TRIJONG EAI HQC VINHTRIIONG THPT CHUYTN of xrrAo sAr cnArr,ugr\c t 6p tzr,An 3, NAna zorr m0n: TOAN; Thli gian lim bhi: IBA phrtt r.. Gieiohuongtrinh lsinZx-cos2xtanr* sin3x =sinr+cosx
Trang 1TRIJONG EAI HQC VINH
TRIIONG THPT CHUYTN
of xrrAo sAr cnArr,ugr\c t 6p tzr,An 3, NAna zorr
m0n: TOAN; Thli gian lim bhi: IBA phrtt
r rHAN cHUNc cHo rAr cA rHi slr.{H 1z,o a$q
1
Ciu I 1Z,O ei6m; Cho him s6 y "4= 1*o -(3m+l)xz +2(m+l), m ldtham s5
l Kh6o s6t sg bitin thi6n vd vC dO thi hdm sb c16 cho khi z = 0
2 Tinr llr A6 A6 fti ham sii da cho co 3 tli6m cgc ti l$p thanh mQt tam gf6c e6 trgng t6m ld g6c toa d0
CAu II (2,0 tti6m)
I Giai phuorg trinh 2Iogo(l a ,l2y a1= logz (5 - r) + log , (3 - x)
2
2 Gieiohuongtrinh lsinZx-cos2x)tanr* sin3x =sinr+cosx.
Ciu III (1,0 di6m) Tinh thc tich khdi trdn xoay tu":iah khi quay hinh phang gidi han boi d6 thi hem
1(:.1
')L-s6 y =
-l-t-4r ttl
Ciu IV (1,0 di6m) Cho hinh Hng tru dtmg ABC.A' B'C' c6 AC = a, BC =2a, ZACB = 1200 vd <tudmg ttrang A'C t4ovoim{tphdng (ABB'A') g6c 300 GgiMldtrungdi6m BB' TinhthCtichk*r6i
teng trU dd cho vd khoang c6ch gita hai ttuong theng AM, CC' theo a
CAu V (1,0 iti6m) Tim a dO hq phucrrg trinh sau c6 nghiQm
II PHAN nrtNc e,o iti6m)
a Theo chuong trinh Chu6n
CATVIa (a0 di6n$
l*'
^[y a -2xy -2x =1
ft' -rr-: x! = a+2
Thi sinh chi tlugc ldm mQt trong hai phdn (phin a, hoic b)
l Trong mat phdng tga ilQ Oxy, chodudrng thang d:2x+y+3=0 vd elip (E) '41 ,t*t-=l Virit phuong trinh dudrng theng A vu6ng g6c voi d vit cht (D t+ihai di6m A, B saocho diQn tich tam gifuc OAB bing 1.
2 trong kh6ng gi* tqu dg oxyz, cho m{t pheng Q):2x-y+22+9:0 vd hai di6m Ae;-l;2),
B(1;- 5; 0) Tim tqa d0 cta diOm MthuQc (P) sao cho ffi.uE d4t gid tri nh6 nhAt
Cf,u VIIa (1,0 tli6m) Vitit ng6u nhi€n mQt s6 tw nhi6n ch8n gdm.4 ght s6 eoi mQt kh6c nhau l€n bang
Tinh x6c su6t e6 si5 vira vii5t th6a mdn trong s6 eo m5i crrt so dAu lon hcrn chit s6 e,mg tru6c n6
b Theo chrorrg trinh Ning cao
Cf,u VIb (z,o ai6n;
1 Trong mat ph6ng tga d0 Oxy, cho parabol (P): y' = 4x c6 ti€u diOm F Gqi M h <ti6m th6a man didu kiQn Ffr = -3fu; d ld ttuong th5"g U6t ti tli qua M, d cgt(P) tai hai di6m phdn biQt A vit-y.
