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ĐẠI SỐ SƠ CẤP_ÔN THI ĐẠI HỌC

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L Ngc Sn_H Quan Bng_Nguyn Xun Tun_Nguyn Quc Vit Lp S Phm Ton K07 _ H Ty Nguyn- 1 - TI LIU N THI I HC (i s s cp) PHN I: PHNG TRNH V BT PHNG TRNH Phng trnh v bt phng trnh bc 2, = = 0)2( 0; ax bx c a . Cu 1: Tmm phng trnh: =22( 3) 4 12 0 x m x mc hai nghim cng ln hn -1. Gii pt cho c hai nghim cng ln hn -1 th m tha mn h sau: ' 21 20 ( 3) 4 12 071. ( 1) 0 1 2( 3) 4 12 0 3.21 312m mf m m mx x m'11' 11^ 111111 1= = < < ! !1 11 1 < 1 11 1+ Phng trnh tr thnh 2 3 4 3222( ) 1 0 1 01 10 ( 0)1 12 0 t m t mt t mt mtt mt m tt tt mtt t+ + = + + = + + = =| | | | + + = ||\ . \ . L Ngc Sn_H Quan Bng_Nguyn Xun Tun_Nguyn Quc Vit Lp S Phm Ton K07 _ H Ty Nguyn- 26 - t 1u tt= . Phng trnh tr thnh 22 0 (3)u mu + + =+ Pt c 2 nghim 28 0 2 2 m m A = > >+ Ta c 211 0 u t t utt= =: c 2 nghim tri du tha mn 1 20 m t t < < < Suy ra ng vi mi t th pt 2x t m = +c 2 nghim x Vy2 2 m>l gi tr cn tm Bi 9 Gii h 2 2 4 4 6 22 3 32 (1 )1 ( ) (3 ) 0xy xy y x xx y x y x = + + + + + s Gii H 2 2 2 6 22 3 31 (1 ) (1 )1 ( ) (2 )(1)(2)xy y x xx y x y x = + + + s + Ly (1) (2) ta c 2 2 2 2 6 2 3 3 32 2 2 2 6 2 32 2 2 2 3 21 (1 ) 1 ( ) 21 (1 ) 1 ( ) 21 (1 ) 1 ( ) ( )xy x y y x x xy xxy x y y x xyxy x y x y + + s + + + + + s + + + + s + Ta thy0; 0 0 VP VT VT VP > s = =2 2 2 2 3 21 (1 ) 1 ( ) ( ) 0 xy x y x y + + = + =332 2 2 22 21111 (1 ) 1 ( )11xx yx y yx yxxy x yxyy = = = =

=

= = + +

==

L Ngc Sn_H Quan Bng_Nguyn Xun Tun_Nguyn Quc Vit Lp S Phm Ton K07 _ H Ty Nguyn- 27 - Th li thy (1; -1) l nghim ca h Bi 10 Cho17 a b c + + > . CMR h 210 1ax bx cx+ + >s s c nghim Gii Gi s h v nghim th | |21; 0,1 ax bx c x + + s eCho 1 1112 4 20 1x a b ca bx cx c= + + s= + + s= s Xt h 2 4 23 44 2C a b ca A B Ca bB c b A B Cc AA c= + += + = + + = + == Khi 2 4 2 3 42 4 2 3 4 17 a b c A B C A B C AA B C A B C A+ + = + + + +s + + + + + + s Mu thun vi gi thit. Vy h c nghim Bi 11. Gii v bin lun h 2 22 22 2(1)(2)(3)x m y m ky n z n kz p x p k + = + = + = L Ngc Sn_H Quan Bng_Nguyn Xun Tun_Nguyn Quc Vit Lp S Phm Ton K07 _ H Ty Nguyn- 28 - Vi 3, , 02km np m n p > . + + =Gii + Gi s( ; ; ),( '; '; ') xy z x y zl 2 nghim phn bit ca h v' x x >T (1): 2 2 2 2' ' ' x m y m k x m y m y y + = = + < < ( ) f t l hm s ng bin. M( ) ( ) f x f y x y = =+ Thayx y =vo (2) ta c: 6 4 264 96 36 3 0 x x x + =tcos 0;2;x u u| |= e |\ .. PT tr thnh: 6 4 246cos 96cos 36cos 3 0 u u u + =

