TRUONG DAI HQC KHOA HQC TU NHIENnE rlrr tnU DAr Hoc NAvt zoro Mon: TOAN Ddt s Thdi gian ldm bd,i: 180 phi,t, khOng *i4 tnal gi.an phd,t di c^.. Tinh th6 tich l5ng trg.. Ciio da giSc ddu
Trang 1TRUONG DAI HQC KHOA HQC TU NHIEN
nE rlrr tnU DAr Hoc NAvt zoro
Mon: TOAN (Ddt s) Thdi gian ldm bd,i: 180 phi,t, khOng *i4 tnal gi.an phd,t di
(c^).
CAu I Cho hbm s5 y: 2x-2m*7
r-rn-I
1) Kh6o s6t srt bi6n thi€n vb, ve dd thi ciia hd,m s6 da cho vdi m: l.
Z; Cto A(1,2) Tim c6,c gi6 tri crtamsao cho tdn tai mOt tlrrdng thing qua A c6t
dd thi (C^) tai hai di6m phan biqt M,N mb" c5,c ti6p tuy6n ta M,I/ cria dd thi song song vdi nhau l
Cdu II.
1) Giei phrrong trinh
tanz t:* 9 cor2 * n\31t! : r+
2) Giei phuong trinh
log! 3z * log, 2r :lo1z3r * Iog, SrLogr2r.
Clu III.
1) Tim gi5, tri ldn nh6t cria hb,m s6
a: {i+{T-;
sinrdr
2) Tinh nguyOn hA,m
t:l sin2z-3cos2lc-7
CAU IV.
i) 1 Cho ISng tru ABC.A'B'C',bi6t A'ABC Ib, ch6p tam gi6c ddu c6 canh d5,y a
vL khoring c5.ch girla canh b€n vd canh dey d6i diqn bbng ,k Tinh th6 tich l5ng trg.
2 Trong hq tqa dQ Oraz cho //(1;3;2) Vict phrtong trinh m5.t ph8,ng (P) di qua
H c6t Ox,Og,Oz tai A,B,C sao cho 1{ I}, trUc tdm tam gi6c ABC.
3 Tlong he tqa dQ Oxy cho drrdng trdn (C) t (r - 1)' + (g + 1)2 : 25 Vi6t phrrong trinh dr-tdng th&ng qua M(7;3) cdt (C) tpi A, B sao cho M A: 3M B
CAu V Ciio da giSc ddu 12 dinh H6i c6 bao nhi€u tam gi6c ttr c6 dinh ld 3 dinh cua rla gi6r: <14, r:tro.
- 11fi1_
$A rHAl'H TCf i ilT:091229i145fi
Trang 2oAp Aw DoT 3 (rom tdt)
L
Cdu I t)-Vdi rn : I ta c6 y : T+ * V' : ffi a 0 vb lim"-aaoog : 2,lim"-2+ g :
*m,lim,rr- U : -oo TCD r : 2,TCN g : 2.
HLm s6 nghich bi6n tr€n cac khoing (-"";2),(2;*oo) vb kh6ng q6 cuc tri Dd thi nhdn I(2,2) lhm tdn d6i xrlng.
f'(r)
f(") ,\
-oo
o*-
. 2
2) Tac6 y' :6jry Gi6,stl M(q,ur),N(rz,ar) e C^(qfr2).Ti6p tuy€n t4i M vit N song song <+ 6;*,;y : i;*ry # rt * rn - ! : - (*, - rn - l){* ri + rz : 2m +2 (I)
MAt khac do A,M,N thing hbng + #: fi* me h-2:W - 2:;$1 n€n
':athu dugc (o1 - 1)("t -rn*L):(s2-L)(cz-rn- 1) vbchri i,rr-m-L: -(rr- nr,- 1)
sriy raol - 1: -(rr- 1) =+21 +.rz:2 Cingvdi (1) suy ra m:0.
