.l-l-',l

,.-lI .--.1-_ :

--ll;

t--.t_:i---.jk-;l

Tir.y theo c6ch. vi6t cta mQt s6 tU nhi&r ta c6th6 x6c dinh s6 cht s6 cua n6.f . Sii tr; nhi6n vi6t durii d4ng diy sii theomQt quy lu$t nio tl6Dti tinh si5 cht s6 cria s6 t.u nhi6n d4ng niy, tac6n tinh s6 cht s6 cria c6c s6 c6 1 cht s6, 2cht sri, 3 cht si5... rdi tinh ti5ng c6c cht'sti tl6.*Thi du l. Viit tdc

.sii nr nhiin ld li1n fi6p il'I dn 20ll ta duoc s6 A - 135791 1...2011. Hoisi A c6 bao nhiu chrt s6?

. t."LOt guL ^et

cac so leTir I -+ 9 c6 5 cht s5.Trill-+99c6((99

- t r) : 2 + r).2 = 90 (cht s6).

Tri 101 -+ 999 c6({eee

-toty : z + t).3 = 1350 (chr sii).Tt 1001 -+ 2011 c6(zot t

- toog: z +t).4 =2024 (chr sri).

Vay sii I c65 + 90 + l35O + 2024:3469 (cht s6). O*Thi du 2. viit coc sd tu nhian fiin nhau ki ti21 ,22 ,2r ,...,210 ta dtn/c s6 B--24816...1u3741824.Hoi sii B co boo nhiiu chtl s6?Ldi eilrtL Xet c6c sti lfiy thrla cria 2Tri 21 drin 23 c6 3 chf s6.Tir 2a -+ 26 c6 2.3 : 6 (chii s6).Tt 27

-+ 2e c6 3.3 : 9 (cht s6).Tit 2to -+ 213 c6 4.4 = 16 (cht sii).Tt2ta -+2t6 c6 3.5 = 15 (cht s6).Tiu2t7 -->2te c6 3.6 = 18 (cht s6).Tit 22o -+ 223 c6 4.7 : 28 (cht sri).Tit 22a -+ 226 co 3.8: 24 (cht s6).Tir 227 -+ 22e c6 3.9 : 27 (cht s6).56 230 c6 10 cht s6.V{ys6Ac6 3 + 6 + 9+ 16 + 15 + 18 +28 +24 + 27 + 10 : 156 (cht s6). tr

TORN H9C'qudi@

ffiedNffisdffisdt1!

CUA MffiSOTUI\$I{m\n

2. Sii tr; nhi6n vi6t du6,i dgng lfry thilaDi5 tinh sii cht s6 cua si5 tu nhi6n dpng ndy, tav{n dUng diu hiQu sau.Sd q nhiin a, vi6t trong hQ thQp phdn c6k chii sd khi vd chi khi tr-t

I'tOt HUONG CHUNG MINH* *h*fuq rriltc *,erq fl*Ot n*c

rong c6c ki thi veo lop 10 THPT chuy6nvd trong c6c ki thi hgc sinh gi6i rOn knipmidn cua ci nudc, ta gN rilt nhi6u bdi

CQng theo v6 ba bdt _q+b+c (tlpcm).2ab 2bc 2caDEng thirc x6y ra khi vi chi khi a = b: c. E*Thl drt 3, Cho a, b, c ld ba tii duorg,chtlmg minh ring

abc abc abcd +b3 +abc b3+c3+abc c3+a3+abcLdi giAi CQng hai ,i5 BOt (l) v6i abc, tadugc a3 + gt + abc >- ab(a + b + c). Suy ra

1

a3 +b3 +abc ab(a+b+c)Tuong tu c6

s

TmQitoan chimg minh bAt d'ing thric (BDT) dangphdn thric. Bni vi5t niy xin gioi thiQu mQthu6ng chimg minh BDT d6 bing c6ch vfndlmg kiit qui cua bii toifur quen thuQc sau:BAI TOAN. Cho ba s6 duong a, b, c. Ch mgminh riing a3 + b3 > ab(a +.b) (1)(Ban tlqc t.u chtng minh BDT nay).*Thi dtt l..Cho a, b, c ld ba s6 arong,ch ng minh rdng

a3 b3 c'-+-+->

ab+ bc + ca.bcaLdi giAL Chia hai vii cua BDT (1) cho D ab+bc+ca (dpcm).'bcaDing thrtc xiy ra khi a: b = c. D*Thi du 2. Cho a, b, c ld ba tii dnong,chimg minh riing

a3+b3 b3+c3 c3+a3- - +:-----1 +- -- 2a+b+c.2ab 2bc 2ca

Ld,i gidi Chia hai v6 cua BDT (1) cho tich2ab,dusc dll >!

Ldi gidi Y a a, b ld cic $ duong, ta c6a2 +b2 > 2ab; a3 +b3 > ab(a+b).

Nhdn theo vti hai b6t eing thric tren, thu dugcas + b5 > a2b2(a+ b)

Chia hai v6 cua (2) cho b2, ta duoc

**O'ra2(a+b).h2

(2)

Tuong t1r c6

!t- *"r,6116*"); I*o, >cz(c+a).c2"a2C6ng theo v6 circ bdlt d,ing thric trn vd thugon ta duqc

{. ) - l, a2 b + b2c + c2a (dpcm).D'Ding thrlc xiy ra khi vi chi khi a = b : c. A

*fhi aB s. Chting minh biit ddng tk cas b5 c5 a3 b3 c3-+-+-z-+-+-b3 c3 a1 b c a

voi a, b, c ld ba s6 dnong.

Lnn gidi Chia hai v6 BpT (2) cho D3 rtuo. c4*61246*b\=" +o,.b3 b' b

Tuong tU c6b5 ^ b3 .- c5 - cl_+ct>_+bzi

-_- +a2 >_+c2.crcata

CQng tlreo vii cric BDT fiOn vd tru ggn, ta tlugca5 bs c5 a) b, c3-+-+->:.*-a: (tlpcm).b3c3a3bca

Ding thfc xiy ra khi vd chi l&i a: b: c. O*Thi du 6, Chung minh biit ddng thilrc

a3 +b3 b3 +c3 c3 +a3c2 a2 b2

vtri a, b, c ld ba sii &nng.Ldi gi,rti Chiahai vti nOt 121 cno a2bL ta d'vgca3 b3-+->

o+b-b2 a2

Tucmg lu, ta c6b3c3-c3a3..._+-_ > b +c:

-+--_>c+a.c' bt a2 c2CQng theo v6 c6c b6t ding thrlc tr6n, ta duo. cd+E U+C C+d

-*

*- - >2(a+b+c) (tlpcm).c2dl*Eing thric xiy ra khi vd chi khi a : b - c. O*Thi dg 7. Chting minh biit diing th.tc

ab bc L'aas +b5 +ab bs +cs +bc c5+as+ca-"

voi a, b, c ld cac sd drcng thda mdn abc : 1.Ldi ginl C6ng hai v6 cria BDT (2) vd ab, thlduoc a5 +b5 +ab> a2b2(a+b)+ab. Suy ra

I

as +b5 +ab_ a2b2(a+b)+ab,. ab I c

as +bs +ab ab(a+b)+l a+b+c(vi abc : 1). Tucrng ty. ta c6

fuacabE +C +bc- a+b+c' cs +as +ca- a+b+c'

CQng theo v6 c6c b6t ding thrlc tr6n,

TUMOTBAITON'-

i5 tiiit trqc tl4t k6t qun cao, ngoai viQc lgachgn phuong ph6p thich hqp, 6y hgc

sinh lim trung tam, nguoi thAy cdn phni gitvai trd chri dao nhem ph6t huy tinh tich cgc,chri ttong, s6ng t?o, ph6t tritin tu duy chohqc sinh.

Trong s6ch gi6o khoa (SGK) hiQn hanh, c6 r6tnhidu bdi t$p don giAn nhrmg niiu bi6t khaith6c, md rQng c6 th6 tim ra vd s6 bai to6nmoi. Bdi vi6t niy xin gioi thiQu mSt trong ritnhidu c6c bdi torin nhu v$y 0 trong SGK.

BAr TOAN Mo'EAu @di tQp 71 - sdchBdi tQp Todn 9 - TQp hai).Tttn. giac ABC noi tiip drnng Irin lOl. Tiepnvin tai A vcri drdng trdn cdt tia BC d D. Tiaphdn gidc g6c BAC cdt iludng trdn (O) o M.Tia phdn giac cua got ADB cat A.V o l.Chtimg minh ring DI vu6ng g6c vrti AM.Ldi sidi (h. t).

