MINHHAKTHIQUCGIA2015 MễNTON TRNGTHPTTRNPH x- x+ a)Khosỏtsbinthiờnvvth (C)cahmsócho. b)Xỏcnhtagiaoimcath (C)vingthng (D): y=x Cõu2: (1im) a) Giiphngtrỡnh: ( cos x sin x ) + cos x ( 2sin x + 1)=0. Cõu1: (2im)Chohms y = ỡ z+z = 10. b) Tỡmphnthc,phn ocacỏcs phcz,bit: ợ z = 13. Cõu3: (0,5im)Giiphngtrỡnh 52 x-2 - 26.5x- + 1= ùỡ y - x + y + = x + y ( x + xy + y- 1) + Cõu4: (1im)Giihphngtrỡnh: ùợy + y - x = p Cõu5: (1im)Tớnhcỏctớchphõn: I =ũ sin 2x.sin3 x.dx Cõu6: (1im)ChokhichúpS.ABCDcúỏyABCDlhỡnhchnht,bitAB=2a,AD=a Trờncnh a ABlyimMsaocho AM = ,cnhACctMDtiH.BitSHvuụnggúcvimtphng(ABCD)v SH=a.Tớnhth tớchkhichúpS.HCDvtớnhkhongcỏchgiahaingthngSDvACtheoa. Cõu7: (1im)ChohỡnhthangcõnABCDcúAB//CD,CD=2AB.GiIlgiaoimcahai ổ 17ử ngchộoACvBD.GiMlimixngcaIquaAvi M ỗ ữ Bitphngtrỡnh ng ố 3 ứ thngDC:x+y 1=0vdintớchhỡnhthangABCDbng12.Vitphngtrỡnh ngthngBCbit imCcúhonh dng. Cõu8: (1im)Trongkhụnggian vihtaOxyz,chomtcu(S): x + y + z - x + y - z - =0 vmtphng(P):x+y+z+2015=0 a)XỏcnhtatõmIvtớnhbỏnkớnhcamtcu(S).Vitphngtrỡnh ngthngquaIv vuụnggúcvimtphng(P) b)Vitphngtrỡnhmtphng(Q)songsongmtphng(P)vtipxỳc(S) Cõu9: (0,5im)Cú30tmthcỏnhst1n30.Chnngunhiờnra10tmth.Tớnhxỏc sutcú5tmthmangsl,5tmthmangschntrongúch cúduynht 1tmmangs chiahtcho10. Cõu10: (1im)Cho3sdngx,y,zthamónxy+yz+zx=3xyz. xy yz zx Chngminhrng: + + Ê x + y + x z + y z y + z + y x + z x z + x + z y +x 2y ưưưưưưưưưưưưưHTưưưưưưưưưưưư PNMINHHAKTHIQUCGIA2015MễNTON Cõu1. 1. y= (2,0) x- x +1 Tpxỏcnh:D= Ă \{1}. Timcn ngang: y =2 lim y =2 xđƠ Timcnng: x=-1 lim y = -Ơ lim- y = +Ơ x đ-1+ y' = 0,25 x đ-1 >0, "xẻD (x+ 1)2 0,25 Hms tngtrờn (Ơ1),(1+Ơ) Hmskhụngcúcctr. x Ơ +Ơ + + y 0,25 +Ơ y Ơ y 0,25 ư5 ư4 ư3 ư2 ư1 x ư1 ư2 2.Phngtrỡnhhonh giaoimca(C)v(D)l: Cõu 2 x- = x- x2 2x=0 x +1 0,25 x=0hayx=2suyray=ư1hayy=1 0,5 Vyta giaoml(0ư1)hay(21) 0,25 1.Giiphngtrỡnh: ( cos x sin x ) + cos x ( 2sin x + 1)= (1,0) sin x + cos x = sin x - cosx 3 sin x - cosx sin x + cos x = 2 2 sin x cos p + cos x sin sin(2 x + p = sin x cos p p p -cos x sin p ) = sin( x - ) 6 0,25 p p ộ + k2p = + x x (kẻ Â) ờ x + p = p - ( x - p ) + k 2p ờở p ộ x = - + k2p (kẻ Â) ờ x = 5p + k2p ờở 18 0,25 ỡ z+z = 10. Tỡmphnthc,phn ocacỏcs phcz,bit: ợ z = 13. Gisz=x+yi=> z=xyi.(x,yẻIR) ỡù2x =10. ùợ x2 + y2 = 13. 