Upload
toan-vo
View
229
Download
1
Embed Size (px)
Citation preview
8/9/2019 Tong Hop de Thi Cao Hoc Toan
1/25
Nguyn Vn T - Website: violet.vn/nguyentuc2thanhmy
Luyn gii thi cao hc mni s 1
GII THI TUYN SINH CAO HC THNG 8/2008MN C BN: I S V GII TCH
Bi 1:Cho nh x tuyn tnh f : R4 R3 xc nh bif(x1,x2,x3,x4)=(x1+x2,x2+x3,x3+x4) vi mi x=(x1,x2,x3,x4) R
4
M={ (x1,x2,x3,x4) R4: x1-x2=0 v x3-x4=0}
a. Tm ma trn f trong c s chnh tc ca R4v R3 . xc nh Imf v Kerf
b. CM f(M) l khng gian vect con ca R3
. tm dimf(M)Gii :
Tm ma trn f trong c s chnh tc ca R4v R3Trong R4 ta c e1=(1,0,0,0),e2=(0,1,0,0),e3=(0,0,1,0),e4=(0,0,0,1)
Trong R3 ta c e1=(1,0,0),e2=(0,1,0),e3=(0,0,1)Ma trn f trong c s chnh tc l
1100
0110
0011
4321
4321
4321
)(),/( 34
cccc
bbbb
aaaa
Aeef
m f(e1)=(1,0,0)=a1e1+b1e2+c1e3ta tm c (a1,b1,c1)=(1,0,0)f(e2)=(1,1,0) (a2,b2,c2)=(1,1,0)
f(e3)=(0,1,1) (a3,b3,c3)=(0,1,1)
f(e4)=(0,0,1) (a4,b4,c4)=(0,0,1)
Xc nh Imf,Kerf
Kerf ={ xR4: f(x)=0 }
Ta c h
Rx
xx
xx
xx
xx
xx
xx
4
43
42
41
43
32
21
0
0
0
h c nghim tng qut l (-a,a,-a,a)
H nghim c bn l (-1,1,-1,1)Vy dimKerf=1, c s ca Kerf =(-1,1,-1,1)
Tm Imf
Ta c f(e1)=(1,0,0),f(e2)=(1,1,0), f(e3)=(0,1,1),f(e4)=(0,0,1)Nn Imf=
Ta c
000
100010
001
...
100
110011
001
vy c s ca Imf l f(e1),f(e2),f(e3) v dimf=3b.
Bi 2:Gii v bin lun h phng trnh
8/9/2019 Tong Hop de Thi Cao Hoc Toan
2/25
Nguyn Vn T - Website: violet.vn/nguyentuc2thanhmy
Luyn gii thi cao hc mni s 2
1
1
1
4321
4321
4321
xmxxx
xxmxx
xxxmx
Gii: lp ma trn cc h s
mmmm
mm
m
m
m
m
m
m
m
A
1.1200
0.0110
1.111
....
1.111
1.111
1.111
1.111
1.111
1.111
2
vy ta c
1
0)1()1(
1)1()2)(1(
4321
32
43
xmxxx
xmxm
mxmxmm
Bin lun:Vi m=1 h c v s nghim ph thuc 3 tham s x2,x3,x4nghim ca h l (1-a-b-c,a,b,c) a,b,c Rvi m=-2 h c v s nghim ph thuc tham s x3
nghim ca h l (a,a,a,1) a Rvi m khc 1,-2 h c v s nghim ph thuc tham s x4v m
nghim ca h l
ax
m
ax
m
ax
m
ax
2
1
2
1
2
1
a R
Bi 3:Cho chui lu tha
1
1
2.
)2()1(
nn
nn
n
x
a.
Tm min hi t ca chuib. Tnh tng ca chui
Gii:
a.
ta cn
nn
nn
xxU
2.
)2()1()(
1
tnh Cxx
nxU
nn
n
nn
2
2
2
2.
1)(
limlim
theo tiu chun csi nu chui hi t khi C0 v
0yx,0
0,1sin),(
22
22
2 yxyx
xyxf
a
8/9/2019 Tong Hop de Thi Cao Hoc Toan
3/25
Nguyn Vn T - Website: violet.vn/nguyentuc2thanhmy
Luyn gii thi cao hc mni s 3
Tu theo gi tr ca a>0 xt s kh vi ca f ti (0,0), s lin tc ca fx,fyti (0,0)Gii : Tnh cc hr
ti x2+y2>0
aax
yxyx
x
yxxf
2222
3
22
' 1cos2
)(
1sin2
ay yxyx
yxf )(
1cos
22222
2'
ti x=y=0
t
ftff
tx
)0,0()0,(lim
0
'
t
ftff
ty
)0,0(),0(lim
0
'
xt s kh vi ca f ti (0,0) Cn xt : ),(lim0,
tsts
Vi tfsfftsf
tsts
yx)0,0()0,0()0,0(),(
1),( ''
22
Nu ),(lim0,
tsts
=0 th hm s kh vi ti (0,0) ngc li th khng kh vi
xt s lintc ca fx,fyti 0(0,0)nu : )0,0(),( ''
0,lim xx
yx
fyxf
, )0,0(),( ''
0,lim yy
yx
fyxf
th hm s khng lin tc ti
(0,0) ngc li th lin tcBi 5: Cho (X,d ) l khng gian Metric A X khc rngCho f: X R nh bi f(x)=d(x;A)=inf{d(x,y): yA}
a.
CM: f lin tc iu trn Xb. Gi s A l tp ng , B l tp compc cha trong X v AB =t d(A,B)= inf{ d(x,y),x A,y B }CM : d(A,B)>0
Gii :
a.
CM f lin tc iu trn X cn CM )',()'()( xxdxfxf
ta c d(x,y) d(x,x)+d(x,y) ly inf 2 v d(x,A)-d(x,A) d(x,x)tng t thay i vai tr v tr ca x v x nhau ta suy ra PCMvy f lin tc ti x, do x tu nn f lin tc iu trn X
b.
