! !"#$%&'()*+,-) ).*.)) /0&1)23)&456)7)489):%&4)&4;&)<=)>?%)489)@4A))) B4%)#%C#()DE#4$%&2:FG&) ! "! H4"I)1%J%)>K)L!ML)N=O9)P%E)Q)L4RS()/T&1)L4U&4)VE6) DW&()L"I&X)/Y)Z[).\]^*) V1US)#4%)()_]*\].*`^) L4a%)1%E&)$U6)bU%()`_*)@4c#d)24W&1)2e)#4a%)1%E&)1%E")>K) f%g&)4h)>0&1)23)24"I)489)Q)!"#$%&'()*+,-) ).*.)Q)B4%)#%C#()iiiF6E#4$%&2:FG&)) Bj=)`)k.d*)>%e6lF)#$%!$&'!()! y = (x + m)(x −1) 2 (1) *! "* +$,%!( !(/!0123!.$143!5&!56!78!.$9!$&'!()!:";!5<1! m = 0 *! =* #$%!71>'!#:"?@A=;*!BC'!'!7>!:";!DE!$F1!71>'!D/D!.G9!HIJ!(F%!D$%!0F!71>'!HIJI#!.$K3L! $&3L*! Bj=).)k`d*)>%e6lF) F; M1,1!N$OP3L!.GC3$! 2 2 sin x. cos x = 1 *!! 0; BC'!()!N$QD!R!DE!N$S3!.$/D!5&!N$S3!,%!7TU!VOP3L!.$%,!'W3! z = 5,z 2 + z 2 = 6 *!! Bj=)7)k*d^)>%e6lF!M1,1!N$OP3L!.GC3$! log 2 (x + 3) = log 4 x −1 + 2 *! Bj=)\)k`d*)>%e6lF)M1,1!$X!N$OP3L!.GC3$! x 2 + 3x + y + 2 = (x +1)(x + 2) y −1 x 2 +16 − 2 x 2 −3x +4 = y −1−1 ⎧ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ , x, y ∈ ! ( ) *!!! Bj=)^)k`d*)>%e6lF!BY3$!.$>!.YD$!Z$)1!.G[3!\%F]!Z$1!^UF]!$C3$!N$K3L!L1<1!$_3!0`1!D-D!7Oa3L! y = 4 + x 2 .ln x x , y = 2 x ,x = 2 ^UF3$!.GbD!$%&3$*! Bj=)-)k`d*)>%e6lF!#$%!$C3$!D$EN!c*HJ#!DE!'d.!043!cJ#!e&!.F'!L1-D!5Uf3L!Dg3!._1!c!5&!3h'! .G%3L!'d.!N$K3L!5Uf3L!LED!5<1!'d.!N$K3L!:HJ#;I! BC = a 2,ASB ! = CSA ! = 60 0 *!BY3$!.$>!.YD$! Z$)1!D$EN!c*HJ#!5&!Z$%,3L!D-D$!.i!71>'!J!723!'d.!N$K3L!:cH#;*! Bj=),)k`d*)>%e6lF!BG%3L!'d.!N$K3L!5<1!.GbD!.%_!7j!k\]!D$%!.F'!L1-D!HJ#!5Uf3L!Dg3!._1!#*! Ml1!m!e&!.GU3L!71>'!D_3$!H#I!n!e&!71>'!.$UjD!7%_3!HJ!.$%,!'W3! DB = 2DA I!o!e&!$C3$! D$12U!5Uf3L!LED!DpF!n!.G43!Jm*!BC'!.%_!7j!D-D!7q3$!HIJI#!012.!n:A=@r;I! H (− 18 5 ; 24 5 ) 5&!7q3$!J! DE!$%&3$!7j!3LU]43*! Bj=)_)k`d*)>%e6lF!BG%3L!Z$f3L!L1F3!5<1!.GbD!.%_!7j!k\]R!D$%!71>'!s:=@=@?;!5&!'d.!N$K3L! (P ) : 3x + 2y − z + 4 = 0 *!t12.!N$OP3L!.GC3$!7Oa3L!.$K3L!V!71!^UF!s!5&!5Uf3L!LED!5<1!'d.! N$K3L!:u;*!BC'!.%_!7j!71>'!m!.G43!V!(F%!D$%!m!D-D$!7TU!L)D!.%_!7j!5&!'d.!N$K3L!:u;*!! Bj=)+)k*d^)>%e6lF!vLOa1!.F!Vw3L!x!DU)3!(-D$!B%-3I!y!DU)3!(-D$!tz.!e{I!|!DU)3!(-D$!o%-!$lD! :D-D!(-D$!Dw3L!e%_1!.$C!L1)3L!3$FU;!7>!e&'!N$S3!.$O`3L!D$%!}!$lD!(13$!'~1!$lD!(13$!7O•D!$F1! DU)3!Z$-D!e%_1I!.G%3L!}!$lD!(13$!3&]!DE!$F1!0_3!vF'!5&!oOa3L*!BY3$!\-D!