Ly ThuyetBai Tap TOAN 10Tap 2

Ths.LVnonThs.LVnonThs.LVnonThs.LVnon

MC LC Trang

PHN I I S CHNG IV BT NG THC & BT PHNG TRNH ------------------------------------- 1 B BT PHNG TRNH ----------------------------------------------------------------------------- 1 I Bt phng trnh & H bt phng trnh bc nht mt n --------------------------------- 1 Dng ton 1. Gii phng bt trnh bc nht Hai phng trnh tng ng ------ 2 Dng ton 2. Bt phng trnh qui v bc nht H bt phng trnh ---------------- 4 Dng ton 3. Bt phng trnh bc nht mt n cha tham s -------------------------- 10 II Du ca tam thc bc hai & Bt phng trnh bc hai ------------------------------------ 15 Dng ton 1. Xt du & Gii bt phng trnh bc hai ----------------------------------- 15 Dng ton 2. Phng trnh & Bt phng trnh cha cn, tr tuyt i ---------------- 20 Dng ton 3. Bi ton cha tham s trong phng trnh & bt phng trnh --------- 35 CHNG V GC V CNG THC LNG GIC ---------------------------------------------- 47 A H THC LNG GIC C BN ------------------------------------------------------------- 47 B CUNG LIN KT ------------------------------------------------------------------------------------ 52 C CNG THC CNG CUNG ---------------------------------------------------------------------- 62 D CNG THC NHN ------------------------------------------------------------------------------- 69 E CNG THC BIN I --------------------------------------------------------------------------- 77

PHN II HNH HC CHNG III PHNG PHP TA TRONG MT PHNG ------------------------------- 89 A TA VCT & TA IM ------------------------------------------------------------ 89 B PHNG TRNH NG THNG ------------------------------------------------------------ 97 Dng ton 1. Lp phng trnh ng thng & Bi ton lin quan -------------------------- 100 Dng ton 2. Cc bi ton dng tam gic S tng giao Khong cch Gc --------- 105 C PHNG TRNH NG TRN --------------------------------------------------------------- 133 D PHNG TRNH NG ELP ---------------------------------------------------------------- 177 E PHNG TRNH NG HYPERBOL ------------------------------------------------------- 197 F PHNG TRNH NG PARABOL --------------------------------------------------------- 211 G BA NG CONIC -------------------------------------------------------------------------------- 224 H NG DNG TA GII TON I S & GII TCH --------------------------------- 234

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cng hc tp mn Ton 10 Tp II Ths. L Vn on

"Cn c b thng minh" Page - 1 -

iu kin ca bt phng trnh

iu kin ca bt phng trnh l iu kin m n s phi tha mn cc biu thc hai v ca bt phng trnh c ngha. C th, ta c ba trng hp:

+ Dng iu kin c ngha: .



Hai bt phng trnh tng ng

Hai bt phng trnh c gi l tng ng vi nhau nu chng c cng mt tp nghim.

Phng php gii bt phng trnh v h bt phng trnh bc nht mt n

a/ Gii bt phng trnh bc nht mt n Phng php:

Bc 1. t iu kin cho bt phng trnh c ngha (nu c) Bc 2. Chuyn v v gii. Bc 3. Giao nghim vi iu kin c tp nghim S.

b/ H bt phng trnh bc nht mt n Phng php:

Bc 1. t iu kin cho h bt phng trnh c ngha (nu c). Bc 2. Gii tng bt phng trnh ca h ri ly giao cc tp nghim thu c. Bc 3. Giao nghim vi iu kin c tp nghim S.

Gii v bin lun bt phng trnh bc nht dng: .

iu kin Kt qu tp nghim

B BT PHNG TRNH

I Bt phng trnh v h bt phng trnh bc nht mt n

Lu : Ta c th gii tng t cho cc trng hp:

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Ths. L Vn on Chng 4. Bt ng thc v Bt phng trnh

Page - 2 - "All the flower of tomorrow are in the seeks of today"

BABABABAI TI TI TI TP AP AP AP AP DUP DUP DUP DU NGNGNGNG Bi1.Bi1.Bi1.Bi1. Tm iu kin c ngha ca cc phng trnh sau

1/ 1 11x x 1<

+. 2/ 3 2x2 x 1 x 1

x 1 +

+.

3/ 3 x3x 2x 3x 2

+ > +

. 4/ x 3 16 2xx x 3

.

5/ ( )2x 1

x 1x 2

+< +

. 6/ 23

2

1 x2x 1

x 3x 2

+

+.

7/ x x 4 1 x 4+ < + . 8/ 2 x x 2 x + < + .

9/ ( )2 2

x 2 41

1 x x 2

+ +

+ . 10/ x 1 2x 2

x 3 x 4

+ >

.

11/ ( )( )

x 1 1 3

x 3 x 4x 1 6 x

++

+ . 12/ 2x 3 13 4x

x 1 x 2 x 6

+

+ +.

Bi2.Bi2.Bi2.Bi2. Chng minh cc phng trnh sau v nghim

1/ 2x x 8 3+ + . 2/ x 6 3 x 4 + .

3/ 3 x x 5 10 + . 4/ 2 21 x 2 x 1+ + > .

5/ ( ) ( )4x 3 x 10 x 1 x 5 > + . 6/ ( ) ( )( )

25 x 4 x

x 4 x 5x 10 x 2

+. & ( )23x 5 7 x 1 > + .

5/ 12x 3 x 4x 5

<

. & 2x 3 x 4 < .

6/ 1 1x 3 2x 7 x 7

+ < + +

. & x 3 2+ < .

7/ 4x 8 1 x+ < . & ( )( ) ( )( )2 218 x 2x 4x 8 18 x 2x 1 x+ + < + .

8/ 3x 1 x 3+ < + . & ( ) ( )2 23x 1 x 3+ < + .

9/ x 5 0x 1

+ . & x 1 2+ > . Bi 4.Bi 4.Bi 4.Bi 4. Gii cc bt phng trnh sau

1/ ( )3 2x 73

2x5 3

+ > . 2/ 2x 1 33 x

5 4

+ > + .

3/ ( ) ( )5 x 1 2 x 1

16 3

+ < . 4/

( )3 x 1 x 12 3

8 4

+ + < .

5/ 3x 1 x 2 1 2x2 3 4

+ < . 6/ x 1 x 2 x2

2 3 6

+ + < + .

7/ 10 3x 2x 79 2x2 4

+ > . 8/ ( ) ( )3 2x 2 x 1 4+ + .

9/ ( )( )x x 2 x 3 x 1+ < + . 10/ ( )( )1 x 3 2 1 x 5 1 x 3 + > .

11/ ( ) ( )2x 4 x 1 0 + > . 12/ ( ) ( )2x 2 x 3 0+ > .

13/ x 3 3 x . 14/ x 1 3 x 1 < + .

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15/ x 2 4x 4 x 4

. 16/

( )10 x x 44

x 4

>

.

17/ ( )( )2x 1 x 1 0 + . 18/ x 3 01 2x

.

19/ ( )x 3 x 2 0 . 20/ ( )4 x 5 x 0 .

Dng 2. Bt phng trnh qui v bt phng trnh bc nht mt n H bt phng trnh bc nht mt n

Du ca nh thc bc nht

a/ S dng bng xt du (tri tri phi cng: vi h s a)

b/ S dng trc s

Nu th :

Nu th :

Bt phng trnh tch s

Dng: Trong : l cc nh thc bc nht.

Phng php: Lp bng xt du . T suy ra tp nghim ca .

Bt phng trnh cha n s mu

Dng: Trong : l cc nh thc bc nht.

Phng php: Lp bng xt du . T suy ra tp nghim ca .

Lu : Khng nn qui ng v kh mu.

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BABABABAI TI TI TI TP AP AP AP AP DUP DUP DUP DU NGNGNGNG Bi5.Bi5.Bi5.Bi5. Lp bng xt du ca cc hm s sau

1/ ( )f x x 1= + . 2/ ( )f x 2x 1= + .

3/ ( )f x 2 x= . 4/ ( )f x 2 2 x= + .

5/ ( )f x 3 3x= . 6/ ( ) ( )2f x m 1 x 1= + .

7/ ( ) ( )2f x 4m 1 m 2m 2 x= + . 8/ ( ) ( )2f x 4m 2m 1 x 3m= + + .

9/ ( ) ( )3 2f x m m 3 m 1 x= + + . 10/ ( ) ( )f x 3x 3x 1= .

11/ ( ) ( )( )5x 3

f xx 3 2x 1

=

. 12/ ( ) ( )

x x 1f x

x 2

+=

.

13/ ( ) 21 1

f xx 1 x 1

=

. 14/ ( ) ( )2f x 2x 5= .

15/ ( ) ( )4f x 3 7x= . 16/ ( ) ( )2f x 3x 1= + .

17/ ( ) ( )3f x 2x 7= . 18/ ( ) ( )7

f x 3 x 2= .

19/ ( ) ( )5f x 5x 2= + . 20/ ( ) ( )f x x 8 3x= .

21/ ( ) ( )( )f x 4x 1 x 1= . 22/ ( ) ( )( )f x 3x 7 5 2x= + .

23/ ( ) ( )( )f x 2x 5 3x 7= + + . 24/ ( ) ( )3f x x x 3= .

Bt phng trnh cha n trong du gi tr tuyt i

Tng t nh gii phng trnh cha n trong du gi tr tuyt i, ta thng s dng nh ngha hoc tnh cht ca gi tr tuyt i kh du gi tr tuyt i.

Dng 1. .

Dng 2. .

, ta chia bi ton thnh nhiu trng hp. Trong mi trng hp ta xt du ca qui tc

: c ngha

Lu : Vi , ta lun c v .

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25/ ( ) ( )3f x x 2 7x= . 26/ ( ) ( ) ( )3f x x 1 4 x= + .

27/ ( ) ( )( )5

f x x 3 1 2 x= . 28/ ( ) ( ) ( )3 5f x 2 x 2x 5= + .

29/ ( ) ( )( )( )f x 2x 4 x 1 6 2x= + . 30/ ( ) ( )( )( )f x 4 x x 1 5x 2= + .

31/ ( ) ( )( )f x 3x 2x 7 9 3x= + . 32/ ( ) ( ) ( )3 2f x 1 3x x 1= .

33/ ( ) ( ) ( )2f x 4 x 5x 2= . 34/ ( ) ( )2f x x 2x 3= . Bi 6.Bi 6.Bi 6.Bi 6. Gii cc bt phng trnh sau

1/ ( )( )( )x 1 x 1 3x 6 0+ > . 2/ ( )( )2x 7 4 5x 0 .

3/ ( )2x x 20 2 x 11 > . 4/ ( )( )3x 2x 7 9 3x 0+ .

5/ 2 0x 3

>

. 6/ 3 02 3x

>

.

7/ 1 2x 1

. 8/ x 1x 5 2

.

9/ 2

x0

x x

. 10/ 4x 3 6

2x 5

+

.

11/ 2

x 20

x 4

+. 18/ x 3 x 5

x 1 x 2

+>

+ .

19/ 2x 3 23x 7 3

+

+. 20/ 7x 5 4

8x 3

+.

21/ x 3 1 2xx 5 x 3

.

23/ 2

2

x 2x0

x 4

+

. 24/

2

x 20

x 4

.

25/ ( )( )

2

5

4x 30

2x 5

+

. 26/

2

3 2x0

x

> .

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27/ 2x 5 12 x

. 28/ 2x 5 3x 2

3x 2 2x 5

+

.

3/ 4 1

12x x3 24x 3 2 x

2 3

+ +. 10/

9x 12 4x 15

19 3x 7 5x

+ < +.

11/ 5x 7

3 x3

1 5x3x 4

13

+ < +

. 12/ ( )22

3 2x 15x

2 6

x x 2

+ < +

.

13/ x 3 4 2x

5x 3 4x 1

+ + < . 14/

x 33 x

71 5x

4x 22

+ < +

.

15/ ( )22

5x 2 4x 5

x x 2

< + < +. 16/ ( )( )

7x 5 0

2x 3 x 1 0

. 20/ ( )( )

x 27

x 32x 3 x 3 0

> +

.

21/ ( )( )

2x 31

x 1x 2 2x 4

0x 1

+ +

. 22/

x 1 x 4

1 2x 3 2x2

1x 1

+ >



23/ ( )( )x 1 x 4 0

2 1

2x 1 x 3

+ + +

. 24/

x 1 2x 3

5 3xx 3

23x x 5

< +

.

25/ 2 1

2x 1 3 xx 1

+ .

7/ 5x 12 3 < . 8/ 1 4x 2x 1 < + .

9/ x2 8 7 . 10/ 3x 15 3+ .

11/ x 1x 12

+ > . 12/ 4x

x< .

13/ xx 22

< . 14/ 2x 5 x 1 + .

15/ 2x 1 x+ . 16/ x 2 x 1 > + .

17/ 2 1x 4

>

. 18/ 2x 1 2x 1

>

.

19/ 2 8x 13 9

>

. 20/ x 2 x 4 2< + .

21/ 1 2x 2 x 1

+ . 22/

x 11

x 1

+.

25/ 1 3x 2x 1

14x x 1

>

+. 26/ x 1 2x 6 x 5 + + .

27/ 2x 2 x 2 3x 2+ + +



BABABABAI TI TI TI TP AP AP AP AP DUP DUP DUP DU NGNGNGNG Bi10.Bi10.Bi10.Bi10. Tm tham s m bt phng trnh sau y v nghim

1/ 2 2m x 4m 3 x m+ < + . 2/ ( )2m x 1 m 3m 2 x+ + .

3/ 2mx m mx 4 > . 4/ ( ) ( )23 mx 2 x m m 1 < + . Bi11.Bi11.Bi11.Bi11. Gii v bin lun cc bt phng trnh sau

1/ ( )m x m x 1 . 2/ mx 6 2x 3m+ > + .

3/ ( )m 1 x m 3m 4+ + < + . 4/ 2mx 1 m x+ > + .