0,25 Theo bitacú: ỡ x=5 ợ y = 12 Cõu 0,25 Giiphngtrỡnh 52 x-2 - 26.5x- + 1= (0,5) ột =1 ởt = 25 tt=5x>0.Ptt226t+25=0 ộ x=0 x= Cõu (1,0) ỡù y - x + y + = x + y ( x + xy + y- 1) + Giihphngtrỡnh: ùợy + y - x = 0,25 0,25 ỡ y> (vỡy=0khụngthahpt) ợx + y -1 iukin: (1) -( x+ 1) = ( x + 1)( x - x + 1) + y ( x + 1)( x + y- 1) y + x + y+1 0,25 2 ( x + 1)[ x - x + xy + y - y+ + ] y + x + y+1 ( x + 1)[ x + (3 y - 1) x + y - y+ + ](3) y + x + y+1 0,25 XộtA=x2 +(3y 1)x+3y2 3y+1 D =ư3(y ư1)2 Ê "x ẻR => A 0"x, y ẻR 0,25 (3) x= ư1 Thayx= ư1vo(2)tacú: y + y+ =5 ộ -1 + 17 y= ờ -1 - 17 (l) ờy = 0,25 Vyh phngtrỡnhcúnghim(ư1 Cõu -1 + 17 ) p (1,0) Tớnhcỏctớchphõn: I = ũ sin 2x.sin3 x.dx p I= sin4 x.cosx.dx. ũ 0,25 tt=sinx=>dt=cosxdx I = 2t4dt. ũ 0,25 t5 = = 5 0,25x2 Cõu 6(1,0 im) *TớnhthtớchkhichúpS.HCD: HaitamgiỏcvuụngAMDvDACcú AM AD = = nờnngdng, AD DC ã =DCH ã = 90o ị DHC ã = 90o ã,m ADH ã + HDC Suyra ADH D ADCvuụngtiD: AC = AD + DC ị AC =a Hthclng D ADC:DH.AC=DA.DC Suyra: DH= DC.DA 2a = AC 0,25 D DHCvuụngtiH: HC = DC - DH2 = Doúdintớch D HCD: SHCD = 4a 4a2 DH.HC= 0,25 ThtớchkhichúpSHCD: VS.HCD = 4a SH.SHCD = 15 TớnhkhongcỏchgiaSDvAC: Dng HE ^SD TacúSH ^ (ABCD)nờnSH ^ ACvDH ^ AC,doúAC ^ (SHD) MHE è (SHD)nờnHE ^ AC 0,25 T úHElonvuụnggúcchungcaSDvAC. nờn HE =d ( SD AC ) D SHDvuụngtiHnờn: 0,25 HE = SH + HD Vy d ( SD AC )= HE= ị HE= 2a 2a Cõu 7(1,0 im) M B A H I C D Tacú:tamgiỏcMDCvuụngtiD =>(MD):x y+5=0 =>D(ư23) MD= 0,25 =>HD= MD=2 0,25 GiAB=a=>SABCD = 3a.2 =12=>a=2 2 =>DC=4 0,25 GiC(c1c)=>DC2 =2(c+2)2 =>c=2hayc=ư6(loi)=>C(2ư1) =>B(32) 0,25 =>(BC):3xy 7=0 Cõu 8(1,0 (S): x + y + z - x + y - z - =0 v(P):x+y+z+2015=0 im) a) (S)cútõmI(1ư23)vR=4 ỡ x = + t r ù (D)quaI(1ư23)vcúVTCP u =(111)cúptts: y = -2 + t ùz = + t ợ 0,25 0,25 b) (Q)//(P)=>(Q):x+y+z+D=0(D ạ2015) d ( I , ( Q ) )= D = -2 0,25 0,25 Vy(Q):x+y+z -2 =0 GiAlbinclyc5tmthmangs l,5tmthmangs chntrong úch cú1tmthmangs chiahtcho10. Cõu9: (0,5im) Chn10tmth trong30tmth cú:C1030 cỏchchn Taphichn: 0.25 5tmthmangs l trong15tmmangs l cúC155 cỏchchn. 1tmth chiahtcho 10trong3tmthmangs chiahtcho10,cú:C13 cc 4tmthmangs chnnhngkhụngchiahtcho10trong12tmnhvy,cú: C412 Vyxỏcsutcntỡml:P(A)= Cõu 10(1,0 Chngminhrng: im) xy 3 x +y +x z+y z + yz 3 2 y +z +y x+z x Tacú:xy+yz+zx=3xyz + zx x3 + y3 + x 2z + y 2z ị Ê Ê xy x3 + y3 + x 2z + y 2z 2 Ê 1 + + =3 x y z 1 1 Ê ( + ) x2 +y2 2xy x + y x y xy xy(x + y) + (x + y )z Ê z + x + z y +x y Vix>0y>0z>0tacúx3 +y3 xy(x+y) xy 0.25 C155 C124 C31 99 = 10 C30 667 Ê 0,25 ự 1 xyộ + ỳ xy(x + y) (x + y 2)z ỷ 1ộ 1 xy ự ổ + Ê + ỳ ỗ ữ (x + y) (x + y 2)z ỷ ố (x + y) 2zứ 0,25 ộ ổ 1 ự ổ 1 (1) ỗ + ữ+ ỳ = ỗ + ữ+ ố x y ứ z ỷ 16 ố x y ứ 8z Chngminhtngt: yz ổ 1 (2) Ê ỗ + ữ+ 3 2 y + z + y x + z x 16 ố y z ứ 8x 0,25 zx ổ 1 Ê ỗ + ữ+ (3) 2 z + x + z y + x y 16 ố z x ứ 8y 3 T Cụng(1)(2)(3)theov tacpcm DE TH ITH U DH NE ngthcxyrakhix=y=z=1 0,25