Trudc ti6n ta chring minh c6ng tfrri* C6ng thric l dugc chu-ng minh... Vi6t phuong trinh tluong thang A nim tong mit phang P, vu6ng g6c vsi d vi c6 khoang cach d6n d mQt khoing h= '#...
Trang 1TRLIONC DHSP Ha NOI
nt rru THrt DAr Hec naOn roAN IAN I
trAwt Hec 2oo8 _ 2oo9
(Th6.i gian IB0phtit)
t***
l Khdo s6r vi v€ d6 thi cria hdm sti khi m = - 3
2 v'i gi6 tri ndro cta m, hdrm s5 c6 clrc d?i, cgc tiAủ Ggi x1, x2 li hai di6m cgc tl4i, cgc tiiiu cria hín s5, hdy tirn gid tri lon nhdt crja bi6u thric A = i*r'i-ri"'i -rl f
Cflu 2 (2,0 di€m)
y l Gini phuong trinh :
cos2x * cos5x - sin3x - cosSx = sinl0x.
Z Giili bdt ptru,rng trinh :
cho hinh ldng tr-tr tam giiic d6u ABC.ÁB'C' c6 dO dai canh d6y bing a, g6c gita cluo*g thing AB' vdrn{t ptrang@e,C'C) bing ạ é 6vv
l Tinh d0 đi rtoen thing AB' rheo a vd s.
2' Tinh di$n tich rn{t.ciu ngo4i titip hinh ldng t4r AIIC.Á8,C, theo a vdi ạ
<pCflu 7 (1,0 didm)
\
Trong m{t phing vdi rr€ r-o: dg oxy, cho tam gi6c ABC vdi Ẳ; -1), B(r; -2) vd rrg*g
tdrn G cfra tarn giric niim tr€n duon! tning d: x-+ y - 2 = ọ iray tim tga dQ diem c, bi,it
rang di-6n tich tam gidrc bing j.
i
i\rtt.-t - 2i
! | - - '
, J' L]
\i r I -.2
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6 ĐỀ THI THỬ ĐẠI HỌC ĐHSP 2008-2009
Sưu tầm: Nguyễn Minh Hải
Trang 2oAp AN vA rueNc DrEM
l (1,25 Aiemt
Gidi hqn: limr,-*_ y = + co
, limrr- Jr= - ,b . Su bii5n thi€n: y, = ?x2 - 4{ - ,y:'= io U - O f,,ia x = 2
y'>o* f ;: vd y,<o<+o< x<2.
Do tl6 hdm s6 d6ng biiSn trong nrdi khod.ng G*; 0) vit (2;+oo), nghich biiin hong
Vdim=-3,thi t=:*t Tap x6c dinh : R
khoang (0;2).
Cgc tri : Hdm si5 y dat cgc d4i tai x:
0 vd yc,p = y(0) :
;,
Hdm sti d4t cgc ti6u t4i x:2 vA.!c.r = l(2)= - 13
Beng bitin thi€n
?t,*r":: r !.',' = Of O,c6 y"(l):0 vd y,, dr5i d6u khi di qua x = l, n€n di6m
(l; -; ) le diiSm u6n yd cfing ti tem d6i xring cria d6 thi.
?6 Ai cit trsc tung r4i ai6m qo; ] ).
?o d6 A ton ntrdt bing 3 khi m = - +.
Ta c6 y'= 2*
I::1::"Lgu1:oc ti6u nri vd chi *, ri:0 c6 hai nghiQm ph6n bigt x1, x2
hay a': (m + l)2 -2(m2 * 4m + 3) > 0 (+ m2 * 6*;;-; il:ffi :;.
$eo dinh lf Viet, ta c6 x, 1x2 = - (m + l), xr.xz =lm2 + 4m + 3y
l"l:l A= li(*' r 4m * 3) +2(m + l)l =|l*, * a** z1
Tanhinth6y,vdi m e(-5; -l)thi: - 9S mf+ gm+ 7=(m*4)2_9 <0.
