Đặc sản toán học tuổi trẻ số 4

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Tuy theo c6ch viet cira mdt s6 tu nhi0n ta c6
th6 x6c dinh sO cht s6 cira n6.

t.

SO

2. Sii tU'nhiOn vi6t Ou'rfi d4ng

du

tinh sd chir s6 cua sii t.u nhie., dang ndy, ta
v4n dung d6u hi6u sau.

Sd try nhi€n a" vi€t trong h€ thqp phdn cd
chfr" tii t t i vd chi khi l0k-1
k

(a,k,n€N.).

l. Vi€t cdc sti ttt nhiAn b li€n ti€p tit. *Thi
chi

dg 3. Trong he thap phan so

550 c'r5

Loi gitii. Xet ciic so le

Ldi gidi. Ta c6 5s0 = 535 (53 )5 = 5:s. 1255 ;
1035 = 53s.235 :53s.1285. Suy ra 550 < 1035.

Tril+9c65chts6.

M6t kh6c

so?

,i

Tri

:
1034 = 534.234 :

-+ 99 co
(fqq - fi) : 2 +t).2 =90 (chr s6).
Tt 101 -+ 999 co
((999 -lo1) : 2 + 1).3 : 1350 (chir s6).
Tu 1001 + 2011 c6
(tzot i - 1001) :2+1).4:2a24 (chr s6).
Y6,y s6 A co
5 - 90 + I 350 + 2024 : 3469 (cht s6). tr
11

Suy ra

*Thi

1034

534

53a.625a;

(2e)3 .27

< 550.

Vay

- 534.51 23.129.

1034

<

550

< 1035 hay

dq 4. Trons h€ thqp phan, so

2100

co

chi -so')

Loi gidi. Ta c6

Hoi so B co bao nhiAu ch* s6?
sO tuy thi:a cua2

Tu 21 d6n 23 c6 3 chri s6.
Tri 2a -+ 26 co 2.3: 6 (cht sO).
Tit 27 -+ 2e c6 3.3: 9 (cht s6).
Tt 2ro -+ 213 co 4.4: 16 (chfi s6).
Tu2t4 -+ 216 c6 3.5 : 15 (chir s6).
Tu2t7 -+2te c63.6:18 (cht s6).
Ti 220 -+ 223 co 4.7 : 28 (chfr s6).
Ti224 -> 226 c6 3.8 :24 (cht sd).
Ti 227 -+ 22e c6 3.9: 27 (chir sd).
SO 230 c6 10 chfl s6.
V0ysO B c6 3+ 6 + 9 + 16 + 15 + 18 + 28 +
24 + 27 + 10: 156 (cht sO;. n

:

)a

chii so. D

bao nhiOu

d4 2. Viit cdc s6 fu'nhian liin nhau ke rir
21,2).2'^ .....13') ta durlc ta g:2qU0...1073741824.

Ldi gidi, Xdt c6c

534 (54

550

5so c6 35

*Thi

bao

nhiAu chti s6?

1d€n2011 ta drro'c s,.i-1.: 135791 1...2011. H6i
s6 A co bao nhiAu

lfiy thua

OC

ty nhi6n viOt aurfi dqng dny s6 theo

m6t qu1'luit nilo tI6
Oe tlnh si5 chf, s6 cua sri t"f nhi6n dqngniry,,ta
cin tinh sti chfi s6 ctra c6c s6 c6 t cht #, 2
chfi s6, 3 chfi s6... r6i tinh t6ng c6c cht sO OO.

*Thf

*

2to

=1024<1050

=

211

<2l.lT.

Do cl6 233 <273.106 =9261.106 < 1010
2eq < l0j0
= 2100 < l0I.

=

MEt kh6c 2t0 =1024 > tr03
=
Vpy 1030 < 21oo < 1031.

Suyra

11oo

c6 31 chfi s6.

2100

>

1030.

E

*Thi dtl5. Via hai sd 82012 vd 1252t)12 hin nhau
thi nhqn daqc m6t sil co bao nhi€u chir s6?
LN gidi. Gi6 sri 82012 c6 m chtr s6 vd 1252012
c6 n chfr s6 thi 10.-1 10, 1 <

1252012

10m+n-2

<

_

< 10,. Suy ra

82012.12520t2

<

10u+,2<106036
IAm+n

> m+n_l=6A36.

Dod6 m+n=6037.
VQy s5 nhdn duoc c6 6037

cht

s6.

tl

(Xem ti€p trang 36)

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},IOT HUONG CHUNG MI}{H

@MWWW

lcv IHCS
c6c ki thi vlro lop 10 THPT chuy6n
vd trong c6c ki thi hgc sinh gioi tr6n khip

Trong

I

COng theo

bii

l.

*Thi

to6n quen thuQc sau:

1

chthtg minh rdng

abc
obc r-.--+--<1.

D-'
i--+1--bca

C

dugc

a3 +b3 +

11
a3 +b3

i

2 ah+bc+ca.

a-+

b2

-3

+cr>b2+bc.. !-+a1
ca

b3+c3+abc- bc(a+b+c)'

11

>.c2+ca.

C6ng theo u6Aabdtdlng thirc tr6n vh thu gon

c3+a3+abc ca(a+b+c)
Cdng theo ,16 ba bat ding thfc tr6n vir quy
d6ng v6 ph6i thu dugc

1

-a3b3cl
tadugc
l-+a-ui-> ab+bc+ca (dpcm).
'bca
Ding thric xby rakhi

a3

a: b: c.A

du 2. Cho a, b, c ld ba s6

at+bt bt+cj
i

Zab 2bc
gi,rti. Chia

Zab. dtroc
.

a3

dwong,

hJ+c3 I

Zbc 2'

c3+a3

" 2ca

+ff +abc F +C +abc

C

<-

1

+ci +abc

abc

c3

Ding thfc xiry rakhi vd chi khi
Tucrng tu. ta c6

abc
51
+a3 +abc

(dpcm).

Zca

>!ro+b\.
2'

+--- |

abc
a3 +b3 +abc b3 +c3 +abc
-+

c3+a' >
a+D+c.

2ab

1

abc
:--1---

hai v6 cua BDT (1) cho tich

+F

+

Suy ra

chang minh rong

Liri

..-

+abc ab(a+b+c)

l1

) a2 + ab. Tucrng t.u, ta c6

b
Ll
u-

*rni

abc> ab(a+b+c). SuYra

Tuong t.u c6

Ldi gidi. Chia hai v6 ctra BDT (1) cho b dugc
a3

abc

d +bj -ciltt' F +c'3 +abc c3 +a3 +abc
Ldi gidi. C6ng hai ,C eOt (1) vdi abc, ta

Cho a, b, c ta ba so dtrcng.

'l

a: b : c' 3

dU 3. Cho a, b, c ld ba so daong,

ring

chtrng minh

2ca

Dlng thfc xhy rakhi vi chi khi

(B4n dgc t.u chimg minh BDT ndY).

dB

ding thrlc tr€n, thu tlugc

2ab Zbc

BAI TOAN. Cho ba td dwo'rg a, b, c. Chtng
(1)
minh rdng a3 + b3 > ab(a + b)

*rni

Ua Uat

b3+c^ c..'^,
at+b3+--+
-u 2a+h+c(dpcm).

moi miAn cira ci nudc, ta gap r6t nhi€u bdi
to6n chimg minh b6t tling thirc (BDT) dpng
phAn thric. Bdi vi6t ndy xin gi6i thiQu mQt
hu6ng chimg minh BDT d6 b[ng c6ch vdn
dung ket qu6 cira

116

*fni

a: b - c. J

du 4. Cho a, b, c ta ba s6

daong,

:,

chung nlrnn rang

as b5 c5 a2b+b2c+c2a.
-+- 1--)
b2 c2 a2

1

2

,/e\

-

HQC 6e66asssesE*ss66&&6&*@&66&*r***@66ae6ee6e.5E&s€686&s6*&&&* *****o fi.deg*d4

ctirdiua

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w


Ldi gidi. Yoi a, b ld cic s6 ducrng, ta c6
a2

+bz > 2ab;

Tucrng tu, ta c6

+b3 2 ab(a+b).

a3

b3ci-c3a3

-+->b*c..
c2 b2

Nhdn theo v6 hai bat Oang thric tr6n, thu dugc
as +b5 >

azb2(a+b)

(2)

a3

ta duoc

+F * b3 + c3*+>2(a+b+c)
c3 +.,3

(dpcrn).

c2

a5

-==+bt2a2(a+b).

*fni

Tucrng tU c6

>b'(b*")l I"5 +a3 ) c2(c+a\.
\*rt
ca'

cl

b3c3a3bca

b3bb

^1

1r+b\:u

b3 clugc

CQng theo

v6

cdc bat dang

ab

+
as +bs +ab
13

thirc trOn, clugc

bc
--

+
bs +c5 +bc

cs

---

ca

+as +ca

<1

(dpcm).

+c2.

Ding thfc xtty rakhi vd chi khi

v€ cfucBET t6n vd thu gon, ta duoc

as bs cs a3 b3
b3c3a3bca

1). Tucrng t.u, ta c6

CQng theo

+a2.

cs "^*ar)-

cs

bcacab
/-.
/bs +c5 +bc a+b+c' c5 +a5 +ca a+b+c

Tuong t.u co

b5 "- b3
.*Cr>-+b2.
c3ca3a

ca

CQng hai v6

abc:

(v\

-!-I-

vcti a, b, c ld ba sd drrorg.

A5
C2
1_+b2>"

bc
bs +cs +bc

tht?c

,=
a5 +bs +ab a?b2(a+b)+ab
ablc /
as +bs +ab ab(a+b)+1 a+b+c

dV 5. ChLrng minh btfu ddng thttc

Ldi gi6i. Chia hai v6 eot (2) cho

a: b: c. J

cta BDT (2) vor ab,th:u
ducvc as + bs + ab > a2b2 (a + b) + ab. Suy ra

a5 bs cs
;; + ; + ; ) azb + b2c + c2 a (dpcm).
Ding thric xily rakhi vd chi khi a: b : c. J

-r-

b'

d47. Chtmg minh brit ddng

Ldi gidi.

-eon

-f

a2

+as +ca
vdi a, b, c ld cdc td dt org thda mdn abc : l.

Aing thirc tr6n vd thu

as bs cs \ at b3

cdc Uat Oing

ab
a5 +b5 +ab

A5

CQng theo v6 c6c OAt
ta duoc

vd

Ding thric xtry rakhi vd chi khi

b2

*Thi

thfc tr€n,

CQng theo

Chia hai v5 cua (2) cho b2, ta duoc

-+-2c*a.
a2 c2

a: b:

c.

J

&&&

c3

BAI LUYEN TAP

Ding thric xity rakhi vd chi khi a : b : c. A

*fni dV 6. Chlrng minh biit ddng thtrc
a3+b"+-+
b3+c3 cJ+a3 >2(a+b+c)
c2
a2
D-

Yoi a, b, clit c5c s5 ducrng, chimg minh ring

vbi a, b, c ld ba t6 dorng.

't ................................,...,..'..-r-L-<'l
^ltl

- I

h3

b2

a2

>a+b.

