Đề thi thử toán lần 2 - ĐH Vinh pptx

TRNGIHCVINH KHOSÁTCHTLNGLP12LN2,NM2011 TRNGTHPTCHUYÊN MÔN:TOÁN;Thigianlàmbài:180phút I.PHNCHUNHCHOTTCTHÍSINH(7đim) CâuI. (2,0đim) 1. Khosátsbinthiênvàvđth(H)hàms x 1 y x 2 − + = − . 2. Tìmtrên(H)cácđimA,BsaochođdàiAB=4vàđngthngABvuônggócviđngthngy=x. CâuII(2,0 đim) 1. Giiphngtrình ( ) sin 2x cos x 3 cos 2x sin x 0 2sin 2x 3 + − + = − . 2. Giihphngtrình 4 2 2 2 2 x 4x y 4y 2 x y 2x 6y 23  + + − =   + + =   . CâuIII.(1,0đim).Tính dintíchhìnhphnggiihnbiđthhàms ( ) 2 x ln x 2 y 4 x + = − vàtrchoành. CâuIV.(1,0đim). ChohìnhchópS.ABCDcóđáylàhìnhchnhtviAB=a,AD= a 2 ,gócgiahaimt phng(SAC)và(ABCD)bng60 0 .GiHlàtrungđimcaAB.BitmtbênSABlàtamgiáccân tiđnhSvà thucmtphngvuônggócvimtphngđáy.TínhthtíchkhichópS.ABCDvàbánkínhmtcungoitip hìnhchópS.AHC CâuV.(1,0đim)Chocácsthcdngx,y,zthomãn 2 2 2 x y z 2xy 3(x y z) + + + = + + .Tìmgiátrnhnht cabiuthc 20 20 P x y z x z y 2 = + + + + + + . II.PHNRIÊNG(3,0đim) a.Theochngtrìnhchun CâuVIa.(2,0đim) 1. TrongmtphngtođOxychotamgiácABCcóphngtrìnhchađngcaovàđngtrungtuynk tđnhAlnltcóphngtrìnhx –2y –13=0và13x –6y –9=0.TìmtođB,Cbittâmđng trònngoitiptamgiácABClàI(5;1). 2. TrongkhônggiantođOxyzchođimA(1;0;0),B(2;1;2),C(1;1;3)vàđngthng x 1 y z 2 : 1 2 2 − − ∆ = = − .Vitphngtrìnhmtcucótâmthucđngthng ∆ ,điquađimAvàctmt phng(ABC)theomtđngtrònsaochođngtròncóbánkínhnhnht CâuVIIa.(1,0đim) Tìmsphczthomãn z 3i 1 iz − = − và 9 z z − làsthuno. b.Theochngtrìnhnângcao CâuVIb(2,0đim) 1. TrongmtphngtođOxychođngtròn(C): 2 2 x y 4x 2y 15 0 + − + − = .GiIlàtâmđngtròn(C). ngthng ∆ điquaM(1;3)ct(C) tihaiđimAvàB.Vitphngtrình đngthng ∆ bittam giácIABcódintíchbng8vàcnhABlàcnhlnnht. 2. TrongkhônggiantođOxyzchođimM(1;1;0)vàđngthng x 2 y 1 z 1 : 2 1 1 − + − ∆ = = − vàmtphng (P):x +y+z  2=0.TìmtođđimAthucmtphng(P)bitđngthngAMvuônggócvi ∆ và khongcáchtAđnđngthng ∆ bng 33 2 . CâuVIIb.(1,0đim)Chocácsphcz 1 ,z 2 thomãn 1 2 1 2 z z z z 0 − = = > .Tính 4 4 1 2 2 1 z z A z z     = +         www.VNMATH.com TRNGIHCVINH TRNGTHPTCHUYÊN ÁP ÁNKHOSÁTCHTLNGLP12LN2,NM2011 MÔN:TOÁN;Thigianlàmbài:180phút Câu ápán im 1.