Bi Gia Phong Gio vin trng THPT Trng Vnh K Bn tre. http://giaphong.schools.officelive.com/Trang 1 TI LIU N THI I HC. MUC LUC. PHUONG TRNH LUONG GIC. BI TAP: I/ CC BI TAP CUNG C KIEN THUC.Trang 02. II/ CC BI TAP RN LUYEN K NNG.Trang 02. III/ CC BI TAP C CCH GIAI DAC BIET.Trang 03. IV/ BI TAP TRONG CC DE THI DAI HOC.Trang 03. HUNG DAN GIAI BI TAP: I/ CC BI TAP CUNG C KIEN THUC.Trang 04. II/ CC BI TAP RN LUYEN K NNG.Trang 05. III/ CC BI TAP C CCH GIAI DAC BIET.Trang 07. IV/ BI TAP TRONG CC DE THI DAI HOC.Trang 09. PHUONG TRNH, BT PHUONG TRNH, HE PHUONG TRNH DAI S. BI TAP: I/ PHUONG TRNH & BT PHUONG TRNH.Trang 10. II/ HE PHUONG TRNH.Trang 12. HUNG DAN GIAI BI TAP: I/ PHUONG TRNH & BT PHUONG TRNH.Trang 14. II/ HE PHUONG TRNH.Trang 17. Bi Gia Phong Gio vin trng THPT Trng Vnh K Bn tre. http://giaphong.schools.officelive.com/Trang 2 PHUONG TRNH LUONG GIC (N THI DAI HOC). I/ CC BI TAP CUNG C KIEN THUC. Giai cc phuong trnh sau dy: 1)sinx sin2x sin3x 0 + + = . 2)cos x.cos3x sin2x.sin6x sin4x.sin6x 0 = . 3) 2 2 23sin x sin 2x sin 3x2+ + = . 4) 4 4sin x cos x 1(tan x cot x)sin2x 2+= + . 5) 4 4sin x 7cos x 4 0 + = . 6)sin2x 2cot x 3 + = . 7) 3 2 3sin x 2sin x.cos x 3cos x 0 + = . 8)cos7x cos5x 3(cos5x sin7x) = . 9)3cos x cos2x cos3x 1 2sin x.sin2x + + = . 10) 4 43 cos6xsin x cos x4+ = . II/ CC BI TAP RN LUYEN K NNG. Giai cc phuong trnh sau dy: 1) 4 6cos2x sin x 8cos x + = . 2) 6 4 4 6sin x sin x cos x cos x = . 3) 8 81sin x cos x8+ = . 4)3 cos x 1 cos x 2 + = . 5) 32sin x cos2x cos x 0 + =6) 2 22tan x 3tan x 2cot x 3cot x 3 0 + + = . 7) 32sin4x 16sin x.cos x 3cos2x 5 + + = . 8) 2sin2x 2cos x 1cos xcos x sinx cos3x sin3x+ = +. 9) 2(1 t anx)sin x 3(cos x sinx)sinx 3 + = + . 10) 1 1 2cos x sin2x sin4x+ = . 11) 2 22cot x tan x32cos 2xcos2x= . 12) 3231 cos xtan x1 sin x=. Bi Gia Phong Gio vin trng THPT Trng Vnh K Bn tre. http://giaphong.schools.officelive.com/Trang 3 III/ CC BI TAP C CCH GIAI DAC BIET. Giai cc phuong trnh sau dy: 1) 7 8cos x sin x 1 + = . 2) 10 10 4 42 2sin x cos x sin x cos x2 2cos 2x sin 2x+ +=+. 3) 2 2 21sin x sin 3x sinx.sin 3x4+ = . 4) 2 23 3 23 3x 1 x 1 81sin cos cos 4xx x2 2 4sin cos2 2| | | | ||+ + + = || ||\ . \ .. 5)6sin x 3sin2x 8 + = . 6) 252sin x.sin2x cos x4= + . 7) 24 2cos4x cos6x 4cos 2x cos10x + + = . IV/ BI TAP TRONG CC DE THI DAI HOC. 1)Tmnghimthuckhoang(0; 2 ) cuaphuongtrnh: cos3x sin3x5 sinx 3 cos2x1 2sin 2x+ | |+ = + |+\ ..(KhiA nm 2002). 2)Tmnghimthucdoan[0; 14] nghimdngphuongtrnh:cos3x 4cos 2x 3cosx 4 0 + = .(KhiD nm 2002). Giai cc phuong trnh sau dy: 3) 2 2 2 2sin 3x cos 4x sin 5x cos 6x = . (Khi B nm 2002) 4) 2cos2x 1cot x 1 sin x sin 2x1 t anx 2 = + +. (Khi A nm 2003) 5) 2cot x tan x 4sin 2xsin 2x + = . (Khi B nm 2003) 6) 2 2 2x xsin . tan x cos 02 4 2 | | = |\ .. (Khi D nm 2003) 7) 25sin x 2 3(1 sin x) tan x = . (Khi B nm 2004) 8)(2cos x 1)(2sin x cos x) sin 2x sinx + = . (Khi D nm 2004) 9) 2 2cos 3x.cos2x cos x 0 = . (Khi A nm 2005) 10) 1 sin x cos x sin 2x cos2x 0 + + + + = . (Khi B nm 2005) 11) 4 43cos x sin x cos x .sin 3x 04 4 2 | | | |+ + = ||\ . \ .. (Khi D nm 2005) 12) 6 62(cos x sin x) sinx.cos x02 2sin x+ =. (Khi A nm 2006) 13) xcot x sin x 1 tan x. tan 42| |+ + = |\ .. (Khi B nm 2006) 14) cos3x cos2x cos x 1 0 + = . (Khi D nm 2006) 15) ( ) ( )2 21 sin x cos x 1 cos x sin x 1 sin 2x + + + = + . (Khi A nm 2007) 16) 22sin 2x sin 7x 1 sinx + = . (Khi B nm 2007) 17) 2x xsin cos 3 cos x 22 2| |+ + = |\ .. (Khi D nm 2007) Bi Gia Phong Gio vin trng THPT Trng Vnh K Bn tre. http://giaphong.schools.officelive.com/Trang 4 I/ CC BI TAP CUNG C KIEN THUC. 1)sinx sin 2x sin3x 0 + + =(sinx sin3x) sin 2x 0 + + =2sin 2x.cos x sin 2x 0 + =sin 2x(2.cos x 1) 0 + =sin 2x 0(2.cos x 1) 01cos x2=

+ =

=

x k22x k23 =

= +

(k Z) 2) 1cos x.cos3x (cos4x cos2x)2= + ; 1sin 2x.sin6x (cos4x cos8x)2= ; 1sin 4x.sin6x (cos2x cos10x)2= . PT d cho tuong duong voi:cos8x cos10x 0 + =2cos 9x.cos x 0 =cos x 0cos9x 0=

=

x k2x k18 9 = +

= +

. 3) 2 2 23sin x sin 2x sin 3x2+ + =1 cos2x 1 cos4x 1 cos6x 32 2 2 2 + + =cos2x cos4x cos6x 0 + + =cos4x(2cos 2x 1) 0 + =cos4x 01cos2x2=

=

x k8 4x k3 = +

= +

. 4)Diu kin:sinx.cos x 0 sin 2x 0 . Voi diu kin trn, phuong trnh d cho tuong duong voi: 211 sin 2x12sin 2x sin 2x= sin 2x 0 =(khng thoa diu kin)phuong trnh v nghim. 5)Dt 2t sin x = (0 t 1) 28t 14t 3 0 + =1t4=(3t2=loai).