Hinh IGoi Nli giao di&n ciaAMv,bi BC,tac6

AND =sdAC+sdBM sdAC +sdCM

2

-sdfrtu -675.2

TORN HOC'cfudife

Suy ra tam gi6c,4ir'D cdrtAi D, c6 Dlli tluongph6n g6c IDI(n&r DI L AM (dproa). ON\Qn nit. Ntiu ggi P vd Q ldn luqt li giaodi6m cria l.B vit AC voi DI thi tam giirc APQcdn tai l, khi d6 tu gi6c lPflg ld htnh thoi.Kiit qun d6 thay d6i nhu th6 ndo kni dii,im Indm 0 bn ngodi tludng trdn? Ta c6 bdi torin Isau

EQS c6n tai E,lai c6 EP ld phnn gi6c cua g6cpES nn EP h trung tnJc cria QS (l )Tuong tg ta chimg minh dugc MQb 6uimgtrung tryc cua PR Q)Tir (1) ve (2) suy ra nr gi6c P0RS ld hinb thoi(

of; ru rur&,r mw r0 o mry r" HA rSCAIL(2,5 didn)l) Cho bi6u tntrc ,e=E+l. Tinh giri trr

Jx +2cria bi6u thuc I khi.r : 36.2) Rrit ggn bi6u ttlic

(Thdi gian lhm bii : f20 ph[t)

(voi.r 2 0, .r + 0).3) V

GIA TRI LON NHAI GIA TRI NH6 NuAT*ngf*nxx;kn&nop

r. r,i rnuvtrCho hdm s6 71"1=*'+bx+c(a+0).Niiu a > 0 thi hnm s6 /(r) nghich bi6n trOn

( -t\ -r ..ikhodng | ---;- l. tl6ng bi6n h6n khodng\ za)

t/L\/r\li*)'u fl;)rd si{i tri nh6 nr6t "tuhdms6/(x)trcn R.N6o o < 0 thi hnm s6lx) nehich bi6n trCn

(-h \khori,ng I

-:;+* l. d6ng bi6n h6n khoing\2a )/ _A\ / _A\l-*';)'u fl;)re si6 tri ron nhat criahnm s6flr) tr6n R.NhQn xito Ki hi0u max /(;r) dnng d6 chi gi6 tri lon' D" "ntr5t cua ham s5 f @)tr4'ntlq D.o Ki hiu min /(x) dune d.i chi gie t{ nnd. o"', "nh6t cria hem si5 /(x) trn t{p D.Cho him s6 71*1= *' +bx+c(a * 0).Ta c6cric ki5t qu6 sau:

TOAN I,IQC'qufiEa

xeuvEn lnx v0(GV THPT Nguydn Binh Khiem, Binh Dlnh)

Trudng hqp 1. a> 0.

l) Ndu -: s a < P thi mnf (x\ = f (a) vitryxf@)= f(0.

,-h2) Nu a .9

) u6,, ,.t'.: p *t ggf {n= t(*)*fl ffi "f

(r) =

max{ f (a); f (fi!.

Trudng hqp 2. a < O.

"-h1) NCu ;

*Thi dq 3. Tim gia rri nho nhar. gid tr! lonnhdt ct)a hAm s6

"f(:r) = si{r';r - cosx + 5.Ldi gilfii Ta c6./(r) = sin2x - cos.r + 5= -cos2, - cos, + 6.D{t cosr= 1, ,e[-1;l] thi-cos2-r - cosx+ 6 = -t2 - t + 6.X6t hdm sti g(t) = -t2 -t+6 vot re[-1;1].Ta c6

-1.:4=-l .t. do d6 theo TH22a2ming(r1= minlr(

-l); s(l)l= s(-l) = 4 vil.l;llf l\ 2s _maxs(,)=s[-, )= o. uouo

)2.* Thi dU 3. Tim rn dA phtxng trinhtan2,r+cot:,t = r?(tanr+ cot-{) c6 nghiim.Ldi giAi D4t t =tanx+cot.r.Ifti 2.PT dd cho trd thdnh t2

-2 = mt .Ta c6 thd giei tiiip bing c6ch xdt mi6n giri tri

.1 ncfuham s6 t=' ", va ltl>Z cirrg co th6 ttmtz d0 phrng tinh t.O

Chrt j, L a) MOt s(i PT can ditr = cosx-iinx= O*r(,.X) n vr = sin.r-cosx = r[rr"(r-ol). xr,r d6 v6n c6t .l-Ji;Ji)nhrmg cosxsinx =+b) MOt s5 PT cen d{t r = tanx - cotx .

sin-r cosr sin2x -

cos2 rKhiil6 r= -'-"---i-'- =---',cos.r sln, cos.rsrn,_

"

cos2x-sin2x ,rcos2x-

-= -2.cot2x.2sin.rcos.r sin2x

Vi cot2.r e (--co,+oo) ndn r e (--

vi,.[-Do tl6 r=

Khi d6 ham s6 da cho xic ditrh ve kh6ng arLhon nira

y2 =2+2(cosx+sinx)+2@.

' rt\Det, =cos"x+sinx=.Ecosl x-- I.t, 4)

Hon nta y2 =2+2t+ +2t + 4.t' -l1y2 =2+2t +2 2t2 +2t _t = f (t).

TOAN HOC'qudi@

r a c6 f (Jiy - e6 + Jb, r$:f =,. .6.Mit kh6c

"t'ttl=z*ft,u2,_1r0,o,.[$'f )Turl6 y.6 =l+Ji vit y. =z'!t1 /[z.aChrt j' 3. Trongcirc PI dang 3 va 4 ta tlidtra vioc6c hp thric luqng gi6c co banoda+sin2a=1,tana.cota = I dii dat in phuThlrc ra c6 nhidu PT phdi dga vdo nhtngc6ng thirc luqng gi6c phirc t4p hon m6i chen

Ldi giAL Chn f rirng sin3.r = 3sin.r - 4sin3 x sina e [OJ] n6n cosa =*Thi dg 7. Tinh tich phdnl =1 tIl' d,jl + cosx

= sinx(3 - 4sin2.r) = sin-r(4cos2x - l)nOn bing c6ch tl6i bi6n x thinh biiin I = cos.rta c6 & = -*inrdr vd do tl6

r ='f!A,--'f!:t\,j t+l j t+ll- l-l

=4lU-tW+ l-41o o,_tl

= 2t2 - 4t +3lnlt + r ll = -2 ..3112. g

3. PHI.IONG PHAP L[IqNG GIAC H6APhin cdn lai trinh bdy mQt s6 co sd tt6 chuy6nbii torin tlai s6 vd bdi toiim luong gi6c de biOtc6ch gi6i.3.1. Dtm vdo tinh chit Mn i tu.orA gtuic co ban.. Ta bitit reng hem s6 y = g6s, x6c ttinh voimgi x nhQn gi6 tri tr6n do4n [-lJ] va tuanhodn voi chu ki 2n.Do d6 vdi mgi z e [-t;t],chgn duo. c a duy nh6t sao cho cosa = z vdia efo;xl.

JrThi dB 8. Gidi phrong trinh,:a1ft-v:Y = xJ2lt-r').

Ldri gidl DIt .r = cosa voi a e [O;r] . rhi

*o f, = cosa, f, = sinc,, a efo;zn)t'nyx= 4cosa, y = 4sha, a ef0;2n).Tuong tp lt z2 +t2 =9 suy ra chqn dugc /duynh6t sao cho z=3m,f,t=Kn{, pelan).Thay vdo xt+ yz>12 c6lZcosasirB +l2sinacosp > 12 haysin(a + p) > 1

= sin(d + B) = |

= a+ B =I + k.2n va k eZ ndo d6. Th6 thi,2

x+z =4$,a+3cff'P =zlcma+}ina= 5o(a-p)hons d6

"o", =! .

"ir, =1., .( 0,I) . sru5' 5'\2)m grd fi lon nhrt cua x+z ld 5,

sAr nmOnv mCIAN rn0 rleuErureuvEr xAr cxAu

(Ph6 Vr1 truSng W Oieo dqc Trung hoc,Gien ddc Chudng tinh Phdt triSn Gi6o duc Trung hoc, Bd GD&DT)

DE TRAC NGHIEM KHACH QUAN(Thtri gian ldm bdi: 40 phrtt)

C6u 1. TQtp xitc dinh cua him s6 y : siu ldr-l

A. R r. m.tl1l 'lzl'

Cdu 2. Hitm s6 ndo sau tl6y nghich bi6n tr6n

*"r'(o'1)'A.2 : sinx;C.y=tatrl-;CAa 3. Dudng cong tr6n hlnh ndo sau tlAy lddd ttr! crla nam s6 y = ,;o, Orr,

-Qt khoing?

CAU 5. Trutng THPT Theng Iong nim nayc6 120 hec sinh kh6i 10, 115 hac sinh khisi I Ivd 110 hgc sinh kh6i i2. N6u cri m6t hoc sinhcta Trudng THPT Thing Long ndm nay thamgia dqt khao s6t thi c6 bao nhi6u c6ch khricnhau?4.225;c.345;Cdu 6. Tit circchfi s5 l, 2, 3 vd 4 c6 thii theftlfp ctuo. c bao nhi6u s6 tu nhi6n c6 4 cht s6kh6c nhau?A. l; 8.4; C.l2; D. 24.CAu 7. Tir circ cha s6 t, z,I vn 4 c6 the thenhhp tluqc bao nhi6u si5 t.u nhien c6 2 cht s5khdc nhau?A. 1; 8.2;A. l; 8.2; C. 12; D.24.CAu 8. N4rdi ddng xu c6 hai m4t ld s6p (.9) vdnCua (1V). Ntiu phep thir ld gieo cing l c haiddng xu thl ta dunc khdng gian miu lil. {s;N};c. {ss;sN;ilN};

n. {ss;rw};D. {ss;sN;Ns;Nnr}.

CAu 9. MQt con sric sic li mQt htnh lipphuong c6 6 met, m6i m{t dugc khic c6cch6m, s5 chirm tr6n m6i m4t li khric n}au, tuI tti5n 6. N6u ph6p thri li greo mgt con stic siic

GIdI

Ki r LoP 11 - NAM 2011

CAU 4. Tdp nghiQm cria phuong trinhsinx =- I lda. {{*mi,*. z; B. {_!+k2^\*ez:[2 ) 12 ]'C. {n+kn\,keZ; D. {n+k2r,},keZ.c. mr$.r,),kez; D. R\{kr.},kez.