Gi s tri li d(A,B)=0Khi ta tm c cc dy (xn) A, (yn)B sao cho limd(xn,yn)=0Do B compc nn (yn) c dy con knky )( hi t ve y0B
Ta c ),(),(),( 00 yydyxdyxd kkkk nnnn
M 0),(0),(),( 00 limlimlim
yxdyydyxdkkkk n
kn
knn
k
Do A l tp ng dy knkx )( A, 0)( yx knk nn y0A
iu ny mu thun vi gi thit AB =.Vy d(A,B)>0
GII THI TUYN SINH CAO HC THNG 9/2007MN C BN: I S V GII TCH
Bi 1: Tm min hi t ca chui lu tha
8/9/2019 Tong Hop de Thi Cao Hoc Toan
4/25
Nguyn Vn T - Website: violet.vn/nguyentuc2thanhmy
Luyn gii thi cao hc mni s 4
nn
n
xn
n 2
0
232
1
Gii : t X=(x-2)2 k X 0
Ta tm min hi t ca chui nn
n
Xn
n
0 32
1 xt
32
1
n
nun
Ta c 2
1
32
1
limlim
n
nul
n
n nn
21
l
R nn khong hi t l (-2,2)
Xt ti X= 2, X= -2
Ta c chui
n
n
n
n
n
n2
32
1)1(
0
n
n
n
n
n
0 32
22)1(
0132
22limlim
n
nu
n
nn
n
nn chui phn k
vy min hi ttheo X l (-2,2)min hi t theo x l 222222 xx
Bi 2:Cho hm s
0yxkhi0
0yxkhi1
sin)(),(
22
22
22
yxyx
yxf
Chng t rng hm s f(x,y)c o hm ring fx,fykhng lin tc ti 0(0,0)Nhng hm s f(x,y)kh vi ti 0(0,0).Gii :
Tnh cc hr ti (x,y) (0,0) va ti (x,y)=(0,0)
Ti (x,y) (0,0)
Ta c
222222
' 1cos21
sin2yxyx
x
yxxfx
222222
' 1cos21
sin2yxyx
y
yxyfy
Ti (x,y)=(0,0)
1
t
1sindo0
1sin
)0,0()0,(2
0
2
2
00
'
limlimlim
t
t
tt
t
ftff
ttt
x
1t
1sindo0
1sin
)0,0(),0(2
0
2
2
00
'
limlimlim
tt
tt
t
ftff
ttty
CM : fx,fykhng lin tc ti 0(0,0) Ta CM : 0'0,
lim
xyx
f v 0'
0,lim
y
yx
f
Hay CM : )0,0(),( ''
0,lim xx
yx
fyxf
, )0,0(),( ''
0,lim yy
yx
fyxf
Ta c :
8/9/2019 Tong Hop de Thi Cao Hoc Toan
5/25
Nguyn Vn T - Website: violet.vn/nguyentuc2thanhmy
Luyn gii thi cao hc mni s 5
Do
0 xkhi,221
cos.2
11
cos
0xkhi,021
sin2,1yx
1sin
,1
cos.21
sin.2),(
22222222
22
22220,
220,
'
0,limlimlim
xyx
x
yxyx
x
yx
xyx
x
yxyx
x
yxxyxf
yxyxx
yx
nn )0,0(),( ''
0,lim xx
yx
fyxf
tng t ta CM : c )0,0(),( ''0,
lim yyyx
fyxf
vy fx,fykhng lin tc ti 0(0,0) Ta CM : f(x,y)kh vi ti 0(0,0). Cn CM : 0),(lim
0,
tsts
Vi tfsfftsfts
ts yx )0,0()0,0()0,0(),(1
),( ''22
)1ts
1sin(do01sin.),(2222
22
0,0,limlim
ts
tstststs
vy f(x,y)kh vi ti 0(0,0)Bi 3: Cho RR*1,0: l mt hm s lin tcCMR : Hm F: C[0,1]R xc nh bi
1
0
))(,()( dttxtxF khi x(t) 1,0C l hm s lin tc trn C[0,1]
Gii: C nh x0, CM f lin tc ti x0
t M=1+sup )(0 tx , t 1,0C
Cho 0
lin tc trn tp compac D= [0,1]*[-M,M] nn lin tc u trn Dtn ti s 1 >0 sao cho
)','(),(',')','(),,( 11 ststssttDstst
t ),(1,0:),1min( 01 xxdx m MMtxtxtx ,)(1)()( 00
)()())(,())(,())(,())(,( 01
0
00 xFxFdttxttxttxttxt
ta CM c ))(),((),(:0,0 00 xFxFdxxd vy F lin tc ti x0
Bi 4:Cho nh x tuyn tnh 34: RRf xc nh bif(x1,x2,x3,x4)=(x1-2x2+x4,-x1+x2+2x3,-x2+2x3+x4)
1.
Tm c s v s chiu ca kerf, Imf2.
f c phi l n cu , ton cu khng?