(1,0đim) a.Tpxácđnh: }.2{\R =D b.Sbinthiên: *Chiubinthiên:Tacó 2,0 )2( 1 ' 2 ≠ ∀ > − = x x y . Suyrahàmsđngbintrêncáckhong )2;(−∞ và );2( ∞ + . *Giihn: 1 2 1 limlim − = − + − = +∞ → +∞ → x x y xx và 1 2 1 limlim − = − + − = −∞ → −∞ → x x y xx ; +∞ = − + − = − − → → 2 1 limlim 22 x x y xx và −∞ = − + − = + + → → 2 1 limlim 22 x x y xx . *Timcn:thcóđngtimcnnganglà 1 − =y ;đngtimcnđnglà 2 =x . 0,5 *Bngbinthiên: x ∞ − 2 ∞ + 'y + + y ∞ + 1 − 1 − ∞ − c.th:  thh àms cttrcho ànhti(1;0), ct trctung ti ) 2 1 ;0( − và nhn giao đim )1;2( −I cahaitimcnlàmtâm đixng. 0,5 2.(1,0đim) Vì đ ngthng ABvuônggócvi xy = nênphngtrìnhcaABlà mxy + − = . HoànhđcaA, Blànghimcaphngtrình mx x x + − = − + − 2 1 ,hayphngtrình 2,012)3( 2 ≠ = + + + − xmxmx (1) Dophngtrình(1)có mmmmm ∀ > + − = + − + = ∆ ,052)12(4)3( 22 nêncóhainghim phânbit 21 ,xx vàchainghimđukhác2.TheođnhlíViettacó 12;3 2121 + = + = + mxxmxx 0,5 I. (2,0 đim) Theogithitbàitoántacó 16)()(16 2 12 2 12 2 = − + − ⇔ = yyxxAB .130328)12(4)3( 84)(8)(16)()( 22 21 2 21 2 12 2 12 2 12 − = ∨ = ⇔ = − − ⇔ = + − + ⇔ = − + ⇔ = − ⇔ = − + + − + − ⇔ mmmmmm xxxxxxmxmxxx *Vi 3 =m phngtrình(1)trthành 23076 2 ± = ⇔ = + − xxx .SuyrahaiđimA, Bcntìmlà )2;23(),2;23( − − + . *Vi 1 − =m tacóhaiđim A,Bcntìmlà )22;21( − − + và )22;21( + − − . VycpđimTM: )2;23(),2;23( − − + hoc )22;21( − − + , )22;21( + − − . 0,5 1.(1,0đim) II. (2,0 iukin: π π  kxx + ≠ ⇔ ≠ 62 3 2sin và ., 3 Z ∈ + ≠ kkx π π  x O 1 1 − 2 y I www.VNMATH.com Khiđópt 32sin2)sin2(cos3cos2sin − = + − + ⇔ xxxxx 0)2cos3)(sin3cos2( 0)2cos3)(3cos2()3cos2(sin 03cos2cos3sin32sin = − + + ⇔ = − + + + ⇔ = − − + + ⇔ xxx xxxx xxxx 0,5       + = + ± = ⇔       =       + − = ⇔ π π π π π  2 6 2 6 5 1 3 sin 2 3 cos kx kx x x ichiuđiukin,tacónghimcaphngtrìnhlà Z ∈ + = kkx ,2 6 5 π π  . 0,5 2.(1,0đim) H      = + + = − + + ⇔ 236)2( 10)2()2( 2 222 yyx yx t .2,2 2 − = + = yvxu Khiđóhtrthành    = − = + = = + ⇔    = + + = + ⇔    = + + + − = + 67,12 3,4 19)(4 10 23)2(6)4)(2( 10 2222 uvvu uvvu vuuv vu vvu vu 0,5 đim) TH1. 67,12 = − = + uvvu ,hvônghim. TH2.    = = + 3 4 uv vu ,tacó    = = = = 3,1 1,3 vu vu *Vi    = = 1 3 v u tacó    = ± = ⇔    = = 3 1 3 1 2 y x y x *Vi    = = 3 1 v u tacó    = − = 3 1 2 y x ,hvônghim. Vynghim(x,y)cahlà ).3;1(),3;1( − Chúý:HScóthgiitheophngphápth 2 x theoytphngtrìnhthhaivàophng trìnhthnht. 