KV THI KSCL TTII DAI Hq}C xAIvI IOTIL,AN rr*tI l XA nn rnr u0n : ToAr\, KI{or A Thdi gian ldm bdi; 180 phtit, kh6ng ka tha gian phat di nA tni g6rn: 02 trang *.||',' '|. . ffi-r*6 fr€' . gh-".% psAw cguNc cno TAT ce Tui swn ( 7,0 arcm 1 Cdu I. 12,0 dihml: 1. Khio s6t sp bi6n thi6n vA v€ dO rhi (C) ctra hdm s6 y _2*+rl . x-1 2. X6c dfnh m d6 duong thEng: y = x -2mc6t (C) tai hai di6m phAn bier M, N sao cho MN :4112. Cffu II. (2,0 iti6m): : 1. Giai phucrng trinh: cos2x*5: Z(Z-cosx)(sinx-cosx) 2. Giai phuong trinh: x'- x * 1006 [+ 8048x = 1006 ciu trrl. (1,0 di€m): Tinh gi6i han : I : ,' **o - J 20L?I - -204i x+l X-l c6u IV. (1,0 ctihm): cho hinh ch6p S,ABCD cir d6y ABCD ld hinh chfr nh6t v6i AB : a , AD : 2a . Canh SA vu6ng g6c v6i mat ph&n g diry, c?nh b€n SB tao vcyi mat phang clay m6t g6c 600. TrOn canh SA I6y di6m M sao cho AM :a.6 ,-X+ ^r.i* =:i, mat phang ( tsCM) c6t cpnh SD tpi N . Tinh th€ tfch kh6i chop S.BCNM cf,uv. (1,0iti€m): chox,y, zldbasOthucthoam6n: x'-xy+y'=1.,Tim giitrildnnh6t vd gi6 tri nho nhAt ctra bi6u thilc: p : xo + yo - 4 . x. +y'_3 PHA|{ RIENG ( 3,0 tti,m) Th{ sinh cht itwgc ldm mpt trong hui phan ( phhn A hoqc B) A.Theo chwons trinh Chudn C6u VIa. (2,0 ctiA@ 1. Trong mdt phing vdi h€ toa d6 oxy, cho ducyng tron (c): x2 +yz -2x+6y -15 : 0. \ri6t phuong trinh ducmg thing A vu6ng goc v6i ducrng thing : 4x-3y+2:0 vd c6t duong rr-on (C) tpi A, B sao cho AB : 6. 2. "Irong m4t phdng v6i hd truc toa dO Oxy cho hinh chfr nh4t ABCD c6 di6n tich bing 12, vd, c6 tdm I ld giao di6m ctra dudng thing d, :x-y-3=0 vdi cl, :x +y-6:0. Trung di6m cua mQt cpnh li giao diiirn cria d1 vdi truc Ox. Tim toa c16 c6c dinh ctra hinh chfr nhat. cdu vlra . (1,0 di\n) : cho n ld so tq nhi6n l6n hcyn 2, tinht6ng : S=Cl, +2.Ci+3.C] +a.Cl + +n.Cil Trang 1 12 I B,Theo chwons trinh Ndns cao Cfiu VIb. (2,0 dihm): 1. Trong m{t pheng vdi hQ to4 d$ Oxy, cho hinh binh hdnh ABCD c6 di€n tich bing +. Bi6t A(1;0), B(0;2) vd giao didm I cua hui c,ro'g chdo nim tr6n ducrng thil; y f ;. ffi;;; c6c dinh C vd D. 2. Trong m4t phing vdi hQ top d0 Oxy cho parabol (p): y : x2 -2x vitelip (E): ** r, :r. Chirng minh ring (P) giao @) tai b6n di6m phdn biQt cung nim tr6n m6t ducrng tron. l Vi6t phuong trinh ducrng trdn di qua 4 di6m d6. cdu vlrb. (1,0 rtihm): cho n ld s6 ru nhi6n l6n hon 2, tittht6ng : s = cl, * 2.C:+ 3.c] * 4.ci+ + (-l)"-'.n.ci ' -,t -________-H6t_-____-_-_ , : (Thf sinh kh6ng dwqc sir &,mg tdi liQu. Cdn b6 coi thi kh6ng gidi thich gi th6m ) i kang2 /2 oAp Ax - BrEU ornvr CAU DAP AN DIEM I (2"r9 ili6m) 1.