Đề-đáp án thi thử ĐHSPHN-lần 5

w TRU'0NG DHSP HA Nol KH6I THPT CHUVEN sE THI THIIFJA.I IIQC LAN V I-,IAM 2009 Mdn fhi: To6n Thdi gian ldm bdi: 180 ph0t Cflu [. (2 dii3m): Cho hdm sii y: x3 + 3x2 + rnx + I (l) (m Id tham si5) ' ' L Khio s6t vi vE d6 thi hdm sii 1t ; thi m = 0. 2. TimmA6euO'ngthingy= lcatd6tlri hdmsO(t)tai 3 di6m phdnbiQt: I(0;,1),A vdB. Vd'i gi6tri nio cria m, c6c tiiip tuy6n crla d6 thi hdm sO (t ) tai c6c di6m A vd B vu6ng g6c v6i nhau. CAu II. (2 di6m). l. Giiiphuongtrinh: 8cosx.cos($-x).cos($ +x)+ I :0' ((x-a)(x+1)-y(y+s) JA 2. Gieihephuongtrinh: i loro_r,0 +D=T c6u rrr. (l.di6m). Tinh tich phdn : t = i"vz -G#r), , Cdu IV. (l di€m). Cho hinh ch6p tir gidc d6u S,ABCD c6 d0 ddi c4nh driy bing a, cAnh b€n tao vdi m4t driy mQt g6c bing 30". Tinh diQn tich mgt cAu ngoai titlp hinh ch6p. Cffu V. (l ditim). C6c s6 thgc duong thay dili x, y, z th6a md.n : x2 + y2 + z2 = 3. Tim gi6tri nh6 nhAt cira bi6u thti'c , P = -i- * + * + xy+1 yz+L zx+\ CAu VI. (2 di€m). L l) Trong mflt phing v6{ he tqa d9 O*y, cho hai di6m A(5; - 2), B(- 3; 4) vd dud'ng thing d c6 phuong \ / trinh: x -2y + I = 0. Tim tga'dQ di6m C tr€n dudng thing d sao cho tam gi6c ABC vu6ng tai C. 2) Trong kh6ng gian v6i hQ tqa dp Oxyz cho hai duong thing , ( *=1+: x+4 y-8 z-B d,:{ v=1-t vA d,:- ( z=-Z+Zt ' 2 1 -1 a) Chung minh ring hai du'dng thing d1 vd d2 chdo nhau. b) Gpi MN ld duo'ng vu6ng g6c chung crla d1 vd d2 (M e d1 , N e d2). Hdy viiit phuong trinh mat cAu duong kinh MN. CAu VII. (l di€m). Tinh t6ng ' s = '4= *=+ *;+- + + *+ + -: . 2!.20071 41.20051 6!.2003! 2006!.3! 20o8t.1! - http://wwww.violet.vn/haimathlx oAp AN mox roAN r,AN v cAu r. 1z,o a;em;. l. (1,0 didm;. Voi rn = 0, ta c6 y : x3 + 3x2 + l. ' T?R x6c dinh : D = R' . Subi6nthi6n: y'=3x:+6x, y'=0c+x=0 hoflc x:-2. Tac6y(0):1,y(-2)=5. Bdng bi6n thi€n : yco: y(-2): 5, ycr : Y(0) : l. . Dd thi : ( hqc sinh qu vE hinh ). 2. (1,0 di6m). Eudng thing y = I cit d6 tlri hdm sd tai 3 didm phdn biet khi vd chi khi pt sau c6 3 nghiQrn ph6n biqt : xj + 3x2 + mx + l = I <.