Các dạng bài tập lượng giác 11

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Chuyªn
®Ò 1 Ph-¬ng tr×nh l-îng gi¸c A. C«ng thøc l-îng gi¸c cÇn nhí I. Mét sè
c«ng thøc l-îng gi¸c cÇn nhí EMBED Equation.DSMT4 EMBED
Equation.DSMT4 . C«ng thøc céng: EMBED Equation.DSMT4 C«ng
thøc nh©n ®«i: sin2x = 2sinxcosx cos2x = cos2x Ŕ sin2x = 2 cos2x Ŕ 1 =
1 - 2 sin2x C«ng thøc h¹ bËc: EMBED Equation.DSMT4 C«ng thøc nh©n
ba: Sin3x = 3sinx Ŕ 4sin3x; cos3x = 4cos3x Ŕ 3cosx. C«ng thøc biÓu diÔn
theo tanx: EMBED Equation.DSMT4 . C«ng thøc biÕn ®æi tÝch thµnh
tæng: EMBED Equation.DSMT4 C«ng thøc biÕn ®æi tæng thµnh tÝch:
EMBED Equation.DSMT4

B. Mét sè d¹ng bµi tËp vª ph-¬ng tr×nh l-îng gi¸c D¹ng 1. Ph-¬ng tr×nh
bËc hai. Bµi 1. Gi¶i çc ph-¬ng tr×nh sau: 1) 2cosx - EMBED
Equation.DSMT4 = 0 2) EMBED Equation.DSMT4
tanx Ŕ 3 = 0 3) 3cot2x + EMBED Equation.DSMT4 = 0
4) EMBED Equation.DSMT4 sin3x Ŕ 1 = 0 5) EMBED
Equation.DSMT4 cosx + sin2x = 0 Bµi 2. Gi¶i çc ph-¬n tr×nh sau:
1) 2cos2x Ŕ 3cosx + 1 = 0 2) cos2x + sinx + 1 = 0
3) 2cos2x + EMBED Equation.DSMT4 cosx Ŕ 2 = 0 4)
cos2x Ŕ 5sinx + 6 = 0 5) cos2x + 3cosx + 4 = 0 6) 4cos2x
- 4 EMBED Equation.DSMT4 cosx + 3 = 0 7) 2sin2x Ŕ cosx + EMBED
Equation.DSMT4 = 0 8) 2sin2x Ŕ 7sinx + 3 = 0 9)
2sin2x + 5cosx = 5. Bµi 3. Gi¶i çc ph-¬ng tr×nh: 2sin2x - cos2x - 4sinx
+ 2 = 0 3) 9cos2x - 5sin2x - 5cosx + 4 = 0 3) 5sinx(sinx - 1)
- cos2x = 3 4) cos2x + sin2x + 2cosx + 1 = 0 5) 3cos2x + 2(1 +
EMBED Equation.3 + sinx)sinx Ŕ (3 + EMBED Equation.3 ) = 0 6)
tan2x + ( EMBED Equation.3 - 1)tanx Ŕ EMBED Equation.3
= 0 7) EMBED Equation.3 8) EMBED Equation.3 9)
EMBED Equation.3 . D¹ng 2. Ph-¬ng tr×nh bËc nhÊt ®èi víi sinx vµ
cosx Bµi 1. Gi¶i çc ph-¬ng tr×nh sau: 1) 4sinx Ŕ 3cosx = 2
2) sinx - EMBED Equation.DSMT4 cosx = 1 3) EMBED
Equation.DSMT4 sin3x + cos3x = 1 4) sin4x + EMBED
Equation.DSMT4 cos4x = EMBED Equation.DSMT4 5) 5cos2x Ŕ
12cos2x = 13 6) 3sinx + 4cosx = 5 Bµi 2. Gi¶i çc ph-¬ng
tr×nh: 1) EMBED Equation.3 2) EMBED Equation.3
3) EMBED Equation.3 4) EMBED Equation.3 5)
EMBED Equation.3 D¹ng 3. Ph-¬ng tr×nh ®¼ng cÊp
bËc hai ®èi víi sin vµ c«sin. 1) sin2x + 2sinxcosx + 3cos2x - 3 = 0
2) sin2x Ŕ 3sinxcosx + 1 = 0. 3) 4 EMBED Equation.3 sinxcosx
+ 4cos2x = 2sin2x + EMBED Equation.DSMT4 . 4) EMBED Equation.3
EMBED Equation.3 . 5) a) EMBED Equation.3 ; b)
EMBED Equation.3 . 6) cos2x Ŕ 3sinxcosx Ŕ 2sin2x Ŕ 1 = 0 7) 6sin2x
+ sinxcosx Ŕ cos2x = 2. 8) sin2x + 2sinxcosx - 2cos2x = 0 9)
4sin2x + sinxcosx + 3cos2x - 3 = 0. 10) EMBED Equation.DSMT4 . D¹ng
4. Ph-¬ng tr×nh ®èi xøng ®èi víi sinx vµ cosx: Bµi 1. Gi¶i çc ph-¬ng
tr×nh sau: 1) EMBED Equation.DSMT4 (sinx + cosx) Ŕ 2sinxcosx = 2
EMBED Equation.DSMT4 + 1 2) 6(sinx Ŕ cosx) Ŕ sinxcosx = 6 3)
3(sinx + cosx) + 2sinxcosx + 3 = 0 4) sinx Ŕ cosx + 4sinxcosx + 1 = 0 5)
sin2x Ŕ 12(sinx Ŕ cosx) + 12 = 0 Bµi 2. Gi¶i çc ph-¬ng tr×nh: 1) EMBED
Equation.3 (sinx + cosx) - sinxcosx = 1. 2) (1 Ŕ sinxcosx)(sinx +
cosx) = EMBED Equation.3 . 3) EMBED Equation.3 . 4) sin3x +
cos3x = EMBED Equation.3 . 5) sinx Ŕ cosx + 7sin2x = 1. 6) EMBED
Equation.3 . 7) EMBED Equation.3 . 8) EMBED Equation.3
. 9) 1 + tgx = 2 EMBED Equation.3 sinx. 10) sinxcosx + 2sinx +
2cosx = 2. 11) 2sin2x Ŕ 2(sinx + cosx) +1 = 0.

