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Phd tbng hidn tiip: - Iyg6 Dar Tir
fi:{ I THANG mS,r Kt
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i}ir- ltii: 52825
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(€ P€sn s&ng Vigr f{qm
GS fr& men v& v{!n$ hEy
khdngi ddn rQc Vig*
Horn c* dugc v& verag
s6nh vsi e *r €ry&ng
gude ndlm rh&u dugc
hery kh$ng, eh{nh E A
nh# rn*f ph&rr $6'cr &
c&ng hg. r€p eriq $&e
chdu r>
Trirlr llur uua &dt fli
grii /rlc sin.ft nftdn ngdr;
l:lni trtrlrtg nitt: 1945
'! :. ,:.=,:.-:'.
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::::i r.' .:l ll: tjr.:rlfl
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l?:
Nbt chuyhn obi edc ban sr& y6u tad*
d-T tri,, I'o6n !tuc vi tribi lri., s6 i (139). nenr
}/ i984 trong b*i : ( 5<i r'& l{:nh r, tsi c1[ gi$i :1
thi$Lr vtii t:f;c han mSt clicir uli.in ntti'i l'E
tirh hai sd tiiuc, r'i d$ nhu tich il'{ = 1? tili
lhln (3* lir nrr.it lgrrh /,, <4r l&'vAl ehlu lCrrh
Ir v[ ( 12 )i ki:t quii Q stu khi lOnlr dE lhi hi.nlr.
Vrvi cncI nhi;li ti{r tiri ticit *frs hai lo^nh }{-t' liieo
.*rdt iirir lqr nio Li$ l5 mOi lf nh Cul- nhSt cho
ki:l grri qiJrrl v niru l;lri lhrt': hiSn hai I{nlr iI5.
r:ho theo th'.1 !.rr tlir cl.ioo nbrr iii:h i:rirr lr*i lt"lnh
(r3}) \,i <5r.i iA lgnh n 15*. S:r.u t16, e'hrirrg t:l ilfl
in$ rsng /,. V, 0 ra ngoti phsrn vi c{e s6 tirr;e, ,
t{;in e$c c,}p s5 .thqrc tl$ c<, nhDng lili: i, gbm
_ /,! ir\
lr*i c{p s6 thrrr s{l thAnh rnCt },&ng{' :,)
NGUYf,H CANH TOJ{I{
churig nhd ,.1t1 guir r'ir rlng. qu6 ttr0]r d"ir, tu&ng
ehung nhu mgi ruan Lr! mil di tluqc ridn len.
Cc duqe nhitng pli:it ilinh nltu v+,v ii nhd
nhfrng bQ 6e r{t ( kh6 linh >, rtit k h0ng < dl,
tiri;,t rttltt )). Ri'n itr-r Qtl cittr nrinir tttQt bd drc nh'r
Yir\. tlirii .lii cirrt ttiu iiltu i,.ln llllrJn sau uiry trj
tliirnh nr{t rhri to6n }roc' {-lrinh vi r-i-v mir sau
biii: <S,5 vi lQrrhn r:hir;rg-'i:r e<';:r lrblli ti6p tpt:
lrao il'bi.
Nio, bsy.giir chitng ta ltlr; cd trl ioi hai 0$rr
h6i d+t ra trcn kia. {lhdlg ta dEu bi6t ndu cho
mQt trgc s6 thi cO nrit str ttrtrng rtng mQt ddi
mdt git* clc sd thuc vi' e6e diBrn tr6n trqc sd
d6. N6u O li g6c tqa c1Q tr0n trqc s6 iI6 thi ta
clng c6 thb n6i.6 11rr'rt tucrng irng nr6t ddi mQt
gita c6.e sd ttrgc vi. cilc vecto g6e O trOn trrlc
s6 d6. B{y gid thi ta cd thb nhin tieh 3.4 = 1!
khirc di ntdt ch'it : ((:, ' ii iFnh I, vdctu OM lir
' *
var t' r'.i veclcr 0]' in k6l ,1ui 0 {hinh 1).