+ x' + 3xt + mx : 0 <+ x(x2 + 3x + m) = 0 e+ [ -r * r-i=*o= 0 (2) (= pt (2) c6 2 nghiQm phin biet ktritc 0, hoy [a = l-*oT ) 0 = {|; I (*) Gii sri A(x1; l), B(xe; l). Khi d6 h9 sil g6c cua ti6p tuy6n tai A li : ke=y'(xe) =3x2l-+ 6x4 + 6:3(xl + 3xa+61 -3xr- 2m= -3xa-2rn. Tuong f.u, ta cr)ng c5 : ke : - 3xs - 2ln . C6c titip ruytin t4i A vd B vu6ng g6c vdi nhau khi vi chi khi : ke.ka:- 1 e (3x6+ 2m)(?B+2m): - l e gxaxs*6m(xs+xs)+4m2+ l=0 (3). Theo h€ thirc Viet, ta c6 : x1 + xe : 3, xaxs : m, thay vdo (3) ta duoc : f,n=:tG 4m2 -9m+ i = 0 <= | ,_h .U hai giri n'i niy cria m thoa mdn dk (*) Lm = -;- cAu tt. (2,0 tii6m) I (r'0 di€m)' *"*'-:: -",il::-;:,Tl4cosx ( 2cos?x - | I + r : o <= ' 3 2' 2(4cos3x-3cosx)=- I c"a cos3x: - 1r*"=++ **, Urz. 293 2. (1,0di6m).Ei6uki€n x-2> 0.x-2* l,y+ 2>0,y+0. Ta c6 : (x - 4)(x+ l) =y(y+5) e (x +y+ lXx-y-4)= 0.= [.::].:-n. IX=-y-l . Vdi x=-y- I e x-2 y-3>0=+ y<-3. lt2 http://wwww.violet.vn/haimathlx Theo di6u kign thi y > - 2, n€n truong hq p ndy bi lo4i. . V<ii x = y+4;rhayvAoptrhirhai tacluoc : logr.*r(y+D=*aV2 = [; : | , *.'.o diiu kiQn chi co x = 6 th6a mdn. Vay : NghiQm cta phuong trinh li x= 6,y =2. :c6 [: f t%- d*-= ltoVZ *td* J1 x(xro+1)2, .r1 x10(x10+1)z' D4t t=xro, vdi x= I thi t= l; x:'W thi t=2,vd dt= l0xedx. Khid6 ,_ r i2 dt 1,2f1+t-t)dt t 12 dr t.2 dt ro,r t(t+l)z: iJr i(*LF = GJr (r-r) - -Jr Gf = L*r:(i-#dt +;fu1?:*''1,*l l?.*(i - )=frrrn+-rn:)-fr cAu ry. ( r,o di6nr). Coj iI {C tim cua hinh vu6ng ABCD, thi SH 1 (ABCD) vi IiE = 30". Do AC: uVZ, ,uy ra SA = t' "u rn = * Tdm O qia cAu ngo4i ti6p hinh ch6p sC fa'giuo Aie,r, giE"u m4t phing trung truc cia doan thing SA vi dudng thdng SH. Gqi M.ld trung di6m cria doqn thing SA, ta c6 4 di6m A, M, H, O cirng nirl tr€n dudng tr'6n dudng kinh AO, n€n A SA.SM = sH.so <+ R: so = s1.s,V = 4 . huu uen sH 3 ' ' '- kfrrh cia mqt ciu ngo4i tiiip hinh ch6p li R = + . 3 Vay, Sr. = 4nR2 = cAu v. ( t,o di6m ). Tru6ch6ttachirng minh bdtdingtlrfic ' 1*1*1- -1*-, vdia, b,cduong. a b c-a+b+c' +b+sx:.;.fr=,.(l . f;).(:.r;). (l * f,):r* z+z+z=s. I119 Suvra:. - + -+ -> :-,ddubingxdyrakhia=b=c. (ttpcrn) ' a b c-a+b+c Ap dung Udt eing thric tr6n vd bdt ding thuc a2 + bz > 2ab. 7LL93 T^ ^: - D- - y-+y!+L y"!r. +t x.+2. +1- x2+y2+22+3 2, 222 Ddu bing xdy ra khi vi chi khi x:Y =z: l. 3 Vey, P*:;,khix=y=z=1. -y-2=t,-[t;; @ (^i\ /.