3 2 4 22324 (4cos ) 24cos 9cos 3 04 4cos 3cos 2 12 2cos 3 1 11cos6218 3u u uu uuuku ( + = ( = ( = == + Do 5 70; ; ;2 18 18 18u u u u | |e = = = |\ . Vy h cho c 3 nghim l 5 5 7 7cos ;cos ; cos ;cos ; cos ;cos18 18 18 18 18 18 | | | | | | |||\ . \ . \ . L Ngc Sn_H Quan Bng_Nguyn Xun Tun_Nguyn Quc Vit Lp S Phm Ton K07 _ H Ty Nguyn- 31 - Bi 14 Gii h3 23 22 2(13 ) (2 2 2 26) 5 7 7 30 0(17 ) (2 2 2 26) 3 2 011 28 0 (1) (2) x x y z x y z yz yz y zx x y z x y z yz yz y zx x+ + + + + =+ + + + = + s (3) Gii + T (3) ta c :4 7 x s s(*) + t u y zv yz= + = th (1) v (2) tr thnh 3 23 22 3 22 2(13 ) (2 2 26) 5 7 30 0(17 ) (2 2 26) 3 2 0(2 7) (5 ) 13 26 30 0(1 2 ) ( 2 3) 17 26 2 05 55 1 5 1x x u x u v v ux x u x u v v uu x x v x x x xu x x v x x xu x y z xv x yz x + + + + =+ + + = + + + + = + + + == + + = + = + = + Suy ra y, z l nghim ca pt : 2( 5) 5 1 0 X x X x + + + =PT c nghim 2310 21 0 77(do (*))xx x xx s A = + > =

> Thay x = 7 vo pt ta c y = z = 6 Vy nghim ca h l (7;6;6) Bi 15 Cho h 2220; ax bx c yay by c z aaz bz c x + + =+ + = =+ + =. CMR:( )21 4 0 b ac + + > + + > Suy ra (*) v nghim hay h cho v nghim Bi 16 Tm a sao cho: 2 22 212 713 10 5 2(1) (2)ax xy yax xy y + >++ s c nghim Gii + Ly (1) nhn 2 tr cho (2) ta c : 24( 3 ) 11x y aa+ s < + + Khi , ly a bt k: a < -1 th 111aa< +. Lc ta xt h 2 22 22 7 13 10 5 2(3) (4)x xy yx xy y + = + = Ly 2 nhn (3) tr (4) ta c: 2( 3 ) 0 3 x y x y + = = Thay x = - 3yvo (3) ta c pt: 2 219 6 7 12y y y y = = Vi a < -1 thay 3 32 21 12 2x xy y = = v = = vo h cho (tha) Vy a < -1 l gi tr cn tm L Ngc Sn_H Quan Bng_Nguyn Xun Tun_Nguyn Quc Vit Lp S Phm Ton K07 _ H Ty Nguyn- 33 - Bi 17. Gii h 2 2 2 23 3 3 3tan tan tantan tan tanx y z mx y z m + + =+ + = Gii K 222x ky kz k = += + = + ttan ; tan ; tan u x v y w z = = =Ta quy v h sau: 2 2 2 23 3 3 3u v w mu v w m + + =+ + = + Nu m = 0 th 2 2 20 0 tan tan tan 0 u v w u v w u v wx ky lz p+ + = = = = = = == = = + Nu0 m = th6 2 2 2 2 2 2 4 4 42 4 4 44 4 4 4( . . . ) ( )( )( ) = m u u v v ww u v w u v wm u v wu v w m= + + s + + + ++ + + + > Mt khc ta c 4 2 2 2 2 4 4 4 2 2 2 2 2 24 2 2 2 2 2 2( ) 2( )2( )m u v w u v w u v uw v wm u v uw v w= + + = + + + + +> + + + L Ngc Sn_H Quan Bng_Nguyn Xun Tun_Nguyn Quc Vit Lp S Phm Ton K07 _ H Ty Nguyn- 34 - 2 2 2 2 2 200 0 00u vu v uw v w uv uw vw v ww u= =

