PHƯƠNG TRÌNH VÀ BPT VÔ TỈ HỆ PT VÀ HỆ BPT

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WWW.VNMATH.COM Chuyªn ®Ò: ph−¬ng tr×nh, bÊt ph−¬ng tr×nh v« tØ, hÖ ph−¬ng tr×nh vμ hÖ bÊt ph−¬ng tr×nh QUA C¸C §Ò THI §¹I HäC PhÇn I: Ph−¬ng tr×nh v« tØ Ph−¬ng ph¸p 1:Ph−¬ng ph¸p gi¶i d¹ng c¬ b¶n: g  x   0 1/ f  x   g  x    f  x   g  x  2 2/ f  x   g  x   h  x  B×nh ph−¬ng hai vÕ 1-(§HQGHN KD-1997) 16x  17  8x  23 2-(§H C¶nh s¸t -1999) x 2  x 2  11  31 3-(HVNHHCM-1999)  x 2  4x  2  2x 4-(§H Th−¬ng m¹i-1999) Gi¶i vμ biÖn luËn pt: m  x 2  3x  2  x 5-(§HC§ KB-2006) T×m m ®Ó pt sau cã hai nghiÖm thùc ph©n biÖt: x 2  mx  2  2x  1 6-(§GKTQD-2000) 5x  1  3x  2  x  1  0 7-(§HSP 2 HN) x  x  1  x  x  2   2 x 2 8-(HVHCQ-1999) x  3  2x  1  3x  2 9-(HVNH-1998) 3x  4  2x  1  x  3 10-(§H Ngo¹i th−¬ng-1999) 3  x  x2  2  x  x2  1 Ph−¬ng ph¸p 2: ph−¬ng ph¸p ®Æt Èn phô: I-§Æt Èn phô ®−a pt vÒ pt theo Çn phô: a b D¹ng 1: Pt d¹ng: ax 2  bx  c  px 2  qx  r trong ®ã  p q C¸ch gi¶i: §Æt t  px 2  qx  r §K t  0 1
WWW.VNMATH.COM 1-(§H Ngo¹i th−¬ng-2000)  x  5 2  x   3 x 2  3x 2-(§H Ngo¹i ng÷ -1998)  x  4  x  1  3 x 2  5x  2  6 3-(§H CÇn th¬-1999) (x  1)(2  x)  1  2x  2x 2 4x 2  10x  9  5 2x 2  5x  3  18x  5  3 9x 2  9x  2 2 3 4- 5- 18x 6- 3x 2  21x  18  2 x 2  7x  7  2 D¹ng 2: Pt D¹ng: P(x)   Q(x)   P(x).Q(x)  0    0  P  x   0 C¸ch gi¶i: * NÕu P  x   0  pt   Q  x   0 Qx * NÕu P  x   0 chia hai vÕ cho P  x  sau ®ã ®Æt t  t0 Px 3 x 1  m x 1  2 x2 1 4 1-(§HC§ KA-2007) T×m m ®Ó pt sau cã nghiÖm: 2-   2 x 2  3x  2  3 x 3  8 3-   2 x 2  2  5 x3  1 D¹ng 3: Pt D¹ng :   P  x   Q  x     Px  Qx   2 P  x  .Q  x     0  2  2  0  t  P  x   Q  x   t 2  P  x   Q  x   2 P  x  .Q  x  C¸ch gi¶i: §Æt 2 1-(§HQGHN-2000) 1 x  x2  x  1 x 3 2-(HVKTQS-1999) 3x  2  x  1  4x  9  2 3x 2  5x  2 3-(Bé quèc phßng-2002) 2x  3  x  1  3x  2 2x 2  5x  3  16 4- 4x  3  2x  1  6x  8x 2  10x  3  16 5-(C§SPHN-2001) x  2  x  2  2 x 2  4  2x  2 2