(1,0 tli6m) +) Tap x6c tlinh : n \ {t} +) Su bi6n thi6n - Chi6u bi6n thi6n ' y'= *h < 0, Vx e (-"o;1) u (1;+*) 0r25 Him s6 nghich bi6n tr€n c6c kho6ng Coo;l) vd (1;+oo) -Hdm sii kh6ng c6 cpc tri -Gi6i han vh tiQm cpn : lim v: lim y:2 : lim v = -oo vd lim v: *co x-++.o" x-+-"oJ ' x-+l-' x +l*' OO tfri hdm s5 c6 tiQm cin ngang:y:2 vi ti6m cfln dimg : x :,1 0,25 BAng bi6n thi€n 0,25 +)E6 thi : 1 pO ttri hdm sd clt trpc Oy tai di€m ( 0 ; -1 ), clrtruc Ox t4i di6m (-l ; O ) 2 Giao di6m cria hai tiQm cQn ld tdm dOi xring cua d6 thi. 0r25 2.0,0 tli6m) Dudng thdng Y = x -2mcat (C) tpi hai di6m phdn biet M, N e phuong trinh : 2x+l x-l <>Phuong trinh r,' -(3 + 2m)x +2m-1 = 0 (*) .o 2 nghiOm phan biQt kh6c 1 0,25 Trang 1/7 n (4,0 iIi6m) 1. (1.0 di6m * [:::x-:1tr I rt t"u".,"-,'". : ;i; ;;;* ;';-,r' = o ; ;.lz"i"ft.olj*'tu(":fr;:;i4 vdyphuong trinh dd cho c6 nghi€m : x: f+k2n;x =n +k2n (kez) 2. (t,0 tti6m ,_-,[o= (3 + 2^)' - 4(2 - ft'-(:* 2m)t+2m l4*' + 4m+ l3 > 0 (+{ [*3*0 m-1)>0 -1+0 thoA m6n VmER. cqi ip; d6;iil iiidrn M "tN i[i rraii,lil: i,")lrvt.;H:]*) . Khi ciit;;;* li hai nghiOm cria phuong trinh (*). Theo gi6 thirit : MN - 4"lt VOy: i\ , ) l2i cfic gia tri cdn tim. Ei6u kiQn x6c dinh cira phucrng trinh , >< t - 1 8048 Phuong trinh : D4t 1+ JG 8048x =2y, y > 1 tu d6 ta duoc : "/ 2 1+ 8048x = 4y' - 4y +1 <) yr - y - 2012x Mdt kh6c, thay I + Ji+ 8048x = 2y vao (*) ta duoc : x, - x _2012y [,n=-1 e4m2+4m-3-oel 2 It lm=- L2 0r25 4,25 Trang2/7 t (;t: ; = t0 itt-iii - . . .' . ' VdytacohQ : {'t. n- z-vL/'! rrtr liictroiti" a"q;lGryjG;FioTli= o " "-'-'.' €x=y (dox+y+2011>0) Thay vdo (2) ta duoc : y' -2013y=Oo[t =:o'3 lo?inghiQmy: 0 ui vr1 LY=u r J 2 voi'p2oitt"4",; :iiili#"r*,t - : ' Vdy phuong trinh dd cho c6 nshiCm : x =201? 