(U€.!7>!vF'!5&! oOa3L!DE!N$S3!.$O`3L!L1)3L!3$FU*! Bj=)`*)k`d*)>%e6lF!#$%!FI0ID!e&!D-D!()!.$/D!VOP3L!.$%,!'W3! a(a −c )+ b(b − c ) = 0 *!BC'!L1-!.G9! 3$•!3$€.!DpF!01>U!.$QD! P = a b 3 + c 3 + b c 3 + a 3 + c 2 + ab 2 − a + b c ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ 2 *! ) mmm!nLmmm) ! !"#$%&'()*+,-) ).*.)) /0&1)23)&456)7)489):%&4)&4;&)<=)>?%)489)@4A))) B4%)#%C#()DE#4$%&2:FG&) ! =! M!oV)LpB!)qrV!)fstV)/uM)uV) Bj=)`)k.d*)>%e6lF)#$%!$&'!()! y = (x + m)(x −1) 2 (1) *! "* +$,%!( !(/!0123!.$143!5&!56!78!.$9!$&'!()!:";!5<1! m = 0 *! =* #$%!71>'!#:"?@A=;*!BC'!'!7>!:";!DE!$F1!71>'!D/D!.G9!HIJ!(F%!D$%!0F!71>'!HIJI#!.$K3L! $&3L*! "* olD!(13$!./!L1,1*! =* BF!DE‚! y ' = (x −1) 2 + 2(x + m)(x −1); y ' = 0 ⇔ x = 1 x = 1−2m 3 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ *! ƒ;!„>!78!.$9!$&'!()!DE!$F1!71>'!D/D!.G9!Z$1!5&!D$q!Z$1! 1− 2m 3 ≠ 1 ⇔ m ≠ −1 *! +$1!7E!.%_!7j!=!71>'!D/D!.G9!e&! A(1;0), B( 1− 2m 3 ; 4(m +1) 3 27 ) I!.F!DE‚! ! AC ! "!! = (9;−2),AB ! "!! = ( −2(m +1) 3 ; 4(m +1) 3 27 ) //(−9;2(m +1) 2 ) *! tz]!HIJI#!.$K3L!$&3L!Z$1!5&!D$q!Z$1! AB ! "!! , AC ! "!! Dw3L!N$OP3L! ! ⇔ −9 9 = 2(m +1) 2 −2 ⇔ m = 0 m = −2 ⎡ ⎣ ⎢ ⎢ (t /m) *! HC#)$=;&(!tz]! m = −2;m = 0 e&!L1-!.G9!DS3!.C'*!!! Bj=).)k`d*)>%e6lF) F; M1,1!N$OP3L!.GC3$! 2 2 sin x. cos x = 1 *!! 0; BC'!()!N$QD!R!DE!N$S3!.$/D!5&!N$S3!,%!7TU!VOP3L!.$%,!'W3! z = 5,z 2 + z 2 = 6 *!! F; „1TU!Z1X3!N$OP3L!.GC3$!DE!3L$1X'‚! sin x > 0 *! +$1!7E!0C3$!N$OP3L!$F1!52!DpF!N$OP3L!.GC3$!.F!7O•D‚! ! 8sin 2 x.cos 2 x = 1⇔ 2sin 2 2x = 1 ⇔ cos4x = 0 ⇔ x = π 8 + k π 4 ,k ∈ ! *! ƒ;!J1>U!V1…3!.G43!5[3L!.G[3!eO•3L!L1-D!.F!7O•D!D-D!3L$1X'!.$%,!'W3!‚! ! x ∈ π 8 + k2π, 3π 8 + k2π, 5π 8 + k2π, 7π 8 + k2π,k ∈ ! ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ ⎫ ⎬ ⎪ ⎪ ⎭ ⎪ ⎪ *!! 0; „d.! z = x + y.i (x, y > 0) I!.$†%!L1,!.$12.!.F!DE‚! ! x 2 + y 2 = 5 (x + yi) 2 + (x − yi) 2 = 6 ⎧ ⎨ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⇔ x 2 + y 2 = 5 x 2 − y 2 = 3 ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ ⇔ x 2 = 4 y 2 = 1 ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ ⇔ x = 2 y = 1 ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ (do x, y > 0) *! tz]! z = 2 +i *!!! Bj=)7)k*d^)>%e6lF!M1,1!N$OP3L!.GC3$! log 2 (x + 3) = log 4 x −1 + 2 *! „1TU!Z1X3‚! −3 < x ≠ 1 *! u$OP3L!.GC3$!.OP3L!7OP3L!5<1‚! ! !"