5/ ( )m x 2 x m x 16 3 2

++ > . 6/ ( ) ( )23 mx 2 x m m 1 < + .

7/ 2mx m 2x 4 > . 8/ x 2m 2 mx+ > + .

9/ 2m x 1 x m + . 10/ 22x m mx 3m 2+ + .

11/ ( )m x 2 2mx m 1 + . 12/ 2 225m x m x 5 < .

13/ ( ) ( )32 x m m 1 3 mx + . 14/ ( )( ) 2m 1 m 2 x m 4+ .

15/ ( )2m 3m 2 x m 1 + . 16/ 2x 25m 5mx 1+ + .

17/ ( )2 3m 2m x 8 4mx m+ + < + . 18/ ( )m x 1 1+ > .

19/ ( )( )m 1 mx 1 2+ > . 20/ ( )( )2 2m 3m 2 mx 1 m 1 + .

21/ ( )2m 3m 2 x m 1 + . 22/ ( )x x m 0 .

23/ ( )( )x 1 x m 0 + . 24/ ( )( )x 3 6m 12 x 0 .

25/ ( )( )2x 6 x m 1 0 + . 26/ x 3 0x 2m 1

>

+ +.

27/ x 4m 02 x

>

. 28/ x 4m 0

4 x

.

29/ ( )( )x m x 1 m 0+ + > . 30/ ( )( )2x m x 2 m 0 + .

31/ m x 0m 2 x

+ +. 32/ x 4m 0

2x m 4

+>

+.

33/ ( )2m x 1 m 4mx 3x < . 34/ 2x m 1 0x 1

+ >

+.

35/ mx m 1 0x 1

+ .

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Page -12- "All the flower of tomorrow are in the seeks of today"

Bi12.Bi12.Bi12.Bi12. Gii v bin lun h bt phng trnh

1/ 2x 1 0

3x 1 m

+ + . 2/

4x 1 0

x 3m 0

+ .

3/ ( )( )x 1 4 x 0x m 1 0

> + . 4/ ( )( )x 5 7 x 0

x m 1 0

.

5/ 3 4

1 x x 1x m 1 0

> +

. 6/ 2 5

1 x 1 2xx m 1 0

>

.

7/ ( ) ( )2 2x 2

4x 82

x m 1 x m 1

< + >

. 8/ x 1 0

mx 2 0

+ > . 10/

x m 1

mx 2 m

+ + .

Bi13.Bi13.Bi13.Bi13. Tm tham s m h bt phng trnh sau c nghim

1/ 4x 5 3x 2

3x 2m 2 0

> + + + + + + . 4/

12

x 1x m 2

> +

.

5/ 1

2x mx 1 2

+

. 6/ 2

1m xx 3

.

7/ x 7 0

mx m 12

+. 8/ ( )

2x 1 0

3m 2 x m 0

> >.

9/ ( ) ( )2x 1 x 2

m m 1 x 1 m 2 x 3m 7

< + + + > + +. 10/ ( )

3x 3 2x

x 1 2m x 2m

+ > .

11/ x m 1 0

3m 2 x 0

+ > >. 12/ ( )

mx 1 0

3m 2 x m 0

> >.

13/ 2x 4m 2mx 1

3x 2 2x 1

+ + + > . 14/

7x 2 4x 19

2x 3m 2 0

+ +


"Cn c b thng minh" Page-13-

1/ 2x 7 8x 1

m 5 2x

+ < + < . 4/ ( )2

4x 5 11 x

m mx 3 3 x

+ > > .

5/ 8

13 xx 3 mx

> >

. 6/ x 7 0

mx m 12

+.

7/ ( )( )x 1 x 2 0mx 1 2x m

.

9/ ( ) ( )( )

2 2

2

3x 5 x 1

x 2 x 1 9

m x 1 3m 2 x m

+ + + + > +

. 10/ ( ) ( )2 21 1

x 1 x 2x x 3m x m 3

x D 2;4 = .

5/ ( )2m 1 x 4+ < x D 1;2 = .

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6/ 5x m 2 ( )x D 0;4 = .

7/ m 3x 2 ( )x D 3;1 = .

8/ ( )3x 1 2 x 1 < + (x D m 1;4 = .

9/ x 2m 1 ( )x D ;2 m = .

10/ 1 2x 0 x D m 1;2 m = .

11/ x 2m 1 0 + > ( )x D m 1;2m 3 = + .

12/ ( )( )x 1 x 3 0 + < )x D m;1 m = .

13/ ( )( )2x 1 x 4 0 + > (x D 2m 1;m 3 = + .

14/ ( ) ( )2m 1 x m x 3 1 0+ + + > x D 1;2 = .

15/ mx 2 x m+ + x D 0;2 = .

16/ ( )2 m 1 x m 0 + > ( )x D 1;3 = .

17/ ( )m 1 x 3m 0+ x D 1;2 = .

18/ mx 3m 2 0 + > ( )x D 0; = + . Bi17.Bi17.Bi17.Bi17. Tm tham s m h bt phng trnh sau c nghim duy nht

1/ x m

2x 1 0

+ . 2/ ( )

2x m 0

2 x 1 3

.

3/ ( )( )x 1 x 2 0x 2m 2

+ . 4/ ( )( )x 2 x 4 0

x 2m 1

+.

5/ ( )2

2x 3 x 7x 1

2m 8 5x

+ + +. 6/

x y 0

mx m 12

+.

7/ ( )

2x 3x 1

xm x 1 2

+ +

. 8/ ( )

2x x 3x

xm x 1 2

+

.

9/ ( )( )( )( ) ( )

2

2

x 1 x 2 0

x m 1 2m 1 3m 2 x

+ . 10/ ( )2m x 1 x 3

4mx 3 4x

+ + + .

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II Du ca tam thc bc hai v bt phng trnh bc hai

Dng 1. Xt du ca tam thc bc hai Gii bt phng trnh bc hai

Du ca tam thc bc hai

: cng du vi a.

: cng du vi a.

: Trong tri.

: Ngoi cng.

Gii bt phng trnh bc hai

Bc 1. Cho tm nghim (nu c).

Bc 2. Lp bng xt du ca da vo du ca tam thc bc hai.

Bc 3. T bng xt du, suy ra tp nghim ca bt phng trnh.

Gii bt phng trnh bc hai dng: hoc

Bc 1. Tm iu kin xc nh nu c.

Bc 2. Cho tm nghim .

Bc 3. Lp bng xt du Du ca v .

Bc 4. T bng xt du tp nghim S1. Vy tp nghim bt phng trnh: .

Gii h bt phng trnh bc hai mt n dng:

Bc 1. Gii c tp nghim tng ng l .

Bc 2. Nghim ca h l .

Lu

Hai bt phng trnh v c gi l tng ng nu v ch nu

.

Cn s dng thnh tho cc php ton trn tp hp.

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BABABABAI TI TI TI TP AP AP AP AP DUP DUP DUP DU NGNGNGNG Bi18.Bi18.Bi18.Bi18. Lp bng xt du ca cc hm s sau

1/ ( ) 2f x x 3x 2= + . 2/ ( ) 2f x 3x 2x 1= + .

3/ ( ) 2f x x 4x 5= + + . 4/ ( ) 2f x x x 12= .

5/ ( ) 2f x 2x 5x 2= + . 6/ ( ) 2f x 4x 12x 9= + .

7/ ( ) 2 9f x 2x 6x2

= + . 8/ ( ) 2f x x x 1= + + .

9/ ( ) 2f x 2x 7x 8= + . 10/ ( ) 2f x 3x 2x 8= .

11/ ( ) 2f x x 2x 1= + . 12/ ( ) 2f x 2x 7x 5= + .

13/ ( ) 2f x 3x 2x 5= + . 14/ ( ) 2f x x 5x 6= .

15/ ( ) 2 1f x 3x 2x3

= + . 16/ ( ) 2 9f x 2x 6x2

= + .

17/ ( ) ( )2f x 2x 2 1 2 6 x 2 6= + + + . 18/ ( ) ( )2f x 6x 2 3 3 2 x 6= + + .

19/ ( ) 2f x 3x 8 2x 16 2= + + . 20/ ( ) ( ) ( )2 2f x x 4 x 8x 11 4= + + + + .

21/ ( ) ( )( )2 2f x x 5x x 5x 10 24= + + . 22/ ( ) ( )( )2f x 4 2x x 5x 4= + .

23/ ( ) ( )( )2f x 3x 10x 3 4x 5= + . 24/ ( ) ( )( )2 2f x 3x 4x 2x x 1= .

25/ ( )2

2

2x x 3f x

4x x

=

. 26/ ( )

( )( )2 22

3x x 3 xf x

4x x 3

=

+ .

27/ ( ) 2x 1 1

f xx 1 x 1

+=

. 28/ ( ) ( )2f x mx 1 2m x 2= + .

29/ ( ) ( ) ( )2 2f x m 1 x 2 m 1 x 4= + + . 30/ ( ) ( )2 2f x mx m 1 x m= + + . Bi19.Bi19.Bi19.Bi19. Gii cc bt phng trnh sau

1/ 2x 4x 3 0 + . 2/ 22x 5x 3 0 + .

3/ 27x 4x 3 0 < . 4/ 2x 6x 9 0 + > .

5/ 23x x 1 0+ + . 6/ 2x 7x 10 0 + .

7/ 22x 4x 3 0+ + < . 8/ 22x 5x 2 0 + .

9/ 25x 4x 12 0 + + < . 10/ 216x 40x 25 0+ + > .

11/ 22x 3x 7 0 + . 12/ 23x 4x 4 0 + .

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13/ 2x x 6 0 . 14/ 2

2

3x x 40

x 3x 5

+>

+ +.

15/ 2

2

4x 3x 10

x 5x 7

+ >

+ +. 16/

2

2

5x 3x 80

x 7x 6

+

+ +.

21/ 4 3 2

2

x 4x 2x0

x x 30

+>

. 22/

2x 6x 70

x 7

+

+. 28/

2x 3x 40

1 2x

.

29/ 2

2

x 10

x 3x 10

+

+.

Bi20.Bi20.Bi20.Bi20. Gii cc bt phng trnh sau

1/ 2

5x 11

x 3

+.

3/ 2

2

4 3x1

x x 1

>

+ +. 4/ x 1 5 x

x 1

+

+.

5/ 2

x 2 1

2x 1

. 14/ 1 2 3x 1 x 3 x 2

+

+ + +.

19/ ( )( )2 218x 18

x 2x 2x 2 0x 2x

. 20/ ( )( )2 2

32x 48x 3x 2x 3 0

x 3x

++ +

+.


1/ 4 2x 5x 4 0 + < . 2/ 4 2x 2x 63 0 .

3/ 4 2x 3x 2 0 + > . 4/ 6 3x 19x 216 0+ .

5/ ( )( )2 2x 3x 1 x 3x 3 5+ + + . 6/ ( )( )2 2x x 1 x x 7 5 + . 2/

2

2

x x 5 0

x 6x 1 0

+ + .

3/ 2

2

3x 8x 3 0

6x 17x 7 0

+ + . 4/

2

2

5x 7x 6 0

5x 13x 6 0

+ + >.

5/ ( )( )x 1 2x 3 0x 1 0

. 6/

2

2

x 4x 3 0

x 6x 8 0

+ + . 8/

2

2

x x 3 3 0

x 2x 2 2 2 0

>

.

9/ ( )2

2

2x 2 3 1 x 3 1 0

5x 8x 3 0

+ + + + . 12/

2

2

2x 5x 4 0

x 3x 10 0

+ .

13/ 2

2

x 4x 7 0

x 2x 1 0

+ + + + >.

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17/ ( )2

2

4x 4x 1 0

3x 9 2 x 3 2 0

+ +

. 18/ ( )( )2

2 2

2x 9x 7 0

x x 6 x 2x 2 0

+ + > + + . 20/

( ) ( )

2

2

x 4x 3 0

2x 1 4x x 1

+ < +

.

21/ ( )( )( )x x 5 4x 2

2x 1 x 3 4x

+ + + . 22/

2

2

x 2x 74 1

x 1

+.

23/ 2

2

1 x 2x 21

13 x 5x 7

+. 24/

2

2

10x 3x 21 1

x 3x 2

<

. 32/ ( )( )

2 3

1 2 2x 3

x 1 x x 1 x 12x 3 4x 2 0

+ + + + + + >

.

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Dng 2. Phng trnh Bt phng trnh cha cn, cha du tr tuyt i

Phng trnh Bt phng trnh cha n trong du gi tr tuyt i gii phng trnh, bt phng trnh cha n trong du gi tr tuyt i, ta thng s dng

nh ngha hoc tnh cht ca gi tr tuyt i kh du gi tr tuyt i. Xem li cch gii phng trnh tr tuyt i (Chng 3. Phng trnh v h phng trnh).

a/ Dng 1.

b/ Dng 2.

c/ Dng 3.

Lu

. .

Vi , ta c: v .

. .

Phng trnh Bt phng trnh cha n trong du cn gii phng trnh, bt phng trnh cha n trong du cn ta thng dng php nng lu tha

hoc t n ph kh du cn. Xem li cch gii phng trnh c du cn (Chng 3. Phng trnh v h phng trnh).

a/ Dng 1.

b/ Dng 2.

c/ Dng 3.

Lu : i vi cc phng trnh, bt phng trnh, khng c dng chun nh l thuyt, ta thc hin:

Bc 1. t iu kin cho cn c ngha. Bc 2. Chuyn v sao cho 2 v u

khng m. Bc 3. Bnh phng 2 v kh cn.

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BABABABAI TI TI TI TP AP AP AP AP DUP DUP DUP DU NGNGNGNG Bi23.Bi23.Bi23.Bi23. Gii cc bt phng trnh sau

1/ x 1 1 . 2/ 2x 1 2 .

3/ 4x 2 3+ . 4/ x 3 1 .

5/ x 2 x 4 . 6/ 2x 1 1 3x+ .

7/ 2x 1 x . 8/ 5 5x 2 x 4 + .

