Cac Dang Toan Trong de Thi 2002 2015 (1)

7/24/2019 Cac Dang Toan Trong de Thi 2002 2015 (1)

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HONG NGC TH

PHN DNG

THI I HC

MN TON(2002 - 2015)


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PHN DNG THI I HC MN TON

T nm 2002 n nm 2015

Hong Ngc Th

Ngy 20 thng 7 nm 2015

2


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Li ni u

Ti liu nh ny gii thiu thi H mn ton t nm 2002 (nm utin ton quc thi chung) n nm 2015. Cc thi c phn dng v

sp xp theo cc ch ln:

1. Kho st hm s

2. Lng gic

3. Phng trnh, h phng trnh, bt phng trnh

4. Tch phn v ng dng

5. Hnh hc tng hp trong khng gian

6. Bt ng thc

7. Phng php ta trong khng gian

8. Phng php ta trong mt phng

9. S phc10. T hp - xc sut

mi ch , bi c sp xp theo nm thi v c p n hoc hngdn i km gip bn c d theo di v kim tra kt qu ca mnh. Bnc nn t lm cc thi sau so snh vi p n. lm c cc thi ny, i hi bn c cn c mt qu trnh n tp kin tr v c hiu qu.

Trong qu trnh tng hp vi vng, hn l s c nhiu thiu st. Rtmong nhn c s ng gp ca cc bn.

Hong Ngc Th

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1 Kho st hm s

1. (A-2002)Cho hm s

y= x3

+ 3mx

2

+ 3(1 m2

)x + m

3

m2

(1)a) Kho st s bin thin v v th hm s (1) khi m= 1.b) Tm k phng trnh:x3 + 3x2 +k3 3k2 = 0c 3nghimphn bit.c) Vit phng trnh ng thng qua 2 im cc tr ca hm s (1).

A: b) 1< k


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A:12

< m 0

6. (B-2003)Tm gi tr ln nht, gi tr nh nht ca hm s

y=x +

4 x2

A:max y= 2

2; min y= 2

7. (D-2003)Cho hm s

y= x2 2x + 4

x 2 (6)

a) Kho st s bin thin v v th ca hm s (6).b) Tm iu kin ca tham sm th hm s (6) v ng thngdm:y =mx + 2 2mct nhau ti hai im phn bit.

A: m >1

8. (D-2003)Tm gi tr ln nht, gi tr nh nht ca hm s

y = x + 1

x2 + 1

trn [1;2]

A: max[1;2] y= 2; min[1;2] y= 0

5


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9. (A-2004)Cho hm s

y=x2 + 3x 3

2(x 1) (7)

a) Kho st s bin thin v v th ca hm s (7).b) Tm m th hm s (7) v ng thng y =mct nhau tihai im phn bit A, Bsao cho AB = 1

A: m=15

2

10. (B-2004)Cho hm s

y =1

3x3 2x2 + 3x (8)

a) Kho st s bin thin v v th ca hm s (8).b) Vit phng trnh tip tuyn d ca th hm s (8) ti imun. Chng minh dl tip tuyn c h s gc nh nht.

A: y= x +83

11. (B-2004) Tm gi tr ln nht, gi tr nh nht ca hm s: y= ln2 x

xtrn on[1; e3]

A:max[1;e3]

y= 4

e2; min[1;e3]

y= 0

12. (B-2004) Tm iu kin ca tham s m phng trnh sau cnghim:

m

1 + x2

1 x2 + 2

= 2

1 x4 +

1 + x2

1 x2

A:

2

1

m

1

6


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13. (D-2004)Cho hm s

y=x3 3mx2 + 9x + 1 (9)

a) Kho st s bin thin v v th ca hm s (9) khi m= 2.b) Tm m im un ca th (9) thuc ng thng y =x + 1.

A: m= 0;2

14. (A-2005)Cho hm s

y=mx +1

x (10)

a) Kho st s bin thin v v th ca hm s (10) khi m=1

4 .b) Tm m hm s (10) c cc tr v khong cch t cc tiu n

tim cn xin bng 1

2.

A: m= 1

15. (B-2005)Cho hm s

y= x2 + (m + 1)x + m + 1

x + 1 (11)

a) Kho st s bin thin v v th ca hm s (11) khi m= 1.b) Chng minh rng vi mi gi tr ca m, hm s (11) lun c cci, cc tiu v khong cch gia hai im bng

20.

16. (D-2005)Cho hm s

y=1

3x3 m

2x2 +

1

3 (12)

a) Kho st s bin thin v v th ca hm s (12) khi m= 2.b) Gi M l im thuc th ca hm s (12) c honh bng1. Tm m tip tuyn ca th hm s (12) ti Msong songvi ng thng5x y= 0.

A: m= 4

7


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17. (A-2006)Cho hm s

y= 2x3 9x2 + 12x 4 (13)

a) Kho st s bin thin v v th ca hm s (13).b) Tm iu kin ca tham s m phng trnh sau c 6nghimphm bit:

2|x|3 9x2 + 12|x| =m

A:4< m 154 , m = 24

8


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21. (A-2007)Cho hm s

y= x2 + 2(m + 1)x + m2 + 4m

x + 2 (16)

a) Kho st s bin thin v v th ca hm s (16) khi m= 1.b) Tm gi tr ca tham s m th hm s (16) c im cc i,cc tiu v hai im to vi gc ta Omt tam gic vung tiO.

A: m= 4 2

6

22. (A-2007) Tm iu kin ca tham s m phng trnh sau cnghim:3

x 1 + mx + 1 = 2 4

x2 1

A:1< m 13

23. (B-2007)Cho hm s

y= x3 + 3x2 + 3(m2 1)x 3m2 1 (17)

a) Kho st s bin thin v v th ca hm s (17) khi m= 1.b) Tm gi tr ca tham s m cc im cc i v cc tiu cahm s cch u gc ta O.

A: m= 12

24. (D-2007)Cho hm s

y= 2x

x + 1 (18)

a) Kho st s bin thin v v th ca hm s (18).b) Tm ta im Mthuc th sao cho tip tuyn ca thti Mct hai trc to ti A, B sao cho din tch tam gic OAB

bng 1

4 .

9


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A: M1

1

2;2

, M2(1; 1)

25. (A-2008)Cho hm s

y= mx2 + (3m2 2)x 2

x + 3m (19)

a) Kho st s bin thin v v th ca hm s (19) khi m= 1.b) Tm gi tr ca tham s m gc gia hai tim cn ca thbng 450

A: m=

1

26. (A-2008) Tm cc gi tr ca tham s m phng trnh sau cng hai nghim thc:

4

2x +

2x + 2 4

6 x + 26 x= m

A:2

6 + 2 4

6 m 3) u ct th hm s (21) ti ba im phn bitI , A , B ng thi Il trung im AB.

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29. (A-2009)Cho hm s

y = x + 2

2x + 3 (22)

a) Kho st s bin thin v v th ca hm s (22).

b) Vit phng trnh tip tuyn ca th hm s, bit tip tuynct Ox, Oyti A, Bv tam gic OAB cn ti O.

A: y= x 2

30. (B-2009)Cho hm s

y= 2x4

4x2 (23)

a) Kho st s bin thin v v th ca hm s (23).b) Tm iu kin ca tham s msao cho phng trnh sau c ng6nghim thc:

x2|x2 2| =m

A:0< m


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A:13

< m


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A: y= 6x + 10

37. (A-2011)Cho hm s

y=x + 1

2x 1 (28)

a) Kho st s bin thin v v th ca hm s (28).b) ng thng y = x+m ct th hm s ti A, B. Tip tuynti A, B c h s gc ln lt l k1, k2. Tm mk1+ k2ln nht.

A: m= 1

38. (B-2011)Cho hm s

y=x4 2(m + 1)x2 + m. (29)

a) Kho st s bin thin v v th ca hm s (29) khi m= 1.b) Tm m th hm s (29) c 3 im cc tr A, B, C sao choOA = OB, Athuc trc tung cn B, Cl hai cc tr cn li.

A: m= 2

2

2

39. (D-2011)Cho hm s

y =2x + 1

x + 1 (30)

a) Kho st s bin thin v v th ca hm s (30).b) Tm iu kin ca tham sk ng thngy=kx + 2k + 1ct th hm s trn ti hai im phn bit A, Bsao cho khong cch

tA, Bn trc honh bng nhau.

A: k= 3

40. (D-2011)Tm gi tr ln nht, gi tr nh nht ca hm s

y =2x2 + 3x + 3

x + 1

trn [0; 2].

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A:min y= 3; max y=17

3

41. (A-2012)Cho hm s

y=x4 2(m + 1)x2 + m2 (31)a) Kho st s bin thin v v th ca hm s (31) khi m= 0.b) Tm iu kin ca tham s m hm s (31) c 3 cc tr l banh ca tam gic vung.

A: m= 0

42. (B-2012)Cho hm sy=x3 3mx2 + 3m3 (32)

a) Kho st s bin thin v v th ca hm s (32) khi m= 1.b) Tm iu kin ca tham s m hm s c hai cc tri A, B saocho din tch tam gic OAB bng 48.

A: m= 2

43. (D-2012)Cho hm s

y=2

3x3 mx2 2(3m2 1)x +2

3 (33)

a) Kho st s bin thin v v th ca hm s (33) khi m= 1.b) Tm iu kin ca tham s m hm s c hai im cc trx1, x2sao cho x1x2+ 2(x1+ x2) = 1.

A: m=2

3

44. (A-2013)Cho hm s

y= x3 + 3x2 + 3mx 1 (34)a) Kho st s bin thin v v th ca hm s (34) khi m= 0.

b) Tm iu kin ca tham s m hm s (34) nghch bin trong(0;+).

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A: m 1

45. (B-2013)Cho hm s

y= 2x3

3(m + 1)x2

+ 6mx (35)a) Kho st s bin thin v v th ca hm s (35) khi m= 1.b) Tm iu kin ca tham s m th hm s (35) c hai imcc tr AvBsao cho ng thngABvung gc vi ng thngy =x + 2.

A: m= 0, m= 2

46. (D-2013)Cho hm s

y= 2x3 3mx2 + (m 1)x + 1 (36)

a) Kho st s bin thin v v th ca hm s (36) khi m= 1.b) Tm iu kin ca tham s m ng thng y= x + 1ct th hm s (36) ti ba im phn bit.

A: m 8

9

47. (D-2013NC) Tm gi tr ln nht, gi tr nh nht ca hm s

y =2x2 3x + 3

x + 1 trn on[0; 2]

A:min y = 1; max y= 3

48. (A-2014)Cho hm s

y= x + 2

x 1 (37)

a) Kho st s bin thin v v th ca hm s (37).b) Tm ta im Mthuc th hm s sao cho khong cch tMn ng thngy = xbng

2

A: M(0;2), M(2;0)

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49. (B-2014)Cho hm s

y=x3 3mx + 1 (38)

a) Kho st s bin thin v v th ca hm s (38) khi m= 1.b) Cho im A(2; 3). Tm m th hm s (38) c hai im cctr Bv Csao cho tam gic ABCcn ti A.

A: m=1

2

50. (D-2014)Cho hm s

y=x3 3x 2 (39)a) Kho st s bin thin v v th ca hm s (39).b) Tm ta im Mthuc th sao cho tip tuyn ca thti Mc h s gc bng 9.

A: M(2;0), M(2;4)

51. (2015)Kho st s bin thin v v th hm s y=x3 3x.

52. (2015)Tm gi tr ln nht v gi tr nh nht ca hm s y =x +4

xtrn [1; 3].