C. Bµi tËp tù luyÖn Bµi 1. Gi¶i çc ph-¬ng tr×nh sau: 1) sin3x =
EMBED Equation.DSMT4 11) sin(2x - 3) = sin(x + 1) 2)
cos2x = - EMBED Equation.DSMT4 12) tan(3x + 2) + cot2x = 0
3) tan(x + 60o) = - EMBED Equation.DSMT4 13) sin3x =
cos4x 4) cot EMBED Equation.DSMT4 = EMBED Equation.DSMT4
14) tan3x.tanx = 1 5) sin2x = sin EMBED Equation.DSMT4
15) sin(2x + 50o) = cos(x + 120o) 6) tan EMBED
Equation.DSMT4 = tan EMBED Equation.DSMT4 16) EMBED
Equation.DSMT4 - 2sin2x = 0 7) cos(3x + 20o) = sin(40o - x) 17)
2cos EMBED Equation.DSMT4 - EMBED Equation.DSMT4 = 0 8)
tan EMBED Equation.DSMT4 = - cot EMBED Equation.DSMT4 18)
3tan EMBED Equation.DSMT4 + EMBED Equation.DSMT4 = 0 9)
sin(2x - 10o) = EMBED Equation.DSMT4 víi -120o x 90o 19)
2sinx - EMBED Equation.DSMT4 sin2x = 0 10) cos(2x + 1) = EMBED
Equation.DSMT4 víi - ( x ( 20) 8cos3x - 1 = 0 Bµi 2. Gi¶i
çc ph-¬ng tr×nh: 1) sin2x = EMBED Equation.DSMT4 11)
sin2x + sin22x = sin23x 2) cos23x = 1 12) sin EMBED
Equation.DSMT4 tan2x = 0 3) sin4x + cos4x = EMBED
Equation.DSMT4 13) (2sinx + 1)2 - (2sinx + 1)(sinx - EMBED
Equation.DSMT4 ) = 0 4) sinx + cosx = 1 14) sinx + sin2x +
sin3x = 0 5) cosx.cos3x = cos5x.cos7x 15) cosx + cos2x + cos3x
+ cos4x = 0 6) cos2x.cos5x = cos7x 16) 1 + sinx + cos3x =
cosx + sin2x + cos2x 7) sin3x.cos7x = sin13x.cos17x 17) cos7x +
sin22x = cos22x - cosx 8) sin4x.sin3x = cosx 18) sinx +
sin2x + sin3x = 1 + cosx + cos2x 9) 1 + 2cosx + cos2x = 0 19)
sin3x.sin5x = sin11x.sin13x 10) cosx + cos2x + cos3x = 0 20) cosx
- cos2x + cos3x = EMBED Equation.DSMT4 Bµi 3. Gi¶i çc ph-¬ng
tr×nh: 1) 2sin2x - 3sinx + 1 = 0 2) 4sin2x + 4cosx - 1 = 0 3)
tan EMBED Equation.DSMT4 + 2cot EMBED Equation.DSMT4 - 3 = 0
4) EMBED Equation.DSMT4 5) cot2x - 4cotx + 3 = 0 6)
cos22x + sin2x + 1 = 0 7) sin22x - 2cos2x + EMBED
Equation.DSMT4 = 0 8) 4cos2x - 2( EMBED Equation.DSMT4
- 1)cosx + EMBED Equation.DSMT4 = 0 9) tan4x + 4tan2x + 3 = 0
10) cos2x + 9cosx + 5 = 0 11) EMBED Equation.DSMT4
+ 3cot2x = 5 Bµi 5. Gi¶i çc ph-¬ng tr×nh sau: 1) 3sinx + 4cosx =
5 2) 2sin2x - 2cos2x = EMBED Equation.DSMT4 3) 2sin EMBED
Equation.DSMT4 + sin EMBED Equation.DSMT4 = EMBED
Equation.DSMT4 4) EMBED Equation.DSMT4 5) 2sin17x +
EMBED Equation.DSMT4 cos5x + sin5x = 0 6) cos7x - sin5x =
EMBED Equation.DSMT4 (cos5x - sin7x) 7) 4sinx + 2 cosx = 2 +
3tanx Bµi 6. Gi¶i çc ph-¬ng tr×nh: 1) 2(sinx + cosx) - 4sinxcosx -
1 = 0 2) sin2x - 12(sinx + cosx) + 12 = 0 3) sinx - cosx + 4sinxcosx
+ 1 = 0 4) cos3x + sin3x = 1 5) 3(sinx + cosx) + 2sin2x + 2 = 0
6) sin2x - 3 EMBED Equation.DSMT4 (sinx + cosx) + 5 = 0
7) 2(sinx - cosx) + sin2x + 5 = 0 8) sin2x +
EMBED Equation.DSMT4 sin(x - 45o) = 1 9) 2sin2x + EMBED
Equation.DSMT4 (sinx + cosx( + 8 = 0 10) (sinx - cosx)2 + (
EMBED Equation.DSMT4 + 1)(sinx - cosx) + EMBED Equation.DSMT4
= 0 Bµi 7. Gi¶i çc ph-¬ng tr×nh 1) sin2x - 10sinxcosx + 21cos2x = 0
2) cos2x - 3sinxcosx + 1 = 0 3) cos2x - sin2x - EMBED
Equation.DSMT4 sin2x = 1 4) 3sin2x + 8sinxcosx + (8 EMBED
Equation.DSMT4 - 9)cos2x = 0 5) 4sin2x + 3 EMBED
Equation.DSMT4 sin2x - 2cos2x = 4 6) 2sin2x + (3 + EMBED
Equation.DSMT4 )sinxcosx + ( EMBED Equation.DSMT4 - 1)cos2x =

1 7) 2sin2x - 3sinxcosx + cos2x = 0 8) cos22x - 7sin4x +
3sin22x = 3 Bµi 8. Gi¶i çc ph-¬ng tr×nh 1) 4cos2x - 2( EMBED
Equation.DSMT4 + 1)cosx + EMBED Equation.DSMT4 = 0 2) tan2x
+ (1 - EMBED Equation.DSMT4 )tanx - 3 = 0 3) cos2x + 9cosx + 5
= 0 4) sin22x - 2cos2x + EMBED Equation.DSMT4 = 0 5) 2cos6x
+ tan3x = 1 6) EMBED Equation.DSMT4 + 3cot2x = 5 Bµi 9. Gi¶i çc
ph-¬ng tr×nh 1) sin2x + sin2xsin4x + sin3xsin9x = 1 2) cos2x -
sin2xsin4x - cos3xcos9x = 1 3) cos2x + 2sinxsin2x = 2cosx 4) cos5xcosx
= cos4xcos2x + 3cos2x + 1 5) cos4x + sin3xcosx = sinxcos3x 6) sin(4x +
EMBED Equation.DSMT4 )sin6x = sin(10x + EMBED Equation.DSMT4
) 7) (1 + tan2)(1 + sin2x) = 1 8) tan( EMBED Equation.DSMT4 -
x) + tan( EMBED Equation.DSMT4 - x) + tan2x = 0 Bµi 10. Gi¶i çc
ph-¬ng tr×nh 1) (1 - cos2x)sin2x = EMBED Equation.DSMT4 sin2x 2)
sin4x - cos4x = cosx 3) EMBED Equation.DSMT4 4) 1 - (2 + EMBED
Equation.DSMT4 )sinx = EMBED Equation.DSMT4 5) tan2x =
EMBED Equation.DSMT4 6) 2(sin3x + cos3x) + sin2x(sinx + cosx) =
EMBED Equation.DSMT4 7) cosx(1 - tanx)(sinx + cosx) = sinx 8) (1
+ tanx)(1 + sin2x) = 1 + tanx 9) (2sinx - cosx)(1 + cosx) = sin2x Bµi
10. Gi¶i çc ph-¬ng tr×nh 1) sinx + cosx - EMBED Equation.DSMT4 -
1 = 0 2) (1 + EMBED Equation.DSMT4 )(sinx + cosx) - sin2x - ( 1 +
EMBED Equation.DSMT4 ) = 0 3) tanx + tan2x = tan3x 4) EMBED
Equation.DSMT4

D. Mét sè Bµi thi ®¹i häc vª ph-¬ng tr×nh l-îng gi¸c Bµi 1. Gi¶i c¸c
ph-¬ng tr×nh 1) (1 + tanx)cos3x + (1 + cotx)sin3x = EMBED
Equation.DSMT4 2) tan2x - tanxtan3x = 2 3) EMBED
Equation.DSMT4 = 1 - 2cosx 4) cos3xtan5x = sin7x 5) tanx +
cotx = 4 6) EMBED Equation.DSMT4 + 2cosx = 0 7) 2tanx +
cotx = EMBED Equation.DSMT4 8) tanx + cotx = 2(sin2x + cos2x) 9)
2sin3x(1 - 4sin2x) = 1 10) EMBED Equation.DSMT4 11)
cosx.cos2x.cos4x.cos8x = EMBED Equation.DSMT4 12) cos10x +
cos24x + 6cos3xcosx = cosx + 8cosxcos23x 13) sin2xcosx = EMBED
Equation.DSMT4 + cos3xsinx 14) sin6x + cos6x = cos4x 15)
sin4x + cos4x = EMBED Equation.DSMT4 cot(x + EMBED
Equation.DSMT4 )cot( EMBED Equation.DSMT4 - x) 16) EMBED
Equation.DSMT4 17) sin3xcos3x + cos3xsin3x = sin34x 18)
2sin3x - EMBED Equation.DSMT4 = 2cos3x + EMBED
Equation.DSMT4 19) cos3xcos3x + sin3xsin3x = EMBED
Equation.DSMT4 20) EMBED Equation.DSMT4 (tanx + cotx) 21)
1 + tanx = 2 EMBED Equation.DSMT4 sinx 22) cosx - sinx =
EMBED Equation.DSMT4 cos3x 23) EMBED Equation.DSMT4 24)
sin3x + cos3x + sin3xcotx + cos3xtanx = EMBED Equation.DSMT4
25) (2cosx - 1)(sinx + cosx) = 1 26) 2sin(3x + EMBED
Equation.DSMT4 ) = EMBED Equation.DSMT4