B. y: cosr ;D. Y = -"ot*.

8.235;D. 1518000.

c. t2; 24.

roRN HOC.. .......4 ------.%wtd4

'cfuoiUe V

vd xlt M ll bi6n c6 "mdt xuh hiQn cia sicsdc c6 s6 chiim ld sii chdn " thi

C6u 14. Cho hinh binh hdnh MNP). Phfutinh ti6n theo vecto ffi Ai6n di6m Q thdnhdii5m ndo?A.M B.N; C.P; D. Q.

A.M= {2;4;6};C.IuF{4\;

B. M= {6};D. M:12\.

5*3 -3(r+3)lLt=- 1

2{n +3)5r+3 _3n +3U--t=-.

2n+1

Cda 10. MQt diy sti c6 si5 h4ag t6ng qu6t ldu, =t' ^

tn , voi r = l, 2, 3,... thi si5 hang"2n

thri n+3 cua diy t16 liA. z-, =

5'-3(n+3), *.

2(n+3){' 1'

C. u-.,=" "" +3: D.2n

CAu 11. Ddy s6 nio dudi t16y ld mQt dey si5giim,van=1,2,3,...?t. u, = (-1)";C. u,=L;

n

B. u, =2" '

D. u" =J.

Cdu t2.N6ttmQt c6p s5 nh6n c6 s5 hang

C. Kh6ng cit nhau thi ch6o nhau.D. Kh6ng c6 tlitim chung thi ch6o nhau.

Cha 20. I

B. M6i khi thuc hien li6n tiiip ph6p vi tu tAmO ti s6 /. (vcri /r* 0 ) vi ph6p quay t6m O g6cquayathi c6 mQt ph6p ddng d4ng.C. M6i khi thlrc hiQn 1i0n ti6p ph6p v! t.u t6mO ti s6 k (voi t+ 0 ) vd ph6p tinh tii5n theovecto i thi c6 mQt ph6p ddng d4ng.D. M6i ph6p bi6n hinh ttdu ln m6t ph6p ddngdang.

Cdu 29. Khing dlnh ndo sau tldy ld

Glru6'n [!Gho [ithit6t nghiOp I}IPIvi thi vdoOai hoc

ssrwweAuoP TRONG DE ITIITUYEN SINI{ DAI HOC

,\2I{HOI A. A1, NAM 2OL2ouoxooUc xAo

GV THPT Huong Khe, He Tinh)

cau s. (Khi;i A - Al ndm 2012). Cho hinhch6p S.ABC co dir ABC la nm gidc tliucqtnh o. Hinh chiu vu6ng g6c c a S trin nqitphang (ABC) lit cli1m H thuoc canh AB saocho HA =2H8. G& giiizr dLing thdryg SC ritmQt phdng (ABQ bdng 60". Tinh th0 tich criakhoi chop S.ABC vit khoong cach gi ct haidLrong thdng SA vd BC.Hvtng ddn c Tinh Vs.nc.Gei 1ld tnrng di&nAB, g6c gita duong thing 5C vd m{t phing@Bq h HCS. Xet tiln giirc HIC ta c6ru =i:rc =S=HC =*il, + IC; =&=

SH = HCtan60' = a

Goi 1( li trung tlidm AE,kt: HFIIBK (F e AE)vit III LSF= HI L(AES)*d(H;(AES))=H1.

-

. d(B:(AES\\la co -:-------------3_d(H;(AES))

BA3HA2

X6t tam gi6c w6ng SHF ta c6l=1*l>HJ=a

HJ2 HSz HF2d (BC,LA)

= d (B{AEEI =|a e *eua> sCAch 2. (h. 2).Xric tlinh mQtmit phingchrla BC misong song vdiSl nhu sau.I;Ay C'd6i xringC. qa B; A'tl6i xung I quaB; H, e BA,;BH'= AH.K6 BB'/IAS;BB'= AS thitan gjirc BA'C'd6u; IBB'^S lahinh thoi canh a;B'II ll ffI

= B'Il I(BC'A\

= AS il(BCB') t

ffo"'*

s,* =* -r,.,* =lo. t,* =ff g*1.

c Tinh d(SA;BC).Cdch 1. (h. l)..Tax6c tlinh m[t phdngchta Sl song songvoi BC nhu sau. K6AE//BC; AE=N=athilCBf ldhinhthoi c4nh a nOnBCil(AES).Dod6 d(SA;BC)= d(sA;(AES))= d(B;QqES)).

TORN HQC'cludiUd

Hinh I

Itr ". \'r ---1.

il)i,tl t

t Y v+'-

= d(AS ; BQ -- d (AS;(BCE)) = d (A;(B C' B \).

Mn kh6c d(A';(BC'B\) _ A'B =3 . "u, ^' d(H':(BC'B)) H'B 2'

1d\AS;BC)==d(H';(BC'B'D. Ke H'tlIA'J,trong d6 ./ ld trun g di6m BC' ,1r1 ,'1 = "f ,ru=21te'=ft,tao6 Hl LBC'.Ke HKLIB'thi IIK L@eB). Trong tam gi6c vu6ngIH'B'tz c6

1 -

1 + I =

H'K'=7a'H'K2 H'12 H'B'2 24rrld6d(rs;Bc) =ir'-=*

x6c itinh thotu1g c6ch tu m$t Ai6m U6t m trentluong theng niy tln mlt phlng kia. Th6ngthudng ta van dlng hQ thr?c luqng trong tamgi6c vu6ng vi k6t qrue sau:l. T* di(n OABC cd OA, OB, OC itoi m|tvu6ng g6c thi hinh chiu cia O tAn(ABC) trilng voi truc fim tam giac ABC. Khid6 d(o;(ABC))=oH va

1111oH2 OA2 OB2 OC2

C6 th6 tinh khoang c6ch girin tiiip th6ng qua:2, Cho mdt phdn7 @) vd ilvdng thing Lcdt(P) tqi O; Mvd N ld.hai didm trn L: H vd Kl,in lrot ld hinh chiiu ttong img c a M vd NtuAn @) khi d6

d(M;(P)) MH Mod(N;(P)) NK NO

Truong hqp kh6ng vfln.dpng tlugc vio tudin vudng, ta st dgng k6t qua sau.3. Cho fii drtqn OABC cd OA L(OBC), dA tun

C6ch 3. (h. 3)Chan he taa d0sao cho O=I1;Ox//IC;Oy chAaHB;Oz chfta HS.Tia Hxcht sctpi rthi o.M=NI=?aH3>nr=7Ct I3A

hinh chiAt aia O tuAndagOKIBC;OH LAK.

p&g @BQ nd6 drr@Bq

rt&Khi

n=(,,?"8),*=(t,7E = (0;a;0). Do d6

= f . o" d6 H(o;o;o); r[f ,o,o),B(o;;;o); s[o;o;, E), ^(0,+s) s'v *

'-fo),ttRfrl.dl_"k,

d(BC;AS:) = /17?1.llD = Er -=-:""'='- --.ILtsF] 8

4Nnpn x* o Ngoii ra, ci6 tinh khoang,c6chgita hai tluong thing chdo nhau ta c6 th6 guyvd x6c ilinh khoang c6ch gita hai mflt phengsong song chira hai thlng tl6. Tir tl6 quy v6

va d(o;(ABC))=oH.. Ed vin dgng tlugc phuong ph6p tga dQ vioviQc giai c6c bdi toriLn tin} khodng c6ch tli6uquan trqng li phii chgn duqc h tqa d0 thichhqp, dd hr d6 d xric tlinh c6c tga tlQ c6c tli6mhay phuong ninh c6c tludng thlng.

BAI TAP1. Cho hinh chbp S.ABCD, dby ABCD h hinh binhhixrh c6 ,48=o,N:2a, fu=tZrp; &4=,SC, S8=SD.Gqi M vd f hn tuqt ld Aung diiim lD ve BC. Heytlih d(sMiND), bi6t rixLg d(s;(ABCD))=3a.2. Cho hinl li.q phvong ABCD.A'B'C'D' c4rfi a, G littrqng 6m tam gi c BCD.lr' ld trung 6im AB. Heytitrh d(GC';A' N) theo a.3. Cho hinh ch6p S. ABCD cb 66y ABCD ld htnh btnhhifi, AB =2BC =2q,SA=3a, M I(ABCD),ABC =6tr.Hdy tfith d(D:(SBC)) theo a.4. Cho hinh ch6p tam $6c.SABC, diy ABC ld t^mgiric tlu canh a; Hai m[t phnng (V-8) ra (S4C) wonggoc voi tliy. goc gita hai m{t phltrg (SAB) viL @BAbane 60.. .ep = !,la r p e,qa t:,qo = 1.4c t o e.a c l.