Gii : 1. Tm c s v s chiu ca kerf
Vi x=( x1,x2,x3,x4)
8/9/2019 Tong Hop de Thi Cao Hoc Toan
6/25
Nguyn Vn T - Website: violet.vn/nguyentuc2thanhmy
Luyn gii thi cao hc mni s 6
Ta c : 0)(:ker 4 xfRxf
f(x1,x2,x3,x4)=(x1-2x2+x4,-x1+x2+2x3,-x2+2x3+x4)=0
02
02
02
432
321
421
xxx
xxx
xxx
lp ma trn
00001210
1021
12101210
1021
12100211
1021
A
vy Rank(A)=2
ta c
Rxx
xxx
xxx
43
432
421
,
2
2
nn dimKerf=2
nghim c bn l (1,1,0,1),(4,2,1,0) v l c s ca Kerfdo dimKerf =2 0 nn f khng n cu Tm c s , s chiu ca Im fIm f l khng gian con ca R3sinh bi h 4 vectf(e1)=(1,-1,0) vi e1=(1,0,0,0)f(e2)=(-2,1,-1) vi e2=(0,1,0,0)f(e3)=(0,2,2) vi e3=(0,0,1,0)f(e4)=(1,0,1) vi e4=(0,0,0,1)ta tm hng ca 4 vect trn
xt ma trn
000
000
110
011
110
220
110
011
101
220
112
011
B
Rank(B)=2, , dim Imf =2 , c s ca Imf l f(e1),f(e2)Do , dim Imf =2 nn f khng ton cuBi 5:Cho '':,': VVgVVf l nhng nh x tuyn tnh sao cho gf kerker Hn naf l mt ton cu . CMR tn ti duy nht mt nh x tuyn tnh ''': VVh sao cho h.f=g
Gii:
Bi 6: Cho dng ton phng trn R3f(x1,x2,x3)= 3121
2
3
2
2
2
1 222 xaxxxxxx
a.
a dng ton phng v dng chnh tc bng phng php Lagrangeb.
Vi gi tr no ca a th f xc nh dng, khng mGii : a. f(x1,x2,x3)= 3121
2
3
2
2
2
1 222 xaxxxxxx =
= 2322
32
2
321
61
62
3
4
22 x
ax
ax
axxx
8/9/2019 Tong Hop de Thi Cao Hoc Toan
7/25
Nguyn Vn T - Website: violet.vn/nguyentuc2thanhmy
Luyn gii thi cao hc mni s 7
t
33
322
3211
33
322
3211
6
32
6
42
yx
ayyx
ayyyx
xy
axxy
axxxy
ta c c s f chnh tc l u1=(1,0,0),u2=(-1/2,1,0),u3=(-a/3,a/6,1)
ma trn trong c s chnh tc l
1006
10
32
11
a
a
Tu
b. f xc nh dng khi 6606
12
aa
f xc nh khng m khi 60612
aa
GII THI TUYN SINH CAO HC THNG 5/2007MN C BN: I S V GII TCH
Bi 1:Cho u=u(x,y), v=v(x,y) l hm n suy ra t h phng trnh
021
.
012.
xv
uey
uvex
vu
vu
tm vi phn du(1,2), dv(1,2) bit u(x,y)=0, v(x,y)=0Gii :l thuyt : cho hm n
0),,,(
0),,,(
vuyxG
vuyxF xc nh bi u=u(x,y), v=v(x,y)
Tnh cc o hm ring ca hm nT h trn ta c
0
0
''''
''''
vvuuyyxx
vvuuyyxx
dGdGdGdG
dFdFdFdF
v
u
vvuuyyxx
vvuuyyxx
d
d
dGdGdGdG
dFdFdFdF
''''
''''
Tnh
)2,1(
)2,1(
v
u
d
d
Ta c :
Bi 2: Tm min hi t ca chui lu tha
2
2)1(
)(ln
1
n
nxnn
Gii :t X= x+1 ta c
22)(ln
1
n
nXnn
Xt212
))1)(ln(1(
1
)(ln
1
nn
u
nn
u nn
Ta c : 2
21
)1ln()1(
)(lnlimlim
nn
nn
u
uL
nn
n
n
8/9/2019 Tong Hop de Thi Cao Hoc Toan
8/25
Nguyn Vn T - Website: violet.vn/nguyentuc2thanhmy
Luyn gii thi cao hc mni s 8
Tnh )1ln(
ln.
1
1
1).1ln(.2
1.ln.2
)1ln(
)(lnlimlimlim
tan
2
2
nn
n
n
nn
nn
n
n
nn
lopi
n
Tnh 1
11
1
)1ln(
lnlimlim
tan
n
n
n
n
n
lopi
n
Nn 11
L
R , khong hi t l (-1,1)
Ti X= 1 ta c chui
2
2)1(
)(ln
1
n
n
nn
T ta c
1)1ln()1(
)(ln2
21
limlimnn
nn
u
uL
nn
n
n
Chui phn k , MHT theo X l (-1,1)
MHT theo x l (-2,0)Bi 3:Cho X l khng gian metric compac f: XX thod(f(x),f(y))0Khi g(f(x0))=d(f(x0),f(f(x0)))< d(x0,f(x0))=g(x0)iu ny mu thun vi s kin g(x0)=min(g(x))
Vy g(x0)=d(x0,f(x0))=0 hay x0=f(x0)CM tnh duy nht ca x0.Gi s c y0X sao cho y0=f(x0)Khi d(x0,y0) =d(f(x0),f(y0))
8/9/2019 Tong Hop de Thi Cao Hoc Toan
9/25
Nguyn Vn T - Website: violet.vn/nguyentuc2thanhmy
Luyn gii thi cao hc mni s 9
Hn na do A1=f(X)X nn A2=f(A1) f(X)=A1Gi s An+1 An ta c An+2=f(An+1) f(An)=An+1Vy An+1 NnAn , nA l h c tm cc tp ng trong khng gian compac