Chtmg minh reng tam gi6c OABliLtam gi6c vu6ng
2 Trong kh6ng gian tga dO Oxyz,cho dudrng thang d,**=l ='.4 =l vitc6c <tii5m Ae;2;7),
-2t2
B(l; 5; 2), C(3;2; 4) Tim tqa d0 di6m MthuQc d sao ebo MA2 - MBz - MCz dat gi6 tri lcrn nh6t
Ciu VIIb (1,0.tli6m) Hai ban An vd Binh thi it6u voi nhau mQt t'{n b6ng ban Hq quy u6c choi v<yi nhau nrlAu.nh6t 5 s6c, ai theng tru6p 3 s6c li ngyd th*g cuQc vn tren rliu k6t trtir rinfr:<a r"aida trat
d6u t6t ttnic sau s6c thf tu, bitit rang xac su6t An thing trong m8i s6c ld 0,4 vd s6c ndo ctng c6 nguoi
th5"g
.L
d-
/n-n6t
Ghi chrt: L Bfg s€ trd bdi vdo cdc ngdy 21, 22/05/2AI I DA nhSn iluqc bdi thi, thi sinh phdi ngp lqi
phidu du thi cho BTC
trongxuanht@yahoo.com sent to www.laisac.page.tl
Trang 2-['ltu'*i\i{i S,{i l-it}C !'i}ili
'trR{i'#.}iil T'l-{P I' il}"ILryEN
DAP EN+ BE K}IAO
M$F{:
$AT CIL-{T L.u-'#io{G L{3P 12 LAN 3, i\.4,.h,{ ?{}11
TO,LN; Thcri gian ihm bii: J8 i) pltfit
I,
(?,0
rr^ \
{ttcm)
Khi m - 0 harn sd trcr thanh y =+t
4
a T?p xic dinh : D = ffi ;1l le hirm sO
b Su bi0n thi€n:
* ChiAu bi6n thi6n: Ta c6 y'- xt * 2
[x=0
y,=0 e \r7 I
Lt = +Ji'
Suy ra hdrn sd ddng bi0n tr0n c6c
o
'* r' +2
chiu
t-x > J, [' -Ji
y'>0<+l ,_ ;y'<0el
r
l-J1<x<o/'/ Lo.x<Ji khoing (-.D; O) vir d1;+ oo); him sO nghich bi6n trOn cAc
0'5
khoing (-*; - Ji) uit (O; Jz).
*Cuctr!:Hdmstidatcgctl4it4i x=0 vdi y.u=2;hdmsi5datcqctitiutqi x= Ji ve r=-Ji
vbi yr, =1.
c" Dd thi:
ss-tfri 'itdiii s# nir$n trgc tung limn trui: d6i
xr?ng.
Hilm s& dd sho c6 3 di6m cgc tri € 3 nghiQm phfin bipt
{=} x3 - 2(3 ln + l)x - 0 c6 3 nghiQm
Khi d6 3 di0rn cuc tri cria
.I'= 0 c6 phAn bipt
d6 thi
1
QM .a
J
le AQ;2m+2),
B(-(1)
-9nr2 - 4m+ 1) va
gi6c ABC a !a+2!n - 0.
I(€t hqp vo'i (1) suy ra gifttri cua m|d m -+.
3
Diou kiqn: 1= x < 3.
2
Khi do phuong trinh dd cho e logr(1 +
e logz(1 +
JZx -_I) - log, (5 - x) - logz (3 - x)
Jzx-r)-log r= 3-x
-#e^lTx-l - ?
f m Zl3
3m-2- 0 e I
L* -Il3
fidn tin
1 (1,0 diint
2" ffn# di6m)
L (1,0 ili€m
/
I
I
I
-"{;
6nt + 2;
2x -I
e1+
Trang 3I
I
I
I
I
i
I
ta
{x *3}'{?x * 1} - { c* {x * t}{?x: -* I tx + 13} * * E=*
X.6t hqp di0u kign ta co nghigm cua phuong trinh la x: l, 1t-Jr?
'r,-4
1 o , Jo *
., r i t Vt /
l'r' -{-tt 4) t'u
A T
tlrJ
2 $,8 iti€nt) DiAu kiEn: cos.r * 0 +> x *L* ')" kn,.k eZ.
