-
LI GII MT S BI TP
TON CAO CP 2
Li gii mt s bi tp trong ti liu ny dng tham kho. C mt s bi tp do
mt s
sinh vin gii. Khi hc, sinh vin cn la chn nhng phng php ph hp v n
gin
hn. Chc anh ch em sinh vin hc tp tt
BI TP GII V BIN LUN THEO THAM S
Bi 1:Gii v bin lun:
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
3 2 5 4 32 3 6 8 5
6 9 20 114 4 2
x x x xx x x x
x x x xx x x x
Gii:
-
1 3
11( 2) 2 21( 3) 3 3
11( 4) 4 34
3 2 5 4 3 1 6 9 20 112 3 6 8 5 2 3 6 8 51 6 9 20 11 3 2 5 4 34 1
4 2 4 1 4 2
1 6 9 20 11 1 60 15 24 48 270 20 32 64 360 25 40 80 46
h h
h h hh hh h h
A B
2( 1) 3 3 42( 5) 4
1 2 3 4
2 3 4
9 20 110 5 8 16 90 5 8 16 90 25 40 80 46
1 6 9 20 11 1 6 9 20 110 5 8 16 9 0 5 8 16 90 0 0 0 0 0 0 0 10 0
0 1 0 0 0 0 0
6 9 20 11(1) 5 8 16 9
h h h hh h
x x x xx x x
4
1
2
3
4
1 2 3 4
2 3 4
(2) 1
1 3 451 9 8 16
1) 0 : (2) 5
1
1 6 9 20 112) 0 : (3) 15 24 48 27 :
0 1
x
tx
txKhi t R
x t
x
x x x xKhi x x x
he vo nghiem
-
Bi 2:
Cho h phng trnh:
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
2 3 4 54 2 5 6 76 3 7 8 9
4 9 10 11
x x x xx x x xx x x x
mx x x x
a) Tm m vi h phng trnh c nghim
b) Gii h phng trnh khi m = 10
Gii:
a) Ta c:
1 4 1
1( 2) 21( 3) 3 2( 2) 31( 4) 4 2( 3) 4
2 1 3 4 5 1 4 3 2 54 2 5 6 7 2 6 5 4 76 3 7 8 9 3 8 7 6 9
4 9 10 11 4 10 9 11
1 4 3 4 5 1 40 2 1 0 30 4 2 0 60 6 3 8 9
c c c
h hh h h hh h h h
A B
m m
m
3 4
3 4 50 2 1 0 30 0 0 0 00 0 0 8 0
1 4 3 4 50 2 1 0 30 0 0 8 00 0 0 0 0
h h
m
m
Ta thy: : 4m R r A B r A . Suy ra h c nghim vi mi gi tr cu m
b) Gii h khi m = 10:
Bin i s cp trn hng ta c:
-
11 2 3 4
22 3 4
33 4
4
2 1 3 4 5 2 1 3 4 54 2 5 6 7 0 1 6 10 14
/ ...6 3 7 8 9 0 0 2 4 6
10 4 9 10 11 0 0 0 0 0
02 3 4 5
4 2(1) 6 10 14
3 22 4 6
A B
xx x x x
x tx x x t R
x tx x
x t
Bi 3
Gii v bin lun h phng trnh sau theo tham s :
1 2 3
1 2 3
21 2 3
1 1
1
1
x x x
x x x
x x x
Gii:
Ta c
3 2 1
1( 1) 2 21( 1) 3
1 1 1 3 3 3 1 1 11 1 1 1 1 1 3 1 1 11 1 1 1 1 1 1 1 1
1 1 13 0 0 3
0 0
h h h
h h
h h
D
21
1( ) 22 21( ) 3
2 2 2
2 2 2 2 3 3 2
1 1 1 1 1 11 1
1 1 0 1 1 11 1
1 1 0 1 1
1 1 1 1 1 2 2
h hx h h
D
-
2
1 3
2 2
1( 1) 22 21( ( 1)) 3
2 2
2 2 3 2 2 3 2
2
1 1 1 1 1 11 1 1 11 1 1 1
1 1 11
0 1 11 2
0 1 2
1 2 1 2 2
2 2 1
c cx
h h
h h
D
3
1 2
2 2
21( ( 1)) 2 2
21( 1) 32
2
2 3 2
1 1 1 1 1 11 1 1 11 1 1 1
1 1 12 1
0 2 1 11
0 1
2 11 1
2 1 1 2 1
c cx
h h
h h
D
Ta thy:
(1) 23
3 00
D
Khi h c nghim duy nht:
2 21
1 2
22 2
3 23
3
2 23 3
2 1 2 13 3
2 13
DxxD
DxxD
DxxD
(2) Nu 3 th 1
3(2 9) 21 0xD : H v nghim
(3) Nu 0 th h tr thnh:
-
1 2 3
1 2 3
1 2 3