Bµi 2. Gi¶i çc ph-¬ng tr×nh 1) sin4 EMBED Equation.DSMT4 + cos4
EMBED Equation.DSMT4 = EMBED Equation.DSMT4 2) 4sin3x +
3cos3x - 3sinx - sin2xcosx = 0 3) cos3x - sin3x - 3cosxsin2x + sinx = 0
4) EMBED Equation.DSMT4 5) sin2x(tanx + 1) = 3sinx(cosx -
sinx) + 3 6) cos6x + sin6x = EMBED Equation.DSMT4 Bµi 3. Gi¶i çc
ph-¬ng tr×nh 1) EMBED Equation.DSMT4 2) EMBED
Equation.DSMT4 3) EMBED Equation.DSMT4 4) sin4x = tanx 5)
cos2x + sin2x 2cosx + 1 = 0 6) sin3x + 2cos2x - 2 = 0 7)
cos2x + cos22x + cos23x + cos24x = EMBED Equation.DSMT4 8) 2 +
cos2x + 5sinx = 0 9) 3(tanx + cotx) = 2(2 + sin2x) 10)
4cos3x + 3 EMBED Equation.DSMT4 sin2x = 8cosx Bµi 4. Gi¶i ph-¬ng
tr×nh l-îng gi¸c 1) cosx + EMBED Equation.DSMT4 sinx = 3 - EMBED
Equation.DSMT4 2) 3sin3x - EMBED Equation.DSMT4
cos9x = 1 + 4sin33x 3) cos7xcos5x - EMBED Equation.DSMT4 sin2x
= 1 - sin7xsin5x 4) 4sin2x - 3cos2x = 3(4sÜnx - 1) 5) 4(sin4x +
cos4x) + EMBED Equation.DSMT4 sin4x = 2 6) 4sin3x - 1 =
3sinx - EMBED Equation.DSMT4 cos3x 7) EMBED Equation.DSMT4
sin2x + cos2x = EMBED Equation.DSMT4 8)
2 EMBED Equation.DSMT4 (sinx + cosx)cosx = 3 + cos2x 9) cos2x -
EMBED Equation.DSMT4 sin2x = 1 + sin2x Bµi 5. Gi¶i çc ph-¬ng tr×nh
(biÕn ®æi ®-a vÒ d¹ng tÝch) 1) sin3x - EMBED Equation.DSMT4 sin2x
= 2sinxcos2x 2) sin22x + cos28x = EMBED Equation.DSMT4 cos10x 3)
(2sinx + 1)(2sin2x - 1) = 3 - 4cos2x 4) cosxcos EMBED Equation.DSMT4
cos EMBED Equation.DSMT4 - sinxsin EMBED Equation.DSMT4
sin EMBED Equation.DSMT4 = EMBED Equation.DSMT4 5) tanx
+ tan2x - tan3x = 0 6) cos3x + sin3x = sinx - cosx 7) (cosx -
sinx)cosxsinx = cosxcos2x 8) (2sinx - 1)(2cos2x + 2sinx + 1) = 3 -
4cos2x 9) 2cos3x + cos2x + sinx = 0 10) sin3x - sinx = sin2x 11)
EMBED Equation.DSMT4 12) sinx + sin2x + sin3x + sin4x + sin5x +
sin6x = 0 13) cos4 EMBED Equation.DSMT4 - sin4 EMBED
Equation.DSMT4 = sin2x 14) 3 - 4cos2x = sinx(2sinx + 1) 15)
2sin3x + cos2x = sinx 16) sin2x + sin22x + sin23x = EMBED
Equation.DSMT4 17) cos3x + sin3x = sinx - cosx 18) sin3x + cos3x =
2(sin5x + cos5x) 19) sin2x = cos22x + cos23x 20) sin23x - sin22x -
sin2x = 0 21) 1 + sinx + cosx = sin2x + cos2x = 0 22) 2sin3x - sinx =
2cos3x - cosx + cos2x 23) 2sin3x - cos2x + cosx = 0 24) cosx + cos2x +
cos3x + cos4x = 0 25) 2cos2x = EMBED Equation.DSMT4 (cosx -
sinx) 26) 4cos3x + 3 EMBED Equation.DSMT4 sin2x = 8cosx 27) sin3x
+ sin2x = 5sinx Bµi 6. Gi¶i çc ph-¬ng tr×nh 1) EMBED Equation.DSMT4
= cos2x + sin2x víi 0 x 2( 2) sin(2x +
EMBED Equation.DSMT4 ) - 3cos(x - EMBED Equation.DSMT4 ) = 1 +
2sinx víi EMBED Equation.DSMT4 x 3( 3) cos7x -
EMBED Equation.DSMT4 sin7x = - EMBED Equation.DSMT4
víi EMBED Equation.DSMT4 Bµi 7. T×m gi¶ trÞ lín
nhÊt, gi¸ trÞ nhá nhÊt cña: 1) y = 2sin2x + 3sinxcosx + 5cos2x 2) y =
EMBED Equation.DSMT4 trong kho¶ng ( -( ; () 3) y =
4sin2x + EMBED Equation.DSMT4 4) y = sinx - cos2x + EMBED
Equation.DSMT4 Bµi 8 (Çc ®Ò thi §H, C§ míi). 1) A_02. Gi¶i ph-¬ng
tr×nh: 5 EMBED Equation.DSMT4 = cos2x + 3 2) D_02. T×m çc
nghiÖm thuéc [0; 14] cña ph-¬ng tr×nh: cos3x - 4cos2x + 3cosx - 4 = 0 3)
A_03. Gi¶i ph-¬ng tr×nh: cotx - 1 = EMBED Equation.DSMT4 +
sin2x - EMBED Equation.DSMT4 sin2x 4) D_03. Gi¶i ph-¬ng tr×nh:
sin2( EMBED Equation.DSMT4 - EMBED Equation.DSMT4 )tan2x -
cos2 EMBED Equation.DSMT4 = 0 5) D_04. Gi¶i ph-¬ng tr×nh: (2cosx

- 1)(sinx + cosx) = sin2x - sinx 6) A_05. Gi¶i ph-¬ng tr×nh:
cos23xcos2x - cos2x = 0 7) D_05. Gi¶i ph-¬ng tr×nh: cos4x + sin4x +
cos(x - EMBED Equation.DSMT4 )sin(3x - EMBED Equation.DSMT4 )
- EMBED Equation.DSMT4 = 0 8) A_05_dù bÞ1. T×m nghiÖm trªn kho¶ng
(0 ; () cña ph-¬ng tr×nh: 4sin2 EMBED Equation.DSMT4 - EMBED
Equation.DSMT4 cos2x = 1 + 2cos2(x - EMBED Equation.DSMT4 ) 9)
A_05_dù bÞ 2. Gi¶i pt: 2 EMBED Equation.DSMT4 cos3( x -
EMBED Equation.DSMT4 ) - 3cosx - sinx = 0 10) D_05_dù bÞ 1. Gi¶i pt:
tan( EMBED Equation.DSMT4 - x) + EMBED Equation.DSMT4
= 2 11) D_05_dù bÞ 2. Gi¶i pt: sin2x + cos2x - 3sinx - cosx - 2
= 0 12) A_06_dù bÞ 1. Gi¶i pt: cos3xcos3x - sin3xsin3x = EMBED
Equation.DSMT4 13) A_06_dù bÞ 2. Gi¶i pt: 4sin3x + 4sin2x +
3sin2x + 6cosx = 0 14) B_06_dù bÞ 1. Gi¶i pt: (2sin2x - 1)tan22x +
3(2cos2x - 1) = 0 15) B_06_dù bÞ 2. Gi¶i pt: cos2x + (1 +
2cosx)(sinx - cosx) = 0 16) D_06_dù bÞ 1. Gi¶i pt: cos3x + sin3x +
2sin2x = 1 17) D_06. Gi¶i pt: cos3x + cos2x - cosx - 1 =
0 18) A_07. Gi¶i ph-¬ng tr×nh: (1 + sin2x)cosx + (1 + cos2x)sinx = 1 +
sin2x 19) B_07. Gi¶i ph-¬ng tr×nh: 2sin22x + sin7x - 1 = sinx 21)
D_07. Gi¶i ph-¬ng tr×nh: (sin2 EMBED Equation.DSMT4 + cos2
EMBED Equation.DSMT4 )2 + EMBED Equation.DSMT4 cosx = 2 22)
C§_07. Gi¶i ph-¬ng tr×nh: 2sin2( EMBED Equation.DSMT4 - 2x)
+ EMBED Equation.DSMT4 cos4x = 4cos2x - 1 23) A_08. Gi¶i ph-¬ng
tr×nh: EMBED Equation.DSMT4 24) B_08. Gi¶i ph-¬ng tr×nh:
sin3x - EMBED Equation.DSMT4 cos3x = sinxcos2x - EMBED
Equation.DSMT4 sin2xcosx 25) D_08. Gi¶i ph-¬ng tr×nh: 2sinx(1 +
cos2x) + sin2x = 1 + 2cosx 26) C§_08. Gi¶i pt: sin3x - EMBED
Equation.DSMT4 cos3x = 2sin2x

Chuyªn ®Ò 2 §¹i sè tæ hîp A. Mét sè d¹ng to¸n th-êng gÆp I) quy t¾c céng
vµ quy t¾c nh©n: Bµi 1: Víi çc ch÷ sè 1, 2, 3, 4, 5 cã thÓ lËp ®-îc bao
nhiªu: 1) Sè lÎ gåm 4 ch÷ sè kh¸c nhau? 2) Sè ch½n gåm 4 ch÷ sè bÊt
kú? Bµi 2: Cã 4 con ®-êng nèi liÒn ®iÓm A vµ ®iÓm B, cã 3 con ®-êng
nèi liÒn ®iÓm B vµ ®iÓm C. Ta muèn ®i tõ A ®Õn C qua B, råi tõ C trë vÒ A
còng ®i qua B. Hái cã bao nhiªu çch chän lé tr×nh ®i vµ vÒ nÕu ta kh«ng
muèn dïng ®-êng ®i lµm ®-êng vÒ trªn c¶ hai chÆng AB vµ BC? Bµi 3: Cã
5 miÕng b×a, trªn mçi miÕng ghi mét trong 5 ch÷ sè 0, 1, 2, 3, 4. LÊy 3
miÕng b×a nµy ®Æt lÇn l-ît c¹nh nhau tõ tr¸i sang ph¶i ®Ó ®-îc çc sè gåm
3 ch÷ sè. Hái cã thÓ lËp ®-îc bao nhiªu sè cã nghÜa gåm 3 ch÷ sè vµ trong
®ã cã bao nhiªu sè ch½n? Bµi 4: Cho 8 ch÷ sè 0, 1, 2, 3, 4, 5, 6, 7.
Tõ 8 ch÷ sè trªn cã thÓ lËp ®-îc bao nhiªu sè, mçi sè gåm 4 ch÷ sè ®«i
mét kh¸c nhau vµ kh«ng chia hÕt cho 10. Bµi 5: Mét ng-êi cã 6 çi ¸o,
trong ®ã cã 3 ¸o säc vµ 3 ¸o tr¾ng; cã 5 quÇn, trong ®ã cã 2 quÇn ®en; vµ
cã 3 ®«i giµy, trong ®ã cã 2 ®«i giÇy ®en. Hái ng-êi ®ã cã bao nhiªu çch
chän mÆc ¸o - quÇn - giµy, nÕu: 1) Chän ¸o, quÇn vµ giµy nµo còng
®-îc. 2) NÕu chän ¸o säc th× víi quÇn nµo vµ giµy nµo còng ®-îc;
cßn nÕu chän ¸o tr¾ng th× chØ mÆc víi quÇn ®en vµ ®i giµy ®en. II)
ho¸n vÞ - chØnh hîp - tæ hîp: Bµi 1: Cã n ng-êi b¹n ngåi quanh mét bµn
trßn (n 3). Hái cã bao nhiªu çch s¾p xÕp sao cho: 1) Cã 2 ng-êi Ên
®Þnh tr-íc ngåi c¹nh nhau. 2) 3 ng-êi Ên ®Þnh tr-íc ngåi c¹nh nhau
theo mét thø tù nhÊt ®Þnh Bµi 2: Mét ®éi x©y dùng gåm 10 c«ng nh©n
vµ 3 kü s-. §Ó lËp mét tæ c«ng t¸c cÇn chän 1 kü s- lµm tæ tr-ëng, 1 c«ng
nh©n lµm tæ phã vµ 5 c«ng nh©n lµm tæ viªn. Hái cã bao nhiªu çch lËp tæ
c«ng t¸c. Bµi 3: Trong mét líp häc cã 30 häc sinh nam, 20 häc sinh n÷.
Líp häc cã 10 bµn, mçi bµn cã 5 ghÕ. Hái cã bao nhiªu çch s¾p xÕp chç
ngåi nÕu: a) Çc häc sinh ngåi tuú ý. b) Çc häc sinh ngåi nam cïng
1 bµn, çc häc sinh n÷ ngåi cïng 1 bµn Bµi 4: V‘i çc sŽ: 0, 1, 2, …,
9 lËp ®-îc bao nhiªu sè lÎ cã 7 ch÷ sè. Bµi 5: Tõ hai ch÷ sè 1; 2
lËp ®-îc bao nhiªu sè cã 10 ch÷ sè trong ®ã cã mÆt Ýt nhÊt 3 ch÷ sè 1 vµ
Ýt nhÊt 3 ch÷ sè 2.

Bµi 6: T×m tæng tÊt c¶ çc sè cã 5 ch÷ sè kh¸c nhau ®-îc viÕt tõ çc ch÷
sè: 1, 2, 3, 4 , 5 Bµi 7: Trong mét phßng cã hai bµn dµi, mçi bµn cã
5 ghÕ. Ng-êi ta muèn xÕp chç ngåi cho 10 häc sinh gåm 5 nam vµ 5 n÷. Hái
cã bao nhiªu çch xÕp chç ngåi nÕu: 1) Çc häc sinh ngåi tuú ý. 2)
Çc häc sinh nam ngåi mét bµn vµ çc häc sinh n÷ ngåi mét bµn. Bµi 8:
Víi çc ch÷ sè 0, 1, 2, 3, 6, 9 cã thÓ thµnh lËp ®-îc bao nhiªu sè chia
hÕt cho 3 vµ gåm 5 ch÷ sè kh¸c nhau Bµi 9: Tõ çc ch÷ çi cña c©u:
Tr-êng THPT Lý Th-êng KiÖt cã bao nhiªu çch xÕp mét tõ (tõ kh«ng cÇn
cã nghÜa hay kh«ng) cã 6 ch÷ çi mµ trong tõ ®ã ch÷ T cã mÆt ®óng 3
lÇn, çc ch÷ kh¸c ®«i mét kh¸c nhau vµ trong tõ ®ã kh«ng cã ch÷ £
Bµi 10: Cho A lµ mét tËp hîp cã 20 phÇn tö. a) Cã bao nhiªu tËp hîp
con cña A? b) Cã bao nhiªu tËp hîp con kh¸c rçng cña A mµ cã sè phÇn
tö lµ sè ch½n? Bµi 11: 1) Cã bao nhiªu sè ch½n cã ba ch÷ sè kh¸c
nhau ®-îc t¹o thµnh tõ çc ch÷ sè 1, 2, 3, 4, 5, 6? 2) Cã bao nhiªu
sè cã ba ch÷ sè kh¸c nhau ®-îc t¹o thµnh tõ çc ch÷ sè 1, 2, 3, 4, 5, 6
nµ çc sè ®ã nhá h¬n sè 345? Bµi 12: Tõ çc ch÷ sè 1, 2, 3, 4,
5, 6 thiÕt lËp tÊt c¶ çc sè cã 6 ch÷ sè kh¸c nhau. Hái trong çc sè ®·
thiÕt lËp ®-îc, cã bao nhiªu sè mµ hai ch÷ sè 1 vµ 6 kh«ng ®øng c¹nh
nhau? Bµi 13: Mét tr-êng tiÓu häc cã 50 häc sinh ®¹t danh hiÖu ch¸u
ngoan B¸c Hå, trong ®ã cã 4 cÆp anh em sinh ®«i. CÇn chän mét nhãm 3 häc
sinh trong sè 50 häc sinh trªn ®i dù §¹i héi ch¸u ngoan B¸c Hå, sao cho
trong nhãm kh«ng cã cÆp anh em sinh ®«i nµo. Hái cã bao nhiªu çch chän.
Bµi 14: Víi çc ch÷ sè 1, 2, 3, 4, 5, 6, 7, 8, 9 cã thÓ lËp ®-îc bao
nhiªu sè cã ba ch÷ sè kh¸c nhau vµ kh«ng lín h¬n 789? Bµi 15: 1)
Cho çc ch÷ sè 0, 1, 2, 3, 4. Hái cã thÓ thµnh lËp ®-îc bao nhiªu sè cã
b·y ch÷ sè tõ nh÷ng ch÷ sè trªn, trong ®ã ch÷ sè 4 cã mÆt ®óng ba lÇn,
cßn çc ch÷ sè kh¸c cã mÆt ®óng mét lÇn. 2) Trong sè 16 häc sinh cã
3 häc sinh giái, 5 kh¸, 8 trung b×nh. Cã bao nhiªu çch chia sè häc sinh
®ã thµnh 2 tæ, mçi tæ 8 ng-êi sao cho ë mçi tæ ®Òu cã häc sinh giái vµ
mçi tæ cã Ýt nhÊt hai häc sinh kh¸. Bµi 16: Sè nguyªn d-¬ng n
®-îc viÕt d-íi d¹ng: n = EMBED Equation.3 Trong ®ã (, (, (, ( lµ
çc sè tù nhiªn 1) Hái sè çc -íc sè cña n lµ bao nhiªu? 2) ¸p
dông: TÝnh sè çc -íc sè cña 35280.

III) to¸n vÒ çc sè EMBED Equation.3 , EMBED Equation.3 ,
EMBED Equation.3 : Bµi 1: Gi¶i bÊt ph-¬ng tr×nh: EMBED Equation.3
Bµi 2: T×m çc sŽ ¨m trong d·y sŽ x1, x2, …, xn, … v‘i: xn =
EMBED Equation.3 Bµi 3: Cho k, n lµ çc sè nguyªn vµ 4 ( k ( n;
Chøng minh: EMBED Equation.3 Bµi 4: Cho n ( 2 lµ sè
nguyªn. Chøng minh: Pn = 1 + P1 + 2P2 + 3P3 + … + (n - 1)Pn - 1 Bµi
5: Cho k vµ n lµ çc sè nguyªn d-¬ng sao cho k n. Chøng minh r»ng:
EMBED Equation.3 VI) nhÞ thøc newton: Bµi 1: Chøng minh r»ng:
EMBED Equation.3 Bµi 2: Khai triÓn vµ rót gän çc ®¬n
thøc ®ång d¹ng tõ biÓu thøc: EMBED Equation.3 ta sÏ ®-îc ®a
thøc:P(x) = A0 + A1x + A2x2 + … + A14x14 H·y x¸c ®Þnh hÖ sè A9 Bµi
3: 1) TÝnh EMBED Equation.3 (n ( N) 2) Tõ kÕt qu¶ ®ã
chøng minh r»ng: EMBED Equation.3 Bµi 4: Chøng minh r»ng:
EMBED Equation.3 Bµi 5: TÝnh tæng S = EMBED Equation.3
(n ( 2) Bµi 6: Chøng minh r»ng: EMBED Equation.3 Bµi
7: T×m hÖ sè cña x5 trong khai triÓn cña biÓu thøc sau thµnh ®a thøc:
f(x) = EMBED Equation.3 Bµi 8: Trong khai triÓn cña
EMBED Equation.3 thµnh ®a thøc: P(x) = EMBED Equation.3
H·y t×m hÖ sè ak lín nhÊt (0 ( k ( 10) Bµi 9: T×m sè nguyªn d-¬ng
n sao cho: EMBED Equation.3 . Bµi 10: CMR: EMBED
Equation.3

Bµi 11: Víi mçi n lµ sè tù nhiªn, h·y tÝnh tæng: 1) EMBED
Equation.3 2) EMBED Equation.3 Bµi 12: Cho ®a thøc
P(x) = (3x - 2)10 1) T×m hÖ sè cña x2 trong khai triÓn trªn cña P(x)
2) TÝnh tæng cña c¸c hÖ sè trong khai triÓn trªn cña P(x) Bµi 13:
BiÕt tæng tÊt c¶ c¸c hÖ sè cña khai triÓn nhÞ thøc: EMBED Equation.3
b»ng 1024 h·y t×m hÖ sè a (a lµ sè tù nhiªn) cña sè h¹ng a.x12 trong
khai triÓn ®ã. Bµi 14: Trong khai triÓn nhÞ thøc: EMBED Equation.3
h·y t×m sè h¹ng kh«ng phô thuéc vµo x biÕt r»ng: EMBED Equation.3
Bµi15: Chøng minh: EMBED Equation.3 Bµi 16:
T×m sè h¹ng kh«ng chøa x trong khai triÓn cña biÓu thøc: EMBED
Equation.3 x ( 0 Bµi 17: Khai triÓn nhÞ thøc: EMBED
Equation.3 BiÕt r»ng trong khai triÓn ®ã EMBED Equation.3 vµ
sè h¹ng thø t- b»ng 20n, t×m n vµ x Bµi 18: Trong khai triÓn: EMBED
Equation.3 T×m sè h¹ng chøa a, b cã sè mò b»ng nhau.

B. Bµi tËp tù luyÖn Bµi 1. Víi çc ch÷ sè 0,1,2,3,4,5, cã thÓ lËp ®-îc
bµo nhiªu sè cã 5 ch÷ sè kh¸c nhau? Bµi 2. Dïng 5 ch÷ sè 2,3,4,6,8 ®Ó
viÕt thµnh sè gåm 5 ch÷ sè kh¸c nhau. Hái: a. B¾t dÇu bëi ch÷ sè 2. b.
B¾t ®Çu bëi ch÷ sè 36 c. B¾t ®Çu bëi ch÷ sè 482 Bµi 3. Dïng 6 ch÷ sè
1,2,3,4,5,6 ®Ó viÕt thµnh sè tù nhiªn gåm 4 ch÷ sè kh¸c nhau. Hái: a. Cã
bao nhiªu sè nh- vËy b. Cã bao nhiªu sè b¾t ®Çu bëi ch÷ sè 1 Bµi 4. Cho 8
ch÷ sè 0,1,2,3,4,5,6,7. Hái cã thÓ lËp ®-îc bao nhiªu sè cã 6 ch÷ sè kh¸c
nhau trong ®ã nhÊt thiÕt ph¶i cã mÆt ch÷ sè 4. Bµi 5. Víi çc ch÷ sè
0,1,2,3,4,5,6 cã thÓ lËp ®-îc bao nhiªu sè cã 5 ch÷ sè kh¸c nhau trong ®ã
nhÊt thiÕt ph¶i cã mÆt ch÷ sè 5. Bµi 6. Tõ çc ch÷ sè 1,2,3,4,5,6,7,8,9
thiÕt lËp tÊt c¶ çc sè cã 9 ch÷ sè kh¸c nhau. Hái trong çc sè thiÕt lËp
®-îc cã bao nhiªu sè mµ ch÷ sè 9 ®øng chÝnh gi÷a. Bµi 7. Cho A =
{0,1,2,3,4,5} cã thÓ lËp ®-îc bao nhiªu sè ch½n, mçi sè cã 4 ch÷ sè kh¸c
nhau. Bµi 8. a. Tõ çc ch÷ sè 4,5,6,7 cã thÓ lËp ®-îc bao nhiªu sè cã
çc ch÷ sè ph©n biÖt. b. Tõ çc ch÷ sè 0,1,2,3,4,5 cã thÓ lËp ®-îc bao
nhiªu sè ch½n gåm 5 ch÷ sè ®«i mét kh¸c nhau? Bµi 9. Cho tËp E =
{0,1,2,3,4,5,6,7,8,9} Hái cã bao nhiªu sè tù nhiªn gåm 5 ch÷ sè kh¸c nhau
chia hÕt cho 5? Bµi 10. Mét tËp thÓ gåm 14 ng-êi gåm 6 nam vµ 8 n÷, ng-êi
ta muèn chän 1 tæ c«ng t¸c gåm 6 ng-êi. T×m sè çch chän sao cho trong tæ
ph¶i cã c¶ nam vµ n÷? Bµi 11. Mét nhãm häc sinh gåm 10 ng-êi, trong ®ã cã
7 nam vµ 3 n÷. Hái cã bao nhiªu çch xÕp 10 hoc sinh trªn thµnh 1 hµng
däc sao cho 7 häc sinh nam ph¶i ®øng liÒn nhau? Bµi 12. Cã mét hép ®ùng 2
viªn bi ®á, 3 viªn bi tr¾ng, 5 viªn bi vµng. Chon ngÉu nhiªn 4 viªn bi
lÊy tõ hép ®ã. Hái cã bao nhiªu çch chän ®Ó trong sè viªn bi lÊy ra
kh«ng ®ñ 3 mµu? Bµi 13. Mét líp cã 20 häc sinh trong ®ã cã 2 çn bé líp.
Hái cã bao nhiªu çch cö 3 ng-êi ®i dù héi nghÞ sinh viªn cña tr-êng sao
cho trong 3 ng-êi cã Ýt nhÊt mét çn bé líp? Bµi 14. Mét ®éi v¨n nghÖ cã
20 ng-êi trong ®ã cã 10 nam vµ 10 n÷. Hái cã bao nhiªu çch chän ra 5
ng-êi sao cho: 1. Cã ®óng 2 ng-êi nam trong 5 ng-êi ®ã 2. Cã Ýt nhÊt 2
nam vµ Ýt nhÊt 1 n÷ trong 5 ng-êi ®ã Bµi 15. Cã 5 nhµ To¸n häc nam, 3 nhµ
To¸n häc n÷ vµ 4 nhµ VËt lý nam. LËp mét ®oµn c«ng t¸c cÇn cã c¶ nam vµ
n÷, cÇn cã c¶ nhµ To¸n häc vµ nhµ VËt lý. Hái cã bao nhiªu çch?

Bµi 16. Mét líp häc cã 30 häc sinh nam vµ 15 häc sinh n÷. Cã 6 häc sinh
®-îc chä ra ®Ó lËp mét tèp ca. Hái cã bao nhiªu c¸ch chän kh¸c nhau.
1. NÕu ph¶i cã Ýt nhÊt 2 n÷. 2. NÕu ph¶i chän tuú ý. Bµi 17.
Mét tæ häc sinh gåm 7 nam vµ 4 n÷. Gi¸o viªn muèn chän 3 häc sinh xÕp vµo
bµn ghÕ cña líp, trong ®ã cã Ýt nhÊt 1 nam. Hái cã bao nhiªu c¸ch
chän? Bµi 18. Chøng minh r»ng: INCLUDEPICTURE
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MERGEFORMATINET . Bµi 19. Chøng minh r»ng: INCLUDEPICTURE
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MERGEFORMATINET Bµi 20. Víi n lµ sè nguyªn d-¬ng, chøng minh hÖ thøc
sau: INCLUDEPICTURE
http://www.onthi.com/images/ct/33f788eb6fe935ca0769614426c7e216.jpg *
MERGEFORMATINET Bµi 21. Chøng minh r»ng: INCLUDEPICTURE
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MERGEFORMATINET Bµi 22. TÝnh tæng: INCLUDEPICTURE
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MERGEFORMATINET Bµi 24. Chøng minh r»ng: INCLUDEPICTURE
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MERGEFORMATINET Bµi 25. Cho n lµ mét sè nguyªn d-¬ng: a. TÝnh :
I = EMBED Equation.3 b. TÝnh tæng: INCLUDEPICTURE
http://www.onthi.com/images/ct/d2da814a153265e060585460d9788553.jpg *
MERGEFORMATINET Bµi 26. T×m sè nguyªn d-¬ng n sao cho:
INCLUDEPICTURE
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INCLUDEPICTURE
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MERGEFORMATINET Bµi 28. T×m sè tù nhiªn n th¶o m·n ®¼ng thøc sau:
INCLUDEPICTURE
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http://www.onthi.com/images/ct/34e2844473e00be9b239c3e68c564ab5.jpg *
MERGEFORMATINET , biÕt r»ng, víi n lµ sè nguyªn d-¬ng:
INCLUDEPICTURE
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INCLUDEPICTURE
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MERGEFORMATINET

Bµi 31. T×m hÖ sè cña x8 trong khai triÓn thµnh ®a thøc cña:
INCLUDEPICTURE
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MERGEFORMATINET Bµi 32. Gäi a3n - 3 lµ hÖ sè cña x3n - 3 trong khai
triÓn thanh ®a thøc cña:(x2 + 1)n(x + 2)n. T×m n ®Ó a3n - 3 = 26n Bµi
33. T×m hÖ sè cña sè h¹ng chøa x26 trong khai triÓn nhÞ thøc Newton cña
EMBED Equation.3 BiÕt r»ng: EMBED Equation.3 Bµi 34. T×m çc
sè h¹ng kh«ng chøa x trong khai triÓn nhÞ thøc Newton cña:
INCLUDEPICTURE
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MERGEFORMATINET víi x 0 Bµi 35. T×m sè h¹ng thø 7
trong khai triÓn nhÞ thøc: INCLUDEPICTURE
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MERGEFORMATINET ; INCLUDEPICTURE
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MERGEFORMATINET Bµi 36. Cho : INCLUDEPICTURE
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MERGEFORMATINET Sau khi khai triªn vµ rót gän th× biÓu thøc A sÏ gåm
bao nhiªu sè h¹ng? Bµi 37. T×m hÖ sè cña sè h¹ng chøa x8 trong khai triÓn
nhÞ thøc Newton cña INCLUDEPICTURE
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MERGEFORMATINET ,

b i ¿-t
r ±-n g : I N C L U D E P I C T U R E
h t t p : / / w w w . o n t h i . c o m / i m a g e s / c t / e 4 b 7
c 6 a 6 0 2 e 0 c 0 0 f 1 0 5 8 7 8 3 b 0 c 8 2 3 5 0 a . j p g *
M E R G E F O R M A T I N E T B µ i 3 8 . k h a i
t r i Ó n b i Ó u t h ø c ( 1 - 2 x ) n t a ® - î c ® a
t h ø c c ã d ¹ n g : I N C L U D E P I C T U R E
h t t p : / / w w w . o n t h i . c o m / i m a g e s / c t / 6 c 2 5
3 4 b 0 e 6 5 e 5 7 0 9 d 4 8 6 a f 5 5 c 0 4 7 7 5 c 0 . j p g *
M E R G E F O R M A T I N E T . T ì m h Ç- s Ñ- c ç-a
I N C L U D E P I C T U R E
h t t p : / / w w w . o n t h i . c o m / i m a g e s / c t / b 1 c d
1 0 6 a f a 3 7 8 f f 9 e 8 c c 4 2 b 5 3 8 0 b 4 e 1 8 . j p g *
M E R G E F O R M A T I N E T , b i ¿-t a o + a 1 + a 2 =
7 1 B µ i 3 9 . T × m h Ö s è c ñ a x 5 t r o n g
k h a i t r i Ó n ® a t h ø c :
I N C L U D E P I C T U R E
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* MERGEFORMATINET Bµi 40. T×m sè h¹ng kh«ng chøa x trong khai triÓn
nhÞ thøc EMBED Equation.3 BiÕt r»ng: INCLUDEPICTURE
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MERGEFORMATINET Bµi 41. Gi¶i çc ph-¬ng tr×nh: INCLUDEPICTURE
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MERGEFORMATINET INCLUDEPICTURE
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MERGEFORMATINET Bµi 42. Gi¶i çc hÖ ph-¬ng tr×nh: INCLUDEPICTURE
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MERGEFORMATINET Bµi 43. Gi¶i c¸c bÊt ph-¬ng tr×nh: INCLUDEPICTURE
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MERGEFORMATINET

chuyªn ®Ò 3. Ph-¬ng ph¸p quy n¹p To¸n häc Bµi 1. Chøng minh r»ng a) 1.2 +
2.5 + 3.8 + ... + n(3n - 1) = n2(n + 1) víi n ( N* b) 3 + 9 + 27 +
... + 3n = EMBED Equation.DSMT4 (3n + 1 - 3) víi n ( N* c) 12 + 32
+ 52 + ... + (2n - 1)2 = EMBED Equation.DSMT4 víi n ( N* d)
13 + 23 + 33 + ... + n3 = EMBED Equation.DSMT4 víi n ( N* e) 12
+ 22 + 32 + ... + n2 = EMBED Equation.DSMT4 víi n ( N* f)
EMBED Equation.3 víi n ( N* g) EMBED Equation.3 víi
n ( N* h) EMBED Equation.3 víi n ( N* i) EMBED
Equation.3 víi n ( 2 k) EMBED Equation.3 víi n (
N* Bµi 2. Chøng minh r»ng víi mäi n ( N* ta cã: a) n3 + 2n chia hÕt cho
3 b) n3 + (n + 1)3 + (n + 2)3 chia hÕt cho 9 c) n3 + 11n chia hÕt cho
6 d) 2n3 - 3n2 + n chia hÕt cho 6 e) 4n + 15n - 1 chia hÕt cho 9 f) 32n +
1 + 2n + 2 chia hÕt cho 7 g) n7 - n chia hÕt cho 7 h) n3 + 3n2 + 5n chia
hÕt cho 3 Bµi 3. Chøng minh c¸c bÊt ®¼ng thøc sau a) 2n + 2 2n + 5
víi n ( N* b) 2n 2n + 1 víi n ( N*, n ( 3 c) 3n n2 + 4n
+ 5 víi n ( N*, n ( 3 d) 2n - 3 3n - 1 víi n ( 8 e) 3n - 1 n(n +
2) víi n ( 4

Chuyªn ®Ò 4: d·y sè D¹ng 1. X¸c ®Þnh mét sè sè h¹ng cña d·y sè. X¸c ®Þnh
sè h¹ng tæng qu¸t Bµi 1. ViÕt 5 sè h¹ng ®Çu cña d·y sè sau: a) un =
EMBED Equation.DSMT4 b) un = EMBED
Equation.DSMT4 b) EMBED Equation.DSMT4 (n 2) c)
un = EMBED Equation.DSMT4 d) EMBED Equation.DSMT4 (víi k (
1) e) u1 = 2; un + 1 = EMBED Equation.DSMT4 (un + 1) g) un =
cos EMBED Equation.DSMT4 h) nsin EMBED
Equation.DSMT4 + n2cos EMBED Equation.DSMT4 Bµi 2. T×m sè h¹ng
tæng qu¸t cña d·y sè a) (un): 1; 2; 4; 8; 16; … b) (un): EMBED
Equation.DSMT4 ; c) (un):
… EMBED Equation.DSMT4 (víi n (
1) d) (un): EMBED Equation.DSMT4 ; Bµi 3. Cho d·y sè (un): u1 =
…
EMBED Equation.DSMT4 , un+ 1 = 4un + 7 víi n ( 1 a) TÝnh u2, u3, u4,
u5, u6 b) Chøng minh r»ng: un = EMBED Equation.DSMT4 víi n (
1 Bµi 4. Cho d·y sè (un): u1 = 1; un + 1 = un + 7 víi ( 1 a) TÝnh u2, u3,
u4, u5, u6 b) Chøng minh r»ng: un = 7n Ŕ 6 Bµi 5. Cho (un): u1 = 2; un +
1 = 3un + 2n Ŕ 1 Chøng minh r»ng: un = 3n - n D¹ng 2. XÐt tÝnh ®¬n ®iÖu
cña mét d·y sè Bµi 6. XÐt tÝnh ®¬n ®iÖu cña c¸c d·y sè sau a) un =
EMBED Equation.DSMT4 ; b) un = EMBED Equation.DSMT4
c) un = EMBED Equation.DSMT4 d) un = EMBED
Equation.DSMT4 e) un = EMBED Equation.DSMT4
f) un = EMBED Equation.DSMT4 g) un = EMBED
Equation.DSMT4 h) un = EMBED Equation.DSMT4 D¹ng 4. XÐt
tÝnh bÞ chÆn cña d·y sè Bµi 7. XÐt tÝnh bÞ chÆn cña c¸c d·y sè a) un = 2n
Ŕ 1 b) un = EMBED Equation.DSMT4 c) un =
3.22n Ŕ 1 d) un = EMBED Equation.DSMT4 e) un = EMBED
Equation.DSMT4 f) un = EMBED Equation.DSMT4 bµi tËp
tù luyÖn Bài 1. tìm các
gi Û-i
h ¡-n s a u :

E M B E D E q u a t i o n . D S M T 4 E M B E D
E q u a t i o n . D S M T 4 E M B E D
E q u a t i o n . D S M T 4

B à i 2 . t ì m c á c g i Û-i h ¡-n s a u :