Theo tnh cht phn giao hu hn ta c A=
n
n
A1
CM: f(A)=A cn CM : f(A)A (1) , f(A) A (2)
CM : f(A)A (1)
Do A Annn f(A) f(An)=An+1vi mi n, l dy gim nn
f(A) AAnn
1
1
f(A) A (2)
ly tu xA cn CM x f(A)v x An+1=f(An) vi mi n=1,2 tn ti xnAn: x=f(xn)do X compact nn c dy con (x
nk)
k: ax
knk
lim
khi xxfkn
k
)(lim , do f lin tc nn afxf knk
()(lim
) ta cn CM a A
c inh n ta c nnnnn AxAAx kkk khi nk n
do Anng nnk
Aaxk
lim
vy a An vi mi n=1,2 do a A, x=f(a) f(A)
vy ta CM c f(A)=A
Bi 4:Gii v bin lun h
1
1
1
4321
4321
4321
xmxxx
xxmxx
xxxmx
Gii :Ta c ma trn m rng
1.1111.111
1.111
mm
m
A i ch d1, d3, bin i ma trn v dng
1.1)2)(1(00
0.0110
1.111
mmmm
mm
m
A
bin lun nu m=1 h c VSN ph thuc 3 tham s x2,x3,x4v RankA=1
nghim ca h l x1=1-a-b-c, x2=a,x3=b,x4=c
nu m=-2 h c VSN ph thuc tham s x3v RankA=3nghim ca h l x1=x2=x3=a,x4=1
nu m 1v m -2 th h c VSN ph thuc vo tham s x4va tham s m
8/9/2019 Tong Hop de Thi Cao Hoc Toan
10/25
Nguyn Vn T - Website: violet.vn/nguyentuc2thanhmy
Luyn gii thi cao hc mni s 10
nghim ca h l2
11
m
ax ,
2
12
m
ax ,
2
13
m
ax , Raax ,4
Bi 5:Trong R3cho c s :u1=(1,1,1), u2= (-1,2,1), u3=(1,3,2)
cho nh x tuyn tnh f: R3 R3xc nh bif(u1)= (0,5,3), f(u2)=(2,4,3), f(u3)=(0,3,2)
tm ma trn ca f trong c s l ma trn cho ho cGii :b1. Tm ma trn ca f trong c s u
Ta c h
)3()(
)2()(
)1()(
3322113
3322112
3322111
ucucucuf
ubububuf
uauauauf
T (1) ta c (0,5,3)=a1(1,1,1)+a2(-1,2,1)+a3(1,3,2)
1
1
0
02
032
0
3
2
1
321
321
321
a
a
a
aaa
aaa
aaa
Tng t t ( 2) ta c b1=1,b2=0,b3=1Tng t t (3) ta c c1=1,c2=1,c3=0
Vy ma trn A trong c s f l
011
101
110
333
222
111
)(/
cba
cba
cba
A ufA
B2. Tm GTR- VTR ca A v ca f (GTR ca A chnh l GTR ca f)
Xt ma trn t trng2
)(1023
11
11
113
m
kepmmm
m
m
m
A c 2 gi tr ring, nn f c 2 gi tr ring m=-1, m=2Tm VTR ca A t suy ra VTR ca f
vi m=-1 ta c 0000
000
111
111
111
111
VTR ca A c dng
bx
ax
baxxx
Rxx
xxx
3
2
321
32
321
,
0 a,bR
Dng VTR ca A l (-a-b,a,b)Vy A c 2 VTR (-1,0,1),(-1,1,0)T VTR ca f c dng n= x1u1+x2u2+x3u3=(-a-b)u1+au2+bu3=
=(-a-b)(1,1,1)+a(-1,2,1)+b(1,3,2)=(-2a,a+2b,b)
vy f c 2 VTR LTT vi a=1,b=0 :VTR l n1=(-2,1,0)vi a=0,b=1: VTR l n2=(0,2,1)
vi m=2 ta c 0000
330
211
112
121
211
211
121
112
8/9/2019 Tong Hop de Thi Cao Hoc Toan
11/25
Nguyn Vn T - Website: violet.vn/nguyentuc2thanhmy
Luyn gii thi cao hc mni s 11
VTR ca A c dng
ax
axx
aaaxxx
Rx
xx
xxx
3
32
321
3
32
32122
033
02
aR
Dng VTR ca A l (a,a,a),Vy A c VTR (1,1,1)
T VTR ca f c dng n= x1u1+x2u2+x3u3=au1+au2+au3==a(1,1,1)+a(-1,2,1)+a(1,3,2)=(a,6a,4a)vy f c VTR l n3=(1,6,4)
b3 : KL vy f c 3 VTR LTT n1,n2,n3do 3 VTR n1,n2,n3lm thnh 1 c s caR
3v ma trn ca f trong c s l ma trn cho ho c
ta c :
200
010
001
)/(nfA
GII THI TUYN SINH CAO HC THNG 9/2006MN C BN: I S V GII TCH
Bi 1:Cho
0yx,0
0y x,1
sin),(
22
22
2
yx
yxyxf
a. Xt s kh vi ca f ti (x,y)R2c bit ti (0,0)b.
Xt s lin tc ca cc HR '' , yx ff ti (0,0)
Gii :
Ti (x,y) (0,0) Ta c
22222
3
22
'
22222
2'
1cos.
)(
21sin.2
1cos.
)(
21
yxyx
y
yxyf
yxyx
xyf
y
x
Do '' , yx ff lin tc ti mi (x,y) (0,0) nn f kh vi ti mi (x,y) (0,0)
Ti (x,y)=(0,0)Ta c
1)0,0()0,()0,0( lim0
' t
ftfft
x
1)t
1sin(do0
1sin
.)0,0(),0(
)0,0(2
22
00
'
limlim
t
ttt
ftff
tty
Tnh ),(lim0,
tsts
Ta c 22
2
22
''
22
1sin..
