1. ri rru ruyBx srxn r,op to rgpr
NAU ugc 2ot9 -2020
M6n thi : TOAN
Ngdy thi : 02 thdng 6 ndm 2019
Thdi gian ldm bii : 120 philt
Biit (2
Cho hai
4 J".r)
25-x
vitB= 15-J;
-+
r -25
t
-T= _
vu+t)
Vz + 1
-T=-
!u - b
vo,r>0,r+25.
l) Tinh gi6 tri cria bi6u thric ,4 khi o = 9.
2) Rrit gqn bi6u thric B.
3) Tim t6t ci gi6 tri nguyCn cria o d6 biAu thrlc P = A.B dgt gi6 tri nguyOn lon nhdt.
BnifiQ,5 diiim)
1) Gidi bai bdn sau bing cdch lQp phtrong tinh hoc hQ phtrong trinh :
,
Hai dOi cdng nh6n cirng ldm chung mQt c6ng viQc thi sau 15 ngdy lim xong. Ni5u ttQi thri
nhAt lim ri6ng trong 3 ngay rdi drmg lai vi <IQi tht hai ldm ti6p c6ng viQc 116 trong 5 ngdy thi ci
hai tl6i hoan thanh itugc 25vo c6ng viQc. Hdi niiu m5i dQi ldm ri€ng thi trong bao nhi€u ngdy moi
xong c6ng vi€c trdn ?
2) MQt b6n nu6c inox c6 d4ng mQt hinh trg voi chiAu cao L,75 m vitdiQn tich dity ld 0,32m2.
Hdi b6n nu6c ndy dyng dAy duqc bao nhi6u mdt kh6i nu6c ? (86 qua b,! ddy cua b6n nudc).
Birillr (2,0 didm)
1) GiAi phuong trinh c{ - 7 a2 - 78 = 0.
2) Trong mIt phing tga dQ Oq,cho duong thdng (d):y = 27.,o - *'? + I vd parabol (P):y = ,z
a) Chrmg minh (d) lu6n cit (P) ai hai di6m phan biQt.
b) Tim tdt ce gi6 tri cria rn itil (d) cit (p) tai hai di6m ph6n biQt c6 hodnh dg q,r, th6a
_1t -2man-+-=-+1.
0, tr, arr,
BniIV Q,0 di1m)
Cho tam giitc ABC c6 ba g6c nhqn (,48 <,4C) nQi ti6p duong trdn (O). Hai dudng cao
BE vd CF cua tam gi6c ABC c*tnhau tai diem 11.
1) Chwrg minh b6n di5m B,C,E,F ctng thu6c mQt tludng trdn.
2) Chrmg minh <ludmg thing OA w6ng g6c voi <ludng thing EF.
3) Gpi K ld trung <li0m cira do4n thing BC. Dudng thing AO cit duong thing BC tai
di6m.I, tluongthing EF citduong thing AH tai di6m P. Chrmgminhtam giitc ApE ddng
d4ng voi tam gidc AIB vit<luong thlng /(11 song song voi duong thing .IP.
Bniv @,5 diim)
Cho bii3u thric P = aa +ba -ab, v6i a,b lit cAc s6 thgc th6a min a2 +b' +ab =5. Tim gie tri
lcvn nh6t vd gi6 tri nhd nhdt cria bi6u thric P.
Cdn bQ coi thi kh6ng gidi thich gi tham.
Ho t6n thi sinh 56 b6o danh
Ho tdn, chir ki cua crin bQ coi thi s6 1 :
HCt
H9 t€n, cht ki cria c6n bQ coi thi s6 2 :
so GrAo DUC vA DAo rAo
HA NOI

2. so GIAO DUC vA EAo rAo
HAN
Ki Tr{r ruyEN srNH vAo Ldp t0 THpr NAM 2019
oAp AN - THANc DIEM
Mdn thi: TOAN
rDap itt Thungt Jietn gom [)-! rrattgt
+(.6. r)
I
Irii '
Tinh gi6 tri cta bi6u thri'c A = khi:r=9
/) - _t:
1 6*rThay z : 9 vdo bi6u thuc ,4 =
25-9
Tinh duoc ,4 = 1
Rrit gqn bi6u thri'c B - I f- va r>o.t *2b.