- 4 " ' 7 '-Hey 6r'h d(PQ;SC) theo a.

tlllllll

i!t'=

tj:-

ToANHec.. ....4 ......"......fuwre4'cftroife V

hA ttri ruyn sinh vdo Dai hoc m6n Vdt lil) ndm 2012 vd nQi dung tap trung chri y6utrong chuong trinh Vat fi ftrp 12, chi c6 haicdu sri dgng d6n ki6n thric lop I 1. C6u tnic ddthi trai ddu cric phin. Sti luqng cdu m5i phingiting.nhu c6u tnic t16 c6ng b6. D.e thi nqnnay vO clQ kh6 cf,ng tuong duong i16 thi tuyonsinh viro D4i hgc nlm 201I, nhmg kh6 hay.^! ^ .: .r -.:56 ciu d6 liy tlim, tuc li chi odn hi6u lithuyi5t ho{c thuQc c6ng thric rdi 6p dung chok6t quri ngay chi6m khoing 35%. 56 ciu h6iphdi tinh torln nhmg kh6ng ddi chi6m kho6ng25%. 36 c6u cdn l4i cdn c6 s1r tu duy vi cAntinh to6n ddi hon, trong tl6 c6 khoing 15% si5c6u ddi h6i tinh to6n dii ddng. MQt s6 c6u h6ikh6 tip trung vio phdn co vd tliQn. Hoc- sinhphni hgc ki li thuyiit vd rdn luyQn nhidu kinlng tinh to6n moi c6 th6 lim t6t tluo. c bdithi. DA thi nim nay c6 ph6n ndi dung ve tUxoay, g6c xoay. NQi dung ndy dE kh6ng ratrong tl6 thi d4i hgc nhi6u n[m nay n6n thisinh thudng chri quan vi b6 qua, hoic chi hgcso siri n6n cflng kh6 ldm tlu-o. c. DAc biet, ddthi n[m nay c6 tli6m moi ld mQt s6 cdu h6ibi6t gin voi thuc t6 hon, vi dg nhu cAu h6i vAs6ng tliQn tu, truydn tii tli6n ning tli xa,...Day h d6 thi ddi h6i thi sinh kh6ng nhtnghi6u li thuy6t, bi6t suy luin md cdn phai c6 kinEng tinh to6n. nhanh vd chinh x6c. Thi sinhphii nit xuat sic moi c6 th6 dpt tli6m 10. Hgcsinh gi6i kdm voi viQc rdn luyQn nhidu bdi t|pthi c6 th6 dat tu 8 - 9 di6m. Hqc sinh kh6 thucsg c6 th6 d4t tu 6 - 7 atiiSm. Do phAn H thuy6ttuong diSi d6 ndn hgc sinh. nim vitng li thuy6tcflng c6 th6 c14t tu 3 - 4 tli6m.

Du6i cl6y chring t6i sE gini thiQu vd hu6ngddn mQt sri d4ng ciu h6i cria dd thi clei hqcniy theo Mn dd 196.

1. DANG CAU HOr Vt DAO DQNG,S6NGvACOCAn 1. M6t con ldc ld xo gim 16 xo nhe c6 d0crlzg 100N/m vd vdt nh6 kh6i luqng m. Conliic dao dQng diiu hda theo phuong ngangvoi chu ki T. BiA d thoi dim t vQt c6 li ilQ5cm, 6 thdi dim t +L v,it cd tiic d6 50cnt1s.

4Gid tri crta m bingA. 1,2kg.C. 1,0kg.

B.0,8kg.

D.0,5kg.

Hudng ddn Ta c6 phucmg trinh cria daotl6ng ctidu hda

x= Acos(cu+p)= u*"ff-r)^ d-r 2x. (2nt )SUVra v =- = -4.-srtrl

-+(D I' dt r lr ')rheo gi6thii5t m"ff * v)=s"

(r,(,*L\ ).2n.1 I 4) I

-^_,4._,-sinl \ ./ +(o l=50cm/s,T I T 'Ir. l

/ ^-- \hav ,4.acosl "'"" + rl = 50cm/s' r \7 ')

tntc r =ucf ,

Tt (l) vd (2) suy ra r = Itrl. Ap dsng c6ng' 5'thay gi6 tri cria ft vi I ta

ctuo.czr= l(kg).D6p 6a C ilung. tr

(l)

(2)

TO6N HOC'qudi[$ fuwd4

A.9--nrad.

c- Q =o rad.

Ciu 11. Hai dao dQng citng phtmng tiin botc6 phrong trinh x1= Z,"orl nr +11("*)\ 6r'

/ -\

uri x, = 6gq5l rr - i l(cm). Dao dQng dng\ z)hqp c{ta hai dao dQng nay c6 phuong trinhx=Acos(nt+e)bm) Thay d6i Ar cho dinkhi biAn ib A dqt gid tri cr.rc tidu thi

B. o=-!rud

D. q=-L.rud

luqt ld iliQn dp hiu dung gi*a hai ddu din' .^ , . ',:trn. gi'ia hai ddu cuqn cdm vd giiia hai ifiu trtr

diQn. Trudng hqp ndo sau ddy, diQn dp tilrcthoi gifra hai diiu doqn mqch c ng pha vaidiQn tip tric thoi gifra hai ttiu itiQn trd?A- Thay diii C dd U R^*.B. Thay ddi R dA Uc,*,,.

c, Thay diii f dd Uc*.rD. Thay itdi L de U L*,..

Hfing d6* Tr) bi6u thtc vd d0 lQch pha_1Lo--

gita I vir i: tanq = -----e-a, ta th6y rAngR

n6u u vd i cing pha thi p=6, L1. 1;u-!=o uy tat=! (tic tr6 tr6nCa'Cadoqn mgch c6 cQng hudng). Trong trudnghop ndy i =

il ='{ vi dign 6p hi6u dung tr6nzR

diQn fi0 R ld lon nh6t (U" = U".*) Nhu viyphni thayd6i C.D6p 6n A dung. tlCAu 47. (PhAn riOng cua chuong trlnh chuin)M\t it\ng c

Hi-Ou sulSt cria ttQng co bing@11

H = :!L = --: = 87 ,5o/o.q88D6p 6n B tlung. OCitu 26. Ti mot trom phdr diin xoay chiium6t pha ddt tai vi tri M, diAn ndng duoctruyin tdi d6n ntti tiu thu N, cdch M l80km.Biit drong ddy co didn to bng c6ng 80 {)(coi ddlt tdi din ld ding chiit, c6 diQn trd ti 16,, ' ':voi t hieu ddi cua ddy). Do su c6, drrrng ddy

bi do didn tai didm Q thai ddy rai didn bi noidt bdi.mil vQt cd diQn trd cd giti tti xdc dinhR). D xdc dinh vi tri Q, trudc en ngutii tangdt dudng ddy kh6i mdy ph/tt vit tai fieu thu,sau Ll6 d ng nguon didn khong doi l2Y. dintro trong khnng tldng ki. n6i vdo hai dtiu ciahoi ddy tdi di1n tui M. Khi hai diu ddy tqi Ndi ho thi cuong dQ d6ng didn qua ngu6n ld0,40A, cdn khi hai clAu day tqi N duqc niii dtboi mor doan day cd didn trtr kh6ng ddng kdrhi cudng dp din7 didn qua ngudn ld i.42A.Khodng cdch MQ ldA. 90km.

C. l35km.B. 167km.

D, 45km.

Khi /f ni5i voi r'thi0,42 =

Rrro+R,P+Rd =122R*+ R1d ti6y R7 lir di6n trd tuong tluong:

111 l1&a R R u+Rpp R 2Ra,crlc phuong trinh (l), (2), (3) vn

dugc Ry7: l0Q. Theo gia thi5t dity MN c6ttiQn trd 40O vi ddi l80km. VQy rto4n diyMQ c6 diQn trb l0O s6 dai 45km.D6p 6n D tlung. D

3. D4NG cAu HOr vii sdNG DrEN rr),s6Nc ANU sAuc.Cit 32. Tai Hd N6i, mQt may dang phdt s6ngdie.n *. Xdt mdt phuong tuyin co phurngrhang d mg huong lin rrin. Vdo rhoi didm r.Iai diin M trin phrong truyAn, vecto camng ti dang cd d0 lon cqc dqi vd hunng vi

phia Nam. Khi il6 vecto ctdng d0 dintnntng c6A. dO lon cqrc dai vd huong vi phia Tdy.B. dO lon cqc ilai vd huong vi phia D6ng.C. dO lon crc dai vd hrong vi phia Bdc.D. dO lon bing kh6ng.

Hwimg ddn Chr[ f reng khi i c6 tt6 ldn clrctlai thi E cine c6 d6 l6n cuc tlai vd ba vectoR=o -, ,v,E,B t4o thanh m$t tam diQn thu{n. VectoE c6 phuong ru6ng g6c voi mdt phing chrta

vi B, c6 chidu theo quy t'ic v[n nrit chai.Quay c6i vf,n nrit chai theo chidu O B tl6n vtheo g6c nh6 nh6t thi chi6u tith cua cai v{n

ring Rg,v = Rnr - Ruo = 40 - Rw , ta

Q)

(3)

chi ftim

Ti

Hrdng dAn Day h bdi tflp g6n voi thgc tii vili6n quan dlin ki6n thric vd ddng tliQn kh6ngd6i trong chuong trinh Vat li 11.Ta c6 scr ttd hinh vE

Do ttiQn trd ti l0 voi chidu dai cria diy vd tuhinh vE ta c6

MQ=rP,8N=PF.Suy ra R 9 =Rn, Rat=Rw.Khi N chua nrii v6i Fthi

0,4 =

TORN HOC, c[ir6iUA

v

12

nft chai lir chi6u ctla vecto E. Tri tl6 ta th6yvecto E c6 chi6u hu6ng vC phla Tdy.