1)0,0()0,0()0,0(),(
1),(
stt
tstfsfftsf
tsts yx
2222
2
0,0,
1sin.),( limlim tststts
tsts
0
8/9/2019 Tong Hop de Thi Cao Hoc Toan
12/25
Nguyn Vn T - Website: violet.vn/nguyentuc2thanhmy
Luyn gii thi cao hc mni s 12
do 11
sin22
ts
nn f kh vi ti (0,0)
b.Xt s lin tc ca cc HR '' , yx ff ti (0,0)
Xt s lin tc ca cc HR '' , yx ff ti (0,0) ta tnh ),(),,('
0,
'
0,limlim yxfyxf y
yxx
yx
nu )0,0(),(),0,0(),( ''
0,
''
0,
limlim yyyx
xx
yx
fyxffyxf
th '' , yx ff lin tc ti (0,0)
'', yx ff khng lin tc ti (0,0)
chn )0,0(0,1,
nyx nn ta c
0)0,1(
1)0,1(
'
0,
'
0,
lim
lim
nf
nf
yyx
xyx
chn )0,0(2
1,
2
1','
nnyx nn ta c
),(
),(
'''
0,
'''
0,
lim
lim
nny
yx
nnxyx
yxf
yxf
vy '' , yx ff khng lin tc ti (0,0)
Bi 2:Cho (X,d )l khng gian mtric compac, f: XX tho mn:d(f(x),f(y))0khi h(f(x0))=d(f(x0),f(f(x0)))
8/9/2019 Tong Hop de Thi Cao Hoc Toan
13/25
Nguyn Vn T - Website: violet.vn/nguyentuc2thanhmy
Luyn gii thi cao hc mni s 13
Nn gnlin tcDo Xxxgxfxdxffxfdxfxdxg n
nnn
n ),())(,()))((),(())(,()( 001
01 (gn(x))dy
gim khng m nn hi tt a= limgn(x)Gi s a>0, do X compac dy fn(x)cha dy con hi t k
nxf k )(
t )(lim
xfy kn
k
Ta ckn
xgk
)(1 l dy con ca nn xg )( nn )()( 1limlim xgxga kk nk
nk
0),())(,()( 00limlim
yxdxfxdxga kk
n
kn
k
)(1lim xga knk
ayxdyfxfdxfxdk
n
k
k
),())(),(())(,( 001
0 limlim
mu thun vy Xxxgnn
,0)(lim
c. CM (gn)nhi t iu v 0 trn Xvi 0 t ),()(: 1 nnn gxgXxG l tp m
do gn(x) >gn+1(x)nn GnGn+1ta c nn
GX
1
do X compac nn c n0:0
0
1
nn
n
n
GGX
vy 0nn,)(0 khiXxxgn vy (gn)nhi t iu v 0 trn XBi 3 Cho V l khng gian vect , f: V V l nh x tuyn tnh tho mn f2=f CM:Kerf+Imf=V v 0Imker ff
Gii
CM: Kerf+Imf=V ta cn CM Kerf+ImfV (1), Kerf+Imf V (2)
CM Kerf+ImfV (1) hin nhin
CM: Kerf+Imf V (2)
Ly tu xV cn CM x Kerf+ImfTa c x= x-f(x)+f(x) m f(x) Imf cn CM (x-f(x)) Kerf cn CM f(x-f(x))=0Xt f(x-f(x))=f(x)-f2(x)=f(x)-f(x)=0 nn (x-f(x)) Kerf hay xKerf+Imf
Vy Kerf+Imf VT (1),(2) ta c Kerf+Imf=V
CM 0Imker ff
Ly y tu y: y ff Imker cn CM y=0Do y ff Imker khi c xV : f(x)=y v f(y)=0Do f2=f nn y=f(x)=f2(x)=f(f(x))=f(y)=0
Vy y=0 hay 0Imker ff Bi 4: Cho f: R4 R3nh bif(x1,x2,x3,x4)=(x1-x2+x3,2x1+x4,2x2-x3+x4)
a. Tm c s v s chiu ca Kerf, Imf
b.
Tm u R4 sao cho f(u)=(1,-1, 0)
Gii :a.
Tm c s s chiu ca KerfVi x=(x1,x2,x3,x4)
8/9/2019 Tong Hop de Thi Cao Hoc Toan
14/25
Nguyn Vn T - Website: violet.vn/nguyentuc2thanhmy
Luyn gii thi cao hc mni s 14
02
02
0
0)(:
432
41
321
4
xxx
xx
xxx
xfRxKerf
ta c ma trn m rng
0.1120
0.1002
0.0111
0.0100
0.1220
0.0111
bin i ta c h
ax
x
ax
ax
Rx
x
xxx
xxx
2
00
022
0
4
3
2
1
4
3
432
321
l nghim tng qut ca h
ta c dimKerf =1
c s ca Kerf l (1,1,0,2) Tm c s v s chiu ca Imf
ta c f(e1)=(1,2,0), f(e2)=(-1,0,2), f(e3)=(1,0,-1), f(e4)=(0,1,1)Imf=(f(e1),f(e2),f(e3),f(e4))
Ta c
110
101
201
021
000
100
200
021
Nn dim Imf =3
Vy c s ca Imf l (f(e1),f(e2),f(e3))
b. Tm u
R
4
sao cho f(u)=(1,-1, 0)ta c : f(u)=(1,-1, 0) =(x1-x2+x3,2x1+x4,2x2-x3+x4)
ta c h
ax
x
xx
xx
xxx
xx
xxx
2
1
2
1
2
1
2
1
2
1
02
12
1
4
3
42
41
432
41
321
(a R)
lp ma trn m rng bin i gii h trn ta c u=(x1,x2,x3,x4)Bi 5
: Tm GTR- VTR v cho ho ma trn
A=
221
221
115
Gii :Xt a thc t trng
3
6
0
0189
221
221
11523
a
a
a
aaa
a
a
a
vy A c 3 GTR a=0, a=6, a=3
tm VTR
8/9/2019 Tong Hop de Thi Cao Hoc Toan
15/25
Nguyn Vn T - Website: violet.vn/nguyentuc2thanhmy
Luyn gii thi cao hc mni s 15
vi a=0 :ta c
000
990
221
115
221
221
221
221
115
ta c h
ax
ax
x
ax
xx
xxx
3
2
1
3
32
321 0
099
022
suy ra VTR (0,a,a) vi a=1 th VTR (0,1,1)
vi a=6: ta c
000
330
111
421
241
111
c h
ax
ax
ax
ax
xx
xxx
3
2
1
3
32
321 2
033
0
suy ra VTR (-2a,-a,a) vi a=1 th VTR (-2,-1,1)
vi a=3: ta c
000330
121
112221
121
121211
112
c h
ax
ax
ax
ax
xx
xxx
3
2
1
3
32
321 3
033
02
suy ra VTR (3a,a,a) vi a=1 th VTR (3,1,1)
ma trn cn tm l T=
111
111
320
v T-1
AT=
300
060
000
GII THI TUYN SINH CAO HC THNG 9/2005MN C BN: I S V GII TCH
Bi 1:Cho hm s0y xkhi0
0yxkhi1
sin)(),(
22
22
22
yx
yxyxf
CMR hm s f(x,y ) c cc o hm ring '' , yx ff khng lin tc ti (0,0) nhng f(x,y)
kh vi ti (0,0)Gii :
Tnh cc hr '' , yx ff
ti (x,y) (0,0)
ta c222222
' 1cos21
sin2yxyx
x
yxxfx
222222
' 1cos21
sin2yxyx
y
yxyfy
ti (x,y)=(0,0)
0
1sin
)0,0()0,( 22
00
'
limlim
t t
t
t
ftff
ttx
8/9/2019 Tong Hop de Thi Cao Hoc Toan
16/25
8/9/2019 Tong Hop de Thi Cao Hoc Toan
17/25
Nguyn Vn T - Website: violet.vn/nguyentuc2thanhmy
Luyn gii thi cao hc mni s 17
a.CMR : M l tp ng khng rng v b chn trong khng gian metric C([0,1]) vimtric d(x,y)=max{ )()( tytx : t 1,0 } vi x(t),y(t) )1,0(C
b. xt RCf )1,0(: xc nh bi f(x)= 1
0
2 )( dttx
CM : f lin tc trn M nhng f khng t c GTNN trn M t suy ra M khngphi l tp compc trong C([0,1])Gii : a.