WWW.VNMATH.COM D¹ng 4: Pt D¹ng: a  cx  b  cx  d  a  cx  b  cx   n Trong ®ã a, b,c,d, n lμ c¸c h»ng sè , c  0,d  0 C¸ch gi¶i: §Æt t  a  cx  b  cx ( a  b  t  2  a  b  1-(§H Má-2001) x  4  x 2  2  3x 4  x 2 2- 3  x  6  x   3  x  6  x   3 3-(§HSP Vinh-2000) Cho pt: x 1  3  x   x  1 3  x   m a/ Gi¶i pt khi m2 b/T×m c¸c gt cña m ®Ó pt cã nghiÖm 4-(§HKTQD-1998) Cho pt 1  x  8  x  (1  x)(8  x)  a a/Gpt khi a  3 b/T×m c¸c gt cña a ®Ó pt cã nghiÖm 5-TT §T Y tÕ tphcm-1999) T×m c¸c gt cña m ®Ó pt cã nghiÖm x  1  3  x  (x  1)(3  x)  m 6-(§H Ngo¹i ng÷-2001) x  1  4  x  (x  1)(4  x)  5 D¹ng 5: Pt d¹ng: x  a  b  2a x  b  x  a 2  b  2a x  b  cx  m 2 Trong ®ã a, b,c, m lμ h»ng sè a  0 C¸ch gi¶i : §Æt t  x  b §K: t  0 ®−a pt vÒ d¹ng: t  a  t  a  c(t 2  b)  m 1-(§HSP Vinh-2000) x 1  2 x  2  x 1  2 x  2  1 2-(HV BCVT-2000) x  2 x 1  x  2 x 1  2 3-(§HC§ KD-2005) 2 x  2  2 x 1  x 1  4 x 5 4-(§H Thuû s¶n -2001) x  2  2 x 1  x  2  2 x 1  2 x 3 5- x  2 x 1  x  2 x 1  2 3
WWW.VNMATH.COM xm 6- XÐt pt: x 6 x 9  x 6 x 9  6 a/ Gi¶i pt khi m  23 b/ T×m c¸c gt cña m ®Ó pt cã nghiÖm II-Sö dông Èn phô ®−a pt vÒ Èn phô ®ã ,cßn Èn ban ®Çu coi lμ tham sè: 1- 6x 2  10x  5   4x  1 6x 2  6x  5  0 2-(§H D−îc-1999)  x  3 10  x 2  x 2  x  12 3-(§H D−îc-1997) 2 1  x  x 2  2x  1  x 2  2x  1 4-  4x  1 x  1  2x  2x  1 5- 2 1  x  x  x  1  x  3x  1 2 2 2 2 6-(§HQG-HVNH KA-2001) x 2  3x  1  (x  3) x 2  1 III-Sö dông Èn phô ®−a vÒ hÖ pt: D¹ng 1: Pt D¹ng: x n  a  b n bx  a  x  by  a  0 n C¸ch gi¶i: §Æt y  bx  a khi ®ã ta cã hÖ:  n  y  bx  a  0 n 1-(§HXD-DH HuÕ-1998) x2 1  x 1 2- x  x  5  5 3- x  2002 2002x  2001  2001  0 2 2 4- (§H D−îc-1996) x 3  1  2 3 2x  1 ax  b  r  ux  v   dx  e trong ®ã a, u, r  0 2 D¹ng 2: Pt D¹ng: Vμ u  ar  d, v  br  e uy  v  r  ux  v 2  dx  e C¸ch gi¶i: §Æt uy  v  ax  b khi ®ã ta cã hÖ:  ax  b   uy  v  2 1-(§HC§ KD-2006) 2x  1  x 2  3x  1  0 2- 2x  15  32x 2  32x  20 3- 3x  1  4x  13x  5 2 4- x  5  x 2  4x  3 5- x  2  x  2 2 6- x 1  3  x  x2 4
