Đề-đáp án thi thử lần 2-ĐHSP

@ '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. 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 -/. : cau 6 (2 dirim). / :./ 1) Trong m{t phing voi hQ tga dQ Oxy, cho elip @) c6 phuong rriot, r { *.* = ,. 9 '4 : MQt g6c 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[' http://wwww.violet.vn/haimathlx M4t 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 rac6 r= 1€$ff= Jfxzdx - J€x\Fldx 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. http://wwww.violet.vn/haimathlx cAU 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, Do Me(E)ndn' xft*Yil-, 4 OM?cos2c OM2sinza , ;-= i . 1 coszrt sin2cr -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 : -, t-2+3+21 li l; sne = JE!t7- = {; + Ianp = J14- @ http://wwww.violet.vn/haimathlx Ta 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 : o.=#=ffi ea-6=*#, 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 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 '5 http://wwww.violet.vn/haimathlx . 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. ?). , 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. _ 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