2' Timtdt citcircgi6tri cirantd6drrongtirang y:mx_ 2+2citci6rhi Ctai hai di6rnphAnbi€t A, B sao cho doan AB c6 dQ dii'nho nhAt... Hoc sinh tu sidi.. Gi6i phLro-ng trinh.. dreu Klgn Dal t
Trang 1.fRUONG
D}.iSP I-IA NOI
TR.UOI\G TT{PT' CHUVEF{ }}FISP
pn :r'm rFIU'F]Ar ri-oc LAI{ Ir ruAnr zolt
lvidii ilii : TOAI{
7'hd'i gian ldm hdi : I B0 phtit, khong kA thdi gian phfrt di
CAu 1 ( 2,0 diAnt )
Cho hdnr sd y: ?+
I I(hAo s6t su bi6n rhi6n vA v6 c16 thi (C) cfra hArn s6.
2' Timtdt citcircgi6tri cirantd6drrongtirang y:m(x_ 2)+2citci6rhi (C)tai hai di6rnphAnbi€t
A, B sao cho doan AB c6 dQ dii'nho nhAt.
Cdrr 2 (2,0 diem)
l Giei phLlo-ng trinh : sin2x.(l + tanx) : 3sinx.(cosx _ sinx) + 3
2 Gi6i bat phuong rrinh : 3*= - 4> 5.3#
Cflu J (, 1,0 cliem )
-.V3ln\fizTJ
Tinh tich phAn I : J, " il-J '6*
Cf,u 4 ( 1,0 diem )
Cho hinh lAp plruo'ng ABCD.A'B'C'D' cci do cldi canh bing a vd cii6rn M t5Lr6c ca'6 CC, sao cho
cM :
? *Ut phang @) diqLra A, M vd song song v6'i BD chia kn6i tap phu.ong thdnh hai l<h6i
da cti6n Tinh th€ tich hai I<h6i da diQn d6
Cdtr 5 ( t,0 dietn )
tla s6 clu'ong thay d6i a, b, c thu6c doan [', B] md F - o {2 crrfi.ng mi'h rdrig :
Gilf +/66a1+rtra+f > a*b*c.
Cflu 6 ( 2,0 didm )
l' Trong rndt phlng toa d0 oxy, cho tam gi6,eABC c6 c( I : 2), hai dildng cao xu6r ph6t tLr A vd B
lAn lu'otc6 phLLongtrinh IA x + y : 0 vd 2x-y + I = 0 Tinh cli6n tic6 ta'r gi6cABC.
2 Trong l<h6ng gian toad6 Oxyz, cho mdtphing (p) c6 phLrong trinh: x*2y 1,22+ l= 0 vd rndt cAu (S) c6 phLrong trinh : x2 + yz + z2 - 4x + gy + 6z+ 17 :0.
Tinr toa dd tam vd b6n kinh crla du'o'ng tron (c) la giao cira niir plidng (p) vd rnat cAir (S)
Cf,u 7 ( t,0 ttient )
Giei he phLLong trinh :
nat
t*t + xyz :40y
Lyt+x'y=10x
DqE kiare ki tki thft fr$i hpc rftn tratu s sE dwgc t6 chs?c vda rcgdy rg,z#/s/zL!i
Trang 2sap Ani - THANc DIEM
rnr rrrtl oH r.An rnU nar NAivr zor r
I
7z AiAml
L (1.0 man Hoc sinh tu sidi
i (t,o aiiiml Tim c6c gi|tri m
Euong thang y = m(x-2) + 2 cttd0 thi (C) tai hai di'5m phdn bi6t <+
c6 hai nghiQm phdn bi€t <+ pt nrx' - 4mx + 4m - 5 = 0 (*) c6 hai
2x+1 Pt;=m(x-2)+z nghiQm phdn biQt khbc2
0,25
(m*0
c+ Jo' : 4m? -m(4m-s) > o o m >o
I
\+m-Bm*4m-5+o
0,2s
Gi6 srlr A(x,,y,), B(xr;yr) trong d6 xr, Xz ld hai nghiQm c0a (E).
Khi d6 yr = lnXr -Zm*Z vit y2= lrlx2 - 2m+2
Tac6 AB2 = (*, - 4,)t + (y, -y,)2 = (xz - xr)2(m2 + l) : [(x2 + x1)2 - 4x1x2](m2 + l)
0,2 5
4(4m-5) 20(m2 + 1) 2o,2m : Il6 -=;= X*t + l) = :-:il"-:1/ Z"#:40 vdi mgi m > 0 Eing thri'c xdy ra
khi vd chi khi m : L Vdy, v6'i rh: I thi AB ngdn nhat Uing V40 .
0,25
II
(2 cti6m)
l ( 1,0 clilnt\ Gi6i phLro-ng trinh .
Ei6u kiQn : cosx f 0 Phuong trinh dd cho tuong duong v6i pt :
# (tun* + 1):3:lla(l- tanx) + c+
tan2x (l+ tanx):3tanx(l-tanx) + 3(l+ tan x,t
cos'x' ' cosx cos'x
0,50
(+ tan2x (l+ tanx) = 3(l+ tanx) e
).
dreu Klgn Dal toanJ.
