TRNGTHPTNGUYNCễNGTR THI TH K THI THPT QUC GIA 2015 - S 75 THIMINHThi HAư Klm THITHPTQUCGIA NM2015 gian bi 180 phỳt oOo -MễNTON TG:180phỳt Cõu1.(2,0im)Chohms y = - x - 2mx + m +m (1) a) Khosỏtvvth(C)cahms(1)khim=2 b) Tỡmttccỏcgiỏtrthccamthhms(1)ctOxti4imcúhonhlpthnhmt cpscng. Cõu2.(1,0im)Giiphngtrỡnh sin 2x - cos x + 2sin x = cos 2x +3sin x Cõu3.(1,0im) a) GiXltphpcỏcstnhiờngmsỏuchsụimtkhỏcnhauctothnhtcỏcchs 1,2,3,4,5,6,7,8,9.ChnngunhiờnmtsttphpX.Tớnhxỏcsutscchnch chabachsl. b) Tỡmtphpcỏcim Mtrongmtphngphcbiudinchosphczbitrngsphc w = 2(i + z )- iz - z lsthuno. Cõu4.(1,0im)Gii phngtrỡnh -2 x + 10 x - 17 x + = x x -x3 (x Cõu5.(1,0im) Tớnhtớchphõn I = ũ e + 1)ln x + x2 + dx + x lnx Cõu6.(1,0im)ChohỡnhchúpS.ABCDcúABCDlhỡnhthoitõmOcnha,gúcABDbng1200, SAvuụnggúc(ABC),gúcgiacnhSCv(ABC)bng600 TớnhthtớchkhichúpS.ABCDv khongcỏchgiahaingthngSAvBMviMltrungimcnhSD. Cõu7.(1,0im) TrongmtphngOxy,chohỡnhchnhtABCDcúB,Cthuctrctung,phng trỡnh ngchộoAC:3x+4y 16=0.XỏcnhtanhA,B,C,Dbitrngbỏnkớnhngtrũn nitiptamgiỏcABCbng1. Cõu8.(1,0im)TrongkhụnggianOxyz,chocỏcimA(010)B(222)C(-234)vngthng x - y + z + dcúphngtrỡnh = = TỡmMthucdsaochothtớchkhi tdinMABCbng3. -1 Cõu9.(1,0im)Choa,b,cl3sdngthamón iukina3 +b3 =c3.Tỡmgiỏtrnhnhtca biuthc P= a + b - c2 ( c - a )( c -b ) Cm nthyNguynThnhHin( https://www.facebook.com/HIEN.0905112810)óchiasờn www.laisac.page.tl 469 PNTHANGIM CU Cõu1 (2,0 im) PN IM a) 1,0im m=2=>y=x4 +4x2 +2 TX:D=R 0,25 lim y = lim y = -Ơ x đ+Ơ xđ-Ơ y==4x3 +8x ộ x = y ' = ởx = 0,25 BBT x -Ơ y - +0 y 0+ -Ơ +Ơ 0,25 -Ơ Hmstngtrờn (-Ơ, - 2) v (0, 2) Hmsgimtrờn (- 2, 0) v ( 2, +Ơ) imcci ( 2, 6) ,imcctiu(0,2) 0,25 470 thnhnOylmtrcixng b) 1,0im Phngtrỡnhhonh giaoimcathhms(1)vOxl: 0,25 2 - x - 2mx + m + m =0 (2) tt=x2 ( t ),(2)thnh: t + 2mt - m - m =0 (3) 0,25 YCBT úpt(2)cú4nghimphõnbitlpthnh cpscng 0,25 úpt(3)cú2nghimphõnbitdngt1,t2 tha9t1 =t2 (0 t1t = -m - m ùS = -2m > ù t + t = -2m ù9t = t ù ợ ùợ9t1 = t 0,25 ỡ ù-1 < m< - ù ỡ -1 < m< ù t = - m ù -25 ù ù m= 34 ù ù 9m = - m - m 9m t = ù ùợ 25 ù ùợt1t = -m - m Cõu2 (1,0 im) sin 2x - cos x + 2sin x = cos 2x +3sin x 2sin x cos x + 2sin x = 2cos x - + 3(1 - cos x) +cos x 2sin x(cos