I

a+b b+c

a3 b3 c3 2a2b2 2b2c2

c+a
2c2a2

a+b+1 b+c+I c+a+l

Ldi gifiL Chia hai v6 eOt (2) cho a2b2 taduoc

a3

I

-a5 bs+-cs )a3+b3+c3.
3.:+"
bc ca ab
,A\

TOAN
HQC r*r***6sq**$s&8&*866reea*&r*"'.dm%U+f
-

Gltudiua

W

ce6ess.c***a*&6*'&*&6E'&

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ffiffiffims

TT}MOT

'1

BAITOAN,

,, (GV' rHCS Thqch

Suy ra tam gi6c AND clntqi D, co Dl ld ctuong
phdn g6c ADNndnDI LAM(dPcm). tr

ti6t hoc dat ktit quA cao, ngodi viQc lga
chon phuong ph5p thich hqp, ldy hgc
sinh lim trung t6m, nguoi thAy cdn phAi gifi
vai trd chii clao nhdm ph6t huy tinh tich cgc,
chir dQng, s6ng tpo, ph6t tri6n tu duy cho

f\e
L-l

NhQn xda N6u gqi P vd Q ldn lugt li giao
di6m cta AB vd AC v6i DI th\ tam gi6c APQ
cAn tpi A,khi d6 tu gi6c APNQ ld hinh thoi.

Trong s6ch gi6o khoa (SGK) hiqnhdnh, c6 rdt
nhiil bii tflp don gi6n nhmg ni5u bi6t khai
thac, rn& rQng c6 th6 tim ra vd s6 bdi to6n
mdi. Bdi vitit ndy xin gi6i thiOu mgt trong r6t
nhiAu c6c bdi to6n nhu vay o trong SGK'
Srictr

Bdi tQp Todn 9 - TQp hai).
Tam giac ABC ndi ti€1t dwmg trr)n (O). Ti€p
tut:An tui A vo'i clu'ong trdn cdt tict BC o D. Tia
phan giac goc BAC cdt drdng trdn (O) ri M.
Tia phan giac cua goc ADB cat AM ri I.

Cht\ng minh rdng

Ldi gidi" (h.

_.r

Nld

\/_) =

ni*

t16 thay dOl

sau ddy.

ftnai to6n 1. Cho tttr giac ABCD nQi ti€p
dtri.mg n'dn (O). Goi E ld giao di€m cua BC
vdi DA, M td giao di€m ct)a BA vrti CD. Phdn
giac g6c AEB cdt CD vd AB t,heo thtb tu tai P
t'd R. Phdn gidc goc AMD citt CB vir AD thir
tu tai Q va i. Chu'ng minh rdng fi'gidc PRQS
la hinh

thoi.

L.

Ldi gidi. (h. 2).

DI vu6ng g6c vcti AM.

1).

Hinh
Ggi

nhu th6 ndo khi di0m I
o bdn ngoiri duong trdn? Ta c6 bdi to6n 1

K6t quri

hqc sinh.

BAI TOAN Mo DAU (Bdi tQp 71 -

Pxnn,$lux RttH
Km, Ldc l"l#'..Hd Trnh)

giao diiim
sdAC

I

ciaAMvoi BC,taco

+sdBM
2

=frtri.

-sdfrfu
2

sdAC +sdCM

Hinh 2

Tln

dryt

e6c ngodi tam Eiitc,ta co

EQS:2cM:!t;

ESQ: sAM + Mz.
md QCM = SAM (cirng bir v6i g6c BAD);
Mt = Mz, n€n EQS = ESQ. SuY ra tam gi6c

A.
TOAN HQC
*

6&eB6sss*48&s&eEss&&6sqe&s*u*".@%r6dr$@o&&+6s*s*ssas*6aE*&*&

cIUdiUr+

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fue'sa*sd4


EQS c6,ntqi E,lai c6 EP ld phdn gi6c cira g6c
(1)
QES n1n EP lir trung truc cua QS

Tucrng tu, ta chimg minh dugc tam gi6c KRS
cdn t4i K, suy ra PQ li trung truc cria R,S (4)

Tuong t.u ta chimg minh duoc MQ liL ducrng
(2)
trung tr.uc cin PR

Tt

Tt

ftit

(1) vd (2) suy ra tu gi6c PQRS ld hinh thoi
(dpcm). D

K6t qu6 niy thay OOi nfru thti ndo k1ri di6m I
) ,.^
ndm o bdn trong rludng trdn? Ta c6 bdi to6n 2
sau ddy.

fteai fi6n 2. Cho rtr gidc ABCD n6i ti€p

cirotis rron (O). Gpi K, M theo thtr tLr ld giao
J:iiri t'trcr AC vd BD; AB vd DC. Phdn giac
sic .1-\[D ch AC vd BD tan lucrt tai R vd S.
Phan gidc goc AKB cti AB vd CD thu'tl,r tai
P va Q. Chtmg minh ring rtlr giac PRQS td

(3) vd ( ) suy ra ttr gi6c PRQS ld hinh thoi
(dpcm). il
trqp Bdi to6n 1 vdi Bdi tobn 2 ta c6 bdi
to6n 3 sau ttdy.

fieal

tor{n 3. Cho ttr giac ABCD nQi ti€p
dadng tron (O), M la giao diiim cua BC vdi
AD, I ld giao di€m cua hai dad'ng cheo AC vd
BD. Phdn gidc g6c AIB cfu CB o Q, ,at O.l A
P. Chtmg minh ring tam giac MPQ cdn o M.

Loi gifli.

(h. 4).

Itirth thoi.

Ldi sidi. (h. 3).

Hinh 4
K6 phdn gi6c goc AID cit AD vd BC theo thir
t.u tai N vd K. Ke phdn gi5c g6c AMB cht OC

tpi -I. I(lti d6 lvII LNK (theo Bdi to6n 2).

M[t kh6c QI L NK (phdn

gi6c cua hai g6c kC

bir), do d6 IuIIllQI.Tac6

^

Hinh 3

MQP

Gqi N, 7ld giao cli6m cria PQ voi dudng trdn
(O). Ta c6

2QPM=frAN
=

(tofr

\/

--i=Gafu+sa&)+soG
+sdBC

+ 1057) +

\

soG

:

i

sdSi

+

:

M2(soletrong); )PM =M1 (ddng vi)

frr=ff,
^d
gi6c MPQ

@p=1Ffu, hay tam
cdn t4i M (dpcm). d
n€n

sdffi

:2PQM.
PQM hay tam gi6c POM cdn
6 M,l1i co MR li phdn gi6c g6c PMQ n€n

E6 h m6t vdi g6c nhin tir Bdi 4p 71 - Sdch

Bdi tQp Todn 9. Chic chin cdn r6t nhi6u di6u
thir vi nta chua duoc kh6m ph6, mong c6c

MRIiL trung truc cua PQ

bpn tlgc ti6p tpc bO sung.

Suy ra

)PM :

(3)

TORN HOC

, cludiE6

n *

@s&{sase*6e**e&*sese6se@sae

***

*

6do 9nrd4


ffi# xffi

ffiffi&

ffiffi$ffiffi

K#

(Thdi gian lirm

Ciu

A=E*4.

1) Cho bi6u thric
cira bi6u thttc A khi

Tinh si6 tri

Jx+2

r:

36.

2) Rlit gon bi6u thric

4 )l.- x+i6
,:(*-*
IJx+a Ji-+) Ji+z
(v6ix>0,x+0).
3) V6i c6c bi6u thirc A vd .B n6i tr6n, hdy

tim c6c gi6 tri nguyOn cira x cl6 gi6 tri cua

- 1) ld sO nguy6n.

Ciu2. (2 diAm)
GiAi bdi toan sau biing cdch
trinh ho\c

hQ

W

ffi

xwxx&

bii : 120 phtit)
2) Cho phuong trinh

1. (2,5 di€m)

bi6u thric B(A

xffi

lQp phuong
viQc

12glo thl xong. Neu
, ..., :.

mol nguol lam
trong
-)
mdt minh thi thdi gian d6 ngudi thri nh6t
hodn thdnh c6ng viQc it hcvn ngudi thri hai ld
2 gid. Hoi n6u ldm mQt rninh thi m6i ngudi
phii ldm trong bao nhi6u gio d6 xong cdng

Cflu 4. (3,5 di€m)
Cho duong trdn (O;fi) cluong kinh AB. Bdn
kitlh CO vu6ng g6c voi AB, MliLCiCm t5t ki
fr6n cung nh6-ic (Mkhdc A vd Q, BM cbt
AC taiA. Cqi K h hinh chi6u cua f1 tr€n AB.

1) Chnng.minh ring

1) Giai h6 phucrng trinh

lzr
l- +' :2
[xy

grde

CBKII ld tb

trei{:lek.

3) TrCn doan thing BM I6y di6m E sao cho
BE : AM. Chimg minh rdng tam gi6c ECM

llr tam gi6c

w6ng clntqi

C.

a) Ggi d ldlil1c:p tuy6n ctradudng trdn (O) tai
al:6m,q. Cho P ld mQt di6m nim tr6n d sao
cho hai di6m P, C ndm trong cirng mot nira
mat

ph[ng bir AB ua

APtY
MA

= n.

Chimg minh ring ilucrng thing PB
trung di6m cua do4n thdng HK.

nh6t cua bi€u

thfc M -x2

Y

I

tri

tli

qua

Cffu 5. (0,5 di€m).YOix,yldc6c s6 ducmg
th6a mdn diOu kiQn x22y, tim gi5 tri nh6

Cflu 3. (1,5 di€m)

16
l:-::)

(an;r).

T\m m dO phucrng trinh c6 hai nghiOm ph6n
bi6t x1, xz thoa mhn cli6u kiQn xl +xj=7.

2) Chrmg minh ring

Hai ngudi ctng ldm chung mQt cdng

lx
i

-2m:0

gi6c nQi ti€p.

phtrong trinh:

viec?

*z -14m-l)x+3mz

+ Y2

xy
THANH LOAN

l.

(Suu tdm va gi6ithi6u)

D6p sd - Huri,ng dAn gif,i

tl a=1 :2\ B=*r3) 14. 15. 17. 18.CAu2.4gio.6gio.Ciu3. l)(x: v)=\2;t);2t l.-f
5. | = r..2r - . -r2+4y2 !- 3y' =.f * ,,,rf' ., =)*!=1n6n
c6u
lauJ'
"'""'' :] w-,;r:zr.
"''^ maxM
cdu

l.

M

2lr2-yz1= 41xzarz1-

4* 4uz*11= +* 41414r'r-2" zo 5

2

TOfiN
HOC ss&B**6.*&eE*&s*e&*6**?&ra---".dM...."o.o.c.&a.**.*'.&r.r* on"*oo fido&*d4
-

cfudiU6

W


giai bdi todn tim Sto t i ton nhh $flry, gia rr!
f/ie,
7 nh6 nhdt (GTNN) cua m7t hdm s6 d6i voi hoc sinh

ct0PBANoNT{P
GIA TRI LON

?r**.

lop 12 ,ld d€ ddng hon ddi voi hoc sinh lop l0 vd lop I l.
Bdi vi€t nq, sA cung cdp mQt phaong phap, md phdn ndo
sd giup giai mfit s6 bdi todn loqi nay thudn lqi hrm.

rlHAt

GIA rRI NH6

xuAr

,ltnt Tn€il,,m T poftll
(un HAffi sO fifrt
atl
NGUYEN ANH VO

(cV THPT Nguy1n Binh Khiem, Binh Elnh)
TrudnghW 1.a>0.

I. LI THUYET

:

Cho hdm sO 11"r;

ax2 + bx + c (a + 0)

,-h
l) Neu

.

N6u a > 0 thi hdm sO /(x) nghfch bi6n tr6n
r-\
I/
khodng | -*; -t) 1, d6ng bi6n tr6n khoing

pa;f

2o=o
(x) = f

(13)

th\

iffi"f

(x): f(a)

vit

.

\

- \

(
I

2a)

-t \

--1;+m I vd

\2a )

1-rr
-"

"fl\zotl ,u

ham sdfix) tr6n

gi6 tri nh6 nh6t cira

iR.

N6u a < 0 thi hdm sd flx) nghich bi6n tr6n

(t

khodng

\

l, COng bi6n tr6n khoing
)
( -n\
( _h\
vd
| -*:j
I
fl
'2a) " -a l ld si6 tri lcrn nh6t cua

l*.**
\/a

\

\2a)

hdm sO/(x) tr6n

1R..

a
th\

pi6f@=f(P)vit

fr*f f'l= "f (a)'

NhQn xdt

.1'

o Ki hieu maxf(x) dtng dC chi gi6 tq
nh6t cira hdm s6

f

lcrn

(x)tr€ntap D.

o Ki hiOu mlnf (x) dnng dC chi gi6 tri nho
D"

nh6t cira hdm s6
Cho hirm s6

,_h

2) Neu

f (x)tr1.r tp D.

/(u.l

/(

Pi

t)\
,l, b:
\

:0,

71r; = ax2 + bx + c (a * 1i).Ta c6

, ,i
cac Ket qua sau:

TORN
u

HQC

elirdiUe

@8*ss*$eeq6*6@asesees&ssees&eaa

*osm,d4


(

,_h
3) N6u o.-.p

1a;f

-b\
tht minf (x) = fl;),u

(x) =max{f ( a);f

(D\

,-h

o.

thi pa;f {x) =t(*)"u
zo< P
(x):min{f (a);f @)}.

3) NCU

ff!ftf

v

/(L)

tw)

.fi4

J@t

J@)

rl-h\
'\2ol

Bx

a_bp

P th\ pi6f @: f (B) va

il. cAc vi DV Ap nuNc
*fni dV l. Tirn gia tri nhd nhdt, gid tri l6n
,:
:
nhdt cia ham so f(*)=x2-4x+10 tuAn

q_b

2q

2o

Tradng hW 2. a < 0.

l)

,-h
N0u

fr?fif

2o=o

<

(x)= f (a).

doqn [3;51.

"v

Lfi gruL la co,b :2<3 <5,do d6 theo
-ZA
trudng hqp 1 (TH1) min/(x)
= f(3\=7 vit
"

,l-h\
'\2al
t@)

[3:5]

max
[];5,l

/(x):

*Thi

-b dp
-_h

2) Neu d
pa;f (x)= f

vir
ff.*f f*l = f @)

(P).

)

,l-h\
'\20t
tvt)
/(a)

/(5) :15. tr

2. Tim gia tr! nhd nhdt, gia tri lon
nhh cila him sii .f@)=xa-3x2+5 ffAn
doqn

dq

l-l;ll.
f

Ldi gidl Ddt t :

thi

/

-3x2

+ 5= tz

-3t

+5

Doxe [-1;1] n6n / e [0;1].
X6t hdm s6 g(r)

:

t2

-3r

Ta c6 0<1<-6-3

2a 2'

+

5 tr6n do4n [0;1].

do d6 theo THl

3 vd
Tlisfrl = s(l) =
t?1gtrl

=

g(0):5

ra 3 < "f (r) = x4 -3x2 + 5<5,Vx
md /(0):5 vd f(-I):3
Suy

e[

'

-1;1];

=min/(x):f(-l)=3
1_lJJ

vit max f
f-ut

TORN HOC *6..f*raa*irrr..*.?os.ilns.or.*
u

clLdiEs

(x): /(0) :5

.O

6dos*64


p:14;ll.

*fni

X6t him sO g(x) =-xz

Ldi gidi. Ta c6

Ta c6 g(x):-x2-3x+4>0Vxe[-4;1] vd
_A _2

dq 3. Tim gia tri nhd nhdt, gid tri ktn
nhat ct)a hdm sti .f (x) : sin2x - cosr + 5.

: sin2x - cos,r + 5: -cos2x - cosx + 6
D[t cosr:t, t e [-1;1] thi
"f

(")

-cos2x-cos.r + 6 = -t2 * t + 6.
X6thdmsO g(l) =-t2-t+6 voi
_4

ming(x) = min{g(-4); g(l)}
f-4Jl -

/e[-l;1].

: min{s(-

1); g(1)}

:

s(-1)

:

4 vd

)5
/ t\ a.
maxg(r) = El
Do do
l=
*\
^
-'
f-r:rl
2) 4

R" /(x)

:4:max f (x\:25 . A
R"
4

dg 4. Tim gid tri nhd nhdt, gia tri lrrn
nhdt cua hdm sti -f (*):sinn+cosjr- 2snLx+2.
Ldi gidi. Ta c6

"f(r):sinx+cosx-4sinx.cour+2
:sinx
:" t
+ cos.r

(-J, =,=Jr)

=

sinx.cosx

: i. D
l-4;ll"
2
*Thi dU 6. Tim gia tri ktn nhAt ctia biiju
thil'c P : a *b+ c -(ab +bc + ca), trong dd
a, b, c td cdc sti thac thoa mdn cac diiu ki€n
a+b+c>0 vd a2+b) +cr =1.
min l'(y) = Q

f-arll"

-

2

Khi d6

Tri

s(r):

Do d6

*Thi

min{s(

tr@n do4n

do tl6 theo TH2

-JlyilJiy: -Ji

c2)

:3=

r<

..5

(re to;",61).

do d6 theo TH2 ta c6

b+c

-

(ab + bc

*

ro\.7.
8

b:r*,":+

thi p

= Z.

vit
YQy

gi|tri lon nh6t cua P la 1.1
8

*Thi

qn/(r) :-J1;maxl(x) :4. J
dq 5. Tim gia tri nhci nhdt, gid tri lon

J-* 1x + 4

Ldi gidi. Tap xhc dinh ctra hdm
a;1]

o.-J= 1..'5,

2az

Khi a :0,

=-

nhar cua ham s6 .f @) =

D:[ -

<3(a' +bz +

.

,-'' 222
=l =-!7,*r*]

P=a*

max g(t):'(;)
r Ji;lVl

(t > 0)

'l

Ta c6 -Ji <- : ! r. Ji,
2a4

r+r4

a+b+c

/r)I=-.Suvra
i maxf(t\:fl"
ro;J31" \2) 8

.

1^

max /(x)

Xdthdmsdflr)=-,
i P +t +l v6i r e lo;",,51.
.
2
Ta c6

X6t hdm sO g(t) = -2t2 + t + 4

'

(a + b + c)z

vir P=

sinx+ cosx -4sinx.cosr+ 2 = 22 + t + 4.

r- Ji;Jit

g(l) = 0 vi

='(;) =+ Do d6

Ldi gidi. Df;t t :

*Thi

Ddt t

TfrI8(x)

:

1

L-Hl

min

1r6n

2a2

.

Ta c6 -1a --' - -' < 1 , do d6 theo TH2
2a2
qr4g(r)

-3x+4

dg 7. Xdt ba so thtrc a,b,c kh6ng dm
sao cho a+b+c=3. Tim gia tr! lon nhdt,
gid tr! nh6 nhtit ct)a bi€u thtrc
P : (r, + b2 + rr)' - 4(ab + bc + ca) +3.

LN gidi. D[t

t = ab + bc + ca(r >

0), ta

c6

.l

sO ld

ab+bc+ca3'

.

suy

ra 0
,ae\

TOfiN
HgC
- qhdiga

s*ss6*ee*$q*e&@&s&**ae*$6?***,*1ffia.s*6s.c6a*es*****a&*&*6&6

**uuuo fuoulvd4


Khi d6 P = (9 -Zt)' - +r +3 = 4tz *

+84

4Ot

.

Xdt hdm sO f (r) = 4tz - 4Ot +84 tren [0;3].
_h
Ta co 0 < 3 <:a = 5, do d6 theo THl
2a
min /(r) = f (3) = 0 vd max "f (t):
- ".f (0):84
[oi]l'
I0il

+0< P<84. Mdv6i a:b: c:1thi P:0
vd v6i a: b :0 vd c : 3 thi P : 84.
Vfly GTLN

*fni
.f'

(x)

P Id 84 vd GTNN le 0.

crta

fl

+ 6x

- 8 - 2Jx - 2 -

2.{a

-t

vit

2

thi

xy+12+o-"-2
2

+t - 1.

/(rl =!t'
2

Xdt hilm sO

.

io = J;la2 +b2 :2
-. i '__ -> ila>o,b>o
[a:J4-r

+t

,_b
la co

2a
---tnwf(t\=ftJq:2+i6,

ttdn (6;,frl.

-l

theo THI ta c6
suy

(o;J61"

K}i d6 J7+6r-8 -2^tx-2-2{4-x = fr'4a+b).
i_)

I

o
'22:a+b=>ab:' )ab-?1a+b\:1P -2t -1.
I

Vi a)0b)0 vd (a+b)?<2(d+bz):4

n6n

0
Mdv6i

o=b-r-!

ta co P

ra P < z* Je.
:2+

J6

3

YQy gi6tq icrn nhAt cria Plil

2+G^ D

III. BAI T,!P

l. Tim gia,ti nh6 nh6t vd gi6 tri 16n nh6t cua
c6c hdm s6

Xdt him s6 g1r; =
c6 0 < 23-J :
2a

Lrr,

-2,-1

2" do d6 theo

tr6n 10;21. Ta

THI

ta c6

t

lo;2

f (*)>-3,

Vx el2;4) vh ta

ra min "/(,r) =

__3.

l2;4

a) y=sin2x+6sinx-9:
b) y :sinx-cos.r+ sinx.cosx+
,
T-'=---:c) y= V-.x'+)x-6.

2

;

2. Tim gi6 try nhd nh6t, gi6 fr

g(2)=-3.
ming(t):
j
Suy-

t=x+y+z (l>0)

vir P = Lt'

DAt

Do d6

P:x+y+z+xy+yz+zx.

Ddt

Ldi gifiL Tdp x6c dfnh cua hdm sd 1;p:12;4).

Ddt

=J;i

Ta c6 (x+y+z)z <3(x2 +y2 +22)=6=il
dq8. Tim gia tri nho nhdt cua hdm so

=!f7

Ldi gi,rti. Dit x=Jo+b, y =Ji+",,
thi x > 0,y > 0,2 ) 0,xz + yz + z2 =2

c6l3)

:

-.

hdm s6 .f

-3.