CM : M l tp ngLy dy (xn) M : limxn=x cn CM xMTa c 1)1(,1,0,1)(0 nn xttx
Cho n ta c 1)1(,1,0,1)(0 xttx nn xMVy M l tp ng
b.
CM f lin tc trn M
Xt tu x )1,0(C , (xn) M : limxn=x cn CM limf(xn)=f(x)Ta c Nxxdxxdtxtxtxtxtxtxtx nnnnn ),().,()(2)()(.)()()()(
22
Vi N= 1,0,)(2sup ttx
Nxxdxxdtxtxxfxf nnnn ),().,()()()()(1
0
22
do limd(xn,x)=0 nn t y ta c limf(xn)=f(x)vy f lin tc trn M
CM f khng t GTNN trn M
Trc tin ta CM inff(M)=0, nhng khng tn ti x
M f(x)=0t a= inff(M) ta c f(x) Mx ,0 nn a 0 Vi xn(t)=t
nta c xnM
n0121
12)()(
1
0
121
0
2
1
0
2 khinn
tdttdttxxfa
nn
nn
vy a= inff(M)=0 khng tn ti xM f(x)=0
gi s tn tai xM f(x)=0 ta c )(,0)(,0)( 221
0
2 txtxdttx lin tc trn [0,1] suy
ra x(t)=0 vi mi t [0,1] iu ny mu thun vi x(1)=1 vi mi xMvy khng tn ti xM f(x)=0t y ta suy ra M khng l tp compcgi s nu M l tp compc , f lin tc th f t cc tiu trn M tc l c x0M saocho f(x0)=inff(M)=0 iu ny mu thun vi khng tn ti xM f(x)=0vy M khng l tp compcBi 4: Cho 33: RRf l mt php bin i tuyn tnh xc nh bif(u1)=v1, f(u2)=v2, f(u3)=v3
u1=(1,1,1),u2=(0,1,1), u3=(0,0,1)
v1=(a+3,a+3,a+3),v2=(2,a+2,a+2), v3=(1,1,a+1)a.tm ma trn f vi c s chnh tc e1=(1,0,0), e2=(0,1,0), e3=(0,0,1)
b. Tm gi tr ca a f l mt ng cu
8/9/2019 Tong Hop de Thi Cao Hoc Toan
18/25
Nguyn Vn T - Website: violet.vn/nguyentuc2thanhmy
Luyn gii thi cao hc mni s 18
c. khi f khng l mt ng cu hy tm c s v s chiu ca Imf v Kerfd. vi a=-3 f c cho ho c khng trong trng hp f cho ho c hy tm mtc s ma trn f voi c s c dng cho .Gii :
Bi 5:Cho dng ton phng323121
2
3
2
2
2
1321 2222),,( xxxaxxxxxxxxxf
a. a dng ton phng v dng chnh tc
b. Vi gi tr no ca a th f l xc nh dng v na xc nh dngGii : a. ta c
2322
32
2
321
323121
2
3
2
2
2
1321
)22()1()(................
.......2222),,(
xaaxaxaxxx
xxxaxxxxxxxxxf
t
33
322
3211
33
322
3211
)1(
)21(
)1(
yx
yayx
yayyx
xy
xaxy
axxxy
c s f chnh tc l u1=(1,0,0),u2=(-1,1,0),u3=(1-2a,a-1,1)
ma trn
100
1102111
aa
Tu
b.f xc nh dng khi -2a2+2a>0 10 a f na xc nh dng khi -2a2+2a=0 1,0 aa
GII THI TUYN SINH CAO HC THNG 9/2004MN C BN: I S V GII TCH
Bi 1: Tm min hi t ca chui hm lu tha
nnn
n
xn
n )1(
1 1
2
Gii :
Xt
nn
nn
u
1
1
11
Ta c L= en
u
n
n
nn
n
1
1
11limlim
NneL
R11
, khong hi t l
ee
1,
1
Xt tai 2 u mt x=e
1
8/9/2019 Tong Hop de Thi Cao Hoc Toan
19/25
Nguyn Vn T - Website: violet.vn/nguyentuc2thanhmy
Luyn gii thi cao hc mni s 19
Ta c chui n
n
n
nnnnn
n enen
n
11
1
11
1
1
2
1
1)1(
1
011.