WWW.VNMATH.COM D¹ng 3: PT D¹ng: n a  f x  m b  f x  c u  v  c C¸ch gi¶i: §Ætu  n a  f  x  , v  m b  f  x  khi ®ã ta cã hÖ:  n u  v  a  b m 1-(§HTCKT-2000) 3 2  x  1 x 1 2- 3 x  34  3 x  3  1 3- x  2  x  1  3 3 4- 4 97  x  4 x  5 5- 18  x  x  1  3 4 4 Ph−¬ng ph¸p 3: Nh©n l−îng liªn hîp: D¹ng 1: Pt D¹ng: f x  a  f x  b  f  x   a  f  x   b C¸ch gi¶i: Nh©n l−îng liªn hîp cña vÕ tr¸i khi ®ã ta cã hÖ:   f  x   a  f  x   a b 1- 4x 2  5x  1  4x 2  5x  7  3 2- 3x 2  5x  1  3x 2  5x  7  2 3- 3- (§H Ngo¹i th−¬ng-1999 ) 3  x  x2  2  x  x2  1 4-(§H Th−¬ng m¹i-1998) x 2  3x  3  x 2  3x  6  3 1 1 5-(HVKTQS-2001)  1 x4 x2 x2 x D¹ng 2: Pt D¹ng: f  x   g  x   m  f  x   g  x  x 3 1-(HVBCVT-2001) 4x  1  3x  2  5 2-(HVKTQS-2001) 3(2  x  2)  2x  x  6 Ph−¬ng ph¸p 4:Ph−¬ng ph¸p ®¸nh gi¸: 1- x  2  4  x  x 2  6x  11 x2  x 1  x  x2 1  x2  x  2 2- 3-(§HQGHN-Ng©n hμng KD-2000) 4x  1  4x 2  1  1 4-(§H N«ng nghiÖp-1999) x 2  2x  5  x  1  2 5
WWW.VNMATH.COM Ph−¬ng ph¸p 5:Ph−¬ng ph¸p ®k cÇn vμ ®ñ: 1-T×m m ®Ó pt sau cã nghiÖm duy nhÊt: x  2x  m 2- T×m m ®Ó pt sau cã nghiÖm duy nhÊt x 5  9 x  m 3- T×m m ®Ó pt sau cã nghiÖm duy nhÊt 4 x  4 1 x  x  1 x  m Ph−¬ng ph¸p 6: Ph−¬ng ph¸p hμm sè (Sö dông ®¹o hμm) 1-(§HC§ KB-2004) - T×m m ®Ó pt sau cã nghiÖm : m   1 x2  1 x2  2  2 1 x4  1 x2  1 x2 2- - T×m m ®Ó c¸c pt sau cã nghiÖm : 1*/ 4  x  mx  m  2 2 2*/ x 1  x  1  5  x  18  3x  2m  1 3--(§HC§ KA-2007) T×m m ®Ó pt sau cã nghiÖm: 3 x 1  m x 1  2 x2 1 4 4-(§HC§KB-2007) CMR m  0 pt sau cã 2nghiÖm pb: x  2x  8  m(x  2) 2 5- 1*/ x  x  5  x  7  x  16  14 2*/ x  1   x 3  4x  5 3*/ 2x  1  x 2  3  4  x 6-(HVAn ninh KA-1997)T×m m ®Ó pt sau cã nghiÖm: x 2  2x  4  x 2  2x  4  m 6
WWW.VNMATH.COM PhÇn II: BÊT Ph−¬ng tr×nh v« tØ Ph−¬ng ph¸p 1: Ph−¬ng ph¸p gi¶i d¹ng c¬ b¶n:  g(x)  0  f (x)  0 g(x)  0   1/ f (x)  g(x)   2/ f (x)  g(x)  f (x)  0  g(x)  0   f (x)  g (x) 2  f (x)  g (x) 2 3/ f (x)  g(x)  h(x) B×nh ph−¬ng hai vÕ bpt 1-(§HQG-1997)  x 2  6x  5  8  2x 2-(§HTCKT Tphcm-1999) 2x  1  8  x 3-(§H LuËt 1998) x  2x 2  1  1  x 4-(§H Má-2000) (x  1)(4  x)  x  2 5-(§H Ngo¹i ng÷) x 5 x 4  x 3 6-(§HC§KA-2005) 5x  