21sin x4=1sin x2= x k265x k26 = +

= +

. 6)Diu kin:sinx 0 . Nucos x 0 =cos xcot x 0sin x= =Khng thoa. Khisinx 0 vcos x 0 :sin 2x 2cot x 3 + =22tan x 231 tan x tan x+ =+ 3 23tan x 4tan x 3tan x 2 0 + =

2(tan x 1)(3tan x tan x 2) 0 + =tan x 1 =x k4= + . 7) 3 2 3sin x 2sin x.cos x 3cos x 0 + =( cos x 0 =khng thoa; chia hai v cho 3cos x )

3 2tan x 2tan x 3 0 + = 2(tan x 1)(tan x tan x 3) 0 + + = tan x 1 =x k4= + . 8)cos7x sin5x 3(cos5x sin7x) = cos7x cos 5x 3 cos5x cos 7x2 2| | | | = ||\ . \ . 2sin x .sin 6x 2 3sin x .sin 6x4 4 4 4 | | | | | | | | + = + ||||\ . \ . \ . \ . sin 6x sin x 3sin x 04 4 4 | | | | | | + + = |||\ . \ . \ .sin 6x sin x 3 cos x 04 4 4 | | | | | | + + = |||\ . \ . \ . sin 6x sin x .cos sin .cos x 04 4 3 3 4 | | | | | | + + = |||\ . \ . \ .

Bi Gia Phong Gio vin trng THPT Trng Vnh K Bn tre. http://giaphong.schools.officelive.com/Trang 5

sin 6x 04sin x 012 | | = | \ .

| | =|\ . 6x k4x k12 = +

=

x k24 6x k12 = +

= +

. 9)3cos x cos 2x cos3x 1 2sin x.sin 2x + + = 23cos x (2cos x 1) cos3x 1 cos x cos3x + + =

22cos x 2cos x 0 + =cos x(1 cos x) 0 + = cos x 0cos x 1=

=

x ( /2) kx (2k 1)= +

= +

(Hoc giai bng cch p dung : 3cos3x 4cos x 3cos x = ;sin 2x 2sin x.cos x = ) 10) ( )24 4 2 2 2 21 1 1 cos4x 3 cos4xsin x cos x sin x cos x 2sin x.cos2x 1 sin 2x 12 2 2 4 + | |+ = + = = = |\ .. 4 43 cos6xsin x cos x4+ =cos4x cos6x = cos4x cos( 6x) =

4x 6x k24x 6x k2= +

= + +

10x k22x k2= +

= +

x k10 5x k2 = +

= +

. II/ CC BI TAP RN LUYEN K NNG. 1) 4 6cos2x 4sin x 8cos x + =2 31 cos2x 1 cos2xcos2x 4 82 2 + | | | |+ = ||\ . \ .. Dtt cos2x = ( t 1)

2t(t 2t 4) 0 + + =t 0 =cos2x 0 =2x k2= + x k4 2 = + . 2)Cch 1: 6 4 4 6sin x sin x cos x cos x = 4 2 4 2sin x(sin x 1) cos x(1 cos x) =

2 2 2 2sin x.cos x(cos x sin x) 0 + =21sin 2x 02| |= |\ . sin 2x 0 =2x k = x k2=3) 8 81sin x cos x8+ =( )24 4 4 41sin x cos x 2sin x.cos x8+ =

( )222 2 2 2 41sin x cos x 2sin x.cos x 2(sinx.cos x)8 + =

2 421 1 11 sin 2x 2 sin 2x2 2 8| | | | = ||\ . \ . Dt 2t sin 2x = (0 t 1) 221 1 11 t 2 t2 16 8| | = |\ . 2t 8t 7 0 + =t 1 = ( t 7 =loai)

2sin 2x 1 =sin 2x 1 = 2x k2= + x k4 2 = + . 4)3 cos x 1 cos x 2 + =( ) ( )2 23 cos x 2 1 cos x = + + 3 cos x 5 4 1 cos x cos x = + + +2 1 cos x 1 cos x + = 21 cos x 04(1 cos x) (1 cos x) + = + cos x 1 = x (2k 1) = + . 5) 32sin x cos2x cos x 0 + =3 22sin x 1 2sin x cos x 0 + + =22sin x(sin x 1) cos x 1 0 + + =

22(1 cos x)(sin x 1) (1 cos x) 0 + =[ ] (1 cos x) 2(sin x 1)(1 cos x) 1 0 + + = [ ] (1 cos x) 1 2sin x.cos x 2(sin x cos x) 0 + + + =2(1 cos x) (sin x cos x) 2(sin x cos x) 0 + + + = (1 cos x)(sin x cos x)(sin x cos x 2) 0 + + + =cos x 1sin x cos xsin x cos x 2=

=

+ =

cos x 1tan x 1=

=

x k2x k4=

= +

. Bi Gia Phong Gio vin trng THPT Trng Vnh K Bn tre. http://giaphong.schools.officelive.com/Trang 6 6) 2 22tan x 3tan x 2cot x 3cot x 3 0 + + =

2 22(tan x cot x) 3(tan x cot x) 3 0 + =22(tan x cot x) 3(tan x cot x) 1 0 + =

2t tan x cot x2t 3t 1 0= + = t 1t 1/2=

=

( )( )tan x 1 5 / 2tan x 1 17 / 2

=

=

x arc tan (1 5)/2 kx arctan (1 5)/2 k

= +

= +

. 7) 32sin 4x 16sin x.cos x 3cos 2x 5 + + =22sin 4x 8sin x.sin 2x 3cos 2x 5 + + =

1 cos2x2sin 4x 8 sin 2x 3cos 2x 52 | |+ + = |\ . 2sin 4x 4sin 2x 2sin 4x 3cos 2x 5 + + =4sin 2x 3cos 2x 5 + =4 3sin 2x cos 2x 15 5+ =(v 2 24 315 5| | | |+ = ||\ . \ .; dt 4 3cos ; sin5 5 = = ) sin(2x ) 1 + =2x k22+ = + x k4 2 = + . 8)cos x sinx cos3x sin3x (cos x cos3x) (sin3x sinx) 2sinx(sin2x cos2x) + = + = +2sin 2x 2cos x 1cos xcos x sinx cos3x sin3x+ = + 2sin 2x 2cos x 1cos x (1)2sinx(sin2x cos2x)+ =+. Diu kin: sinx 0sin 2x cos2x 0 + Voi diu kin trn, PT (1) tuong duong voi: sin 2x cos 2xcos x2sinx(sin2x cos2x)+=+ 1 2sinx cos x =sin 2x 1 =x k4= + (thoa diu kin). 9) 2(1 t anx) sin x 3(cos x sinx) sinx 3 + = +Voi DK:cos x 0 , PT d cho tuong duong voi: 2 2(1 t anx) tan x 3(1 t anx) t anx 3(1 tan x) + = + + 3 2t t 3t 3 0t t anx =

