TRU,ONG EHSP HA NOI TRTIONG THPT CUUVNTV - EHSP DE THI rnu DAI HQC t AN I Narr ZOrt Mdn thi : TOAN Thdi gian tdm bdi : IB0 phtit, kh6ng ke thdi gian phdt di Ciu 1. ( 2,0 dilm ) Cho hAm s6 .y: Irt - 1**' + (*2 -3)x, trong do m littham s6. 32 I . I(hAo s6t sg bi€n thiOn vi v€ d6 thi cta hdm s6 khi z = l. diri cdrc canh g6c vudng cira mQt tam giirc vudng c6 dQ ddi c4nh huy€n birg F . "{2 Cdu 2. (2,0 diem) 1. Giii phuong trinh : 2. Gidi phuo-ng trinh : CAu 3. ( 2,0 di1m ) 1. Tim nguyOn hdm cria hdm sO f1x; : tanx.tan(x + ];.tan1x - l;. 3" 3', 2. Tim c6c gi6 tri cta m d6 phuong trinh sau c6 nghiQm duy nh6t : g-lz-xl _ 43-|2-xl= p. Ciu 4. ( 1,0 didm, , Cho, hinh ch6p tu gi6c ttdu S ABCD tt6 UU ddi canh ddy bing a, cgnh b6n bing t' ttnn goc tao boi '2 mdt b€n v6'i mdt d6y vi the tictr t<hOi cau ngopi ti6p hinh ch6p d6. Cfru 5. ( 2,0 diem ) .3 Trong mdt phdng Oxy : L Chohaidi€m A(2;1), B(-l;-3)vdhai cluongthing d1: x+y+3 =0; cl2: x-5y- 16:0. Tim tga dQ c6c cti6m C, D lin luE thuQc clr vd dzsao cho t6'gi6c ABCD ld hinh binh hdnh. 2. Vi6tphuongtrinh tluo'ngthingtiOpxircv6'i duongtron (C), xt+ y?-zr] x+4y+4=0vdtao v6'i trr,rc tung mdt g6c bing 60". CAu 6. ( 1,0 diem ) Xdtcdctam thirc bgc hai f(x):ax2 + bx+ c,trotlgd6 a< bvnf(x)> 0 vd.i moi x € R. Hay tiin gi6 tri nho nh6t cira biriu thric M = TT Het 2. Tim t6t cd cdc gi6 tr! oba m ae ham s6 c6 cuc d4i t4i x6p, c!.rc ti6u t4i xcr ddng thd'i xss, .rcr lh dO z+^[s-x, +12 ,[s-*,1 sinal3x*|l * sinal3x -?=: x2 -l Dy kiAn ki ttti thrb Egi hgc l6n 2 sE ihrqc t6 chrbc vito ngdy 26,27/2/2011 www.laisac.page.tl nAp AN - THANG omirn rru fiillon IAN rntl Nnar nAu zotl Ciu DAP AN DIEM I e aidm) (t,O a$ni. Hoc sinh tu ei6i. t. 2. (1,0 itiiinr) . Tim c6c gi|trim - y'=x'-mx+m2-3 . Hdm sti c6 CD, CT vd Xco ) 0, xcr > 0 e pt y' = 0 c6 hai nghiQrn duo'ng phdn biQt 0,25 (L:m2-4(m2-3)>0 ._^ (+ l" -i'= '"';^; * [";l!'r' o o t ;=m2-3>o ( m>o .,13 < m <2. 0,25 ld dO dei c6c cqnh g6c vudng cta tam gi6c vu6ng c6 canh f,ufAn Uine f ".,=1 €(xco+xcr)2-z*"u'*.t: i (*) ' Xcor xct e xzru+ 0,25 . Theo dinh l)t Vi-et, thi (*) trd thdnh : m' - 21m'- 3) = .l; t **=J;. (fi,2) 0,25 il (2 iliim) l. ( 1,0 iti€m\ . Gi6i phuong trinh . a EiAukiQn:-3< x<3, Phuong trinh trd thenh xl0. Dat 1=1f$Qz +xt= 9-t2,t>0,1+3 g-tz \ -+-=l 3+t 4(3-t) a 0,50 (+ 4Q-q2 -4Q-t)+ I : 0 e 2(3-t)= I <+ *:9 - E o *=*E( th6a mdn diOu kiOn). 5 t- - 2 e 0,50 2. (1,0 iti€nt). Gidi phuong oinf . Phuongtrinh tuongduongv6i phuongtrinh: cosa(:*-l ) +sin4(3x-T l:i ' 4 4' 2 0,25 e [cos2(3x -l I *sin213x-i lt' -2 cos2(3x -[ ).rin'{:* -on) = } 0,25 (+ t-]sin'{ox- sin26x = 0 <+ ?I 1 -)=- z' z kn x: - 6 <+ sin2(6x - , kez. lt -)=l <+ 2' cos26x = I (+ 0,50 ilI (2diim) l. (1,0 iliAm) . Tim nguyOn hdm a f(x): sinx.sin(x + l).sin (x - T) .or* or(* + i).costx - i) slnx.(cosf - coszx) _ sinx.(zsin'x - i) sin3x : cos3x .orr.(.orf + .orz*) .os*.