V6i di€u kiQn d6 phuong trinh tuong cluong v6i
sin2.xsinx - cos2;rsinx + sin 3x = cosx(sinx + cosx)
e sin 2xsinx - cos2-rsinx + sin 2xcosx + cosixsinx = cosr(sinx + cosx)
e cosx(2sinx - l)(sinx * cosx) = Q
e (2sinx - l)(sinx + cosx) = 0, vi cosx ;t 0
0'5
015
* 2sinx-l = 0 <+ sinx=!o * =L + k2nv x =5T + kyn
266
* sinx+cosx - 0 e tan.x = -l e x = -! + kn
4
Vfy nghiem cta phuong trinh li , =[+ k2n, =+ + k2r;, = -L + kn, k eZ.
Chri f: HS c6 tnc viet nghiQm cria PT: r : (- ry
r[["
t1,0
iIi6m)
r-m , vxg'
Ta cd + € x - 0" Suy ra hinh phing da cho la
st+1 ixe' Y- -,!:0,x-0 ve x:t"
r
et +l
r
Do d6 &$ tfch kiroi triiii xilay le V - n'{-g ^ dx
i {-n -i- 1}'
hinh thang cong dugc gi6i han bdi c6c du'crng
(1)
015
015
Edt u: x dv = "' = d* Khi d6 du = dx.u :
-l-.
Theo c6ng thirc tich phAn tung phAn ta c6
'fG,+tt'* -
"' *11'
- J"' *t - ".1- J[' -'\1 f'
c.
=-!*rl' -,n1", *rJ' : :-_tn "l l.
Thay vio (l) ta iluqc th6 tich kh6i trdn xoay li v =' ,( '"[e+1 "' z - " tn ' * 1.).)' IV.
(1ro
di€rn
+) Ke CH L AB Vi AA'-L(ABC) n6n
AA,LCH * CH LTABB,A,)
* /,CA' H = (A'C, {ABB, A,)): 300.
+) S* dUng dinh li cosin ve cdng thirc diOn tich cho LABC ta c6
AB - aJ| , CH -ZS AB ffi-:ol,'nuc - a'}o'sinl}}o - tr
+C4=1CH =z"E + AA'= -a
+ ) Th€ tich ldng trg li V - AA'.5 ABC - a az Jj _ a3Jl os
214
7
:
7
0'5
\u-<A' +) MAt phlng (ABB' A') chua AMvi song song
= d(AM,CC') = d(C,(ABB' A')) = CH - c 5
'11
-CC'
,ln
7
0'5
Trang 4(1ro
di0m
{ t _ " xz? * I D+t, = Jtn I r 2 0 rrQ trcr rrranir
l;,
t
:utr; == o + Z
Rd ran g z S khOng thoa rnfln hg VEi e > 0, d?t x - tz hQ tro thantr
f tt U' *zt1 -1
fr'(r';3r)= a+Z
(1) (2)
Suy ra BBT
DUa viro BBT suy ra he c6 nghiqm hay
0n chin g6m 4 chf s0 duo-c viet ra th6a mdn m6i chfr sd 16n hon chfi' s
dung tru6c n6 Khi d6
Q = {abcd ' a + 0, d e {0, 2, 4, 6,8\\;
-:-
3
S,5-fa>4
I
l1
la<
L2
lo*2>6
I
a+2<-L2
f'(t)
\ (1,0 di6m.
(?,0
iIi6m)
+) Tqa dQ A, ,B le nghiQrn
d cit (r1 tai hai diom A, B
+) Ggi A(2yt-m; !t), B(Zyz mi !) trong d6 h' lz ldnghiQm ctia (1)
+
!,:!,:+'!:!: #- -: - -.- .-;j.: . -,
* ABz = 5(yz - yr)' =5[(vr + vr)' - 4vrvr)- 8P ] AB ='li E7
+) Euong cao oH =d(qL)=#- roou=loH"e'n-I.'YA =1 € m2 =4
€ m=t2 (th6a man (*)) suy ra phuong t inh A.t-2y* 2 = 0 hoa" t
g x-2y + m=0.
lx -2y +
, ,A I
cua hQ 1 x'' )
l-+
Y-t4 J
€ h9 c6 2 nghiQm
m-q
[,r_ Zy-m (+{
-1 ls"ut -4my+m2-4:o (1) phen biet€ 3 2- 4m' > 0 e -zJz <m <zJt (*)
2 (7,0 ttidm.