1 (1,0 di6m)
CAU
II Phuong trinh dusc viiSt ve ftng
cosSx - cos2x + sin3x + sinl0x _ cos5x = 0
<+ - 2sin5x.sin3xI sin3x + 2sin5x.cos5x_cos5x=0
c+ cos5x(2sin5x_ l) _ sin3x(2s
@ in5x-1)=0
Trang 3Voi sinsx= 1 z 0 [ u*=r+Zkr
[s*=n_]*zkn (+
Vdi cos5x = sin3x (+ cos5x = cos( _ fx;
J4f tfln nglri0m crlaphuong uinh la S ={ t30' I + 3II
suyra togr(x-il= 2ex-*-no *>?.
Vly t4p nghiQm cria bdt phyone trinh H S = tf,; + o).
lcosSx = sin3x
;
Ellr.
i t' tl:
1 Gqi I vi I'lAn lugt li t6m hai ttriy ABC vi A,B,C,
Klti CO, rtng di6m O cria n, lA tankh6i ciu neoei
-tifo Utotr langtru .
Trang 4va oA2=ot2+tAr=
#(g-+sin2a)- f
Gqi R ld brin kinh m[t ciu, thi *i= #; (z _ +sinz a) +
Khi d6 dign tich mflt c6u ngo4i ti6p hinh lang trr.r ld :
?23
s = +z' (# cs - + rin'"1 + *) = ara2ffi + 5.
Tt he phuong trinh suy ra x ) 0, y, 0
Cfing tu hQ phuongtrinh vd theo Uit aang
th&c C6si, ta c6 .
6V3 = 2^/7 + y =,/F + lF + y >3W = 3\/74= 6iE
Deng thric xdy ra khi vd chi khi G =y =2W:+ x = \876.
Vdy nghi€m cria h€ phuong trinh la' x: ffi vity - 21/i.
Trudc ti6n ta chring minh c6ng tfrri*
C6ng thric (l) dugc chu-ng minh.
Ap dpng (l) vdri k di rir O Adn ZOOS;G
Trang 5VII
Tir gin thi6t ta suy ra Segc = 3Snea =+ Sesc :1 uU dO dei AB = r,E.
Phuong trinh tluong thit g AB : x-y - 3 : 0.
0ls
Gii sri G(xc; 2 - x6), khi tt6 khoang cich tir G ttdn AB la 1 : l2xq'sl
' ,tz suyra segc =i* n + l2xc-51 : t *
V6i xc: 3 thl yc = -1, kfii d6 thay sd ta ttirgc xc : 6: Yc: 0.
V$y c6 hai <tiAm C th6a mfln bii to6n: C1(3; 3) vn Cz(6; 0).
Trang 6'rRU'a,NG DHSp r{A NOl Dt THr rrrrl DAr Hgc rAN rr NAvr zoog
Thoi gian ldm bii: 180 ph0t
lr**
clu I (2 di€m): Cho hdm sd r =-lP f rl
1) KhAo s6t vi ve d6 thi (C) crha him s5 khi m = 0
?) Tim nr AE A6 tbi hAm sO (t) cit tryc Ox t+i hai di,im phdn biQt c6 hoinh d0 ldn luqt li x1, x2
sao cho r = I xr - x2 I dat gii tri nh6 nhAt
C6u 2 (2 di6nr)
i GiAi phuong trir*r :
2sin2 (x - 5 = 2sin2x - tarx .
2 V6'i gi6 tri ndo cia m, phuong trinh sau c6'nghiQm duy nhdt :
2log' (mx + 28) = - log5(12 -4x - x2)
Cdu 3 (l di€nr) Tinh tich phan :
Cnu 4 (1 di€m)
-a'- Tan gi6c MNP c6 dinh P nim trong mflt phang (a), hai dinh M vi t'f nirn vB mQt phia cia (o)
c6 hinh chiiiu vu6ng g6c tren (s) Dn luqt li M' vi N' sao cho PM'N' li tam gi6c dAu canh a.
CiAsirMM'= 2NN'= a.
j
- Tinh diQn tfch tam gi6c PMN, tu d6 suy ra gi6 tri eua g6c gita hai mflt pheng (c) vA (MNP)
- : Ciu 5 (l ditim) Cho tlp hpp A c6 l0 phan * H6i c6 bao nhi€u cich chia tfp hqp A thenh hai tip
vu6ng tOv quay xung quanh di6m O c6 c6c canh Ot vi ov cit (E) lan luqt t4i M viN.
chil11113ng mrnn rang: 6F " ON, =
36 .Trong kh6ng gian v6i hQ toa d0 Oryz, cho ducrngthang O' T=?= | tamit phturg
CI) : x + ! + z- 3 = 0 Vi6t phuong trinh tluong thang A nim tong mit phang (P), vu6ng g6c
vsi d vi c6 khoang cach d6n d mQt khoing h= '# .