[tanx = -1 [* =
Itan2x=3 H
l*: (kez) ( th6a rndn
-i+tn
*I+kn
-3
0,50
2 (1,0 cli1nt) Giai bat phuong trinh .
DiAuki€n:xl
BAt phu'o'ng trinh dE cho tuong duo'ng v6'i bpt
3sx-, - 4,5.3"-t*u
')
AfJ
D{t1=3sx-2, t>0.'Bpttr6ntrdthanh t2 -4t 45>0 + t:9(dot> 0) 0,25
'isx-z> ge
-5x-z
27
>2 e -<x< - 5 -9 Ddp sd :
a1
III
(1,tti€nl
(1,0 diAm) Tinh tich phAn
rrc6 r- - I,fitnVTJ?d1= -11nur1 1p lf -llr"1otrn.,rTT7;
Trang 3Tas€tinh.l =f*clx, ddt x:tanr =r dt:-+ dr:(l of,)d,
, x= r/3 thi
IV
(1 ili1nt)
o (0,50 ilid@ Dung thirit diQn cira mat phing di qua A, M va song song v6.i BD
Gqi o=AC o BD, o' =A'c' n B'D' va I =AM n oo' eual k6tluo.ngth6ngsong
song v6'i BD cit BB' vri DD' lan luo.t t4i K, N Khi d6 AKMN la thi€t di€n can dung.
DAt Vr = Ve.scvr* Ve,.oouru , Vz= Veaco.a,s.c.o,- Vt
t (0,50 tlie@.racO ff =#= 1 =+ DN=BK=Ot =;a* =; .
Hinh ch6p A.BCMK c6 chi€u cao ld AB = a, I
ddy ld hinh thang BCMI(
1
Suy ra vo.scrr, ::aB o - AB Bc(BK+cM) a3
3 '.recvrc:T z :T B' Hinh ch6p A.CDNM c6 chi€Lr cao ld AD = a,
tf6y ld hinh thang CDNM
1 - AD CD(ND+CM) a3
Suy ra Ve.coN" =:AD
3 '.Jcolrlr:T , =; r(
V?y, Vr '333 =-:- , Vz: ai -a' -2a" B
t (1,0
TtLgi6thi6tsuyra lg llSF-o <2 +(a-b)t< 4=+(a+b)2 _4abs 4
suy ra a* b < 2lr+a, tlro'ngtu'tacfingc6 : b+c <2fi1fi , a+ c szrlr+ ac
Dod6: a*b*c<V1 +ab+/1 +bc+/1 +a;(dpcnr)
(1,0 ttiAm) Tinh di6n tich tarn gihc
VI
(2 tli€n)
DLld'ng thdng BC c6 vecto chi phLro-ng il la vecto phap cira dLLdtrg thing x +y : 0, n€n
d=(l; l) Phuo'ngrrinh cira*a' [i:1il tu, ra B(l+t;2 +t) B rhu6cdr-rongcaoxu6t
phSttrrB n€ntead6th6amdrr phuongtrinh:2(l +t)-z-r+ I -0 +t:- l vay B(0; I) ruo'ng ts, phuong trinh AC ,
[; = i:i' vd A(- 5; 5)
Tac6: BC=J2
Duo'ng thing BC vitit va d4ng t6ng qurit : x - y +
l-s-s+r I 9 Goi AH ld dudng cao, ta c6 AH - :-: - I -j
-
Trang 4(1,0 tti€m) 'lim toa d0 tdrn vd bAn kinh .
Mat cAu (S) c6 tdrn IQ; - 3; -3) vd bdn kinh R : \i5 .
I(ho6ngc6chtiL1t1iinmp(P)lith=w=l<R=V5,n€nmp(P)cdt
rn{t cAu (S) theo mQt tfud'ng trdn c6 b6n kinh bing r : ,lP-.P = 2.
0,50
Tdm K cria dudng trdn (C) ld giao di€m cria mp(P) vdi dub'ng thingd tli qua di€m 1vd vudng
g6c v6{ mp(P) Vecto chi phuong.c0a duong thhng d Li vecto phrip tuytin cria mp(p), n6n
ui:rt;1 :2)vi phuongtrinh cria o,[;='-{ir, Dod6 K(2+t ;-3 -2t;-3 +zt)
\z: -3 { 2t
Tqad0tem K cira(C).th6amdn phuongtrinh: 2+t+6+4t-6+4t + I :0 + t =-+.,
\ 7 11
Vay tAm I((il ;- 'J', 3' 3" ) bdnkinh r:2
0,50
WI
,Q iliem)
(1,0 cli€nt) Giei he phLro'ng trinh
N6u x=0thi y:0 =+ (0;0) ld mQtnghiQm crhahQ phuongtrinh
Ntiu x I 0 D?t y = tX, khi d6 hQ pt tro thAnh :
fx3 + x3t2 = 4otx fx3(t + tt) : 4otx
tt3x3+*tt=10* :
["t1rt+tj=1gx (+
(x.(t +t') :4ot ( "t = # (1)
I *'(r' * t) = 1s €
lt## : to (z)
Tt(l) + t>0,k6tho.pv6'i(2) - t::.Thay t= ) ,eo(l)raduo-c x=+4 + y=r2.
Drip sti : HQ pt c6 3 nghiOm (0; 0), (4; Zj , e a; - 2).
I,00