\\t / "'', t'.,H - / / \ j- L \ r\ / \ : \- : ' \ | \ http://wwww.violet.vn/haimathlx cAu vL ( 2,0 tlitirn). I) (1,0didrn).Cdchl GidsfLdi€mCed,khid6C(2t-l:),Te (2t-6;t+2)vAEC1zr+2;t-4). G6cIeB:90'eTe .Ee :0 e (2t-6)(2t+2)+(r +2){t-4)=0 +.+ t2-2.t-4:0<+t=l +1/6 Vayc6haidi6mCrCndthoamdrryduciubiitorln: C1(1+ ZrlS;l+r/5)uaC:( I -26; f -r/5). Cdch 2. Gid su di6m C € d, khi d6 C(2t - i ; t). G6c TeB :90' suy ra di6m C nirn tr€n dudng trdn dudng kinh AB. Ta c6 trung di6m I cria do4n thing AB c6 tga tI$ (l; l) vA AB : 10. N6n : IC = 5 c=+ (2t-42 + ( t- l)2=25. Gidi ptndy ra s€ dugc k6t qun nhu cdch l. 2) (1,0 di6m). a) (0,5 di6m). Duongthingd1diquaA(I; I; -2)6c6uectochiphuongE=tf ;- I;2). Euong thing d2 tti qua B(-4; 8; 8) vA c6 r,ecto chi phusng @' = 1l; l; - l). Ta c6 ffi,41= (-l; 5;\, TE = (- 5; 7; l0). Suy ratL;f,.,6). af, = 1-l)(-s) + s.7 + 3.10:70 + 0. Vfly hai dulng thing d1 vd d2 chdo nhau. b) (0,5 di6m). DitlmM e dl + M(l+ t; l- t; -2+ 2Q. Di€m N e d2+ N(- 4 + 2t'; 8 + t'; 8-t'). Suy ra MF : (-5 - t+ 2t':7 + t + t'; 10 - 2t- t'). Tac6Mfi tuleMrt.q.=0<+6t+t'= 8; Mfr r7;,eMfr.q=0<+t+ 6t'= 13. Giaihe {6rt!=8_ e il=1 _ f M(Z;o;o) .rt+6t,=13 tt,=2 1 lru(O;fO;O) Suy ra mit cAu dudng kinh MN c6 tArn I(l; 5; 3), bdn kinh R: tr. Phuong trinh cria n6 Ii : (x- l)t + (y - 5)2 + (z- l)2 = 35 cAu vu. ( l,o di6m). rac6 2ooe!.s : ffi-;##.##+ . +ffi .ffi : C\.oog+ Cloos+ CSoog + + Ci333 * Crt333 . & Bing crich khai tli6n 1l + x ;200e ue 6hqn X : -1, ta dnoc ding thirc : Cloos+ C3.oog+C!.o0"+ *C3333* C3333:Cloor+C3oor+Clooe+ *C3331* C3333, NgoAi ra, chgn x : I ta cluo.'c tling thri'c : cloos+ cztoos+ C3.oog + * C:339 * ci333 = 2200e Ttrd6 suy ra : C|oo, + C\.oor+ Cloor+ C!oo, + * C3333 + Ci333 - 22008 -200s ^ L -I v av ?r, - 2009! http://wwww.violet.vn/haimathlx http://wwww.violet.vn/haimathlx . : D = R' . Subi6nthi6n: y'=3x:+6x, y'=0c+x=0 hoflc x:-2. Tac6y(0):1,y(-2) =5. Bdng bi6n thi n : yco: y(-2): 5, ycr : Y(0) : l. . Dd thi : ( hqc sinh qu vE. = 5 c=+ (2t-42 + ( t- l)2= 25. Gidi ptndy ra s€ dugc k6t qun nhu cdch l. 2) (1,0 di6m). a) (0 ,5 di6m). Duongthingd1diquaA(I; I; -2)6c6uectochiphuongE=tf ;- I;2). Euong thing. vE d6 thi hdm sii 1t ; thi m = 0. 2. TimmA6euO'ngthingy= lcatd6tlri hdmsO(t)tai 3 di6m phdnbiQt: I(0;,1),A vdB. Vd'i gi6tri nio cria m, c6c tiiip tuy6n crla d6 thi