TRU'ONG EHSP HA NQITRUONG THPT CTTUVBN.. KhAo s6t sq' biOn thi€n vi vE dO thi C cira hirm s6 ftng vdi gi6 tri cria m tirn dugc.. Trdn duong thing vudng g6c vdi m{t phing MBC tai M l6y ha
Trang 1TRU'ONG EHSP HA NQI
TRUONG THPT CTTUVBN DHSF
Bfr rm rruu'DAr Hgc LAN vr Nann zorr
Mdn thi : TOAN Thdi gian tdm bdi : I 80 phfit, khong ki thdi gian phdt dd
Cho hdm so Y : x3-3mx2 + (nr- l)x+ 2
a
1 Tim rr d6 hdm sO d3t cgc ti6u tai x = 2 KhAo s6t sq' biOn thi€n vi vE dO thi (C) cira hirm s6 ftng
vdi gi6 tri cria m tirn dugc
2 BiQn ludn theo ftsO nghiQm c[ra phuong trinh :
)-_k x'-2x-2=
'.
Ciu 2 (2,0 di6m)
l Ciei phLLong trinh : 6sinx - 2cos3x = 5sin2x.cosx
2 ciai hQ phuong tri'h :
.:
CAu 3 (1,0 di|m )
Tinh tich phan r = l.€$ a
-a
Cho tam gi6c cdn MBC c6 EMe : l20ovd duo'ng cao MH : a^12 Trdn duong thing vudng g6c
vdi m{t phing (MBC) tai M l6y hai di€m A vi D vC hai phia cria di6m M, sao cho ABC le tarn giilc
dAu vd DBC le tam gi6c vu6ng cdn t4i D Tfnh th€ tfch kh6i cau ngoqi ti€p til diQn ABCD
Cffu 5 (1,0 diem )
Cho c6c sd duong a, b, c thay ddi ludn th6a m6n a + b * c = 1 Chri'ng minh ring :
abbcca3
.+"b + r*b *'b+ca 2
-Cdu 6 (2,0 diim )
,1
l Trong m4t phdng Oxy, cho hinh chir nhat ABCD c6 giao cli6m c0a hai duong chdo ld M(;; O),
phuongtrinh dud'ngthing AB ldx- 2y+2 = 0 vdAB = 2AD Tim toa d0 c6cdinh A, B, C, D ;
biiit ring dinh A c6 hoinh d6 duo.ng.
2 Trong khdng gian Oxyz, cho mit phing (a) : 3x + 2y - z* 4:0 vA di€m M(2;2;0) X6c dinh toa
d0 didm N sao cho MN vu6ng g6c vdi (a) clOng thd'i N c6ch dAu gOc toa dQ O vd rrrit phing (a)
Cdu 7 (1,0 diem )
Cho cdc si5 phirc z1: -{3 * i, 22: eosl - i,sin i .
oo
Hiy bidu diSn sd ptiLlc z : (?\t'ou.di cl4ng ctai s6.
\L2/
tsqr ki6n ki tlri tfe# Bai leoc ldm fid-lk 7 sE dwvte fA rh{pn ttAn n*}n, t p , a/K./rni t
Trang 2Dat f(x) = x' - 3xt
X6t phuong trinh
EAP AN _ TTIANG DIENI THI TI{TIDII LAN TH{T SAU - NAM 2011
-p 2:1x-1X* -zx-21.
x'-zx-Z= ffi o lx t lix'? -2x-2)
ra c6 | x-r I qx'? -zx-r):{y.;1.r;1'"- 1) = r(x)
n6u x > 1"
l-(x' - Lx- l)ix - 1) = *f(x) n6u x ( 1
Suy ra dii thi c0a hdm s6 v : I x-l I (*t - 2x-2)tr0n midn R\{l } le
l,a0
SO nghiQm cira pt(*) bang se' giao cli€m ( v6'i hodnh dQ giao di€m kh6c
voi dl thi hiin to v: I x-t l1x2 -zx-21'
Tr) d6 thi tr6rr ta slry fa :
- N€u k<-zthi Pt(*) v6 nghiQm
- Nliu k=-|ho[c k> 0 thi p1(*) c6 2 nghiOra phAn biQt
- N€u -2 < 1'-< 0 thl pt(*) co 'l irghi€m phi"n biQi'
I
.^ =.; ,
{z drcln)
Ta c6-y, :?:t -f*-.151^* -0,*;i,^irc 1iA,r tni v = ? khi d6,t'e,\: 0 c+ m = l.