1.
1
11
1
limlim
ee
enu
n
n
nn
n
vy MHT ca chui hm lu tha l
ee
1,
1
Bi 2 :Cho hm s f:R2R xc nh bi
(0,0)y)(x,khi0
(0,0)y)(x,khi2
),(22
yx
xy
yxf
a.
Xt s lin tc ca f trn R2b.
Tnh cc o hm ring ca f trn R2Gii :Ch : nu 0)0,0(),(lim
0,
fyxfyx
th hm s lin tc
Ti mi (x,y) (0,0) th hm s lin tc v l hm s cp
Xt s lin tc ca f trn R2ti (0,0)
Tnh22
0,0,
2),( limlim
yx
xyyxf
yxyx
Chn dy )0,0()1,1(),(
nn
MyxM nnnn khi n
Ta c 0111
12
)(
22
2
nn
nMf n , )0,0(012
22
),(
2
2
220,0,
limlimlim f
n
n
yx
xyyxf
nyxyx
vy hm s khng lin tc ti (0,0) Tnh cc hr '' , yx ff
Ti (x,y) (0,0)
ta c222
22'
)(
)2(2)(2
yx
xyxyxyfx
222
22'
)(
)2(2)(2
yx
xyyyxxfy
Ti (x,y)=(0,0)
ta c 0)0,0()0,(
lim0
'
t
ftff
tx
0)0,0(),0(
lim0
'
t
ftff
ty
Bi 3:Tnh tch phn D
dxdyyxI )2(
Vi D l na trn ca hnh trn c tm ti im (1,0) bn knh 1Gii :
Phng trnh ng trn tm I(1,0) bn knh R=1 l (x-1)2+y2 1 x2+y2 2x
8/9/2019 Tong Hop de Thi Cao Hoc Toan
20/25
Nguyn Vn T - Website: violet.vn/nguyentuc2thanhmy
Luyn gii thi cao hc mni s 20
i sang to cc
t
0,r
sin
cos
ry
rx1 chu k
Ta c x2+y2 2x ta c r2
2rcosnn cos2r
Vy ta c
20
cos20
r
Vi
rdrddxdy
ry
rx
sin
cos
Vy
2
0
cos2
0
22
0
cos2
03
2
16...)sincos2()sincos2()2(
drrdrdrdrrdxdyyxID
Bi 4:Cho tp hp cc s t nhin N vimi m,n N
t
nmneu0
nmneu1
1),( nmnmd
a. CM d l metric trn N
b. CM (N,d ) l khng gian metric y Gii :a. d l metric trn N
d(m,n) Nnm ,,0
d(m,n)=0 m=n
),(0
11
nmneu0
nmneu1
1
),( mndmnnmnmd
CM d(m,n) d(m,l)+d(l,n) (1) Nnml ,,
TH1 : nu m=n,m=l,n=l th (1) ng
TH2 : nu m n thnm
nmd
1
1),(
nu m l thlm
lmd
1
1),(
nu l n thnl
nld
1
1),(
th VT ca (1) 2 , VP ca (1) 2 nn (1) ngb. (N,d ) l khng gian metric y gi s (xn) l dy cauchy trong (N,d) ta CM xn x ddo (xn) l dy cauchy trong (N,d) nn ta c 0),(lim
,
nmnm
xx
),(:,:,0 00 nm xxdnnmn
chn 000 ,.0),(2
1),(,:,
2
1nnmxxxxdxxdnnmn nmnmnm
vy xxnnxxNx nn 0:: trn d
8/9/2019 Tong Hop de Thi Cao Hoc Toan
21/25
Nguyn Vn T - Website: violet.vn/nguyentuc2thanhmy
Luyn gii thi cao hc mni s 21
Bi 5:tnh nh thc
000021
000073
217667
125115
850042
640031
Gii :
Bi 6:Cho nh x tuyn tnh f: R4 R3c ma trn trong cp c s chnh tc l
3502
1132
1201
xc nh nhn v nh ca f , Hi f c n cu , ton cu khng? V sao?Gii :t mae trn tac nh x
f(x1,x2,x3,x4)=(x1+2x3+x4,2x1+3x2-x3+x4,-2x1-5x3+3x4)Xc nh nhn v nh ca f tc l tm c s v s chiu ca Imf, Kerf
Tm c s v s chiu ca Kerf
Ta c
0.3502
0.1132
0.1201
0.5100
0.1530
0.1201
Ta c h )(,
315
26
33
3
5
1
3
5
2
05
053
02
4
3
2
1
4
43
432
431
43
432
431
Ra
axax
ax
ax
ax
xx
xx
xx
xxx
xx
xxx
xxx
f c 1 n t do nn dimKerf = 1 v Kerf c c s l (-33,26,15,3)Vy f khng n cu v dimKerf = 1
Tm c s, s chiu ca ImfTa c
B=
0.3110.512
0.030
0.221
0.5100.150
0.030
0.221
0.1500.030
0.050
0.221
0.26000.1500
0.510
0.221
0.000
0.39000
0.510
0.221
Vy Rank (B)=3 nn dimImf=3 v Imf c 1 c s gm 3 vect(f(e1),f(e4),f(e2))f khng ton cu v dimImf=3
Bi 7:Cho ma trn
133
153
131
A
a. Tm GTR-VTR ca A
8/9/2019 Tong Hop de Thi Cao Hoc Toan
22/25
Nguyn Vn T - Website: violet.vn/nguyentuc2thanhmy
Luyn gii thi cao hc mni s 22
b. Tnh A2004
.
