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7/24/2019 Cac Dang Toan Trong de Thi 2002 2015 (1)
1/94
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
3
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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
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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 .
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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.
10
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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
4. (A-2003)Gii phng trnh:
cot x 1 = cos2x1 + tan x
+ sin2 x 12
sin 2x
A: x=
4+ k
5. (B-2003)Gii phng trnh:
cot x tan x + 4 sin 2x= 2sin2x
A: x= 3
+ k
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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
8. (B-2004)Gii phng trnh:
5sin x 2 = 3(1 sin x)tan2 x
A: x=
6
+ k2; x=5
6
+ k2
9. (D-2004)Gii phng trnh:
(2cos x 1)(2 sin x + cos x) = sin2x sin x
A: x= 3
+ k2; x= 4
+ k
10. (A-2005)Gii phng trnh:
cos2 3x cos2x cos2 x= 0
A: x= k
2
11. (B-2005)Gii phng trnh:
1 + sin x + cos x + sin 2x + cos 2x= 0
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A: x= 4
+ k; x= 23
+ k2
12. (D-2005)Gii phng trnh:
cos4 x + sin4 x + cos
x 4
sin
3x 4
3
2= 0
A: x=
4+ k
13. (A-2006)Gii phng trnh:
2
cos
6 x+ sin
6 x sin x cos x2 2sin x = 0
A: x=5
4 + 2k
14. (B-2006)Gii phng trnh:
cot x + sin x
1 + tan x tan
x
2
= 4
A: x=
12+ k; x=
5
12+ k
15. (D-2006)Gii phng trnh:
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
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17. (B-2007)Gii phng trnh:
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
19. (A-2008)Gii phng trnh:
1
sin x+
1
sin
x 3
2
= 4 sin74 x
A: x=
4
+ k; x=
8
+ k; x=5
8
+ k
20. (B-2008)Gii phng trnh:
sin3 x
3cos3 x= sin x cos2 x
3sin2 x cos x
A: x=
4+ k
2; x=
3+ k
21. (D-2008)Gii phng trnh:
2sin x(1 + cos 2x) + sin 2x= 1 + 2 cos x
A: x= 23
+ k2; x=
4+ k
22. (A-2009)Gii phng trnh:
(1
2sin x)cos x
(1 + 2 sin x)(1 sin x)= 3
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A: x= 18
+ k2
3
23. (B-2009)Gii phng trnh:
sin x + cos x sin2x +3cos3x= 2(cos 4x + sin3 x)
A: x= 6
+ k2; x=
42+ k
2
7
24. (D-2009)Gii phng trnh:
3cos5x
2sin3x cos2x
sin x= 0
A: x=
18+ k
3; x=
6+ k
2
25. (A-2010)Gii phng trnh:
(1 + sin x + cos 2x)sin
x +
4
1 + tan x
= 1
2cos x
A: x= 6
+ k2; x=7
6 + k2
26. (B-2010)Gii phng trnh:
(sin2x + cos 2x)cos x + 2 cos 2x sin x= 0
A: x=
4+ k
2
27. (D-2010)Gii phng trnh:
sin2x cos2x + 3 sin x cos x 1 = 0
A: x=
6+ k2; x=
5
6 + k2;
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28. (A-2011)Gii phng trnh:
1 + sin 2x + cos 2x
1 + cot2 x =
2sin x sin2x
A: x= 2
+ k; x= 4
+ k2
29. (B-2011)Gii phng trnh:
sin2x cos x + sin x cos x= cos 2x + sin x + cos x
A: x=
2
+ k2; x=
3
+ k2
3
30. (D-2011)Gii phng trnh:
sin2x + 2 cos x sin x 1tan x +
3
= 0
A: x=
3+ k2
31. (A-2012)Gii phng trnh:
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
33. (D-2012)Gii phng trnh:
sin3x + cos 3x sin x + cos x= 2cos2x
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A: x=
4+ k
2; x=
7
12+ k2; x=
12+ k2
34. (A-2013)Gii phng trnh:
1 + tan x= 22sinx + 4A: x=
4+ k; x=
3+ k2
35. (B-2013)Gii phng trnh:
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
38. (B-2014)Gii phng trnh:
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
10. (A-2004)Gii h phng trnh:
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]
14. (B-2005)Gii h phng trnh:
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
22. (A-2007PB)Gii phng trnh:
2log3(4x 3) + log 13
(2x + 3) 2
A: 3
4
; 3
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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
27. (A-2008)Gii h phng trnh:
x2 + y+ x3y+ xy2 + xy= 54
x4 + y2 + xy(1 + 2x) = 54
A: 354
;
32516 ,1;
3
2
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28. (A-2008PB)Gii phng trnh:
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)
35. (B-2009)Gii h phng trnh:
xy+ x + 1 = 7yx2
y2
+ xy+ 1 = 13y2
A:
1;1
3
, (3; 1)
36. (D-2009)Gii h phng trnh:
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
38. (A-2010)Gii h phng trnh:
(4x2 + 1)x + (y 3)5 2y= 04x2 + y2 + 23 4x= 730