0,25 tr'?5 IN (1,0 tli€m) v*_1 0r25 0,25 {},25 _:_ ___,_ __- 0,25 II/ (1,"0 tti6m) G " *ff,_\ .7" k'\ A a : ,'.':B, vl"-AD7/BC;fi ( BcM)il At "y ;t.p( BcMt;ti.mpasADi ih;" sd; tuy6n MN // AD , MN//BC, SA r (ABCD;= Se r eC , fecIAB Iaco: { =BCI(sar)=BCl_BM. LBC I SA :- rri giric BCMNithl"riiri*s il6;s ;6 BMUa.'t*e.;; :-&t'ithid[;hi6; ru6ng g6c ctra SB l6n (ABCD) do vdy g6c hcro bdi SB v6i (ABCD) ld ffi Ia c5 SA : AB tan,ffi :AB tan60o : u.l! l i / I I o,,u f I Trang3/7 MN-sM .MN sA-AM a\6'* z 4a Ivil\ DIVI IVIN JA - AJVI I 7 4.n .T=:-(=)-= AD SA 2a SA u.,6 j BM:",ffi[AM'-+ {3 (^' '4a\ secxrr,r - BC tMN BM : l'+l+ : ry* [ , )r' tJi Hp SH IBM ,vi BC I(SAB) =+ BC r SH. -Y-?y_.$_H__+._LP_-CN\0 = .9"1{_ 1.?.4s.ge__'_eg. s_ye_l_+& sb_Qp _s_,F_c_NM .^=;=; AM "-5 :-=: ^^n :a:: n l tunffiF =+=++Affi = 300 > m= 300, sB = 2a, SH =153 =u AB 3 's'vll-z" Goi V 1A th€ tich ch6p S.BCNM ta c6 1sH.s".r, :ry Ta c6 : x' -xy+y' -l g7 1+3xy=(x+y)' >0=xy=-] J MiI |.h.fP.:g j 1 : x' - xy + y' = (* - y)' * xy ) xy Cfrngtir : x' -xy+ y' =r= x' *y' =xy+l={*' +y' -3 =xy -2 - [xo *yo -4- -("y)' +2xy_3 suyra: P= (xY)'+2xY-3 xy, -2 Xdt hdm so : f(t) ( :2:-3 r,6i t t*2 . 3 -(t_ z\' f'(t)- \-:z'f'(t;= "'' (-z)' ) Ta c6 bing bi6n thi6n : 0,25 0,25 t; VJ t; VJ . [-1,r-l L 3') ft-z- oel It-Z+ |vtalf(t) =f (r)=2, L-t"l -f 1€'t' 5) = zJi -z |{iq r(t) = f (2 - L-;r j I V (1ro rli6m) Trang 4/7 -/ *k Tir d6 ta c6 : MaxP =2 uri,[*=Y=1 Ix=Y=-1 r r E | | *_Vt-lJi -"JJl-r,,_"17_:JI+J"r6_i i l^-T't=T I I_t I*-Jz-rS*J.,6-i ,,_!t t"1-t-JFL I MinP-zJj-z ru'il - 2 'r-T I | " = -Jz-rS -J 6-r _ -#-tT+JG{ l^ ^_ l^ 2 't:T 1o3s / Fr = i | * _ -Vz-rJr +VJ:-i _ -,lt -tJi -JS-r j l^ ,Y=f i Wa (ro 1. (1,0 di E\,-:* \:'t"" tIi6m) :-3-y__l_1,6__lrurr \ \.,, uu raril r( r;_J ); oan Klnn t(:). =9g"lg b glu,s*-q-i o* 4P / : thi AH=3 vd IH eAB suy ra IF{ = ./nt - ALII _ 4 -or-?5- 0,25 0,25 0,25 vr ouCIng rhang A vu6ng g6c v6i dudng th6ng : 4x_3y+2:o "e"piiucrngiiinh cira A c6 dang : 3x*{y+g:g , lc-el [c 11 4=IH= d(I; A)= t- -t <>1"- rr - { l^-to '-j-r-: : l-u t' V0y A : 3x*4y*)e:0 hodc l: :xi+v_i-i:0 * ,. \rr\, rtttttl, Ta c6: a, nO Do vai trd A,B,G,D la nhu nhau n€n gi6 sri M rd trung di€m canh AD M = d, nox suy ra M( 3; o). Q '- -' ra c6 : AB:2rM=rtre?T =3Jj rheo gia trri6t , Soncn:AB.AD =IZ=AD:+S" = 12 :t^6 AB 3Jr-', " Vi I vd M cung thu6c duong thing dr 3 dr I AD Ducrng thdng AD diqua M ( 3; 0) vd vu6ng g6c v6i cl1 n6n AD nhfln i(t;t) dm v6c to phdp tuytin. Phucrng trinh cua AD: x * y _ 3 = 0 f. _g f*-y-3=0 i;;:o€)1 : vov'(Z'ij ' tv t"2 I .l m r rang 5/'/ ng thuQc AD n6n: di€m A, D *y-3:0 [y:-x+3 [y:-x+3 l]:'-* c>l e], .j €)]lx-3:1 <) [(*-l)'*y2 =r - l(" -t)'*(3-x)2 -2* ll,. ,: , lL Lai c6: MA:MD:$-O 2 Toa d6 A, D ld nghiQm cua h€ : Y-ey A( ?;.0 P( $ :_!) fqong tu I cfing lir trung di€m cira BD n6n ta c6 B( 5; a) V0y toa d0 c6c dinh cua hinh cht nhdt ld: (Z; t ), (S; +), ( Hodnh dd giao di cua (E) vd (P) li nghi6m "t i phucrnf t.inh . -2x)2 =1c)9xa.36x3 +37x2-9:0 (x) VA Ix l L! t; (x-t)2 +rz ll":' llu=t ir.' ll": o llv=-t L L" 0,25 X1*,*' ?ry_lt," !{y_itr_gs?!: {11911: ci p"C1,*,'.;.;t::1.;: _" _._ _ ,-_ Chonx : 1 = n.(1 +1)'-' =Cl + 2Ci + +.nCi Vdy: S=n.2n-r 1. (1,0 di6m VIb Q$ tIi6m) 3:-"-, S-:-tI-?)=AB = J5 ' Ph.'o''g trinh AB: z I e (d) : y : x si(ilit. I itiffis A;ffi ;il AC ;tBit ;a;; C(zt - 1 ; 2t), D (2t; Zt - Z) Trang 6/7 X6ti(;i=da_3o"5_h7;i:q;-ir"iiic"ir,.t'o"n"o'-r(riifi0i.0 (0X(1) < 0,,(1X2) .0, "(2X(3) < 0 suy ra (*) c6 4 nghiQm phdn bi€t, D"ge -(Pl q41-e) Bi-l_4ieql_p_b4s_b_iet _-_,_ __ ._ i " I Y=x2 -2x Top d6 giao di€m cua (E) ve (P) tho6 mdn h€ i *, , | -=-*Y2:1 Ie [8*t-t6x:8v . ^ ^ <> {-; _ ";- ' > 9x'+9yz -l6x-8y-9:0 lx'+9y' :9 <+ x2 +y'-**-*y-t = O (**) ' g g' \ / (**) te phur:ng trinh <tuong hon c6 t6m r: [3,f ), b6n kinh R : g Do d6 4 giao di€m cua (E) vd (P) cung nim tr6n ttudrng trdn c6 phuong trinh : xz+y2-f"-fv-r:o 0,25 0,25 4,25 v'ub (1ro ili6m) X6t khai tri6n: (1+ x)" = Cl + xCl + x'Cf, + + x"Cl Ar25 Dao him hai vti ta dugc: n.(1+ ")n-t = Cl + 2xcl + + nxo-rCi 0'?5 Chgnx : : 1 + n.(l-1)"-t - Cl - ZCI+ + (-l)".nCi 0,25 Vdv : S:0 0.25 Chri i : - Gi6m kh6o tru6c khi chAm thdng nh6t <hp An. - Thi sinh ldm theo c6ch khdc md dring vdn cho cti6m t5i cla. - Cdu V thi sinh c6 thti lam theo cdch x6t : % "" - %.r.* - SM'SN - q'{ -i- %.*o \.u.o sA.sD sD - C6u VIIa vd Cdu VIIb thi sinh c6 th€ cti chimg minh,cOng thirc : kcl = ncl-i Sau d6 thay lAn lucrt k :1,2,3, . . ,n Trang 717 . coi thi kh6ng gidi thich gi th6m ) i kang2 /2 oAp Ax - BrEU ornvr CAU DAP AN DIEM I (2"r9 ili6m) 1.(1,0 tli6m) +) Tap x6c tlinh : n {t} +) Su bi6n thi6 n - Chi6u bi6n thi6 n. :,1 0,25 BAng bi6n thi n 0,25 +)E6 thi : 1 pO ttri hdm sd clt trpc Oy tai di€m ( 0 ; -1 ), clrtruc Ox t4i di6m (-l ; O ) 2 Giao di6m cria hai tiQm cQn ld tdm dOi xring cua d6 thi. 0r25 2.0,0. " Vi I vd M cung thu6 c duong thing dr 3 dr I AD Ducrng thdng AD diqua M ( 3; 0) vd vu6ng g6c v6i cl1 n6n AD nhfln i(t;t) dm v6c to phdp tuytin. Phucrng trinh cua