0,5 III. (1,0 đim) Tacóphngtrình    − = = ⇔ = − + 1 0 0 4 )2ln( 2 x x x xx .Suyrahìnhphngcntínhdintíchchính làhìnhphnggiihnbi cácđng .0,1,0, 4 )2ln( 2 = − = = − + = xxy x xx y Dođódintíchcahìnhphnglà .d 4 )2ln( d 4 )2ln( 0 1 2 0 1 2 ∫ ∫ − − − + − = − + = x x xx x x xx S . t x x x vxu d 4 d),2ln( 2 − − = + = .Khiđó 2 4, 2 d d xv x x u − = + = . Theocôngthctíchphântngphntacó .d 2 4 2ln2d 2 4 )2ln(4 0 1 2 0 1 2 1 0 2 ∫ ∫ − − − + − − = + − − + − = x x x x x x xxS 0,5 www.VNMATH.com t .sin2 tx = Khiđó ttx dcos2d = .Khi ; 6 ,1 π − = − = tx khi .0,0 = = tx Suyra .3 3 2)cos(2d)sin1(2d 2sin2 cos4 d 2 4 0 6 6 0 0 6 2 0 1 2 − + = + = − = + = + − = ∫ ∫ ∫ − − − − π π π π  ttttt t t x x x I Suyra . 3 322ln2 π − + − =S 0,5 +)Tgithitsuyra ).(ABCDSH ⊥ V )( ACFACHF ∈ ⊥ ACSF ⊥ ⇒ (đnhlíbađngvuônggóc). Suyra .60 0 = ∠SFH K ).( ACEACBE ∈ ⊥ Khiđó . 32 2 2 1 a BEHF = = Tacó = = 0 60tan.HFSH . 2 2a Suyra . 3 . 3 1 3 . a SSHV ABCDABCDS = = 0,5 IV. (1,0 đim +)Gi J, r l nl tlàtâmvàbánkínhđngtrònngo itiptamgiác AHC .Tacó . 24 33 2 .. 4 .. a S ACHCAH S ACHCAH r ABCAHC = = = K đng thng ∆ qua J và .// SH ∆ Khi đó tâm I ca mt cu ngoi tip hình chóp AHCS. làgiaođimcađngtrungtrcđon SHvà ∆ trongmtphng(SHJ).Tacó . 4 2 2 22 r SH JHIJIH + = + = Suyrabánkínhmtculà . 32 31 aR = Chúý:HScóthgiibngphngpháptađ. 0,5 Tgithittacó .)( 2 1 )()(3 222 zyxzyxzyx + + ≥ + + = + + Suyra 6 ≤ + + zyx . 0,5 V. (1,0 đim Khiđó,ápdngBTCôsitacó 2 2 11 4 2 8 2 8 )2( 88 )( −         + + + +         + + + + + +       + + + + + = yzxyy y zxzx zxP .26 2 28 222 )2)(( 8 1212 4 ≥ + + + + ≥ − + + + + ≥ zyxyzx Duđngthcxyrakhivàchkhi 3,2,1 = = = zyx . VygiátrnhnhtcaP là26,đtđckhi 3,2,1 = = = zyx . 0,5 1.(1,0đim) Ta có ).8;3( − −A Gi M là trung đim BC AHIM// ⇒ .Tasuyrapt .072: = + − yxIM SuyratađM thamãn ).5;3( 09613 072 M yx yx ⇒    = − − = + − 0,5 VIa. (2,0 đim) Ptđngthng .011205)3(2: = − + ⇔ = − + − yxyxBC ⇒ ∈ BCB ).211;( aaB − Khiđó 0,5 B A H M I C B A S D C E F J I K H www.VNMATH.com    = = ⇔ = + − ⇔ = 2 4 086 2 a a aaIBIA .Tđósuyra )7;2(),3;4( CB hoc ).3;4(),7;2( CB 2.(1,0đim) Tacó ).3;1;2(),2;1;1( − − − ACAB Suyrapt .01:)( = − − − zyxABC Gitâmmtcu ⇒ ∆ ∈I )22;2;1( tttI + − .Khiđóbánkínhđngtrònlà .2 3 6)1(2 3 842 ))(,( 22 22 ≥ + + = + + = − = ttt ABCIdIAr Duđngthcxyrakhivàchkhi .1 − =t 0,5 Khiđó .5),0;2;2( = − IAI Suyraptmtcu .5)2()2( 222 = + + + − zyx 0,5 t ).,( R ∈ + = babiaz Tacó |1||3| ziiz − = − tngđngvi |1||)3(||)(1||)3(| aibibabiaiiba − − = − + ⇔ − − = − + 2)()1()3( 2222 = ⇔ − + − = − + ⇔ babba . 