9/ ( )22x x 6 2 x 2+ + + . 10/ 2x x 2 x 3 + .

11/ 22 x x 2 1 2x+ + . 12/ 2 2x 1 2x 2x + .

13/ 2x 2 3x x 1+ + . 14/ 2x 1 x 3 + .

15/ 2 2x x 2 x 2x 3 + + . 16/ 2 24x 3 2x 2x 1 + .


1/ x 1 x 3 > . 2/ x 5x 4< .

3/ x 3 2x 0 < . 4/ 5 x 2x 7 .

5/ x 1 2x 1+ < . 6/ x 2 x 3 3 .

7/ 3x 2 1 x + + < . 8/ 7x 11 x 1 0+ + + .

9/ 2x 5 x 1 0 + . 10/ 2x 2x x 1 + .

11/ 2x 8 x 4+ . 12/ 2x 4x x 3 > .

13/ 2x 4x 3 x 1 + < + . 14/ 2x 3x 2 2x 1 + < .

15/ 2x x 6 x 2+ + . 16/ 22x 3x 5 x 1 < .

17/ 2x 7x 6 3 2x + < + . 18/ 2x 6x 5 8 2x + > .

19/ 23x 13 2x 1+ + < . 20/ 2x 4x 5 2x 3 + + .

21/ 2x 6x 3 x 1+ < + . 22/ 22x 6x 1 x 2 0 + + > .

23/ 2x 4x 1 0+ > . 24/ 23x 5x 5x 2 .

25/ 2x x 12 7 x < . 26/ 21 x 2x 3x 5 0 + .

27/ 2x 3x 10 x 2 < . 28/ ( )23 x x 6 2 1 2x + + > .

29/ 23x 13x 4 2 x 0+ + + < . 30/ 22 3x x 2x 1+ .

31/ 22x 6x 20 2 x 0 + > . 32/ 2x 2x 15 x 3 .

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33/ x 2 x+ . 34/ 2x x 6 x 1+ < .

35/ 2x 1 2x 3 . 36/ ( )2x 8x 2 x 1+ + .

37/ 2x 9x x 4+ > + . 38/ 22x 1 1 x > .

39/ 2x 4x 12 x 4 > . 40/ 2x 5x 14 2x 1 .

Bi25.Bi25.Bi25.Bi25. Gii cc phng trnh sau

1/ 4 2x 4x 3 0 + = . 2/ 4 2x 10x 9 0 + = .

3/ 4 2x 3x 4 0 = . 4/ 4 2x x 12 0 = .

5/ 4 2x x 3 0 + = . 6/ ( )( )2 21 x 1 x 3 0 + + = .

7/ 3x 2 2x 1 = . 8/ 2x 3 x 3 = .

9/ 24 6x x 4 x = + . 10/ 5x 10 8 x+ = .

11/ x 2x 5 4 = . 12/ 23x 9x 1 x 2 + = .

13/ 23x 9x 1 x 2 + = . 14/ ( )2x 3x 2 2 x 1 = .

15/ 3x 7 x 1 2+ + = . 16/ 2x 2x 4 2 x+ + = .

17/ 2 2x 9 x 7 2+ = . 18/ 2 2x 3x 2 x 3x 4 + = .

19/ 2 2x 6x 9 4 x 6x 6 + = + . 20/ 2 2x 3x 3 x 3x 6 3 + + + = .

21/ ( )( )3 x 6 x 3 x 6 x 3+ + = + + . 22/ ( )( ) 2x 1 x 4 3 x 5x 2 6+ + + + = .

27/ x 3 4 x 1 x 8 6 x 1 1+ + + = . 28/ ( )( ) 2x 1 x 2 x 3x 4+ + = + .

29/ x 2 2x 5 x 2 3 2x 5 7 2 + + + + = .

30/ x 5 4 x 1 x 2 2 x 1 1+ + + + + = .

31/ 2x 2 2x 1 2 2x 3 4 2x 1 3 2x 8 6 2x 1 4 + + + = .

32/ 2

2 2x 4356 x x x 4356 x 5x

+ + + = .

33/ 21 x 21 x 21x21 x 21 x

+ + =

+ . 34/ 3 3 3x 5 x 6 2x 11+ + + = + .

35/ 33 3x 1 3x 1 x 1+ + + = . 36/ 3 31 x 1 x 2+ + = .

37/ 3 3 3x 1 x 2 x 3 0+ + + + + = . 38/ 22x 1 1 x > .

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39/ 2x 4x 12 x 4 = . 40/ 2 23x 5x 8 3x 5x 1 1+ + + + = .

41/ 33 5x 7 5x 13 1+ = . 42/ 3 39 x 1 7 x 1 4 + + + + = .

43/ 3 324 x 5 x 1+ + = . 44/ 4 447 2x 35 2x 4 + + = .


1/ 1 x 1 2x

+ . 2/ 1 8x 3 4

4x

.

3/ 21 1 4x

3x

< . 4/

23x 16x 52

x 1

+

.

5/ 22 3x x 4

2x

+ + +< . 6/

21 21 4x x 1

x 4 2

. 8/

2 26 x x 6 x x

2x 5 x 4

+ +

+ +.

9/ ( ) 2 2x 3 x 4 x 9 . 10/ ( ) 22x 1 x 1 4x 1+ + < .

11/ ( ) 2 2x 2 x 3x 4 x 4+ . 12/ 2x 1 x 8 3+ > .

13/ 1 4x 2x 1 + . 14/ x 1 x 0 < .

Bi27.Bi27.Bi27.Bi27. Gii cc phng trnh sau

1/ 2x 2

2x 1

=

. 2/ 2x 5x 4 x 4 + = + .

3/ 2 2x 8x 12 x 8x 12 + = + . 4/ 2 2x 5x 4 x 6x 5 + = + + .

5/ 2 2x 1 x 2x 8 = + . 6/ 22x 5x 2 0 = .

7/ 2x 1 x 1 + = . 8/ 2 22 3x 6 x 0 = .

9/ 2x 2x 3 x 1 + = + . 10/ 2 x x 3 3 = .

11/ 2x 4 2x x 2 1 + = + + . 12/ ( )

2x 1 x 12

x x 2

+ +=

.

13/ x 3 4 x 1 x 8 6 x 1 1+ + + = . 14/ x 14x 49 x 14x 49 14+ + = .

14/ 2 2 x 1 1 3 = . 16/ ( )2 2x 1 x 2 2x 1+ = .


1/ 4 3x 8 . 2/ 2x 4 x 12 + .

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3/ 4xx

< . 4/ 2x 4x 5 < .

5/ 2x 2x x < . 6/ 2x 2x 3 3x 3 < .

7/ 2x 3x x 2 0 + < . 8/ 2x 4 2x 4 + < .

9/ 2x x 1 2x 5 + . 10/ 2

2

x 5x 41

x 4

+

.

11/ 2

2

x 41

x x 2

+ +. 12/ 2 2x x x 1 .

13/ 2x 6 x 5x 9 > + . 14/ 2 2x 4x 3 x 4x 5+ + > .

15/ x 3 x 1 2 + < . 16/ 3x 2 7 .

17/ x 3 3x 15 > + . 18/ 22x 5x 3 0 < .

19/ 2x 2x 8 2x > . 20/ 2x 2x 3 3x 3 .

21/ 1 4x 2x 1 > + . 22/ 23 x 1 x 7 0 + > .

23/ 2 2x 3x 2 x 2x 0+ + + + . 24/ 2x 8 x 3x 4 > + .

25/ x 1 x 2 3 + + < . 26/ 2 x 3 3x 1 x 5 + + .

27/ 2x 1 2x 0 < . 28/ 2x 5x 3 x 2 < .

29/ 2x 7x 12 x 4 + > . 30/ 3x 1 1 x .

31/ 34x 3x 1 . 32/ 2x 6 x 5x 9 > + .

33/ 2x 2x 3 2 2x 1 > . 34/ 2 x 1 x 2 3x 1+ < + + .

35/ 2 1x 4

>

. 36/ 2x 1 2x 1

>

.

37/ 2

2

x 4x1

x x 2

+ +. 38/ 2x 5 1 0

x 3

+ >

.

39/ 2

x 23

x 5x 6

+. 40/

2

2

x 3x 21

x 3x 2

+ +

+.

41/ 2

2

x 5x 41

x 4

+

. 42/

2

2x 1 1

2x 3x 4

+.

9/ 2x 2 2 x 1 x + + . 10/ x 1 2x 6 x 5 + + .

11/ 2x 2 x 2 3x 2+ + + . 18/ 2x 4 7x x 12 + + .

19/ 2x 6 x 6x 7+ + . 20/ 2x 3x 4 2 x 3 2 0+ + + > .

21/ x 1 2x 4 3x 2 3 + + . 22/ 2

24x 1

x x 12x 1

< +

.

23/ 2

2

x 5x 41

x 4

+

. 24/

x 2 x2

x 3 x

+ + .

3/ 22 x 10 x 2 < . 4/ 2 2x 9 x 7 2+ + .

5/ x 2 x 1 x+ + . 6/ 2x 1 2 x x 3+ .

7/ x 3 x 1 x 2+ < . 8/ x 3 7 x 2x 8+ .

9/ x 3 x 1 x 2+ < . 10/ x 2 2x 3 4x 7+ .

11/ x 11 x 4 2x 1+ + . 12/ x 3 2 x 3+ + .

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13/ x 4 5 4x 1+ . 14/ 2x 1 5 x 3 2 x + > .

15/ 2x 3 x 1 1+ + > . 16/ 1 x 4 x 1 + + > .

17/ 1 x 4 x 3 + + > . 18/ x 4 2x 6 1+ .

19/ 3x 7 1 x 2+ + > . 20/ 2 25 x 5x 1 4x + > .


1/ 2 25x 10x 1 7 2x x+ + . 2/ ( )( ) 2x 1 x 4 5 x 5x 28+ + < + + .

3/ ( )( ) 24 x 6 x x 2x 12+ . 4/ ( )( ) 24 4 x x 2 x 2x 12 + .

5/ ( ) 2x x 3 x 3x 6+ + . 6/ ( )( ) 2x 1 x 2 x 3x 4+ + < + .

7/ 2 2x 2x 2x 4x 3+ > + . 8/ x x 12. 3x 1 x

+ >

+.

9/ 3x 1 x2. 1x 3x 1

+

. 10/ 6x 1 2x2. 1

x 6x 1

< +

.

11/ 2x 1 x 3x1x 2x 1 2x 1

+ + >

. 12/

( )2 x 1x x 12. 3

x 1 x x

+

.

13/ 2x x 1 3x 33. 4. 10x 1 2x 2x

+ +

. 14/ x 3 2x 12 8x5. 5

3 2x x x

+ > +

.

15/ 5 15 x 2x 42x2 x

+ < + + . 16/ 2 2

1 3x1

1 x 1 x>

.

Bi32.Bi32.Bi32.Bi32. Gii cc bt phng trnh sau (nhn lin hip)

1/ ( ) ( )( )22

4 x 1 2x 10 1 3 2x+ < + + . 2/ ( )

2

2

xx 4

1 1 x

++

.

5/ ( )

( )

2

2

3x 2x 2

4x 1 x 1

+< +

+ + . 6/

( )2

2

25xx

6x 3 x 3

+ + +

.

7/ ( )( )x 4 x 5 x 1 x 4 3+ + + > . 8/ ( )( )x 1 x 2 x 6 x 3 3+ + + < .

9/ ( )

( )2

2

16x4 3x 2

4x 1 1

+

. 10/ ( )

2

2

9x4x 5

5x 1 2x 1

+

.

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11/ ( )

( )

2

2

x 2x 8

3x 1 2x 1

+ +

+ . 12/ 2 23x 5x 7 3x 5x 2 1+ + + + .

13/ ( )x 8 x 3 x 3+ + . 14/ ( )x 1 x 3 8 x 2x 11 .

15/ ( )x 2 3x 5 2x 3 x 8 + . 16/ ( )2x 3 x 1 x 2 1 .

17/ ( )2x 8 x 3 7 x 2x 4 + + > . 18/ ( ) ( )x 3 2x 8 7 x 3 x 5+ > .

19/ ( )5x 1 3x 2 2x 3 1 x + > + . 20/ ( )2x 4 5x 1 x 1 4x + < .

21/ ( )3x 5 x 2 2x 3 5 x + + < . 22/ ( )1 2x x 4 1 x 2x 3 + + < + .

23/ ( )( )3x 6 3x 3 3x 1 3x 2 3+ + + . 24/ ( )( )x 12 x 6 x 2 x 4 6+ + + . Bi33.Bi33.Bi33.Bi33. Gii cc bt phng trnh sau

1/ ( )22 x 16 7 x

_ x 3x 3 x 3

>

. 2/

2 26 x x x x 6

2x 5 x 4

+ + +

+ +.

3/ ( )2 2x 3x 2x 3x 2 0 . 4/ 13 x x 2 94 x 2 x

+

.

5/ ( )2x 1 x 3 2 x 3 2x 2 + + . 6/ 3 13 x 2x 72x2 x

+ < + .

7/ 3x 8

x 2x

+> . 8/

2xx 1 x 1 2

4+ + .

9/ ( )( )x 1 3 x x 1 3 x 2+ + + . 10/ ( ) ( )22x x 4 x 4x x 2 2+ + + < .

11/ 21 x x 1 x + < . 12/ 2 2 2x 2x x x x 7x+ + + .

13/ 2 2x 3x 2 x 4x 3 x 4x 4 + + + .

14/ 2 2 2x 8x 15 x 2x 15 4x 18x 18 + + + + .

15/ 27x 7 7x 6 2 49x 7x 42 181 14x+ + + + < .

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Bai tp qua cac ki thi Bi34.Bi34.Bi34.Bi34. Cao ng Bn Cng Hoa Sen khi D nm 2006

Gii BPT: 2x 4x x 3 > . S: ( ) 9x ;0 ;2

+ .