A:max y= 5; min y= 4

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2 Lng gic

1. (A-2002)Gii phng trnh:

5

sin x +cos 3x + sin 3x1 + 2 sin 2x

= cos 2x + 3

A: x= 3

+ 2k

2. (B-2002)Gii phng trnh:

sin2 3x

cos2 4x= sin2 5x

cos2 6x

A: x= k

9 ; x=

k

2

3. (D-2002)Tm xthuc on [0; 14]nghim ng phng trnh:

cos3x 4cos2x + 3 cos x 4 = 0

A: 2;3

2 ;5

2 ;7

2


cot x 1 = cos2x1 + tan x

+ sin2 x 12

sin 2x

A: x=

4+ k


cot x tan x + 4 sin 2x= 2sin2x

A: x= 3

+ k

17


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6. (D-2003)Gii phng trnh:

sin2x

2

4

tan2 x cos2 x

2 = 0

A: x= + k2; x= 4

+ k

7. (A-2004)Cho tam gic ABCkhng t tha mn iu kin:

cos2A + 2

2cos B+ 2

2cos C= 3

Tnh ba gc ca tam gic.

A: A= 90o, B =C= 45o


5sin x 2 = 3(1 sin x)tan2 x

A: x=

6

+ k2; x=5

6

+ k2


(2cos x 1)(2 sin x + cos x) = sin2x sin x

A: x= 3

+ k2; x= 4

+ k


cos2 3x cos2x cos2 x= 0

A: x= k

2


1 + sin x + cos x + sin 2x + cos 2x= 0

18


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A: x= 4

+ k; x= 23

+ k2


cos4 x + sin4 x + cos

x 4

sin

3x 4

3

2= 0

A: x=

4+ k


2

cos

6 x+ sin

6 x sin x cos x2 2sin x = 0

A: x=5

4 + 2k


cot x + sin x

1 + tan x tan

x

2

= 4

A: x=

12+ k; x=

5

12+ k


cos3x + cos 2x cos x 1 = 0

A: x= k; x= 23

+ k2

16. (A-2007)Gii phng trnh:1 + sin2 x

cos x +

1 + cos2 x

sin x= 1 + sin 2x

A: x=

4+ k; x=

2+ k2; x= k2

19


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2sin2 2x + sin 7x 1 = sin x

A: x=

8+ k

4 ; x=

18+ k

2

3 ; x=

5

18+ k

2

3

18. (D-2007)Gii phng trnh:sin

x

2+ cos

x

2

2+

3cos x= 2

A: x=

2+ k2; x=

6+ k2


1

sin x+

1

sin

x 3

2

= 4 sin74 x

A: x=

4

+ k; x=

8

+ k; x=5

8

+ k


sin3 x

3cos3 x= sin x cos2 x

3sin2 x cos x

A: x=

4+ k

2; x=

3+ k


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

A: x= 23

+ k2; x=

4+ k


(1

2sin x)cos x

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

20


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A: x= 18

+ k2

3


sin x + cos x sin2x +3cos3x= 2(cos 4x + sin3 x)

A: x= 6

+ k2; x=

42+ k

2

7


3cos5x

2sin3x cos2x

sin x= 0

A: x=

18+ k

3; x=

6+ k

2


(1 + sin x + cos 2x)sin

x +

4

1 + tan x

= 1

2cos x

A: x= 6

+ k2; x=7

6 + k2


(sin2x + cos 2x)cos x + 2 cos 2x sin x= 0

A: x=

4+ k

2


sin2x cos2x + 3 sin x cos x 1 = 0

A: x=

6+ k2; x=

5

6 + k2;

21


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1 + sin 2x + cos 2x

1 + cot2 x =

2sin x sin2x

A: x= 2

+ k; x= 4

+ k2


sin2x cos x + sin x cos x= cos 2x + sin x + cos x

A: x=

2

+ k2; x=

3

+ k2

3


sin2x + 2 cos x sin x 1tan x +

3

= 0

A: x=

3+ k2


3sin2x + cos 2x= 2 cos x 1

A: x=

2+ k; x= k2; x=

2

3 + k2

32. (B-2012)Gii phng trnh:2

cos x +

3sin x

cos x= cos x

3sin x + 1

A: x=2

3 + k2; x=

2

3


sin3x + cos 3x sin x + cos x= 2cos2x

22


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A: x=

4+ k

2; x=

7

12+ k2; x=

12+ k2


1 + tan x= 22sinx + 4A: x=

4+ k; x=

3+ k2


sin5x + 2 cos2 x= 1

A: x= 6+ k 23 ; x= 14+ k 2736. (D-2013)Gii phng trnh:

sin3x + cos 2x sin x= 0

A: x=

4+ k

2; x=

6+ k2; x=

7

6 + k2

37. (A-2014)Gii phng trnh:sin x + 4 cos x= 2 + sin 2x

A: x= 3

+ k2


2(sin x

2cos x) = 2

sin2x

A: x= 34

+ k2

39. (2015)Tnh gi tr ca biu thc P = (1 3cos2)(2 + 3 cos 2)bit sin =

2

3.

A:

14

9

23


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3 Phng trnh, h phng trnh, bt phng trnh

1. (A-2002)Cho phng trnh:

log23 x +

log23 x + 1 2m 1 = 0 (40)a) Gii phng trnh vi m= 2b) Tm m phng trnh (40) c t nht mt nghim thuc on

1; 33

A:a) x= 33; b)0

m

2

2. (B-2002)Gii h phng trnh:3

x y= x yx + y=

x + y+ 2

A:(1; 1),3

2;1

23. (B-2002)Gii bt phng trnh: logx(log3(9

x 72)) 1

A: (log973; 2]

4. (D-2002)Gii h phng trnh:

23x = 5y2

4y

4x + 2x+1

2x + 2 =y

.

A:(0;1), (2; 4)

5. (D-2002)Gii bt phng trnh: (x2 3x)

2x2 3x 2 0

A:;12 {2} [3;+)

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6. (A-2003)Gii h phng trnh:

x 1x

=y 1y

2y=x3

+ 1

A:(1; 1),

1 + 5

2 ;

1 + 52

,

1 5

2 ;

152

7. (B-2003)Gii h phng trnh:

3y=

y2 + 2

x2

3x= x2 + 2

y2

A:(1; 1)

8. (D-2003)Gii phng trnh: 2x2x 22+xx2 = 3

A: x= 1, x= 2

9. (A-2004)Gii bt phng trnh:2(x2 16)

x 3 +

x 3> 7 xx 3

A: x >10

34


log 1

4

(y x) log41

y = 1

x2 + y2 = 25

A:(3; 4)

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11. (D-2004)Tm m h phng trnh sau c nghim:x +

y= 1

x

x + y

y= 1

3m

A:0 m 14

12. (D-2004)Chng minh rng phng trnh sau c ng mt nghim:

x5 x2 2x 1 = 0

13. (A-2005)Gii bt phng trnh:

5x 1 x 1> 2x 4

A: [2; 10]


x 1 +2 y= 13log9(9x

2) log3 y3 = 3

A:(1;1), (2; 2)

15. (D-2005)Gii phng trnh: 2

x + 2 + 2

x + 1 x + 1 = 4

A: x= 3

16. (A-2006)Gii h phng trnh:x + y xy = 3

x + 1 +

y+ 1 = 4

A:(3; 3)

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17. (A-2006PB)Gii phng trnh:

3.8x + 4.12x 18x 2.27x = 0

A: x= 1

18. (B-2006PB)Gii bt phng trnh:

log5(4x + 144) 4log520, h phng trnhsau c nghim duy nht:

ex

ey

= ln(1 + x) ln(1 + y)y x= a

21. (D-2006PB)Gii phng trnh:

2x2+x 4.2x2x 22x + 4 = 0

A: x= 0; x= 1


2log3(4x 3) + log 13

(2x + 3) 2

A: 3

4

; 3

27


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23. (B-2007)Chng minh rng vi mi gi tr dng ca tham s m,phng trnh sau c hai nghim thc phn bit:

x2 + 2x 8 = m(x 2)24. (B-2007PB)Gii phng trnh:

2 1x

+

2 + 1x 22 = 0

A: x= 1

25. (D-2007) Tm gi tr ca tham s m h phng trnh sau cnghim thc:

x +

1

x+ y+

1

y = 5

x3 + 1

x3+ y3 +

1

y3 = 15m 10

A: 7

4m

2hoc m

22

26. (D-2007PB)Gii phng trnh:

log2(4x + 15.2x + 27) + 2 log2

1

4.2x 3= 0

A: x= log23


x2 + y+ x3y+ xy2 + xy= 54

x4 + y2 + xy(1 + 2x) = 54

A: 354

;

32516 ,1;

3

2

28


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log2x1(2x2 + x 1) + logx+1(2x 1)2 = 4

A: x= 2; x=5

4

29. (B-2008)Gii h phng trnh:x4 + 2x3y+ x2y2 = 2x + 9

x2 + 2xy = 6x + 6

A:4;17

4

30. (B-2008PB)Gii bt phng trnh:

log0,7

log6

x2 + x

x + 4


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A: x= 2

34. (A-2009NC)Gii h phng trnh:

log2(x

2

+ y

2

) = 1 + log2(xy)3x

2xy+y2 = 81

A:(2; 2), (2;2)


xy+ x + 1 = 7yx2

y2

+ xy+ 1 = 13y2

A:

1;1

3

, (3; 1)


x(x + y+ 1) 3 = 0(x + y)2 5

x2+ 1 = 0

A:(1; 1),

2;32

37. (A-2010)Gii bt phng trnh

x

x

1 2(x2 x + 1) 1A: x=

352


(4x2 + 1)x + (y 3)5 2y= 04x2 + y2 + 23 4x= 730


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A:

1

2; 2


3x + 16 x+3x214x8 = 0

A: x= 5

40. (B-2010NC)Gii h phng trnhlog2(3y 1) =x4x + 2x = 3y2

A:1;1

2


42x+x+2 + 2x

3

= 42+x+2 + 2x

3+4x4

A: x= 1; x= 2

42. (D-2010NC)Gii h phng trnh:x2 4x= y + 2 = 02log2(x 2) log2 y= 0

A:(3; 1)

43. (A-2011)Gii h phng trnh:5x2y 4xy2 + 3y3 2(x + y) = 0xy(x2 + y2) + 2 = (x + y)2

A:(1; 1), (

1;

1),2

10

5

;

10

5 ,

2

10

5

;

10

5

31


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3

2 + x 62 x + 4

4 x2 = 10 3x

A: x=6

5


log2(8 x2) + log 12

1 + x +

1 x 2 = 0

A: x= 0

46. (D-2011)Tm m h phng trnh sau c nghim:2x3 (y+ 2)x2 + xy=mx2 + x y= 1 2m

A: m 2

3

2


x3 3x2 9x + 22 =y3 + 3y2 9yx62 + y2 x + y=1

2

A:

1

2;3

2

,

3

2;1

2

48. (B-2012)Gii bt phng trnh: x + 1 +

x2 4x + 1 3x

A:

0;1

4

[4;+)


xy+ x 2 = 02x3 x2y+ x2 + y2 2xy y= 032


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A:(1; 1),

1 + 5

2 ;

5

,

1 5

2 ;

5

50. (A-2013)Gii h phng trnh:x + 1 + 4

x 1

y4 + 2 =y

x2 + 2x(y 1) + y2 6y+ 1 = 0

A:(1;0), (2; 1)

51. (B-2013)Gii h phng trnh:2x2 + y2 3xy+ 3x 2y+ 1 = 04x2 y2 + x + 4 =

2x + y+

x + 4y

A:(0;1), (1; 2)

52. (B-2013NC)Gii h phng trnh:

x2 + 2y= 4x 12log3(x 1) log3(y+ 1) = 0

A:(3; 1)


2log2 x + log 12

(1 x) =1

2log2(x 2x + 2)

A:1 + ln 2


x12 y+ y (12 x2) = 12x3 8x 1 = 2y 2

33


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A:(3; 3)


(1 y)x y+ x= 2 + (x y 1)y2y2 3x + 6y+ 1 = 2

x 2y

4x 5y 3

A:(3; 1),

1 +

5

2 ;1 + 5

2


log2(x 1) 2log4(3x 2) + 2 = 0

A: x= 2

57. (D-2014)Gii bt phng trnh:

(x + 1)

x + 2 + (x + 6)

x + 7 x2 + 7x + 12

A: [2;2]

58. (2015)Gii phng trnhlog2(x2 + x + 2) = 3.