100
x x xx x xx x x
H v nghim
Bi 4
Gii v bin lun h phng trnh sau theo tham s :
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
5 3 2 4 34 2 3 7 18 6 5 97 3 7 17
x x x xx x x xx x x xx x x x
Gii
2( 1) 12( 2) 32( 1) 4
1( 4) 2 2 31( 3) 4 2( 1) 4
5 3 2 4 3 1 1 1 3 24 2 3 7 1 4 2 3 7 18 6 1 5 9 0 2 7 19 77 3 7
17 3 1 4 10 1
1 1 1 3 20 2 7 19 70 2 7 19 70 2 7 19 7
h hh hh h
h h h hh h h h
A B
4 3
1 1 1 3 20 2 7 19 70 0 0 0 00 0 0 0
1 1 1 3 20 2 7 19 70 0 0 00 0 0 0 0
h h
H phng trnh tng ng vi h:
-
1 2 3 4
2 3 4
3 22 7 19 7
0
x x x xx x x
Ta thy:
(1) Khi 0 th h v nghim
(2) Khi 0 th h tr thnh:
1 2 3 4
2 3 4
2 3 4
1 3 4 3 4 1 3 4
3 2 (1) 2 7 19 7 (2)
7 19(2) : 72 27 19 5 13(1) 7 3 2 52 2 2 2
x x x xx x x
x x x
x x x x x x x x
Vy nghim ca h khi l:
1 3 4
2 3 4
3 4
5 13 52 27 19 72 2
,
x x x
x x x
x x
tuy y
Bi 5
Gii v bin lun h phng trnh sau theo tham s
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
3 2 5 4 32 3 6 8 5
6 9 20 114 4 2
x x x xx x x x
x x x xx x x x
-
Gii
Ta c:
3 1
11( 2) 2 3 21( 3) 3 41( 4) 4
3 2 5 4 3 1 6 9 20 112 3 6 8 5 2 3 6 8 51 6 9 20 11 3 2 5 4 34 1
4 2 4 1 4 2
1 6 9 20 11 1 60 15 24 48 270 20 32 64 360 25 40 80 46
h h
h h h hh hh h
A B
9 20 110 5 8 16 90 15 24 48 270 25 40 80 46
2( 3) 3 3 42( 5) 4
1 6 9 20 11 1 6 9 20 110 5 8 16 9 0 5 8 16 90 0 0 0 0 0 0 0 10 0
0 1 0 0 0 0 0
h h h hh h
Khi :
(1) Nu 0 th 3 4r A B r A : h c v s nghim (tm nghim nh bi
trn)
(2) Nu 0 th :
3
2
r A Br A B r A
r A
: h v nghim
Bai 6
Gii v bin lun h phng trnh sau theo tham s
21 2 3
3 21 2 3
4 31 2 3
1 3
1 3
1 3
x x x
x x x
x x x
-
Gii
Ta c:
3 2 1
1( 1) 2 21( 1) 3
1 1 1 3 3 3 1 1 11 1 1 1 1 1 3 1 1 11 1 1 1 1 1 1 1 1
1 1 13 0 0 3
0 0
h h h
h h
h h
D
1
2
2
3 2 2
4 3 3 2
1( ) 22 21( ) 3
2 2
2 2 2 2 3
3 1 1 3 1 1 1 1 13 1 1 3 1 1 3 1 13 1 1 3 1 1 1 1
1 1 11 1
3 0 1 1 3 11 1
0 1 1
3 1 1 1 3 1 1
x
h h
h h
D
3 2 23 2 3 2
2
2
3 2 2
4 3 3 2
1 3 1( 1) 2
1( ( 1)) 32 2 2
2 2
1 3 1 1 3 1 1 1 11 3 1 1 3 1 3 1 11 3 1 1 3 1 1 1
1 1 1 1 1 13 1 1 3 0 1
1 1 0 1 2
13 1
1 2
x
c c h h
h h
D
2 2
3 2 2 3 2
2 2
3 1 2 1
3 2 2
3 2 3 2 1
-
3
2
3 2 2
4 3 3 2
1 2 1( ( 1)) 2 21( 1) 3
2 2
22
2
1 1 3 1 1 3 1 1 11 1 3 1 1 3 3 1 11 1 3 1 1 3 1 1
1 1 1 1 1 13 1 1 3 0 2 1
1 1 0 1
2 13 1 3
1
x
c c h h
h h
D
2
2 2 2 3 2
2 11 1
3 2 1 1 3 2 1
Ta thy:
(1) Khi: 0
03
D
. Suy ra h c nghim duy nht:
2 221
1 2
22
2 2
2 3 23 23
3 2
3 22
3
3 2 12 1
3
3 2 12 1
3
DxxD
DxxD
DxxD
(2) Khi 0
03
D
v 1 2 3
0x x xD D D suy ra h c v s nghim
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