CAu II 1) Chf !, 2cot2r: cotz - tanc
"e #* : tano * cotc Phrrong tr)nh tuong duong
tan2 r * 9 cot2 c * (cot c - tan r) * 2 (tan r * cot r) : 16
<+ (tano * 3cotz)2 f (tanr*3cotr) - 20:0
I tanz * 3cotc :4e tanr : lhocLanr:3
o
I tunz f 3cotc : -5 <+ tanx : (-s * r/n) 12.
2) Dk z > 0 Ptrinh ++ (log, 3u - log, 2n) + (logr3r log3 2r - logl3r) : g
<+ (log2 3o - 1) (log2r - log2 3o) : fl
I Sr :2 + r:213
o
L tor, 2x :lo1zSa : t i 2r :3t,3r : 2t + st+L - 2t+L=+ t : :1 + r : Ll6
CduIII i) Dk0 ( r ( 1 Khid6theob5,tdingthircBunhiacopxki (Srr+ W4)2 <z(r/i + \/7i4)
"i, (,/i + ^E-;)2 < 2 @ * 2 - a) : 4 + {, + {r-S 2, d6u bbng xAy ra <+ r : 1.
MErkh6cdo0 < {i,<E- < 1=+ Ji+ {T-" > u +1 -x:!+ T,#= ( 1, d6u b5,ng x8 v r& e r: t ho[c r :0 Vd.y maxy - 2 khi r = 1.
2) Ta c6 sin2:c-3cos2,x - 1 :2 (sinzcosu - 2cos2r +sin2 r) =Z(sinc - cosr) (sinr * 2cosz)
Bidu di6n sin tr : A (sinc - cos r)+B (sin r * 2 cosa) + A+ B : l, -A+28 : 0 .+ A * 213, B : 1/3 Do cl6 61 = J;;*-!t-* + Jm#i* : t#xJ + J#ft, : ,fttnltan(! -;)l+ l/5t" ltan (f + 5.)l+ c ( trong d6 sin p:2/t/i,co,p: l/JE).
Cdu IV 1) Gqi M li trung d\6n BC, MK L AA',
{, t (ABC).
C6 M K.AA' : A' H.AM E K.\f A'2 + * : A' H.+
* A'H: ffi,sell: + 1vlss.1's'"': ffiw(dvtt).
2) II Id tr{c t6,m, OH L (ABC) n€n m{,t ph&ng (P) qua Il(1;3;2) c6 vecto phSp tuy6n Id OII : (1;3; 2) c6 phrrong trinh: z * 3g * 2z - 14 0.
3) Gqi .IJ li brung diiim .4,B (C) c6 tam I(1;-1),R: 5 C6 MA.MB : MIz - R2 e 3MB2 :
27 + MB :3,MA: 9 =+ MH :6 + IH :4.
L6p A qua M(7;3) c6fi.: (A;B),A2+82l 0 cach l mOt dopn bling4: A(r-7)+B(9-3) :0
d(I, L) : 4 e l3A + 2Bl : 2t/pl@ # A -0 hotc A : -12,8 : 5.
Phuong trinh A : y * 3 : 0 holc -12r *59 * 69 : 0.
Cdu V X6t 1 dinh .4 cria da giac ddu, ta s€ tim s5 c5c tam giSc bir c6 g6c ti tai dinh ,4 DSnh
s6 c6c dinh ti6p theo cria da gi6r: ddu theo chi6u kim d6ng hd li 2,3,4, ,12 Chf f Id vdi hai
dinh o, b mh la - bl : e thi ob ld drrdng ch6o chinh vb A.4ab vuOng tq.i A Khi d6 tam gi6c Aab
(2<o<b <12) lb, tir tai Aeb- a > 6 + 8 < o*6 < b < 12 suy rac6 c! cach chgn c5c dinh
a,b (a t 6 vd, b nhQn 2 trong 5 gi6 tri 8,9, 10, 11, 12) Vdy s6 tam giac cdn tim IiL 12CZ : I20