0,5 VIIa. (1,0 đim) Khi đó 4 )262(5 4 )2(9 2 2 9 2 9 2 23 2 + + + − = + − − + = + − + = − a iaaa a ia ia ia ia z z là s o khi và chkhi 05 3 = − aa hay 5,0 ± = = aa . Vycácsphccntìmlà iziziz 25,25,2 + − = + = = . 0,5 1.(1,0đim) ngtròn (C)cótâm ),1;2( −I bánkính .52 =R Gi H làtrungđim AB.t ).520( < < = xxAH Khiđótacó 2 4 1 . 8 20 8 2 (ktm vì ) 2 x IH AB x x x AH IA =  = ⇔ − = ⇔  = <  nên .24 = ⇒ = IHAH 0,5 PtđngthngquaM: )0(0)3()1( 22 ≠ + = + + − baybxa .03 = − + + ⇔ abbyax Tacó baabaa ba ba IHABId 3 4 00)43(2 |2| 2),( 22 = ∨ = ⇔ = − ⇔ = + + ⇔ = = . *Vi 0 =a tacópt .03: = + ∆ y *Vi . 3 4 ba = Chn 3 =b tacó 4 =a .Suyrapt .0534: = + + ∆ yx Vycóhaiđngthng ∆ thamãnlà 03 = +y và .0534 = + + yx 0,5 2.(1,0đim) Gi(Q)làm tphngqua M vàvuônggócvi ∆ .Khiđópt .032:)( = − + − zyxQ Tacó ).1;1;1(),1;1;2( PQ nn − TgithitsuyraA thucgiaotuyn dca(P)và(Q). Khiđó )3;1;2(],[ − = = QPd nnu và dN ∈)1;0;1( nênptca      − = = + = tz ty tx d 31 21 : . Vì dA ∈ suyra ).31;;21( tttA − + 0,5 VIb. (2,0 đim) Gi Hlàgiaođimca ∆ vàmtphng(Q).Suyra ). 2 1 ; 2 1 ;1( −H Tacó 7 8 1016214 2 33 ),( 2 = ∨ − = ⇔ = − − ⇔ = = ∆ ttttAHAd . Suyra )4;1;1( − −A hoc ). 7 17 ; 7 8 ; 7 23 ( −A 0,5 VIIb. (1,0 đim) t w z z = 2 1 tađc 0|||||| 2222 > = = − zwzzwz .Hay 1|||1| = = − ww . Gis ),( R ∈ + = babiaw .Khiđótacó 0,5 M H B I A www.VNMATH.com 1)1( 2222 = + = + − baba hay . 2 3 , 2 1 ± = = ba *Vi . 3 sin 3 cos 2 3 2 1 π π  iiw + = + = Tacó 3 4 sin 3 4 cos 4 π π  iw + = và . 3 4 sin 3 4 cos 1 4 π π  i w − =       Dođó 1 3 4 cos2 − = = π  A . *Vi iw 2 3 2 1 − = ,tngttacngcó 1 − =A . Chúý: HScóthgiitheocáchbinđitheodngđiscasphc. 0,5 www.VNMATH.com . mxxmxx 0,5 I. (2, 0 đim) Theogi thi tbài toán tacó 16)()(16 2 1 2 2 1 2 2 = − + − ⇔ = yyxxAB .1303 2 8)1 2 (4)3( 84)(8)(16)()( 2 2 2 1 2 2 1 2 1 2 2 1 2 2 1 2 − = ∨ = ⇔ = − − ⇔ = + − + ⇔ =. − − = − + ⇔ − − = − + 2 )()1()3( 2 2 2 2 = ⇔ − + − = − + ⇔ babba . 0,5 VIIa. (1,0 đim) Khi đó 4 ) 26 2 (5 4 ) 2 (9 2 2 9 2 9 2 2 3 2 + + + − = + − −. . VycpđimTM: ) 2 ; 2 3(), 2 ; 2 3( − − + hoc ) 2 2 ; 2 1( − − + , ) 2 2 ; 2 1( + − − . 0,5 1.(1,0đim) II. (2, 0 iukin: π π  kxx + ≠ ⇔ ≠ 6 2 3 2 sin và ., 3 Z