/.
0 i't a.
+*+-l.f#-t+-l-'-:'i -- " ---" --'; - *i-'
"-&3!'a
Uirtlt 1 : if .O; =, CI;":}
.:
' , II$t 6iEu tili:g lihi lrr nhi*.n li ti! clc vecio
lrifl trilc s6, ta rnlr i't)lt{ t;t t:ic r'e.to trorlg rnilt
ph&ng (hinh 2); l(hi dri {'ng viri m6i teeto trong
rnil phirtg l* (c, nl0t c[p hr"i sii 11ro* nhu veeln
o*? f, (ninl *l *ng vrli'hai sd thgc im, rr). Cho.
n6u vi$c tr:& rilr:g tir sd thqc slng t'fip sd thgc
ring r:i!! r;iQc lirur ktr{ tr,r rhi&n i:i tr} r'e cts tr€n
Dern l*ntr
tili r-1rrg*
'ptr!-m vi
!'tt b\,
l" aloP
eSp sd thge
dung vio ' { p sd thrrc {:i', y)
(ar: * br7, r.r * ,l3i). Ti'ong
,,r; iinh r.:qng/" 1i, elrirlgiac!i tirrir
"\c dI
xuf,t !ri$n nhiru rliBu k)' l+. Oqe bAi l:.61o iriini
ctlic clir,: ban tlr€y nirrlug diBu k.i' l;r $y riruli
r,it di: hitu. NSu ,.Jfi tr.hs,r e$c d&ng rl:r'rc rlrl ririn
rlln c.ire ri!Bir k{z tq rtii, tlri b&n. tirAn rnSi d:ing
tliir,: vir viir: chr",,,hn tdr riilug ttritr: nil qua iiSng
thLrc s*.u I:i r{t r0 rfi.ng. Nlrrrng n6rr la tr drit
c$ rr ir,i.i ; < Tr3i sn.o Iq.i q& rSng tu sii :ang ' ; 1r
/
sd n"i ktr&ng nr& rQng thecl hrrring lctr{e 'l Bin
i:h.{t cfra c6r: p.rh$n iiv -kj- la nhrr }iry iinir, tfly
diine tn gi ? thi ttisy ring edn rrhiBii tlrBu *hun
<<d6 hiEu n v:\ 'e!:ra a th6a ln6u >. l<PE hiBu v;
(116 lli,ia nrfru Blll mQt hgnh t$i ky ddi vtli nh'&*g
nh'& kha:r irg* n$i ehrrng, to{n hgc n6i rifng.
'l'rong lich *fr, toirn hoc, dA r'o niiiilu phit rninh
quan trqng nly iiinh Lrl nh*rg vflirl fih t*&ag
:l
t
,u
*
kr
vdo ON (hi;h 3) vd hiEn nhi6n !I duqe vecto
il r;op ruc 6p dung tenn {l ,l) ,u,, cho r-eero
-t
O thi di.rh.l0n v5n c6 k6t quiL li, vecto0. yi
,'.,, (: J)': (3 ;) ",, rrroi s6 ru nhi6n n>z.