#$%&'()*+,-) ).*.)) /0&1)23)&456)7)489):%&4)&4;&)<=)>?%)489)@4A))) B4%)#%C#()DE#4$%&2:FG&) ! ‡! ! log 2 (x + 3) = log 2 4 + log 2 x −1 ⇔ log 2 (x + 3) = log 2 4 x −1 ⇔ x + 3= 4 x −1 ⇔ (x + 3) 2 = 16 x −1 ⇔ x >1 (x + 3) 2 = 16(x −1) ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ x <1 (x + 3) 2 = −16(x −1) ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⇔ x = 5; x = 8 2 −11 *! HC#)$=;&(!u$OP3L!.GC3$!DE!$F1!3L$1X'! x = 5;x = 8 2 −11 *!!! Bj=)\)k`d*)>%e6lF)M1,1!$X!N$OP3L!.GC3$! x 2 + 3x + y + 2 = (x +1)(x + 2) y −1 x 2 +16 − 2 x 2 −3x +4 = y −1−1 ⎧ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ , x, y ∈ ! ( ) *!! Phân%tích%lời%giải:!„>!{!N$OP3L!.GC3$!.$Q!3$€.!DpF!$X!DE! x 2 + 3x + 2 = (x +1)(x + 2) V%!5z]!7>! 7P3!L1,3!.F!7d.! a = (x +1)(x + 2) I!N$OP3L!.GC3$!.G`!.$&3$‚! a + y = a y −1 ⇔ a ≥0 (a + y)( y −1) 2 = a 2 ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ !*! „OF!5T!N$OP3L!.GC3$!0zD!$F1!DpF!F‚!! a 2 −( y −1) 2 a − y( y−1) 2 = 0,Δ a = (y −1) 4 + 4y( y −1) 2 = (y 2 −1) 2 *! n%!7E! a = ( y −1) 2 + (y 2 −1) 2 = y 2 − y;a = ( y −1) 2 − ( y 2 −1) 2 = −y +1 *! BQD!e&! x 2 + 3x + 2 = y 2 − y x 2 + 3x + 2 = −y +1 ⎡ ⎣ ⎢ ⎢ ⎢ ⇔ x 2 + 3x +1+ y = 0 x 2 + 3x − y 2 + y + 2 = 0 ⎡ ⎣ ⎢ ⎢ ⎢ ⇔ x 2 + 3x +1+ y = 0 (x + y +1)(x − y + 2) = 0 ⎡ ⎣ ⎢ ⎢ ⇔ x 2 + 3x +1+ y = 0 y = −x −1 y = x + 2 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ *! Lời%giải:%%%% „1TU!Z1X3‚! x 2 + 3x + y + 2 ≥ 0, y >1 *!u$OP3L!.GC3$!.$Q!3$€.!DpF!$X!.OP3L!7OP3L!5<1‚! x +1 ( ) x + 2 ( ) ≥ 0 x 2 + 3x + y + 2 = x +1 ( ) 2 x + 2 ( ) 2 y −1 ( ) 2 ⎧ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ *! ! !"#$%&'()*+,-) ).*.)) /0&1)23)&456)7)489):%&4)&4;&)<=)>?%)489)@4A))) B4%)#%C#()DE#4$%&2:FG&) ! r! ⇔ x +1 ( ) x + 2 ( ) ≥ 0 y −1 ( ) 2 x +1 ( ) x + 2 ( ) + y ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = x +1 ( ) 2 x + 2 ( ) 2 ⎧ ⎨ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⇔ x +1 ( ) x + 2 ( ) ≥ 0 x 2 + 3x + y +1 ( ) x + y +1 ( ) x − y +2 ( ) = 0 ⎧ ⎨ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ *! ƒ!BF!DE! y >1⇒ x 2 + 3x +1+ y > x 2 + 3x + 2 = x +1 ( ) x + 2 ( ) ≥ 0 *! ƒ!;!v2U! y = −x −1 ⇒ −x −1 >1 ⇔ x < −2 *! B$F]!5&%!N$OP3L!.GC3$!.$Q!$F1!DpF!$X!.F!7O•D! x 2 +16 − 2 x 2 − 3x + 4 = −x −2 −1 ⇔ x 2 +16 +1 = 2 x 2 − 3x + 4 + −x − 2 *! +$1!7E‚! VT = x 2 +16 +1 ≤6,∀x ∈ −3;−2 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ , VP = 2 x 2 −3x + 4 + −x −2 ≥ 2 14 > 6,∀x ∈ −3;−2 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ *! n%!7E!N$OP3L!.GC3$!5f!3L$1X'*! !ƒ!;!v2U! y = x + 2 .$F]!5&%!N$OP3L!.GC3$!.$Q!$F1!DpF!$X!.F!7O•D‚! x 2 +16 − 2 x 2 − 3x + 4 = x +1 −1 *! u$OP3L!.GC3$!3&]!DE!71TU!Z1X3‚! x ≥−1 *!! B$/D!$1X3!3$g3!e143!$•N!.F!DE‚! ⇔ −3x 2 +12x x 2 +16 + 2 x 2 −3x +4 = x x +1 +1 ⇔ x = 0 x 2 +16 + 2 x 2 −3x +4 = −3 x −4 ( ) x +1 +1 ( ) (1) ⎡ ⎣ ⎢ ⎢ ⎢ *! M1,1!N$OP3L!.GC3$!:";!0h3L!D-D$!Z2.!$•N!5<1!N$OP3L!.GC3$!7SU!DpF!$X!.F!7O•D!:!ˆ†'!.$4'! #U)3!“%Bài%giảng%chọn%lọc%Phương%trình%–%Bất%phương%trình%vô%tỷ”!Dw3L! D!L1,;*! x 2 +16 + 2 x 2 −3x +4 = −3 x −4 ( ) x +1 +1 ( ) x 2 +16 − 2 x 2 −3x +4 = x +1−1 ⎧ ⎨ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⇒ 2 x 2 +16 = 13−3x ( ) x +1 − 3x +11 ⇔ 2 x 2 +16 −5 ( ) + 3x −13 ( ) x +1 − 2 ( ) + 9 x −3 ( ) = 0 ⇔ x − 3 ( ) 2 x +3 ( ) x 2 +16 +5 + 3x −13 x +1 + 2 + 9 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = 0 ⇔ x − 3 ( ) 2 x +3 ( ) x 2 +16 +5 + 5 + 9 x +1 +3x x +1 + 2 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = 0 ⇔ x = 3 ! B$‰!e_1!.$€]!.$%,!'W3*!cU]!GF! x; y ( ) = 0;2 ( ) ; 3;5 ( ) *! ! !"#$%&'()*+,-) ).*.)) /0&1)23)&456)7)489):%&4)&4;&)<=)>?%)489)@4A))) B4%)#%C#()DE#4$%&2:FG&) ! "! HC#)$=;&(!#$%!&'!(&)*+,! /+&!01!&23!+,&3'4!56!! x; y ( ) = 0;2 ( ) ; 3;5 ( ) 7!!! Cách%2:!89-! t = x 2 + 3x + 2+ y ⇒ (x +1)(x + 2) = t 2 − y 7! :&)*+,! /+&!-&;!+&<-!0=2!&'! >!-&6+&?! ! t = t 2 − y y −1 ⇔ t 2 −( y −1)t − y = 0 ⇔ (t − y)(t +1) = 0 ⇔ t = y (do t ≥ 0) 7! #/!@$%! x 2 + 3x + 2+ y = y ⇔ y ≥ 0 x 2 + 3x + 2+ y = y 2 ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ ⇔ y ≥ 0 ( y − x −2)(y + x +1) = 0 ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ 7! A2!01!BC-!DEF!-)*+,!-G! H+7!!!! BI=)J)KLM*)>%N6OF!AI+&!-&J!-I0&!B&K3! L+!MN2%!B&3!DE2%!&/+&!(&O+,!,3P 3!&Q+!R>3!0S0!T)U+,! y = 4 + x 2 .ln x x , y = 2 x ,x = 2 DE2+&! V0!