A: x= 2; x= 3

59. (2015)Gii phng trnh trn tp s thc:

x2 + 2x 8x2 2x + 3 = (x + 1)(

x + 2 2)

A: x= 2, x=3 +

13

2

34


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4 Tch phn v ng dng

1. (A-2002)Tnh din tch hnh phng gii hn bi cc ng:

y = |x2

4x + 3|; y=x + 3

A: S=109

6

2. (B-2002)Tnh din tch hnh phng gii hn bi cc ng:

y= 4x2

4

; y= x2

42

A: S= 2+4

3

3. (D-2002) Tnh din tch hnh phng gii hn bi th hm s

y =3x 1

x 1 v hai trc ta .

A:1 + 4 ln43

4. (A-2003)Tnh tch phn I=

23

5

dx

x

x2 + 4dx

A: 14ln5

3

5. (B-2003)Tnh tch phn I=

40

1 2sin2 x1 + sin 2x

dx

A: 12ln 2

35


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6. (D-2003)Tnh tch phn I=

20

x2 x dx

A:1


21

x

1 +

x 1 dx

A: 11

3 4 l n 2


e1

1 + 3 ln x ln x

x dx

A: 116

135


32

ln(x2 x)dx

A:3ln3 2


2

0

sin2x + sin x

1 + 3 cos xdx

A: 34

27


2

0

sin2x cos x

1 + cos x dx

36


37/94

A:2 ln 2 1


20

esinx

+ cos x

cos xdx

A: e +

4 1


2

0

sin2xcos2 x + 4 sin2 x dx

A: x=2

3


ln 5

ln 3

dx

ex + 2ex

3

A: ln3

2


10

(x 2)e2xdx

A: 5 3e2

4

16. (A-2007)Tnh din tch hnh phng gii hn bi cc ng:

y= (e + 1)x; y= (1 + ex)x

A: e

2 1

37


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17. (B-2007)Cho hnh phng Hgii hn bi cc ng:

y=x ln x, y= 0, x= e

Tnh th tch ca khi trn xoay to thnh khi quay Hquanh trcOx.

A: V = (5e3 2)

27


e1

x3 ln2 xdx

A: 5e4 1

32


60

tan4 x

cos2xdx

A: 1

2ln

2 +

3 10

9

3


40

sin

x x4

dx

sin2x + 2(1 + sin x + cos x)

A: 4 32

4


21

ln x

x3 dx

A: 3 2 l n 216

38


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20

cos3 x 1 cos2 xdx

A: 8

15

4


30

3 + ln x

(x + 1)2dx

A: 14

3 + ln27

16


31

dx

ex 1 dx

A: ln(e2 + e + 1)

2


10

x2 + ex + 2x2ex

1 + 2ex dx

A: 1

3+

1

2ln

1 + 2e

3

26. (B-2010)Tnh tch phn I=e

1

ln x

x(2 + ln x)2dx

A:13

+ ln3

2


e1

2x

3

x

ln xdx

39


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A: e2

2 1


40

x sin x + (x + 1) cos xx sin x + cos x

dx

A:

4+ ln

2

2

4

+ 1


30

1 + x sin x

cos2 x dx

A:

3 +2

3 + ln(2

3)


40

4x

1

2x + 1 + 2 dx

A: 34

3+ 10ln

3

5


3

1

1 + ln(x + 1)

x2 dx

A: 2

3+ ln 3 2

3ln 2


10

x3

x4 + 3x2 + 2dx

A: ln 3 32

ln 2

40


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40

x(1 + sin 2x)dx

A: 2

32+

1

4


21

x2 1x2

ln xdx

A: 52ln 2 32


10

x

2 x2dx

A: 2

2 13


10

(x + 1)2

x2 + 1 dx

A:1 + ln 2

37. (A-2014)Tnh din tch hnh phng gii hn bi ng cong y =x2 x + 3v ng thng y= 2x + 1.

A: 1

6


21

x2 + 3x + 1

x2 + x dx

A:1 + ln 3

41


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40

(x + 1) sin 2xdx

A: 3

4

40. (2015)Tnh tch phn I= 10

(x 3)exdx

A:4 3e

42


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5 Hnh hc tng hp trong khng gian

1. (A-2002) Cho hnh chp chp tam gic u S.ABCc cnh ybng a. Gi M, Nln lt l trung imSB, SC. Mt phng(AM N)

vung gc vi (SBC). Tnh theo adin tch tam gic AM N.

A: a2

10

16

2. (B-2002) Cho lp phng ABCD.ABCD cnh a. Gi M, N, P ln lt l trung im BB, C D , AD. Tnh khong cch gia ABv BD. Tnh gc gia M Pv CN.

A: a6

, 90o

3. (D-2002) Cho t din ABCD c AD vung gc vi mt phng(ABC), AC = AD = 4cm, AB = 3cm,BC = 5cm. Tnh khongcch tAn(BCD).

A: 6

34

174. (A-2003)Cho lp phng ABCD.ABCD . Tnh gc phng nh

din[B, AC, D]

A: 120o

5. (B-2003)Cho lng tr ng ABCD.ABCD c y l hnh thoicnh a, BAD = 600. Gi M, N ln lt l trung im AA, CC.

Chng minh rng B, M , D , N ng phng. Tnh AA theo a BM N Dl hnh vung.

A: AA =a

2

6. (D-2003)Hai mt phng (P)v (Q)ct nhau theo giao tuyn .Trn ly AB = a. Trong (P) ly C, trong (Q) ly D sao choAC=BD=AB v AC, BDcng vung gc vi . Tnh bn knh

cu ngoi tip t dinABCDv khong cch tAn (BC D)theoa.

43


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A: R= a

3

2 , d=

a

2

2

7. (B-2004) Cho chp t gic u S.ABCD, cnh y bng a. Gc

gia cnh bn v y l gc nhn. Tnh tang ca gc gia hai mtphng (SAB)v (ABCD). Tnh th tch khi chp S.ABCD theoa.

A: tan =

2tan , V =a3

2tan

6

8. (A-2006PB) Cho hnh tr tm y l O, O. Bn knh y bng

chiu cao hnh tr v cng bng a. Trn ng trn(O)l imA,trn ng trn(O)l im Bsao cho AB= 2a. Tnh VOOAB.

A: V = a3

3

12

9. (B-2006PB)Cho chp S.ABCD c y l hnh ch nht, AB =SA = a, AD = a, cnh bn SAvung gc vi (ABCD). Gi M, N

ln lt l trung im AD,SC. ng thng BMct ng thngACtiI. Chng minh hai mt phng(SAC)v(SM B)vung gc.TnhVANIB

A: V = a3

2

36

10. (D-2006PB)Cho hnh chp S.ABCc y l tam gic u cnh a,

cnh bn SA= 2av SAvung gc vi y. Gi sM, N ln ltl hnh chiu vung gc cu Aln SB, SC. Tnh VA.BCMN

A: V =3a3

3

50

11. (A-2007PB)Cho hnh chpS.ABCDc y l hnh vung cnha.Tam gicSADu v nm trong mt phng vung gc vi y. Gi

M , N , P ln lt l trung im SB,BC,CD. Chng minh AM BPv tnh VCMNP

44


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A: V = a3

3

96

12. (B-2007PB)Cho chp t gic u S.ABCD c cnh y bng a.

Gi E l im i xng vi D qua trung im ca SA; M, N lnlt l trung im ca AE,BC. Chng minh M N BD v tnhd(MN,AC)

A: d= a

2

4

13. (D-2007PB) Cho chp S.ABCD, c y ABCD l hnh thang,ABC=

BAD= 90

0

, BA = BC=a,AD= 2a, cnh bn SAvunggc vi y, SA= a2, Hl hnh chiu ca Aln SB. Chng minhSCDvung v tnh d= d(H,(SCD)).

A: d= a

3

14. (A-2008PB)Cho lng tr ABC.ABCc cnh bn bng 2a. Tamgic ABCvung ti A, AB = a,AC= a

3. Hnh chiu ca A ln

mt phng (ABC)l trung im Hca BC. Tnh VA.ABCv cosingc gia hai ng thng AAv BC.

A: V = a3

2;cos =

1

4

15. (B-2008PB) Cho chp S.ABCD c y l hnh vung cnh 2a,SA = a, SB = a

3, (SAB)vung gc vi y. Gi M, N ln lt

l trung im AB, BC. Tnh VSBMDNv cosin gc gia hai ngthngSM,DN.

A: V = a3

3

3 ;cos =

5

5

16. (D-2008)Cho lng tr ngABC.ABCc y l tam gic vung,AB = BC = a, cnh bn AA = a

2. Gi M l trung im BC.

Tnh th tch khi lng tr v khong cch gia hai ng thngAM,BC

45


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A: V = a3

2

2 , d=

a

7

7

17. (A-2009) Cho chp S.ABCD, c y l hnh thang vung ti A

v D, AB = AD = 2a,CD = a, gc gia (SBC)v mt y bng600. Gi Il trung im AD. Hai mt phng (SBI)v(SCI)cngvung gc vi (ABCD). Tnh VABCD.

A: V =3a3

15

5

18. (B-2009)Cho lng tr ABC.ABCc BB =a, gc gia cnh bn

BBv mt y bng 600

, tam gic ABCvung tiC,BAC= 60

0

.Hnh chiu ca B ln (ABC) l trng tm tam gic ABC. TnhVAABCtheo a.

A: V =9a3

208

19. (D-2009)Cho lng tr ngABC.ABCc tam gic ABCvung

tiC,AB =a,AA = 2a,AC = 3a. GiMl trung imAC,AMct ACti I. Tnh VIABCv d(A,(IBC)) theo a.

A: V =4a3

9 ; d=

2a

5

5

20. (A-2010) Cho chp S.ABCD c y l hnh vung cnh a. GiM, N ln lt l trung im AB, AD; H l giao im ca CN v

DM. ng thng SH vung gc vi y v SH = a3. TnhVS.CDMNv d(DM,SC) theo a.

A: V =5a3

3

24 ; d=

2a

3

19

21. (B-2010)Cho lng tr tam gic u ABC.ABC c AB =a, gcgia hai mt phng(ABC)v(ABC)bng600. GiGl trng tm

tam gicABC. TnhVABC.ABCv bn knh mt cu ngoi tip tdin GABC.

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A: V =3a3

3

18 ; R=

7a

12

22. (D-2010)Cho chpS.ABCDc y l hnh vung cnha, SA= a.

Hnh chiu ca S ln mt phng y l im H trn AC sao choAH=

AC

4 .CMl ng cao ca tam gic SAC. Chng minh rng

M Ml trung im ACv tnh VSMBC.

A: V = a3

14

48

23. (A-2011)Cho chp S.ABC, c tam gic BACvung cn, AB =BC = 2a. Hai mt phng (SAB)v (SAC) cng vung gc vi(ABC); Ml trung im AB, mt phng i qua SMv song songvi BC ct AC ti N. Gc gia (SBC)v (ABC)bng 60o. TnhVS.BCNMv d(AB,SN)theo a.

A: V =a3

3; d=2a

39

13

24. (B-2011) Cho hnh lng tr ABCD.ABCD c y ABCD lhnh ch nht, AB = a,AD = a

3. Hnh chiu ca A ln mt

phng (ABCD)trng vi giao im ca ACv BD. Gc gia haimt phng(ADDA)v(ABCD)bng60o. Tnh th tch khi lngtr cho v khong cch t im Bn mt phng (ABD).

A: V =

3a3

2 ; d=

a

3

2

25. (D-2011)Cho chp S.ABCc y l tam gic vung ti B, BA=3a,BC = 4a. Mt phng (SBC) vung gc vi y. Bit SB =2a

3, SBC= 30o. Tnh th tch khi chp v khong cch tBn(SAC).

A: V = 2a3

3; d=6a

7

7

47


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26. (A-2012) Cho hnh chp S.ABCc y l tam gic u cnh a.Hnh chiu vung gc ca Sln mt phng (ABC)l im Hthuccnh AB sao cho HA = 2HB. Gc gia ng thng SCv mtphng (ABC) bng 60o. Tnh th tch khi chp S.ABC v tnh

khong cch gia hai ng thngSAv BCtheo a

A: V = a3

7

12 ; d=

a

42

8

27. (B-2012)Cho hnh chp tam gic u S.ABCvi SA= 2a,AB =a.GiHl hnh chiu vung gc ca Aln cnhSC. Chng minhSCvung gc vi mp(ABH). Tnh th tch khi chp S.ABH.

A: V =7a311

96

28. (D-2012) Cho hnh hp ng ABCD.ABCD c y l hnhvung, tam gic AACvung cn, AC=a. Tnh th tch ca khit dinABBCv khong cch t im An mt phng

BC D

theo a.

A: V = a32

48 ; d=

a66

29. (A-2013)Cho hnh chp S.ABCc y l tam gic vung ti A,ABC = 30o, SBCl tam gic u cnh av mt bn SBCvunggc vi y. Tnh theo ath tch khi chp S.ABCv khong cchtCn mt phng(SAB).

A: V = a3

16; d=

a3913

30. (B-2013) Cho hnh chp S.ABCD c y l hnh vung cnh a,mt bn SAB l tam gic u v nm trong mt phng vung gcvi mt y. Tnh theoath tch ca khi chpS.ABCDv khongcch t im An mt phng (SCD).

A: V = a336

; d= a217

48


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31. (D-2013)Cho hnh chpS.ABCDc y l hnh thoi cnh a, cnhbn SA vung gc vi y, BAD = 120o, M l trung im cnhBCv SM A= 45o. Tnh theo ath tch ca khi chpS.ABCDvkhong cch t im Dn mt phng(SBC).

A: V = a3

4; d=

a

6

4

32. (A-2014) Cho hnh chp S.ABCD c y ABCD l hnh vung

cnh a, SD = 3a

2 , hnh chiu vung gc ca S trn mt phng

(ABCD)l trung im ca cnhAB . Tnh theoath tch khi chp

S.ABCDv khong cch tAn mt phng (SBD).

A: V = a3

3; d=

2a

3

33. (B-2014)Cho lng tr ABC.ABCc y l tam gic u cnh a.Hnh chiu vung gc ca A trn mt phng (ABC)l trung imca cnhAB, gc gia ng thngACv mt y bng 60. Tnh

theo a th tch ca khi lng tr ABC.ABC v khong cch tim Bn mt phng (ACCA).

A: V =3a3

3

8 ; d=

3a

13

13

34. (D-2014)Cho hnh chpS.ABCc yABCl tam gic vung cnti A, mt bn SBCl tam gic u cnh av mt phng (SBC)

Vung gc vi mt y. Tnh theo ath tch ca khi chp S.ABCv khong cch gia hai ng thngSA, BC.

A: V = a3

3

24 ; d=

a

3

4

35. (2015)Cho hnh chp S.ABCDc y ABCDl hnh vung cnha,SAvung gc vi mt phng (ABCD). gc gia ng thngSC

v (ABCD)bng 450

. Tnh theo ath tch khi chp S.ABCD vkhong cch gia SB, AC.

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A: a)a3

2

3 ; b)d=

a

10

5

50


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6 Bt ng thc

1. (A-2003)Cho x, y, zl cc s dng tha mn x + y + z 1. Chngminh rng:

x2 + 1

x2+

y2 +

1

y2+

z2 +

1

z2

82

HD:Dng phng php vector

2. (A-2005)Cho x, y, zl cc s thc khc 0tha mn

1x

+1y

+1z

= 4

Chng minh rng:

1

2x + y+ z+

1

x + 2y+ z+

1

x + y+ 2z 1

HD:S dng BT 4

a + b1

a +1

b

3. (B-2005)Chng minh rng vi mi x, ta c:12

5

x+

15

4

x+

20

3

x 3x + 4x + 5x

HD:S dng BT Cauchy cho hai s

4. (D-2005)Cho ba s thc dng x, y, z tha mn xyz = 1. Chngminh rng:

1 + x3 + y3

xy +

1 + y3 + z3

yz +

1 + z3 + x3

zx 3

3

HD:S dng BT Cauchy:1 + x3 + y3

3xy

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5. (A-2006)Cho x, ykhc0v (x + y)xy=x2 + y2 xy. Tm gi trln nht ca A=

1

x3+

1

y3

HD:t x=1

a , y=1

b , ta c A= (a + b)2; max A= 16

6. (B-2006)Tm gi tr nh nht ca

A=

(x 1)2 + y2 +

(x + 1)2 + y2 + |y 2|

HD:S dng phng php ta . Xt hm sf(y) = 21 + y2 + 2 y; min A= 2 +

3

7. (A-2007)Cho ba s thc dng x, y, ztha mn xyz = 1. Tm gitr nh nht ca:

P = x2(y+ z)

y

y+ 2z

z+

y2(z+ x)

z

z+ 2x

x+

z2(x + y)

x

x + 2y

y

HD:nh gi x2(y+ z)

2x2

yz = 2x

x; min P = 2

8. (B-2007)Cho x, y, zl ba s thc dng. Tm gi tr nh nht ca

P =x

x

2+

1

yz

+ y

y

2+

1

zx

+ z

z

2+

1

xy

HD:Bin i P =

x2

2 +

x2

xyz. Dng BT Cauchy. Xt hm s

f(t) = t22

+1t

; min P =92

9. (D-2007)Cho cc s thca b >0. Chng minh rng:

2a + 1

2a

b

2b + 1

2b

a

HD:lnhai v. Xt hm s f(x) =ln(1 + 4x)x

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10. (B-2008)Cho cc s thc x, y tha mn x2 +y2 = 1. Tm gi trln nht, gi tr nh nht ca

P = 2(x2 + 6xy)

1 + 2xy+ 2y2

HD:Quy v 1 bin. t x= ty;6 P 3

11. (D-2008)Cho x, y l cc s thc khng m. Tm gi tr ln nht,gi tr nh nht ca biu thc

P = (x y)(1 xy)

(1 + x)2

(1 + y)2

HD:14 P 1

4

12. (A-2009)Cho x, y, xl s thc dng vx(x + y + z) = 3yz . Chngminh rng:

(x + y)3 + (x + z)3 + 3(x + y)(x + z)(y+ z)

5(y+ z)3

13. (B-2009)Cho cc s thc x, y tha mn: (x + y)3 + 4xy 2. Tmgi tr nh nht ca

A= 3(x4 + y4 + x2y2) 2(x2 + y2) + 1

HD:t t= x2 + y2 v xt hm s f(t) =9

4t2

2t +1;min A=

9

16

14. (D-2009)Cho x, y 0v x+y = 1. Tm gi tr ln nht, gi trnh nht ca

S= (4x2 + 3y)(4y2 + 3x) + 25xy

HD:t t= xy. Xt hm s f(t) = 16t2 2t + 12; 252 S 191

16

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15. (B-2010)Cho a, b, cl cc s thc khng m v a + b + c= 1. Tmgi tr nh nht ca

M= 3(a2b2 + a2c2 + c2b2) + 3(ab + ac + ca) + 2a2 + b2 + c2HD:t t= ab + bc + ca. Xt hm s f(t) =t2 + 3t + 2

1 2t

trn

0;1

2

; min M= 2

16. (D-2010)Tm gi tr nh nht ca hm s

y = x2 + 4x + 21

x2 + 3x + 10

A:min y =

2

17. (A-2011) Cho ba s thc x, y, z thuc on [1; 4] tha mn: xy, x z. Tm gi tr nh nht ca P = x

2x + 3y+

y

y+ z+

z

z+ x

HD:p dng 1

1 + a+

1

1 + b 2

1 + abv xt hm s

f(t) = t2

2t2 + 3+

2

1 + ttrn [1; 2]; min P =

34

33

18. (B-2011)Cho a, bl cc s thc dng tha mn

2(a2 + b2) + ab= (a + b)(ab + 2)

Tm gi tr nh nht ca biu thc:

P = 4

a3

b3 +

b3

a3

9

a2

b2 +

b2

a2

HD:t t=a

b+

b

av xt hm s f(t) = 4t3 9t2 12t + 18;

min P = 234

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19. (A-2012)Cho cc s thc x, y, ztha mn x + y+ z = 0. Tm gitr nh nht ca biu thc:

P = 3|xy|+ 3|yz|+ 3|zx| 6x2 + 6y2 + 6z2

HD:p dng 3t t + 1,t 0; min P = 3

20. (B-2012)Cho cc s thc x, y, z tha mn x+y + z = 0v x2 +y2 + z2 = 1. Tm gi tr ln nht ca biu thc

P =x5 + y5 + z5

HD: yz =x2 12

; xt hm s f(x) = 2x3 xtrn6

3 ; 6

3

;

max P =5

6

36

21. (D-2012)Cho cc s thcx, ytha mn

(x

4)2 + (y

4)2 + 2xy

32

Tm gi tr nh nht ca biu thc

A= x3 + y3 + 3(xy 1)(x + y 2)

HD:t t= x + yv xt hm s f(t) =t3 32

t2 3t + 6trn

[0; 8]; min A=17 55

4

22. (A-2013)Cho cc s thca, b, ctha mn (a + c)(b + c) = 4c2. Tmgi tr nh nht ca biu thc:

P = 32a3

(b + 3c)3+

32b3

(a + 3c)3

a2 + b2

c

HD:nh gi P (x + y 1)3

(x + y)2 + 2(x + y) 6,min P = 1255


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23. (B-2013)Cho a, b, cl cc s thc dng. Tm gi tr ln nht cabiu thc:

P = 4

a2 + b2 + c2 + 4 9

(a + b)(a + 2c)(b + 2c)HD:t t=

a2 + b2 + c2 + 4. Xt hm s f(t) =

4

t 9

2(t2 4)trn(2;+); max P =5

8

24. (D-2013)Cho x, yl cc s thc dng tha mn xy y 1. Tmgi tr ln nht ca biu thc:

P = x + yx2 xy+ 3y2 x 2y6(x + y)

HD:t t= x

y. Xt hm s f(t) =

t + 1t2 t + 3

t 26(t + 1)

trn0;

1

4

; max P =

5

3 +

7

30

25. (A-2014)Cho x, y, zl cc s thc khng m tha mn

x2 + y2 + z2 = 2

Tm gi tr ln nht ca biu thc:

P = x2

x2 + xz+ x + 1+

y+ z

x + y+ z+ 11 + yz

9

HD:t t= x + y+ z. Kho st hm s f(t) = t

t + 1

t2

36

trn

[0;

6]; max P =5

9

26. (B-2014)Cho a, b, cl cc s thc khng m tha mn

(a + b)c >0

. Tm gi tr nh nht ca biu thc:

P = a

b + c + b

a + c + c

2(a + b)

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HD:nh gi

a

b + c 2a

a + b + c; min P =

3

2

27. (D-2014)Cho x, y l cc s thc tha mn 1 x 2; 1 y 2.Tm gi tr nh nht ca biu thc:

P = x + 2y

x2 + 3y+ 5+

y+ 2x

y2 + 3x + 5+

1

4(x + y 1

HD:t t= x + yv xt hm s f(t) = t

t + 1+

1

4(t 1) trn

[2; 4]; min P =7

828. (2015)Cho cc s thc a, b, cthuc on[1; 3]va + b + c= 6. Tm

gi tr ln nht ca biu thc

P = a2b2 + b2c2 + c2a2 + 12abc + 72

ab + bc + ca 1

2abc.