I

*fni dg 9. T?m gia tri tctn nhtit cua bidu thac
p=rfa6anl6,r"+t[o+o+tZo+Lr+rlO+*+r7r+ott,

(x):
-

16n nh6t cua
+3xz
+10
tr6n
dopn [0;3].
-xa

3. Tim gi6 trl lon nhat cira bi6u thric
P = (a + b + c)2 * (o' * 6z a sz)z + 6
trong d6 a, b, c ld cdc sO thgc duong th6a
mdn ab'l bc + ac:3.
4. Tim gil,talonnhAt crha hdm s6

trong do a, b, c td ba sd thryc dactng thoa mdn

f (x) = J2-, +',1; +2 + Ja -

a*b-rc:1.

P

.

eh#-fiff, t$S,ffi E0qrt@, ,ffi-l$ t
tI7

.tft

sa$fiGffi3$FffiW*,:ffi&ffitfi,

uclirsiUe

TOfiN HQC

.&'&

d,:*$

6se4ss6a&&6&se6*es6aas**e&*****@ffi%rstsa6&ee*oe6&&*&Eea&&&E6a&E u*u*ou

W

.:,,,.

@fuvd4


frW*6far::rr:*ry**:
fGYSi-f Wfffi, lVSrfle,4n)

yiz

Ldi gidi. a) Vi6t PT di cho du6i

p{y,td mg,1 ffuAi'
"n*"q
*{;yAQ co bdn.q,hi giAi rca". nat
ffi

srfraeltg

'idi nhin

ta

(cos2x + 3cosx)(cos2x + 3cosx + 2) = m

dang

.

iet sd kinh nghfu-Edt t = X2 +3X voi X = cosx e [-1;t] ta c6
phg
fi
khi gifri m\t sb dWe'
I
todn lrgag gitic. :'i, ,r-'* 1 :
t el-z;+1. pT da cho tr& thenh qt +2): m
1i
-''
I. PHT.O\'G PUTP DAT .{\ PHTI DOI hay t2 +2t=m . Tim miAn gi6 trf hdm sri
\ or PHUoxc rniNn Lugrc

-f (t)=tz +2t voi t e[*z;+1, ta suy ra didu
Bmtc l. O6t an phu vd ndu didu ki€n 6n phu.
ki6n d6 PT de cho c6 nghiCm ld -1 < m < 24 .
Btdc 2. Dra phuong tinh (PT) di cho ve pt An
I
phr. Gi6i fT vcri ap phu tim ilugc. OOi cnieu voi b) DAt I = cosx +:. Vi cosx € [- t;t1
2L
dieu ki€n An phu de tim ga tq thich hqp.
1
3-l
nen t e | -"-. - - 1. PT de cho tr6 thdnh
Brbc 3. Tro vA c6ch dit de tim An chinh.
Eiffi.iniQu

sfc dyng y0u

cric

i

^

1 )'.)
L

Trong ba bu6c tr6n, bu6c 1 quan trong nh6t
loi gi6i. N6u dat An
phu kh6ng thich hqp thi sE d6n t6i PT m6i
kn6 gi6i hcrn PT xuAt ph6t.
quy6t dinh thdnh c6ng cua

Sau c16y ld m6t sO du.rg PT dcrn gi6n
duoc bdng phucrng ph6p tr6n.

giii

D4NG 2.

Cdch

(X *

a)a

gidi. Ddt

-3

+ nX

* cd): 7n

sO -f

-

rdi

+12A2 Bz + 284

Cdch

dii dua PT da cho vA PT rrung phuong An r.

l.

-

nmgyoz cosx vd sinx.

PT

+ sinx

: .,,D"or[,
'--""( -1).

4)

,.1-J1;Jl)
.

=

t2

-1

2

da cho trd thdnh

.

o." -l

+ bt

+c

:

2

*Thi

trinh sau co nghi€m.

b) cos4x+(cosx + l)o :

8

-)

gidi. Edt t :cos;r

Di6u kien

Tint diiu ki(n cua m dA phu.o'ng

+ sin3 x) = sin2x

Ldi gidi. Ddt I : cosx+sinx

m.

0.

dg2. Giai phu.crng trinh

O(cos'r

a) cos"r(cos"r + 1)(cosx + 2)(cos"r + 3) = m.

+!

+3t2

X6t phucrng trinh acosxsirur+ b(cosx* sinx) :.r.

dat

(l)

(t):2ta

re|

DA.\'G 3. Phrong trinh d6i

r=X*o!b roi dirng hing
2"
:2Aa

8

8

* (X + b)+ = ry.

B)a

|

i- r r-l
I l -tt

l1]u do Cosirstnx

(A + B)4 + (A

dq

Tim midn gi6 tri hdm

L

dang thuc

*Thi

hay
J 2t4+3t2+1=m
-

pT
I-a;j
) )tL sry' ra didu ki€n dd de
choconehiemld l< m
Crich gidi. ViCt PT dd cho du6i d4ng

(X' + nX + ab)(Xz
t=X2+nX.

I

-J

r t)' +lt+-l:m
/ l\4
lr-:l
\ 2) \ 2)
voi

t.

(.\' - a)( X + b)(X + c)(X + d) = m.
tron-e do ,Y 1a hanr so lu,mg gi6c vd a, b, c:, tl
thoa mdn didu kiQn cr * b = c + d = n.

D.1\'G

-

.

: O.or[r-I).

\4)

,A
TOfiN HOC

'clLdiEe

s&e6*e&*666q6*s*6aq&a*@&6e****.ffirs6a6s666ee+66a&e&6s&@se6&a *n***u

fidog*d4


Khi d6 , .l-'[l;Jil vd
n6n sin2x = tz

-l

cosx sinx

i 1. a) MOt sO Pr cAn dflt
n-)

nuY
r : cosir - sinx : Jiror( **
\.4,/

Chrt

=4

vd

cos3x+sin3x

:

pT da cho tro thdnh 3r -f = J2U, -1)

Gi6
t- e

tri

/3

*I,ft

b) Mgt

hay

bi loai vi kh6ng thoa mdn di6u kiqn

L

Pt

cAn

cosxsinx

c6

=+

d[t t = tanx - cotx

.

sinrx-cos2x
[: sinx cosx
=cos.r slnx cosxslnJ
_ _r.cos2 x - sin2 x : _2"?r?* = -2.cot2x.
2sinxcosx sin2-rr

_l c.
L,\,tt;f
L-\

sO

Klri d6 vin

Khido

-Ji .t.

=

\4)

t.l-J|;Ji.] *t**

/ 12-l\ 3t-t3
1l--.
lr))

:11

F + ^uEt2 _3t _Jl = O.
Do d6 t1= J2,6 = -J,

=Jlrin(*-l)

/=sinx-cosn

+ sin2 x)

- cosxsinx

(cosx + sinx)(cos2 x

Vi

).

cot2x e (-oo,+m) n6n I e (-m.+co).

ring tir phdp bi6n d6i trCn, ta nhdn

Chir i
k.2n, voi mqt k e Z '
Ji thi * =L*
duoc c6ng thuc tana=cot&-2cotZa.
4
Ntfu I :l-Ji thi x= Lto+k.2n v6i moi *-fhi dqr 1. finr tliitt kien di phuo'ng trinh
4

N6u r :

k e Z,trong d6

a

ia;l:r .i.

xdc dinh bdi

t-J, (" <-A <. -\
-r
: ------l
ft l. LJ
l't l?

/
\tL
\4

COSA

q'

a_;

t

q!1

1;

'rr

I'

',

t2

)

cho c6 nghiQm khi vh chi khi

gp,"1 !t..i :l i :

khi
t2

{-}.

4.

Do d6

lrl, Z

.

PT da cho trd thdnh

a(t2
tltnl

-2)+bt *c
--r -i-

=

j

:-':t

;'lthtiry"

Ldi girti. Ddt t :tanx+cotx. Khi d6 | t | > 2'

PT da cho trd thdnh tz

-2

= mt

.

Ta c6 th6 gi6i ti6p bing c6ch xdt midn gi6 tri
i_1
voi lrl >2, ctingc6 th6 tim
cua hdm to y:j,

"t

lz d6 phuong nixh dA cho vd nghiQm, hay trrong
ducrng vdi diAu kiQn tam thfic f(t):tz -mt-2
c6 hai nghiQm thu6c khoine e2;2) (chu ;f
tam thric f (t) c6 nghiQm v6i moi z). Tri d6
nhdn

dusc

lml>-t.l

0

c6 nghiOm, nghia la khi vd chi

Chrt j,2. Phuong ph6p dit An pfur AC Oai sO h6a

PT dA cho cflng thuong dugc su dung d6.tim gi6
lon nhAt ve nho nhAt cin ctrc hirm s6 lugng
gia". frry nhi6n, cdn nh6 reng c69 hdm s6lugng
itdc c6 frrft .nat t ran hodn vd d6 thi cua chring
Ihrr*g c6 tinhch6t dOi xrmg (qu9 Am hoic^qu1
trr,rc) n6n chi cAn xetchchdm s6 Ay kong mQt s6
khoAng thich hqp ndo d6.

d

0 vbiltl>2.

ijo1.:.r .- i'j{iirjt.\ -: i'rl,i.r

mt + 2 =

PT

A 0 hay m2 _8> 0 . Do d6 m>ZJi
hodc m<-2Jr.J

gidi.DLt t =tanx+ cot;r thi
4tanxcotx =

cot-r) c6 nghi€rn.

PT da cho tro thanh f + 2 = mt hay
tz - mt + 2 = 0. Vi r e (-.o,+-) n0n PT d5

tii"r. riri. r'r r ;1ri.1

i:iit1-:ll r' .i-.i-i-:l:l i,-, t;ilitil \'- irrr{-I} . /- ..
Cdch

rrl(tt111.\.-

Ldi gidi. a) D4t t = tanx - cotr thi
tan2x+cotl x=t) +2.

)

;r.' r.{r J. i'7';.'r ,
1 i.ilt.l..
,...r.L li

- r'trtl .: =

*.ffr; rlq 5. firt

gia

tri

lt'nt nltat va gid tri

nki nkat t'titr Jiitnt so
',.,
= ..,''1+ 2.in, + Jl +2co.s-r.
Ldi girti. Vi hdm sO da cho tudn hodn vdi chu
ky ix, n6n ta chi cdn x6t hhm s6 d6 ttong

mQt chu ki [-t;n]. Ei6u kien dO hirm sO x6c

[l+2sinx>o[ n 2n1
hayxel-.;
dinhldi
L 6 3l
ll+2cosx>0

1.

,;'

h{#(.
T*ffih{
s&RSt{H#

/,.dili:|**
'!,rir-qeqi..i'

r,
"

"

r

s 4 3 n pa*

r

4

os*4se

g s & ts

*

&$ s $ E

**

*

fidp sam sd4


dffi

Khi d6 hdm s6 dE cho xac
honnta

vd kh6ng

AnL

nghiQm

+2ll+

2(cosx + sinx) + 4cosxsinx

rr I sn
I x 2n-lnen
^ x--el--.-1.
vl r€l--'-l
L 6 3_t

(=_J )(i!_.=+) +2:ohay
(l+12 t+rr/(t
-P t-r, )'
(l - 2t - t')(l + 4t + t2) + 2(l - to) :0, hay
3t4 +6t3 +8tz -2t-3=0.
l.x
In)
I u do / = f---; hav tan- = ta nl +- l.
l, 6/
J:2

.

