x) =
{x2 5x+ 6 x 1x
x 1x < 1
, f(0), f(1), f(2)
6.5 i. f(0) f(x) =
x
x 1, 0 < 1
f(0) =0
0 1= 0
ii. f(1) f(x) = x2 5x+ 6, 1 1 f(1) = 12 5 1 + 6 = 2
iii. f(2) f(x) = x2 5x+ 6, 2 1 f(2) = 22 5 2 + 6 = 0
6.6 x, f(x) = |x2 8| 7.
6.6 f(x) = 7 |x2 8| = 7
x2 8 = 7
x2 8 = 7
x2 = 15
x2 = 1
x =
15
x = 1
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-
6.1.3
1.
i. f(x) =2
x 3+ x ii. g(x) =
x2 + 1
x3 + x2
iii. h(x) =x2 16 iv. t(x) =
x2 4x+ 3 + 2
x3 4x2
v. k(x) =x+ 3 +
x2 25 vi. b(x) =
x
x 1x2 3x
+2
x2 4x2.
i. f(x) =
2 |x 1|
x
x 2 1 ii. g(x) =
x+ 10 6
x+ 1
3.
i. f(x) =
16 x2x4 8x
ii. g(x) =
2
2 + x x2
4. R
f(x) =x2 (5+ 2)x 8
( 1)x2 2( 1)x+ 3 R.
5. f(x) ={x3 x < 12x+ 3 x 1
, f(0), f(1), f(2)
6. x, f(x) =x2 16x2 4x
2.7.
i. f(x) = xx2 4 ii. g(x) = x
2 + x+ 1
x2 + x+ 28. f(x) = 2x A = [1, 2].
.
9. f(x) ={x 1 x < 0 2x2 x 0
, R, :f(0) = 3 f(2) = 1. :
A = 2f(0) 3f(1) 3f(12
) + 5f(1
2)
10. f(x) ={x+ 2 x < 1x 2 x 1
x : f(x) = 12
11. f(x) ={
1, x 0, x
i. f(0), f(
3), f(3500), f()ii. |x+ 1 f(
3)| < f(1821)
12. f(x) = 2x+ 3, :i. f(+ ) = f() + f() 3ii. f(
+ ) f(
) + f(
) 3
13. f(x) = x2 2x+ 3, f(x+ 1) 2f(2x) + 3f(1) = 14
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-
14. Z , f(x) = 3(1)x + 2(1)x+1
i. f(x) ={
1, x 1, x
ii. f(23), f(55), f(0), f(1000), f(28)
- 149
-
6.2
6.2.1 6.8 f Oxy; f ;
f , , M(x, f(x)), x A. y = f(x) (x, y) f
A(xo, yo) f yo = f(xo)
6.1:
6.9 M(x, y) , O(0, 0) y = x
xx M (x,y) yy M (x, y) M (x,y) y = x M (y, x)
6.10 2 A(x1, y1) B(x2, y2)
, (AK) = |x2 x1| (BK) = |y2 y1| :
(AB)2 = (AK)2 + (BK)2
= |x2 x1|2 + |y2 y1|2
= (x2 x1)2 + (y2 y1)2
- 150
-
(AB) =
(x2 x1)2 + (y2 y1)2
6.11 ;
f [, ), xx
6.2:
f [, ), yy
6.3:
- 151
-
xx
xx f(x) = 0 f(x) xx (, ) (, ), f(x) > 0 f(x) xx (, ), f(x) < 0
6.4: xx
(, ) (0, ) x1, x2 Df x1 < x2 f(x1) < f(x2) (, 0) (, ) x1, x2 Df x1 < x2 f(x1) > f(x2)
6.5:
- 152
-
- f() = 0 f(0) = xo x xo f(x) f(x0) f() = <
!!! !
6.6:
- 153
-
- f() = f() = xo x xo f(x) f(x0) f() = f()) =
6.7:
- 154
-
6.12 - ;
x1, x2, x3 f(x) = g(x)
6.8: 2
f g f g (x1, x2) (x3, ) f(x) > g(x)
6.9: 2
- 155
-
f g f g (, x1) (x2, x3) f(x) < g(x)
6.10: 2
- 156
-
6.11:
- 157
-
6.2.2
6.7 A(2, 5), :i. xx ii. yy
iii. 0(0, 0) iv. 1 3 y = x
6.3 M(x, y) xx M (x,y) x y yy M (x, y) x y M (x,y) x y y = x M (y, x) x y
6.7 i. A(2, 5), xx (2,5)ii. A(2, 5), yy (2, 5)iii. A(2, 5), 0(0, 0) (2,5)iv. A(2, 5), y = x (5,2)
6.8 R xx A(1,5) B(1,
+ 1)
6.8 xx
.
1 = 1
5 =+ 1
25 = + 1 = 24
6.9 A(2, 2), B(1, 1), (3,6) - .
6.4 , , , 2 .