Bi35.Bi35.Bi35.Bi35. Cao ng K Thut Y T I nm 2006

Gii BPT: 2x 4x 5 2x 3 + + . S: 2x ;3

.

Bi36.Bi36.Bi36.Bi36. Cao ng Kinh T Cng Ngh Tp. H Ch Minh khi A nm 2006

Gii BPT: 2x 4x 3 x 1 + < + . S: )1x ;1 3;3

+ .

Bi37.Bi37.Bi37.Bi37. Cao ng S Phm in Bin khi A, B nm 2005

Gii BPT: 1 x 1 2x

+ . S: 3x 0;

4

.

Bi38.Bi38.Bi38.Bi38. Cao ng iu Dng H chnh quy nm 2004

Gii BPT: x 11 x 4 2x 1+ + . S: x 4;5 .

Bi39.Bi39.Bi39.Bi39. H Cao ng khi T, M nm 2004 trng i hc Hng Vng

Gii BPT: 2x x 6 x 2+ + . S: (x ; 3 . Bi40.Bi40.Bi40.Bi40. Cao ng Ti Chnh Qun Tr Kinh Doanh khi A nm 2006

Gii BPT: x 3 2 x 1+ > . S:

Bi41.Bi41.Bi41.Bi41. Cao ng S Phm Vnh Phc khi D, M, T nm 2006

Gii BPT: x 1 x 3 > . S: Bi42.Bi42.Bi42.Bi42. Cao ng S Phm Tp. H Ch Minh khi A nm 2006

2 2x 2x 5 4 2x 4x 3+ + + + . S:

Bi43.Bi43.Bi43.Bi43. Cao ng S Phm Yn Bi khi B, D1, M nm 2006

Gii BPT: 2x 2 x 1 0 . S:

Bi44.Bi44.Bi44.Bi44. Cao ng S Phm Lo Cai khi A nm 2006

Gii BPT: 5 2x 2 x 2x 5 + . S:

Bi45.Bi45.Bi45.Bi45. Cao ng Giao Thng Vn Ti nm 2006

Gii BPT: 2 2x 4x 3 2x 3x 1 x 1 + + . S:

Bi46.Bi46.Bi46.Bi46. Cao ng S Phm Cn Th khi A nm 2006

Gii BPT: ( )( ) ( )x 5 3x 4 4 x 1+ + > . S:

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Bi 47.Bi 47.Bi 47.Bi 47. Cao ng Kinh T K Thut Thi Bnh nm 2004

Gii BPT: ( )22 x 1 x 1 + . S: ( )x 1 x 1;3= . Bi48.Bi48.Bi48.Bi48. Cao ng S Phm Tp. H Ch Minh nm 1998

Gii BPT: 2x

1 x 1 x 24

+ + . S: x 1;1 .

Bi49.Bi49.Bi49.Bi49. Cao ng Hi Quan nm 1999 H phn ban Gii BPT: 2 2 2x 3x 2 x 4x 3 2 x 5x 4 + + + + . S: x 1 x 4= .

Bi50.Bi50.Bi50.Bi50. Cao ng A, B 2009

Gii BPT: x 1 2 x 2 5x 1+ + + S: 2 x 3 .

Bi51.Bi51.Bi51.Bi51. Cao ng S Phm Nh Tr Mu Gio TWI nm 2000

Gii BPT: ( )2 2x x 2 2x 1 0+ < . S: Bi52.Bi52.Bi52.Bi52. Cao ng Nng Lm 2000

Gii BPT: ( )( ) 24 4 x 2 x x 2x 8 + . S: Bi53.Bi53.Bi53.Bi53. Cao ng S Phm Kinh T nm 2002

Gii BPT: x 6 x 1 2x 5+ > + + . S:

Bi54.Bi54.Bi54.Bi54. Cao ng Giao Thng Vn Ti nm 2005

Gii BPT: 2x 2x 15 x 2+ < . S:

Bi55.Bi55.Bi55.Bi55. Cao ng S Phm Vnh Long khi A, B nm 2005

Gii BPT: 2x 6x 5 8 2x + > . S: 3 x 5< .

Bi56.Bi56.Bi56.Bi56. i hc Dn Lp Phng ng khi A nm 2000

Gii BPT: 2 22x 4x 3 3 2x x 1+ + > . S: x 3;1 .

Bi57.Bi57.Bi57.Bi57. i hc Dn Lp Duy Tn khi D nm 2000 Gii BPT:

1/ 2

2

x 3x 21

x 3x 2

+

+ +. S:

2/ ( )2x 1 2 x 1+ . S: Bi58.Bi58.Bi58.Bi58. i hc Dn Lp K Thut Cng Ngh Tp. H Ch Minh (Hutech) nm2000

Gii BPT: x 6 x 1 2x 5+ > + + . S:

Bi59.Bi59.Bi59.Bi59. Hc Vin Chnh Tr Quc Gia Phn Vin Bo Ch Tuyn Truyn nm 2000

Gii BPT: 2x x 4x 1+ + > . S:

Bi60.Bi60.Bi60.Bi60. Hc Vin Ngn Hng Tp. H Ch Minh nm 2000

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Gii BPT: ( ) ( )22x x 4 x 4x x 2 2 + + < . S: ( ) x 2 3; 2 3 + . Bi 61.Bi 61.Bi 61.Bi 61. i hc Bch Khoa Tp. H Ch Minh nm 2000

Gii BPT: 2 2 2x 3x 2 x 6x 5 2x 9x 7+ + + + + + + .

Bi62.Bi62.Bi62.Bi62. i hc Thi Nguyn nm 2000

Gii BPT: 3 13 x 2x 72x2 x

+ < + . S: 3 7

0 x 42

3 7x 4

2

< < > +

.

Bi63.Bi63.Bi63.Bi63. i hc Thy Li nm 2000

Gii BPT: 2 14 x 2x 22xx

+ < + + . S:

Bi64.Bi64.Bi64.Bi64. i hc Ngoi Thng khi D nm 2000

Gii BPT: x 3 7 x 2x 8+ . S: ) (x 4;5 6;7 . Bi65.Bi65.Bi65.Bi65. i hc Thy Li H Ni i hc Thng Long nm 2000

Gii BPT: x 2 3 x 5 2x+ < . S:

Bi66.Bi66.Bi66.Bi66. i hc Bch Khoa H Ni khi D nm 2000

Gii BPT: 2 2 2x 3x 2 x 6x 5 2x 9x 7+ + + + + + + .

Bi67.Bi67.Bi67.Bi67. i hc Dc H Ni nm 2000

Gii BPT: 2 2 2x 8x 15 x 2x 15 4x 18x 18 + + + + . S: x 0 x 5< > .

Bi68.Bi68.Bi68.Bi68. Hc Vin Quan H Quc T nm 2000 Gii BPT: ( )( ) 2x 1 x 4 5 x 5x 28+ + < + + . S: ( )x 9;4 .

Bi69.Bi69.Bi69.Bi69. i hc Anh Ninh khi A nm 2000

Gii BPT: 27x 7 7x 6 2 49x 7x 42 181 14x+ + + + < .

HD: t 7x 7 7x 6 0 ......= + + . S: 6 x 67 < .

Bi70.Bi70.Bi70.Bi70. i hc M a Cht H Ni nm 2000

Gii BPT: ( )( )x 1 4 x x 2+ > . S: 7x 1;2

.

Bi71.Bi71.Bi71.Bi71. i hc Dn Lp Hng Bng nm 1999

Gii BPT: x 3 x 1 2x 1+ > S: 31 x2

.

Bi72.Bi72.Bi72.Bi72. i hc Dn Lp K Thut Cng Ngh (HuTech) khi A, B nm 1999

Gii BPT: 2x 3 x 2+ . S: 3x ;3 2 22

+

.

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Bi 73.Bi 73.Bi 73.Bi 73. i hc Dn Lp K Thut Cng Ngh (HuTech) khi D nm 1999

Gii BPT: 2x 1 8 x . S: 1x ;52

.

Bi74.Bi74.Bi74.Bi74. i hc M H Ni khi A, B, R, V v D4 nm 1999

Gii BPT: x 1 x 12 3x x

. S: 1x ;0

12

.

Bi75.Bi75.Bi75.Bi75. i hc Thy Sn nm 1999

Gii BPT: x 2 x 1 2x 3+ . S: 3x ;22

.

Bi76.Bi76.Bi76.Bi76. i hc Hu khi D, R h cha phn ban nm 1999

Gii BPT: 2 26 x x 6 x x

2x 5 x 4

+ +

+ +. S: x 2; 1 x 3 = .

Bi77.Bi77.Bi77.Bi77. i hc Hu khi D, R, T h phn ban nm 1999

Gii BPT: 2 212 x x 12 x x

x 11 2x 9

+ +

. S: x 2;4 x 3 = .

Bi78.Bi78.Bi78.Bi78. i hc Ti Chnh K Ton Tp. H Ch Minh nm 1999

Gii BPT: 2x 1 8 x . S:

Bi79.Bi79.Bi79.Bi79. i hc An Ninh H Ni khi D nm 1999

Gii BPT: 5x 1 4x 1 3 x . S: 1x ;4

+ .

Bi80.Bi80.Bi80.Bi80. i hc M a Cht nm 1999

Gii BPT: ( )

2

2

2xx 21

3 9 2x

< + +

. S: { }9 7x ; \ 02 2

.

Bi81.Bi81.Bi81.Bi81. Hc Vin Ngn Hng nm 1999

Gii BPT: 3x 2 x 1 x 2 x 12

+ + > . S: )x 1; + .

Bi82.Bi82.Bi82.Bi82. i hc Bch Khoa H Ni nm 1999

Gii BPT: x 1 3 x 4+ > + . S: x 0> .

Bi83.Bi83.Bi83.Bi83. i hc Quc Gia Tp. H Ch Minh t I khi D Hc Vin Ngn Hng nm 1999

Gii BPT: ( ) ( )22x x 4 x 4x x 2 2 + + < . S: ( ) x 2 3; 2 3 + . Bi84.Bi84.Bi84.Bi84. thi chuyn Ton Tin i hc Quc Gia H Ni nm 1988

Gii BPT: ( ) ( )3

2

4 2 2x 1

x x 1 x x x 1x

++ + + + . S: x 0> .

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Bi 85.Bi 85.Bi 85.Bi 85. i hc Giao Thng Vn Ti Tp. H Ch Minh nm 1998

Gii BPT: 2 23x 6x 4 2 2x x+ + < S: 2 x 0 < < .

Bi86.Bi86.Bi86.Bi86. i hc Nng Nghip nm 1998

Gii BPT: 21 1 4x

3x

< . S: 1 1x

2 2 .

Bi87.Bi87.Bi87.Bi87. i hc Lut nm 1998

Gii BPT: 4 2x 2x 1 1 x + > . S: ( ) ( ) { }; 2 0; \ 1 + . Bi88.Bi88.Bi88.Bi88. i hc Ti Chnh K Ton nm 1998

Gii BPT: 251 2x x

11 x

. S: 2 x 10 .

Bi95.Bi95.Bi95.Bi95. i hc A 2004

Gii BPT: ( )22 x 16 7 x

x 3x 3 x 3

+ >

S: x 10 34> .

Bi96.Bi96.Bi96.Bi96. i hc D 2002

Gii BPT: ( )2 2x 3x 2x 3x 2 0 . S: 1x2

hoc x 3 .

Bi97.Bi97.Bi97.Bi97. i hc Ngoi Thng nm 2001

Gii BPT: 1 x 1 x x+ . HD: Lin hip 0 x 1 .

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Bi 98.Bi 98.Bi 98.Bi 98. i hc Dn Lp Phng ng nm 2001

Gii BPT: 7x 13 3x 9 5x 27 S: 299 26304 x 2359

+ .

Bi99.Bi99.Bi99.Bi99. i hc Y H Ni nm 2001

Gii BPT: 2 22x x 5x 6 10x 15+ > + S:

5 53x

25 53

x2

+ > + +

S: 1 x 8 < .

Bi102.Bi102.Bi102.Bi102. i hc Dn Lp Ngoi Ng Tin Hc nm 1997

Gii BPT: ( ) 2 2x 3 x 4 x 9 S: 13x 3 x6

.

Bi103.Bi103.Bi103.Bi103. D b i hc khi D nm 2004

Gii BPT: 2 2x 2x 4x 3 6 2x+ + + . S: x 2 x 0 .

Bi104.Bi104.Bi104.Bi104. D b i hc khi D nm 2005

Gii BPT: 28x 6x 1 4x 1 0 + + . S: 1 1x x ;4 2

= + .

Bi105.Bi105.Bi105.Bi105. D b i hc khi B nm 2005

Gii BPT: 2x 7 5 x 3x 2+ . S:

Bi106.Bi106.Bi106.Bi106. i hc Cn Th khi D nm 2001

Gii BPT: ( ) 2x x 1 x x 4 2 0+ + + + . S: x 1 x 0 . Bi107.Bi107.Bi107.Bi107. i hc Kinh T Quc Dn nm 2001

Gii BPT: ( )( ) ( )x 5 3x 4 4 x 1+ + > . S: ( 4x ; 5 ;43

.

Bi108.Bi108.Bi108.Bi108. i hc Giao Thng Vn Ti Tp. H Ch Minh khi A nm 2001

Gii BPT: x 1 x 12 3 0x 1 x 1

+ +

. S:

Bi109.Bi109.Bi109.Bi109. i hc Kin Trc H Ni nm 2001

Gii BPT: 2 2x 4x 3 2x 3x 1 x 1 + + . S:

1x 1 hay x

2= .

Bi110.Bi110.Bi110.Bi110. i hc khi A nm 2010

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Gii BPT: ( )2x x

11 2 x x 1

+.

HD: Mu s 0< , chuyn v, p dng k du " "= ca BT B.C.S 3 5x2

= .

Bi 111.Bi 111.Bi 111.Bi 111. i hc S Phm Tp. H Ch Minh nm 1994

Gii BPT: 22x 6x 1 x 2 0 + + . S: 3 7x x 32

> .

Bi112.Bi112.Bi112.Bi112. i hc Nng Lm Tp. H Ch Minh nm 1995

Gii BPT: x 3 x 1 x 2+ < . S: 28x3

> .

Bi113.Bi113.Bi113.Bi113. i hc Quc Gia Tp. H Ch Minh nm 1996 Gii BPT: 2 2 2x 3x 2 x 4x 3 2 x 5x 4 + + + + .

HD: S dng min nghim ca iu kin. S: x 1 x 4= .

Bi114.Bi114.Bi114.Bi114. i hc Dn Lp Vn Lang nm khi B, D nm 1997

Gii BPT: ( ) 2 2x 3 x 4 x 9 + . S: 5x x 36

.

Bi115.Bi115.Bi115.Bi115. i hc Quc Gia Tp. H Ch Minh nm 1997

Gii BPT: ( )2

2 2x 1 x x 2 1+ + + . S: S 0; 2 1

=

.

Bi116.Bi116.Bi116.Bi116. i hc Lut Tp. H Ch Minh nm 1996

Gii BPT: 5 15 x 2x 42x2 x

+ < + + . S: 3 2 2 3 2 20; ;2 2

+ + .

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Dng 3. Phng trnh Bt phng trnh H phng trnh c cha tham s

Tam thc khng i du trn

T nh l v du ca tam thc bc hai, ta suy ra kt qu sau: Cho

(iu kin bt phng trnh c nghim )

a/

b/

T , ta c th suy ra iu kin v nghim ca bt phng trnh nh sau:

c/ bt phng trnh v nghim .

d/ bt phng trnh v nghim .

e/ bt phng trnh v nghim .

f/ bt phng trnh v nghim .

Lu

Nu c cha tham s. , ta cn xt:

Trng hp 1. . Trng hp 2. .

Nu c cha tham s. , ta cn xt:

Trng hp 1. . Trng hp 2. .

Gii v bin lun bt phng trnh bc hai

Bc 1. Xt nu h s a c tham s. Bc 2. Lp v tm nghim (nu c nghim, th lc ny nghim

l ). Bc 3. Lp bng xt du a v trn cng mt bng xt du (bin s l m). Bc 4. Da vo bng xt du, bin lun nghim ca bt phng trnh.

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BABABABAI TI TI TI TP AP AP AP AP DUP DUP DUP DU NGNGNGNG

Bi117.Bi117.Bi117.Bi117. nh m biu thc ( )f x sau lun dng vi mi x

1/ ( ) ( )2f x x m 2 x 8m 1= + + + . 2/ ( ) ( )2f x mx 2 m 1 x 4m= + .

3/ ( ) ( ) ( )2f x m 1 x m 5 x m 1= + . 4/ ( ) ( ) ( )2 2f x m 1 x 2 m 3 x 1= + + + + .

5/ ( ) ( ) ( )2f x 3m 1 x 3m 1 x m 4= + + + + . 6/ ( ) ( ) ( )2f x m 1 x 2 m 1 x 3m 3= + + .

Bi118.Bi118.Bi118.Bi118. nh m biu thc ( )f x sau lun khng dng vi mi x (lun lun m)

1/ ( ) ( )2f x 2x 2 m 2 x m 2= + + . 2/ ( ) ( ) ( )2f x m 4 x m 1 x 1 2m= +

3/ ( ) ( ) ( )2f x m 1 x 2 m 2 x m 6= + + + . 4/ ( ) ( ) ( )2f x m 2 x 2 m 2 x 2= + + .

5/ ( ) ( ) ( )2 2f x m 4m 5 x 2 m 1 x 2= + + . 6/ ( ) ( ) ( )2f x m 4 x m 1 x 2m 1= + + + . Bi119.Bi119.Bi119.Bi119. Chng minh cc phng trnh sau lun c nghim vi mi gi tr ca m

1/ ( )24x 4 m 2 x 3m 2 0+ + + + = . 2/ ( )2mx 5m 6 x m 1 0+ + + = .

3/ ( ) ( )23 2m x 3m 2 x m 1 0 + + = . 4/ ( ) ( )23m 1 x 3 m 1 x 1 0 + + + = . Bi120.Bi120.Bi120.Bi120. Chng minh cc phng trnh sau v nghim

1/ ( )2 2x 1 m 3 x m m 3 2 0+ + + + + = . 2/ ( )2 2x 1 m 3 x m m 3 2 0+ + + = 3/ ( ) ( )2 22m 7m 10 x 2 m 3 x 1 0 + + + = . 4/ ( ) 2m 1 x mx 7m 0+ + + = .

Bi121.Bi121.Bi121.Bi121. Tm tham s m cc bt phng trnh sau c v s nghim (BPT tha x ) 1/ 2x mx m 3 0 + + . 2/ ( )23x 2 m 1 x m 4 0+ + + > .

3/ ( )2 2x 3m 2 x 2m 5m 2 0 + . 4/ ( )2x m 1 x 2m 7 0+ + + + > .

5/ ( )2x 2 1 m x 9 0 + . 6/ ( )2x m 2 x 8m 1 0 + + > .

7/ ( )22x m 2 x m 4 0+ + > . 8/ ( )2mx m 1 x m 1 0+ + < .

9/ ( )2mx 4 m 1 x m 5 0 + + > . 10/ 2mx mx 5 0 < .

11/ 25x 4mx m 0 + . 12/ 2mx 2mx 1 0+ < .

13/ ( )24x 2 m 1 x m 0 + + > . 14/ ( ) 21 3m x 2mx 1 m 0 + .

15/ ( )2 21 x 2 m 2 x m 0m

+ + > . 16/ ( ) ( )2 2 21 m x 3 m 1 x 1 0 + .

17/ ( ) ( )2 2m 3 x 2 m 1 x 1 0+ + + + > . 18/ ( ) ( )2m 1 x 2 m 1 x m 2 0+ + + > .

19/ ( ) ( )2m 4 x m 1 x 1 2m 0+ < . 20/ ( ) ( )2m 1 x 2 m 1 x 4m 0 + < .

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21/ ( ) ( )22m 1 x m 1 x m 0 + + . 22/ ( ) 2m 2 x 4mx m 1 0 + + .

23/ ( )2

2

x 8x 200

mx 2 m 1 x 9m 4

+

+ + + .

25/ 2

2

x mx 11

2x 2x 3

+ .

9/ ( ) ( )2m 1 x 2 m 1 x 3m 2 0 + + > . 10/ ( ) ( )2m 2 x 2 m 1 x 4 0+ + < .

11/ ( ) ( )2m 2 x m 2 x m 0 + + . 12/ ( ) ( )2m 2 x 2 m 2 m 4 0 + + + .

13/ ( ) ( )2m 3 x m 2 x 4 0 + + > . 14/ ( ) ( )2m 3 x 2 m 1 x 4m 0+ + + < .

15/ ( ) ( )2m 4 x m 1 x 2m 1 0 + + + . 16/ ( ) ( )2m 4 x m 4 x 2m 1 0+ + < .

17/ ( ) ( )23 m x 2 2m 5 x 2m 5 0 + > . 18/ ( ) ( )23m 1 x 3m 1 x m 4 0+ + + + .

19/ ( ) ( )2m m 8 x 2 m 8 8m 1 0+ + + + . 20/ ( ) ( )2 2m 2m 3 x 2 m 1 x 1 0+ + + < . Bi123.Bi123.Bi123.Bi123. Tm tham s m cc bt phng trnh sau c nghim

1/ ( )2mx 2 m 1 x m 2 0+ + + . 2/ ( ) ( )2m 2 x 2 m 2 x m 4 0+ + + + + .

3/ ( ) ( )2m 2 x 2 m 2 x 2 0 + + . 4/ ( ) ( )2m 1 x 2 m 2 x m 6 0 + + + .

5/ ( )22x 2 m 2 x m 2 0 + + < . 6/ 2x 3x m 1 0 + + < .

7/ ( )2x m 2 x 8m 1 0+ + . 8/ ( ) ( )2 2m 1 x 2 m 3 x 1 0+ + + + .

9/ ( ) ( )2m 2 x 2 2m 3 x 5m 6 0 + + = . 10/ ( ) 2m 5 x 4mx m 2 0 + = .

11/ ( ) ( )23 m x 2 m 3 x m 2 0 + + + = . 12/ ( ) 21 m x 2mx 2m 0+ + = .

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13/ ( ) ( )2 22m 3 m x 2 2 3m x 3 + = . 14/ ( ) 2m 2 x 4mx 2m 6 0 + = . Bi124.Bi124.Bi124.Bi124. Gii v bin lun cc bt phng trnh sau

1/ 2x mx m 3 0 + + > . 2/ ( ) 21 m x 2mx 2m 0+ + .

3/ 2mx 2x 4 0 + > . 4/ ( )2 2x 2 m 3 x m 4m 1 0 + + + + > .

5/ ( )2 2x 2 m 3 x m 4m 1 0 + + + + < . 6/ 23mx mx 1 0 + .

7/ ( ) 2m 1 x 4mx 4 0 + < . 8/ ( )2mx m 3 x m 3 0+ + .

9/ 2mx 4mx m 6 0+ + + . 10/ ( ) ( )2m 1 x 2 m 1 x 3m 3 0+ + .

11/ ( ) 21 m x 2mx m 1 0 + + + . 12/ ( )22x 2 m 1 x 1 0 + + .

13/ ( ) ( )2m 3 x 4 m 3 x m 0 + . 14/ 2 2x 2x 2m m 0 + .

15/ 2x 2mx 3m 0 + < . 16/ ( ) ( )2m 1 x 3 m 1 x 5 0 + + .

17/ 2 2x 2mx m 1 0 + . 18/ ( ) 2m 1 x 2mx 4m 0+ + > .

19/ ( ) ( )2 2m 1 x 2 m 3 x 1 0+ + + + . 20/ 1 1 m 1 mxx 1 m 1 m

++ +

+ .

Bi125.Bi125.Bi125.Bi125. Tm tham s m bt phng trnh sau tha iu kin x cho trc

1/ ( )2x 2 m 1 x m 3 0+ + + x 0 .

2/ ( )2x 2 m 1 x m 3 0+ + + x 0 .

3/ ( )2x m 1 x 1 0 + + > x 0 > .

4/ ( ) ( )23 m x 2 m 1 x 1 0 + + > x 0 < .

5/ ( )2x 2 m 2 x m 2 0 + x 0;1 .

6/ ( ) ( )2m 2 x 2 m 2 x 1 0 + x 0;1 .

7/ 2x 2mx 3m 2 0 + > ( )x 1;2 .

8/ ( ) ( ) ( )2m 1 x 2 2m 1 x 3 1 2m+ < ( )x 1;1 .

9/ ( )( )x m 1 x m 3 0 + + < ( )x 1;2 .

10/ ( )( )x 3 2m x 3m 2 0+ + < x 2;3 .

11/ ( )( )x 2m x m 1 0 + > )x 3; + .

12/ ( )2

2

x 2x 42

x m 1 x 4

+ + & 2 4x 2x 1 m 0 + .

2/ 2x 8x 7 0+ + > & 2mx 2x 4 0 + > .

3/ 22x 5x 3 0+ + < & 2x 3x m 1 0 + + < .

4/ 2x x 20 0 & ( )2x 2 m 1 x 1 0 + + .

Bi128.Bi128.Bi128.Bi128. Tm tham s m bt phng trnh: ( ) 21 m x 2mx m 6 0 + + 1/ C nghim.

2/ C duy nht mt nghim.

3/ C nghim l mt on trn trc s c di bng 1.

Bi129.Bi129.Bi129.Bi129. Tm tham s m bt phng trnh: ( )2mx 2 m 1 x m 5 0 1/ C nghim.

2/ C duy nht mt nghim.

3/ C nghim l mt on trn trc s c di bng 2.

Bi130.Bi130.Bi130.Bi130. Tm tham s m nghim cc h sau tha yu cu bi ton

1/ ( )2

2 2

x 6x 5 0

x 2 m 1 x m 1 0

+



4/ 2 2

2

x 2mx 2m m 2 0

x 2mx 2m 0

+ + = + > C nghim duy nht.

5/ ( )2 2

2 2

x m 4

x 5m 2 x 4m 2m 0

+ = + + + + C nghim.

12/ ( ) ( )

( )2

2

m x 2 m 1 x 2 mx

m 1 x x 2 0

+ + > + + + + > C nghim.

13/ ( )2

2

x 2m 1 x 4m 0

x 2mx 3 2m 0

+ + C nghim duy nht.

14/ ( )2 2

2

x 4 m 1 x 4m 8m 3 0

x 5x 4 0

+ + + + + > C nghim.

15/ ( )( )2

2

x m 1 x 1 0

m 1 x x 1 0

+ + + + + + C nghim duy nht.

16/ 2

2

2x 3x 2 0

mx 2mx 1 0

+ + V nghim.

Bi131.Bi131.Bi131.Bi131. Tm tham s m cc h sau c tp nghim l

1/ 2

2

3x mx 66 9

x x 1

< + + = C nghim duy nht.

3/ ( )2 2

4 2

x 2m 1 x m m 2 0

x 4x 4 0

+ + + + = +



Bi137.Bi137.Bi137.Bi137. Cho h bt phng trnh: 2

2

x 2x 4m 0

x 4x m 0

+

1/ Tm m h c nghim.

2/ Tm m h c nghim duy nht.

Bi138.Bi138.Bi138.Bi138. Cho h bt phng trnh: ( ) ( )2

2

m 1 x m 1 x m 1 0

mx 2x 2m 1 0

+ + + +

1/ Tm m h c nghim.

2/ Tm m h c nghim duy nht.

Bi139.Bi139.Bi139.Bi139. Xc nh m h sau c nghim tha yu bi ton

1/ 2x 2mx 0

x 1 m 2m

+ C nghim duy nht.

2/ 2

3 2

x 3x 4 0

x 3x x m 15m 0

C nghim.

Bi140.Bi140.Bi140.Bi140. Gii v bin lun h bt phng trnh sau

1/ 2mx 1 0

x 3x 10 0

+ . 2/ ( )

2

2

x x 6 0

x m 1 x m 0

+ + >.

3/ ( )( )( )( )x 1 x 2m 0

x 2 x m 0

+ + . 4/ ( ) ( )

2

2

x 2x 8 0

x 1 m x 2 m 1 0

+ + + . 6/ ( ) ( )( ) ( )

2

2

x 1 3m x m 1 2m 0

x m 4 x 2 m 2 0

+ + .

Bi141.Bi141.Bi141.Bi141. Tm tham s m cc h sau c nghim tha yu cu ca bi ton

1/ ( ) ( )( ) ( )

2 2

2 2

x 1 y 3 4

x m y 3 1

+ +

C nghim.

2/ ( ) ( )( ) ( )

2 2

2 2

x 1 y 1 m

x 1 3 y m

+ + + +

C nghim duy nht.

3/ ( )( ) ( ) ( )

22

2 2 2

x y m 1 4

x m y 1 m 2

+ + + + +

C nghim.

4/ ( ) ( )2 2

x 1 y 1 2

x y m 0

+ + = Nghim ng x 0;2 .

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Bi142.Bi142.Bi142.Bi142. Cho h bt phng trnh: ( )

( )( )2 2x 2 m 3 x m 6m 5 y

x y m 2 x y m 5 0

+ + + +

1/ Gii h bt phng trnh khi m 1= .

2/ Tm tham s m tp nghim ca h cha on 2;4 .

Bi143.Bi143.Bi143.Bi143. Cho h bt phng trnh: 2y x x 1 0

y 2 x 1 1 0

+ +

1/ Gii h khi y 2= .

2/ Tm cc nghim ca h.

Bi144.Bi144.Bi144.Bi144. Cho bt phng trnh: ( ) mx x 3 m 1 +

1/ Gii bt phng trnh ( ) khi 1m2

= .

2/ Vi gi tr no ca tham s m th bt phng trnh c nghim.

Bi145.Bi145.Bi145.Bi145. Cho bt phng trnh: ( ) ( ) m 2 x m x 1+ +

1/ Gii bt phng trnh ( ) khi m 1= .

2/ Tm tt c gi tr ca m bt phng trnh ( ) c nghim x tha: 0 x 2 .

Bi146.Bi146.Bi146.Bi146. Tm tham s m 0> bt phng trnh: x x 1 m > c nghim.

Bi147.Bi147.Bi147.Bi147. Tm tham s m : ( )( ) ( )21 2x 3 x m 2x 5x 3+ > + + tha 1x ;32

.

Bi148.Bi148.Bi148.Bi148. Tm nghim ca bt phng trnh: 2 2x 1 x x 1 x+ < trn on 0;1 .

Bi149.Bi149.Bi149.Bi149. Cho bt phng trnh: ( ) ( ) 22 2x 1 m x x 2 4+ + + +


2/ Tm tham s m bt phng trnh ( ) c tha x 0;1 .

Bi150.Bi150.Bi150.Bi150. Cho bt phng trnh: ( ) x 2 x 1 m 1 +


2/ Vi gi tr no ca tham s m th bt phng trnh ( ) c nghim.

Bi151.Bi151.Bi151.Bi151. Cho x 0;1 . Chng minh rng: 4 4x 1 x x 1 x 2 2 2+ + + + . ng thc

xy ra khi no ?

Bi152.Bi152.Bi152.Bi152. Gii bt phng trnh: 2 22x a x 0+ > vi a l s cho trc.

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Bi153.Bi153.Bi153.Bi153. Tm m bt phng trnh: x m x 1 m 1 > + c nghim.

Bi154.Bi154.Bi154.Bi154. Gii bt phng trnh: 21 x x 1 0 + > . Suy ra bng xt du ca hm s

( ) 2f x 1 x x 1= + .

Gii bt phng trnh: 21 x x 1 x + < .

Bi155.Bi155.Bi155.Bi155. Cho a thc: ( ) 3 2P x 2x mx nx p= + + + Trong m,n,p l hng s.

1/ Xc nh m,n,p ( )P x chia ht cho a thc ( ) 3 2Q x 2x 3x 1= + .

2/ Vi cc gi tr tm c ca m,n,p , gii bt phng trnh: ( )P x 1 0 > .

Bi156.Bi156.Bi156.Bi156. Vi gi tr no ca tham s m th bt phng trnh: 2 2x 2 x m x m x 1 0+ + + c nghim.

Bi157.Bi157.Bi157.Bi157. Tm tham s m bt phng trnh: 2x x m 3+ < c nghim m.

Bi158.Bi158.Bi158.Bi158. Cho bt phng trnh: ( ) ( ) 2x 2 x m 1 x 2x 3 + + +

1/ Tm m bt phng trnh ( ) c nghim.

2/ Tm m di min nghim ca bt phng trnh ( ) bng 2.

Bi159.Bi159.Bi159.Bi159. Cho bt phng trnh: ( ) ( ) 2x 6 x x 6x m 2 + +

1/ Tm tham s m bt phng trnh ( ) c nghim.

2/ Tm tham s m min nghim ca bt phng trnh ( ) thuc on 2;4 .

Bi160.Bi160.Bi160.Bi160. Tm cc gi tr ca m sao cho bt phng trnh: x x 3 m < c tha x 1;4 .

Bi161.Bi161.Bi161.Bi161. Tm tham s m h sau c nghim

1/ ( )( )

2

2

x m 2 x 2m 0

x m 7 x 7m 0

+ +



Bi163.Bi163.Bi163.Bi163. Cho h bt phng trnh: ( )4 2

2 2

x 5x 4 0

x 2m 1 x m m 2 0

+



2/ Vi gi tr no ca tham s m th bt phng trnh c nghim ?

S: 1/ 3 x 7 . 2/ 1 3m4

+ .

Bi 171.Bi 171.Bi 171.Bi 171. i hc Ti Chnh K Ton Tp. H Ch Minh nm 1995

Cho h phng trnh: 2y x x 1 0

y 2 x 1 1 0

+ + .

1/ Gii h bt phng trnh khi y 2= .

2/ Tm tt c cc nghim nguyn ca h.

S: 1/ 1 5 x 02

. 2/ ( ) ( ){ }S 0;2 , 1;3= .

Bi172.Bi172.Bi172.Bi172. i hc Y Dc Tp. H Ch Minh nm 1996

Tm a bt phng trnh x x 1 a c nghim vi a l tham s dng.

S: 0 a 1< < . Bi173.Bi173.Bi173.Bi173. i hc Ngoi Thng C S 2 nm 1999

Vi gi tr no ca m th h bt phng trnh ( )2

2 2

x 8x 7 0

x 2m 1 x m m 0

+ + + + c nghim ?

Xc nh m h bt phng trnh c mt nghim duy nht ? Bi174.Bi174.Bi174.Bi174. Hc Vin Quan H Quc T nm 1997

Tm m h ( )( )

2

2

x m 2 x 2m 0

x m 7 x 7m 0

+ +



Chng


Bi1.Bi1.Bi1.Bi1. Chng minh cc ng thc sau

1/ 2 2 2cos x sin x 1 2 sin x = . 2/ 2 22cos x 1 1 2 sin x = .

3/ 2 23 4 sin x 4 cos x 1 = . 4/ sin x cotx cos x tan x sin x cos x+ = + .

5/ 4 4 2 2sin x cos x 1 2 sin x cos x+ = . 6/ 4 4 2 2cos x sin x cos x sin x = .

7/ ( )( )24 cos x 3 1 2 sin x 1 2 sin x = + . 8/ ( )( )2 2 21 cosx sin x cosx cos x sin x+ + = . 9/ 4 4 2 2sin x cos x 1 2cos x 2sin x 1 = = . 10/ 3 3sin x cos x sin x cos x sin x cos x+ = .

11/ 2 2 2 2tan x sin x tan x sin x = . 12/ 2 2 2 2cot x cos x cot x cos x = .

Bi2.Bi2.Bi2.Bi2. Chng minh cc ng thc sau

1/ 1tan x cot xsin x cos x

+ = . 2/ 1 cos x sin xsin x 1 cos x

=+

.

3/ 1 1 11 tan x 1 cotx

+ =+ +

. 4/ 21 11 1 tan x 0cos x cos x

+ + = .

5/ 2

2

2

1 sin x1 2 tan x

1 sin x

+= +

. 6/ tan x tan ytan x tan y

cotx coty

+=

+.

7/ 42 4

2 11 cot x

sin x sin x = . 8/ cos x 1tan x

1 sin x cos x+ =+

.

9/ ( )( )2 11 cos x 1 cot x1 cos x

+ =+

. 10/ 1 cos x 1 cos x 4 cotx1 cos x 1 cos x sin x

+ =

+.

11/ sin x 1 cos x 21 cos x sin x sin x

++ =

+. 12/ sin x cos x 1 cos x

sin x cos x 1 1 sin x

+ =

+ +.

13/ 2

2

sin x 2cos x 1 cos x

1 cos x2 cos x cos x

+ =++

. 14/ sin x cos x 1 2cos x1 cos x sin x cos x 1

+ =

+.

GC V CNG THC LNG GIC 5555

A H THC LNG GIC C BN

H thc lng gic c bn

. .

. ..

.

.

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Ths. L Vn on Chng 5. Cung gc lng gic Cng thc lng gic


Bi 3.Bi 3.Bi 3.Bi 3. Chng minh cc ng thc sau

1/ 6 6 2 2sin x cos x 1 3 sin x cos x+ = .

2/ ( )( )6 6 2 2 2 2sin x cos x sin x cos x 1 sin x cos x = .

3/ ( )28 8 2 2 4 4sin x cos x 1 2 sin x cos x 2 sin x cos x+ = .

4/ ( )( )8 8 2 2 2 2sin x cos x sin x cos x 1 2sin x cos x = . Bi 4.Bi 4.Bi 4.Bi 4. Chng minh cc ng thc sau

1/ ( )( )1 sin x cos x tan x 1 cos x 1 tan x+ + + = + + .

2/ ( )( )1 tan x 1 cotx sin x cos x 1 2 sin x cos x+ + = + .

3/ ( ) ( ) ( )22 21 tan x cos x 1 cot x sin x sin x cos x+ + + = + .

4/ 2 2sin x tan x cos x cot x 2 sin x cos x tan x cot x+ + = + .

5/ 2 2 2 2 2sin x tan x 4 sin x tan x 3 cos x 3+ + = . Bi 5.Bi 5.Bi 5.Bi 5. Chng minh cc ng thc sau

1/ 2 2sin cos tan 1

1 2 sin cos tan 1

=

+ +. 2/

2 26

2 2

tan sintan

cot cos

=

.

3/ 2 2cos sin

1 sin cos1 tan 1 cot

+ =

. 4/

( )3tan sin 1

sin cos 1 cos

=

+ .

5/ 2 22 2

1tan cot 2

sin cos= + +

. 6/

22

2 2

1 3 tantan 1

cos cos

= +

.

7/ 2 2 2 2

2 2 2 2

tan tan sin sin

tan tan sin sin

=

. 8/

( )22

1 cos1 cos1 2 cot

sin sin

+ =

.

9/ ( )( )2 11 cos 1 cot1 cos

+ =+

. 10/ 2 33

sin cos1 tan tan tan

cos

+ + + + =

.

11/ 2

2

sin cos 1 cot

sin cos cos sin 1 cot

+ =

+ . 12/

2 2 4

2 2 2 2

tan 1 cot 1 tan.

1 tan cot tan cot

+ + =

+ + .

13/ 2

21 sin 1 sin 4 tan1 sin 1 sin

+ = + . 14/

2

21 cos 1 cos 4 cot1 cos 1 cos

+ = + .

Bi 6.Bi 6.Bi 6.Bi 6. Rt gn cc biu thc sau

1/ 4 2 21

P sin sin cos= + . 2/ 4 4 22

P sin cos cos= + .

3/ 2 2 23

P sin sin cot= + . 4/ 2 2 24

P cos cos cot= + .

5/ ( )2 2 25P 1 sin cot 1 cot= + . 6/ 2 26P sin tan cos cot 2sin cos= + + .

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7/ 2

7

2cos 1P

sin cos

=

+ . 8/

8 2

1 cos 1P

1 cossin

=

+ .

9/ 9

cot cotP

tan tan

+ =

+ . 10/

10

cosP tan

1 sin

= +

+ .

11/ 11

sin x tan xP sin x cot x

tan x

+= . 12/

12 2

cos x tan xP cot x cos x

sin x= .

13/ 2 2

13 2 2

cos x cot xP

sin x tan x

=

. 14/

2 22

14 2

1 sin x cos xP cos x

cos x

= .

15/ 3 2

15

sin x sin x cos x cos xP

1 2sin x cos x

+ =

. 16/

( )216

sin x cos x 1P

tan x sin x cos x

+ =

.

17/ 2 2 2

17 2 2 2

sin x sin x tan xP

cos x cos x tan x

+=

+. 18/

2 22

18 2

cot x cos xP 1 cos x

cot x

= + .

19/ 2 2

2 219 2 2

cos x sin yP cot x cot y

sin x sin y

= . 20/

20

1 sin x cos x cotxP

sin x 1 cotx 1 tan x=

+ +.

21/ 21

1 cosx 1 sinxP tanx cotx

sinx cosx

= + + + . 22/

22

1 1P 1 cotx 1 cotx

sinx sinx

= + + + .

23/ ( ) ( )2 223P 1 tanx cos x 1 cotx sin x= + + + . 24/ 242 2

1P , 0 x

2sinx cot x cos x

pi = <



14/ 3 2N 2cos x 2cos x sin x 1= + + .

15/ 3 3 2O cos x sin x 2sin x 1= + + .

16/ ( )( ) 2P 2sin x 1 2 cos x 2 sin x 1 3 4 cos x= + + + .

17/ ( )( )Q 2cos x 1 sin x cos x 1= + .

