Tim cdc di6m M thuQc C c6 hodnh dQ l6n hon I sao cho khodng cdch tir M d6n giao di6m cira hai duong tiQm c{n cira c nh6 nh6t... oAp AN ivtoN roAN IAN rvl.. CT crla tl6 thi vd Exe, yE ld
Trang 1rnu0Nc EHSp HA Nor
KHor rHPT csuytN
o4.t
Hgc t
Aw lv
tlAtvt
zooq
Mdn thi: To6n
"': ::::l1T.?::: ::: ono',
cau r (2 di6m): cho hdm sri , = ff t'i
',,'l' Tim t6t cA circ giittri cia m d6 hdm st5 c6 cyc d4i, cuc titiu Chrlmg minh ring trung di6m cria do4n thing
n6i cric dii3m cgc d4i, cgc ti6u cria d6 thi hdm s0 in{c6 O;ntr khi m thay d6i
,12 Kf hiQu (C) la dd thi cria hdm sd ilng vdi m = 2 Tim cdc di6m M thuQc (C) c6 hodnh dQ l6n hon I sao cho khodng cdch tir M d6n giao di6m cira hai duong tiQm c{n cira (c) nh6 nh6t
Ciu 2 (2 <ti6m).
,/ l ciai b6r phuong trinh :
ri
I
@or2* ffi=o
CAu3(l di'3m)" Tinhdi€ntich hinh phing gi6i h4n boi hai parabol : y = I - x2, !=axz v6i a>0.
Cnu * , 4 (1 di6m) Cho hinh lap phuong ABCD.A'B'C'D' c6 dd ddi canh bing a Cgi K ia trung di€m cua cqurh
/ tC vd I Id tdm cia hinh vu6ng CC'D'D Tinh th6 tich cria c6c khoi da diQn cto m4t phing (Aklt) chia ra tr€n hinh l4p phuong
icau 5(1 di6m) chrmgminh ringphuongtrinh
2x3 - 3x - 6\6F=x+ 1+6=0kh6ngc6nghigm iim
^7
CAu 6 (2 di€m)
' l) Trong m{t phing oxy, cho elip (E) ,
+ 'i = t ,udirim M {J; t) viiir phuomg trinh cdc dudng thdng
v
tli qua M vd c6t (E) t4i hai tti€m A vd B sao cho M ld trung tli€m cua AB /
2) Trong k:h6ng gian oxyz, cho cric di€m S(0; 0; 2), A(0; 0; 0), B( r ; z; 0), c(0; 2; 0) Gqi E vd F ran rugt
m4t cdu Virlt phuong trinh m4t cAu d6
CAu 7 (l <li€m)
Cloro - Clo.o + Cloro - + (- rlC35iot + - CiBl6 + CrzBlB = z,uu'
Dtr kiiin itgt thi thii Ifrn sau vdo cdc ngdy t6,17/s/200g.
8,3x 3x + zx 9(3x- Zx) - 3x ,
@
Trang 2oAp AN ivtoN roAN IAN rv
l (1,0 diem) TSp xdc dinh : R\ { I }.
Tac6y' = 6r- =t y' =Q:[*, _ Zx * 2 _m = 0 (1)
Hdm si5 c6 cgc d4i, cgc tiiiu khi vd chi khi pt ( l) c6 hai nghiQm phdn bigt khric i.
ual:
[4, = I_ (2_m) > o e+ n]> l.
Gid sri A(xr, yr), B(xz, yz) Id c6c di6m CD CT crla tl6 thi vd E(xe, yE) ld trung di6m crla AB.
Khid6 xl,X2tdnghigmcrla(l)vdxs= |t*, *xz)=l.Suyradi6m EthuQcduongthing x= lc6tllnh.
2 Voi m=2 Phuongtrinhctia(C)duo.c vi6tthdnh : y: x- I * + x-1
D6thi(C)c6tiQmc{ntltmg x=l vAti€mcflnxiOn y= x-l.GiaocriahaitiEmcdnldl(l;0).
Di6m M e (C) <=+ M( ' x*; xy- I * fr I xM-l Nh4n xdt : IM nhd nh6t khi vd chi khi IM2 nh6 nhdt Tac6, IM2 =(xv- l)2+(x"- I +# )2 = 2(xru- t)t +o;h +z> 2^12+2, dlubingxiy ra
khivdchikhi 2(xy-l)'=
*hae'(xr,r-l)'= i =*" =l++(vi xv>l).
)'
1 (1,0 di6m) Bdt phuong trinh dd cho du-oc Uiiln dOi thdnh :
''(;) - (i).*'
;m-,i:
^r-'
'l\z/ ^l \21
Dpt 1= 0- r 0, t + l Khi d6 bdt phuong trinh trd thdnh : ,(r-r) s ;
€t+l- 8t >o<+ t2-s>oe[,tt3 ,*i 6-=
t e(r_l)-; r(r_1) Lo<r<t
lo.(f)-2 (1,0 diCm) EiAu ki€n sin4x * = +
Khi d6 pt tuong duong vdi pt : 2sin4x - \E - 2sin2x + Zt[1 cos2x = 0
<+ 4sinZx.cosZx-2sin2x +2{1cos2x'- VS=O e Zsin2x(Zcos2x- l)+ Vg(Zcos2x- t;:g
<=+(2cos2x-r)(2sin2x+V:)=0,=[ ZcosZx- 1=o *=r [ .t:tz" =t!,^
r [ 2x =:* 2kn
Cos2x - - <=+ I
z " lZx= -n* Zkn' ,tr [2x = -I* 2kn
rSin2x '"erl 3
z - l2x= +* zkn'
,Ttzft
E6ps6: x= -*kTr, *=T*nt,
k6t ho p v6'i diAu kiQn suy ra x: +kn, keZ.
k6t trqp v6i di6u ki6n suy ra x : 4 * on, o * r i
ke Z
' t;'' \:;
t.,,, T ).'t,i
:itr.::ii,
@
3 [* > log13
<r I xco.
lt
6
Trang 3CAU UI ( 1,0 di6m).Hoinh dQ giao ctidm cria hai parabol li nghiQm cta phuong trinh :
1
7'.