Ir n\ . '*
I.gnh t ' " I bi6rr vecto OM rhAnh hinh ehi6u
\o ol i -------- ':--'"-
vu0ng g{rc cira n6 xudng trrlc hoAnh; l4i Ap dung
ti6p Idnh d6 v{o lrinh ehirlu nfr-v thi ta vfr.n aluoc
hin! crri6u,rio. chinh vi r,{.v -r(; :)" = (i l)
vdi nrei s6 rtp nirien rt21",. t*,,r(: -;) bi6n
*+
vecto o:If cir t9a d+J":;,1:l,u"n vecto oN c6
toa dd (-n, rrr): ver (" ;l ctrtnrr tn lQnrr
_-
( que-\' r ecto U,lf di 90e ngrroc chiEu kim tlbng
hOD (hirih r). 'fhqc hifn li€n ri6p hai igsh dl
tn -r\2
chinh la qut.r' I80a cho nen l" I I = * l. Va.v
\ I 0l
i2 = - I c6 nghia lA quay g0o iien tidli liai lh.n
," ,:,r (l ,) o,un n..,o
Lhi ciing nhu quay l8i
4
'lr'lt (rrr, ri) thinlr vecto ON (n, nt) la vecto d6i
. "ung t 6i Oll qua phen gii.r: thr'r nb{t. Thuc
hign li0n ti6p hai phep'ridi*xung OO *Ao -daf tl,i
oi nrrien lai ciutrc riar "s". {f ;)'= ,. '
i'
truc s6 chuylu sang vecto trong mEt phing. M6r
ienh d+ng (€q :) o*,, "..tu 6t , ,r rqa a0 (;1, n)
+
thdnh vceto 0.Y cti toe d0 (an j bir, t:tir * rln).
Ndu 0:c = 0 lvA a = rJ thi ffi "o toa dQ lir
(ar;r, an); tuc i*1,, - u, iifcho ndn ,r",, ([ :)
ddng nhdt v0i lgnh rr. B&n chrit cria ntri?ng dittr
k! la cnng rdr 16 rang. Lerh (f l) u,0,, r,ecro
1-
O,!1 thinh veeto c6 tga dO (n, o) (hinh.3) B6 l{
vecto[O.,\ mA ta c6 duoc b{ag ul.cir t:hi{u vuOng
,6'
L
t:t
:ti
I
;s
g6c OY*rudng truc triug dE eri veeto OM' rb
quay OM' di 90! tleo e hiEu kirn ding hd (trintr S).
v,1." r+"r,(l l) ,* -,,.0
e ia hei ldnh: n clid*
ru6ng trUc tung ) vi ( quay di g0o theo chiBu
kim dbng i,t'r, thy tlreo lhu tg d6. Ap dung lQnh
(; li' vao ri.ir rne ra ap dpng le,h (: l)
t
I
t
i
t-
!
- Iltnh I
B[n chdt nroi sn viQc k j-
$trg. Bgn c6- lht drle theo
d&y mI tn minh ph6t triBn
la lai tr& th&nh rd
.nhfrng f ki€n tr€n
di ra hqn.
lllnft 2
L*...;'€ >
I/i*ft S
i\[HIflU Lr{i tof,n b{.e cao ia c6 thB gili duoe
I r bing c*.cn rirra vB phuoag'trinh bs*. hai.
-Sarr il&y lA m*t e6 phuong ph&p itrru*ng
Ct}ng.
\ l. Idu la c6 piruong trinh bic 4 dpng
,,,r4+h.rs +c*? * r/.r'1e = o. rr* o
rnA tbn lq.i mQt e6 i f 0' sao cho rJ : bf,. e = ak?
thi ta co th'b dAr yB phuoag trinh bAe h:ii,
Phrong tririh riJry gq'i ln phtroag lrinA llil, vi'
ndu clbi biln;r - TtlS tlii phuong trii:ir theo r1
iroan toirn giiug phrrirng Iliuh thecl it:.
'f,a gi.{i ;riruoug tr"!t}r lhi nlru satr i
Viir.r=0 lih0rg ptrii i& nghi$m ciia plir,rng Lrinh
{n6u lrrli lai ta cd e = 0 rniiti,.n€n chia 2 I'S
ciro e-2 .,! 0 ia duoc :
u.rt2 .- ir.r.* t: t .{1 + *"t'= o
'.E &'-
Thu-v el :.: |r[, e = u?rz t,& nh6m e rlc s6 trgng
cirung:
, atrz +Lalnz) +r(* * 1/r)*c= o
n5t i = s'* hit,, kiri dd 12 * xZ + ?"2 ltz +- Zh,
ta c<5 oi2 '! bt + r: * 2e1L = 0"
Chi a-j Ndu plruong trinh cd dgng/(rz + \rzxz''
* l- Llr): 0, tiri t16i bi6n I - r :h Lir ta ddn
ptruong trinh vE d4ng phrrong trlnh b&c hai.