6.9 AB =
(2 + 1)2 + (2 1)2 =
10A =
(3 2)2 + (6 2)2 =
65
B =
(3 + 1)2 + (6 1)2 =
65 A = B
6.10 A(1, 1), B(3, 1), (1, 3) - .
6.5 , , , 2 , ,
- 158
-
6.10 AB =
(3 1)2 + (1 1)2 = 2A =
(1 1)2 + (3 1)2 = 2
B =
(3 1)2 + (3 + 1)2 =
8 A = AB B2 = 8, AB2 +A2 = 4 + 4 = 8 B2 = AB2 +A2 .
6.11 (-1, 1) - f(x) =
x
x2 + 1
6.6 A(xo, yo) y = f(x) yo = f(xo)
6.11 (-1, 1) f(x) =x
x2 + 1 f(1) = 1.f(1) = 1
(1)2 + 1= 1
26= 1 (-1, 1) f(x) = x
x2 + 1
6.12 f(x) = 5x 1. R, A(2,).
6.12 f A(2,) :f(2) = 5 2 1 = = 9
6.13 f(x) = x2 4 xx.
6.7 f yy, f(0). f xx, f(x) = 0. f xx, f(x) > 0, xx, f(x) < 0
6.13 f(0) = 02 4 = 4. yy A(0,4)f(x) = 0 x2 4 = 0 x = 2. xx B(2, 0) (2, 0)f(x) > 0 x2 4 > 0 x2 > 4 |x| > 4x (,2) (2,+)
6.14 f(x) = x2 5x+ 4 g(x) = 2x 6. :i. Cf , Cgii. Cf Cg
- 159
-
6.8 f g, f(x) = g(x). f g, f(x) > g(x).
6.14 i.f(x) = g(x) x2 5x+ 6 = 2x 6
x2 7x+ 12 = 0, = 1
x1,2 =5 1
2 x1 = 3, x2 = 2
Cf , Cg A(3, g(3)) = (3, 0) B(2, g(2)) = (2,2)ii. f(x) > g(x) x2 7x + 12 > 0
:x 2 3 +
x2 7x+ 12 + 0 0 + Cf Cg x (, 2) (3,+)
- 160
-
6.2.3
1. , xx, yy, O(0, 0) 1 3 y = x, :
i. A(1, 2) ii. B(2, 3)iii. (3,1) iv. (0, 5)
2. (1, 2), (4, -2) (-3, 5) .
3. (1, -1), (-1, 1) (4, 2) .
4. , , xx:
i. f(x) = x+ 3 ii. g(x) = 2x+ 8
iii. h(x) = x2 3x iv. k(x) = x2 xx 2
5. f(x) = 2x3 + ( 1)2, R. R, O(0, 0).
- 161
-
6.3 f(x) = x+
6.3.1 6.13 ;
f(x) = x+ , = = yy A(0, )
6.12:
6.14 ;
, xx
6.13:
- 162
-
6.15 , ;
, xx
. (0o 90o) ,
6.14:
xx . (90o 180o).
6.15:
- 163
-
, xx, = 0o.
6.16: 0
!!! , xx = 90o .
6.17:
- 164
-
6.16 f(x) = x ;
= .
6.18: y = x
6.17 A(x1, y1) B(x2, y2);
=y2 y1x2 x1
, x1 6= x2.
6.19: 2
- 165
-
6.18 ;
2 , : -
12 1 = 2
6.20:
6.19 ;
2 , :
12 1 2 = 1
6.21:
- 166
-
6.3.2 6.15 y = x 5, :
i. .ii. xx.iii. .iv. , .
6.15 1. y = x + , - x. = 1
2. , = = = = 1 = = 45
o
3. xx y = 0, 0 = x 5 = x = 5. A(5, 0) yy y = 0, x = 0 5 = y = 5. B(0,5)
4. :
E =1
2(OA)(OB) =
1
2|5|| 5| = 25
2..
6.16 A(2, 1) B(3,1).
6.16 y = x+
y = x+ A(2,1)
==== 1 = 2+ y = x+
B(3,1)===== 1 = 3+
: = 2 :1 = 2+
=2==== 1 = 6 + = = 5
y = 2x+ 5.
6.17 A(2, 4) B(2, 0) xx.
6.17 =y2 y1x2 x1
, x1 6= x2A(2,4)
=====B(2,0)
4 02 + 2
= 1
45o
= 1 = = 45o
6.18 A(2, 3) y = 4x 3.
6.18 y = x+ . , . y = 4x 3 4, = 4. y = 4x+
A(2,3)==== 3 = 4 2 + = = 5.
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-
y = 4x 5.
6.19 A(2, 1) y = 2x+ 5.
6.19 y = x+ . , - ( 1). y = 2x+ 5 2,
2 = 1 = 12
y = 12x+
A(2,1)===== 1 = 1
2 (2) + = = 0
y = 12x.
- 168
-
6.20 R y = (2 + 3||)x+ 2 (1) y = 4x+ 5 (2).
6.20 , ., 2 + 3|| = 4 ||2 + 3|| 4 = 0 || = :
2 + 3 4 = 0
1 = 1