&N6+&7! WX!:&)*+,! /+&!&N6+&!TY!,32N!T3J4?! 4 + x 2 .ln x x = 2 x ⇔ 4 + x 2 .ln x = 2 ⇔ x 2 .ln x = 0 ⇔ x = 0(l ) x =1 ⎡ ⎣ ⎢ ⎢ 7! #/!@$% V = π ( 4 + x 2 .ln x x ) 2 − 4 x 2 dx = 1 2 ∫ π ln x dx 1 2 ∫ 7!!! WX!89-! u = ln x dv = dx ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ ⇒ du = dx x v = x ⎧ ⎨ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ 7! WX!ZE%!.2?! V = π(x ln x 2 1 − dx 1 2 ∫ ) = π(2ln 2− x 2 1 ) = π(2ln 2−1) ![T@ X7! BI=)-)KLM*)>%N6OF!\&N!&/+&!0&1(!Z7]^\!01!49-!RH+!Z^\!56!-24!,3S0!@E_+,!0`+!-Q3!Z!@6!+a4! N+,!49-!(&O+,!@E_+,!,10!@P3!49-!(&O+,![]^\Xb! BC = a 2,ASB ! = CSA ! = 60 0 7!AI+&!-&J!-I0&! B&K3!0&1(!Z7]^\!@6!B&NF+,!0S0&!-c!T3J4!^!TC+!49-!(&O+,![Z]\X7! ! WX!de3!f!56! E+,!T3J4!^\b!-&gN!,3F!-&3C-?! ! SH ⊥ BC (SBC ) ⊥ (ABC ) ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ ⇒ SH ⊥ (ABC ) 7! #6!-24!,3S0!Z]^\!@E_+,!0`+!01! SH = BH = CH = BC 2 = a 2 2 7! A24!,3S0!Z]^!@6!Z]\!01!Z]!0&E+,b! SB = SC ,ASB ! = ASC ! = 60 0 +H+! Ra+,!+&2E7! hN!T1! AB = AC @6!-24!,3S0!]^\!0`+!-Q3!]b!T9-! SA = x 7! i(!jV+,!Tk+&!5l!f64!mK!0_m3+!0&N!-24!,3S0!Z]^b!(3-2,N!0&N!0S0!-24!,3S0!Z]fb!]f^!01?! ! !"#$%&'()*+,-) ).*.)) /0&1)23)&456)7)489):%&4)&4;&)<=)>?%)489)@4A))) B4%)#%C#()DE#4$%&2:FG&) ! n! ! AB 2 = SA 2 + SB 2 − 2SA.SB cos60 0 = x 2 + a 2 − ax, AH 2 = AB 2 − BH 2 = x 2 + a 2 − ax − a 2 2 , SA 2 = SH 2 + AH 2 = a 2 2 + (x 2 + a 2 − ax − a 2 2 ) = x 2 ⇒ x = a 7! hN!T1! AH = a 2 2 ,S ABC = 1 2 AH .BC = 1 2 . a 2 2 .a 2 = a 2 2 7! #/!@$%! V S .ABC = 1 3 SH .S ABC = 1 3 . a 2 2 . a 2 2 = a 3 2 12 [T@ X7! WX!A2!01?! d (B;(SAC )) = 2d(H ;(SAC )) 7! op!fo!@E_+,!,10!@P3!]\!-Q3!ob!op!fq!@E_+,!,10!@P3!Zo!-Q3!q!-&/!! ! HI ⊥ (SAC ) ⇒ d(H ;(SAC )) = HI 7! A24!,3S0!@E_+,!]f\!@6!Zfo!01! ! 1 HI 2 = 1 SH 2 + 1 HK 2 = 1 SH 2 + 1 HC 2 + 1 HA 2 = 2 a 2 + 2 a 2 + 2 a 2 = 6 a 2 ⇒ HI = a 6 6 7! #$%! d (B;(SAC )) = 2HI = a 6 3 7! BP94).(!AI+&!-&gN!-&J!-I0&!@/!j3'+!-I0&!-24!,3S0!Z]\!-I+&!T*+!,3F+! A2!01? S SAC = 1 2 SA.SC.sin60 0 = a 2 3 4 ⇒ d (B;(SAC )) = 3V SABC S SAC = a 3 2 4 a 2 3 4 = a 6 3 7!!! QR&4)$=;&(!\&r!l!,3F!-&3C-!R63!-NS+!-2!-I+&!T)s0! HA = 1 2 BC ⇒ ΔABC @E_+,!0`+!-Q3!]7!!! QS%)#;@)#<T&1)#U)V)\&N!&/+&!0&1(!Z7]^\!01!49-!RH+!Z^\!56!