ĐỀ THI KSCL ĐẠI HỌC NĂM 2012 LẦN 1 MÔN TOÁN KHỐI D ppt

rct rsr KSCL THr DAr Hoc xAu zltz rAx rH{I1 on rnr ivr0x, ioAN, xu6r u Thdi gian ldm bdi : IB0 philt, kh6ng k€ thdt gian giao di PA tU 96*' 02 trang rHAN cHUNG cno rAr cA THi srNH (7,0 tliam) , 2x-I /- CAu I. (2,0 ctiiim) Cho hdm s6: y -'4 (H) x-l \ 1. Kh6o s6t ss bi6n thi6n vd vE dO thi (H) crtahdm s6. 2. Tim cfrc giStri cria * dQ duong thing ! =mx-m+z cht OO ttri @) tai hai di6m phAn bil.t a,B sao cho dopn AB c6 dO dei nho nhAt. C6u II. (2,0 cfiAm) 1. Gi6i phucmg trinh: sin'x(sinx + cosx) + cos'x(cosx - sinx; * I 4 2. Giei h9 phucrng trinh: ft+x*xy=5y ft*"'y'-5y' tl" *7 - J;+ 3 Ciu III. (1,0 eli€m) Tinh gi6i h4n ; 7 = pnlT r=:-: ' x+l X'-3X+2 C6u IV. (1,0 ititm) Cho hinh ch6p S.ABCD co d6y ABCD le hinh chfr nhflt, s,l v6i m[t phing d5y, SC tpo v6i m{t phdn g d6y g6c 450 vir tpo v6i m{t phlng :00. Bitit dO dai c?nh AB = a . Tinhthd tich khdi chop S.ABCD tr\eo a. Cffu V. (1,0 diAd Tim gi6 tri nh6 nhdt ctra hdm sd y - x+./r' + ! (x > 0) Yx : 2 PHAN RIENG (3,0 iti€m) vuong goc (srB) g6c I Thi sinh chi ctwqc ldm mQt trong hai phdn (phfrn A hogc B) A. Theo chucrng trinh Chuin Cfiu VI.a.(2,0 itihm) 1. Cho tam gi6c ABC cdn tai d,bi6t phuong trinh ducmg x+2y-5=0 vd 3x -y+7 =0. Vi6t phucrng trinh ducrng thdng di6m D(1;-3). 2. Trong mdt phdng v6i hQ trpc to4 dQ Oxy, cho dulng tron (C) co phucrng trinh: *'+ y'*2x-6y+6=0 vd di6m M(-3;r).Gqi A vir B IdctrctiOp di6m ke tir M ddn e). Tirn to4 dQ diOm H ldhinh chi6u vudng g6c cua di6m M tr}n AB . Cf,u VII.a. (1,0 ili€m) Tim sO hang chira UiCt n9 s5 cria si5 h4ng thf 3 bing 36. B. Theo chutrng trinh Nfing cao Cdu VI.b. Q,0 meryl 1. Trong m{t phSng v6i hQ truc to4 d6 Oxy, eho tam giSc ABC, dinh B ndm trOn ducrng thdng (A): zx -3y+14=0, c?nh .4C song songvdi (A), dudrng cao AH cophuongtrinh: x-2y-1= 0. Gqi M(-3;0 lir trung di6m cria c4nh BC .X6c dinh toa dQ c6c dinh A,B,C . I thing AB,BC lAn lugt li: .qc, bi6t ring ,tc di qua xu trong khai tri6n cta nhi thfc [x'Jr * +)" , www.VNMATH.com Elfp (E) , + * + =1 vd diiSm M thu6c (E). cie str (d) td ttu<rng thsng ti6p xirc v6i (E) tai