=

2(t 1)(t 3) 0t t anx + =

=

t anx 1tan x 3=

=

x (a/4) kx (a/3) k= +

= +

. 10) 1 1 2cos x sin 2x sin 4x+ =Voi DK:sin 4x 0 , PT d cho tuong duong voi: 2sin x 1 2sin 2x 2sin 2x cos 2x+= (2sin x 1)cos2x 1 + = 2sin x cos 2x 1 cos2x = 22sin x cos 2x 2sin x =sinx(cos2x sinx) 0 = cos2x sinx 0 = (vDKsin x 0 )22sin x sinx 1 0 + = 1sinx2=( sinx 1 = loai vcos x 0 = ) x (a/6) k2x (5a/6) k2= +

= +

. (c th giai:cos2x sinx 0 =cos2x sinx =cos2x cos x2 | |= |\ .) 11) 2 22cot x tan x32cos 2x (1)cos2x= . Diu kin: sinx 0cosx 0cos2x 0 sin 4x 0 . Voi diu kin trn, PT (1) tuong duong voi: 4 422 2cos x sin x32cos 2xsin x.cos x.cos2x=

2 222 2cos x sin x32cos 2xsin x.cos x.cos2x=22 2132cos 2xsin x.cos x=2 2 232cos 2x.sin x.cos x 1 =

22132cos 2x sin x.cos x 12| |= |\ . 2 28cos 2x.sin 2x 1 =21sin 4x2=1 cos8x 12 2=cos8x 0 =x k16 8 = + . Bi Gia Phong Gio vin trng THPT Trng Vnh K Bn tre. http://giaphong.schools.officelive.com/Trang 7 12) 3231 cos xtan x1 sin x=.Diukin: cosx 0sinx 1 cos x 0 .Voidiukincos x 0 ,phuongtrnhdcho tuong duong voi: 2 32 31 cos x 1 cos x1 sin x 1 sin x = 221 cos x 1 cos x 1 cos x cos x01 sin x 1 sin x 1 sin x sin x| | + + + | | = || + + +\ .\ .

2 2(1 cos x)(1 cos x)(1 sin x sin x) (1 sin x)(1 cos x cos x) 0 + + + + + + + =(1 cos x)(cos x sin x)(sin x cos x sin x.cos x) 0 + + =1 cos x 0cos x sin x 0sin x cos x sin x.cos x 0 =

=

+ + =

1 cos x 0cos x sin x 0 =

=

hoc 2t sinx cos x ( t 2)t 2t 1 0= + + = cos x 1tan x 1=

=

hoct 2 sin x 1 24 | |= + = |\ .

x k2x k4=

= +

hocx k243x k24 = +

= +

(voi 1 2sin2 + = ) III/ CC BI TAP C CCH GIAI DAC BIET. 1) 7 8cos x sin x 1 + = . Ta c: 5cos x 1 7 2cos x cos x ; 6sin x 1 8 2sin x sin x

7 8 2 2cos x sin x cos x sin x 1 + + = . 7 8cos x sin x 1 + =7 28 2cos x cos xsin x sin x == 2 52 6cos x(1 cos x) 0sin x(1 sin x) 0 = =

22 6cos x 0sin x(1 sin x) 0 = = hoc 52 6cos x 1sin x(1 sin x) 0 = =

cosx 0sinx 0= = hoc cosx 0sinx 1= = hoc cosx 1sinx 0= = hoc cosx 1sinx 1= = x k2x k2 = +

=

. 2) 10 10 4 42 2sin x cos x sin x cos x2 2cos 2x sin 2x+ +=+ 210 102 211 sin 2xsin x cos x22 2(1 sin 2x) sin 2x+= +

210 1022 sin 2xsin x cos x2 sin 2x+ = 10 10sin x cos x 1 + =(Giai tuong tu bi tp 5). Hoc giai tip: 2 5 2 5(sin x) (1 sin x) 1 + = . Dt 2t sin x = (0 t 1)

5 2 3 4 5t (1 5t 10t 10t 5t t ) 0 + + + =2t(t 1)(t t 1) 0 + =t 0t 1=

=

sinx 0sinx 1=

=

x k2= . 3)Cch 1: 2 2 21sin x sin 3x sinx.sin 3x4+ =2 2 4 4 24sin x 4sinx.sin 3x sin 3x sin 3x sin 3x 0 + + =

2 2 2(2sin x sin 3x) (1 sin 3x) sin 3x 0 + =22 22sin x sin 3x 0(1 sin 3x) sin 3x 0 = =

22sin 3x 02sin x sin 3x 0 = = hoc 22sin 3x 12sin x sin 3x 0 = = sin3x 0sin x 0= = hoc 2sin 3x 11sin x2 == Bi Gia Phong Gio vin trng THPT Trng Vnh K Bn tre. http://giaphong.schools.officelive.com/Trang 8 sinx 0 =hoc 1sin x2=x k = hoc x k265x k26 = +

= +

. Cch 2: 2 2 21sin x sin 3x sinx.sin 3x4+ =2 2 21sin x sin 3x.sinx sin 3x 0 (1)4+ + =4 2 2 2sin 3x sin 3x sin 3x(sin 3x 1) 0, x R = = . (1) 201sinx sin 3x2 = = 22sin 3x 01sinx sin 3x2 == hoc22sin 3x 11sinx sin 3x2 ==(tuong tu cch 1) 4)Diu kin: xsin 02xcos 02 sinx 0 . 2 23 3 6 63 3 6 6x 1 x 1 x x 1 1VT sin cos sin cos 4x x x x2 2 2 2sin cos sin cos2 2 2 2| | | | | | |||| |= + + + = + + + + ||||\ . |||\ . \ . \ . 6 6 2 266 6x x 1 x x 64VT 4 sin cos 1 4 1 3sin cos 1x x2 2 2 2 sin xsin cos2 2| | || | | || |= + + + = + + ||| |\ . \ .\ . |\ . (v 36 6 2 2 2 2 2 2x x x x x x x xsin cos sin cos 3sin cos 1 3sin cos2 2 2 2 2 2 2 2| |+ = + = |\ .v 6 6 6x x 1sin cos sin x2 2 64= )

3 81VT 4 1 (1 64)4 4| | + + = |\ .; 281 81VP cos 4x4 4= . Phuong trnh d cho tuong duong voi: 81VT481VP4== 22sin x 1cos 4x 1 == 2x k2cos 4x 1 = + = x k2= + . 5)Xt hm sf (x) 6sin x 3sin 2x = + . f(x) l hm s tun hon voi chu k2 nn ta chi cn xt trn doan[ ; ] . 2f '(x) 6cos x 6cos 2x 6(2cos x cos x 1) = + = + . f '(x) 0 =cos x 11cos x2=