+ + = + + = = =

= = TH1: tan tan 00tanarctanx kx yu v w m y lz mz m p== = = = = = = = + TH2: tan tan 00tanarctanz kz yw v u m y lx mx m p== = = = = = = = + TH3: tan tan 00tanarctanx kx zu w v m z ly my m p== = = = = = == + Vy h c nghim x ky lz p== =; arctanx kz ly m p= == +v cc hon v Bi 18 Gii h 3 2 22 2(2 )(3 2 ) 33 3 263 (1) (2) (3) (4)x x z zy y x xy z zz = + = ++ = s GiiT (3) 2 26 0 0 6 y z z z = > s sT (1)23 2(3 ) 3 3 0 x zx z + + + = . Phng trnh nay c nghim khi 203 03zz zz s A = > > Khi ta c: 300 630 3zzzzz zs= s s

= > v > L Ngc Sn_H Quan Bng_Nguyn Xun Tun_Nguyn Quc Vit Lp S Phm Ton K07 _ H Ty Nguyn- 35 - + Vi 100xzy= = = + Vi 233xzy= = = Vy h c nghim l (1;0;0); (2;-3;3) Bi 19 Gii h; , , 0 (1) (2) (3)a bc xzx zb ca xy a b cy xc ab yzz y = = > = Gii Ly (1).c + (2).a +(3).b ta c:2 2 22 2 2( ) ( ) ( )0 (*)ac bc ab ac bc abb yz a xy c xzx z y y z za b c axy byz czya b c axy byz czy + + = + + = + + + + = + + Quy ng pt trong h ta c 2 22 22 2az bx cxz xzbx cy axy xycy az byz yz = = = Cng cc pt ta c 2 2 2 2 2 22 2 2 2 2 22 2 2 2 2 20(**)az bx bx cy cy az axy byz cxz yz xy xzaxy byz cxz yz xy xzaxy byz cxz yz xy xz + + = + + = + + + + = + + Ly (*) + (**) ta c L Ngc Sn_H Quan Bng_Nguyn Xun Tun_Nguyn Quc Vit Lp S Phm Ton K07 _ H Ty Nguyn- 36 - 2 2 2( ) ( ) ( ) 0yx aa xy b yz c xz yz bzx c=