Didu ki€n cdn : Gi?t su nam so cl4' t'gu rruu (+r
EiAukiQnd0,fVeu*: IthiY':3x2=6x:3x(x-2) = y' = 0<+x=0hodcx=2'
'Fi' !.,inc l.,iAn thiAn suv tra x-: 2 lit dl€m cgc ti€u cfra him
l Ll UiallB ulltr (r'rvtr eeJ 'e "
Chri f : C6 ih€ ki6m tra cuc tri bing da' o hdrn c6p hai'
r ru
oa_-"
y5i nl 1 11'ti y = x' -3* +2' ( Hqc sinh tu ve dO tlii)
2.{1,0 elia@"
1) cira dud'ng thdngY : k
Trang 3'2 ili4m)
l ( 1,0 iIiA@ GiAi phuong trinh
Do gi6 tri x md cosx = 0 khdng ld nghiQm cria phuong trinh, n6n xdt cosx l0'
Khi d6 phLro-ng trinh duoc viiit thdnh :
0,50
€) 3tanx(l + tan2x) - 5tanx -1 = 0
<+ (tanx - l)(3tan2x * 3tanx + l) = 6 o
<+3tan3x-2tanx-l =0
tanx: I <+ x=I+kn ,kez.4 0,50 Z4,O tidm) Clai trffiuong trinh
HQ pr da cho dusc vir5t thdnh , { , J',1,= ? - lx + zvl
[(x + 2Y)z - 2(x- Y) = 4L D{r u= JT1 20,v= lx+2ylzo tnihetrdthanh {"r:;;, ="n, * {;=?
0-50
rir d6 ta c6 hQ :
[;J- {r, _],
III
Q didm)
(1,0 tli6m) Tfnh tich phdn
rac6 r:- frr-z + ro(i)=-vFilf I,ri;#ill
Gz+1 l./3 : l
Xdt hdni sO f1x; = ln(x + ,IWI) thi f(x)
fi,*
suy.a lf ;711 = ln(x + .p t, 1,
lf = ln[(2 +fry.E -ry1.
2
vay t=^[7- 6 +ln[(2+/3XV7-tlt.
Ltut it de Urtn Jf # c(i thd itgt x = tant.
0,50
IV
(1 di€nl
Tu gi6 thiiit suy ra ABMC cdn tai M+ EffF[ = 6Qo.
Vi MH : afr n€n MB = MC = 2^l2avdBC:2rl-6a.
Do AABC d€r.r n€n AB = AC = BC =2{6a
Do ADBC vr-rdng cdn ndn DB = DC :BC+ =2,13a.
2
y716 = ^/ 46T tr4p = 4a vd gg = ,ffifiT frfi = 2a.
Suyra AD=MA+MD=6a vd
AB2 + BD2 = ADz "+ TED:90o .
Tuong tu ta cfing c6 frD = 90o
Vdy m4t cAu ngoai tiOp trl'dien ABCD c6 dLrbng kinh ld AD, n€n V1p =
4_
-'irRt = 36nat.
J
1,00
2
Trang 4L (1,0 aiiiml Chirng minh
1 ili€m\
Tuongtqr cirngc6 a*bc: (a+ b)(a+ c), b + ac= (b + a)(b + c) Khi d6 bat ddngthrlcddcho
tr6' thdnh :
G+c)G+c)
+
ab(a+b) +bc(b+c) +ca(c+a) t1 o (a+b)(b+c)(ctqi - ?ab9:1
t4* (a+b)(b+c)(a+c) -4 (a+b)(b+c)(a+c) - 4 (a+b)lD+cj(
€t r - ;lnlt6*ti1'ta3\= 4 - (a+b)(b+c)(c+a) - 8'
\'-' _ji\- _
-,\-MdtkhectheoBDTC6- si tac6 : a +b>2{{b, b+cZ2rffi' c+ aZ 2'lla''
Tird6suyraBDTduqcchfuigminh.EingthricxAyraklria=b=c'
W
2 ttidmJ
hodnh dQ duong.n€n'a > 1'
ViAthuQcdu'bngth6ng:x-2y+2:U nenA(/a-z;a),\t\)
AUUrruatrrtuvuuvrrE!rwl"s-Do N4 ld trung di6m crha AC n€n C(3 -7a; a) Vi BC I .A'B ndn ffi -6 = (2;'I)'
Suy ra phuong rrinli BC : 2x* y + 5a- 6 : 0 Do B ld giao cira AB vi BC n€n B(2 -2E .2 - a)'
Vi M cfing ld trung di€m cria BD n€n D(Za - l; a'2)
D"
^B:t^D *(* -4)'+(2-2a)2 =zo azaaz -.40a=0 * [:
=
2 =u- z(.via> l)'
Thaya:2vitotgad6cacdi6m,tatimdu'-occ6cdinh A(2;2),8(-2;0), C(4;a) vdD(3;0)'
' -re" ffi 4td = ,'" i*a N rd
[i=-lit;, \z: _t
:'
+(2+2t)2 +tz vikhoiLngc6chtilN d€n(ci) ld:d(t\ (u)=lja lt+ t I
GqiN(x; y; z), do MN
Ta c6 ON2 = 72 + 3t)2
Do do ON: d(1V, (u))
7 1 3, vey N(- ;;;;1)
{2+3t)2 +(2+2t)2 +t2:14(t+ l)'?
(l,O aiiimj Bitlu di6n s6 phiic
-w!
I Arcfitl
Ta c6 zt = 2(-* ir: 2(cos 11 + ;.sinT) vit z2 cos(- fl + ;ti" t- f)
N€n a=ztcos(f *f l*rsin(f *ilt :z(cosff+i.r,nff)
siv,* J i3)\a ,
\L2 /
13ft LrtL \ ^1"
= ?':r'cns 2 * i Sin- i = - / -.t