RMa + R+ Rrp 2R*+R (1) D6p rin A dung. D

greertsd4

Citt .4. Trong thi nghiQm Y-dng v giao thoa

dnh sdng, nguin sdng phdt ra dnh sdng don: ., ,sd, co hmtc s6ng ,)-r. Trn man quan sat, trdndoan thdng MN ddi 20cm ftu6ng g6c vbi hQvdn giao thoa) co 1O van 6i, M vd N ld vi tr{ciia hai tdn sang. Thay dnh sang trOn bingdnh sdng drm sdc cd buctc s6ng I =1s., *i

5

tai M la vi tri cua mot vtin gioo thoa. so vonsang rrdn doan MN lic ndy ldA.5. 8.6. C.7. D.8.Hwing ddn Tr6n tlo4n il0y'c6 l0 ven ti5i meM vit N ldhai ven sAng, nen tren tlo4n nay c6I I v6n s6ng. M vfly trn tlogn rt$r' c6 l0khoing v6n s6ng, m5i khoang ld 2cm. Ta c6khoing v6n

, _hD _n*. . _)4D _5kD .\=-=zcmirr=;=1;;. 5. 10Su] rx 12 =

tr1 = Tcm.

vay s6 khoeng vefl sang mn iraf Uing ff = O.3

Tr6n

ffiffim+iemffinAr r4r odNc pnAN TRoNG no,q. H{ru co

f\6ng phdn ld khdi niQm #t co bAn trongLl nda hru co vd *u6t hign hiu h6t trongc6c bii toan hoi htu co.. Vidc xric dinh duqicdu tao cua cric chdt dgng p4an th6ng quaviQc lin hO vdi tinl chdt (nhAt ld tinh cbdth6a hqc) giirp ich rdt t6t cho viQc cring c6,nim viing vd vin dgng linh ho4t ki6n thrictrong hqc tip, phuc vu vigc thi tiit nehiep, thltuy6n sinh vdo Dpi hgc, Cao t16ng.

r. KHAr NrEMvt odxc pnAxViQc dAu tin li nguoi hgc cin phdn biQt vdhi6u 16 mot sii kh6i niqm 1i6n quan toi hiQntuqng ddng phdn vd cric ch6t ddng ph6n:

- Trudc day, ddng phin

lai c6ng thtrc tinh L cho cdc hqp chdt hiru codqng CxHy, C;rHyO, vd CyHyO7N1.

Hrdmg ildru Do oxi c6 ho6 tri 2 ndn (ns -

2)= 2

- 2 = 0, vi v{y st5 nguy6n tr? oxi kh6ng

inh hudng tl6n gi6 tri A. C6c c6ng thfc dpgClHy vd C;HyOz c6 c6ng thric tinh A chungh, L=2+2!-Y. N6u nito c6 ho6 tri 3, thi

2c6ng th&c tinh A cho c6ng thric dang

-. 2+2x-v+tU.H.U,NI ta A= '2

*Thi dB 2.Itit c6ng thtlc cdu qo vd tn cfiacdc hidrocacbon c6 c6ng th rc phdn a CCH1.Vai dQ khgng no bdng 1, cac hidrocacbonnay c6.th thuAc bqi anken Qfidn ni c6 mQtliAn kt d6i C: C) hodc xicloankan @hdn t*c6 mQt vdng no).Hwdng ddn Nhu vfy c6c ttiing phdn c5u t4odddyh-

Ctic anken:

CH2:CHCH2CH:; CH:CH:CHCH:;but-l-en bu!2-en

CH2=c(cH)22-metylpropen

- C6c xicloankan g6m:

D-cr,metylxiclopropan

*Thi dB 3. Vid c6ng thtic cdu qo vd gqi ftn, ,,1

cdc chdt cd c ngc6ng thrir phdn ni CaH19O.Hudmg ddn Can nfr4n tfr6y cOng thtc tren c6thnnh phdn CnH2n+2O voi tlO kh6ng no bnng0. VSy cdng thrtc ndy irng v6i cic ch6t thu6clo4i ancol ho{c ete no, don chic mach hd.Tir. d6 vi6t duo. c c6ng thric c6u t3o cira c5cchat tuong tng.-

Ancol m4ch kh6ng nh6nh

CH3CH2CH2CH2OHbutan- 1-ol

oxiclobutan

CH3CH2CH(OH)CH3butan-2-ol

-

Ancol mach phin nhri,nh(CH3)2CHCH2OH2-metylpropan-1-ol

(CH3)3CoH2-metylpropan-2-ol

- Ete

CH3CH2CH2OCH3; (CH3)2CHOCH3;metyl propyl ete metyl isopropyl ete

CH3CH2OCH2CH3.dietyl ete

*Thi dU 4. C6 bao nhiAu hqp chiit don chucil6ng phdn cia nhau cd citng c6ng th*c phdnttiC3H6O?Hrhng ddn Ya cdu hdi niy cin xdc clinlrduoc: cic chit trn c6 rtQ kh6ng no bdng 1, dotl6 c6 th6 li andehit ho{c xeton no tlon chuc,hoic ancol kh6ng no don chr?c ho[c aacol nomach vdng.

C6c ch6t d6 h CH:CHzCHo, CH3CoCH3,CH2:CHCH2OH vd xiclo-C3H5OH.Trudng hqp ndy chua xu6t hiQn cting phdnhinh hoc.*Thi dg 5. Cd bao nhiAu chtit ddng phdn,phdn t& c6 mQt loqi nh6m ch*c, c6 citng c6ngthtlc phin hi C4HgO?Htrtng ddn Ll.ng voi dQ kh6ng no bing I,phin tu cric cttit c6 thi! c6 li6n k6t d6i C=Cho{c li6n kiit d6i C:O hof,c vdng no. Vi nhuvdy c6 th6 c6 ci ddng phan hinh hoc xungquanh li6n k6t tl6i C=C ho[c vdng no. Phdntu c6 mdt nguyCn tu O n6n c6 th6 c6 nh6mantlehit hodc xeton ho{c ancol don chirc, etedon chlc.Tir d6 c6 th6 viiit

- Xeton no, tlon chirc m4ch hd

cH3 -cH2 -c-cH3

ilo

- Ancol kh6ng no, don chrlc m4ch hd

CH3CH2CH2COOH; (CH3)2CHCOOHvir ddu ld cric axit no, mach hd.

- Niiu ld ancol R(OH)r:

CHepOH

:IIaOH:CHI(==r

o

xiclo-CH3CCH. OHI

t6c d\mgthuQc lo4i

cacboxylic don chuc R{OOH. Voi g6c R c6Mn : 43 img voi nh6m C3H7, cdc axit tl6 ld

TORN HOC, c[u6i$e

CH3CH:CHCH2OH ung voi c{p tl6ng phtnhinh hgc:

H\ ,,H CHr.- -H

-C:C.- vi ,C:C\cH3 cH2oH H CH2OHCH3CH:CHCH2OH; CH2=CHCH2CH2OHCH, = 6-6111611

I

cHr-

Ancol no, don chric mach vdng

xiclo-C4H7OH :

Go,xiclo-C3H5CH2OH :

}_orron

-Ete: CH3-0{2H5.*Thi du 6. H6n hqp x gin m6t sd chilt

..,',..mach ho, d6ng phdn cua nhau. phdn ri chiaC, H, O vd chi co mAt bai nhom chm, dintd, clung dwtc rrti narri. Ti kh6i hoi tia X sovoi hidro biing 44. Viit c6ng thuc cdu tqo cuacdc chdt trong X.Hnfng dAn Voi M - 88 g/mol,dugc v6i natri chimg t6 c6c ch6taxit cacboxylic hofc ancol.

- N6u l} axit thi do nh6m {OOH c6 I't6i

luqng mol li 45 g/mol n6n chi li c6c axit

+Y,6in= 1,g6cRc6Mp=71 tuong img voig6c hitlrocacbon c6 h6a a'f I (C5H11OH):g6m 8 ddng phin cdu tao (ban tlqc tU viistc6ng thuc ceu t4o);+ Y(i n:2, Mn : 54 img voi giic h6a tri II

-

C4H6- tuong img voi ancol C+Ho(OH)2, g6mc6c ancol:Ancol 2 chrtc, kh6ng no, m4ch hd:

CHr:CHCH-CH, CH,:C-CH.OH- | t' '|

oH oH cH2oHHOCH2CH=CHCH2OH ring v6l hai tlingphdn hinh hpc:

H\ -H HOCHr- -H

-C-C- vd --C:C-HOCH2 -CH2OH H- -CH2OH

* Voi c6c ch6t c6 M = 88 g/mol, nilu thay d6itinh ch6t cia cric chAt t o,ig t6n herp (thi du:c6c chet taa dgng tluoc v6i dung dich NaOH:khi tt6 g6m cic axit cacboxylic don chirc vieste

Tfiol;uMffim(Thdi gran ldm bdi : 180 phrtt)

PHAN

(l)

pnAN nrtNc(Thi sinh chi iluo. c lam mdt trong hai phdn A hodc B)

CHUNG

CAu III. O di4m)!

Tinh tich phen , =\"--- 4-;V2+srn.x-cosr

CAu IV. (l dien)Cho hinh leng hU dimgABC.A'B'C'c6 dAy ABCh tam gi6c vu6ng tai L Khoing ciah n AA' denm{t pheng \BCB'C) bdng a, l.hoang c6ch tu Cil6n mat phdng (ABqbeng 6 va g6c gita haimPtphing (ABC\vit@Bqh p Tffith6tichkh6i hngt]}l-Bc. A'B'C'thf, 9,avdb.CauV.O di\m)Cho ba sii duong x, y, z thba mdn x+yrz=1.Chimg minh ring

Caur, (2 diAm).1

Cho him sd v=1x3 -2x2 +3x'3

1) Khno s6t sy biiln thi6n vi vC dd thi (C) ctahem s6 (l).2) Vitit phuong ntnh tiiip tuyiin cria tid thl1q, tl6t ti6p tuy6n niy

I-Di TRj,C NGHIEM KHACHQUAN

oAp s6 - ntldNc oAN crAroi xuAo sAr u6N roAN ugc xi r r.,6p rr - NAU zorr

@A .fing 0 fiang 16)

M6i cdu tluqc 1 diiim.