HD:t t= ab + bc + ca,t

[11; 12]. Xt hm s:

f(t) = t2 + 5t + 144

2t , max P =160

11

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7 Phng php ta trong khng gian

Cc bi ton sau u xt trong khng gian vi h ta Oxyz.

1. (A-2002)Cho hai ng thng:

1 :

x 2y+ z 4 = 0x + 2y 2z+ 4 = 0 , 2 :

x= 1 + t

y= 2 + t

z= 1 + 2t

.

a) Vit phng trnh mt phng (P)cha1v song song vi 2.b) Cho im M(2;1;4). Tm ta im Hthuc 2 M Hnh

nht.

A: H(2;3;3)

2. (D-2002)Cho ng thng

dm: (2m + 1)x + (1 m)y+ m 1 = 0mx + (2m + 1)z+ 4m + 2 = 0

(ml tham s) v mt phng (P) : 2x y + 2 = 0. Tm m ngthngdmsong song vi mt phng (P).

A: m= 12

3. (A-2003) Cho hnh hp ch nht ABCD.ABCD c A(0;0;0),B(a;0;0), D(0; a; 0), A(0; 0; b)vi a >0, b >0.a) Tnh theo av bth tch khi t din BDAMb) Tm t s

a

b hai mt phng (ABD)v (M BD)vung gc.

A: V = a2b

4 ; a= b

4. (B-2003)Cho cc im A(2;0;0), B(0;0;8)vAC= (0;6;0). GiIl trung im BC. Tnh khong cch tIn ng thngOA.

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A: d= 5

5. (D-2003)Cho ng thngdk :

x + 3ky z+ 2 = 0kx

y+ z+ 1 = 0

(kl tham

s). Tm k ng thngdkvung gc vi mt phng

(P) :x y 2z+ 5 = 0

A: k= 1

6. (A-2004)Cho hnh chpS.ABCD, c yABCDl hnh thoi tml gc to O. Bit A(2;0;0), B(0;1;0), C(0;0;2

2), Ml trung

im cnhSC.a) Tnh gc v khong cc gia hai ng thng SA, BM.b) ng thngSDct mt phng (ABM)ti Ntnh th tch khichp S.ABMN

A: a)30o,2

6

3 ; b)

2

7. (B-2004) Cho im A(4;2;4)v ng thng d:

x= 3 + 2ty= 1 tz= 1 + 4t

.

Vit phng trnh ng thng i quaAct v vung gc vi ngthngd.

A: d :

x=

4 + 3t

y= 2 + 2tz = 4 t

8. (D-2004)Cho lng tr ngABC.ABCcA(a;0;0),B(a;0;0),C(0;1;0), B(a; 0; b)via >0, b >0.a) Tnh theo a, bkhong cch gia hai ng thng BC,AC.b) Khi a+ b = 4 tm a, b khong cch gia hai ng thngBC,ACln nht.

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A: a) aba2 + b2

; b)a= b = 2

9. (D-2004)Cho ba imA(2;0;1),B(1;0;0),C(1;1;1)v mt phng

(P) : x+y + z 2 = 0. Vit phng trnh mt cu qua A, B, C vc tm thuc mt phng (P)

A: (x 1)2 + y2 + (z 1)2 = 1

10. (A-2005)Cho ng thng

d: x 1

1

= y+ 3

2 =

z 31

v mt phng(P) : 2x + y 2z+ 9 = 0.

a) Tm im Ithuc ng thng dsao cho khong cch tInmt phng (P)bng 2b) Tm giao im A ca d v (P). Vit phng trnh tham s cang thng nm trong mt phng (P), i qua Av vung gc

vid.

A: a)I1(3;5;7), I2(3;7;1); b)

x= t

y= 1z = 4 + t

11. (B-2005)Cho lng tr ngABC.ABCcA(0;3;0),B(4;0;0),C(0;3;0), B(4;0;4).a) Tm ta A, C; vit phng trnh mt cu tm Av tip xcvi mt phng (BC C)b) Gi M l trung im AB vit phng trnh mt phng (P)iquaA, Mv song song viBC. GiNl giao im caACv(P).Tnh di M N.

A: a)x2 + (y+ 3)2 + z2 =576

25;

b)(P) :x + 4y 2z+ 12 = 0, M N= 172

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12. (D-2005)Cho 2 ng thng:

d1 :x + 1

3 =

y+ 2

1 = z+ 1

2 ; d2 :

x + y z 2 = 0x + 3y 12 = 0 .

a) Chng minh hai ng thng d1, d2song song.b) Vit phng trnh mt phng (P)cha d1, d2.c) Gi A, B ln lt l giao im ca d1, d2 vi mt phng (Oxy).Tnh din tch tam gic OAB .

A: b)15x + 11y 17z 10 = 0; c)S= 5

13. (A-2006) Cho lp phng ABCD.ABCD, c A(0;0;0), B(1;0;0),D(0;1;0), A(0;0;1). Gi M, Nln lt l trung im AB,CD.a) Tnh khong cch gia hai ng thngACv M N.b) Vit phng trnh mt phng(P)chaACv to vi mt phng

(Oxy)mt gc bit cos = 1

6.

A: a) 1

2

2; b)(P1) : 2x y+ z 1 = 0, (P2) :x 2y z+ 1 = 0

14. (B-2006)Cho im A(0;1;2)v hai ng thng:

d1 :x

2 =

y 11

= z+ 1

1 , d2 :

x= 1 + t

y= 1 2tz= 2 + t

.

a) Vit phng trnh mt phng i qua Av song song vi hai n

thngd1, d2.b) Tm M, Nln lt thuc d1, d2sao cho A, M , Nthng hng.

A: a)x + 3y+ 3z 13 = 0; b)M(0; 1;1), N(0;1;1)

15. (D-2006)Cho im A(1;2;3)v hai ng thng ng thng

d1 :

x

2

2 =

y+ 2

1 = z

3

1 , d2 :

x= 1 t

y= 1 + 2tz = 1 + t .

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a) Tm ta Ai xng vi Aqua d1.b) Vit phng trnh ng thng di qua Avung gc vi d1 vct d2.

A: a)A(1;4; 1); b)d: x 11 = y 23 = z 35

16. (A-2007)Cho 2 ng thng

d1 : x

2 =

y 11 =

z+ 2

1 , d2 :

x= 1 + 2ty= 1 + t

z = 3

a) Chng minh d1v d2cho nhau.b) Vit phng trnh ng thng dvung gc vi mt phng

(P) : 7x + y 4z = 0

v ct c hai ng thng d1, d2.

A:

x= 2 + 7ty =t

z = 1 4t.

17. (B-2007)Cho mt cu

(S) :x2 + y2 + z2 2x + 4y+ 2z 3 = 0

v mt phng(P) : 2x y+ 2z 14 = 0

a) Vit phng trnh mt phng (Q)cha trc Ox v ct (S)theomt ng trn c bn knh bng 3.b) Tm ta im Mthuc mt cu (S)sao cho khong cch tMn mt phng (P)ln nht.

A: a)y 2z = 0; b)M(1;1;3)

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18. (D-2007)Cho hai im A(1;4;2), B(1;2;4)v ng thng:

: x 11 =

y+ 2

1 =

z

2

a) Vit phng trnh ng thng d i qua trng tm G ca tamgic OAB v vung gc vi mt phng (OAB).b) Tm ta im Mthuc ng thng sao cho M A2 + M B2

nh nht.

A: a)x

2 =

y 21 =

z 21

; b)M(1;0;4)

19. (A-2008)Cho hai im A(2;5;3)v ng thng:

d: x 1

2 =

y

1=

z 22

.

a) Tm ta hnh chiu vung gc ca im Atrn ng thngd.b) Vit phng trnh mt phng ()cha dsao cho khong cch tAn()ln nht.

A: a)H(3;1;4); b)x 4y+ z 3 = 0

20. (B-2008)Cho ba im A(0;1;2), B(2;2;1), C(2;0;1).a) Vit phng trnh mt phng i qua ba im (ABC).b) Tm ta ca im Mthuc mt phng 2x + 2y + z 3 = 0saochoM A= M B =M C.

A: a)x + 2y 4z+ 6 = 0; b)M(2; 3;7)21. (D-2008)Cho bn im A(3;3;0), B(3;0;3), C(0;3;3), D(3;3;3).

a) Vit phng trnh mt cu i qua bn imA, B, C, D.b) Tm ta tm ng trn ngoi tip tam gicABC.

A: a)x2 + y2 + z2 3x 3y 3z= 0; b)H(2;2;2)

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22. (A-2009CB)Cho mt phng

(P) : 2x 2y z 4 = 0

v mt cu(S) :x2 + y2 + z2 2x 4y 6z 11 = 0

Chng minh rng mt phng (P)ct mt cu (S)theo mt ngtrn. Xc nh ta tm v bn knh ng trn .

A: H(3;0;2), r= 4

23. (A-2009NC)Cho mt phng (P) :x2y + 2z1 = 0v hai ngthng

1 :x + 1

1 =

y

1 =

z+ 9

6 , 2:

x 12

= y 3

1 =

z+ 1

2 .

Xc nh ta im M thuc 1 sao cho khong cch tMn2v khong cch tMn mt phng (P)bng nhau.

A: M

18

35;53

35;

3

35

24. (B-2009CB)Cho t dinABCDc cc nhA(1;2;1),B(2;1;3),C(2;1;1) v D(0;3;1). Vit phng trnh mt phng (P) i quaA, Bsao cho khong cch tCn(P)bng khong cch tDn(P).

A:4x + 2y+ 7z 15;2x + 3z 5 = 0

25. (B-2009NC) Cho mt phng hai im A(3;0;1), B(1;1;3) vmt phng

(P) :x 2y+ 2z 5 = 0Trong cc ng thng i qua A v song song vi (P), hy vit

phng trnh ng thng m khong cch t B n ng thng l nh nht.

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A: x + 3

26 =

y

11=

z 12

26. (D-2009CB) Cho cc im A(2;1;0), B(1;2;2), C(1;1;0) v mt

phng (P) :x + y+ z 20 = 0Xc nh ta imDthuc ng thngAB sao cho ng thngCDsong song vi (P).

A: D

5

2;1

2;1

27. (D-2009NC)Cho mt phng

(P) :x 2y+ 2z 1 = 0

v ng thng

:x + 2

1 =

y 21

= z

1Vit phng trnh ng thng dnm trong mt phng (P)sao chodct v vung gc vi .