18/ 3 3 2R 4sin x 3cos x 3sin x sin x cos x= + .

19/ ( ) ( )2S 1 sin x tan x 1 cos x= + + .

20/ ( ) 2T 2 5 sin x 3 1 sin x tan x= + .

21/ 2U 2sin x cos x 2sin x 3sin x cos x 1= + .

22/ ( )V tan x 3 cot x 4 sin x 3 cos x= + . 23/ X 3sin x 2cos x 3 tan x 2= + .

24/ ( ) ( )Y 2 tan x sin x 3 cotx cos x 5= + + .

25/ ( ) ( )Z 3 cot x cos x 5 tan x sin x 2= . Bi 8.Bi 8.Bi 8.Bi 8. Chng minh cc biu thc sau khng ph thuc vo bin x

1/ 4 4 2A cos x sin x 2sin x= + .

2/ 4 2 2 2B sin x sin x cos x cos x= + + .

3/ 4 2 2 2C cos x sin x cos x sin x= + + .

4/ ( ) ( )4 2 4 2D cos x 2 cos x 3 sin x 2 sin x 3= + . 5/

6 6 4 4 2E sin x cos x 2sin x cos x sin x= + + .

6/ 1 1

F sin x. , 0 x1 cos x 1 cos x 4

pi = + <



Bi 9.Bi 9.Bi 9.Bi 9. Chng minh cc biu thc sau khng ph thuc vo bin x

1/ 2 cot x 1

Atan x 1 cot x 1

+= +

.

2/ ( )

22

2 2 2

1 tan x 1B

4 tan x 4 sin x cos x

= .

3/ ( )( )2

2 21 tan xC 1 tan x 1 cot xtan x

= + + .

4/ ( )

22

2 2 2

1 cot x 1D

cot x sin x cos x

= .

5/ 6 2

6 2

1 sin x 3 tan xE

cos x cos x

= .

6/ 2 2 2 2

2 2

tan x cos x cot x sin xF

sin x cos x

= + .

7/ 2 2

2

cot x cos x sin x cos xG

cotxcot x

= + .

8/ 4 4

6 6

sin x cos x 1H

sin x cos

+ =

+.

Bi 10.Bi 10.Bi 10.Bi 10. Cho ( ) ( )f x 2 1 cos x ,= ( )g x 2 cos x 3.sin x,= + ( )h x 2 cos x 3.sin x= + + . Chng minh cc i lng sau khng ph thuc vo bin: ( ) ( ) ( )2 2 2A f x g x h x ,= + +

( ) ( ) ( )4 4 4B f x g x h x ,= + + ( ) ( ) ( ) ( ) ( ) ( )2 2 2 2 2 2C f x .g x g x .h x h x .f x= + + . Bi 11.Bi 11.Bi 11.Bi 11. Cho a sin x sin y b cos x cos y 0 = . Chng minh rng biu thc

2 2 2 2

1 1Q

a sin x b cos x a sin y b cos y= +

+ + khng ph thuc vo bin.

Bi 12.Bi 12.Bi 12.Bi 12. Cho ( )6 6 4 4Q sin x cos x m sin x cos x= + + . Tm tham s m biu thc Q khng ph thuc vo x v tnh gi tr ca Q vi m va tm c (nu c). (HKII Chuyn Trn i Ngha nm 1998)

Bi 13.Bi 13.Bi 13.Bi 13. Cho 4 4sin x cos x 1

a b a b+ =

+. Chng minh rng:

1/

( )

8 8

3 3 3

sin x cos x 1

a b a b+ =

+ (HKII Chuyn L Hng Phong nm 2001)

2/

( )

10 10

4 4 4

sin x cos x 1

a b a b+ =

+ (HKII H Ni Amsterdam nm 2007)

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B NG TRN LNG GIC CUNG GC LNG GIC CUNG LIN KT

I Gi tr lng gic ca gc (cung) lng gic 1. nh ngha cc gi tr lng gic

Cho . Gi s .

.

.

.

.

Nhn xt

.

tan xc nh khi .. cot xc nh khi .

. .

. .

2. Du ca cc gi tr lng gic

Phn t Gi tr lng gic I II III IV

cos + +

sin + + tan + + cot + +

3. Mt s lu

Quan h gia v raian: v .

Vi th v ngc li .

cosin O

cotang

sin

tan

g

H A

M K B S

T

di l ca cung trn c s o bn knh R l . S o ca cc cung lng gic c im u A, im cui l B l

.

Mi cung lng gic ng vi mt gc lng gic v ngc

li.

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II Cung gc lin kt

Gc i nhau Gc b nhau Gc ph nhau

Gc hn km Gc hn km

rad 0

00 300 450 600 900 1200 1350 1800 2700 3600

sin

cos

tan

cot

Cn nm vng h thc c bn

. .

. .

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BABABABAI TI TI TI TP AP AP AP AP DUP DUP DUP DU NGNGNGNG MI LIN H GIA N V V RADIAN

Bi14.Bi14.Bi14.Bi14. i s o cc cung sau ra , pht, giy: 2 3 5 7 13 3 15, , , , , , , 2, ,3 4 9 5 3 3 4 7 15

pi pi pi 9pi pi pi pi pi pi .

Bi15.Bi15.Bi15.Bi15. i s o cc gc sau ra radian ( )rad : 0 0 0 0 0 0 0 0 090 , 36 , 15 , 72 , 270 , 240 , 540 , 750 , 21 . Bi16.Bi16.Bi16.Bi16. i s o ca cc cung trn sau thnh s o radian (chnh xc n phn nghn):

0 0 021 30 ', 75 54 ', 12 36 ' 0 0 0, 15 30 ', 105 45 ' 30 '', 27 38 ' 49 '' .

Bi17.Bi17.Bi17.Bi17. s o radian ca cc cung trn sau ra s o (chnh xc n pht): ( )2 radpi

v 2,5pi .

Bi18.Bi18.Bi18.Bi18. in cc gi tr thch hp vo trng

00 015 030 045 060 075 090 0120 0150 0180

16

pi

8

pi

6

pi

5

12

pi

2

pi

2

3

pi

3

4

pi

5

6

pi

pi 43

pi

Radian 74

pi

12

pi

5

6

pi

2

pi

2

3

pi

3

4

pi

0135 050 0810

TNH DI CUNG TRN Bi19.Bi19.Bi19.Bi19. Mt ng trn c bn knh ( )10 cm . Tnh di cc cung c s o l

( ) 0 02 3 7 13 3 3, , , , , 2 , , 45 , rad , 753 18 5 3 6 4 2

pi pi pi pi pi pi pi .

Bi20.Bi20.Bi20.Bi20. Kim gi ca ng h di ( )7 cm , kim pht di ( )10 cm . Tnh qung ng kim pht, kim gi i c trong 30 pht.

Bi21.Bi21.Bi21.Bi21. Bnh xe c ng knh (tnh c lp) l ( )55 cm . Nu xe chy vi vn tc ( )/40 km h th trong mt giy bnh xe quay c bao nhiu vng ?

Bi22.Bi22.Bi22.Bi22. Kim pht v kim gi ca ng h ln nh Bu in H Ni theo th t di ( )1,75 m v ( )1,26 m . Hi trong 15 pht, mi kim pht v mi kim gi vch trn cung trn c di bng

bao nhiu mt ?

BIU DIN NGN CUNG LNG GIC

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Bi 23.Bi 23.Bi 23.Bi 23. Biu din cc gc sau ln ng trn lng gic gc A: 0 0 0 0 030 , 45 , 120 , 120 , 330 ,

0 0 0 0 07 4 15 2010630 , 750 , 1250 , , , 750 , 1125 , ,4 3 2 4

pi pi pi pi .

Bi 24.Bi 24.Bi 24.Bi 24. Xc nh im cui ca cung c s o: ( ) k2 , k , k , k , k , k2 3 4

pi pi pipi pi .

Bi 25.Bi 25.Bi 25.Bi 25. Xc nh im ngn ca cc h nghim sau y trn ng trn lng gic vi k .

1/ x k= pi . 2/ x k22

pi= + pi .

3/ x 4k= pi . 4/ x k3

pi= + pi .

5/ x k26

pi= + pi . 6/ x k

4

pi= + pi .

7/ kx3

pi= . 8/ k2x

3 3

pi pi= + .

9/ k3x4 4

pi pi= + . 10/ kx

6 3

pi pi= + .

Bi 26.Bi 26.Bi 26.Bi 26. Cho hai im M v N sao cho s AM6

pi= v ks AN

798

pi= vi k . Tm k :

1/ M trng vi N. 2/ M i xng vi N qua tm O.

Bi 27.Bi 27.Bi 27.Bi 27. Hai gc lng gic c s o radian 353

pi v m

5

pi vi m l s nguyn c th c cng tia u v

tia cui hay khng ?

GA TR LNG GIC CA MT CUNG (GC) Bi28.Bi28.Bi28.Bi28. Xc nh du ca cc gi tr lng gic hoc biu thc

1/ sin x, cos x, tan x, cotx vi 3x2

pipi < < .

2/ 3sin x , cos x , tan x , cot x4 2 2 2

pi pi pi pi + + vi 0 x

2

pi< < .

3/ ( )0 0A sin40 .cos 290= . 4/ ( )0 0B sin 25 .cos170= .

5/ ( )0 0 0C sin225 .tan130 .cot 175= . 6/ ( )0 0 0D cos195 .tan269 .cot 98= .

7/ ( )0 0E sin50 .cos 300= . 8/ 0 21F sin215 .tan7

pi= .

9/ 3 2G cot .sin5 3

pi pi = . 10/ os 4 4 9H c .sin . tan .cot

5 3 3 5

pi pi pi pi= .

Bi29.Bi29.Bi29.Bi29. Tnh gi tr cn li ca gc x, bit

1/ 1sin x2

= vi 0 090 x 180< < . 2/ 4sin x5

= vi 0 0270 x 360< < .

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3/ 3sin x5

= vi 3x2

pipi < < . 4/ 1cos x

4= vi 0 x

2

pi< < .

5/ 3cos x5

= vi 00 x 90< < . 6/ 5cos x13

= vi 0 0180 x 270< < .

7/ 2cos x5

= vi x 02

pi < < . 8/ 4cos x

5= vi 0 0270 x 360< < .

9/ 5sin x13

= vi x2

pi< < pi . 10/ 1sin x

3= vi 0 0180 x 270< < .

11/ tan x 3= vi 3x2

pipi < < . 12/ tan x 2= vi x

2

pi< < pi .

13/ 1tan x2

= vi x2

pi< < pi . 14/ cotx 3= vi 3x

2

pipi < < .

15/ 3tan x4

= vi 3x2

pipi < < . 16/ tan x 2= vi x

2

pi< < pi .

17/ 2cotx3

= vi 0 x2

pi< < . 18/ cotx 3= vi x

2

pi< < pi .

19/ cotx 3= vi 3 x 22

pi< < pi . 20/ 0cot15 2 3= +

Bi30.Bi30.Bi30.Bi30. Tnh gi tr ca cc biu thc lng gic sau

1/ Cho tan x 2= . Tnh:

1 2

5 cot x 4 tan x 2 sin x cos xA , A

5cot x 4 tan x cos x 3 sin x

+ += =

.

2/ Cho cotx 2= . Tnh: 1 2

3 sin x cos x sin x 3cos xB , B

sin x cos x sin x 3cos x

= =

+ +.

3/ Cho cotx 2= . Tnh: 1 2 2

2 sin x 3cos x 2C , C

3 sin x 2 cos x cos x sin x cos x

+= =

.

4/ Cho tan x 2= . Tnh: 1

2 sin x 3 cos xD ,

4 sin x 5 cos x

+=

2 3 3

3 sin x 2cos xD ,

5 sin x 4 cos x

=

+

3 3

3 2

sin x cos xD ,

sin x sin x cos x

=

+

4 3 3

sin x 5cos xD ,

sin x 2 cos x

+=

3 3

5 3

8 cos x 2 sin x cos xD ,

2 cos x sin x

+=

3 3

6 3 2

sin x 2cos x 2 sin x 3cos xD

sin x cos x 5 sin x

+ +=

+.

5/ Cho 3sin x , 0 x5 2

pi= < < . Tnh cot x tan xE

cotx tan x

+=

.

6/ Cho 0 01sin x , 90 x 1803

= < < . Tnh 28 tan x 3cotx 1

Ftan x cotx

+ =

+.

7/ Cho 2cos x3

= . Tnh cotx 3 tan xG2cotx tan x

+=

+.

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8/ Cho

02sin x , 0 x 903

= < < . Tnh 1 2 2

tan x cos x tan x cos xH , H cos x cotx

cotx sin x

= = .

9/ Cho

4cos x , x

5 2

pi= < < pi . Tnh

1 2

cotx tan x sin xI , I cot x

cotx tan x 1 cos x

+= = +

+.

Bi31.Bi31.Bi31.Bi31. Cho 5sin x cos x4

+ = . Hy tnh gi tr ca biu thc sau

1/ A sin x.cos x= . S: 9A32

= .

2/ B sin x cos x= . S: 7B4

= .

3/ 3 3C sin x cos x= . S: 41 7C128

= .

Bi32.Bi32.Bi32.Bi32. Cho tan x cotx 3 = . Hy tnh gi tr ca biu thc sau

1/ 2 2A tan x cot x= + . S: A 11= .

2/ B tan x cotx= + . S: B 13= .

3/ 4 4C tan x cot x= . S: C 33 13= . Bi33.Bi33.Bi33.Bi33. Cho sin x cos x m+ = . Hy tnh theo m gi tr cc biu thc

1/ A sin x cos x= . 2/ 3 3B sin x cos x= + .

3/ 4 4C sin x cos x= + . 4/ D sin x cos x= .

5/ 2 2E tan x cot x= + . 6/ 6 6F sin x cos x= + . Bi34.Bi34.Bi34.Bi34. Tnh sin x, cos x, tan x, cotx . Bit rng

1/ sin x cos x 2+ = . 2/ sin x cos x 2 = .