+ + = =

= Nhn cc pt trong h ta suy ra 2 2 2xyz abc xyz abc = = T ta c nghim ca h lac acx xb bab aby yc cbc bcz za a = = = v = = = ( V x,y,z cng du) Bi 20 Gii h 222222x xy yy yz zz zx x + =+ =+ = Gii Ta thy x = y =z =0 l mt nghim ca h Mt khc1; 1; 1 x y z = = = khng phi l nghim ca h Khi ta a h v dng: 222212121xyxyzyzxz= == t 22tantan tan 2 tan 4 tan81 tantan tan87ax a y a z a x aakx a a a= = = = == = = L Ngc Sn_H Quan Bng_Nguyn Xun Tun_Nguyn Quc Vit Lp S Phm Ton K07 _ H Ty Nguyn- 37 - Th li (tha) Vy h c nghim (0;0;0); 2 4tan ; tan ; tan7 7 7k k k | | |\ . PHN II : BT NG THC I. BT NG THC C BN Bi 1: Cho A, B, C l 3 gc ca mt tam gic bt k. Tm GTLN ca biu thc: 4cos 4cos 3cos F A B C = + +Khi qut: Cho0 2 M N < < . Tm GTLN cacos cos cos F M A M B N C = + +Gii V A, B, C l 3 gc ca mt tam gic nn ta c cos sin2 2 2 2 2A B C A B C + +| |= = |\ . 2 2224cos 4cos 3cos 4(cos cos ) 3cos8cos cos 3 1 sin 6sin 8sin cos 32 2 2 2 2 22 8 8 176 sin cos cos 3 32 3 2 3 2 3 3F A B C A B CA B A B C C C A BC A B A B = + + = + ++ | |= + = + + |\ . | |= + + > + = |\ . Du = xy ra khi 2cos 122sin cos2 3 2A BC A B == L Ngc Sn_H Quan Bng_Nguyn Xun Tun_Nguyn Quc Vit Lp S Phm Ton K07 _ H Ty Nguyn- 38 - cos 1 22 22 2sin cos sin2 3 2 2 3cos 1 22 22 2sin cos sin2 3 2 2 3(loai)A B A BkC A B CA B A BkC A B C = = = = = = + = = 0( 0 0)2 22arcsin 2arcsin3 3 v A B k A B C A BC C = = + + = = = = Vy GTLN ca F l 173 Khi qut tng t trn, ta c 222 22cos cos cos 2 sin 2 sin cos2 2 22 sin cos cos2 2 2 2 2 2C C A BF M A M B N C N M NC M A B M A B MN N NN N N= + + = + + | |= + + s + |\ . Vy GTLN ca F l 22MNN + . t c khi 2arcsin2A BMCN= = Bi 2 Cho 2 2 2 2 2 20 p q a b c d + > . Cmr2 2 2 2 2 2 2( )( ) ( ) p a c q b d pq ab cd s Tng qut cho nhiu cp s Cho 2 2 2 21 10n nk kk kp q a b= =+ > . Cmr 22 2 2 21 1 1n n nk k k kk k kp a q b pq a b= = =| || | | | s | ||\ .\ . \ . Gii L Ngc Sn_H Quan Bng_Nguyn Xun Tun_Nguyn Quc Vit Lp S Phm Ton K07 _ H Ty Nguyn- 39 - + Nu 2 2 2 2 2 2( )( ) 0 p a c q b d th kt hp vi k bi ton ta c 2 2 2 2 2 2 2 2 22 2 2 2 2 2 2 2 2( )( ) 0 000 0p a c q b d p a cpqp q a b c d q b d > > = + > > Xt tam thc bc hai 2 2 2 2 2 2 22 2 2 2 2 2 2 2 22 2 2( ) ( ) 2( )2 2 2( ) ( ) ( )

f x p a c x pq ab cdx q b dpx pqx q ax abx b cx cdx dpx q ax b cx d= + = + + + = 2 2 2( ) 0pp a c fq| | < |\ . ( ) 0 f x =c nghim 2 2 2 2 2 2 2( ) ( )( ) 0 pcmpq ab cd p a c q b d A = > Khi qut tng t ta xt 2 21( ) ( ) ( )nk kkf x px q a x b== c nghim suy ra kt lun Bi 3 Choa b c < < . Cmr 2 2 23 3 a a b c a b c ab bc ca c s + + + + sGii Xt tam thc 2( ) 3 2( ) f x x a b cx ab bc ca = + + + + +c nghim 2 2 21,23a b c a b c ab bc cax+ + + + =M ta li c 223 ( ) 3 2( )( ) ( )( )( ) 03 ( ) ( )( ) 0

(gt)(gt)f a a a b c a ab bc caa ab ac bc a a b c a ba b a cf c c b c a= + + + + += + = = >= > L Ngc Sn_H Quan Bng_Nguyn Xun Tun_Nguyn Quc Vit Lp S Phm Ton K07 _ H Ty Nguyn- 40 - V 3a b ca c+ +< + + = + +> = = Du bng xy ra khi 2 2 242 2 2 232 4 , , 032x z yx z y y xz yz xyxyz x z xyzxyz = == = = = = = = = > = Vy 24yx z= = = l nghim ca h Bt ng thc( ) ( ) ( ) ( ) ( ) ( ) FA FB FC GA G B G C + + > + + Bi 1. Cmrx e, ta c 12 15 203 4 55 4 3x x xx x x| | | | | |+ + > + + |||\ . \ . \ . Khi no du ng thc xy ra L Ngc Sn_H Quan Bng_Nguyn Xun Tun_Nguyn Quc Vit Lp S Phm Ton K07 _ H Ty Nguyn- 41 - a)Cho, , 0 a b c > . Cmr 2 2 2a b c a bc b ca c ab + + > + +b)Cho, , 0 xy z >tha mn1 xyz = . Cmr 2 2 231 1 1 2x y zy z x+ + >

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