Ciu 7 (l di€m) C6c s6 thpc x, y thay d6i sao cho x* y = 2.
Hdy tim gi6 tri lon nh6t cua bi€u thric : P = 1x3 + 4(f + 4.
- ,.li xdx
l=l
' J1 x+y;l['
Trang 7M4t khdc lim*-s+ f(x) = + oo vi lim*_e- f(x) = _ - ,
Tac6f(x)>0v6i -4>x> -6vdf(x)<0voi x e(-4;0) u (0;2) .
Bing bi€n thi€n :
Nhu vfy, tu bing biiin thi€n suy ra phuong trinh (3) hay ciing Ii phuong tdnh (2) c6 nghiQmduy nhAt thuQc ( - 6; 2) \ {0} khi vA chi khi :
= t'lf -itf r.,- r)ia(*, - 1) =+ -*ic.,-,)-lf
cAu rv ( 1,0 di6m) K6o dAi MN cit M'N'tai E,
khi d6 NN' li duong
trung binh trong AEMM', mi M,N' = pN'= a n€n
EN' = 4 suy ra APEM' li tam giac n?ng tei p
vi EP = rGMryffi7? = 816 , d6ng tiroi Ep .t- pM
Trong tam.gi6c vu6ng c6n pMM', c6 pM = a.,12
,
nAn FP PM = a.E "^17- : ^2-17
Ta c6 966p = 2Suup + Suup= | fe.ru =l*^f, .
Viy S,r.1up =I^'#
Vi EP la giao tuy6n cria hai mpt pheng (a) ve @Ia}Q
vi EP 1PM, n€n g6c a giiia t.rai m{t phing nay bAng
g6c frFFf = 450
Cht )t ; C6 th6 tinh g6c a bing c6ch sri dsng
c6ng thrlc SpM,N, = Spyy.cos g.
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Trang 8cAU v ( 1,0 didm) GiA sri k li sii'cdch chia r{p A rh6a man y€u c6u bAi to6n Ta nhin th{y ring,
'''si mdi c6ch chia ta dugc hai tip con kh6c r6ng cua A Suy ra s6 cdc t6p con kh6c r6ng cria
bing 2k Tri d6 ta c6 :
2k = Cls*Crzo* +Cio =zta -Z s ft=2e- I =511
Viy, si5 c6ch chia theo y€u cAu bii toan bang 5l L
cAu u ( 2,0 di,im)
l) (1,0 di6m) Dat @;ffi1=a (0 S oS2Tr) vA (d; }]f)=c+l
Tac6: Mf*"=oMcosc "'tYr,r = OMsina,
usrg rU', ra cung co & = T * T
^ 7 1_ _ coszd sinzc * sin2c , cos2c :1 - 1
Ke AB J A, B e A Ggi C ld giao di6m cria m(e) voi d
vi g ld g6c giGa d vA (P) thi g = ffie ,tac6 fi(l; l; t) h
mQt vdc to ph6p tuy6n cria (p), Khi d6 :
sne =
JE!t7- = {; + Ianp =
J14-@
Trang 9Ta c6 BC li duong vu6ng g6c chung cira d vi A, d6ng thoi dg dai troqn BC = h = '# .
Do A nim trong mf@), n€n A li giao ruyiin cua hai mft pheng (P) va (e.
T6m lai ta c6 hai rtuong theng A th6a m6n bAi torin lA:
tv'i :
[-zx+3y+ 2z*6*#= o ua (a)'[-zx+3yr
zz* 6_#= o
cAu ylr ( l,o di6m).
Tac6 P =x3y3 *2(x3 +f; + 4=*tf +Z(x+yXxz-xy +yt1++
= x3y3 + 2(x + y)[(x +y)2 _ 3xy ] + +
Theo giithitit x + y = 2 n€n p = x3y3 - l2xy +2A.