M vit (Q chttruc ox, oy ldnluqt t4i A, B. Tim top d0 di6m M dC dien tich tam gi6c AoB nho nh6t. C6u VII. b. (1,0 ifiAm) Tim x bi€t rang trong khai tritin "tu(J7*-J )' r )., , \ s6 cta 3 s6 hpng cu6i bdng 2?,t6ngc6c s6 hang thf 3 vd thri 5 bdng 135. , t6ng c6c hq 2 www.VNMATH.com EAp AN-IHANc orfm xV nu KscL THr DAr Hec NAnn zal: - lAn thrn, I MOn: Tofn; ftr6it n (D,ip dn - thang dtd*t gdm 07 trang) Ciu D6p 6n Di6m I 2rA t dlem I ! *Q,Q*4i?r0 - - 1r T{p x6c dinh : D: m t {t} 2. Su bi6n thi6n I a) Chi6u bi6n thi6n. Ta c6 : y' * - < 0. Vx e D x -l)' Hdm sd dE cho nghich bi6n trdn c5c khoAng (-*;r) vd (f +o) 0,25 b) Cgc tri: Hdm sd kh6ng c6 cgc trf c) Gi6i h4n vd ti6m cfln: tliy - Z; IY_! = 2, d6 thi cria hdm s6 c6 tiQm cfn ngang ld ducmg thdng ! =2. limy - +oo; Limy = -@ , dd thi ctra hdm sd c6 tiQm can dimg h :+1" t+l- ducrngthing x=1 4,25 d) Bing bi6n thi6n I X l-oo -tr I ll _ll +oo v 2 0,25 thi a J. 0,25 ?, $.'.0*f-Q@. X6t phucmg trinh holrnh dQ giao di€m cria dudng thdng dd cho v6i -, 2x-l fx+l dd thi @): -=mx- m+ 2 (1) e { x-l ' l**'-2mx+m-1=0(2) l- 0,25 www.VNMATH.com Eulng thlng y=mx-m+2 cdt (H) tqi hai ili6m ph6n biQt e (1) c6 hai nghiQm ph6n biQt e (2) c6 hai nghiCm phAn biQt kh6c 1 € m>0. Ysi mr0, (2) c6 hai nghiQm phdn biQt, gi6 sri x,,x, . ' lY'=ffixt-m+2 Df;t A(x.yr1, B(xr,!z), ta c6: j ".' lYr=ffixz-m+2 Khi d6 AE =(xr-\)'+nf (xr-x,)' (*,-*r)'("f +l) =[(", + \)' - +4x,]1ni + t1 Theo Viet: fxr+xr-z 1 r_ I *-1= AB' =4(m+l)= 8,Ym>0 =+ AB>2J2,Ym>0 lxrx, = tn lm MinAB =2J, khi ln = 1. Yatv m= I ldr niJoi .An-,irn. 0,25 0,25 0,25 CflU II 2ra drem l. (1,0 didm) sin3 x(sinx + cosx) + cos'(cosx - sinx) - <> sino x * cos4 x + sin3 Jccos.r - cos' xsinx a J 4 .t J 4 0,25 I - 2sin' xcos' x - sinxcosx(cos' x - sint; : l I - !rin' 2" - lsin2xcos 2* =1 113 -:n- cos4x) - lsin 4x = I 4' 4 4 e 0,25 <+ sin4x - cos 4x - 0e sin(4x - 4) = O 4' 7t _7r <+x :-+k keZ 16 4' V4y phucmg trinh c6 nghiQm 1A x: * + t: 164 ,keZ. 0,5 0,25 4 www.VNMATH.com /i I Ddt: x+-=S v (t - lx+-=-) lS=-5 I Vdil- -=1 ,Y ,hQvOnghiQm LP = lo 1".! - to Lv [ 1 [s=3 l"*;-t rx-2 lx=t v6t t; =;'l*r'=, *t; =i " b ; LY 1 VQy h€ phucrng trinh c6 nghiQm 1a.: (x;y) - (2;I) vi (x;y) = (t;t). 