,/Ga
Dohai parabol ddu nhdn tryc oy lAm trycd6i xirng vd I -x2 > ax2, v * (-#, ;ft i' nen
s=2IJ'(1 -xz -axz)dx=z.l? - f,' .ul*'lt =2xr-3,,*u1 d= #-#"=#
cAuw ( l,otli6m).
cttcc'vd DD' tan tuqt tei M vA N.
M[t phing (AKJ) chia hinh lflp phucrng thAnh hai
t<trtii aa diQn ln mr6i ctrOp cut tam giric ADN.KCM
Vi KB: KC n€n CF = AB, do tl6 CF = CD Trong
AFC'D, FI vi C'C lA cic ilulng trung ruy€n n€n M
li trgng tdm cria tam gi6c d6
11
Vi I le trung di6m cta CD' n€n D'N : CtU = ] a
3 I -xt=axt <=+(1 +a)xz= 1 <+
2 Vay, DN :; a
Ta c6 :Vr :VeDN.Kcvr:1 n," + B' + y'E-E
;,trong tl6 h : CD : a;
a a2 az a2 zas
Ii€n V'= -t- ,t 3'3 rz 6) 3l
Ggi Vz li th6 tictr cria kh6i da diQn cirn l4i, khi dd : Vz : a3 - Vr : at
-cAu v ( l,o di6m ).
11
Dat f(x)=:x*- ; x-V5xz - x* 1+ l,hAms6xrictlinhvoimgi x e R
32 Tac6 f(x)= *'-1- 2 # 2{Sxz-x+7
lsGit:;;T-+g:!1
vd f'(x) = 2x - 2(5x2- - x zJsxz*x+t +1)
-r*-F
B : SnoN - -; $,': Srccrur :
E
zas z9a3
35
36
L9
4(5x2- x+r1rF x+r Nhfn thdy f '(x) < 0 vdi mqi x S 0, n6n f (x) nghich biiln trong khoring (- c"; 0)
Suy ra f (x) > f(0) = 0 v6i moi x < 0 Vay hdm s6 f(x) d6ng bi6n trong khodng (- *; 0)
Do il6 (x) < f(0) = 0 voi mgi x < 0 V4y, phuong trinh di cho kh6ng c6 nghi€m im.
:j
^it:
Trang 4cAu vI ( z,o ei6m).
l) (1,0 tti€m) Duong thing x = I di qua M cit (E) tai haidi6m o,t,
f ) vn B(l; -* I
Rd rdng M kh6ng lA trung di€m cria AB
' Xdt tludng thing (d) di qua M c6 hQ sti g6c k Ta c6 phuong trinh cua (d) : y : k(x - I ) + I ( I ).
Thay (l) viro phuong trinh cfia (E) ra duqc :
4x? +9J11x- l) + Il'z= 36 <=+ (9k2+4)x2 + l8k(l -k)x+9(l -kF-36= 0 (2) -Dulng thing (d) cit (E) tai trai ei6m A, B thda mdn MA = MB khi vd chi khi phuong trinh (2) c6
hoiXy: , -6=lerk= .
rac6 9(l -k)'-36 =9(l *il'-36<0 c6n 9k: +4>0,
Vd'i k:- ;,4
Dod6,voi k: -! pt(Z)c6 hainghi€mphAn bi6txa,xsrhoamdn: xy:*ol-t =,.
T6m lai, c6 m6t dudng thing di qua M thoa mdn y6u ciu cira bdi torin ld d : 4x+ 9y - l3 = 0.
2) (l,0 di€m)
+ AS 1(ABC) vi AC J- BC + BC t_ (SAC)
(theo gt), n€n n5m iliiim A, B, C, E, F cirng nim
tr€n mQt m4t ciu duong kfnh AB.
Gqi I ld trung di6m cria AB thi I (1; l; 0) lA tdm
t;
vd b6n kinh R = IA :
l-{+'
Vdy phuong trinh 1 m{t ciu ld : }
s
(*-;) +(y-l)'+z==;.
CAU Vn ( 1,0 di6m) Theo khai tri6n nhi thric Niu-Tsn, ta c6:
(l+r" =cg+icfi +lcf,+ +r'Cff = cfi+,cl_ci-jCfl+cf +icfi_Cf -rcfr,.r
=(l -ci + c* -cf + )+r(c* -c; + c; -c; + ).
Met khdc (l+ i)" tlugc viiit v6 d4ng luong gidc :
(r+ i)": rF lcosf + isinf) : @*"7 + t,l7:sinT .
Theo tfnh ch6t cria hai sr5 phrlc bing nhau, r{p dl,lng cho n : 201 0, ta suy ra:
cloro - cSoro + c!o.o - + (-r)k't.c3[;J * -cZEiI+ cSBlB :.,[ffi"i]llf" :2roos
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