015
+) Gpi / le tt""g di6m AB.Khi d6 I(2;- 3; 1) vi fr,*78 =
0-+) Ta c6 fr4.M8 = (Mi +fr>fffi +787 = tui + u>fat - u) = MI' - IA2
+ffi.uE datgi6trinhonhAt e MInh6nhAtldo u' =lf khongd6i)'
.? 4-44.bit'I qt'i9r-v'l-ole eeg gll+-{lt-"1-(1)'- - -
-lx=2+2t
I
+) Chqn G =6 =(Z;-t;2) + phuong trinh tU:ll =-3-t Thay vio phucrng trinh (P) suy ra
lz =l+2t
t=-2* M(-2;-1;-3).
Trang 5i ri ltltxl,] n.{ ={r;l;*.r.t':0<{r <h {{- 1dt,
De tinh ifil ta xrit cac tnrcvng hqp sau
+) d =0 Trudng hqp niy c6 , j s6 :
+) d e{2,4,6,8} Truong hqp niy
"6 (4 - efi.+ sA.
suyralol=4 ++(4-4>72221 _ _ ._ .-._. _- _
pe tintr lQn I ,u x€t c6c trubng hqp sau +) d - 4 Truoug hqp niy cd I s6.
+) d = 6 Truung hgp nay cd Ci s6.
+) d :8 Truo'ng hqp nay c6 C; sS.
Suyra lCInl=t *Cl.+Cl
-46-Do do P(A)=l# -/ : :? r= 0,02,
lCIl 2296 '
d ln s* cl-i1n ) .
\.-l (1,0 tti6m.
Ii}' | .l (P) : yz - 4x c6 p :2 + ti6u di€m r(l; 0)
olai"l
l.rl.t€u d r ox*pt d:x=e.Tthe {f
|'
^rYs
[x-4
=ffi.og: 16- 16 - o + aoB: 9oo.
+)NCu d LOx* pt
d:y-k{x-4)-* M(4;
la$;
=+{
LB(A;
0), 4)
-4)
Tqa 6Q A, B la nghiQm cria h$
li;la;G;'a;aist{uii,;iaiffi-pifi;i,i-ditii-(it;ltt;si;Gil;i'd;tilct c+l';-d:
_.2 -.,2
Gii sri LtLq; yr), B(?; yr) trong d6 /rr v, li nghi$rn cria (2) ) !t!z = -i6.
Ta c6 d.oE = 1Wz'rz '4 + lrtz = ?4)' -16 = 0 = AOB= 900.
Suv ra O,4 vu6ne e6c v6i OBhay tam gi6c OAB vu}ngtrong mgi trudmg hqp Tu
"9 !pg!n.
{v-b-4k l*-Y:
i'^ <+{ 4
Ly'==4x ' , A., Itty'-4y-r6k-o 1r (1)
2 (1,0 iti€nr
@thfcsau4s6c;Anbi6nc6Anlingudithdngchung,cuQc;l;1ibi6n
;6 A; itt"ne te"rhi t; B auii5n
"6 Binh li ngudi thdng chung cu6c vi.a, tiui6n c5 ninn thing sdc
tht i, i :1,2,3,4 Khi cl6 ta c6
H=AwB;
A : "Trong3 s6c tliu nn *ring 2 sdc vi s6c thfr tu An thing"
= ([Azh w ArBrA, w BtArAt) Ao ;
B : "Trong 3 s6c ciiu Binh thing 2 s6c vd s6c thri tu Binh thing"
= (B1B2A3 v BrArB, w ArBrBt) B o.
'
tt ;# iildt ;l,r *
- "' -'
-Theo c6ng thirc tinh x6c su6t ta c6
P(l) = 3.19 ,412 .0,6.0,4:0,1152,
P (B) =3'10,6;2'0'4'0'6 = 0'2592'
Suy ra P(H) - PtA)+ P(B) =
0,3744-YIIb.
(1,0
tIi6m)