4,25 0,5 Cfiu ilI 1'0 tli6m 1=lim x+l <tx+? ,6+3 x'-3x+2 :,1,t+? -2- J71j +z = lim x+l 'x' *3x +2 ,. (:,1.*t-z G;3-2) - rlrtl I - : | x-'r ["' - 3x+2 x' -3x+2 ) 0,5 1 6 0,5 Qnu ry I 1'0 Vi SAL(ABD)*SCA= f BC L(SAB)+&-300 0,25 Gqi S,4 =x (x > 0). A&4C vu6ng t4i A, na< i .l I I .! .l I j 5 www.VNMATH.com c6 SC) - 450 n}n AC = SA =r vd ,SC - xJi nsaS ".'*::: i : 11 e :::: ::: -" :-1r " : * LABC vu6ng tu B , c6 2 AB' + BC' - AC' e o' *\- y2 e *= oJi, 2 SA-oJi. BC=a 0,25 t17 a ! z (dvtt). -\/ J VOy, Vr.nuro = l* " '"o'= 0,25 CAu V 100 (Irenr l" 1 !=x*{"'+a tren (o;+.o) , I L&7 ,.f -1, X- )/ -L' f : 2^lx'* t Vx tf-' !'=oo{ -zx=2^lr'+! xYx <+ 1- 2xt = 2x' ^lx' + I eI- 2xt Vx h-z*'> o <+1, .r2 gle= f(t-zx')'=4*'(r'+1) 1r 2 X6t hlm s6 Tir b&ng bi6n thi6n cho k6t qu6: Minv-2 khi "=f (0;+o)" 2 0,5 0,5 Cfiu I VI.a l 2rA i GIEIn L, Qr-q 4i-Q4- Gqi,vdc tcr phfp tuyiin cna AB tiii;ij; ililr'uptuy6ffi';a-it I fit:;-fl vd v6c to ph6p tuy6n cua AC n ,t (a;b),(o, *b, *A) | -9g 449-9 g-T-L?ij,.193e.le s9e g-,-g *s*-yl lgle$s* '"y_p ] 0,5 6 www.VNMATH.com l,\,\l _|,\il cos-B-cosc<+ffi -t=-t= lryllryl lryllryl t lta-ul <+f -J e22d +2b' LSab=0(*) ./S 'ld +b' Giei (*), ta dugc 2a =b ho{c tta =2b . - V6i 2a=b, chgn a=l suy ra b=2 thi ,tr(UZ). Do D e AC n6n phucrng trinh AC ld: l(x-l) +2(y+3) = g hay x+2y+5 = 0 ( loai do AC ll AB) - Voi lIa = zb, chgn a = Z svy ra b =l 1 thi ,a1Z;tt1 . Do D e AC n€n phucrng trinh AC ld: z(x -L)+ il(y + 3) : 0 hay 2x +lly +31 = 0 (nhan). Vfly, AC: 2x+lly+31:0. 7 ?'-&-o-gj Eulng trdn (C) c6 t6m MI -zJi > z= R= M 1(1;3) vd b6n kinh n = Z. nim ngodi ducrng tron (C). 0,25 Ggi H(x;y). X6t thdy t, M, H thdng hdrrg n€n7fr(a;-2) '1 v-3 phuong u6i Ifr(x -1;y- 3) e + = E€) x - Zy - -s cirng ta c6 0,5 Lpi c6 NAM - NHA= IA' = IM.IH md IM.IH - IMJH , IM.IH = IA' e -4(x- 1) - 2(y -3) = 4 e Zx * y =3 ' _ ___ ___ _-T to4 tlQ di6m H r}roh mdn hO phuong trinh: f _ _ r I:-r,= _rol" -t = I/[,l,li'] l2x+!=3 1 _13 \.5'5) l"-T i Y$y H(+,9.] \) 5 / 0,25 Ta c5 www.VNMATH.com o tr0 tem :ZCI* k=0 [,'v;'- :)' =fc: (*, J])r[*)' " n 5k =lClxT.