II -

DT TII LUANBNi I.l. x=I+kn:,x=!+kt k eZ.63 /\J,2 . Bi6n d6i thenh sinl 2x-1 l=

-

-.\6/2Ddp s6. x = -! + kn;, =)Ja * Pn' 1, . v'24243. o Giii phuong trinh cos2 x+ 1,5cosx=0,5tlm duoc x = +L ', P2n' 1 .r.'3o Tim c6c gi6 tri x thuoc nua tloen [-+,rr),L 3', )'Dio s6. x nhin cit" ,ie fii

-fr :T ,5n ,7 n

.3'3'3'3'Bii II. t. sti l6p tluqc c6 dan g n: iF& t ongd6 a, b, c, d, e k}rlc nhau vi 6y tr) t|pA= {o;l;2;3;4;s} . ointrtcn e:ovda:5;c6c cht sti b, c, d ntan c6c gi6 tr! 1;2;3; 4ilusc xiip vdo 3 t tri cdn Qi, c6 A] c6ch xiip.Theo quy uic nhan ta c6 1. A1 .l = 24 s6 c,6 5cht s6 kh6c nhau l6y tn tQp A lon hcrn 50 000vi chia h6t cho 10.

TO6N HOC'qufi[E

2, 36 cich chgn c6c c6u cdn lai cria hpc sinh cl6ld Cff =326976s.

Bni III.- lut+2us=0 l3ur+8d=0 [a, =8r. {

I.?+ni ,nY"ffiifltlIfitrff,neuvEn MINH NHreN

t. xAv orrNc npr rr/ MoT pr, HprnoN crAN*Xuiit phtit ti m\t PT tich (hortc PT dqngu' = v') ta khai fidn vd td,ch thdnh hai PThoic tim thAm mQt PT kltdc de hip thdnh HPT.

*rni ag t xuSt phat fi PT tich(f -t)1r'+2y'z-1)=o

e x2y2 +2ya _3yz _rz *1-_O (*)Chon x = 2;y =l ld mOt nghiOm cria (*) r6it6ch vii tr6i cira PT (*) thanh t6ng hai bi6uthric, m6i biu thric nfun gi6 fi bdng 0 khix=2;y=llir(*,y,

-

y, +2x2 -11)+(2y4 -2y2 -3x2 +12).

Diin ctdy ta clugc HPTlxzv2 _ v2 +2x2 _ll

-_O)', - 0)l2yo -2y' -3x2 +12 = 0

Ctich gidl CQng theo vi5.hai PT cria hO G) diitlua v6 PT (*), r6i bi6n tl6i thdnh d4ng PT tichban tl6u.IIPT (I) tuong

Tt d6 dan d6n HPT[t5xt -3oxzy-loy3 =4 (l) fifirf10x3 +30.rry2 +15yt =7 121

Cdch giiri Nhdn hai v6 oia (2) voi 2 r6i trttheo vri voi (1) ta

-a2Wr-Ap {.rng cdngrhuc sna = 2 tafuwHPT

-atanz: + I2

rl,*1'l = u[,*1) = ol.' -, *'*,) -,,,"(^',.1-1'' y)--1,*y t-xy) \""

Ctich giil O6 giai hq (MI) ta dung phuongph6p luqng $6c h6a. Nhfln x6t ring .r,y lu6ncung d6u, nn chi c6n xdt trudng hqpx > 0,y > 0; trudng hqp .r < 0, l, < 0 nrong t.u.

Df;t x=tana,y=tanb voi ol .(O;!\ ru\2)duo. 5 = 6 = .4--'- sin2a sin26 sn(2a +2b)'Tu"6 5 = 6 = sn2b =9sin2asin2a sin2b 5=

cos, Zt = !36cos22a -1 l)25't_

+ cos2b = :J36 cosz 2a -ll.5 0,Vx,y,z,t el-l;17, tting thric xdy ra khi mQtbitin bdng I cdn ba bi6n cdn 1ai beng 0. T6chthanh hai bi6u thric

x2 + ya + z6 +tB =1, x3 + ye + zra +121 =lta ttugc HPT

lx2 + ya + z6 + t8 =1 (1) fDO\x3 + Ys 12u 11zt -1 (2)

Cdch giAi. Tn IT (l) suy ra.r, y, z, t el-l;11.Khi d6 x2 + ya + z6 +t8 >_x3 + ye + zr4 +t21,dAng thirc xdy ra khi mQt bi6n b6ng I cdn babitin cdn l{ bing 0.* Xtnt phur ti trudng ho. p xay ra dung thnc

^ , : ,: , ; ,ctn mQt hat dang thirc, kdt hoo thdm mot PTkhac d duoc HPT.

An = BC = CA e(x -t)'z + 4 = * + f = (r -2)' +ttairuo.cHPr {i_i;==::

TOAN H9C, cltudiue

A{z&cd4

*Thi dB 10. xufu ph6t nr b6t ding th?c2x,[ y a,y a;f2 rJ, - z < s ry.

Deng thric xdy ra khi (x;y) = (4;S) .V6i (x;y) = (a;8) thi

@ - 0,[y a + 0 -tlJ x + z - t Ja.Ki5t hqp lai ta dugc HPT

It, - +1^p *1y -r)Jx +2 =7J6 0),.,,Itzx,!y-t++JzyJx-2=5xy (2)'

Cdch gidl Nh|n thdyt2x,[y

- a,, a,! 2r'[*

- z < s*y.

D6 c6 PT (2) thi (x ;y) : (4 ; 8).Thtr x = 4,y -- 8 vno PT(l) thAy nghiQm dung.

5. xAY DING HPT TTI PT TRt,N TAPSO PHTIC*Itdi m6t PT co bdn (iin z) trn tQp s6 phtic,chi cin thay z = x + i(x,y eF.) vd sti &tngdiiu ki\n biing nhau c a hai s6 ph*c ta nh$ntluot m|t HPT khd kh6.*Thi d+ 11. Ttr PT (2'z-1)(z+t+i)=othay z = x + y(x,y e R) sau tl6 cho phihthuc vd phan 6o bnng 0, ta iluqc HPT

{i--1;^,.-r.'r==!i..1ir---itl,,*,Ctich gi,rtL Tt (1) vd (2) ta c6

x3 -3ry2 - x-l-(y3 -3yx2 + y +1)i

= y2 +2ry - x2 - (x2 +2ry-y2)ia x3 +3x2y +3xy2i2 + y313 _(x+yi)_t_i

= (t + i) y2 + 2xy(1 - i) - x, (1 + i)e (r+l)3

-(x+yi)-r-i=(t*iXy, -2ryi+i2x2)

e (,r+ yi)3 -(x +yi) -r -i = (r +i)(y-ix)'

e z3 +(1+i)22 -z-(t+l) = o.

e 7 =l;2 = -l;,2 = -l-i.Do t16 (.r;rl) = (1 ;0), (-1 ; 0) (-1 ;-l).

TORN HOC'qrtdilF

6. xAY DUNG mr rrlpr LUqNG GrAC*Xuiit phdt ti mQt PT chtla sina,cosa bingviQc dfit x = sina,y = cosa ld ta cd PT moi,kA hqp voi x2 + y2

--l cd HPT*Thi dB 12. Tt pr

2cos2 a+2.Esinacosa+f = :(sina +.,6oosa).Dit x = sina,y = cosa ta dugc HPT

l-2-,,2-1),^', -: _

(x[)l2y2 +2,t3xy -3x-3J3y+ I = 0.'

Ctich gi,rti Ngoii viQc dpt An phg ttua vd PTluong gi6c ban ddu, ta c6 cich kh6c. Tir hQ(XII) ta c6

2yz + 2^l-3xy -

3x -

3.f3y + xz +yz =go (.fsr *,)'

- :(tr:r +,) = o

e .l3y+x = o ho4c r6y +x-3 = 0.K6thqp voi x2 + yz =1d6 tim ra.r, y.*f* m\t c6ng thtic lwng gidc kA hqp voimAt PT fuong giac ta sd dtto. c m6t HPT moi.

*Thi dU 13. Tt c6ng thictan3, =

latn3 t -3tait

vi k6t hqp3tat2 t

-lDSt x = tail ,y = tan3t ,z = ta1;.gt ,var tant = tan27, ta dugc IIPT

l(3x'1 -t)t = x3 -zxl(ty, -t)z = y3 -tylQz' -t)x = z' -zz.