=

xx3=

=

f ( ) 0 = ;f ( ) 0 = ; 9 3f3 2 | | = |\ .; 9 3f3 2 | | = |\ . 9 3max f (x) 82= . 13) 2 2x 17 x x 17 x 9 + + = . 14) 32 x 1 x 1 = . 15) ( )3 3 3 3x 35 x x 35 x 30 + = . 16)10 x 3 x (10 x)(3 x) 11 + + + + = . 17) 3 3 32x 3 x 1 3x 7 + + + = + . 18) 2 23 3 3(2 x) (7 x) (7 x)(2 x) 3 + + + = . 19)2x x 1 1 2x x 1 2 x 1 1 + + + + + = + + . 20)x x 5 x 7 x 16 14. + + + + + =21) 2x 2 4 x x 6x 11 + + . 22)Tm m d phuong trnh: 2x mx 2 2x 1 + + = +c hai nghim thuc phn bit. 23) Cho phuong trnh:2 x 6 x (2 x)(6 x) a + + + = (1). a) Giai phuong trnh khi a = 4. b) Xc dinh a d phuong trnh (1) c nghim. 24) Cho bt phuong trnh: 24 (4 x)(2 x) x 2x a 18 + + (1). a) Giai bt phuong trnh khi a = 6. b) Xc dinh a d bt phuong trnh (1) duoc nghim dng voi moi x [2 ; 4]. 25) Xc dinh m d phuong trnh sau c nghim: ( )2 2 4 2 2m 1 x 1 x 2 2 1 x 1 x 1 x + + = + + . Bi Gia Phong Gio vin trng THPT Trng Vnh K Bn tre. http://giaphong.schools.officelive.com/Trang 11 II/ HE PHUONG TRNH. 1)Giai cc h phuong trnh sau dy: a) 2 22x 5xy 6y 04x 2xy 6x 27 0 + =+ + =b)2 22 2x 2xy 3y 92x 2xy y 2 + + =+ + =c) 2 22 22x 3xy y 12x xy 3y 11 + + = + = 2)Giai cc h phuong trnh sau dy: a) 2 2xy 4x y 28=+ =b) 3 3x y 1x y 61+ =+ = c) 4 4x y 5x y 97+ =+ = d) 2 24 4x xy y 3x y 17 + + =+ = e) 3 34 4x y 1x y 1 + =+ = f) 9 94 4x y 1x y 1 + =+ =

3)Giai cc h phuong trnh sau dy: a) 2 2x xy y 13x y 2 + =+ = b) 2 2x y xy 11x y 3(x y) 28+ + =+ + + = c) 2 22 2 2y xy 6x1 x y 5x + =+ =d) 2 2xy x y 11x y xy 30+ + =+ = e) 2 24 4 2 2x y xy 7x y x y 21 + + =+ + =f) 2 2x y x y 4x(x y 1) y(y 1) 2 + + + =+ + + + = 4)Giai cc h phuong trnh sau dy: a) 2 2x x y y 18x(x 1)y(y 1) 12 + + + =+ + = b) 2 22 2x y 3x 4y 13x y 9x 8y 3 + + = = c) 2 2 22 3x y 2x y 02x 4x 3 y 0 + = + + = d) xx y 5yx(x y) 6y+ + =+ =e) 2 32 2x x12y yx y xy 6| | | |+ = ||\ . \ .+ = f) 2 22 21(x y) 1 5xy1(x y ) 1 49x y| |+ + =| \ .| |+ + = |\ .g) 2 2 2 2(2x y) 5(4x y ) 6(2x y) 012x y 32x y + + =+ + = h) 31 1x yx y2y x 1 = = +i) 1 32xy x1 32yx y+ =+ = j) 2222y 23yxx 23xy +=+= k) 2 2 2xy x 1 7yx y xy 1 13y+ + =+ + = l) 2 22 21 1x y 4x y1 1x y 4x y+ + + =+ + + =m) 22x(x y 1) 3 05(x y) 1 0x+ + =+ + = Bi Gia Phong Gio vin trng THPT Trng Vnh K Bn tre. http://giaphong.schools.officelive.com/Trang 12 5)Giai cc h phuong trnh sau dy: a) 2x y 2 y x 13 2x y x y 10 = + =b) 2 2x 2xy 3y 0x x y y 2 + =+ = c) 44x y 1 1y x 1 1+ =+ = d) 36 3x y 15x 8x y 2y 2 = + =e)x y y x 30x x y y 35+ =+ = f) 4 42 2 2 2x y 144x y x y y =+ = g) 3x y x yx y x y 2 = + = + + h) x 5 y 2 7x 2 y 5 7+ + = + + = i) x y 71y x xyx xy y xy 78+ = ++ = 6)Giai cc h phuong trnh sau dy: a) 2x y z 19xy yz zx 114xz y + + =+ + ==b) x y 2xy1 yz 2y1 zx 2x+ =+ =+ = c) x y z 9xy yz zx 271 1 11x y z+ + =+ + = + + =d) 2 2 2x y z 6xy yz zx 7x y z 14 + + =+ =+ + =e) x xy y 1y yz z 4z zx x 9+ + =+ + =+ + = 7)Giai cc h phuong trnh:a) 3x 2x x 1x2 5y 4y4 2y2 2+ = += +b) 2 2x y x 1x y y x2 2 x y+ + = + = c) 3x 1 y 2 y 3x22 2 3.23x 1 xy x 1+ + + =+ + = +d) ( )2 39 3x 1 2 y 13log 9x log y 3 + = = 8)Giai h phuong trnh sau dy: 2 2 22x y zxyz 64 = += Voi diu kin ba s: logyx; logzy; logxz theo thu tu d tao thnh mt cp s nhn. 9)Cho h phuong trnh: 2 2x y 4x y m+ =+ = . Dinh m d: a) H phuong trnh v nghim. b) H phuong trnh c nghim duy nht. c) H phuong trnh c hai nghim phn bit.10)Dinh m d h phuong trnh: 5(x y) 4xy 4x y xy 1 m+ =+ = c nghim. 11)Tm m d h phuong trnh sau c nghim : x y 1x x y y 1 3m+ =+ = . Bi Gia Phong Gio vin trng THPT Trng Vnh K Bn tre. http://giaphong.schools.officelive.com/Trang 13 12)Dinh m d h phuong trnh: x 4 y 1 4x y 3m + =+ = c nghim. 13)Cho h phuong trnh: 2 2x xy y m 2x y y x m 1+ + = ++ = +.a) Giai h d cho khi m = 3. b) Dinh m d h c nghim duy nht. 14)Cho h phuong trnh: (I)2x xy y 2m 1xy(x y) m m+ + = ++ = +

a) Chung minh rng voi moi m, h phuong trnh (I) lun c nghim. b) Xc dinh m d h phuong trnh d c nghim duy nht. 15)Cho h phuong trnh: 2 2 2x y m 1x y y x 2m m 3+ = ++ = .a) Giai h d cho khi m = 3. b) Dinh m d h c nghim.16)Cho h phuong trnh: 2 2x y x y 8xy(x 1)(y 1) m + + + =+ + =. a) Giai h d cho khi m = 12. b) Dinh m d h c nghim. 17)Cho h phuong trnh: 22xy y 12x xy 26 m = = + a) Giai h phuong trnh voi m = 2. b) Voi gi tri no cua m th h phuong trnh d cho c nghim ? 18)Tm m d h phuong trnh: 3 2 23 2 2x y 7x mxy x 7y my = + = + c nghim duy nht. 19)Xc dinh tham s a d h phuong trnh: 22(x 1) y a(y 1) x a + = ++ = + c nghim duy nht.20)Xc dinh tham s a d h phuong trnh: 22 2x 3 y ay 5 x x 5 3 a+ + =+ + = + + c dng mt nghim. 21)Xc dinh cc tham s a, b d h phuong trnh: 22 2 2xyz z axyz z bx y z 4+ =+ =+ + =c nghim duy nht. 22)Tm m d h bt phuong trnh: 2x x 1 2 x 125 5 2011x 2011x (m 2)x 2m 3 0+ + + + + + + + c nghim.