(zcoszx - ]) 0,50 . Jf(x)dx=-JHo* = lJffi = llnlcos3xl + c. 0,50 http://www.violet.vn/haimathlx Sưu tầm: Nguyễn Minh Hải- THPT Lê Xoay 2. (1,0 . DAt t: t- x , phuong trinh tr6'thdnh 9-ltl - 4. 3-ltl : m (l ) Phuong trinh dd cho c6 nghiQm duy dr6t khi vd chi khi pt (l) c6 nghiQm dr-ry nh6t. + ) EiAu kien cAn : Gid sft (l ) c6 nghiQm duy nhAt t" . khi d6 - to cfrng ld nghi€m. Suy ra to = 0, thay vdo (l ) ta cluo.'c m = - 3. +) DiAukigndrl:Khirn=-3,thi (l)trdthdnh: 9-ltl -4.3-ltl :-3 (2). Suy ra 3-ltl = 1 a:r t = 0, d0y ld nghiQnr duy nh6t cria (2). . E6ps6: m:-3. t (0,5 eti6@. Gqi H ld t6m cta d6y ABCD, ta c6 SH I (ABCD); M td trung di6m cfia BC thi BC I (SHM), do c6c m{t ben t4o v6i mqt d6y ctrng mQt g6c, n€n 5fr8 bing g6c tao bdi m[t b€n voi cl6y. Ta c6 : 311: r/ffi - 4112 + tan sffE= ffi= vr +fliliH=60o. o (0,5 diA@. Hinh ch6p S.ABCD ddu, nOn tdm I cria t<trOi cAu ngo4i tii5p hinh ch6p ld giao <ti6m c0a club'ng thing SH v6'i m{t phing trung trqc cria mQt c4nli b€n ndo d6 cta hinh ch6p. GqiN ld trung cli6m ctra SC, thi IN ld trung trr,rc cria SC. Suy ra: ASNI - ASHC SN.SC 5a =rR=SI=-;fr-=m *l 4 . rzsa3{1i A V0y, V: -tR' : 4n . /::\ rl !1 ,l tl ll I I I rl' I I ,_ 1 t I tr I rl -H' I I I I t t I IV Q iri6m) t. (1,0 V Q di6m\ Gi6 sri ABCD ld hinh binh hdnh, ta c6 Cd = ffi = 13; +; (+ [}3 - ;: = i Vi D ed2 ndn xe-5yo- 16=0 + (xc+3)-5(ys+4)= ViC€d1,n€n xc+yc*3=0. Tu d6 ta c6 hQ phuong trinh : fT l?It-=j= (x' = 3 (xo = 6 - (xc+yc=-3 <+ tv.=-o ttvl=-z Ta c6 : el = (3; a) ve Be = (4; - 3 ) n6n hai vectcv Bf, Be khdng ctrng phuong, tric ld 4 dit5m A, B, C, D kh6ng thdng hing, hay tri gi6c ABCD ld hinh binh hdnh. Edp sd : C(3; -6) vi D(6; -2) a? r-l'f http://www.violet.vn/haimathlx Sưu tầm: Nguyễn Minh Hải- THPT Lê Xoay 2.(1,0 @2)vib6nkinhR:J3; Tir gi6 thii5t suy ra ti6p tuyiin cAn tim t?o v6i trr,rc hodnh mQt g6c bing 30o , n€n ti6p tuy6n c6 h0s6g6cbing ** t€rrptti€ptuy6nc6d4ng:y=*rt x*m e +*-V3y +"13m=0' Khoang cich htir t6m l(r/3; -2) den ti6p tuy€n bing b5n kinh R = v3' ItJs +z\tr + VSml r; TfclA ir=-7=\5 rfm = -5 * |ilill =22 * lfn=-l L[m=1 V{y c6 4 tiiip tuy€n th6a mdn y6u cAu bii to6n : x-rfgy-5r/3=o; "+r/3y+3V3=0; *-fsy-V3=o; x+y'3y-J3=0. Tim gi6 tri nh6 n w Q diAm) Tir giAthi6t suy ra a> 0 vd A = b2 - 4ac<0 + c > -yz a+b+- M># b-a Dat t=b-a>0+MZ +a2 + +a(t+a)+(t+a)z - gaz + 6at+ tz 4at 4at 3 gaz + t2 3 T''lw7 =-a-/-a-=J 2 4at -Z 4at (^_b' Dingthftcxiyrakhi vdchi kf ti = A (+ b=c=4a =+ f(x):a(x+2)2 vhminM = 3' - 3t http://www.violet.vn/haimathlx Sưu tầm: Nguyễn Minh Hải- THPT Lê Xoay . thenh xl0. Dat 1=1f$Qz +xt= 9-t2,t>0,1+3 g-tz -+ -= l 3+t 4(3-t) a 0,50 (+ 4Q-q2 -4 Q-t)+ I : 0 e 2(3-t)= I <+ *:9 - E o *=*E( th6a mdn diOu kiOn). 5 t- - 2 e 0,50 2. (1,0 iti€nt) cosa(:*-l ) +sin4(3x-T l:i ' 4 4' 2 0,25 e [cos2(3x -l I *sin213x-i lt' -2 cos2(3x -[ ).rin'{:* -on) = } 0,25 (+ t-]sin'{ox- sin26x = 0 <+ ?I 1 -) =- z'. : x-rfgy-5r/3=o; "+r/3y+3V3=0; *-fsy-V3=o; x+y'3y-J3=0. Tim gi6 tri nh6 n w Q diAm) Tir giAthi6t suy ra a> 0 vd A = b2 - 4ac<0 + c > -yz a+b +- M># b-a Dat