D6t 1 = xy, do (x + y)2: 4xy n6n t < l.
DAtf(t) =f -lzt+20, te(-oo; U,thif(D=3f - 12=0et=-2.
Ta c6 f( 2) = 36,lim,*-* f(g = - o, f(r) = 9 vi f(t) > 0 voi t < -' 2 c6n f (t) o voi -2 < t < l
Tir c6c t6t qua tr€n, suy ra maxf(t) :36 khi t = - 2.
Vsi i = -2,tac6 hQ phuong trinh :
(x*y=2
IuL-l e x=lt16;y=lTG.
Y 4y, gtdtri lsn nh6t cfra p b6ng 36, khi x = I * 16, y = I _ 16 ho4c x = I _y'3, y = ! * fi .
Dy kiiin k) thi tht? tin sau sE vdo cdc ngdy 2b - 29 thdng 3 ndm 2009
Trang 10TRUONC DHSP HA NQI
rcr6r rHPT cnuvtN
CAu I (2 di€m): Cho hdm sii y =
NT THI THU DAI HOC I,AN III NAN,r ZOOS
Nidn thi, To,in Thdi gian.lAm bdi: 180 phrit
* **.
x2- zmx+ m2
v l X6c dinh tAt ce cac gi6 trf cira m d6 ham s5 d4t cgc ti6u tai x = 2
'Jz Tim c6c gi6 tri cua m d6 tr6n d6 thi crla hdm sii
1t; tdn tei it nh6t mQt di6m mA ti6p tuy6n cria d6 thi teidii5m d6 vu6ng g6c vdi dudng thing y = x
,,1 2 Giai he phuong trinh :
v CAu 3 (l di6m) Tinh tich ph6n : t = [/3
x2+r+.,/@Txf
r/Cau a (i diem) Cho tri diQn SABC c6 g6c AB'C = 90", SA = fift = 2a,BC= a.,,/3 v.d SA vu6ng g6c vdi rn4r
phing (ABC) Gqi M li tlitim trdn duong thing AB, sao cho AM = 2Md
Tinh khodng cdch tir tlitim B dffor mp(SCM)
qo l) 'l rong mflt phing v6i hQ tga dQ Oxy, cho tam gi6c ABC c6 ctinh A(-2;3), duong cho CH nim trdn duong'/ thing : 2x+y -7 =0 viduongtrungtuy6n BM nimtrdndudngthing : 2x-y+l =0.
Hay vi6t phuong trinh c6c cqnh vd tim tga d6 trgng tdm G cira tam gi6c ABC
I z-3i I
f t*t *y=4+,tW ll,*' -ztg2=te1+h
D4r kiiin thi th* Idn tdi vdo ctic ngdy 18,19/4/2A09
Trang 11(Thi thti'DH IAn III - 2009)
CAU I.
xz-2x+2m-m2
1 (t,O Oiem) Tgp xdc dinh: R\ {l} Ta c6 y'=-
(x-1f Gii sir him s6 d4t cgc ti6u tei x = 2, suy ray'(2) = 0 hay
4 - 4 +Zm-mz - 0 <+ m = 0 ho{c m= 2
2, tad6u c6 y' J
#=+ y' =o <+x= o ho4c x=2'
Mat khric y' > 0 khi x e (- o; 0) u (2; + co) vd y' < 0 khi x e (0; l ) u (l; 2)'
Do d6 x = 2 ld tliOm cuc ti6u cria him s5
Viy, d6 th6a mdn bdi to6n thi : m:0 hodc m = 2
Do phuong trir,h (*) c6 it nhAt mgt nghiQm kh6c 1, nCn ta c6 hai trucrng hqp sau :
a) Phuong trinh (*) c6 hai nhiQm phdn bi-Ot' hay
[r,orrioo trong d6 f(x)=2*z - 4x*2m'- m2 + 1, thitr.' 0,5 di6m
CAU il.
l (1,0 c1i6m) Phuong trinh dugc biiSn d6i thdnh :
ci{ - "ixr +,i,{.,o{) =} z+rin")"or*
'oJr)'
V6irinl - "or|=0c+sin(| -fl=0<=+x =|*lkrc,kez'
22La'z
r Voi r + jsinx =-i(t+sinx)(sinj + cos)<+sini + co5 =-;(ptndyv6nghi€m).