Bi Gia Phong Gio vin trng THPT Trng Vnh K Bn tre. http://giaphong.schools.officelive.com/Trang 14 I/ PHUONG TRNH & BT PHUONG TRNH. Huong dn giai & dp s: 1) 2(x 5)(2 x) 3 x 3x + = +2 2x 3x 10 3 x 3x (1) + = + . DK:22x 3x 10 0x 3x 0 + + 5 x 30 x 2

Voi DK trn (1) tuong duong voi: 2 2(x 3x) 10 3 x 3x + + = +22t x 3x 0t 3t 10 0= + + =

2t x 3x 0t 2 = + = ( t 5 = loai) 2t x 3x 2 = + =2x 3x 4 0 + =x 4x 1=

=

. 2) ( )2 2x 3x . 2x 3x 2 0 2x 3x 0 =hoc 22x 3x 2 0 =hoc 22x 3x 02x 3x 2 0 > =

1x2 hocx 2 =hocx 3 . 3)x 3 7 x 2x 8 + x 3 2x 8 7 x (1) + + . Diu kin:4 x 7 . Voi DK trn (1) tuong duong voi: ( ) ( )2 2x 3 2x 8 7 x + + 2 (2x 8)(7 x) 2x 11x 30 0 + x 5 hocx 6 . Kt hop voi DK(1) c tp nghim:S [4; 5] [6; 7] = . 4)1 x 1 x x + 1 x x 1 x (1) + + . DK:1 x 1 . Voi DK trn (1) tuong duong voi: ( )x 2 x 2 1 x 0 (2) x = 1 l nghim. 1 x 0 < : (2) 2 1 x 2 x 2x 0 (khng thoa) 0 x 1 < : (2) 2x 0 (thoa) Tp nghim cua (1):S [0; 1] = . 5)x x 1 m = x m x 1 = + 22m x 1 1 m = 21 m 0 < m > 1 (v m > 0)PT v nghim. 21 m 0 = m = 1 (v m > 0)x = 1. 21 m 0 >0 m 1 < |\ .. 6) 221 x x x 1 x (1)3+ = + . DK:0 x 1 . Cch 1: Voi DK trn (1) tuong duong voi: 2 24 41 x x (x x ) x 2 x(1 x) 1 x3 9+ + = + +

2 22x x x x 1 03| | = |\ .Tp nghim cua (1):S {0; 1} = . Cch 2: Voi DK trn (1) tuong duong voi: 2 2u x 0; v 1 x 0u v 121 u.v u v3 = = + = + = +. 7) 2 2 2x 3x 2 x 4x 3 2 x 5x 4 + + + + .(x 1)(x 2) (x 1)(x 3) 2 (x 1)(x 4) (1) + . DK: x 1x 4

. x = 1 l nghim. x 1 < 2 x 3 x 2 4 x (2) + . V2 x 4 x < v3 x 4 x < nn 2 x 3 x 2 4 x + < x < 1 khng phai l nghim cua (2). Bi Gia Phong Gio vin trng THPT Trng Vnh K Bn tre. http://giaphong.schools.officelive.com/Trang 15 x 4 x 2 x 3 2 x 4 (3) + . Vx 2 x 4 vx 3 x 4 nn x 2 x 3 2 x 4 + x 4 nghim cua (3). Tp nghim cua (1):S {1} [4; + ) = . 8) 2 2 2x 8x 15 x 2x 15 4x 18x 18 + + + +(x 3)(x 5) (x 3)(x 5) 2(x 3)(2x 3) (1) + + . DK: x 5x 3x 5

=

. x = 3 l nghim. x 5 x 5 x 5 2(2x 3) + + 17x3 Nghim: 175 x3 . x 5 17x3 Nghim:x 5 . Tp nghim cua (1):S ( ; 5] {3} [5; 17/3] = . 9)2 x 2 2 x 1 x 1 4 + + + + =( )22 x 1 1 x 1 4 + + + =x 3 = . 10)x 1 2 x 2 x 1 2 x 2 1 + =( ) ( )2 2x 2 1 x 2 1 1 + = x 2 1 x 2 1 1 + = x 2 1 x 2 1 1 + = x 2 1 x 2 = x 2 1 x 2 = 1x 22 =9x4= . 11)Tuong tu bi 6. Tp nghim:S [2; ) = + . 12) 22(x 16) 7 xx 3 (1)x 3 x 3 + > . Voi DK:x 4 , (1)22(x 16) 10 2x (2) > Nux 5 th (2) dngNghimx 5 . Nu4 x 5 th (2)2x 20x 66 0 + (*) c hai nghim phn bitDuong thngy m =ct (C) tai hai dim phn bit m 9/2 . 23)Giai b):2 x 6 x (2 x)(6 x) a (1) + + + = . DK:2 x 6 . Bi Gia Phong Gio vin trng THPT Trng Vnh K Bn tre. http://giaphong.schools.officelive.com/Trang 17 0210 766 0f '(t)f(t)tDt t =2 x 6 x + + voi[ ] x 2; 6 1 1 6 x 2 xt '2 2 x 2 6 x 2 2 x 6 x += =+ + . t = 0x = 2 min t =2 2< t < max t = 42t 8 2 (2 x)(6 x) = + + .(1)2ta t 4 f (t)2= + + =Xt hm s 2tf (t) t 42= + +trn doan2 2; 4 f (t) = t +1.f (t) =0t =1 0 f (t) 2 2 (1) c nghim a 0; 2 2 . 24)Giai b): 24 (4 x)(2 x) x 2x a 18 (1) + + . Dtt (4 x)(2 x) = + ( 2 x 4 0 t 3) . (1)2t 4t 10 a + .Xt hm s:[ ] ( )2f (t) t 4t 10 t 0; 3 = + f '(t) 2t 4 = . f '(t) 0 = t = 2 [ ] 7 f (t) 10, t 0; 3 (1) nghim dng voi moi x [2 ; 4] [ ] f (t) a, t 0; 3 a 10 . 25) ( )2 2 4 2 2m 1 x 1 x 2 2 1 x 1 x 1 x (1) + + = + + . DK:1 x 1 . Dt 2 2t 1 x 1 x = + ( )2 22 2 4x 1 x 1 xx xt '1 x 1 x 1 x + += + =+ ;t ' 0 =x 0 = . x 0 =t 0 = .x 1 = t 2 =0 t 2 . Xt hm s: 2t t 2f (t)t 2 + +=+ 22t 4tf '(t)(t 2) +=+;f '(t) 0 =t 0 =( t 4 = loai)2 1 f (t) 1 PT (1) c nghim 2 1 m 1 . II/ HE PHUONG TRNH. Huong dn giai & dp s: 1)a) 2x 2y14y 6y 9 0= + = hoc 2x 3y14y 6y 9 0= + = Nghim (x = 3; y = 3/2),(x = 9/5; y = 9/10), 9 9 15 3 3 15x ;y14 14| | = = | |\ .. 1b)( ) ( )2 2 2 22 x 2xy 3y 9 2x 2xy y + + = + +2 216x 14xy 3y 0 + + =