Sz-]

4 I t2 12)

,=:.!+k.2n,YkeZ

Suy ra

J

I
rz
y2: 2+2t +2.11+2r
+ 4.' -l

hay

2

ra co f (Ji)

= 4(1 +

Ji),

/ff.ll
(2 )

ki6n.x
'2

=

1

+.6.

*I+kn.A

MQt sd P.T luqng giiic thoat nhin khd phric tap
nhrmg n6u doi bi6n thich hqp sE ituoc PT moi
thu6c d4ng quen thu6c.

*Thi

drJ 6: Giai phLt'rtris rrirrlt
.I

:
-rmin

t+

Ji

v3

Sco:;'l.i

y.* :

Z,!1+.J-Z .

A

Chrt!,3. Trong c6c PT dang 3 vit 4 tadl dga vdo
c6c h€ thric luong gi6c ccr b6nm2a+sin2a:1,
tanacotq = I d6 dflt An phu.

Thuc ra c6 nhidu PT phai dua vdo nlrfng
cdng thric lugng gi6c phric tap hcrn moi chon
duoc c6ch cldt dn R[u thfch ho,p. D6hg chn 1i
Idr cong thric biOu di6n c6c hdm s6 luong gi6c
eua
'

tan{.

Ndu ddt I

tanl

rhi
=
2
2--.
2t
l-t2 tanx
sln,r = -------:-. cos.\ :

l+t2

*Thi

1+t2

=

2t

l-t2

du 5. Giai phu'ong trinh

. (-

{cosr-sin.r)l 2tan r+
\

r
COSJr

\

/

l+2=0.

Ldi gidi.Di6u kiQn cosx + 0 <> x + ! + kn.
2

NhAn xdt ring v6ix= rc+k.2n thi sinx=ta:x=O
vd cosr = -1 n6n PT dd cho h6 thinh

(-1).(-1) +2=0, vO li. VQy x=n*k.2n
kh6ng ph6i ld nghiCm PT. Ta tim nhfrng

TOfiN
HQC s$&s&s*&$sss

- Gfigiift$

&s$s@e&s$&&se

th6a min di6u

2. PIIT:O\G PH.iP DCt BIEI\

Mdt kh6c

Tu d6

x

=tan-

vd dua PT dA cho vd dang

I)
Ddt /I = cos,r
+ sinx = Jiror(
cos,r+sinx
V2cos[x*--;)
\ 4)

Honnfra

c6ch cldt t

2

= 2+ 2(cosx + sin.r)

!2

x*rr+k.2n bing

s***

*

tl

, .u..,,

LN gi,fii. Ve tUi b,iu pf kh6 phfc tap, do d6
ta d6i birin.x thdnh bi6n

t: x*1.

rcri d6 pT

3

da cho ffd thenh 8cos3l=cos(3r-n)
8cos3/

:

-cos3l <> 8cos3,

:

hay

-4cos3l + 3cos/

<+ cosl = 0 ho6c cos2t = 1.

,4
.N6u cos/=0 thi I =I+kn= x- !+m.
26
I
l+cos2l
I
--!
oNdu cos2l=' 1[i
424aI
<> cos2l--' -to"" o t - +l!+ kr
2
3
3 --=2n
=x-frn hodc x=-1+kn.
J
VAy PT dd cho c6 nghiem x=I+kx.x:kn
6
.2n
--.,r
va x=kn,(Yk eZ).A
J

Chfi i 4. Phucrng ph6p d6i bi6n thuong dusc
su dung trong vi6c tinh tich phdn cira hdm si5
lugng gi6c.

,\
*mil**
r:'ro*';

fu,o,sa*fi;4


dq7. Tinh tich phan , =

*:rni

Ldi gi,rti. Chir

-

= sinx(3

i

ring

4sin2

r)

i .i,,:,
Jffi*

sin3x = 3sinx

= sinx(4cos2 x

-

-

4sin3

x

1)

bing c6ch d6i bi5n x thdnh bi6n r:cosn
ta c6 dt =-sinxdr vd do d6

nOn

=';(44:\, ='('-!:\,
'dt+l
jr+l
7

rl.
--1

: olu - r)d/ +

l#,

:

2t2

3.

PHUoNG PHAP LUqNG GrAc

-

4t +31n1 r +r lli

= -2+31n2'

3

3.1.

noA

giii.

Dravdo tinh chdt hdm so laqng giac co ban'

ring hdm s5 y = cosx x6c dinh v6i
msi x nhfln gi6 ffi trCn dopn [-1;1] vd tudn

hodn v6i chu ki 2x.Do d6 v6i mqi m e [--t;t],

'

Ta bi6t

*fni

Chgn

dg8. Gidi phtrong trinh

,' +.[t -r'Y = *JzT- f

).

Liri gidi. Difi x= cosa vbi a e [O;n]. Khi d6
sina e[0;1] ncn r[-l =r[-€a =',ffi-o =titto
PT da cho trd thdnh

:

cosa + sina . Theo Thi dv2 ta c6

+ Ho{c t:J1.Khi d6cosa.sina :":l
22=!
6
x=cosa:sina
="
n6n

vd sinae [Olf]
2
+ Ho[c t

TOAN

-

=!-J7. rm ao

HQC aassee*ate$e**os*s*$i'aass

cfudiH

sao cho

/ --\
el-+,;
tan B = b voi d,P
\ '-/

J.

txtr=a,

rc,i 0o

- ab) - (tana + tan gA't*gYP)(+ a2)(l+b'z) (7+tanz a)(7+tanz P)
(a + b)(1

+ F)
+ F)
.coszacos2
.cos(a

_sin(a

p

cosacosB cosacosP

= sin(a + p)cos(a + F) = l.sin(za + 2p)

,

-t

< stn(2a +20\
.

,,rn\
.li,;)

drnn v6i moi a,

n6n

b.A

3.2. Dva vdo h€ thwc cosz q + sin2 o =l ' Ndu
c6 hai s6 a, b sao cho a2 +b2 =l thi chPn
duqc cx duy nhtit sao cho cosa = a)

sina = b, voi

*Thi

a

c-lO;Ztr).

dU 10. Trong tdt ca cac nghiQm cua h€

lx)

cosrd +sinia = .,/2cosasina haY
J2(cos' a + sin3 a) = sinZa.
Ddt t

a vd B

_l .(a+b)(l-ab)
2- (1+ az\(l + bz)=l2

e [o;n].

.

. Vi hdm s6 y = tartx x6c dinh vdi mqi

vit nhQn gi6 tri ffen iR' nOn v6i
**L+hr
)
m$h ddu chgn dugc a duY nhit sao cho
( nn)
tand=mvorael-):;1.
\ - -/
*Thi dq 9. Chtmg minh bdt ddng thitc
_1 . (a + b)(1 - ab) , L di,nn" t.cti rnoi a, b.
2-11+0210+b2\ 2

Vi
chgn duoc a duy nh6t sao cho cosa =m vfi
a

t2_

LN gidi.

Phdn con lai trinh bdy mQt s6 co s& d0 chuy€n
bdi to6n dai s6 ve Uai to6n luong gi6c d5 biOt
c6ch

l "t 't-li
=[-v/2 v2l
cosa.sina =;='
;"
r--Jz+t
-tlt-"'lz
sina e [0;1] nen cosd =

.

+

Y'=16

I

=9
lxt + Yz >12-

7,'*t2

Hdy tim nghi€m (x,Y,,,t) sao cho x+z
dqt gid tri lon nhdt.
LN gidi. Tt diOu kiQn x2 + Y2 :16 hay
.

\ 2

/

\/

[l] *ffl :t
[+,1

W

[+ ]

chen dusc

'6t'3"ttt"'8tas?&"Bcat6

a

duynh6tsao

t"o"*fidgg'?d4


cho

I = cosa,'44:
4

b) l+3tanx = 2sinZx;

rino,a efo:zn)nay

x = 4cosd, y = 4sina,

a

ti

Tucrng 4r
z, +t2 = 9 suy ra chon dugc p
duy nhAt sao cho z = 3 cuB, t =3srn[, p efa2n)

212 co
l2cosasinp +l2sinacosp

Thay vdo xt + yz

sin(a + p)>l

+

+

sin(a + F)

:

4cos

4

ThC

thi

a + 3sina = Sccr;(a - q)

(r)

SuV

[Ol:.J.

ra gi|tri lon nh6t cira x + z ld 5, dat cluoc khi
vd chi khi cos(a - e):l

e a, - cp+ n.Zn vbi n e Z nin d6. Vay
x = 4cosa : 4c.oscp =*, ! = 4sina : 4" rp =Y,
)5
=3sin|:3cosa = 3cosp :+
)
Vdymax(r * z):5, d4t duoc khi vd chi khi
.

6. Tim gi|
hdm

a)

.y

2)(sinx +3) = 24.

=

J."* + Jrin,
2x
4x
1+-x2 1+x2

= tlt

s1nir

cosx

Bii 1. (

die@. Giili

a)

-tan-2: ..ll-',

ls1]].x

phuong trinh

chc

x2

(cosx+l)(sinx +l)=m.
a) Gi6i phu
;

b) Tim di6u kiQn

.

ci:r-

m:2.

m dC phucmg hinh c6

nghi6m.

- crlc phuong trinh
3. Giei
a)

;

BAi 2. Q diA@. Cho phuong trinh

.

2. Tim m d,6 cdc phucrng trinh sau c6 nghi6m

cosJr

cbc

b) sin2x - tanx :2cot2x.

2

1l
b)

--- --

nho nhdt crta

of rrnnr rRA (as phrit)

1. Giai c6c phucrng trinh

mcosxsinx

lcrn nh6t vd

xy216
l_x2 l_yr'l_22- 2'

BAI TAP

:

ta

sO

.n
s s)
\5."s.2.]2)

a) cos3x + sin3x

Bdi 3. Q diAm). Tim gi6 tr! lnn nh6t vd nho

nh6t cria hdm s6 y = J +.i* + ",h +.orr.

+2:2rrn!;
2

,4.

TOfiN HgC
u

clldiE;e

.

+
") (J - ""* 1XJ + r"s, - 1) = 2.orr;
( r\
1 I
b) -+----" :)l la _ l.
x Vl-x2 ( V3i

(x. y. z,t\ =(
'u

b) sinax+(sinx+11+-1'
8'
c) I + sin3 2x + cosr 2x = 1sin4x

,*+=Ni
--'2 3

Chfng minh ring la@ + d) + b(c - ill< Jr.
8. Cho c6c s6 x, !, Z thu6c khodng (0 ;1) vi
xy * yz + zx: l. Chung minh rang

5

a) sinx.(sinx+l)(sinx+

trinh tanx-

lA nghiOm cria phucrng

7. Cho a2+b2:1vir c2+d2=1.

=3cos| =3sina: 3snq=2.

t

4. Cho tam gi6c ABC cdn vi m6t g6c cria n6

5. Giai c6c phucrng trinh

=l

3
trongdo. coS(p=-, srnp=r.ee

z

)5

Chimg minh ring tam gi6c ABC d€u.

>_12 hay

a+ ,2
0 =I+k.2n voi k e Z ndod6.

x + z = case-+ 3cosp

Zcosr8+ 1 = 3cosE.

c)

elO;Zn).

t.&e6a&&6te*{&a6**€*ee&tsaGu***.ffi.}are$eae{re&ea&s&**!*&&ae

*n****

fid?Mftd4


*T6T THI

ffiAOMTMffiTEffi
xi I rop

11

NAnn

2a11

THUNCHIirut

xcuyEN nAr cHAu

{Phb Vt/ tnt&ng Vu Giao duc Trung hoc'

a.!?:r

oE

B.