1  x  1  2x  4 7-(§H Ngoai th−¬ng-2000) x  3  2x  8  7  x 8-(§H Thuû lîi -2000) x  2  3  x  5  2x 9-(§H An ninh -1999) 5x  1  4x  1  3 x 10-(§HBK -1999) x 1  3  x  4 2(x 2  16) 7x 11-(§HC§ KA-2004)  x 3  x 3 x 3 Ph−¬ng ph¸p 2: Sö dông c¸c phÐp biÕn ®æi t−¬ng ®−¬ng f (x) f (x)  0 f (x)  0 1/ 0 hoÆc  g(x) g(x)  0 g(x)  0 f (x) f (x)  0 f (x)  0 2/ 0 hoÆc  g(x) g(x)  0 g(x)  0 7
WWW.VNMATH.COM B  0 A B  0 A B  0  L−u ý: 1*/ 1  2*/ 1  hay A  0 A  B A  0 2 B B  A  B 2 51  2x  x 2 3x 2  x  4  2 1-(§HTCKT-1998) 1 2-(§HXD) 2 1 x x 1  1  4x 2 2  x  4x  3 3-(§H Ngo¹i ng÷ -1998) 3 4-(§HSP) 2 x x Ph−¬ng ph¸p 2:Nh©n biÓu thøc liªn hîp: x2 2x 2  x  4 2-(§H Má-1999)  x  21   1-(§HSP Vinh-2001) 1   3  9  2x 2 2 1 x 3- 4(x  1)  (2x  10)(1  2 3  2x ) 2 Ph−¬ng ph¸p 3:X¸c ®Þnh nh©n tö chung cña hai vÕ: 1-(§H An ninh -1998) x 2  x  2  x 2  2x  3  x 2  4x  5 2-(§HBK-2000) x 2  3x  2  x 2  6x  5  2x 2  9x  7 3-(§H D−îc -2000) x 2  8x  15  x 2  2x  15  4x 2  18x  18 4-(§H KiÕn tróc -2001) x 2  4x  3  2x 2  3x  1  x  1 Ph−¬ng ph¸p 4: §Æt Èn phô: 1-(§H V¨n ho¸) 5x 2  10x  1  7  x 2  2x 2-(§H D©n lËp ph−¬ng ®«ng -2000) 2x 2  4x  3 3  2x  x 2  1 3-(HV Quan hÖ qt-2000) (x  1)(x  4)  5 x 2  5x  28 4-(§H Y-2001) 2x 2  x 2  5x  6  10x  15 5-(HVNH HCM-1999) x(x  4)  x 2  4x  (x  2) 2  2 3 1 6-§H Th¸i nguyªn -2000) 3 x  2x  7 2 x 2x 8
WWW.VNMATH.COM 2 1 7-(§H Thuû lîi) 4 x  2x  2 x 2x 8-(HV Ng©n hμng 1999) x  2 x 1  x  2 x 1  3 2 9- Cho bpt: 4 (4  x)(2  x)  x  2x  a  18 2 a/ Gi¶i bpt khi a  6 b/T×m a ®Ó bpt nghiÖm ®óng x   2; 4 10-X¸c ®Þnh m ®Ó bpt sau tho¶ m·n trªn ®o¹n ®· chØ ra : (4  x)(6  x)  x 2  2x  m trªn  4;6 Ph−¬ng ph¸p 5: Ph−¬ng ph¸p hμm sè: 1-(§H An ninh-2000) 7x  7  7x  6  2 49x 2  7x  42  181  14x 2- x  x  7  2 x 2  7x  35  2x 3- x  2  x  5  2 x 2  7x  10  5  2x 4- X¸c ®Þnh m ®Ó bpt sau cã nghiÖm: a/ 4x  2  16  4x  m b/ 2x 2  1  m  x 9