-24!,3S0!0`+!-Q3!Zb! SB = a !@6!+a4! N+,!49-!(&O+,!@E_+,!,10!@P3!49-!TS%![]^\X7!^3C-! ASB ! = BSC ! = CSA ! = 60 0 7!AI+&!-&J!-I0&! B&K3!0&1(!Z7]^\!@6!B&NF+,!0S0&!-c! E+,!T3J4!TNQ+!Z^!TC+!49-!(&O+,![Z]\X7!!! BI=),)KLM*)>%N6OF!A.N+,!49-!(&O+,!@P3! V0!-NQ!TY!tM%!0&N!-24!,3S0!]^\!@E_+,!0`+!-Q3!\7! de3!u!56! E+,!T3J4!0Q+&!]\b!h!56!T3J4!-&EY0!TNQ+!]^!-&NF!4v+! DB = 2DA b!f!56!&/+&! 0&3CE!@E_+,!,10!0=2!h! H+!^u7!A/4!-NQ!TY!0S0!Tw+&!]b^b\!R3C-!h[xyz{Xb! H (− 18 5 ; 24 5 ) @6!Tw+&!^! 01!&N6+&!TY!+,E%H+! :&)*+,! /+&!T)U+,!-&O+,!hf!56! x + 2 y −6 = 0 7! 8)U+,!-&O+,!^u!T3!DE2!f!@6!@E_+,!,10!@P3!hf!+H+!01!(&)*+,! /+&! 2x − y +12 = 0 7! A2!0&;+,!43+&!\bfbh!-&O+,!&6+,!@6! CH ! "!! = 3 2 HD ! "!! 7! hN! CH ! "!! = 3 2 HD ! "!! = ( 12 5 ;− 6 5 ) ⇒ C (−6;6) 7!89-! CA = CB = a > 0 ⇒ AB = a 2,BD = 2a 2 3 7! ! !"#$%&'()*+,-) ).*.)) /0&1)23)&456)7)489):%&4)&4;&)<=)>?%)489)@4A))) B4%)#%C#()DE#4$%&2:FG&) ! |! i(!jV+,!Tk+&!5l!&64!mK!\_xm3+!0&N!-24!,3S0!^\h!01! CD 2 = BC 2 + BD 2 − 2BC.BD cos 45 0 = a 2 + 8a 2 9 − 4a 2 3 = 20 ⇔ a 2 = 36 7! de3!^[RzyRW}yX!@P3!R~•!-&EY0!^u!-2!01! ! BC 2 = 36 ⇔ (b + 6) 2 + (2b + 6) 2 = 36 ⇔ 5b 2 + 36b + 36 = 0 ⇔ b = −6(t / m) b = − 6 5 (l ) ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ 7! ZE%!.2! B(−6;0),DA ! "!! = 1 2 BD ! "!! ⇒ A(0;6) 7! HC#)$=;&(!#$%!][•znXb!^[xnz•Xb!\[xnznX7!!! BI=)W)KLM*)>%N6OF!A.N+,!B&_+,!,32+!@P3! V0!-NQ!TY!tM%€!0&N!T3J4!q[yzyz•X!@6!49-!(&O+,! (P ) : 3x + 2y − z + 4 = 0 7!#3C-!(&)*+,! /+&!T)U+,!-&O+,!j!T3!DE2!q!@6!@E_+,!,10!@P3!49-! (&O+,![:X7!A/4!-NQ!TY!T3J4!u! H+!j!m2N!0&N!u!0S0&!T•E!,K0!-NQ!TY!@6!49-!(&O+,![:X7!! WX!8)U+,!-&O+,!j!@E_+,!,10!@P3![:X!+H+!+&$+!@-(-!0=2![:X!564!@‚0!-*!0&w!(&)*+,b!@/!@$%! u d !"! = (3;2;−1) 7!! hN!T1!(&)*+,! /+&!0=2!j!56! d : x = 2 + 3t y = 2+ 2t z = −t ⎧ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ,t ∈ ! 7! WX!de3! M (2+3t;2+ 2t;−t ) ∈ d b!-2!01?! ! d (M ;(P )) = 3(2+ 3t ) + 2(2+ 2t)− (−t )+ 4 3 2 + 2 2 +1 2 = 14 t +1 , MO = (2+ 3t) 2 + (2+ 2t ) 2 + t 2 = 14t 2 + 20t + 4 7! WX!A&gN!,3F!-&3C-!-2!01?! ! 14 t +1 = 14t 2 + 20t + 4 ⇔ 14t 2 + 20t + 4 =14(t +1) 2 ⇔ 4t + 3= 0 ⇔ t = − 3 4 ⇒ M (− 1 4 ; 1 2 ; 3 4 ) 7! HC#)$=;&(!#$%!T3J4!0ƒ+!