*3k-3n k=0 1lk-6n 2* H0 s6 cria s6 heng thf 3 ld 36, ta ducyc Cl, =3A a n- 9 ek=6. n66 Lnx ^ .L .1Ik-54 f U YeU CaU Dal toan, ta Co Z = 6 Vfly s6 hang chria xu trong khai triOn ld 0,5 'Ciu VI.b 2'0 i (Irem (2x-3v-2=a top dO di6m A thobmdn h0 phucrng trinh: 4'^ " !) l*-2y-l=0 "-\-)-/ !,1-1.'8 4i-'.@ Vi BC L AH n6n BC c6 phuong trinh: 2x + y* c = 0 Do M(-3;0) e BC n€n c = 6. Vfly phuong trinh BC Id 2x + y + 6 = 0 Ma B. (A), to4 dQ B tho6 mdn hd phucrng trinh: (2, -3v +I4 = 0 {^ ' - :+B(a;z) [2x+ y*6=0 Y-iY-f.1,.9)1t-tryle {.ri-rp-g.t_T__qL?it) _ Cenh AC ll (A) va di qua C ndn AC co phucmg trinh: 2(x +2) -3(y +2) = 0 hay 2x -3y + 2 = 0 . YQy A(r;0), B(-4;2), C(-2;-2). 0,5 0,5 ?.' Q'.Q Fi-c*) Gei M(xo;%) € (n)* +i', phuong trinh (d): Ij!- + l'' 9t 1 l$i d6 Snou =;OA.OB = O |",y,I l6n nh6t. TG.;bfidds thii; clt , 36 = 44 +9y1' > 6ia, rc-rr xhy ra g 4xt Tt (1) vd (2), ta dugc V4y c6 b6n di6m M thoi *,(+,ll),*,(+ l l khi ] I I l I I I I I I I .I Jrl I ,t ./t suy ra Srou fro nhAt khi vd chi co: S yi = tTlx,yol = lx,yol < 3 9v',(z) 92a 2"0 rdn y6u cdu bdi to6n li Jt), *,(*,rr),*^(-+,- 0,5 0,5 CAU VII.b 1r0 Tdng c6c hQ s6 cria ba s6 hang cu6i bdng 22, n€n 9-i.l :l 9u ' ! 9= :??:9 )9i rly*g jf'h lg duec n = 6 ( , \6 : :.:_,/- ; V'- khi d6, ta c6 kirai tri6n [Jr. . # ) =Lc:(Jr. ) t#J 0,25 0,5 8 www.VNMATH.com f6"e;a;;6-hds iii i 3 ;tttiti i uB",e' it5;e' V$y * = -1 vd x = 2 thod mdn y6u cdu cria bdi to6n. Q!rt-!: Hgc sinh ldm theo cdch khdc drtng phdn ndo thi vfin cho iti6m phdn tuong drng. 0,25 www.VNMATH.com . vd ti6m cfln: tliy - Z; IY_! = 2, d6 thi cria hdm s6 c6 tiQm cfn ngang ld ducmg thdng ! =2. limy - +oo; Limy = -@ , dd thi ctra hdm sd c6 tiQm can dimg h : +1& quot; t+l- ducrngthing. x =1 4,25 d) Bing bi6n thi6 n I X l-oo -tr I ll _ll +oo v 2 0,25 thi a J. 0,25 ?, $.'.0*f-Q@. X6t phucmg trinh holrnh dQ giao di€m cria dudng thdng dd cho v6i -, 2x-l fx+l dd thi. orfm xV nu KscL THr DAr Hec NAnn zal: - lAn thrn, I MOn: Tofn; ftr6it n (D, ip dn - thang dtd*t gdm 07 trang) Ciu D6 p 6n Di6m I 2rA t dlem I ! *Q,Q*4i?r0 - - 1r T{p x6c dinh : D: m t