NGHIBnT xsAcH QUAN
(fhdi gian ldm bdi: 40 Phin)

s6

/:

rR\ll.l

4. Tap nghiQm cua phu:1 ld
sirx
(_l

(-)
B.
eZ;
l-1n
+txl,k
n. ]l
lz k2rl.k ez;
[2 )
C. {;r+ kx},k e Z; D. {n+k2x\,k e Z.
CAu

sinx ld
,

12)

rl

)

c. R\ l!+ml,k.r,;

12)

D.

lR

l{trn},t er'.

Cdu 2. Hdm s6 ndo sau clay nghich bi6n tr0n

/ _\

khoansi o:1
'11lz
\

A.y:

T:v:s !?e: Pe ?P&DT)

rnAc

Cdu 1. TAp x6c clinh cta hdm

A.R

$:glyyrq tul l-!* !r!!t e i:: !v:

L,/

B. y: cosx;
D. v:-cotx.

sinx;

C.y:tanx;

nhau?

CAu 3. Ducrng cong trOn hinh ndo sau ddy lh
sinx tr6n mQt khoing?
dO thi ctra hdm s6 y

:

Cdu 5. Trudng THPT Thlng Long ndm naY
c6 l2O hgc sinh kh6i t 0, 1 15 hoc sinn mOi t t
vd 110 hoc sinh kh6i 12. N6u cir mQt hqc sinh
cira Tru&ng THPT Thbng Long n[m nay tham
gia dqt kh6o s6t thi c6 bao nhi6u c6ch khSc

A.225;

8.235;

c.345;

D.

CAu 6. Ttr c6c chri s6

1s18000.

1,2,3 vd4

co th6 thdnh

lAp dugc bao nhi€u sO t.u nnien c6 4 chir s6

kh6c nhau?

A.

i;

8.4;

C.l2;

D.

24.

Cfru 7. Tir c6c chfi sd 1,2,3 vi 4 c6 th6 thdnh
dugc bao nhi6u sO t.u nhi6n c6 2 cht s6
kh6c nhau?

10p

A.

1;

8.2;

C.12;

D.

24.

Cdu 8. Vt6i OOng xu c6 hai m[t h sAp (,9) vd
ngua (1V). Ntiu phdp thir 1A gieo citng lilc hai
ding xu thi ta tluoc kh6ng gian m6u ld

I

n. {ss;NN};
c. {ss;SN;l'rlr}; o. {.ss;sr;.n/s;,n/l/}.
Cdu 9. M6t con sfc sic ld mQt hinh laP
{s;n};

6 mdt, m5i m6t tluoc kh6c c5c
chdm, sO ch6m tr6n m6i mflt ld kh6c nhau, tri
1 d6n 6. NOu ph6p thu ld gieo m6t con silc sdc

phuong cb

,4.
HQC
TOAN
-

&&a*es*&s&6s6q&&&+$ss&*&*s***r*ffir6seseoee6oe&&&&&s&s*6*6s6

cILriiEA

\*Eiiz'

u*o***fideMnrd4


ve xdt M ld bi6n cd "mdt xudt hi€n ctia suc
J

,

;

,:

suc co so chdn la so chdn

A.

M:

{z;+;a};

c. tt:{+l
C&a

;

"

tW\

B.

ru:

D.

M:{2\.

eliOm nAo?

ia\;

L.

ii

co sd hang tong qudt

10. MQt d6y s6

5n -3n
' = 2tt
- , yot n :

u,,

1, 21 3,... thi s6 hang

thir n+3 cua ddy d6 ld
5"s

-Xn+3)

2(n+3)

.

)

-3n +3
2n+3

5n+3

CAu

IL

Dfly s6 ndo dudi

ctr6y

v6i n: I,2,3,...?
A. u, - (-1), ;
B. tt, = 2,

li

m6t ddy sd

gi6rm,

I

C. lrn:-'

'

D. u,=5.

n

L" uro: 5 + 10.5 ; B. LIo :5 + 9.5 ;
C. u,.o =5+5e; D. un:5.5e.
13"

Quy

khAng pkdi
phing?

A"

tic

Lit

ddt tuong img/ndo sau ddy
mQt phdp bi6n hinh trong m[t

phdng v6i chinh n6.

B. f. DAt hrcrng img mdi di0m M thuQc mflt
,:
,.,a
^ dien
phdng vcvi mot
lt[, voi M ld di€m cho
tru6c.

f: Dlt
,3

m5i di6m M thugc mfr
,.;.
^.
phdng voi mttt dier* M'sao cho O ld trung
di6m cira doan thing MM' voi O ld di6m cho
tucmg fing

5. t

truoc, flcng drem

u rltfoc dat tucrng rmg vor

chinh n6.

D.

/

m6i di€m M thu6c m[t
-ffifu'=ffio,
M' sao cho

DAt tur:ng img

phing v6i m6t di6m
voi O ln di6rn cho trudc.

TOfiN HOC

'clirdiu6

B.//;

C. P

;

hinh
r,,n6ng IWPQ nhu
hinh b6n. Qua phdp
quay tAm P g6c quay
- 90" thi diOm Q bi€n
thinh dirim ndo?

A"M
C.P;

B.N;

D. Q.

Cdu 16. Cho hinh r,.u6ng MNPQ nhu hinh
b6n. X6t phdp doi hinh/c6 dusc bing cSch
thpc hiQn li6n ti€p phdp quay tdm A g6c quay
-90o vd phdp tinh ti6n theo vecta- Mj Qua
phdp doi hinh/thi di6m tuIbidn thdntr di6rn
ndro?

M;

Cfru

8.1/; C.P;

t7: Gqi O la trung

MAt. Phep

D. I

diOrn

.

cta do4n thing

,1

vi tu tdm ,41ti so - bidn iliern N
2

thdnh diilm ndo?

B.l/; C. O;
D. ,\f sao cho ffi':2h l

A"

M

Cfru

li.Ifting

.

dinh ndo sau dAy ld dring?

m[t phing hodn todn xric dinh khi bi6t
A. No di qua ba di6m.
B. N6 chira hai doan thing.
C.. No di qua mQt diem vd chfta mQt rlulng
MQt

f. E& tuong img m6i di6m M cira m5t

C.

M;

C&u 15. Cho

A.

Cdu 12. Niiu mQt c5p sO nhdn c6 sO hpng dAu
ld 5 vd cdng bdi q : 5 thi s6 hurg thir 10 cta
A A -,.,
cap so do
Ia

CAu

Cdu 14. Cho hinh binh hdnh MNPQ. Phdp
tinh tidn theo vectcr ffi aien di6m p thdnh

*&qr*6e*osseoaer6&&ea*&sa6&.*r$

thdng.

D. N6 chua hai cludrg thing song song v6i
nhau.

Cdu tr9. Khing dinh ndo sau d61,ld clung?
Hai ducrng thing phan biet

A. Kh6ng c6 di6m chung vd kh6ng song song
thi chdo nhau.

B. Ndm o hai m5t phing kh6c rrhau thi chdo
nhau.

..r..**a*$6*.r*6$r*r*t&scr

u"

*,

"

u

fido gard4


D. Kh6ng c6 diem chung thi chdo nhau.

Cfru

20.

Khing dfnh ndo sau d6y

li

Cdu 24. Ntiu (a,) li mQt cAp .O nhdn b6t ki,
trong d6 fi : 1, 2, 3, ... vlr k : 5, 6,7, ... thi
c6ng thuc niro sau ddy ld dfng?

dring?

Ducrng thSng kh6ng song song vdi m{t
phlng thi cit vdi m[t phing d6.

A.

B. Eucrng thing kh6ng nim trong mflt ph[ng
thi song song v6i m[t phing d6.
C. Dudng th[ng khOng c6 diOm chung v6i
mQt duong thing nio t16 cira mQt mflt phing
thi n6 song song vdi m[t Phing d6'

:

1

(t*

I

t6

I [6

(n

C.

)

\'lk = Llk-q'Ll*+'

-.,6"ot, :

1

t rzo\u
{-r + *zn\,t .z;
Jl.6

[2

)

I

D. {r + r,zn\u

12

)

,zn\,n.2;
)

iU n kzn\,t .z

I [6

)

Cdu 26. MQt b0 bii Tu-lo-kho co 52 qudn,
trong d6 c6 13 tOn ggi ld 2, 3, 4, 5, 6, 7, 8, 9,
10, i, Q, K, At vitm6i tcn g6m c6 bon crrAt ta
R6, Co, Pich, Nhdp. N6u nguoi ta rut ngdu
nhi6n ttx b0 bai c16 4 qudn thi viqc rut iluoc
4 qudn dAu ld qa6,n Q vdi x6c su6t beng bao

I

)

I

+ k2x\1,k eZ;
i-;
Lo )

D. A.
= (2x

-3y)5. Sau khi

khai tri6n thdnh da thirc thi

A. P(x;y) = 32xs + 240xa Y + 720x3 Y2

-

243ys

.3y +10.2x3.3y2

-10.2x2 .3y3 + 5.2x.3ya

-3ys
+3yt

l:
52

4
B.;270725
A

D.
52

A.NM; B.PN; C.QP;

D. P(x;y) = 2x5 + 5.Zxa .3y +10.2x3 .3yz
+10.2x2.3y3 + 5.2x.3ya

l:
A. 270725

Cdu 27. Cho hinh vudng Ivfr'{PQ nhu hinh
tr0rr. Xdt phdp doi hinh / c6 dugc b[ng c5ch
thuc hi€n liOn ti6p ph6p quay tdm O g6c 90o

vi phdp tinh ti6n theo vecto ,VP. Qua ph6p
ddi hinh f doqn thing NP bi6n thdnh dopn
thing ndo?

B. P(x;y) =32xs --240xay +720x3y2
-1080x2y3 + 9l}rya

nhi6u?

c.

+1080x2y3 +8l0xya + 243ys

C. P(x;y) =2x5 -5.Zxa

{r

+ r,zn\u JI*
t 2 I [6

l

Cdu 22. Cho P(x;y)

D'

c. {j

0 dugc

eZ;
+
B. l1
t) knl.k
\2

C- u'o = Llk-4.1tk+4;

B.

A. l-1 +k2xiU ]]1 +k2trl,t.z;
t*

B. ut = ttk-q'ltk*q

A.a;

trip nghiQm ld

r

A,. uf = Ltk-4.Ltk+ ;

Ch.u 25. Giii phuong trinh sinx
duqc tap nghiQrn ld

D. Dudng thing kh6ng c6 diOm chung vdi
mgi duong thing niim trong mQt mit phing
thi n6 song song v6i mflt phing d6.
Cdu 21. GiAi phuong trinh 2sinx +

B. ur:3n+l ;
D. un: 5n*t.

A. ur:6n;
C. u,:7n - 5;

C. Kh6ng cit nhau thi ch6o nhau.

.

D.MQ.

Cfru 29.Khdng dinhnio saudiy kh6ng drtng?