Cdch gi,ii Dirng phuong ph6p luqng gi6c.Dpt .r = tant, suy ra / = tar8t, z : tan9t,x : tan27 t. Tit d6( kn 3kn gbr)(-r;y;z)=[tan---- : tan

^

:tan- ) (k e [,).Nhu viy, chi cAn xuSt ph6t tt c5c mtii quanh0 cua hai ho{c nhiiiu A6i tuqng trong d4i si5hay hinh hgc, ta c6 th6 xiy dpg tlugc ciicHPT khdng h6 tlon gian. Qua tl6, hi vgng bpndgc sE hi6u duo. c phdn ndo nhrmg f rudng cuat6c gi6 trong qu6 trinh tli tim bdi to6n g6c.

fuaad4

BAI TOAN 4 NCG. Cfto tam gidc ABC voi b) Tim ttoan th6ng c6 dQ dniy b6ng 2MO rottryc fim H, trpng tdm G vd giao clidm O cila chirng minh y = I H. Ch'ingh1rt chon BT = y,ba.dudng trung. tryc. Ch mg minh rdng ba t 9!C !:_ry9 ld cludng trung binh cta tamdiiim H, G, O thdng hing vd GH : 2GO. gi6c CBT (h. 1). Ta cb BTIIMO vir MOIIAH(Euong theng

DHinh 2

nhau t4i trung cti6m M ci:a BC. Tam 916gAHD c6 hai dudng trung f'ryn AM vir HO cetnhau t4i trgng t0m G, suy ra GH:2GO. ADINH HUONG r'. Sri drmg phuqng ph6pvecto d6 ch(mg td iarre OE =lOG rdi suy ratli6u phni chung minh.

o Cdch 4. Tir k6t qun AH =2.MO ta cbffi = Z.OU. Goi "/ld trung di6m AG. Ta c6on =oe+78 =oL+2ofr=d + oG + zofr - oG = zd + zofr - oG=qoG-oG =zoG.Ttd6 q G, HthinghangvnGH:2GO.1NlAn xit Crlc c6ch 1.2, 3 chi cAn sri dungki6n thirc cria Hinh hqc lop 8. C6c bpn saudua ra tlugc nhidu c6ch giai cho bdi niy:l) Trtin Chi nien, llT, THPT chuyn Nguy6nQuang D0r1 TP.Cao Uinh, D6ng Thdp.2) T6 Khdnh Hda, 104, THPT Nam D6ngQuan, D6ng Hrmg, Thii Binh.3) Biti Dinh Hi6u, llAl, THPT Qulnh C6i,Qulnh Phs, Thdi Btnh.Moi c6c ban tim nhidu c6ch gi6i cua BAITOAN 5 NCG sau d6y vn gui ',rd Tda so4nTH&TT tru6c ngdy 30.9.2012.

BAI TOAN 5 NCG. Ching minh biit ddngthtirc sau vtri r, y, z ld ba s6 thqrc drrmgllllll

-+-+->-+-+-.rlr -r,E za x1),3 zl ,t yt r' r3l-] z'

Nguoi rad6: HoANG NGUYEN(NghQ an)

TOAN HOC'cfudife

XAC OnU CHUSO... (rip trang r)*Thi dq 6. Chung minh ring sd chii si c a2102zrot vA 2OOZzrl't + 21001 ld bing nfuit. Tid6 suy ra sii chtr ti

"r;io 211g2zoot n 2zoot .

Ldi gidl Gie st 2002200r c6 z cht sti thi1000200r < 2002100r 6003.(1)

Gia sri 20022001 + 22m1 c6 nhidu hon z chts6, tuc li it nh6t c6 nt + I cht si5 thi

20022001 + 22001 >10n Q)Ti (1) vd (2) suy ra

20022o0t

PHUONG9-@tm T@rtst

MOT PIIUONG PNAP XAC DINHTAM MAT CAU NGOAI TIEP HINH CHOP

BAI TOAN. Gia si co hinh chop S.ABCDEFrii dtrcrng coo SH. tim drong rrdn ngoai riipddy ld O. Xdc dinh tdm t,it tim bon kinh mdtcdu ngoai tiip hinh thdp tAn.Cdch sidi @.1)oQuaOkdtluong thdng dsong song voiSHthi d h tructludng bdn ngopiti6p &ly.o Goi 1 tr6n d ldtim m{t c6u cdntlm, d t OI = x.Khi x>0 thi d

NGUYEN THANH HAI(GV THPT TnQu Son 4, Thanh Hba)

*Thi du l. Cho hinh chop tam gidc diuS.ABC day ABC ti ram giat diu cenh d.

-/i5s -- zT : . Goi D ta diin nii ring , tu B

3qua (. Xo< dinh tdn fi tim hott kinh mir citrngoai ri|p hinh ch6p S.ABD.Ldi sidi (h.2)

Hinh 2Ggi G li trgng t6m tam gi6cSG L(ABC). Do CB=CA=CD

Hinh I

ngugc chiCu.r K6 IK-1 ,S11 th\ IK=OH, sK=lS/{-rlhic t16IS = IC e IK2 + SK2 = IO2 + OC2

e OH2 +(SH -

x)2 = rz *gr'.o Trl t16 tim ra r, suy ra t6m I vd b6n kinh ,Rcria m{t cdu cin tim.

ABC tbtn6n C ld

tdm

Ggi I ed ld t6m mit cAu can tim, det CI =x.Khi x>0 d,i d va 6

".roe chidrr kiti ,

66 =\g =IJER * Ec, =!1;u * ="D .22.2'2Ta c6 IS = IC o IKz + SK2 = IO2 +OC2

5a2 ,- \2 ^ a2 3ae -+l2a-xl

=xz+-90', mtt Mn tao voi tl6ymQt g6c 45". X6c dinh tdm vd tim brin kinhm{t cAu ngopi tiilp trl diQn AA'B'C'.

fu*td4

-,\

'tran Ddc san Todn hgc vdTudt bi Sd J, trang 44)

Luu f reng, hQ tl6 cho ld h0 c6 dang ho6n vivdng quanh. Vai trd cta.r, y, z kh6ng binhding, nn L-h6ng th6 gie sn reng x 2 y 2. zduo. c. Nhu vfly, ldi giAi z = f(x)>-f{y)>-f(z)) y>z>x

-

x=y=2.Trwdng hqp 2. x> z> y= f(x)>f(z)>-"fA)=

y>x>z -

x=y=2.Y$y x, y,z ld nghiQm cua PTt=f

-v+,{A-3, hay 13-l+"[I-:=o 1t;Xdt him s6 g(r.1 =;:-3r+.,[-l-3, ta thiy

Ic'(r)=3(r'z-l)+

-(v/> l).

2.,1t -l

Suy ra him si5 g(r) ttdng bii5n tr6n [;+oo).Lai.+hiiy g(2) = 0, nOn PT (1) c6 nghiQm duynhat , = 2. Tri d6 hQ tltr cho c6 nghiQm duynhdt 1x;y;z) = (2i2:2).

NGoc H'CN

LOT CIAI OA CUUEN CHUA ?Trong DA thi tuy6n sinh vao l@ l0 cua tinh Tir xi

-.r,x, + *Z =( or-or, *lr;)*1r,,Hii Ducmg nIm hoc 40G2007 c6 bii todn ( ' 4 ' ) 4 '

thoa mdn rf + .rr' > 0 ..Nhitu hqc sitb dt,giAi ohu sau : .

^.!.llllus0snrL[tIl

\v

T]lUilII$ GIP

KHrcrA cec eA roAN pHuoNG TRiNx,eAr pHuoNG TRiNlr,uE pHuorto rnixH naO - lOcenr

LE OUC THUAN - cAo vAN DONG

i,5ltog,(r-t)'- t5 = to':(r+ 3)' . log, (S - r)'. 2.3.logr(x+t)+15=logr(x+3)5+loCr(5-"r)s.i,* r*, UOnf.DK -r + -1 vd -3 < x < 5. NguvAn nhdn sa-i lim, l}ty h nhrmg sar PmPT tt5 cho h.rong tiuong v6i

-' cria phan Cl6ng hqc sinh vi 16 ring voi ttiAu

s kiQn cria bii torin thi (.r + 1) d6i diu nong)logz(x +t)' + l5 = log:(x + 3)t + togr(5 -.r)' ldroang (-3;5) . De 5logrl,r+ { + 15 = 5logz(r+ 3)+ 5logz(S -.r) * Thi du 2. Gidi phaong trinh

. | (x+l)=(x+3{5-x) (:-t -

s)logl x + (or -

l9)log,:r -

| I = s.l$x+l)=g+3)(5-x) Ldi gidi itilng. DK.r>0. DIt t =togzx,

| *+ax-t=o [x=l fr=s-\/E PTtrdthdnh (3x-s)t,+(9x-r9)r-12=o (r)el

-1.Vi vfly lcri gidi tr6n {a nm ttrS hgp tip x6ctlinh cria PT tt1 cho, ddn tl6n mit nghiQm.

TORN HQC.........'clirdiE;e

fus*d4

r'(x\=-L+ 12 > o.vxe D.xln3 (3x

-

5)'z

Xiy ra hai khi nlng. N6u o

Ldi gi,rti irnnglx>0. x+l

DK I -' IIPT hrong duong voilY>o Y+t

((fI m,r ll '=Yllon.'* -- / = 0 .= .ll ton,r = -l| -' ! loglx.log;! lL * log,.y[

"='' [ "=.rTrwdng hqp I . v= y ta c6 1S =31efu? *3)=0suy ra .r=y=16.

Trudng hqp 2. log, x = -1.lo8: -/

Til 3:r = /3 =log, (:r) = 1otrr,

0,V/Rt't2nOn tu,(r) =ln suy ra v = y, nhrmg thgc chSt

Ihnms6 /(r)=,-l bi ednaoanBi r=0.t* Thi dB 6. Tim m dii phrong tinh

1g(x2 + 2mx)-tg(; -

t) = o (1)c6 nghiQm duy nhdt.Ldi gidi ilrtng

(v-'l>o (x>lPr (' ) 1; ;; = xsa\r +(z--r):+r =o (2)PT(l) c6 nghiQm duy nh6t khi vi chi h:hiPT(2) c6 nghi0m duy nhat th6a men r > 1.o Tnmng hqp 1. PT (2) c6 nghiQm k6p lonhon 1. Didu niy xiy ra khi vd chi khifa=o l4mz

-4m-3=o)t-z* ol I =mea.l->l lm

HU6NG oAN . oAp sO o6 THUSUC TRU6C ri rul(DA ddns ii tans 29)

Ciu L l) Ban ttqc t.u grii.2) C6 hai ti6p tuy6n th6a m6n ed uai v6iphuong trlnh lAy : 0; y = 3r.Cf,u II. PT di cho hrong

.i. Hlnh nao trong cic hlnh A, B, C, D phu hqp v6i vi Uf cdn khuydt 6 hinh E?