Thi cltr 1: *i&i phuc,rig trinh 14 + .t :5r(:rz -?1.
Gi*i: nt& diir ngo;L,.: vi chuybn v6 ta duqrc
' ,.4.-sr3*Ioc*{=o
D*v li phuong tr:inhltii,virJ = 10=-2(-5)= *:tr,
e -4=( - rlr. i :( -z)zu. Do vfr-v ta giii ;,irrt
salr:
Vin = 0l<.heug phiri.i{ nghi$m c.rra pirrrotg
{riirlr dh ciro iu6u tr'&i l4i ttr c6 4 * 0 vn 1j},
*+"i. *t,iu hei ud cho n'2 ua nnOo, eAe s,5 h+ng
1a rtuqe
x**41:12*5(r-2l,,)ao
.DLt t = x * }tr, khi dd tz * rz * &!x2* 4 r,d
tac612*51+4=0.
Do a*b* c = 0 nOr: phuung trinh c6 nghigm
f =1,1=4. Irdu l=l,tac6 e -Zl r:I.+ftz
*,r -2=0. trii a -b * e = 0, n8n pliucugtrinh
e6 nghi$in J;a*lr fi*I' NSul*{, la e6
*: *1lr: ;a 4 e* rQ *,1* * 2 = 0r 6 E 3 * d;*-,
:rir..1:.1.:::i I a -.
r,fr *iimrx'r'HINH
Thi da 2. (tl+ rhi vi+ dai ho* khdi A n*'m 1976).
Tlm tit ci er,: g6c ?. sr,,r cl:rr' rdi lnoi c = U,
ta cir
(*2 .F :ic?) + (r *. 3*ina)(r * l/-ri * 3sina ) 0
Girii: $iit I = r; -1- tr/.r, Lhi d6 vi r.1/.r ) 0 nan
lIi = lrE{- rlc[ = ;r;*r/!ri >zll;llllil * e
Dilu dang lliiic ^nltrrtr*(l ttr,c lttri vir ulri klri lcl=l
tray r:-"-Fi ; t2 =' .rz +- il.e?-t- ?- 'l'ha-r. viro b5t
dftplr tltii: [:t tluot:
,z +- (1 - il.sinslf * i:jsinn - :> 0 Y ltl*2.
Yd 116i lir tam th#c llAc irrri tii-.o l, lqi c6
ir r- 0 * r: - ti, rrfrt i:irn []rrrc irir nghiQm lir I = 1,
I = 3cinry" - l. Blit yrlrrrong irirrh dirng vrli inoi
gi6 tr! I nirrn ngoAi hai nghi{m, uhung vi ltl } Z,
n&n I * 1viri -3sine -2 phiri n5.rn trong hho&ng
(- f,f l. Hibu nhien i = 1 tlrrrtic irhciilg (-2,3);
nghiinr i : 3sina - 2 tiri-rLrc khoing riry, n6u
-2<ilsinc*-2<(2 hay 0(ilsirre,<J4, suv ra
sina l:> 0, 2A'r .( ,r < (21'+ th.
Thi cia 3. (lrril tlii iihdi h, Dai hoe gi6o dqc,
rniEn Nanr n{rn Igc i9i5-1'976). Chiriig nrintr
i'5.ng iriFu'thr'rr'
.'t1:r2,y? + y2rrl-i - Err/q"r rl,'r) F t0
khfing {m voi mqri giri tri iliue e r}a r: v{ g (hAc
gi{ tr! 0' ..*
G',i,. E{r z = 3'\n2lg2+Lf ln2)-8{.1t1+glr\+t0.
Ta phfii chirng nriirh r&lg Z);0 -7 r, g+{J,
Dgt i = xlu + alr, ta c0 lli ) Z, aeu bi.rrg riiy
rn ltiri rl,J = qla * )t- tr, tric !{ . = :L g.