Trong m[t phlng'
Cau X.N6u mQt c6p so cQng c6 c6c s5 h4ng
thu ba va tht ndm tucrng ung ra 16 vd 30 thi A. M6i ph6p vi t.u t6m o ti s6 k (v6i k+0)
sii hsng tdng qu6t cua cilp sO aO ta
d6u ld mQt phdp d6ng dang.

/At
TOAN HQC
*

ss*s&6&46&es6s*qs*dssa&sa**aae6mo6s6€6aaee

cfuoifte

"W'

06*se6rqs6r*aaoe uor*o*

6-doua'#4


B. M5i khi thuc hi6n li6n ti6p ph6p vi tu tdm

ti

sd k

(vli k+0)

vd ph6p quay tam
q\ay rzthi c6 mQt phdp d6ng dang.
O

O goc

C. Ludn t6n tai hai duong thing a vit b cing
di qua O kh6ng thuQc hai clucrng thing d vd d'
.i
vd chung cung cdt cihai duong thSng d vd d' .

C. M6i khi thuc hiQn 1i0n ti6p ph6p vi.t.u t6m
O ti s6 k (vdi k+0) vd phdp tinh ti0n theo
i,'ecto i thi c6 mQt phdp ddng dang.

D.

D. M5i phdp bi6n hinh d6u ld mdt phdp d6ng

Cfru 30: Cho fu di€n MNPQ c6 MN : NP :
Pg : QM : MP : QN. Gqi O ld trung didrn
cua dopn thing MN. }/rq,t mlt phing (R) di
qua di6m O vd n6 song song v6i cd hai cluong
thing MQ vd,^/P. M4t phing (,R) cit tu diQn
dd cho theo thii5t diQn ld
A. Hinh binh hdnh (r,hung kh6ng ld hinh thoi);

dang.

Cfru 29" IGeng dinh ndo sau ddy ld dring?
Cho hai ducrng th[ng chdo nhau l]r d vit d' .
A. LuOn ti,n tai hai duong thing a vd 6 song

ci hai

v6i nhau vd chirng cung c6t

song

ducrng thdng d vit d'

.

Kh6ng th6 c6 hai duong thing a vd b cit
nlrau vir chfng cung cit cb hai cluong thhng d
vit d' .

B. Lu6n tdn tai hai duong thing a vd b cht B. Hinh thoi;
nhau vir chirng cirng cit cA hai ducrng thdng d C.Tam gi6c (nhrmg kh6ng ld tam gi6c
,.:
-J' )a Lt

d6u);

D. Tam giric d6u.

DE rV'LUAN
(Thdi gian ldm bdi: 80 philt)
ti6n cua n6
c6ng sai d

clAu

Bdi I. (18 diam)

\.ry

l. 6 dien) Giai phuong"2
trinh sin2x 2. $ die@ Gi6i phucrng trinh
..l3rir2r-cos2x =-Jr.
3.

$

diA@ Tim t6t

thudc nua doqn

cd

cdc gi6 tri cria An s6 x

[,r

| -;:3r

\

I thoa rndn
)

Bdi II.

L$

Tt

c6c

cht

s6

0,1,2,3,4,5

c6 thO

nhi6u sO tU ntt6n c6 5 chri sO
kh6c nhau, lon hotr 50 000 vd chiah6t cho tO?

2. $ diA@ C6 30 c6u h6i tric nghiQm kh6ch
quan ducrc d! ra cho m6t tO hgc nh6m. MQt
hgc sinh mudn.chon 20 cdl. H6i, n6u cl6 chon
5 cAu r6i, thi s6 c6ch chon c6c c6u cdn lai cua
hoc sinh d6 ld bao nhi6u?

3. $ die@ Dirng tam gi6c Pascal chimg
rdng C! +C! + Ctr:71.

t6

Bii IIII. (12 diam)
L $ dieQ Cho c5p sd cQng (u,) co s6 hprrg
d6i cua s6 2u5 vd t6ng cua 14 s6 hang

TOfiN HOC

ltr
322

sA,= -sA. sB'=

-sB. sc'=:.sc.

2. $ dietd Chrmg

minh hai dudng thing SA

vd BC'ld hai duong thing ch6o nhau.
3. $ di\@ Gqi 1, .r lAn luqt ld cbc diem dOi
ximg cira A' qua B'vd C'. Chrmg minh ring 1/
song song v6i mit phing @Bq vd ff gi6c

BIJC ld hinh hinh binh hdnh.
BAi V. (6 die@ Cho hai tam gi6c ddu ABC,
ECF sao cho ba di6m B, C, F thing hdng vd
.l
cung ndm tr€n mot nua mflt phdng bo BF. Goi
M, N ldn luot ld trung diem cua BE vit AF.
Hdy xdc dinh phdp quay bi6n di6m A thdnh
di6m B, bi0n di6m F thdnh di6m E. Chimg
minh ring CMNldmQt tam gi6c ddu.

6qas6644e4essa&&6&66e6&s66.*.-.#ffi%06a6+@oqs6&*&ea666*s&&qe*a&

'cfndiffe

hang ddu ur vit

diA@ Tim c6c giao di6m (n6u c6) cria
c6c dulng thlng A'B' vd A' C' voi mp(ABQ.

14p ctugc bao

z1 ld sO

sO

L$

(7 diem)

diA@

14. Tim

2. $ diA@ Chirng minh ring s6 fi +2n lu}n
chia h6t cho sd 3 v6i moi s6 nguydn ducrng zi.
Bii IV. (17 diA@ Cho hinh tri diQn SABC.
Tr6n c6c canh Sl, SB, .lC ta lAn luort 15y c6c
di€mA' , B', C'sao cho

phucrng

LJ
trinh cos2x+ 1,5cosr:0,5.

li

'5W

******

fidoM

sfirMwcAms
DE TI{I

[i

Chudn
cho lri thi

^\

TrtONG

I6t nshi0p THPT
uir thi vio
Oai hoc

SINI{ f}AI I{OC
. - A1, NAM 2,OL2
vt

'

ouoHe Duc HAo
(GV THPT Hudng Kh€, Hit Trnh)

Di thi mydn sinh vdo Eqi hpc khdi A - At

ndm nay c6 CAu 5 hA.n quan ddn
nhils phdn khd co bdn cua hinh hec kh6ng gian dd ld tinh the t{ch kh6i da di€n v.d
t{nh khoaig cdch gitca.hai dmtng chdo nhau. DAy ld dqng todn thwdag g\p trong fhiey

ndm qua. Trong bdi vi/ ndy, chilng t6i xin trinh bdlt mAt vdi suy nghi thAm khi gidi cdc bdi
todn thu$c dqng nay.

ft
I

rons

C6u 5. (KhOi A - A1 ndm 2012). Cho hinh
chrip S.ABC cri day ABC tit tatn giac ditt
can.h a. Hinh chiAu vuong, goc' c'ucr S tr1n mat
phdng (ABC) ld di€m LI thuoc canh AB .scrrt
c'ho HA = 2HB. G(tc gifi'a drd'ng thdng SC lr)
mQr.phdng (ABC) bing 6t't". Tinh rhi ti,'lt ctict
khoi r:hop S.ABC va khoang cach giti'a hai
drong thdng SA vit BC.
Hwting ddn . Tinh Vs ap6.G9i 1ld trung ditim
AB, goc gifra ducrng thing SC vd mflt phing
(ABC)

lil HCS. Xit tam

rtr -a. ,. -{f.+
tu6,rw-----?'

=

Sl1

=

gtdc

H/--

HIC ta c6

^trn,

- ila -ail

3

BA3
HA2

Xdt tam gi6c vu6ng SHF ta co

lttt;
HJ : o^lL.Suy ra
=
HJ 2 I7S2 HF:=
\ Zq
-+-.
-

='F

U:trc$)

8

Cilch. 2. (h. 2).

X5c dinh

o.E,
-"!3'

mat

chua

mOt

phang
BC mir

song song voi
,514 nhu sau.
Ldy C' dOi xtmg

'l'i,,ry.

chaa SA song song
vor BC nhu sau. Kd
AEIIBC; /[-fif -ct

-ln'r-l
r I L+-I .l- t)-

1l? 'r

I

BH, = AH,
K6 BB'IIAS;

BB'=,4,S thi
tam gi6c BA'C'
clAu; ,4BB'S ld

I

/ Lit\

C. qua B; A'
d6i ximg A qua
B; I{' e BA';

Cfch 1. (h. 1)..Ta
x6.c dinh mit phlng

-

d(B;(AES))
d(H;(AES))

'2

,*, =* >v, uuc=!sn s *, =+(dvtt).
. Tlnh d(SA;BC).

TOAN

Ta c6

d(BC,SA) = d(8.(AE87 =)ar

HCtan6O" =

th\ ACBE 1d hinh
thoi c4nh a ndn
BCil(AES).
Do il6 d(SA;BC)
= d(SA;(AES))
= d(B;(AES)).

Gqi Kld tmng di€mAE,ke HFIIBK (F e A$
vd IIJ ISF= ril t (AES)>d(H;(AE\)=HI.

t

,'

/

i'/Y-tl

,\

,'f{

A
Hinh 2

hinhthoi cq,.tha]'

Hinh

B'I{ ll SH * B'It -L(BC'A)

l

A

)

AS l /(BCB')

>

HQCa&ssd$6s4sss*s6sa&&@&ess66*****@[ffisa$e*&eaeeo*&*s*4&*&@6ees&oeeoeo.i--@osapsd4

clijdiui

\e,


+

x6c dinh khodng c6ch tu mQt di6m b6t ki tren
tluong thdng niry d6n mit ph[ng kia. Th6ng
thuong ta vfn dgng hQ thric luqng trong tam
gi6c vu6ng vd k6t qui sau:
l. Tilr diQn OABC cd OA, OB, OC d6i mQt

d(AS;BQ = d(AS;(BCB)) = d (A;(B C' B')).

Met
' kh6c

d(A';(BC'B')) _ A'B

:1. ,rr.u

d(H':(BC'B')) H'B 2'
a
1
:d(H';(BC'B')).
d(AS;BC)'2
Ke H'IlIA'J,
=
t;
trong d6 -rld trung di6m BC' th\ A'J -'!-t,

vudng g6c

2'

g1=?1tr':a.tu

3

"o

J:',

IH'B'ta

1

c6

I

gi6c vu6ng

H'K2 H'12 H'B'Z
Tu d6

d(

AS:BC)

C6 thC tinh khoAng clch gi6nti6p th6ng qua:
2. Cho mdt phting (P) vd d.rdng thang Lcdt
(P) tqi O; M vd N ld.hai di€m tr€n L; H vd K
ldn lrqt ld hinh chi€u trong trng cfia M vd N
tuan (P) khi dd

--)

|

24

:1tl'

28rc =

oJ42

d(M;(P)) _MH
=Mo

.

d(N;(P)) NK No'

C6ch 3. (h. 3)
Chgn hQ toa d6
sao cho O=H;

Tru&ng hqp kh6ng v4n.dgng dugc vdo tu
di6n lu6ng, ta sir dring k6t qui sau.
3. Cho fii, dt)n OABC c6 OA L(OBC), dd tim
hinh chiOu caa O ffen mdt phnns @BQ ta
fumg OK LBC;OH LAK. Khi do OH L@Bq
vd d(O;(ABC)) = OU.