I Vz-5
15- It25
2
Vr+5
15- "'6 2 tl-G*z(
"t
i)
+
t: -25
1
J"-s
.6*s (.'t.s)(.,6-s)BNi I
2,0 diam
l)
r
1 Vr-5
t_t-
V.r-5 V.r+I
Di6m
l)
n )
0,25
0,5
0.25
n )
n )
n 15
1,0
2)
D1
Vr+1
T
nguy
im tat c1i girl tri nguyOn cfra .r d
6n k?n nhAt.
bi u thric P = ,4.8 dat gi6 tri 0,5
3)
+(G.r) 1.4
".f,*r 25-r
P=A,B= n )
25-t'
I-ap lu{n d0 .p dat giii n'i nguy6n lon nhAt bang .1 khi :r: 2.1
llai ddi c6ng nhin ctng lhm chung rnQt cdng vifc thi sau 1l nghl'lirm
xong. N5u dQi thri'nh6t tirm ri6ng trong ;l ngav r6i dnng 14i vir d6i thrir hai
lirm ti6p c6ng viQc d6 trong 5 nghy thi cri hai tlQi hohn thAnh dw1c 25%:
cdng viQc. Htii n6u m6i a6i lhm ri6ng thi trong bao nhi6u nghy m6i rong
c6ng viQc tr6n ?
cqi ihoi gian doi tnu nhdt ti,r, rieng xong.ong
"ie.
li ; ia* ii' ;8ir. .; ; aii
Goi thoi gian ddi thu hai ldm ri6ng xong c6ng vi6c ld y (d<m vi: ngdv, u > 0)
I ngAy dQi rhu nhdt l2rm duoc ] tcong uiect
:t
I ngal dQi thu hai lirm dutvc I (c6ng viec)
v
0,2 5
2,0
o:25
o )5Bii II
2.5 cli€nt
1)
n )1
Hai d6i ctng lim chung mQt c6ng vi6c sau 15 ngdy xong. c6 phuong trinh
L
NQi dung
(

3. 1tt
-+ = (l)
r 1! 15
n )
0,2 5
DQi thu t lim ri6ng trong 3 ngay vi d6i thu hai ldm tlep cong vrec d6 trong
5 ngiy thi ca hai dQi hodn thdnh duoc 257,, c6ng vi6c, c6 phuong trinh
!*!=11:i
rtl 4
Tu ( I ) va (2) ta cti h6 phuong trinh
111
:r ll 15
351
:x?)1
Gidi hC phuong trinh tim duoc
!=40
D i chi6u clidu ki6n vd k€t luAn: d6i thir nhdt I im ri6ng thi sau 24 ngdy xong
n )i
n );
o
n )5
con vi c. cl i thu hai lim ri6 thi sau .10 n a xon c oll VICC.
2)
l)
#
:dG
via
BNi III
2.O dretn
2)
b) Tim tat ce gi{ tri cta m d (d) cIt (P) tai hai.li m phin bift c6c
11 2
n+=+
TT
c
hoinh aI6 :r .:r thiia mi'll
1
r rr"
Theo dinh ly Vi-et: rt + 12 = 2rn vd r,r, = m
tla
'+I:DK: .r{ /Oatnt'l
J' J, r,J',
-1
tich ilrly li 0.32 nr'. tkii bdn nu'6c niv tlung tl6y du'gc bao nhi6u m6t kh6i
nu'oc ? (Bri qua bO diy ciia b6n nu'6c).
56 m6t kh6i nu6c bdn dirng ddy bAng th6 tich crha bdn.
Bdn nu6c drmg duoc s5 m6t kh6i nuoc ld 1. 75 x 0. 32 = 0. 56 (m')
0,25
n );
n )i'l'H1: r" = 2(v6 nghiQm)
TH2: r' - 9e :" = 3 hoac r=-3 n )
9
)
0
0,5
(z' + z)(
NI6t b n nu'6c inox c6 d4ng m6t hinh trg v6i chi u cao 1.75 rrr vh diQn
0.2 s
t,0
T4p nghiQm phuongtrinh S = J
I l
't.rong m{t phing toa d0 Or;u. cho tlu<rng thing (d):'u = 2nt,.r - nt,' + I
vlr parabol (P):u - .r-'.
a) Chring minh (d) lu6n cit (P) tgi hai tli6m phin bi6t.