.l Hinh nio trong c6c hinh A, B, C, D, E khiic vdi cac hinh cdnE

lai?AO-.aac %X O r-l-.@"

ex4u ouYr.rx(Cong ty CP SGD tai TP He Noj)

reHlHAFr/16 mdt

"o, fth vh lg con mdi (m6i con mdi

LUidu'tfri boi mdt chdm trdn) & vi tri tlinh c6c0 vuong nhu hinh v6. Crl m6i fdn niriy

"on frh

chuydn tir ddu nhy sang ddu kia ctra m6J dudngch6o hinh chfr nh6t gdm hai 6 vudng kd nhau. H6isau 19 ldn nhriy thi con fuh c6 thd b6t duo.c hdt19 con m6i kh6ng? Ndu dugc b4n hdy vE dudngdi d6 cfra con Ech'

oAN euYNHHe N,i)

aaaaa

aaa

a

aa

a

o u aAa AaA a o oa

a? tr

o aA a

a

. . . . . .. . . ...

p#RdN..r+9#SJ. HgFt.VgJ. SILTS. 6 IG 3 nhy c6c ban gui ba.,i Tda so4n tl6u cho tl{p dn dring. Ttry0n duong cdc ban c6 ldi giii t& vidud. c nhan tang phdm cria Tda soan: B ili Dtnh Hie'u, 11A 1, THPT Qujmh Coi, Quj,nh Phu, Thrli Binh;Trdn NguyinTry, LOC3A, THPT chuyOn Hing Vrrong, TP, Pleiku, Gia Lai; Dwng Thi Tuydt Vdn,10A5, THPI My HIo, Hmg Yen; D ong Thi Hdi, lIAl, THPT YOn Phong l, Bdc Ninh; NgzydnHoa,KhoaTodnK5l, DHSP Hi NOi.Mong ring Ki 4 c6c ban sE tham gia tich cqc vir nhanh giti bi,i giai vd Toa soan tru6c 30110D012 fue.Sau diy li tl6p 6n Ki 3.

uiuuruggrPHg,[qP'f' Trong hinh G cec mfii rcn 6 m6i o ddu chi theo mQt hu6ng: tn{i, phAi, xudng. M vly chon hinh Dvio vi tri cdn khuydt A hinh G.

'f' T.ong hinh G, hinh A cot thf ba c6 ttuo. c li do ch6ng hinh d cot thrl nhdt len cot thf hai. \n vaychon hinh B lh phi hgp.

gA.uINHgJTHEIuAgaC6 32 miii tOn, sau khi b6 di 4 m0i t6n m6i ldn thi ldn m6r cdn lai 28 mfi En, ldn hai cdn lai 24 mfii tn,vi ldn ba cdn lai 20 miii ten. C6 nhidu phuong 6n s6p xdp mfii ten th6a min yeu cdu bii ra. Ching han

II\N I

rAru Z

LI\N 3

TORN HOC. cli.diw

Kltuug tr=itrq+gu!l?.'Zqfil

NTgaV fii5/2012 BQ Ciriro dsc vii Ddo tao dd raI \ Quviit dinl s6 1866/QD- BGDDT ban hrimhKlrung kii ho4c\ thoi gian n6m hqc ?012-2013cria gi6o dgc mim non, gi6o dpc ph6 th6ng vigi6o duc thudng xuyen. Theo tlo. Nguydn tic xdydpg vi Khung k hoach thdi gian nim hoc2012-2013 nhu sau:NGUYTt,N rAc xAy opNc xE HoACHruot cIlN x.ilr rrec2ot2-zot31. Kt5 hoach thdi gian nim hgc tl6m beo s6 tuanthuc hoc:a) D6i vdi cAp Mim non, c6 it nh6t 35 tu,in thuchs9 (Hs" ki 1 c6 it ntr6t tA tuin, Hgc ki II c6 itnhAt 17 tuan);b) Eiii voi c6p Titiu hgc, c6 it.nhAr 35 tuin thgchgc (Hgc ki I co it nhat 18 tuan, Hqc ki II co itnh6t tz tuin;;c,;.Di5i voi c6p THCS vi cip THPT, c6.it nb6t 37udn thgc hQc (H9c ki I c6 it nhat 19 tuin. Hqc kiII c6 it nhit 18 tuan);d) D6i vdi Giiio duc thuong xuyn

-(THCS vdTHPT), c6 it nhet 32 tuAn thuc hqc (mdi hqc ki c6it nh6t 16 tuin).2. K6 hoach tloi gian nlm hqc pht hqp voi ttirju kicnt.u nhi6n, xe hQi vn tip qu,in cua tlp phuong.3. Cric ngiy nghi G, Tiit dugc thgc hifn theo Quytlinh cia Luit Lao

ilolhenotis ond Youlh [tlugorine

xuir nil rrJ DatDic san s6 4

IM soen: lE 0, fidEians Ui, [a iloi0I U& l0:0t15121507

lI - tax P[6t [inlr Td su : 04.3fl21006Emaih b$m0.lli0c_tr0ltre@rCr0r.c0m,rn

BAN CO VAN KHOA HOCcs. rsffr. rcuvfo:l cArur roAN

cs rsrn rn-fur vAr Nntxcrs NcurEN vAN vo. Nc

Gs. DoAN euiNHpcs rs. rn AN vAr.r ruo

CHIU TRACH NHIEM XUAT BANCht tich Hqi ddng Thlnh vien kiem

Tdng Gi6m d6t NXB Gi6o dqc Viet NalnNG{//. NC,o rR-Al! Ar

I'h6 Tdng Gi6m ddc kicmT6ng bicn tap NXB Gi6o du. c viet Narn

rs. NGUYSN euf THAouOr o6uc afiw rAp

Tdns bian tqp : Ts.YflAM THI BACH NGqC Thu ki Tda soan : rm. ud Quer.lc rtlxrs. rn {N oiNn cHAu, rrs. NouyBr{ ,qNH DONG, rs. rn iN NlM DfrNc, rs. NcuytN urNs o(rc, rs. ucuviNMrNH HA, rs. NcuGu vrpr nAl, pcs. rs. rf, qu6c nAN, rrs. IHAM vAN HiNc, pcs. rs. vcr rnlrrn rHr6r,cs. rsrn. NouvEx vAN M^u,6ng NGUYEN rcrlc urNn, rs. rni.N uutr NAM, pcs. rs. ucuGN DANG pHir,p6s. rs. TA DUy pHrBNG, zs. Ncuyfil rr$ rrncr! cs. rslffi. DANc Ht Nc rn (Nc, pcs. rs IHAN DoAN THoAr,rrs. vo KrM THrJy, pcs. rs. vo DUoNG THUY, Gs. rs(lr. NGo vrBr rRUNc.

Pham Vdn Hoan -

Hu6ng dAn 6n tip mOn Hoiihoc tuydn sinh vlo Dai hoc & Cao ddng.........26N guydnVdnThAng

- Thrl srlc tru6c ki thi ......29

D6p sti - Hu6ng din giii Dd khio sdt mon To6n(thit nghiem) Hoc ki I l6p 1l ndm 201 1 ..........30Di6n dirn day hoc to:inN guyin Minh Nhien

- MO.t s6 ! tudmg xAy drmg

h phuong trinh.. .................31Nhi6u crich giii cho m6t bhi torinViet Hdi

- Chrlng minh mot tinh chat cira duimg

thdng EuIer......... .................35Phuong ph6p giii todnN guydn Thanh Hdi

- Mot phudng phdp xic

dinh tem mat cdu ngoai tidp hinh ch6p ...........37

Don vudn Tor{n.................................................. 40LA Dila ThuQn - CaoVdn Drtng - MOt s6 sai ldmkhi giii cdc btri todn phuong trinh, bdt phuongEinh, hp phuctng trinh mfi - Iogarit..................41Hu6ng dan - Ddp s6 Dd thir srlc tru6c ki thi....44

To6n hoc vui Ki 4..........................................45Ban cdn bi6tThanh Loan

- Khung kd hoach thdi gian nim

hoc 2Ol2 - 2013 ..................47

TRONG S6 ruAVDhnh cho Trung hoc Co s6Tluii Nh4tsd trr nhien

Plutqng -

)({c dinh sd cht sd cua mOt

N guydn Thanh Todn -

Mbt hu6ng chring minhc6c bdt ding thrlc dang phin thrlc.....................2Phan Dinh A* - m mot bai to6n trong S{chgi6o khoa tfirh hqc 9........................................ 4Thanh Loan

- Dd thi tuydn sinh ldp 10 TIIPT,

TP. Hi NOi nam hoc 2012 -

2013.................... 6

Gi[p ban 6n tepN guydn Anh V fr

- Gi6 tri l6n nhdt, gi6 tri nh6

nhdt cta harn s6 bac hai fien mot doqn...........7Lt Qudc Hdtn - Sir dung y6u td phq tronggiii torin lrrgng gi6c.................................. 11N guyin Hdi Chd.u

- Gi& thiQu Dd khio s6t

mOR Todn (thfi nghiem) Hoc ll I 16p 1 1 nam2011 ................. ..............16Chudn bi cho ki thi t6t nghip THPT, thi vioDai hoc & Cao ttingDudng D rc Hrio

- Suy nghi vd Cau 5 trong

Dd thi tuydn sinh Dai hoc Kh6i A - A1,nam2O12........... ................ 20N guyn V'dn ThwQn

-Yd d thi tuydn sinh Daihoc mon vit li - ntun2012.............................22