Gii :
a. Tm GTR- VTR ca A Tm GTR ca A
Xt a thc t trng
Ta c:210)2)(1(
133
153
131
2
aaaa
a
a
a
Vy A c 2 GTR a=1, a=2 Tm VTR ca A
Vi a=1 ta c
000
110
132
330
110
132
033
143
132
Ta c h
11
1
0
032
3
2
1
3
32
321
xx
x
axxx
xxx
vy c VTR (1,1,1)
Vi a=2 ta c
000
000
133
133
133
133
Ta c h
bx
ax
xxx
bx
ax
xxx
3
2
321
3
2
321 33033
Vy c 2 VTR (1,1,0), (-1,0,3)b.
ta c
301
011
111
Q
ma trn cho ca A l
200
020
0011AQQB
(Q
-1
AQ)
2004
=Q
-1
A
2004
Q
vy A2004=QB2004Q-1= 1
2004
200
020
001
QQ = 1
2004
2004
2004
200
020
001
GII THI TUYN SINH CAO HC THNG 9/2003MN C BN: I S V GII TCH
Bi 1:
Bi 2:
Bi 3:Cho (X,d) l khng gian metric compc
a.Gi s Anl h cc tp con ng trong X v An+1 Anmi n N
CMR nu vi mi n N ,An th
1n
nA
8/9/2019 Tong Hop de Thi Cao Hoc Toan
23/25
Nguyn Vn T - Website: violet.vn/nguyentuc2thanhmy
Luyn gii thi cao hc mni s 23
b.Gi s NnRXfn ,: l cc hm lin tc vXxxfxfxf n ........,)(.........)()( 21 CMR nu Nnnn
n
fXxxf
)(,,0)(lim
H t u v 0 trn XGii :
a. gi s vi mi n N ,An CMR :
1n
nA
vi mi n N ly xn Ando nnpn AnenNpnAA pn x,,,
do X compc nn vi (xn)nX c dy con (kn
x )khi t
t x=kn
k
xlim
do nk k nn Ak l tp ng vi mi i dy nn
iikn AxAxx
k
1
vy
1n
nA
b. cn CM :1. 0)( xfn
2. )(xfn
ta c Xxxfxfxf n ........,)(.........)()( 21 v Xxxfnn
,0)(lim nn 0)( xfn
vi 0 cho trc t ),()(: 1 nnn fxfXxF do ),( l tp m, f lin tc nn Fnm
do fn+1(x) fn(x) suy ra fn(x) l dy gim nn 1 nn FF
do XFXxxf nn
nn
1
,0)(lim
do X compc nn c tp J hu hn trong N sao cho XFnJn
t n0=maxJ ta c c 0,)()(0 00 nnxfxfFXF nnnnJn
vy NnnfXx )(, hi t u v 0 trn X
Bi 4:b tm min hi t ca chui hm
1n
n
n
x
dn
n
Gii : xt
2n
ndn
nU
Ta cdn
n
n
n
nn
n e
n
dndn
nUL
11limlimlim
Bn knh hi t R=ed, khong hi t (-ed;ed)Xt ti 2 u mt x=ed,x=-ed
Ta c chui 011)( lim1 1
22
n
nnn n
d
n
nd
n
Uedn
nedn
n
Vy MHT ca chui l (-ed;ed)
8/9/2019 Tong Hop de Thi Cao Hoc Toan
24/25
Nguyn Vn T - Website: violet.vn/nguyentuc2thanhmy
Luyn gii thi cao hc mni s 24
GII THI TUYN SINH CAO HC THNG 9/2002MN C BN: I S V GII TCH
Bi 1 :a. Cho hm s
(0,0)y)(x,khi0
(0,0)y)(x,khi)(
),( 22
22
yx
yxxy
yxf
Xt tnh lin tc ca f(x,y) v cc hr '' , yx ff trn tp xc nhGii :
Ti mi (x,y) (0,0) f(x,y) lin tc v l hm s cp Xt s lin tc ca f ti (x,y)= (0,0)
Nu 0)0,0(),(lim0,
fyxfyx
th hm s lin tc
Ta c :22
3
0,22
3
0,22
22
0,0,limlimlimlim
)(),(
yx
xy
yx
yx
yx
yxxyyxf
yxyxyxyx
Xt 02 22
3
0,
2
22
3
lim yxyxx
yxyx
yx
02 22
3
0,
2
22
3
lim
yx
xyy
yx
xy
yx
t 0)0,0(),(lim0,
fyxfyx
vy f lin tc Tnh cc hr '' , yx ff
Ti (x,y) (0,0)
'xf
'yf
Ti (x,y)=(0,0)
0)0,0()0,(
lim0
'
t
ftff
tx
0)0,0(),0(
lim0
'
t
ftff
ty
b.
Tnh tng ca chui hm
1n
nnx trong MHT ca n
Gii : ta tm c khong hi t l (-1,1)
Ta c
1
1 1..)(n
n
xxnxS
t
1
1
1 .)(n
nxnxS (1)
Ly tch phn 2 v ca (1) trn on [0,x] ta c
1 0 1
1
0
11
1.)(
n
x
n
nn
x
ttdttndttS (2) l CSN
o hm 2 v ca (2) ta c 21 )1(1)(x
xS
8/9/2019 Tong Hop de Thi Cao Hoc Toan
25/25
Nguyn Vn T - Website: violet.vn/nguyentuc2thanhmy
vy
1x:1
1x:)1.(
1
)( 2xxxS
Bi 2:
Bi 3:
Bi 4:b. Tm cc VTR- GTR ca ma trn
A=
702
052
226
Gii :Xt a thc t trng
9
3
6
0)2712)(6(
702
052
2262
a
a
a
aaa
a
a
a
Vy c 3 GTR a=6,a= 3, a= 9 Tm VTR
Vi a=6 ta c
102
012
220
220
012
102
000
110
102
Ta c h
ax
ax
ax
x
ax
xx
xx
2
2
2
2
0
02
3
2
31
3
32
31
C VTR l (-1,2,2)
Vi a=3 ta c
402
022
223
Ta c hC VTR l (,,)
Vi a=9 ta c
202
042223
Ta c hC VTR l (,,)
NH GH THM THNG XUYN: VIOLET.VN/NGUYENTUC2THANHMY