x 3y / 8x y / 2=

=

Nghim ( 1; 2), ( )3/ 17; 8/ 17 . 1c) Nghim (x = 1; y = 2), ( )5/ 3; 1/ 3 . 2)a) Dt S = x + y; P = xy. Nghim ( )3 5;3 5 . 2b)(5; 4) ,( 4; 5) . 2c) ( )2 24 4 2 2 2 2 2 2 2 2 2 2x y x y 2x y (x y) 2xy 2x y (25 2xy) 2x y + = + = + = 2(xy) 50xy 264 0 + = . Nghim(2; 3) ,(3; 2) . Bi Gia Phong Gio vin trng THPT Trng Vnh K Bn tre. http://giaphong.schools.officelive.com/Trang 18 2d) 2 22 2 2 2 2x y 3 xy(x y ) 2x y 17 + = + =. Nghim(1; 2) ,(2; 1) . 2e) 3 34 4x y 1(1)x y 1(2) + =+ =.(2) x 1, y 1 . Nu x < 0: (1) 3 3y 1 x 1 = > y > 1 (v l) 0 x 1,0 y 1. (1) (2)3 4 3 4(x x ) (y y ) 0 + = 3 3x (1 x) y (1 y) 0 + =(*) 33x (1 x) 0y (1 y) 0 = = (0; 0), (1; 1), (1; 0), (0; 1). Th vo hNghim (0; 1), (1; 0). 2f) tuong tu 2e)Nghim (0; 1), (1; 0). 3)a) Cch 1: Phuong php th. Cch 2: Dt S = x + y; P = xy. Nghim( 3; 1) ,(1; 3) . 3b) Dt S = x + y; P = xy. Nghim( 3; 7) ,( 7; 3) ,(2; 3) ,(3; 2) . 3c)(1; 2) ,(1/2; 1) . 3d)(1; 5) ,(5; 1) ,(2; 3) ,(3; 2) . 3e)(1; 2) ,(2; 1) ,( 1; 2) ,( 2; 1) . 3f) 2 22 2x y x y 4x y x y xy 2 + + + =+ + + + = xy = 2. Nghim ( )2; 2 . 4)a) 2 2x x y y 18x(x 1)y(y 1) 12 + + + =+ + =. Dt u = x(x + 1); v = y(y + 1).u v 18u v 72+ = + = .Nghim(3; 2) ,(2; 3) ,( 4; 2) ,(2; 4) ,(3; 3) ,( 3; 3) ,( 3; 4) ,( 4; 3) . 4b) 2 22 2x y 3x 4y 13x y 9x 8y 3 + + = = . Dt u = x2 3x; v = y2 + 4y u v 13u 2v 3+ = =

Nghim 3 3; 02| | | |\ ., 3 3; 42| | | |\ .. 4c) 2 2 22 3x y 2x y 02x 4x 3 y 0 + = + + =

2 22 3(x 1)y 2x2(x 1) 1 y 0 + = + + = 2232xy 1x 11 y 0= ++

1 y 1y 1

y = 1; 2x 1 2x + =x = 1 Nghim (1; 1). 4d) xx y 5yx(x y) 6y+ + =+ =. Dt xuy= ; v = xy.u v 1uv 6+ = = Nghim(2; 1) ,(3/2; 1/2) . 4e) 2 32 2x x12y yx y xy 6| | | |+ = ||\ . \ .+ =. Dt xuy= ; v = xy.2 32u u 12v v 16 + =+ = Nghim(2; 1) ,( 2; 1) . 4f) 221 1x y 5x y1 1x y 49x y | || |+ + + =||\ .\ .| || | + + + = ||\ .\ . Dt1x ux1y vy+ =+ =

2 2u v 5u v 49+ = + = Nghim 7 45; 12| | | |\ ., 7 451;2| | | |\ .. Bi Gia Phong Gio vin trng THPT Trng Vnh K Bn tre. http://giaphong.schools.officelive.com/Trang 19 4g) 2 2 2 2(2x y) 5(4x y ) 6(2x y) 012x y 32x y + + =+ + =.Dt u = 2x + y; v = 2x y2 2u 5uv 6v 01u 3v + =+ = DK: v 0 3v 1uv=4 3 26v 15v 14v 6v 1 0 + + = 21(v 1) v (6v 6v 2) 02| | + = |\ . Nghim( ) 3/4; 1/2 ,( ) 3/8; 1/4 . 4h) 31 1 y xx yx y xy2y x 1 = == +

3y x 02y x 1= = + hoc 41yxx x 2 0= + + = 2y x 0(x 1)(x x 1) 0= + = hoc 2 221yx1 1 3x x 02 2 2= | | | | + + + = ||\ . \ . x y 1 = = ; 1 5x y2 = = . 4i) DK: x.y0222x y x 3y2x y y 3x + =+ = (x y)(2xy + 4) = 0 (1; 1) ,( 1; 1) ( 2; 2) . 4j) 2222y 23yxx 23xy +=+=. DK: x > 0, y > 0.2 22 23yx y 2 (1)3xy x 2 (2) = += +. (1) (2) (x y)(3xy x y) 0 + + =( 3xy x y 0 + + =v nghim v x > 0, y > 0). Nghim (1; 1). 4k) 2 2 2xy x 1 7yx y xy 1 13y+ + = + + =. y = 0 khng thoa21 xx 7y y1 xx 13y y| |+ + = |\ .| |+ = |\ . 21 1x x 20 0y yx 17 xy y| | | |+ + + = ||\ . \ .| |= + |\ . Nghim (1; 1/3), (3; 1). 4l) 2 22 21 1x y 4x y1 1x y 4x y+ + + =+ + + =. Tu PT du tin22 2 22 21 1 1 14 x y 4. x y 4.4x y x y| | | |= + + + + + + = ||\ . \ . 1 1x y 1x y= = = =x y 1 = = . Kim trax y 1 = =thoa h Nghimx y 1. = =Cch khc: 2 22 21 1x y 4x y1 1x y 4x y | || |+ + + =||\ . \ .| || |+ + + = ||\ .\ . 221 1x y 4x y1 1x y 8x y | || |+ + + =||\ .\ .| || | + + + = ||\ .\ . .Dt 1u xx1v yy= += +. 4m) 22x(x y 1) 3 05(x y) 1 0x+ + = + + = 223x y 1 0x5(x y) 1 0x+ + =+ + = 223x y 1x3 51 1 0x x+ = | | + = |\ .