Xdt phuong trinh hodnh d6 giao di6m cta (d) vd (P)
:t' =2ntr -m,' +L e tt -2m^r+m'-l=O
L.=m -(. -r)=1:A >0 Vm
./
Vay (,1) 1u6n cit (P) tai hai di0m phan birlt vrii moi gi6 tri cua rn.
015
0,2 5
0,2 5
0,s
0,5
Gi6i phurrng trinh r'' - 7r' - 18 - 0.
I

4. J. + J. -2+ rl.<+
-=-
. -I
t,*,,= -2+.r,r,rrx, rrr,
n )i
e 2m = -2+ m' -t<= m) -2m-s = 0<+ (rn+ t)(- -:) = o
- nr, = -l (lo4i) hoqc rri, = 3 (tmdk). VAy m=3
0,2 5
t,0
Cho tam giirc .1 /i( ' c6 ba gric nhr.rn ( .,1 /i < . I (') n6i ticp du'irng trdn
(f)). Hai duirng cao tJli ti ('t,'cria tanr giitc ..11)('cit nhau tai tli6m //.
l) Chri'ng minh b6n di6m lj.('. /i. /,' ctng thuQc mQt du'tr'ng trdn.
I
Vd cling hinh diSn y l)
BELAC) BEC=90''
n )5
0,25
l)
IJ + b6n didm B.('.8.F ctng thu6c duong
tron duong kinh BC
0,25
,1i
0.2 5
(
2) Chf'ng minh ilutrng thing O.4 vu6ng g6c r'6i itu'tmg thing EF. 1,0
.----l
Ta co B(.:EI: ld tir giric n6i ti6p + AEF = .4BC:
Biri IV
3.0 diOnr
2) K6 dudng kinh AQ = L.1QC l'u6ng tai C - QAC + AQC =90" 0,25
0,25
0.25
x6t (o)c6 AQC = ABC dAC
I
= -s.)
+ ll.:1.' . l.; l() !r(] = lo ) l.;1:
3) Goi A- li trung diOm cria do4n thing B('. Du'dng thlng -4 O cdt
du'ong thing 1i(' t4i di6m /. dut'ng thing Il-F cit du]ng thnng .11/ tai
di6m P. Chring minh tam gi{c ;l l'.E- ddng d4ng vcri tam giic AIB vir
du'trng thing A1l song song v6'i du'dng thlnEIP
1,0
a
'io
o,.o
tti
4
EAO : HAB (vi ctng phgr voi ABC ) = EAP = IAB n )i
0.2 5
3) A,APE ddngdang A,AIB (g.g) + O.O^
{rrtAB AI' ,
^AEH
ddnsdang LABe (s.s) + #=
**O,
ru (r) . t2y =
A.l
=
o.l- + = +) Pt // He t3)
AI AQ AH AQ
Chnng minh t{r gi6c HCQB ld hinh binh hdnh + H, K.Q thing hdng (4)
Tir (3), (4) + KH /l 1P.
0,25
0.2 5
_T
I
l
Cl' L lll + (rl'l] = !X)"
I
AEP = ABI + LAPE d6ng d4ng LAIB (g.g)

5. Bni v
0,5 di€m
Cho hi6u thri'c /r = a' + b' - ub. voi tt.b lh c6c s6 thuc thria min
o"' + b' + rzb = 3. Tim giri tri l6n nhit vir gii tri nhri nhit cria bi6u thrlc P.
p =(a'+6't)! -2a2b) -ub
= (: - ra)' - 2a)l;' - ab = -.a'b' -Tab +
Tacb a' +b') +2ab = 3+ab+ 3+ab=
3-ab - a' +b' > 2ab - ab <1
e=-[a
, ,)
(a+b)
7
2
85r
4
+
2
+ab>-3
0,5
n ),
!.(4[
vi -3+!=ob*! ,t*1 =222
,7ab+-
2
SI
I
2
1
4
,7o,h+-
2
>-ql =S-' =
p2 85-4+zr> p> r
141 l1
GTLN cua P =2l.xity rakhi a =..6.r, - --..6 toa. u=-Ji, t
GTNNcua P=1, xriyrakhi a=b=1.
.6
0,2 5
ro(; t)t[.l 10,0
u'cl;
,l
I l
hoặc a = b = -1