-/4! M (− 1 4 ; 1 2 ; 3 4 ) 7!!! BI=)+)K*MJ)>%N6OF!„,)U3!-2!j…+,!"!0EK+!mS0&!ANS+b!n!0EK+!mS0&!#$-!5lb!|!0EK+!mS0&!fNS!&e0! [0S0!mS0&!0…+,!5NQ3!-&/!,3K+,!+&2EX!TJ!564!(&ƒ+!-&)>+,!0&N!†!&e0!m3+&!4‡3!&e0!m3+&!T)s0!&23! 0EK+!B&S0!5NQ3b! N+,!†!&e0!m3+&!+6%!01!&23!RQ+!„24!@6!f)U+,7!AI+&!MS0!mE<-!TJ!„24!@6! f)U+,!01!(&ƒ+!-&)>+,!,3K+,!+&2E7! de3!M!56!mK!&e0!m3+&!+&$+!mS0&!ANS+!@6!#$-!ˆl7! de3!%!56!mK!&e0!m3+&!+&$+!mS0&!ANS+!@6!fNS!&e0! de3!€!56!mK!&e0!m3+&!+&$+!mS0&!fNS!&e0!@6!#$-!ˆl! ! !"#$%&'()*+,-) ).*.)) /0&1)23)&456)7)489):%&4)&4;&)<=)>?%)489)@4A))) B4%)#%C#()DE#4$%&2:FG&) ! ‰! A2!01?! x + y + z = 9 x + y = 5 x + z = 6 y + z = 7 ⎧ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⇔ x = 2 y = 3 z = 4 ⎧ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ 7!#$%!0&w!01!y!&e0!m3+&!+&$+!mS0&!ANS+!@6!#$-!5lz!Š!&e0!m3+&! +&$+!mS0&!ANS+!@6!fNS!&e0z!{!&e0!m3+&!+&$+!mS0&!#$-!5l!@6!fNS!&e0![‹X7! WX!o&_+,!,32+!4ŒE!56!mK!0S0&!0&32!mS0&!0&N!†!&e0!m3+&!-&gN!T3•E!B3'+![‹X!01! Ω = C 9 2 .C 7 3 .C 4 4 = 1260 7! de3!]!56!R3C+!0K!y!RQ+!„24!@6!f)U+,!01!(&ƒ+!DE6!,3K+,!+&2Eb!01!0S0!B&F!+•+,! X!L(!\F!y!0…+,!+&$+!mS0&!ANS+!@6!#$-!5lb!B&3!T1!|!RQ+!0L+!5Q3!01!Š!RQ+!+&$+!mS0&!ANS+!@6! fNS!&e0z!{!RQ+!+&$+!mS0&!fNS!&e0!@6!#$-!5l7!ZK!0S0&!(&`+!0&32!56! C 7 3 .C 4 4 = 35 7! X!.(!\F!y!0…+,!+&$+!mS0&!ANS+!@6!fNS!&e0b!B&3!T1!|!RQ+!0L+!5Q3!01!y!RQ+!+&$+!mS0&!ANS+!@6! #$-!5lz!}!RQ+!+&$+!mS0&!ANS+!@6!fNS!&e0z!{!RQ+!+&$+!mS0&!#$-!5l!@6!fNS!&e07! ZK!0S0&!(&`+!0&32!56! C 7 2 C 5 1 .C 4 4 = 105 7!!!! X!7(!\F!y!0…+,!+&$+!mS0&!fNS!&e0!@6!#$-!ˆlb!B&3!T1!|!RQ+!0L+!5Q3!01!y!RQ+!+&$+!mS0&!ANS+! @6!#$-!5lz!Š!RQ+!+&$+!mS0&!ANS+!@6!fNS!&e0z!y!RQ+!+&$+!mS0&!fNS!&e0!@6!#$-!5l7! ZK!0S0&!(&`+!0&32!56! C 7 2 .C 5 3 .C 2 2 = 210 7! #$%! Ω A = 35 +105 + 210 = 350 7! ŽS0!mE<-!0ƒ+!-I+&! P (A) = Ω A Ω = 350 1260 = 5 18 7!!!!!!!! BI=)L*)KLM*)>%N6OF!\&N!2bRb0!56!0S0!mK!-&G0!j)*+,!-&NF!4v+! a(a −c )+ b(b − c ) = 0 7!A/4!,3S! k! +&•!+&<-!0=2!R3JE!-&;0! P = a b 3 + c 3 + b c 3 + a 3 + c 2 + ab 2 − a + b c ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ 2 7! 89-! A = a b 3 + c 3 + b c 3 + a 3 ,S = a b 3 + c 3 + b a 3 + c 3 , !! Z•!jV+,!R<-!TO+,!-&;0!fN5jg.!-2!01? A.S 2 ≥ (a + b) 3 7!! !! Z•!jV+,!R<-!TO+,!-&;0!\2E0&%!