x + 1) = - cos x + cos x +2 0,25 471 2sin x(cos x + 1) = -(cos x - 2)(cos x +1) (cos x + 1)(2sin x + cos x - 2) =0 0,25 ộcos x = -1 (1) ở2sin x + cos x = (2) 0,25 (1) x = p + k2p (k ẻZ) (2) 2 sin x + cos x= 5 Vi sin a = tac ,cosa = 5 ổp cos(x - a ) = sin a = cosỗ - a ữ ố2 ứ ộ ờx = ờx = ờở Cõu3 (1,0 im) p + k2p (k ẻ Z) -p + 2a + k2p 0,25 a) 0,5im Gi lkhụnggianmucaphộpth:Chnngunhiờnmtsttp hpX.Khiú n(W) = A96 =60480. GiAlbinc:Scchnchchabachsl.Khiú + Chnbachslụimtkhỏcnhautcỏcchs1,3,5,7,9:cú C53 cỏch. + Chnbachschnụimtkhỏcnhautcỏcchs2,4,6,8:cú C4 cỏch. + SpxpsỏuchstrờncsthabincA:cú6!Cỏch. 0,25 Khiú n( A) = C53 C4 6! =28800. Vyxỏcsutscchnchchabachsll n( A) 28800 10 P( A) = = = ằ 0, 48. n(W) 60480 21 472 0,25 b) 0,5im Gis z = x + yi vi x, y ẻR w = 2(i + z ) - iz - z w = 2(i + z )- iz - z 2 = (- x - y + x + y ) + (2 - y -x)i lsthuno 1ử ổ - x - y + x + y = x + y - x - y = ( x - 1)2 + ỗ y - ữ = ứ ố Vytphpcỏcim Mtrongmtphngphcbiudinchosphcz 0,25 0,25 ổ 1ử lngtrũntõm I ỗ1 ữ cúbỏnkớnh R = ố 2ứ Cõu4 (1,0 im) Nhnxột:x=0khụngthaphngtrỡnhcho 10 17 Chiahaivcaphngtrỡnhchox3,tac: -2 + x - x + x3 = x -1 t t = ( tạ0),phngtrỡnhtrthnh: -2 + 10t - 17t + 8t = 5t - x ( 2t - 1) + ( 2t - 1) = ( 3 ) 5t - + 5t - f ( 2t - 1) = f ( f ( t )= t + 2t , t ẻR ) 5t -1 ,vi 0,25 Tacú: f ' ( t )= 3t + > 0, "t ẻR nờnfngbintrờn R,vỡvy: f ( 2t - 1) = f ( ) 5t - 2t - = 5t - ộ ờt = (loaùi) 17 + 97 ( 2t - 1) = 5t - 8t - 17t + 6t = ờờt = (nhaọ n) 16 ờ 17 - 97 (nhaọ n) ờởt = 16 t= t= 17 + 97 17 - 97 ị x = 16 12 0,25 0,25 0,25 17 + 97 17 + 97 ị x = 16 12 473 17 97 Vyphngtrỡnhchocú2nghim: x = 12 Cõu5 (1,0 im) e Tớnhtớchphõn I = ũ (x + 1)ln x + x2 + dx + x lnx e e e x ( x ln x+ 2) + ln x + lnx I =ũ dx + ũ dx = ũ x dx + ũ dx = I1 + I2 + x ln x 1 + x ln x 1 + x lnx e Tớnh I1 e x3 e3 - I1 = = 3 0,25 TớnhI2:t t = + x ln x ị dt = (1 +lnx )dx icn: x = ị t =2 x = e ị t = e +2 e+ I = ũ t dt = ln | t| e+ 2 e+ = ln 0,5 e3 - e+ Vy: I = +ln 0,25 Cõu6 (1,0 im) *TamgiỏcABCucnhanờnAC=a, BO= a * SA ^(ABC) nờnAClhỡnhchiucaSCtrờn(ABC) =>GúcgiaSCv(ABC)lgúcSCAbng600 *TamgiỏcSACvuụngcú SA = AC.tan 600 =a * SABCD 0,25 a = 2SABC = Vy VS.ABCD a3 = SA.SABCD = 0,25 *GiNltrungimcaAD 474 CúMN//SA=>SA//(BMN) d(SA,BM)=d(SA,(BMN))=d(A,(BMN)) 0,25 Dng AH ^BN tiHtac: ỡAH ^ BN ị AH ^ (BMN) ị d(A, (BMN)) = AH AH ^ MN (do MN//SA) ợ 7a2 *TamgiỏcABMcú BN = AB + AN - 2AB.AN.cos120 = ị BN= 2 a a SAMN = AB.AN.sin120 = => AH= Cõu7 (1,0 im) 2SAMN a 21 a 21 = => d(SA, BM)= BN 14 14 *ClgiaoimcaACvOy=>C(0,4) *GiB(0,b) *PhngtrỡnhAB:y=b(doABvuụnggúcBC Oy) ổ 16 - 4b , bữ ố ứ *AlgiaoimcaABvAC=> A ỗ 475 0,25 0,25 *GirlbỏnkớnhngtrũnnitiptamgiỏcABC.Tacú: b - 2SABC S = pr ị r = = = b - AB + BC + CA b - + b - + b -4 3 ộ b = ị A(4,1), B(0,1), C(0, 4), D(4, 4) ởb = ị A( -4,7), B(0, 7), C(0, 4), D( -4, -4) 0,25 *r=1 b - = 0,25 0,25 Cõu8 (1,0 im) MẻdịM(1+2m-2-m-3+2m) uuur uuur uuur uuur AB = ( 21 ) AC = ( -2 ) ị ộở AB, ACựỷ = ( -12 6) uuuur AM = ( 2m + - m - 2m- 3) uuur uuur uuuur ộ AB, AC ự AM = 24m + 18 ỷ VMABC = ộ 24m+ 18 = 18 ộ uuur uuur ự uuuur AB , AC AM = 24 m + 18 24m+ 18 = -18 ỷ 6ở ộ m = ị M(1 -2 -3) ờ m = -3 ị M ổỗ -2 -1 -6ửữ ờở 2 ố ứ Cõu9 (1,0 im) Do a , b, c > nờn x = 0,25 0,25 0,25 0,25 a b > y= >0 c c Tacú: x + y = 1& ( x + y ) = x + y + xy ( x + y ) = + 3xy ( x + y ) ChiactvmucaPchoc2 ạ0vthayx,y.Tac: P= x + y2 - ( x + y ) - xy- = (1 - x )(1 - y ) - ( x + y )+ xy +1 Dat t = x + y ị xy= 0,25 t3 - 3t 476 ỡt > ỡt > ù Do x y > nờn < tÊ 4 t ợt Ê ùt 3t ợ ( ) 0,25 Khi : P = t - 3t + t+ = = 1+ = f ( t) t - 3t + 3t - t - t - 0,25 Vi < t Ê nờn f ( t) Vay : MinP = + -1 + -1 0,25 a = b, c = a3 Cm nthyNguynThnhHin( https://www.facebook.com/HIEN.0905112810)óchiasờn www.laisac.page.tl 477 ... Cõu3 (1,0 im) p + k2p (k ẻ Z) -p + 2a + k2p 0,25 a) 0,5im Gi lkhụnggianmucaphộpth:Chnngunhiờnmtsttp hpX.Khiú n(W) = A96 =60480. GiAlbinc:Scchnchchabachsl.Khiú + Chnbachslụimtkhỏcnhautcỏcchs1,3,5,7,9:cú... y -x)i lsthuno 1ử ổ - x - y + x + y = x + y - x - y = ( x - 1)2 + ỗ y - ữ = ứ ố Vytphpcỏcim Mtrongmtphngphcbiudinchosphcz 0,25 0,25 ổ 1ử lngtrũntõm I ỗ1 ữ cúbỏnkớnh R = ố 2ứ Cõu4 (1,0 im)... (1,0 im) 2SAMN a 21 a 21 = => d(SA, BM)= BN 14 14 *ClgiaoimcaACvOy=>C(0,4) *GiB(0,b) *PhngtrỡnhAB:y=b(doABvuụnggúcBC Oy) ổ 16 - 4b , bữ ố ứ *AlgiaoimcaABvAC=> A ỗ 475 0,25 0,25 *GirlbỏnkớnhngtrũnnitiptamgiỏcABC.Tacú: