Vi6t phuong trinh tluong thang A nim tong mit phang P, vu6ng g6c vsi d vi c6 khoang cach d6n d mQt khoing h= '#... cia sir aa dgmg duo.
Trang 1'rRU'a,NG DHSp r{A NOl Dt THr rrrrl DAr Hgc rAN rr NAvr zoog
rliol T'HPT ciruvtlx M6n thi: To6n
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.
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
-/.
: cau 6 (2 dirim) /
:./
1) Trong m{t phing voi hQ tga dQ Oxy, cho elip @) c6 phuong rriot, r { *.* = ,.
9 '4
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 2M4t 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 nghiQm duy nhAt thuQc ( - 6; 2) \ {0} khi vA chi khi :
cAum ( 1,0 di6m)
| -^> t! lm < _L4
l .-;,=l-rT
L-m=-4 L m=4.
3,13- 2,12 -t
3
e.,6-1
lis-,y' = 't-' zfi
= 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.
Trang 3cAU 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 :
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,
Do Me(E)ndn' xft*Yil-,
4 OM?cos2c OM2sinza
-m=J-= 4'
1 sin2q cos2cr
.r
usrg rU', ra cung co & = T * T
^ 7 1_ _ coszd sinzc * sin2c , cos2c :1 - 1
suYra 6fr?*o=il, s r'i+:r=;*;.
1113
_T_
= _
oMz 0N2 36'
2) (1,0 ditim)
cia sir aa dgmg duo c A th6a mEn bii to64 tlf a se nin trong m{t phang (e vu6ng g6c v6i d,n€n m(Q) nh4n vdc t?hi phusnC cria d lA i(-2;3;2) tAm v6c to ph6p tuy6n
Phuong trinh cria *(O g :
-2x+3Y+22+a=0 (1)
GqiA li giao di6m cuad vr5'i mf(fi, thi tga ttg giao diiim cria A tn nghi€in ctia he phucrng
ft+3 v-9 z-6
r-:-:_
trinh : J -2 3 2 e+ A(3: 0: 0) (2)
(x*y*z-3=0
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 4Ta c6 BC li duong vu6ng g6c chung cira d vi A, d6ng thoi dg dai troqn BC = h = '# .
Suy ra: Ac = : Bc
<=+ AC -z'iE' E = g tane 11 ' ,,/ a rry'?'
"1*" AC cfing tA khoang c6ch hr A dAn m(e, n€n tir (r) vi (2) ta c6 :
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:
,",.,.f x+y+z-3= [ x+y+z-3=0
tv'i :
[-zx+3y+ 2z*6*#= o ua (a)'[-zx+3yr
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
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