‘Z0&’2 €!-2!01?! S = a 2 (b + c).(b 2 −bc + c 2 ) + b 2 (c +a)(c 2 − ca +a 2 ) ≤ a 2 (b + c) +b 2 (c +a) ( ) b 2 + a 2 + 2c 2 − c (a +b) ( ) = c 2(ab(a + b) +c(a 2 + b 2 )) = c 2(a + b)(ab + c 2 ) ! ZE%!.2?! A ≥ (a +b) 2 2c 2 (c 2 + ab) 7!! #/!@$%! P ≥ (a +b) 2 2c 2 (c 2 + ab) + c 2 + ab 2 − a + b c ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ 2 ≥ a + b c − a + b c ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ 2 ≥−2 7! ^>3!@/!-&gN!,3F!-&3C-!-2!01?! ! c(a + b) = a 2 + b 2 ≥ 1 2 (a +b) 2 ⇒ a + b c ≤ 2,c(a + b) < (a + b) 2 ⇒ a + b c >1 7!! ! !"#$%&'()*+,-) ).*.)) /0&1)23)&456)7)489):%&4)&4;&)<=)>?%)489)@4A))) B4%)#%C#()DE#4$%&2:FG&) ! †! Cách%2:!A&gN!,3F!-&3C-!-2!01?! a 2 + b 2 = c(a + b) ≥ 1 2 (a +b) 2 ⇒ a + b c ≤ 2 7! Z•!jV+,!R<-!TO+,!-&;0!\2E0&%!‘Z0&’2.€!-2!01?! a b 3 + c 3 + b c 3 + a 3 = a 2 a(b 3 + c 3 ) + b 2 b(c 3 + a 3 ) ≥ (a +b) 2 a(b 3 + c 3 ) + b(c 3 + a 3 ) = (a +b) 2 ab(a 2 + b 2 ) + c 3 (a +b) = (a +b) 2 abc(a +b) + c 3 (a +b) = a + b abc + c 3 = a + b c(ab + c 2 ) 7! hN!T1?! ! P ≥ a +b c(ab + c 2 ) + c 2 + ab 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ − a +b c ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ 2 ≥ 2 a +b c(ab + c 2 ) . c 2 + ab 2 − a +b c ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ 2 = 2(a + b) c − a +b c ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ 2 = f (t) = 2t −t 2 ,t = a +b c ∈ 1;2 ( ⎤ ⎦ ⎥ 7! o&FN!mS-!&64!mK!“[-X! H+!+•2!B&NF+,! 1;2 ( ⎤ ⎦ ⎥ -2!01! f (t) ≥ f (2) = −2 7! A2!01!BC-!DEF!-)*+,!-G7! ! !!!! ! !! ! ! ! ! . 2) y −1 x 2 +16 − 2 x 2 −3x +4 = y −1−1 ⎧ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ , x, y ∈ ! ( ) *!! Phân%tích%lời %giải: !„>!{!N$OP3L!.GC3$!.$Q!3$€.!DpF!$X!DE! x 2 + 3x + 2 = (x +1)(x + 2) V%!5z]!7>! 7P3!L1,3!.F!7d.! . 0 (x + y +1)(x − y + 2) = 0 ⎡ ⎣ ⎢ ⎢ ⇔ x 2 + 3x +1+ y = 0 y = −x −1 y = x + 2 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ *! Lời %giải: %%%% „1TU!Z1X3‚! x 2 + 3x + y + 2 ≥ 0, y >1 *!u$OP3L!.GC3$!.$Q!3$€.!DpF!$X!.OP3L!7OP3L!5<1‚! . ! "! HC#)$=;&(!#$%!&'!(&)*+,! /+&!01!&23!+,&3'4!56!! x; y ( ) = 0;2 ( ) ; 3;5 ( ) 7!!! Cách% 2:!89-! t = x 2 + 3x + 2+ y ⇒ (x +1)(x + 2) = t 2 − y 7! :&)*+,! /+&!-&;!+&<-!0=2!&'!