23x y 1x4 62 0x x+ = + =

11xx y 2=+ = hoc 1 1x 21x y2=+ = x 1y 1= = hoc x 23y2= = . Bi Gia Phong Gio vin trng THPT Trng Vnh K Bn tre. http://giaphong.schools.officelive.com/Trang 20 5)a) 2x y 3x y 1 = = Nghim x 2y 1= = ;x 2y 1= = ; x 4y 5= = ; x 4y 5= = . 5b) 2 2x 2xy 3y 0x x y y 2 + =+ = x yx x y y 2= + = hoc x 3yx x y y 2= + = x yx x 1= = hoc x 3y1y y4= =

2x y 0x 1= < = hoc 2x 3y 01y4= < = x 1y 1= = hoc x 3/2y 1/2= =. 5c) (I)44x y 1 1y x 1 1+ =+ = . Voi DK: x 1y 1 , h (I) tuong duong voi: (II)44(x 1) y 1 0(y 1) x 1 0 + = + = V x 1 0y 1 0 4(x 1) y 1 0 + v 4(y 1) x 1 0 + . Khi d (II) x 1 0y 1 0 = = x 1y 1= =. 5d) Dt: 3u x ,v y 0 = = . Nghim ( )39; 10 . 5e) DK: x 0y 0> >. DtS x y 0 = + > ;P xy 0 = >2SP 30S(S 3P) 35= =

Nghim(4; 9) ,(9; 4) . 5f) DK: 2 2x yy 0 . 2 2 2 2x y x y y + = 2 4 4 22x 2 x y y =2 2y 2x 24 = Nghim( 2 5; 4) ,( 2 3; 0) . 5g) DK: x y 0x y 0 + .( ) ( )( )2 32x y x yx y x y 2 = + = + +Nghim(1; 1) ,(3/2; 1/2) . 5h) Bnh phuong hai v mi PT ri tru hai PTNghim(11; 11) . 5i) x y 7 xy(x y). xy 78+ = ++ =

( )x y xy 7(x y). xy 78+ =+ = Nghim(4; 9) ,(9; 4) . 6)a) 2x y z 19xy yz zx 114xz y + + =+ + == 22x y z 19xy yz y 114xz y+ + = + + == 2x y z 19y(x y z) 114xz y + + =+ + == x 9y 6z 4= == hocx 4y 6z 9= == 6b) x y 2xy (1)1 yz 2y (2)1 zx 2x (3)+ = + =+ =. (2) (3) z(y x) 2(y x) = (z 2)(y x) 0 = . Nghim x 1y 1z 1= == 6c) DK: x 0y 0z 0 . Khi d h d cho tuong duong voi: x y z 9xy yz zx 27xy yz zx xyz+ + = + + =+ + =

x y z 9xy yz zx 27xyz 27+ + = + + ==

2y z 9 xx (y z) xyz 27xxyz 27+ = + + == 2y z 9 xx (9 x) 27 27xxyz 27+ = + == 3y z 9 x(x 3) 0xyz 27+ = == y z 6x 3yz 9+ = == x 3y 3z 3= == 6d) 2 2 2x y z 6xy yz zx 7x y z 14 + + =+ =+ + =

2x y z 6xy yz zx 7 2zx(x y z) 2(xy yz zx) 14 + + =+ + = ++ + + + = 2x y z 6xy yz zx 7 2zx(6) 2(7 2zx) 14 + + =+ + = + + = Bi Gia Phong Gio vin trng THPT Trng Vnh K Bn tre. http://giaphong.schools.officelive.com/Trang 21

x y z 6xy yz 9zx 2+ + = + == y (x z) 6y(x z) 9zx 2+ + = + ==

x 1y 3z 2= == hoc x 2y 3z 1= ==. 6e) x xy y 1y yz z 4z zx x 9+ + = + + =+ + = x(y 1) y 1 2y(z 1) z 1 5z(x 1) x 1 10+ + + = + + + =+ + + =

(y 1)(x 1) 2(z 1)(y 1) 5(x 1)(x 1) 10+ + = + + =+ + =(x 1)(y 1)(z 1) 10 + + + = Nghim: x 1y 0z 4= == ; x 3y 2z 6= = = . 7)a)3x 2x x 1x2 5y 4y4 2y2 2+ = += + 3x 2x2 5y 4y2 y = = 3 2xy 5y 4yy 2 = = 2 2xy 5y 4 0y 2 + == x 0y 1= = hocx 2y 4= = 7b) 2 2x y x 1x y y x2 2 x y+ + = + = 2 2x y x 1x y x y2 2 x y+ = = 2x x 1x y 02 2 0 = = hoc x 1y 1 x2 2 2x 1= =

x 1x y2 2= = hoc x 1y 1 x2 3 2x (*)= = x 1y 1= = hoc x 1y 0= = (PT (*) c nghim duy nhtx 1 = ). 7c) 3x 1 y 2 y 3x22 2 3.23x 1 xy x 1+ + + =+ + = +

3x 1 y 2 y 3x22 2 3.2x 1 03x 1 xy x 1+ + + =+ + + = +

3x 1 y 2 y 3x2 2 3.2x 1x(3x y 1) 0+ + + = + = * x = 028y log11= . * y = 1 3x2(3x 1) 3x 12 62 1 0+ + + = . Dt 3x 11t 24+= (vx 1 ). t 3 8 = +( t 3 8 = loai)21 3 8x log3 2+= ( ) 2y 2 log 3 8 = +7d) ( )2 39 3x 1 2 y 1 (1)3log 9x log y 3 (2) + = =. DK: x 10 y 2 < . Voi DK trn, (2) 3 3log (3x) log y 1 =

33xlog 1y=y x = . Th vo (1)x 1 2 x 1 + =1 x 22 (x 1)(2 x) 0 = x 1x 2=

=

. H PT c hai nghim: x 1y 1= =v x 2y 2= =. 8)DK: 0 x 10 y 10 z 1< < < . Khi d: ( )2 2 22z y x2x y zxyz 64log y log x.log z= +==

( )2 2 22z y2x y zxyz 64log y log z= +== 2 222x 2yxy 64y z ===

x 4y 4z 4= == 9) 2 2x y 4x y m+ = + = 2x y 4(x y) 2xy m+ = + = x y 4xy (16 m) / 2+ = = x, y l nghim cua phuong trnh: 2X SX P 0 + = 216 mX 4X 0 (*)2 + = . m 8'2=a) H PT v nghim (*) v nghim' 0 m 8 > . Bi Gia Phong Gio vin trng THPT Trng Vnh K Bn tre. http://giaphong.schools.officelive.com/Trang 22 10) 5(x y) 4xy 4x y xy 1 m+ = + = x y 4mxy 5m 1+ = = x, y l nghim cua PT: 2t 4mt 5m 1 0 + = . H PT lun c nghim24m 5m 1 0 = + m ( ; 1/4] [1; ) + . 11) x y 1x x y y 1 3m+ =+ = ( ) ( )3x y 1x y 3 xy x y 1 3m+ =+ + = x y 1xy m+ == (1) x ,yl hai nghim khng m cua phuong trnh: 2X SX P 0 + = 2X X m 0 + = . H (1) c nghim 1 4m 0m 0 10 m4 . 12) x 4 y 1 4x y 3m + = + = . DK: x 4y 1 . Dt u x 4 0v y 1 0= = H PT d cho tro thnh: 2 2u v 4u v 3m 5+ = + =

u v 4uv (21 3m) / 2+ = = u, v l nghim khng m cua PT: 221 3mt 4t 0 (*)2 + = . H PT d cho c nghim (*) c hai nghim khng m 1 2' 0t .t 0