A: x + 3

1 =

y 12 =

z 11

28. (A-2010CB)Cho mt phng

(P) :x

2y+ z = 0

v ng thng

: x 1

2 =

y

1=

z+ 2

1GiCl giao im ca (P)vcn Ml mt im thuc . Tnhkhong cch tMn(P)bit M C=a

6

A:

1

6

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29. (A-2010NC)Cho im A(0; 0;2)v ng thng

: x + 2

2 =

y 23

=z+ 3

2

Tnh khong cch t A n . Vit phng trnh mt cu tm Act ti B, Csao cho BC= 8.

A:3; x2 + y2 + (z+ 2)2 = 25

30. (B-2010CB)Cho ba imA(1;0;0),B(0; b; 0),C(0; 0; c)vib, c >0v mt phng(P) :yz +1 = 0. Tmb, csao cho mt phng (ABC)vung gc vi P, v khong cch tOn mt phng (ABC)bng1

3 .

A: b= c =1

2

31. (B-2010NC) Cho ng thng : x

2 =

y 11

= z

2. Tm ta

im Mnm trn trc Ox sao cho khong cch tM n bng di OM.

A: M1(1;0;0), M2(2;0;0)32. (D-2010CB)Cho 2 mt phng

(P) :x + y+ z 3 = 0, (Q) :x y+ z 1 = 0Vit phng trnh mt phng (R)vung gc vi c (P), (Q)v cchOmt khong bng 2

A: x + z 22 = 033. (D-2010NC)Cho hai ng thng

d1:

x= 3 + t

y =t

z =t

; d2 : x 2

2 =

y 11

= z

2

Tm ta im Mtrn ng thngd1sao cho khong cch tMn ng thngd2bng 1.

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A: M1(4;1;1), M2(7;4;4)

34. (A-2011CB)Cho im A(2;0;1), B(0;2;3)v mt phng

(P) : 2x y z+ 4 = 0Tm ta im Mthuc (P)sao cho M A= M B= 3.

A: M1(0;1;3), M2

6

7;4

7;12

7

35. (A-2011NC)Cho im A(4;4;0)v mt cu

(S) :x2 + y2 + z2 4x 4y 4z= 0

Vit phng trnh mt phng(OAB), bit imBthuc(S)v tamgic OAB u.

A: x y+ z = 0, x y z = 0

36. (B-2011CB)Cho ng thng

:x 2

1 =

y+ 1

2 = z

1v mt phng

(P) :x + y+ z 3 = 0GiIl giao im ca v(P). Tm ta im Mthuc(P)saochoM Ivung gc vi v M I= 4

14.

A: M1(5; 9;11), M2(3;7; 13)

37. (B-2011NC) Cho hai im A(2;1;1), B(3;1;2)v ng thng

: x + 2

1 =

y 13

=z+ 5

2

Tm ta im Mthuc ng thng sao cho tam gic M ABc din tch bng 35

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A: M1(2;1;5), M2(14;35; 1)38. (D-2011CB)Cho im A(1;2;3)v ng thng:

d: x + 1

2 =

y

1=

z

3

2Vit phng trnh ng thng i qua im A, vung gc ving thngdv ct trc Ox.

A: x 1

2 =

y 22

= z 3

3

39. (D-2011NC)Cho ng thng

: x 12

= y 34

= z1

v mt phng(P) : 2x y+ 2z = 0

Vit phng trnh mt cu c tm thuc ng thng, bn knhbng 1v tip xc vi (P).

A: (x 5)2

+ (y 11)2

+ (z 2)2

= 1,(x + 1)2 + (y+ 1)2 + (z+ 1)2 = 1

40. (A-2012CB)Cho im I(0;0;3)v ng thng

d: x + 1

1 =

y

2=

z 21

Vit phng trnh mt cu (S)tm Iv ct dti hai im A, Bsaocho tam gic ABIvung ti I.

A: x2 + y2 + (z 3)2 =83

41. (A-2012NC) Cho im A(1;1;2) ng thng d v mt phng(P):

d: x + 1

2 =

y

1=

z 21

, (P) :x + y 2z+ 5 = 0

Vit phng trnh ng thngct dv(P)ln lt ti Mv Nsao cho Al trung im M N

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A: x 1

2 =

y+ 1

3 =

z 22

42. (B-2012CB)Cho hai im A (2;1;0), B (2;3;2)v ng thng

d: x 12

= y1

= z2Vit phng trnh mt cu i qua A, B c tm thuc ng thngd.

A: (x + 1)2 + (y+ 1)2 + (z 2)2 = 17

43. (B-2012NC) Cho A (0;0;3) , M(1; 2; 0). VIt phng trnh mtphng (P)qua A v ct cc trc Ox,Oy ln lt ti B, Csao chotam gic ABCc trng tm thuc ng thng AM.

A:6x + 3y+ 4z 12 = 0

44. (D-2012CB)Cho im I(2; 1; 3)v mt phng

(P) : 2x + y 2z+ 10 = 0Vit phng trnh mt cu tm Iv ct (P)theo mt ng trnc bn knh bng 4.

A: (x 2)2 + (y 1)2 + (z 3)2 = 25

45. (D-2012NC)Cho hai imA (1;

1;2),B (2;

1;0)v ng thng

d: x 1

2 =

y+ 1

1 = z

1

Xc nh ta im M thuc dsao cho tam gic AM Bvung tiM.

A: M1(1;

1;0), M27

3

;

5

3

;2

3

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46. (A-2013CB)Cho im A(1;7;3)v ng thng

: x 63 =

y+ 1

2 =z+ 2

1

Vit phng trnh mt phng(P)i quaAv vung gc vi . Tmta im Mthuc sao cho AM= 2

30.

A: M1(3;3;1), M2

51

7;1

7;17

7

47. (A-2013NC)Cho mt phng(P) : 2x + 3y + z11 = 0v mt cu

(S) :x2 + y2 + z2 2x + 4y 2z 8 = 0

Chng minh(P)tip xc(S). Tm ta tip im ca(P)v(S).

A: M(3;1;2)

48. (B-2013CB)Cho im A(3;5;0)v mt phng

(P) : 2x + 3y z 7 = 0Vit phng trnh ng thng i qua A v vung gc vi (P). Tmta im i xng ca Aqua(P).

A: (1;1;2)

49. (B-2013NC)Cho cc im A(1;

1;1), B(

1;2;3)v ng thng

: x + 1

2 = y 2

1 =

z 33

Vit phng trnh ng thng i quaAv vung gc vi hai ngthngABv .

A: x 1

7

= y+ 1

2

= z 1

4

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50. (D-2013CB)Cho cc im A(1;1;2), B(0;1;1)v mt phng

(P) :x + y+ z 1 = 0

Tm ta hnh chiu vung gc ca Atrn(P). Vit phng trnhmt phng i qua A, Bv vung gc vi (P).

A: x 2y+ z+ 1 = 0

51. (D-2013NC)Cho im A (1;3;2)v mt phng

(P) :x 2y 2z+ 5 = 0

Tnh khong cch t A n (P). Vit phng trnh mt phng iqua Av song song vi (P)

A: x 2y 2z+ 3 = 0

52. (A-2014)Cho mt phng (P) : 2x + y 2z 1 = 0v ng thng

d:

x

2

1 =

y

2= z+ 3

3

Tm ta giao im ca d v (P). Vit phng trnh mt phngcha dv vung gc vi (P).

A: x + 8y+ 5z+ 13 = 0

53. (B-2014)Cho im A(1; 0;

1)v ng thng

d: x 1

2 =

y+ 1

2 =

z

1Vit phng trnh mt phng qua Av vung gc vi d. Tm ta hnh chiu vung gc ca Atrn d.

A: 53

;

1

3

;

1

3

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54. (D-2014)cho mt phng (P) : 6x + 3y 2z 1 = 0v mt cu

(S) :x2 + y2 + z2 6x 4y 2z 11 = 0

Chng minh rng mt phng (P)ct mt cu (S)theo giao tuynl mt ng trn (C). Tm ta tm (C).

A:

3

7;5

7;13

7

55. (2015) Trong khng gian vi h to Oxyz, cho cc im A(1;2;1),B(2;1;3)v mt phng

P :x y+ 2z 3 = 0

Vit phng trnh ng thng AB v tm giao im ca AB vimt phng (P).

A: (0;5;1)

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8 Phng php ta trong mt phng

Cc bi ton sau u xt trong mt phng vi h ta Oxy.

1. (A-2002)Cho tam gicABCvung tiA, hai nhA, Bthuc trcOx, bn knh ng trn ni tip tam gic bng 2, phng trnhng thng BC :

3x y

3 = 0. Tm to trng tm Gca

tam gic ABC

A:

7 + 4

3

3 ;

6 + 2

3

3

,

43 1

3 ;

6 233

2. (B-2002)Cho hnh ch nhtABCDc tm I

12

; 0

, AB = 2AD,

phng trnh ng thngAB:x 2y + 2 = 0. Tm to cc nhca hnh ch nht, bit honh nh Am.

A: A(2;0), B(2; 2), C(3; 0), D(1;2)

3. (D-2002)Cho elip

(E) : x2

16+

y2

9 = 1

Gi M, N ln lt l cc im thuc tia Ox,Oy sao cho M N tipxc vi (E). Tm M, N M Nnh nht. Tnh gi tr nh nht .

A: M(2

7;0), N(0;

21), M N= 7

4. (B-2003)Cho tam gicABCvung cn ti Ac M(1;1)l trungimBC,G

2

3; 0

l trng tm tam gic. Tm to cc nh ca

tam gic.

A: A(0; 2)B(4; 0), C(2;2)

5. (D-2003)Cho ng trn

(C) : (x 1)2 + (y 2)2 = 4

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v ng thngd: x y1 = 0. Vit phng trnh ng trnCi xng vi ng trn (C) qua ng thng d. Tm to giaoim ca (C)v

C.

A: (C) : (x 3)2

+ y2

= 4; A(1; 0), B(3; 2)

6. (A-2004) Cho A(0; 2), B(

3;1). Tm ta trc tm H, tmng trn ngoi tip Ica tam gic OAB

A: H(

3;1), I(

3;1)

7. (B-2004)Cho hai im A(1; 1), B(4;

3)v ng thng

d: x 2y 1 = 0

Tm ta imCnm trn ng thng dsao cho khong cch tCn ABbng 6.

A: C1(7; 3), C2

43

11;27

11

8. (D-2004)Cho A(1;0), B(4; 0), C(0, m), m= 0. Tm ta trngtm Gca tam gic ABC. Tm gi tr ca m AGB= 900

A: G

1;m

3

, m= 3

6

9. (A-2005) Cho hnh vung ABCD c nh A thuc ng thngd1

: x

y = 0, nh Cthuc ng thng d2

: 2x + y

1 = 0. Tmta cc nh ca hnh vung, bit B, Dthuc trc Ox.

A: A(1; 1), B(0; 0), C(1;1), D(2; 0)

10. (B-2005) Cho hai im A(2; 0), B(6; 4). Vit phng trnh ngtrn (C)tip xc Ox ti Av khong cch gia B v tm ca (C)bng 5.