1t t{.i Z - ltt2 * Sr' i { ): ij '* illl>'t,
Z' = -*i *,8l*-{) iitriil = --.'- .'Lap tring bidn thi€n
J
D.r tirr Z)> Zorilr:,, Yt ltlb2.
II. Ndrr phuoiig lrini, r() (lang {ftr) + a)a +
+ (lir)- ii)a-c ttri tlat /(r)l-c ={ * n, I(:'} * tr =
:l -cv. t G, : (a+b)12, ciE rirra vE ilaug ddi x(rng
(t+a)1 +(f -cr)4-c lray 214+l2a2t2l.zaa-c:tt
(phuong 1r'inh trirng phuong), rbi dua vb phuong
lrinlr bflc bai 2Xz + nxzl*4' 2a* * c=0,trong d6 X =
*.8'\
- I d. V-
T' hi tlu 4. GiAi phuong trin h cus4,r,i ( t * cosr)a= I
' Gi&t. Yi (l -eosr)4:(cc,-sr - li4, nOr: te c6
cos4+ -r (.,rr, *- {}n : ,.
Dtrt cosc =t+1i2, cos,x:1,*f * 1/::, tu rItrrl,c
(t+tlz)4+tt -tlz,t!- I iray :fa..l-3/rt r/g=1
${t tz =.X, iihi d6 0 < X < gi4 vi ra cr}
lx2 + ::i"\ * T,/8 :. o.
'Phucrng trinh r:r'i fighrprn -f = l/4,I = * i/i
( 0 (loai) =+ 1 : +-ti,:. o
NdLr I = I/2, cos.*: 1. .r.= Zrit6. lri irtr!l.i!n.
Ndrr / - * li3, co,s.r ---, r)..r. : t;t2t!;r',J,i n{tr.rOn.
III. NhiBu khi ta d<;in rn0t sd nglri6nr hriu tir,
rdi hii birc ph*ong rrinh r.i) bne hai. tllri y rAng,
rr0u tbng cic, hQ s6 e ira phuong trir:h irirng '0,
thi phrrong trinh c6 rn6t n-ghiOrrr'r = I, con ndu
trSng c-Ac lig sd Lrric nllu lring rang "i.. l,+ ".i
l-LC ]6 thi phucr,"g trii.h c6.-nguiFnr .r. = _ i.
lidu khdng th6a mIn irai cliiu t,ign rrO,i thi ia
rloi.n'nghi$nt hfl.u tt lhrr sau :
Tinr r du&i dgng o = p,! q,{,tr, 11) : 1,
. 'l'hay vao plrrron,t lrinh or,t;1,il,o-l r- .. -L
t os-1r * orr : fi r's qlrrI rli)ur:-,rrrirr sd ta rliioc
ogpn * e)pn'tq + ... + uo.-,p,i-, f rrr; ='u.-'
^. \rt rr sri hang di.rr clrirrr lt, rr+n clril,. lr0-t clro p,
0 cfrng chia hdt clro p, rlo v,i-v ar,{o ... p. Ilo
!y;,i:t..duy t.a ao I p._i'u.urg tr ia ,,,1,'rr',," . ,;.,
Ya:' ndu phtrrrng irint, .ait; -1-- a, in--i -1-.. ... +
* an-l;t; * ao:0, vr"ri 116,.rit."... nn nq-uyen, "d
nShiQnr hfl,u ijn" *: 1t!t!. (;,, {i =. t"rt,i-f'ta. u.i"
cria ,6n, con 17 Ia rr*rl ,_.in ,r,r.' Dig blf t ndu oo-
: ,r_ I, thi nghii:nr hi.u ri ia lghii:m'nj,,;Cn v,
la uoe cria I':f s6 tu do c1.