OxllIC;
Oy chfia HB;
Oz chfta HS.

Tia Hx
cit ac
tai Tthi

.

D6 vOn dung duoc phucrng ph6p tqa dO vdo
viQc giii c6c 6ii to6n tinh khoarig c6ch diAu
quan trgng ld phni chon duoc hQ tga dO thich
hqp, i16 fii d6 d6 x6c dfnh cbc tga dQ c6c di6m
hay phucrng trinh c6c ilucmg thdng.

O

THBH2
A BI 3
)
= HT :1CI

:ft

BAI TAP

Dod6 H(o;o;o);

r[f

Ifi

=

E),

* : (a,

1. Cho hinh ch6p S.ABCD, ddy ABCD ld hinh binh
hdnh c6 /g=qBQ=24 fr1=tzP; SA=SC,SB=SD.
Goi M vit ff lin luqt ld trung di€m AD vit BC. Hdy
tinh d(SM;ND), bi6t ring d(S;(ABCD))=3a.

'o'o)'

,(0,;,r''[0,0,",[)' r(r-+,0)

a :(0,!,,

suv,u

2. Cho hinh l6p phuongABCD.A'B'C'D'c4nha, Glir
trong t6m tam [i6c BeD, N 1d trung di6m AB. Hdy

-

tinh d(GC':A'N; theo

: n),

hirh,

d(BC;AS)= d(TB;AS) = lw,n14=ot[42

ItB,,ill

a.

3. Cho hinh ch6p S.ABCD c6 d,6y ABCD ld hinh binh

1O;a;0). Do d6

4Nnpn xdt.

tr€n

_ 1111
_+_+_
oH2= OA2 OB2 OCz

H'I L BC'. Kb ItK LIB'

thi ItK L(BCB). Trong tam

thi hinh chidu cila O

(ABC) rritng vdi trryc tdm tam gidc ABC. Khi
dd d(O;(ABC)) = OH vd

L(ABCD),frd :60'.

HEy tinh d(D;(SBC)) theo a.

4. Cho hinh ch6p tam

gi6c. SABC, ddy ABC ld tam
gi6c tl6u canh a; Hai mflt phAng (SlB) vd (SlQ w6ng
g6c voi d|y, g6c gita hai m[t phlng (SAB) vd (ABQ

8

o Ngodi ra, dd

tinh khodng^c6ch
gifia hai duong thlng cheo nhau ta c6 th6 guy
v0 x6c dinh khoing c6+ gita hai mflt ph[ng
song song chfa hai th[ng d6. Tt d6 quy vO

AB =2BC =2a, SA=3a, SA

.15
60", AP=-47'-

bdng

AB(Pe ABt:AQ=IAC(QeAC).

HEy tinh d(PQ;SC) theo a.

,4.

*cfirdifie

TOf,N HQC

oc&.o.+&&q&6".s{e*er.*$ne***".dm%Dco.cre.cc.&r&i6a*&&&*s6*a*

W

o*nn*.

fidoMtud4


ffiffilr$ffi^r

USI'D-CETHI ITU

HGusylgrTIYtT
{CYSHSPHe

thi tuy0n sinh vdo Dai hqc m6n Vat li
tpp trung chir ytiu
ndm2012 v€ n6i dung-l6p
12, chi c6 hai
trong chucrng trinh Vat f
c6u su {rng d6n ki6n thric lop 1 i. C5u truc dC
thi tr6i ddu c6c phAn. 56 lugng cau m5i phAn
.,4

,
giong nhu cdu truc dE c6ng b6. DC thi n[m
nay v6 d9 kh6 cffng tucrng duong eC tni tuy6n
sinh vdo Dai hgc ndm2}71, nhung kh6 hay.
56 cdu dO tay cli6m, tric ld chi cdn hiOu li
thuy6t hoic thuQc c6ng thirc rOi ap dpng cho
,;
..a
k€t qu6 ngay chiem kho6ng 35%. S0 c6u hoi
phii tinh to6n nhrmg klr6"g ddi chi6m khoing
25%. Sd cdu cdn lai cAn c6 sir tu duy vd cdn
tinh to6n ddi hcrn, trong d6 c6 kho6ng 15% s6
cAu doi h6i tinh to6n ddi ddng. MOt s6 c6u hoi
Lh6 tap trung vho phAn co vd diqn. Hgc sinh
ph6i hgc ki li thuyOt vh rdn luyQn nhi6u ki
ndng tinh to6n m6i c6 th6 ldm t6t clugc biri
thi. DC thi n[m nay c6 phAn nQi dung vd: qr
xoay, g6c xoay. N6i dung ndy dd kh6ng ra
trong dd thi dai hoc nhiiru ndm nay n6n thi
sinh thulng chtr quan vd bo qua, hoic chi hoc
scy sdi nOn ctng kh6 ldm dugc. D{c biQt, de
thi ndm nay c6 di6m m6i ld mQt s6 cdu hoi
bi6t gan v6i thuc t6 hcrn, vi dp nhu cAu h6i vd
s6ng diOn tu, truy6n t6i diQn ndng di xa,...
Ddy ld OC ttrl ddi h6i thi sinh kh6ng nhfing
hi0u li thuy6t, bit5t suy lufln md con phii c6 ki
n[ng tinh to6n nhanh vd chinh x5c. Thi sinh
t.
i. :
,

phdi rat xudt sdc moi c6 th6 d4t di6m 10. Hoc
sinh gioi kdm vdi vi6c rdn 1uy0n nhi6u bdi t4p
thi c6 th6 dat tu 8 - 9 di6m. Hgc sinh kh6 thuc
su c6 th6 clat tu 6 - 7 di6m. Do phAn H thuyi5t
tucrng eOl AC n€n hoc sinh nim vfrng li thuy6t
cflng c6 thO dpt tu 3 - 4 di€m.

fle
Ll

.:,

1. DANG CAU sOr VE DAO DQNG,
SONG VA CO
Cffu 1. M6t con ldc ld xo gim ld xo nhe co de
cang 100N/m vit vat nho khdi ltrqmg m. Con
Itic dao dQng diiu hda theo, phao'ng ngctng
voi chu ki T. BiA 6'thdi diem t rat c'o li d6
| -.-n
^ l+-T val co toc
do 50cm/s.
5cm, a thcti diem
4

Gia tri r:tia m biing

1,2kg.
C. 1,0kg.

A.

B. 0,8kg.
D. 0,5kg.

Hrdng ddn. Ta c6 phucrng trinh ctra dao
dQng didu hoa

x

= Acoslat - e) = sror(\'r
! + rtl.
)

' ra

Suv

v=

dx 2x (2rt (0\
T [r-+ ')
-dt- -1.-Slnl
(2nt
)
a*r/l=5cm

|

Theo gi6 thi6t

.

.-

lcosl

/t--

(2)

TLi (1) vd (2) suy ra 7 = 11r;. Ap dpng cdng

)

thuc 7 :

Zn.E , thay grd ta otn k vd T ta
VT

dusc

ndy theo MA dO 196.

Ddp 6n C dung.

m: l(ke).

il

6&s6+**ea&ea6a*6aaae&s$*e66**o+m.€e6...o.o.&*6e*.&r&e6*arc

cfudific

(l)

\

L)L
I ZILL
] -^
l.-cosl
I =5Ucrn/s
-+@ )

" r \r

hav

')

Ir

Du6i dAy chring t6i sE gi6i thiQu vh hu6ng
,: ^ so:. dqng cdu h6i cira clO thi ttai hoc
d6n m6t

TOfiN HQC
*

N1p,J

"W

nn*o**

fu?w?fi4


CAu 11. Hai dao dQng cilng phuong

c6

phtrcrng

trinlt x, :

(

lin lrqt

(cos(,., * ]'11.*;
\ 6./'

va r:-i)=6cosi rur-fn\.l(cm). Dao dong
t

tong

-,)

hqp cua hai dao dQng nay co phtrong trinh
x: Acos(nt + rp)(cm) Thav ddi A1 cho ddn
khi bi1n d0 A dqt gia tri crc tieu rhi

A.

e=nrad.

B. Q =

C. q= 0 rad.

D. e= -

1l

-6

hqp ndy

. u
i--'zR

*dA?

tane

r0, v{y A

\,6,

-

Ddp 6n A dring. D

\

2)

t.

= -!J,

\.6/

Ia

(D

tL

=

_

-- a tad.

Cdu 47. (PhAn ri6ng cria chuung trinh chuAn)
MQt dQng co diQn xoay chiiu hoqt d6ng binh
thtrdng vcti di€n ap hiQu dAng 220Y, cudng

dQ ddng di€n hi€u dqrng O,5A vd h€ s6 c6ng
:,
suat cua dQng cct /a 0,8. Bi€t rdng c6ng sudt
hao phi ctia dQng cct td llW. Hi€u suAt cila
I .dQng crr (ri so giira c6ng sudt hftu ich vd cdng
suat ti1u thu todn phAfl h

4.90%.
c.92,50 .

J

Ddp 6n B dring. O

2. DANG CAU HOr
XOAY CHIEU

vE DONC DrEN

8.87,50

.

D.80%.

Hrdrng ddn. C6ng su6t ti6u thu todn

phAn

,J,

Dang

4

Ciu

37. Eqt di€n dp u:UycosZvft vdo hai
)
'
-;
dau doqn mach gdm diQn tro thudn R, cu6n
cdm thuin co d0 tU cdm L vd tu di€n c6 di,An
J
:,
dung C mac noi ti€p. G7i UR,UL,U. ldn

=ulcose =220.0,5.0,8

= 88

0I/).

C6ng suAt hiru ich bing

gfr =88*11= 77(W).

TORN HQC rsecra&a*{8*ie6*r**aa66'.*--...d&na..oco..e6+*ae*se**}***e{.

'clirdiE;e

U^-*). Nhu vpy

dat cgctieu tai

:"or[1)
SUY

" vi diQn 6p hiQu dpng tr0n

Un

di€n tro R ld lon nn6t (Uo =
phii thay AOi C.

3.

c6

vir

4An36'

srir[1'']* ori,r[-I)
Ta

u

dopn m4ch c6 cQng hucrng). Trong tnrong

Tai Ar: 3 thi

At:

Lr-

Ar-3

0 suyru 4=3.

i=

iQch pha

Ca . ta th6v rins
i: tano,R=
neu u vit i cirng pha thi e=0, tric h
l'l
La- ^ =0 hav
Lro- ' (hic d6 tr€n
'
Cco
Co
glua

Kh6o s6t Atheo At.

dA

B. Thay doi R dA U.,u*.

Hudng ddn. Tn bi6u thric
e9
"A
|

rad.

A=riA? _6A,+36.

J Al

Thay aAi C de IJo^u*.

D. Thay atji L aij Ur^u*.

Hmimg ddn- Ta c6 bi6n dQ tdng hqp

M
dAt
-:

A.

C. Thal, aiii 1aa uc*,*.

-1tad.
TI

Itror la di(n ap hiQu dang giita hai diu di(n

, -; cu)n cam vd gitra hai ddu tu
tro', giita hai dau
diQn. Trudng hqp ndo sau ddy, di€n ap tac
-: doqn mqch cilng pha v6'i
' dau
thoi girta hai
diQn ap tthc thdi giira hai ddu di€n trd?

W

n*uoun6.fusatysd4