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A:
1
2; 2
39. (B-2010)Gii h phng trnh:
3x + 16 x+3x214x8 = 0
A: x= 5
40. (B-2010NC)Gii h phng trnhlog2(3y 1) =x4x + 2x = 3y2
A:1;1
2
41. (D-2010)Gii phng trnh:
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
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44. (B-2011)Gii phng trnh:
3
2 + x 62 x + 4
4 x2 = 10 3x
A: x=6
5
45. (D-2011)Gii phng trnh:
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
47. (A-2012)Gii h phng trnh:
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;+)
49. (D-2012)Gii h phng trnh:
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)
53. (D-2013)Gii phng trnh:
2log2 x + log 12
(1 x) =1
2log2(x 2x + 2)
A:1 + ln 2
54. (A-2014)Gii h phng trnh:
x12 y+ y (12 x2) = 12x3 8x 1 = 2y 2
33
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A:(3; 3)
55. (B-2014)Gii h phng trnh:
(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
56. (D-2014)Gii phng trnh:
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
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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
7. (A-2004)Tnh tch phn I=
21
x
1 +
x 1 dx
A: 11
3 4 l n 2
8. (B-2004)Tnh tch phn I=
e1
1 + 3 ln x ln x
x dx
A: 116
135
9. (D-2004)Tnh tch phn I=
32
ln(x2 x)dx
A:3ln3 2
10. (A-2005)Tnh tch phn I=
2
0
sin2x + sin x
1 + 3 cos xdx
A: 34
27
11. (B-2005)Tnh tch phn I=
2
0
sin2x cos x
1 + cos x dx
36
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A:2 ln 2 1
12. (D-2005)Tnh tch phn I=
20
esinx
+ cos x
cos xdx
A: e +
4 1
13. (A-2006)Tnh tch phn I=
2
0
sin2xcos2 x + 4 sin2 x dx
A: x=2
3
14. (B-2006)Tnh tch phn I=
ln 5
ln 3
dx
ex + 2ex
3
A: ln3
2
15. (D-2006)Tnh tch phn I=
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
18. (D-2007)Tnh tch phn I=
e1
x3 ln2 xdx
A: 5e4 1
32
19. (A-2008)Tnh tch phn I=
60
tan4 x
cos2xdx
A: 1
2ln
2 +
3 10
9
3
20. (B-2008)Tnh tch phn I=
40
sin
x x4
dx
sin2x + 2(1 + sin x + cos x)
A: 4 32
4
21. (D-2008)Tnh tch phn I=
21
ln x
x3 dx
A: 3 2 l n 216
38
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22. (A-2009)Tnh tch phn I=
20
cos3 x 1 cos2 xdx
A: 8
15
4
23. (B-2009)Tnh tch phn I=
30
3 + ln x
(x + 1)2dx
A: 14
3 + ln27
16
24. (D-2009)Tnh tch phn I=
31
dx
ex 1 dx
A: ln(e2 + e + 1)
2
25. (A-2010)Tnh tch phn I=
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
27. (D-2010)Tnh tch phn I=
e1
2x
3
x
ln xdx
39
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A: e2
2 1
28. (A-2011)Tnh tch phn I=
40
x sin x + (x + 1) cos xx sin x + cos x
dx
A:
4+ ln
2
2
4
+ 1
29. (B-2011)Tnh tch phn I=
30
1 + x sin x
cos2 x dx
A:
3 +2
3 + ln(2
3)
30. (D-2011)Tnh tch phn I=
40
4x
1
2x + 1 + 2 dx
A: 34
3+ 10ln
3
5
31. (A-2012)Tnh tch phn I=
3
1
1 + ln(x + 1)
x2 dx
A: 2
3+ ln 3 2
3ln 2
32. (B-2012)Tnh tch phn I=
10
x3
x4 + 3x2 + 2dx
A: ln 3 32
ln 2
40
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33. (D-2012)Tnh tch phn I=
40
x(1 + sin 2x)dx
A: 2
32+
1
4
34. (A-2013)Tnh tch phn I=
21
x2 1x2
ln xdx
A: 52ln 2 32
35. (B-2013)Tnh tch phn I=
10
x
2 x2dx
A: 2
2 13
36. (D-2013)Tnh tch phn I=
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
38. (B-2014)Tnh tch phn I=
21
x2 + 3x + 1
x2 + x dx
A:1 + ln 3
41
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39. (D-2014)Tnh tch phn I=
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.
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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
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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
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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
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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
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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
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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
14. (D-2006)Cho ng trn
(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)
17. (D-2007)Cho ng trn
(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
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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