3/ 1sin x cos x2

+ = . (HKII, Nguyn Thng Hin nm 2005)

4/ 1sin x cos x5

+ = . S: 4 3 4 3; ; ;5 5 3 4 .

5/ tan x cotx 4+ = . S: 1 2 3; ; 2 3; 2 322 2 3

+

.

Bi35.Bi35.Bi35.Bi35. Cho tan x 2cotx 1 = . Hy tnh

1/ 2 2A tan x cot x= . 2/ 3 3B tan x cot x= + .

3/ 4 4C tan x 2cot x= + . 4/ 5 5D tan x 3 cot x= . Bi36.Bi36.Bi36.Bi36. Tnh gi tr ca cc biu thc sau y khi

1/ Cho 4 4 33 sin x cos x4

+ = . Tnh 4 4A sin x 3 cos x= + . S: A 74

= .

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2/ Cho 4 4 13 sin x cos x2

= . Tnh 4 4B sin x 3 cos x= + . S: B 1= .

3/ Cho 4 4 74 sin x 3 cos x4

+ = . Tnh 4 4C 3sin x 4 cos x= + . S: 7 57C C4 28

= = .

CUNG GC LIN KT Bi37.Bi37.Bi37.Bi37. Biu din cc s o ca cung di dng k2 + pi vi k .

3 5 11 14 21, , , ,

2 4 3 5 6

pi pi pi pi pi .

Bi38.Bi38.Bi38.Bi38. Biu din cc s o ca cung di dng 0k360 + vi k .

0 0 0 0 0370 , 512 , 765 , 1000 , 1234 .

Bi39.Bi39.Bi39.Bi39. Biu din cc s o ca cung di dng k2 + pi vi k .

8 11 7 18 27 25, , , , ,

3 5 2 7 10 4

pi pi pi pi pi pi .

Bi40.Bi40.Bi40.Bi40. Dng cung lin kt (khng dng my tnh), hy tnh cc gi tr sau 1/ 0sin150 . 2/ 0cot135 . 3/ 0cos225 . 4/ 0tan210 .

5/ 0cot225 . 6/ 0sin240 . 7/ 0cos 315 . 8/ 0tan 300 .

9/ ( )0cot 1380 . 10/ cos11pi . 11/ sin13pi . 12/ tan10pi .

13/ 7cot6

pi. 14/ 11cos

3

pi. 15/ 25cot

4

pi. 16/ 17sin

3

pi.

17/ 29cos6

pi. 18/ 45tan

4

pi. 19/ 16cos

3

pi . 20/ 31sin

2

pi .

21/ 19cot4

pi . 22/ 26cos

3

pi . 23/ 159tan

4

pi . 24/ 115sin

6

pi .

Bi41.Bi41.Bi41.Bi41. Din t gi tr lng gic ca cc gc sau bng gi tr lng gic ca gc x

1/ ( )0sin x 90 . 2/ ( )0cos 180 x+ . 3/ ( )0sin 270 x . 4/ ( )0sin x 180 .

5/ ( )0cos x 540+ . 6/ ( )0cot 180 x+ . 7/ ( )0sin x 540 . 8/ ( )0tan 360 x .

9/ ( )0cos 450 x+ . 10/ ( )2 0sin 270 x+ . 11/ ( )3 0cos 90 x+ . 12/ ( )5 0cot 180 x . Bi42.Bi42.Bi42.Bi42. Din t gi tr lng gic ca cc gc sau bng gi tr lng gic ca gc x

1/ ( )cot xpi . 2/ ( )sin xpi + . 3/ ( )tan 2 xpi . 4/ ( )cot 3 xpi .

5/ ( )sin x 7 pi . 6/ ( )tan x 5 pi . 7/ 5sin x2

pi + . 8/ 3cos x

2

pi + .

9/ 3cot x2

pi . 10/ 5cos x

2

pi . 11/ 11tan x

2

pi + . 12/ 7sin x

2

pi + .

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13/ ( )2sin xpi + . 14/ ( )9cos xpi . 15/ ( )11cot x 3 pi . 16/ ( )4cos x 3+ pi .

17/ ( )2cot x 5 pi . 18/ ( )6cos xpi . 19/ 8cos x2

3pi . 20/ 5cos x

2

pi .

21/ 2012sin x2

pi . 22/ 2 9tan x

2

pi . 23/ 2013 7cos x

2

pi + . 24/ 1991 5sin x

2

pi .

25/ 2015 11cos x2

pi . 26/ 2 9cot x

2

pi . 27/ 11 11tan x

2

pi + . 28/ ( )cos x 2012 pi .

Bi43.Bi43.Bi43.Bi43. Rt gn (n gin) cc biu thc

1/ ( )A cos x sin x2

pi = + pi .

2/ B cos x sin x cos x sin x2 2 2 2

pi pi pi pi = + + + .

3/ ( ) 7 3C 2 cos x 3 cos x sin x tan x2 2

pi pi = + pi + .

4/ 3

D 2sin x sin(5 x) sin x cos x2 2 2

pi pi pi = + + pi + + + + .

5/ 7 3

E 2cos x 3cos( x) 5 sin x cot x2 2

pi pi = pi + + .

6/ ( ) ( ) 3F sin 5 x cos x cot 3 x tan x2 2

pi pi = pi + + + pi + .

7/ ( ) 3 11G cos 15 x sin x tan x cot x2 2 2

pi pi pi = pi + + .

8/ ( ) ( ) 3H sin x cos x cot 2 x tan x2 2

pi pi = pi + + pi + .

9/ ( ) ( )3 3I cos 5 x sin x tan x cot 3 x2 2

pi pi = pi + + + pi .

10/ ( ) ( ) ( ) ( )0 0 0 0J cos 270 x 2 sin x 450 cos x 900 2sin 270 x= + + + .

11/ ( )2 2 2 23K sin x sin x sin x sin x4 2 4

pi pi pi = + + + + + + + pi .

12/ 2 2 2 2 2 22 5 7

L sin sin sin sin sin sin3 6 9 9 8 8

pi pi pi pi pi pi= + + + + + .

13/ ( ) ( )2013 2013 2012 2011M cos x cos x .sin x sin x2

pi = + pi + pi + .

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14/ ( ) ( ) ( )6 6 4 4 23N sin x cos x 2 sin x 2 sin x cos x2 2

pi pi = pi + + pi + pi + .

15/

( ) ( )

( )

19tan x .cos 36 x .sin x 5

2O

9sin x .cos x 99

2

pi pi pi =

pi pi

.

16/ ( ) ( )2 285 3P sin x cos 207 x sin 33 x sin x2 2

pi pi = + + pi + + pi + + .

Bi44.Bi44.Bi44.Bi44. Rt gn v tnh gi tr ca biu thc (khng dng my tnh) 1/ 0 0 0 0A cos 0 cos20 cos 40 ...... cos180= + + + + .

2/ 0 0 0 0B cos20 cos 40 cos 60 ...... cos180= + + + + .

3/ 0 0 0 0C cos10 cos 40 cos70 ...... cos170= + + + + .

4/ 0 0 0 0D tan20 tan 40 tan 60 ...... tan180= + + + + .

5/ 0 0 0 0E cot15 cot 30 cot45 ...... cot165= + + + + .

6/ 0 0 0 0F sin 5 sin10 sin15 ...... sin 360= + + + + .

7/ 0 0 0 0G cot195 cot210 cot225 ...... cot345= + + + + .

8/ 0 0 0 0H cot15 .cot35 .cot55 .cot75= .

9/ 0 0 0 0I tan10 . tan20 . tan 30 ...... tan 80= .

10/ 0 0 0 0J tan1 . tan2 .tan 3 ...... tan 89= .

11/ 2 0 2 0 2 0 2 0K sin 28 sin 36 sin 54 cos 152= + + + .

12/ 2 0 2 0 2 0 2 0L cos 2 cos 4 cos 6 ...... cos 88= + + + + .

13/ 2 0 2 0 2 0 2 0M sin 10 sin 20 sin 30 ...... sin 90= + + + + .

14/ 2 0 2 0 2 0 2 0N cos 10 cos 20 cos 30 ... cos 180= + + + + .

15/ 0 0 0 0 0O sin20 sin 40 sin 60 ... sin 340 sin 360= + + + + + .

Bi45.Bi45.Bi45.Bi45. Rt gn v tnh gi tr ca biu thc (khng dng my tnh) 1/ ( )0 0A cos 315 .sin 765= . 2/ 0 0 0 0B sin 32 .sin148 sin 302 .sin122= .

3/ 0 0 0 0C sin 810 .cos 540 tan135 .cot585= + .

4/ ( ) ( )0 0 0 0D sin 825 .cos 15 cos75 .sin 555= + . 5/ 0 0 0E 2 tan 540 2 cos1170 4 sin 990= + + .

6/ ( )0 0

0

0 0

sin 234 cos216F .tan 36

sin144 cos126

=

.

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7/ ( )0 0

0

0 0

sin 324 cos666G .cot36

sin1206 cos 36

+=

+.

8/ ( ) ( ) ( )

( )

0 0 0 0

0 0

sin 328 .sin 958 cos 508 .cos 1022H

cot572 tan 212

=

.

9/ ( )( )

0 0

0

0 0

cos 288 .cot72I tan18

tan 142 .sin108

=

.

10/ ( ) ( ) ( ) ( )0 0 0 0J 2sin 790 x cos 1260 x tan 630 x .tan 1260 x= + + + + .

11/ ( )0 0

0 0 0

2 sin2550 .cos 1881K

tan 368 2cos638 cos98

= +

+.

12/ ( )0 0 0

0 0

0

cos 44 tan226 .cos 406L cos72 .cot18

cos 316

+= .

13/ ( )0 0 0 0

0 0

0

tan 46 .sin 44 cot 136 .sin 404M tan 36 .tan54

cos 316

+ = .

14/ ( ) ( ) ( )

( )

0 0 0 0

0 0

sin 328 .sin 958 cos 508 .cos 1022N

cos572 tan 212

=

.

15/

4 17sin cos

3 6O23 10

sin cos6 3

pi pi+

= pi pi +

.

16/ ( ) ( )

( )( ) ( )

( )sin 4, 8 .sin 5,7 cos 6,7 .cos 5, 8

Pcot 5,2 tan 6,2

pi pi pi pi= +

pi pi.

Bi46.Bi46.Bi46.Bi46. Chng minh rng nu A, B, C l ba gc ca mt tam gic th

1/ ( )sinB sin A C= + . 2/ ( )cos A B cosC+ = .

3/ A B Csin cos2 2

+= . 4/ ( ) ( )cos B C cos A 2C = + .

5/ ( )cos A B C cos2C+ = . 6/ ( )sin A 2B C sinB+ + = .

7/ ( )cot A B C cot2B + = . 8/ 3A B Ccos sin2A2

+ += .

9/ A B 3Csin cosC2

+ += . 10/ A B 2C 3Ctan cot

2 2

+ = .

11/ A 3B Ccos sin B2

+ += . 12/ A 2B C 3Bcot tan

2 2

+= .

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Bi47.Bi47.Bi47.Bi47. Khng dng my tnh. Hy tnh gi tr ca cc biu thc

1/ 0 0 0 0A sin12 .cos 48 cos12 .sin 48= + .

2/ 0 0 0 0B cos 38 .cos22 sin 38 .sin22= .

3/ 0 0 0 0C sin10 .cos 55 cos10 .sin 55= .

4/ 0 0 0 0D sin 36 .cos 6 sin126 .cos 84= .

5/ 0 0 0 0E cos112 .cos23 sin112 .sin 23= .

6/ 0 0 0 0F sin200 .sin 310 cos 340 .cos50= + .

7/ 0 0 0 0 0G cos11 .cos21 cos69 .cos79 cos10= + .

8/ 0 0 0 0 0H cos68 .cos78 cos22 .cos12 sin100= + .

9/ ( ) ( )0 0 0 0I cos 53 .sin 337 sin 307 .sin113= + . Bi48.Bi48.Bi48.Bi48. Tnh gi tr lng gic ca cc cung gc sau

1/ 015 . 2/ 075 . 3/ 0105 . 4/ 0285 .

5/ 1912

pi. 6/ 5

12

pi. 7/ 7

12

pi. 8/ 13

12

pi.

Bi49.Bi49.Bi49.Bi49. Tnh gi tr ca cc biu thc sau

1/ 0

0

1 tan15A

1 tan15

+=

. 2/ 0 0

0 0

tan25 tan20B

1 tan25 tan20

+=

.

3/ 0 0 0 0

0 0 0 0

sin10 cos20 sin20 cos10C

cos17 cos13 sin17 sin13

+=

. 4/

0 0 0

0 0

tan225 cot81 cot69D

cot261 tan201

=

+.

5/ 0 0 0 0

0 0 0 0

sin 73 cos 3 sin 87 cos17E

cos132 cos 62 cos 42 cos28

=

+. 6/

0 o0 0

0 0

cot225 cot79 ot71F

cot259 cot251

=

+.

C CNG THC CNG

.

.

. .

H qu:

. .

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7/ 2 0 2 0G cos 75 sin 75= . 8/ 2 0 2 0 2 0H sin 20 sin 100 sin 140= + + .

9/ 2 0 0 2 0I cos 10 cos110 cos 130= + + .

10/ 0 0 0 0 0 0J tan20 .tan 80 tan 80 .tan140 tan140 .tan20= + + .

11/ 0 0 0 0 0 0K tan10 .tan70 tan70 .tan130 tan130 .tan190= + + .

12/ 0 0 0 0 0 0L sin160 cos110 sin250 cos 340 tan110 tan 340= + + .

13/ ( )( ) ( )( )0 0 0 0 0 0 0 0M cos70 cos50 cos310 cos290 cos40 cos160 cos320 cos380= + + + + . Bi 50.Bi 50.Bi 50.Bi 50. Tnh gi tr ca

Ly ThuyetBai Tap TOAN 10Tap 2

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