A: (C1) : (x 2)2 + (y 1)2 = 1, (C2) : (x 2)2 + (y 7)2 = 49

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11. (D-2005)Cho im C(2; 0)v elip (E) : x2

4 +

y2

1 = 1. Tm ta

cc im A, Bthuc(E)sao cho Bi xng Aqua trc Oxv tamgic ABCu

A:

A1

2

7;4

3

7

, B1

2

7;4

3

7

; A2

2

7;4

3

7

; B2

2

7;4

3

7

12. (A-2006)Cho ba ng thng

d1:x + y+ 3 = 0, d2 :x

y

4 = 0, d3 :x

2y= 0

Tm ta im Mthuc ng thng d3 sao cho khong cch tMn d1bng 2 ln khong cch tMnd2.

A: M1(22;11), M2(2; 1)

13. (B-2006)Cho im M(3;1)v ng trn

(C) :x2

+ y2

2x 6y+ 6 = 0Gi T1, T2 l cc tip im ca tip tuyn k t M n (C). Vitphng trnh ng thngT1T2

A:2x y 3 = 0


(C) :x2 + y2 2x 2y+ 1 = 0

v ng thngd: x y+ 3 = 0.Tm to imMthuc ng thngd ng trn tmMtipxc ngoi vi ng trn (C)c bn knh gp i bn knh ngtrn(C).

A: M1(1; 4), M2(2;1)

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15. (A-2007)Cho tam gic ABC c A(0; 2), B(2;2), C(4;2). GiM, N ln lt l trung im AB,BC, H l chn ng cao h tnhB. Vit phng trnh ng trn i qua ba im H, M, N

A: x2 + y2 x + y 2 = 016. (B-2007)Cho im A(2; 2)v hai ng thng

d1 :x + y 2 = 0, d2 :x + y 8 = 0

Tm to imBthuc ng thngd1, imCthuc ng thngd2sao cho tam gic ABCvung cn ti A.

A: B(1;3), C(3; 5)hoc B(3; 1), C(5; 3)


(C) : (x 1)2 + (y+ 2)2 = 9

v ng thngd: 3x 4y+ m= 0.

Tmm trn ng thngdc duy nht im Pm t k ccc tip tuyn P A, P B n ng trn sao cho tam gic P ABu.

A: m= 19, m= 41

18. (A-2008)Vit phng trnh chnh tc elip(E)c tm sai bng

5

3 ,

chu vi hnh ch nht c s bng 20

A: x2

9 +

y2

4 = 1

19. (B-2008)Cho tam gicABCcH(1;1)l hnh chiu ca CtrnAB, ng thng phn gic trong gc A c phng trnh xy+2 = 0,ng cao nhBc phng trnh 4x + 3y 1 = 0. To nh C.

A:103;34

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20. (D-2008) Cho parabol (P) : y2 = 16x v im A(1; 4). Hai imphn bitB, C(B, CkhcA) di ng trn (P)sao cho BAC= 90o.Chng minh rng ng thng BClun i qua im c nh.

21. (A-2009)Cho hnh ch nht ABCD. ACct BD ti I(6; 2). imM(1; 5)thuc cnh AB, trung im Eca CDthuc ng thng:x + y 5 = 0. Vit phng trnh ng thng AB.

A: y 5 = 0; x 4y+ 19 = 0

22. (A-2009NC)Cho ng trn

(C) :x2 + y2 + 4x + 4y+ 6 =4

5

tm Iv ng thng :x + my 2m + 3 = 0.Tm m ng trn (C)ct ng thng ti 2 im phn bitA, B sao cho din tch tam gic IABln nht.

A: m= 0, m= 8

15

23. (B-2009)Cho ng trn

(C) : (x 2)2 + y2 =45

v hai ng thng

d1 :x

y= 0, d2 :x

7y = 0

Xc nh tm Kv bn knh ng trn (C1)tip xc vi c d1, d2bit Kthuc (C).

A: K

8

5;4

5

; R=

2

5

24. (B-2009NC)Cho tam gic ABC cn ti A(1;4), hai nh B, Cthuc ng thng:d: x y 4 = 0. Tm ta cc imB, Cbitdin tch tam gic ABCbng 18

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A: B1

11

2;

3

2

, C1

3

2;5

2

hoc B2

3

2;5

2

, C2

11

2;

3

2

25. (D-2009)Cho tam gic ABCc im M(2; 0) l trung im AB.

ng trung tuyn, ng cao nhAln lt c phng trnh7x 2y 3 = 0, 6x y 4 = 0

Vit phng trnh ng thngAC.

A:3x 4y+ 5 = 0

26. (D-2009NC)Cho ng trn (C) : (x 1)2

+y2

= 1, tm I. Xcnh ta imMthuc ng trn(C) :sao cho IM O= 300.

A: M

3

2;

3

2

27. (A-2010)Cho 2 ng thng

d1 : 3x + y= 0, d2 : 3x y= 0

ng trn (T)tip xc vi d1 ti A, ct d2 ti B, Csao cho tamgic ABCvung ti B. Vit phng trnh ng trn(T)bit din

tch tam gic ABCbng

3

2 v im Ac honh dng.

A:

x + 1

232

+

y+3

22

= 1

28. (A-2010NC)Cho tam gic ABCcn ti A(6; 6), ng thng iqua trung im ca AB,ACc phng trnh: x + y4 = 0. Tm ta B, C, bit E(1;3)nm trn ng cao qua nh C.

A: B(0;4), C(4;0)hoc B(6;2), C(2;6)

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29. (B-2010)Cho tam gicABCvung tiA, c nhC(4;1), phngtrnh phn gic trong gc A l: x+ y 5 = 0. Vit phng trnhng thng BC. Bit im Ac honh dng v din tch tamgic ABCbng 24.

A:3x 4y+ 16 = 0

30. (B-2010NC) Cho im A(2;

3) v elip (E) : x2

3 +

y2

2 = 1. Gi

F1, F2l cc tiu im ca(E)(F1c honh m). Ml giao imc tung dng ca ng thngAF1vi(E),Nl im i xngvi F2 qua M. Vit phng trnh ng trn ngoi tip tam gic

AN F2.

A: (x 1)2 +

y 2

3

3

2=

4

3

31. (D-2010)Cho tam gicABC, cA(3;7), trc tmH(3;1), tmng trn ngoi tip I(2;0). Tm ta im Cbit im Cchonh dng.

A: (2 +

65; 3)

32. (D-2010NC)ChoA(0; 2)v ng thngi quaO. GiHl hnhchiu ca A ln . Vit phng trnh ng thng bit khongcch tHn trc honh bng AH.

A:5 1 x 25 2y= 0

33. (A-2011)Cho im Mthuc ng thng : x+ y + 2 = 0. vng trn

(C)) :x2 + y2 4x 2y= 0tm I. T Mk cc tip tuyn M A, M B (A, B l tip im) n(C)). Tm ta im M, bit din tch t gic M AIB bng 10.

A: M(2;4), M(3;1)

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34. (A-2011NC)Cho elip(E) : x2

4 +

y2

1 = 1. Tm to cc imA, B

thuc (E)c honh dng sao cho tam gic OAB cn ti O cdin tch ln nht.

A: A

2;2

2

, B

2;

22

hoc

A

2;

2

2

, B

2;

2

2

35. (B-2011)Cho hai ng thng

:x y 4 = 0; d: 2x y 2 = 0

Tm ta im Nthuc ng thng dsao cho ng thng ONct ng thngti im Mtha mn OM.ON= 8.

A: N1(0;2), N2

6

5;2

5

36. (B-2011) Cho tam gic ABC c nh B

1

2; 1

. ng trn ni

tip tam gicABCtip xc vi cc cnh BC, CA, ABtng ng tiD , E , F . ChoD(3; 1)v ng thngE Fc phng trnh:y3 = 0.Tm ta nh Abit Ac tung dng.

A: A3;133

37. (D-2011)Cho tam gic ABCc nh B(4;1), trng tm G(1; 1)v ng thng cha phn gic trong gcAc phng trnh:

x y 1 = 0

Tm ta cc nh A, C.

A: A(4; 3), C(3;1)

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38. (D-2011NC)Cho im A(1; 0)v ng trn

(C) :x2 + y2 2x + 4y 5 = 0Vit phng trnh ng thngct (C)ti hai im M , Nsao chotam gic AM Nvung cn ti A.

A: y= 1; y= 3

39. (A-2012)Cho hnh vung ABCD c M l trung im BC, N l

im trn cnh CD sao cho CN = 2N D. Gi s M

11

2;

1

2

v

ng thng ANc phng trnh 2x

y

3 = 0. Tm ta imA

A: A1(1;1), A2(4; 5)

40. (A-2012NC)Cho ng trn(C) :x2 + y2 = 8. Vit phng trnhchnh tc ca elip (E), bit rng (E)c di trc ln bng 8 v(E)ct (C)ti bn im to thnh bn nh ca mt hnh vung.

A: x2

16+

y2

16

3

= 1

41. (B-2012)Cho cc ng trn

(C1) :x2 + y2 = 4; (C2) :x

2 + y2 12x + 18 = 0

v ng thng d: x y 4 = 0. Vit phng trnh ng trn ctm thuc (C2), tip xc vi dct (C1)ti hai im phn bit A, Bsao cho ABvung gc vi d

A: (x 3)2 + (y 3)2 = 8

42. (B-2012NC)Cho hnh thoi ABCD c AC = 2BD v ng trntip xc vi cc cnh ca hnh thoi l x2 + y2 = 4. Vit phng trnh

chnh tc ca elip (E)i qua cc nh A, B, C, Dca hnh thoi. BitAthuc Ox.

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A: x2

20+

y2

5 = 1

43. (D-2012)Cho hnh ch nht ABCD. Cc ng thng ACv AD

ln lt c phng trnh lx + 3y= 0vx y + 4 = 0; ng thngBDi qua imM

1

3; 1

. Tm ta cc nh ca hnh ch nht

ABCD.

A: A(3;1), B(1;3), C(3;1), D(1;3)

44. (D-2012NC) Cho ng thng d : 2x y + 3 = 0. Vit phng

trnh ng trn c tm thuc d, ct Ox ti Av B. ct Oy ti Cv Dsao cho AB =C D= 2.

A: (x + 3)2 + (y+ 3)2 = 10

45. (A-2013)Cho hnh ch nht ABCDc imCthuc ng thngd: 2x + y+ 5 = 0v A(4;8). Gi Ml im i xng ca B quaC,Nl hnh chiu vung gc ca B trn ng thngM D. Tm ta

cc imBv C, bit rng N(5;4).A: B(4;7), C(1;7)

46. (A-2013NC)Cho ng thng : x y = 0. ng trn (C)cbn knh R =

10ct ti hai im Av B sao cho AB = 4

2.

Tip tuyn ca (C)ti AvBct nhau ti mt im thuc tia Oy.Vit phng trnh ng trn (C).

A: (x 5)2 + (y 3)2 = 10

47. (B-2013) Cho hnh thang cn ABCD c hai ng cho vunggc vi nhau v AD = 3BC. ng thng BD c phng trnhx + 2y 6 = 0v tam gic ABDc trc tm l H(3;2). Tm ta cc nh Cv (D).

A: C(1;6), D1(4; 1), D2(8;7)

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48. (B-2013NC)Cho tam gic ABCc chn ng cao h t nh A

lH

17

5;1

5

, chn ng phn gic trong ca gcAl D(5; 3)v

trung im ca cnh AB l M(0; 1). Tm ta nhC.

A: C(9; 11)

49. (D-2013)Cho tam gic ABCc im M9

2;3

2

l trung im

cnh AB, im H(2;4) v im I(1;1) ln lt l chn ngcao k tBv tm ng trong ngoi tip tam gic ABC. Tm ta nhC.