13m 73 . 13) (x y) xy m 2(x y)xy m 1+ + = + + = + x y 1xy m 1+ = = + hoc x y m 1xy 1+ = + =. a)m 3 = : x y 1xy 2+ = = hoc x y 2xy 1+ = =. Tp nghim{(x; y) / ( 1; 2), (2; 1), ( 1; 1)} . b) x, y l nghim cua PT: 2t t m 1 0 (1) + + =hoc2t (m 1)t 1 0 (2) + + = . H PT c nghim duy nht 1221 4(m 1) 0(m 1) 4 0 = + = = + < hoc 1221 4(m 1) 0(m 1) 4 0 = + < = + = m 1m 3/4=

=

. Cch khc: Nu h c nghim 0 0(x ; y )th 0 0(y ; x )cung l nghim. H c nghim duy nht th nghim duy nht l 0 0(x ; x )m 1 = ,m 3 = ,m 3/4 = . Thu lai chi cm 1 = ,m 3/4 = thoa. 14)DtS x y = + ,P xy = . DK: 2S 4P 0 . Khi d: 2x xy y 2m 1xy(x y) m m+ + = + + = + 2S P 2m 1S.P m m+ = + = + S, P l nghim cua PT: 2 2t (2m 1)t m m 0 + + + =t m =hoct m 1 = + . ChonS m 1 = + ,P m = 2 2S 4P (m 1) 0, m = m , h lun c nghim. H c nghim duy nht 2S 4P 0 = m =1. Khi d nghim duy nht l x = y =1. 15) 2 2 2x y m 1x y y x 2m m 3+ = + + = x y m 1xy(x y) (m 1)(2m 3)+ = + + = + a)m 3 = : tp nghim{(x; y) / (1; 3), (3; 1)}. b) Num 1 = : x y 0xy(x y) 0+ = + = x y 0 + =c v s nghim. Num 1 : x y m 1xy 2m 3+ = + = x, y l nghim cua PT: 2t (m 1)t 2m 3 0 + + = . H lun c nghim v 2 2(m 1) 4(2m 3) (m 3) 4 0, m R = + = + > . 16)Dt 2u x x x(x 1) = + = + , 2v y y y(y 1) = + = + . H tro thnh: u v 8u.v m+ = = u, v l nghim cua PT: 2t 8t m 0 + =28t t m(*) =a) m = 12: tp nghim{(x; y) / ( 3; 2), ( 2; 3), ( 3; 1), (1; 3), (2; 2), ( 2; 2), (2; 1), (1; 2)} . b) H PT c nghim(*) c hai nghim 1 2t , tv PT: 2iz z t + = ( i 1, 2 = ) c nghimBi Gia Phong Gio vin trng THPT Trng Vnh K Bn tre. http://giaphong.schools.officelive.com/Trang 23 - -33160tf(t)f '(t)-1/4 +164 2iz z t 0 + =c nghim it 1/4 . Xt hm s 2f (t) 8t t = voit 1/4 .f '(t) 8 2t = . f ' (t) 0 =t 4 = . Dp s: 33m 1616 .17) 22xy y 12x xy 26 m = = + y(x y) 12 (1)x(x y) 26 m(2) = = + 2y(x y) 12 (1)(x y) 14 m(2) (1) = = + a)m 2 = : tp nghim{(x; y) / (7; 3), ( 7; 3)} . b) H PT c nghim 14 m 0 + >m 14 > . 18)Nu h c nghim 0 0(x ; y )th 0 0(y ; x )cung l nghim. H c nghim duy nht th nghim duy nht l 0 0(x ; x )3 20 0 0x 8x mx = 20 0 0x (x mx m) 0 + = . V phuong trnh 20 0x mx m 0 + =khng c nghim kp 0x 0 =nn d h PT c nghim duy nht th phuong trnh 20 0x mx m 0 + =v nghim m 16 > . Nguoc lai voim 16 > . Tru v theo v hai PT cua h 2 2(x y) x (y 6)x y 6y m 0 + + + = PT: 2 2x (y 6)x y 6y m 0 + + + =v nghim v 23(y 2) 4(m 12) 0 = . 19)Nu h c nghim 0 0(x ; y )th 0 0(y ; x )cung l nghim. H c nghim duy nht th nghim duy nht l 0 0(x ; x ) . Vy h PT c nghim duy nht a 3/4 > . Nghim duy nht( 1/2; 1/2) . 20)Nu h c nghim 0 0(x ; y )th 0 0( x ; y ) cung l nghim. H c nghim duy nht th nghim duy nht l 0 0(x 0; y 0) = =a 3 = . Nguoc lai khia 3 = : 22 2x 3 y 3y 5 x x 5+ + =+ + = + x 0y 0= = (v 2x 3 y 3 + + ) Vy h PT c nghim duy nht a 3/4 > . Nghim duy nht(0; 0) . 21)Nu h c nghim 0 0 0(x ; y ; z )th 0 0 0( x ; y ; z ) cung l nghim. H c nghim duy nht th nghim duy nht l 0(0; 0; z )a b 2 = = . * Khia b 2 = = : 22 2 2xyz z 2 (1)xyz z 2 (2)x y z 4 (3)+ = + =+ + =c nghim(0; 0; 2)v khi (2)(1) xyz(z 1) 0 =.Chonz 1 =2 2xy 1x y 3= + = c thm nghim 0 0(x ; y ; 1) (0; 0; 2) nn h khng c nghim duy nht. * Khia b 2 = = : 22 2 2xyz z 2 (1)xyz z 2 (2)x y z 4 (3)+ = + = + + =c nghim(0; 0; 2) v khi (2)(1) xyz(z 1) 0 =z 0z 1=

=

z 0 =hocz 1 =du dn dn h PT v nghim. Vy h PT c nghim duy nht a b 2 = = . Nghim duy nht(0; 0; 2) . 22) 2x x 1 2 x 125 5 2011x 2011(1)x (m 2)x 2m 3 0 (2)+ + + + + + + + . DK:x 1 . Ta c 2x x 1 2 x 15 5 0, x [ 1; 1]+ + + + (1) ( )x 1 2x 25 5 5 2011(1 x)+ dngx [ 1; 1] v sai khix 1 > . Do d (1)1 x 1 . H bt PT c nghim2f (x) x (m 2)x 2m 3 = + + + c nghimx [ 1; 1] .

x [ 1;1]max f (x) 0 { } max f ( 1); f (1) 0 { } max 3m 6; m 2 0 + + m 2 .