,,'llrol$ thi rll.r i. ri t6ng ,.,, ,., 10 si bic chdn
Dang tdng eic. lid si b+c iri l_ri1g i. n61 phuong
trinh cd nghi{nr i:: - i, r.a c,j tli} vi€it
(r + 1) {i,3 "* 6rt +l 6r, + j; -, g
vi.rlt *S:l?''i- 6ir.1. 4 lihiinli ro rrgl.ri6nt :r=+ l,
ndn ta x€t c6c rishiirr liit tr I.i: li r/:,.: ir-Shi+rn
nguv0n (vi cro = 1) r.i l;i iiic rr$. ,:ri,, .f , iiic ii
|.rL
r - If J. t+. i_]* "u = 2 ia ilghi+ilr.: g._:4,r-
+ 12 + 4 ={.i, ne$ co t}tb ,".iit
(z + 1) (c - ?) (,: - 1,.- 2; = ,,,
cliu, ttr! rluoc 2rB * IJ.r:2 - 3 r: *,2 = 0. Do u *'b *
Sc + d - 0, n€n phuong tlinh cb ngirii.m;,= I,
vA c6 thb vi6t
(t * l) (2,.' + Er * l) *11
Do v&-v plruong irinh eritr c6 2 nghiEin x x -- l!2
vi.r'=:2.
\dtt r == -tlZ, fteory: - l'2 ri hi: cd nghiQrn
(-r'r. -t/2)
r = + 1. la eri ll : - 2 rir hQ co nghiQln (1, ^ 2)
:u = - 2, ta cd !!:1 vn hg c6 ng;hif rn (* f, tl.
Tlti dtt 6. {Db thi viro Dai lr,.,r: s,r irhan-, th..irh
phti }tb Clr{ lfirrh nim hgc 19?6 - 1$??).
,' t;iii phrrsog trinh 4rt * Br * 1 = 0.
6,rii. Ta hii hdc phrrcrng rrinh bing cfeir tim
l, nghi0rrr hiru ti r a pl11, (,n. rt) == I ; rr6e ciia p li
;;. I. criir q ln il, t l. + 4. n0n n6u phuong
trinh cd nghi{rn htru tj., thi eh?c6thB Ii - = ;[l.
i 112. * /4. Phucrng rrinh l.h,jne cri nghiOnr
;r = I. r,i tbng eAc hg sd bina Z+A, cirn tbng
r:5c hQ sd bdc ch5n tt l; b&rg l6ng c6c h6 s6
b.1c 16 li I, n6n phuong lrinh c6 ngtiieni .r::- i
lA co tht vidt
{ic + 1) (4.rt
{r * 1) (:*
\riy phucrng trinh
: 4.r J" lJ;: i) ia3'
*l)z -0
ed nghi0nrc= *7,:r:1/2
(kep).
.
t'h[ du 7 . GiAi phuong tqinh
dos?.rc *.ns23.r, =B cos32r.
Girii. Theo cOng th&c cos?or: (1 + cosle)/2 va
cos3fl = 4,:gs3z - $cosa. [a ,:ii
(l + cos2r)/2 {- (r * cossr)/t = 3.o.?1,"
1 * (l12)eosixi + (ll2)cos3 . 2r = J cosz2c
3cou32, - 3cos22c - cos?t: * 1 sfj".
l')at cosie: = l, - t <l< [.'ta du,:rc
^r3 .,2 ,
, At -Jl- --r?l =ll.
tr'i u6c cila I lA -i- t" ,.,in I ii + l, -l.- I nr:n
ndrr phrrong trinh'cri nghiQm hiiu {i, thiairi cO
rha .iA t l. + l/2. Khi t = tlz ra cb r/+ *sli --,
- 1,2,.i- [ = 0, ntu phrrong trinh e o nghii.rn
t = l./2 ra e6 thb vidt
2(r-1,2)$2*r*1)=0
vl i:riruong_trlnh con c6 nghigrn t*b:! {)lZ,
,=u *r/s )lz (to+i)
A'€u t : 1f2, cosls=1f2=cost;r3 : Zr --l-*lf +
* Lkr
do r-$.,i, phrrnng tr.irrh crin
'fhi du 5. {DA tiri .niiin
trin h
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