A: C(1;6)

50. (D-2013NC)Cho ng trn

(C) : (x 1)2 + (y 1)2 = 4v ng thng :y 3 = 0. Tam gic M N Pc tr tm trng vi

tm ca(C), cc nhNvPthuc, nhMv trung im cnhM Nthuc(C). Tm ta im P.

A: P1(1;3), P2(3; 3)

51. (A-2014) Cho hnh vung ABCD c im M l trung im caABv Nl im thuc on ACsao cho AN= 3N C. Vit phngtrnh ng thngCD, bit rng M(1; 2)v N(2;1).

A:3x 4y 15 = 0

52. (B-2014) Cho hnh bnh hnh ABCD. Im M(3;0) l trungim ca cnh AB, im H(0;1) l hnh chiu vung gc ca BtrnADv imG

4

3; 3

l trng tm tam gic BCD. Tm ta

cc im Bv D.

A: B(2; 3); D(2; 0)

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53. (D-2014)Cho tam gic ABCc chn ng phn gic trong cagc Al im D(1;1). ng thng ABc phng trnh

AB: 3x + 2y 9 = 0

tip tuyn tiAca ng trn ngoi tip tam gic ABCc phngtrnhx + 2y 7 = 0. Vit phng trnh ng thng BC.

A: x 2y 3 = 0

54. (2015) Trong mt phng vi h to Oxy, cho tam gic ABCvung ti A, H l hnh chiu vung gc ca Aln BC, D l im

i xng caBqua H, Kl hnh chiu vung gc ca Ctrn cnhAD. Gi s H(5;5), K(9;3)v trung im ca cnh ACthucng thngx y+ 10 = 0. Tm to im A.

A: A(15; 5)

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9 S phc

1. (A-2009)Cho z1, z2l hai nghim phc ca phng trnh:

z2

+ 2z+ 10 = 0

TnhA= |z1|2 + |z2|2.

A:20

2. (B-2009)Tm s phc ztha mn:|z (2 + i)| =

10v z.z = 25

A: z = 3 + 4i, z = 5

3. (D-2009)Tm tp hp im biu din s phc ztha mn

|z (3 4i)| = 2

A: ng trn tm I(3;4), R= 2

4. (A-2010CB)Tm phn o ca s phcz, bitz = (

2 + i)2(1 i

2)

A:

2

5. (A-2010NC)Cho s phc z, bit

z=

1 3i31 i

Tm modun s phc z+ iz.

A: 8

2

6. (B-2010)Tm tp hp im biu din s phc ztha mn

|z i| = |(1 + i)z|

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A: x2 + (y+ 1)2 = 2

7. (D-2010)Tm s phc z, bit|z| =

2v z2 l s thun o.

A:1 i8. (A-2011CB)Tm s phc z, bit

z2 = |z|2 + z

A:121

2i, 0

9. (A-2011NC)Tm modun s phc z, bit

(2z 1)(1 + i) + (z+ 1)(1 i) = 2 2i.

A: 2

3

10. (B-2011CB)Tm s phc z, bit

z 5 + i3

z 1 = 0.

A:1 i

3; 2 i

3

11. (B-2011NC)Tm phn thc v phn o ca s phc

z=

1 + i31 + i

3

A:2; 2

12. (D-2011CB)Tm s phc z, bit

z (2 + 3i)z = 1 9i

A: z = 2 i

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13. (A-2012NC)Cho s phc ztha mn

5(z+ i)

z+ 1 = 2 i

Tnh m-un ca s phc w= 1 + z+ z2.

A:

13

14. (B-2012NC)Goiz1v z2l hai nghim ca phng trnh

z2 2

3iz 4 = 0

Vit dng lng gic ca z1v z2.

A: z1 = 2

cos

3+ i sin

3

, z2= 2

cos

2

3 + i sin

2

3

15. (D-2012CB)Cho s phc ztha mn

(2 + i) z+

2 (1 + 2i)

1 + i = 7 + 8i

Tm mun ca s phcw=z + 1 + i.

A:5

16. (D-2012NC)Gii phng trnh sau trn tp s phc

z2 + 3 (1 + i) z+ 5i= 0

A:1 2i,2 i

17. (A-2013NC)Cho s phc z = 1 +

3i. Vit dng lng gic cas phcz. Tm phn thc v phn o ca s phc w = (1 + i)z5

A:16(

3 + 1), 16(1

3)

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18. (D-2013CB)Cho s phc ztha mn

(1 + i)(z i) + 2z= 2i

Tm mun ca s phcw= z

2z+ 1

z2 .

A:

10

19. (A-2014)Cho s phc ztha mn iu kin

z+ (2 + i) z= 3 + 5i

Tm phn thc v phn o ca z.

A:2;3

20. (B-2014)Cho s phc ztha m iu kin

2z+ 3 (1 i) z = 1 9i

Tnh mun ca z.

A:

13

21. (D-2014)Cho s phc ztha mn iu kin

(3z z) (1 + i) 5z = 8i 1

Tnh mun ca ca z.

A:

13

22. (2015)Cho s phc z tho mn (1 i)z 1 + 5i = 0. Tm phnthc v phn o ca z.

A: phn thc 3, phn o

2.

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10 T hp - xc sut

1. (A-2002)Cho khai trin:

2x1

2 + 2x

3n

=C0n

2x1

2n

+C0n

2x1

2n1 2x3 +...+Cnn 2x3 n

Tm n, xbit C3n = 5C1nv s hng th t bng 20n.

A: n= 7, x= 4

2. (B-2002)Cho a gic uA1A2...A2n,(n 2, n Z)ni tip ngtrn (O). S tam gic nhiu gp 20ln s hnh ch nht lp t2n

nhA1, A2,...,A2n. Tm n

A: n= 8

3. (D-2002)Tm n NbitC0n+ 2C

1n+ 4C

2n+ ... + 2

nCnn = 243

A: n= 5

4. (A-2003)Tm h s ca s hng cha x8 trong khai trin

P(x) =

1

x3+

x5n

bit Cn+1n+4 Cnn+3 = 7(n + 3)

A:495

5. (B-2003)Tnh tng C0n+22 1

2 C1n+ ... +

2n+1 1n + 1

Cnn .

A: 3n+1 2n+1

n + 1

6. (D-2003)Gia3n3l h s ca s hng cha x3n3trong khai trin(x2 + 1)n(x + 2)n. Tm nbit a3n3 = 26n

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A: n= 5

7. (A-2004)Tm h s cha x8 trong khai trin ca

1 + x2(1 x)8A:238

8. (B-2004)C 30 cu hi khc nhau, trong c 5 cu hi kh, 10cu trung bnh, 15 cu d. C th lp c bao nhiu gm 5 cuc 3 mc v s cu d khng t hn 2.

A:56875

9. (D-2004)Tm cc s hng khng cha xtrong khai trin

P(x) =

3

x + 14

x

7

vix >0

A:35

10. (A-2005)Tm n Nsao cho

C12n+12.2C12n+1+3.23C12n+14.24C12n+1+...+(2n+1)C12n+1 = 2005

A:1002

11. (B-2005)Mt i thanh nin tnh nguyn c 15 ngi gm 12 nam,

3 n. Hi c bao nhiu cch phn cng thanh nin tnh nguyn vgip 3 tnh min ni sao cho mi tnh gm 4 nam v 1 n.

A:207900

12. (D-2005)Tnh M= A4n+1+ 3A

3n

(n + 1)! , bit

C2n+1+ 2C2n+2+ 2C2n+3+ C2n+4 = 149

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A: 3

4

13. (A-2006)Tm h s ca s hng cha x26 trong khai trin

P(x) =

1x4

+ x7n

bitn

k=1

Ck2n+1 = 2020 1

A: C610

14. (B-2006)Cho tp Agm n(n 4)phn t. S tp con gm 4 phnt ca Abng 20 ln s tp con gm 2 phn t ca A. Tm k saocho s tp con gm kphn t ca Aln nht.

A: k= 9

15. (D-2006)i thanh nin xung knh ca mt trng ph thng c12 hc sinh gm 5 hc sinh lp A, 4 hc sinh lp B, 3 hc sinh lpC. C bao nhiu cch chn 4 hc sinh thc hin mt nhim vsao cho 4 hc sinh ny khng thuc qu 2 trong 3 lp trn.

A:225

16. (A-2007)Chng minh rng

1

2C12n+

1

4C32n+ ... +

1

2nC2n12n =

22n 12n + 1

17. (B-2007)Tm h s ca s hng cha x10 trong khai trin (2 + x)n

bitn

k=0(1)k3nkCkn = 2048

A:22

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18. (D-2007)Tm h s ca s hng cha x5 trong khai trin

P(x) =x(1 2x)5 + x2(1 + 3x)10

A:33

C

3

10

19. (A-2008)Cho khai trin (1 + 2x)n =a0+ a1x + ... + anxnv

a0+a1

2 + ... +

an

2n= 4096

Tm s ln nht trong cc s a0, a1,...,an

A: a8

20. (B-2008)Chng minh rng: n + 1

n + 2

1

Ckn+1+

1

Ck+1n+1

=

1

Ckn, k n

21. (A-2012)Cho nl s nguyn dng tha mn5Cn1n =C3n. Tm s

hng cha x5 trong khai trin nh thc Newton ca

nx2

14 1

x

n, x =

0.

A:3516

x5

22. (B-2012)Trong mt lp hc gm c 15 hc sinh nam v 10 hc sinhn. Gio vin gi ngu nhin 4 hc sinh ln bng gii bi tp. Tnhxc sut 4 hc sinh c gi c c nam v n.

A: 443506

23. (A-2013)Gi Sl tp hp tt c cc s t nhin gm ba ch sphn bit c chn t cc ch s 1;2;3;4;5;6;7. Xc nh s phnt ca S. Chn ngu nhin mt s t S, tnh xc sut s cchn l s chn.

A: 37

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24. (B-2013)C hai chic hp cha bi. Hp th nht cha 4 vin bi v 3 vin bi trng, hp th hai cha 2 vin bi v 4 vin bi trng.Ly ngu nhin t mi hp ra 1 vin bi, tnh xc sut 2 vin bic ly ra c cng mu.

A: 10

21

25. (A-2014)T mt hp cha 16 th c nh du t 1 n 16, chnngu nhin 4 th. Tnh xc sut 4 th c chn u c nhs chn.

A: 1

26

26. (B-2014) kim tra cht lng sn phn t mt cng ty sa, ngita phi gi n b phn kim nghim 5 hp sa cam, 4 hp sa duv 3 hp sa nho. B phn kim nghim chn ngu nhin 3 hp sa phn tch nu. Tnh xc sut 3 hp sa c chn c c 3 loi.

A:

3

11

27. (D-2014)Cho mt a gic unnh, n Nv n 3. Tm nbitrng a gic cho c 27ng cho.

A:

7 + 4

3

3 ;

6 + 2

3

3

,

43 1

3 ;

6 233

28. (2015) Trong t ng ph dch Mers- Cov, S t thnh ph chn ngu nhin 3 i phng chng c ng trong s 5 i t trungtm t d phng thnh ph v 20 i ca cc trung tm t cs kim tra cng tc chun b. Tnh xc sut c t nht 2 ica cc trung tm t c s.

A: 209

230

93


94/94

Mc lc

1 Kho st hm s 4

2 Lng gic 17

3 Phng trnh, h phng trnh, bt phng trnh 24

4 Tch phn v ng dng 35

5 Hnh hc tng hp trong khng gian 43

6 Bt ng thc 51

7 Phng php ta trong khng gian 58

8 Phng php ta trong mt phng 73

9 S phc 85

10 T hp - xc sut 89

Cac Dang Toan Trong de Thi 2002 2015 (1)

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