VAy, nghiQm cira phucrng trinh ld
' * = ; +zkn,k eZ'
w.
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Trang 12Phusng trinh (2) <+ lxl = 4 + 2y thay vio pt(l) ta dugc :
cAu rn ( l,o dii!m).
Tac6 ,= [/t xdx -,'o ,G 6qnffixdx
DAt r= ,@-*z ql thi dt= #|v6i x:0thit=l; v6'i1=y'Jthit=2.
Khi d6 t: I:# = frr+r1*aqt* 1) :fr a * t)+,11,:zbf3 - lz).
cAu ry ( 1,0 di6m).
Do M =2W, n€n khoang c6chttrA tliin mdSCM)
beng 2 lAn khodng cSch tu B d6n mp(SCM) .
Tir gia thi6t ta suy ra AM: +, BM = + ,
V{y, ktroing crich tt B ddn mp(SCM) bing - + {43
CAUv ( l,0rli€m) Ttgiathii5tsuyra 0<"s i.e6tdingthricduscbiiSndoithdnh:
-fL/^
a[b(c+e) +d(c+e)] +cd(b+e -u)= * o a@+d)(c+e)+cd(b+e-a)S 2S+
x2+r+.,ftr+xf
Trang 13Theo bAt ding thtic C6-Si, ta c6 : )
,
a(b + d)(c + e) + cd(b + e -a) = ^ (ry=)'* 1.*o*!*._")3= a(r:a)z *,'=}|,',
Xethdms6: f(a) =+*ry,vdi 0<as f .rac6 f(a)=frf - 5a2-4a+1)>0,vutto;f)'
Suy ra (a) d6ng bicn trcn (0;
Jl * it"l s (; ) =f tan'*)
cAu vt ( 2,0 di€m)
- l) (1,0 di6m) Dubng thdng chira c4nh AB vu6ng g6c vsi CH n€n nhen vectq il(2; 1) lim vecto chi piiuong'
Do d6 dudng thing AB c6 phuong trinh : x -2y + 8:0'
Suy ra, tqa d0 diem B ld nghiQm criahQ phuong irinh :
| {"-r, * I = o *1i,.=_3-B(2;s).
^Yr -' -^ *-'-^ '+;A '-? Et thu6c BM'
Gqi C(x; y) thuQc CH, suy ra trung di€m c0a AC Ie M t7"', '
( 2x+ v-7 = o"
o * [;lf +c(:;r).
Tqa d0 cria diSm Cli nghigm cria hQpt,
\, * - # *, Suyra, BC : 4x+y- 13 =0 vd AC : 2x+5y-11 =0'TrgngtdmG(1; 3)'
2) (1,0 di6m) D[t EA= d ,EF =E 'EC = d '
Tac6 EF : d +6+ d, vi MN i/ BD' n€nffi =Ufr
Di€uki€n:x*0vdY*3-D6 dang chimg minh du-oc tinh chAt l:l: #
Suv- l#l=l elx + (v +1)il2=[x * (v-3)il2 e.x2+(v+l)t=x2*(v -3)2-ev=1'
K6t lu{n: Tip hqp c6c di€m trong m{t phdng phrlc bi6u di6n c6c s6 phfc zthbamtutdi|u kign :
Trang 14rnu0Nc EHSp HA Nor
KHor rHPT csuytN DE THI rutl 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 saocho khodng cdch tir M d6n giao di6m cira hai duong tiQm c{n cira (c) nh6 nh6t
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 ratr€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
ld hinh chiliu vu6ng g6c cria A l€n SB vd SC Chung minh ring 5 di6m A, B, C, E, F cirng thuQc mQtm4t cdu Virlt phuong trinh m4t cAu d6.
CAu 7 (l <li€m)
Chung minh tling thric :
Cloro - Clo.o + Cloro - + (- rlC35iot + - CiBl6 + CrzBlB = z,uu'
Dtr kiiin itgt thi thii Ifrn sau vdo cdc ngdy t6,17/s/200g.
9(3x- Zx) - 3x ,
@
Trang 15oAp AN ivtoN roAN IAN rv cAu I (z,o ei6m).
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
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!,^
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