B à i 5 . t ì m c á c g i Û-i h ¡-n s a u : 1 .
E M B E D E q u a t i o n . D S M T 4 2 .
E M B E D E q u a t i o n . D S M T 4 5 . l i m
E M B E D E q u a t i o n . D S M T 4 B à i 6 t ì m
c á c g i Û-i h ¡-n s a u : E M B E D
E q u a t i o n . D S M T 4 2 . E M B E D
E q u a t i o n . DSMT4 6. EMBED
Equation.DSMT4

chuyªn ®Ò 5 . giíi h¹n cña HµM sè Bµi 1: T×m çc giíi h¹n sau (d¹ng
EMBED Equation.DSMT4 ): 1) EMBED Equation.DSMT4
2) EMBED Equation.DSMT4 3) EMBED Equation.DSMT4
4) EMBED Equation.DSMT4 5) EMBED
Equation.DSMT4 6) EMBED Equation.DSMT4 7)
EMBED Equation.DSMT4 8) EMBED Equation.DSMT4
Bµi 2. T×m çc giíi h¹n sau(d¹ng EMBED Equation.DSMT4 ): 1)
EMBED Equation.DSMT4 2) EMBED Equation.DSMT4
EMBED Equation.DSMT4 7) EMBED Equation.DSMT4 8)
EMBED Equation.DSMT4 15) EMBED Equation.DSMT4 Bµi 3. T×m çc
giíi h¹n(d¹ng EMBED Equation.DSMT4 ): 1) EMBED Equation.DSMT4
EMBED Equation.DSMT4 8) EMBED Equation.DSMT4 Bµi 4.
T×m çc giíi h¹n (d¹ng EMBED Equation.DSMT4 ): 1) EMBED
Equation.DSMT4 2) EMBED Equation.DSMT4 3) EMBED
Equation.DSMT4 6) EMBED Equation.DSMT4 Bµi 5.
T×m çc giíi h¹n (( - (): 1) EMBED Equation.DSMT4 2) EMBED
Equation.DSMT4

CHUY£N §Ò 6. ®¹o hµm I. TÝnh ®¹o hµm b»ng ®Þnh nghÜa Bµi 1. Dïng ®Þnh
nghÜa tÝnh ®¹o hµm cña çc hµm sè sau t¹i çc ®iÓm: 1) f(x) = 2x2 + 3x +
1 t¹i x = 1 2) f(x) = sinx t¹i x = EMBED Equation.DSMT4 3) f(x) =
EMBED Equation.DSMT4 t¹i x = 1 4) f(x) = EMBED Equation.DSMT4
t¹i x = 0 5) f(x) = EMBED Equation.DSMT4 t¹i x = 2 6) f(x) =
EMBED Equation.DSMT4 t¹i x = 0 7) f(x) = EMBED Equation.DSMT4
t¹i x = 0 8) f(x) = EMBED Equation.DSMT4 t¹i x = 0 Bµi 2.
Dïng ®Þnh nghÜa tÝnh ®¹o hµm cña çc hµm sè sau: 1) y = 5x Ŕ 7 2) y
= 3x2 Ŕ 4x + 9 3) y = EMBED Equation.DSMT4 4) y = EMBED
Equation.DSMT4 5) y = x3 + 3x Ŕ 5 6) y = EMBED
Equation.DSMT4 + x II. Quan hÖ gi÷a tÝnh liªn tôc vµ sù cã ®¹o
hµm Bµi 3. Cho hµm sè f(x) = EMBED Equation.DSMT4 Chøng minh
r»ng hµm sè liªn tôc trªn R nh-ng kh«ng cã ®¹o hµm t¹i x = 0. Bµi 4. Cho
hµm sè f(x) = EMBED Equation.DSMT4 1) Chøng minh r»ng hµm sè
liªn tôc trªn R 2) Hµm sè cã ®¹o hµm t¹i x = 0 kh«ng? T¹i sao?. Bµi 5.
Cho hµm sè f(x) = EMBED Equation.DSMT4 T×m a, b ®Ó hµm sè cã ®¹o
hµm t¹i x = 1 Bµi 6. Cho hµm sè f(x) = EMBED Equation.DSMT4 T×m
a, b ®Ó hµm sè cã ®¹o hµm t¹i x = 0 Bµi 7. Cho hµm sè f(x) = EMBED
Equation.DSMT4 T×m a ®Ó hµm sè kh«ng cã ®¹o hµm t¹i x = 3. III.
TÝnh ®¹o hµm b»ng c«ng thøc: Bµi 8. TÝnh ®¹o hµm cña çc hµm sè sau:
1) y = EMBED Equation.DSMT4 x3 Ŕ 2x2 + 3x 2) y = - x4 +
2x2 + 3 3) y = (x2 + 1)(3 Ŕ 2x2) 4) y = (x Ŕ 1)(x Ŕ 2)(x Ŕ 3)
5) y = (x2 + 3)5 6) y = x(x + 2)4 7) y = 2x3 Ŕ 9x2 + 12x Ŕ 4
8) y = (x2 + 1)(x3 + 1)2(x4 + 1)3 Bµi 9. TÝnh ®¹o hµm cña çc hµm
sè sau : 1) y = EMBED Equation.DSMT4 2) y = EMBED
Equation.DSMT4 3) y = EMBED Equation.DSMT4 4) y =
EMBED Equation.DSMT4 5) y = EMBED Equation.DSMT4 6) y
= EMBED Equation.DSMT4 7) y = EMBED Equation.DSMT4
8) y = EMBED Equation.DSMT4 Bµi 10. TÝnh ®¹o hµm cña çc hµm
sè sau: 1) y = EMBED Equation.DSMT4 2) y = EMBED
Equation.DSMT4 3) y = (x Ŕ 2) EMBED Equation.DSMT4 4) y
= EMBED Equation.DSMT4 5) y = EMBED Equation.DSMT4
6) y = x + EMBED Equation.DSMT4 7) y = EMBED
Equation.DSMT4 8) y = EMBED Equation.DSMT4 + EMBED
Equation.DSMT4 III. ViÕt ph-¬ng tr×nh tiÕp tuyÕn cña då thÞ t¹i mét
®iÓm Bµi 11. Cho hµm sè y = EMBED Equation.DSMT4 x3 Ŕ 2x2 + 3x
(C) 1) ViÕt ph-¬ng tr×nh tiÕp tuyÕn ( víi ®å thÞ (C) t¹i ®iÓm cã hoµnh ®é
lµ x = 2. 2) Chøng minh r»ng ( lµ tiÕp tuyÕn cã hÖ sè gãc nhá nhÊt Bµi
12. Cho hµm sè y = -x3 + 3x + 1 (C) 1) ViÕt ph-¬ng tr×nh tiÕp tuyÕn (
cña (C) t¹i ®iÓm cã hµnh ®é lµ x = 0 2) Chøng minh r»ng tiÕp tuyÕn ( lµ
tiÕp tuyÕn cña (C) cã hÖ sè gãc lín nhÊt. Bµi 13. 1) ViÕt ph-¬ng tr×nh
tiÕp tuyÕn víi ®å thÞ cña hs: y = x3 Ŕ 3x2 + 2 t¹i ®iÓm (-1; -2) 2) ViÕt
ph-¬ng tr×nh tiÕp tuyÕn víi ®å thÞ cña hµm sè y = EMBED Equation.DSMT4
t¹i ®iÓm cã hoµnh ®é x = 0 IV. ViÕt ph-¬ng tr×nh tiÕp tuyÕn cña ®å
thÞ (C) khi biÕt hÖ sè gãc k. Bµi 14. 1) ViÕt ph-¬ng tr×nh tiÕp tuyÕn víi
®å thÞ cña hµm sè y = EMBED Equation.DSMT4 biÕt hÖ sè gãc cña tiÕp
tuyÕn lµ EMBED Equation.DSMT4 . 2) ViÕt
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