WWW.VNMATH.COM PhÇn III: HÖ Ph−¬ng tr×nh A- mét sè hÖ pt bËc hai c¬ b¶n I-hÖ pt ®èi xøng lo¹i 1 f (x; y)  0 1*/ §Þnh nghÜa:  Trong ®ã f (x; y)  f (y; x),g(x; y)  g(y; x) g(x; y)  0 §Æt S  x  y, P  xy §K: S  4P 2 2*/ C¸ch gi¶i: D¹ng 1: Gi¶i ph−¬ng tr×nh  x  y  xy  11  x y  y x  30 1-(§HQG-2000)  2 2-   x  y 2  3(x  y)  28  x x  y y  35  x  y  xy  11  x  y  xy  7 2 2 3-(§HGTVT-2000)  2 4-(§HSP-2000)  4  x y  y x  30  x  y  x y  21 2 4 2 2  1 1  x  y   5 x y 5- (§H Ngo¹i th−¬ng-1997)  x 2  y2  1  1  9  x 2 y2  x 2  y 2  5  x  y  xy  3 6-(§H Ngo¹i th−¬ng -1998)  7-(§HC§KA-2006)   x  x y  y  13  x  1  y  1  4 4 2 2 4 D¹ng 2: T×m §K ®Ó hÖ cã nghiÖm:  x  y  1 1-(§HC§KD-2004) T×m m ®Ó hÖ sau cã nghiÖm:   x x  y y  1  3m  x  y  xy  a 2- T×m a ®Ó hÖ sau cã nghiÖm:  2 x  y  a 2 x  y  x 2  y2  8 3-Cho hÖ pt:   xy(x  1)(y  1)  m a/ Gi¶i hÖ khi m  12 b/ T×m m ®Ó hÖ cã nghiÖm 10
WWW.VNMATH.COM  x  xy  y  m  1 4-Cho hÖ pt:  x y  y x  m 2 2 a/ Gi¶i hÖ khi m=-2 b/ T×m m ®Ó hÖ cã Ýt nhÊt mét nghiÖm  x; y  tho¶ m·n x  0, y  0  x  y  2(1  m) 2 2 5- T×m m ®Ó hÖ cã ®óng hai nghiÖm:   x  y   4 2  1 1  x   y  5 x y 6-(§HC§KD-2007) T×m m ®Ó hÖ sau cã nghiÖm:   x 3  1  y3  1  15m  10  x3 y3 D¹ng 3: T×m §K ®Ó hÖ cã nghiÖm duy nhÊt.  x  y  xy  m  2 1-(HHVKTQS-2000) T×m m ®Ó hÖ sau cã nghiÖm duy nhÊt  2 x y  y x  m  1 2  x  xy  y  2m  1 2-(§HQGHN-1999) T×m m ®Ó hÖ sau cã nghiÖm duy nhÊt:   xy(x  y)  m  m 2  x 2 y  y 2 x  2(m  1) 3- T×m m ®Ó hÖ sau cã nghiÖm duy nhÊt:  2xy  x  y  2(m  2) D¹ng 4: HÖ pt ®èi xøng ba Èn sè : NÕu ba sè x, y, z tho¶ m·n x  y  z  p, xy  yz  zx  q, xyz  r th× chóng lμ nghiÖm cña pt: t  pt  qt  r  0 3 2 1-Gi¶i c¸c hÖ pt sau :  x  y  z  1 x  y  z  1 x  y  z  9   2  a/  xy  yz  zx  4 b/  x  y  z  1 c/  xy  yz  zx  27 2 2  3  3 1 1 1 x  y  z  1 x  y  z  1 3 3 3 3    1  x y z 11
WWW.VNMATH.COM x 2  y2  z2  8 2- Cho hÖ pt:  Gi¶ sö hÖ cã nghiÖm duy nhÊt  xy  yz  zx  4 8 8 CMR:  x, y, z  3 3 II-HÖ ph−¬ng tr×nh ®èi xøng lo¹i 2 f (x; y)  0 1*/ §Þnh nghÜa  trong ®ã : f (x; y)  g(y; x),f (y; x)  g(x; y)  g(x; y)  0 f (x; y)  g(x; y)  0 (x  y)h(x; y)  0 2*/ C¸ch gi¶i: HÖ pt    f (x; y)  0 f (x; y)  0 x  y  0 h(x; y)  0  hay   f (x; y)  0 f (x; y)  0 D¹ng 1: Gi¶i ph−¬ng tr×nh:  y  x  3y  4 x  x 3  3x  8y 1-(§HQGHN-1997)  2-(§HQGHN-1998)   y  3x  4 x  y  3y  8x 3  y  1 3  2x   y x  x 3  1  2y 3-(§HQGHN-1999)  4-(§H Th¸i nguyªn-2001)   y  1  2x 3 2y  1  3  x y  8   7x  y  0 x 1  7  y  4 x2 5-(§H V¨n ho¸-2001)  6-(§H HuÕ-1997)   y 1  7  x  4 7y  x  82  0  y D¹ng 2:T×m ®k ®Ó hÖ cã nghiÖm:  x  1  y  2  m 1-(§HSP Tphcm-2001) T×m m ®Ó hÖ cã nghiÖm:   y  1  x  2  m 12
WWW.VNMATH.COM 2x  y  3  m 2- T×m m ®Ó hÖ cã nghiÖm:  2y  x  3  m D¹ng 3: T×m ®k ®Ó hÖ cã nghiÖm duy nhÊt  x  12  y  a 1-(§HSP-Tphcm-2001) T×m a ®Ó hÖ sau cã nghiÖm duy nhÊt:  (y  1)  x  a 2  xy  x 2  m(y  1) 2- T×m m ®Ó hÖ sau cã nghiÖm duy nhÊt:   xy  y  m(x  1) 2  x 2  y  axy  1 3- T×m a ®Ó hÖ sau cã nghiÖm duy nhÊt:   y  x  axy  1 2 III - HÖ ph−¬ng tr×nh ®¼ng cÊp: */ HÖ pt ®−îc gäi lμ ®¼ng cÊp nÕu mçi pt trong hÖ cã d¹ng ax  bxy  cy d 2 2 */ C¸ch gi¶i: §Æt x  ty */ L−u ý: NÕu (a;b) lμ nghiÖm cña hÖ th× (b;a) còng lμ nghiÖm cña pt. D¹ng 1: Gi¶i ph−¬ng tr×nh: 2x  3xy  y  12  x 2  2xy  3y 2  9 2 2 1-(§HP§-2000)  2-(§HSP Tphcm-2000)  2  x  xy  3y  11 2x  2xy  y  2 2 2 2  x 2 y  xy 2  30 3-(§H Má-1998)   x  y  35 3 3 D¹ng 2: T×m ®k ®Ó hÖ cã nghiÖm, cã nghiÖm duy nhÊt 3x 2  2xy  y 2  11 1-(§HQG HCM-1998) T×m m ®Ó hÖ sau cã nghiÖm :   x  2xy  3y  17  m 2 2  x 2  2xy  3y 2  8 2-(§HAnninh2000)T×m a®Ó hÖ cã nghiÖm:  2x  4xy  5y  a  4a  4a  12  105 2 2 4 3 2  x  mxy  y  m  3m  2 2 2 2 3-T×m m ®Ó hÖ sau cã nghÖm diuy nhÊt:   x  2xy  my  m  4m  3 2 2 2 B- Mét sè ph−¬ng ph¸p gi¶i hÖ pt : 13
WWW.VNMATH.COM Ph−¬ng ph¸p 1:Ph−¬ng ph¸p thÕ: x  y  m  1 1-(§HSP Quy nh¬n -1999) Cho hÖ pt:   x y  y x  2m  m  3 2 2 2 1/ Gi¶i hÖ khi m  3 2/T×m m ®Ó hÖ trªn cã nghiÖm  x  y  3 x  y  x  y  x  y  2 2-(§HC§KB-2002)  3-(HVQY-2001)   x  y  x  y  2  x  y  x  y  4 2 2 2 2 x 2  y2  1 4-(§H HuÕ-1997) T×m k ®Ó hÖ sau cã nghiÖm:  x  y  k  x  my  m 5-(§H Th−¬ng m¹i-2000) Cho hÖ pt:  2 x  y  x  0 2 a. Gi¶I hÖ khi m  1 b. BiÖn luËn sè nghiÖm cña pt c.Khi hÖ cã hai nghiÖm ph©n biÖt (x1 ; y1 );(x 2 ; y 2 ) t×m m ®Ó : A  (x 2  x1 ) 2  (y 2  y1 ) 2 ®¹t gi¸ tri lín nhÊt x  y  1 6-(SP TPHCM-1999) T×m m ®Ó hÖ sau cã 3 nghiÖm ph©n biÖt:  3  x  y  m(x  y) 3 Ph−¬ng ph¸p 2: ph−¬ng ph¸p biÕn ®æi t−¬ng ®−¬ng:  xy  3x  2y  16 1-(§HGTVT TPHCM-1999)  2 HD:nh©n pt ®Çu víi 2 vμcéng víi pt sau  x  y 2  2x  4y  33  x  xy  y  1 x  y  z  7   2 2-(§HTh−¬ng m¹i-1997)  y  yz  z  4 3-(§HBKHN-1995)  x  y  z  21 2 2 z  zx  x  9    xz  y 2  y  xy 2  6x 2 4-(§HSPHN-2000)  HD:chia c¶ hai vÕ cña2pt cho x2 1  x y  5x 2 2 2 Ph−¬ng ph¸p 3: Ph−¬ng ph¸p ®Æt Èn phô: 14
WWW.VNMATH.COM  x 16  xy    x 2 x 3 y 3 ( )  ( y )  12 1-(§H Ngo¹i ng÷-1999)  2-(§H C«ng ®oμn-2000)  y  xy  y  9 (xy) 2  xy  6   x 2  x y 7    1 3-(§H Hμng h¶i-1999)  y x xy (x  0, y  0)   x xy  y xy  78  x  1  y  1  3 4-(§H Thuû s¶n-2000)   x y  1  y x  1  y  1  x  1  6 15
WWW.VNMATH.COM PhÇn:IV HÖ BÊt Ph−¬ng tr×nh A- HÖ bpt mét Èn sè: f1  x   0(1) Cho hÖ:  (I) Gäi S1 ,S2 LÇn l−ît lμ tËp nghiÖm cña (1)&(2) 2f (x)  0(2) S lμ tËp nghiÖm cña (I)  S  S1  S2 T×m m ®Ó hÖ sau cã nghiÖm:  x 2  (m  2)x  2m  0 1-(HVQH Quèc tÕ-1997)   x  (m  7)x  7m  0 2  x 2  2x  1  m  0  x 2  (m  2)x  2m  0 2-(§H Th−¬ng m¹i-1997)  3-   x  (2m  1)x  m  m  0  x  (m  3)x  3m  0 2 2 2  x 2  2mx  0 4-(§H Thuû lîi-1998)   x  1  m  2m  x 2  3x  4  0 5-(§H Th−¬ng m¹i-1998)   x  3x x  m  15m  0 3 2 T×m m ®Ó hÖ sau v« nghiÖm:  x 2  1  0  x 2  6x  5  0  x 2  7x  8  0 1-  2-  3-  (m  x )(x  m)  0  x  2(m  1)x  m  1  0 m x  1  3  (3m  2)x 2 2 2 2 T×m m ®Ó hÖ sau cã nghiÖm duy nhÊt:  x 2  3x  2  0  x  2x  a  0 2 1-  2-   x  6x  m(6  m)  0  x  4x  6a  0 2 2  x 2  (2m  1)x  m 2  m  2  0 3-   x  5x  4  0 4 2 B- HÖ bpt hai Èn sè: T×m a ®Ó hÖ sau cã nghiÖm: 16
WWW.VNMATH.COM  x  y  2  x 2  y 2  2x  2 1-(§HGTVT-2001)  2-   x  y  2x(y  1)  a  2 x  y  a  0 4x  3y  2  0 3-  2 x  y  a 2 T×m a ®Ó hÖ cã nghiÖm duy nhÊt:  x 2  y 2  2x  1  x  y  2xy  m  1 1-  2-  x  y  a  0  x  y  1 17