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GACHDUOI@

B?I TP HC K? I NM HC 2015 ? 2016

H t?n:......................................

B?I TP HC K? I NM HC 2015 ? 2016

1 Ni dung ?n tp

? Phng tr?nh lng gi?c; ? B?i to?n m - X?c sut - Nh thc Newton; ? T?m giao tuyn, thit din - Hai ng thng song song - ng thng song song vi mt phng.

2 B?i tp

2.1 Phng tr?nh lng gi?c

B?i 1. Phng tr?nh:...................................................................................

1.

sin 3x = -

3 2

2.

2

sin

(x

+

3

)

+

3=0

3.

cos(x + 3) =

1 3

4.

sin(2x - 15) =

2 2

5.

cos(3x - 450) =

3 2

6. cos 3x - sin 2x = 0

10. 4 cos2(6x - 2) + 16 cos2(3x - 1) = 13

11. cos(10x + 12) + 4 2 sin(5x + 6) = 5

12. (sin 2x +

3

cos

2x)2

=

cos(2x

-

6

)

13. (1 + sin 2x)(1 - tan x) = 1 + tan x

14. tan 3x + cos 6x = 1

15. sin 3x = cos2 2x - sin2 2x

7. cos 4x + 1 = cos 2x

16. sin 2x + sin

x

-

4

=0

8.

cos

2x

-

3

cos

x

=

cos2

x 2

9.

cos6

x

+

sin6

x

=

sin

2x

-

3 4

17. sin 5x + 2 sin2 3x = 1 18. cos2 3x + sin2 2x = 1

B?i 2. Phng tr?nh:...................................................................................

1. 5 cos x - 12 sin x = 1

8. 2 cos 6x + 2 sin 6x = -1

2. 2 cos 3x - 6 cos 3x + 6 = 0

9. 3 sin x - 5 cos x = 17

3. 5 sin x + 6 cos x = 7

4. cos 7x - 3 sin 7x = - 2

5.

sin x - cos x =

1- 2

3

6. 3 cos x + sin x = 2

7. 2 sin x - 3 cos x = -1

10. sin2 x + sin x cos x + 2 cos2 x = 4 11. sin2 x - 3 sin x cos x + 1 + cos 2x = 0 12. 3 sin2 x + 5 cos2 x - 2 cos 2x - 4 sin 2x = 0 13. 4 cos2 x + 3 sin x cos x - sin2 x = 3 14. 2 sin2 x - sin x cos x - cos2 x = 2

B?i 3. Phng tr?nh:...................................................................................

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GACHDUOI@

B?I TP HC K? I NM HC 2015 ? 2016

1. sin x + sin 2x + sin 3x = 0

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

3. sin2 x + sin2 2x + sin2 3x + sin2 4x = 2

4. cos 4x + cos 2x + 2 cos x = 0

5. cos 5x + cos 3x = sin 6x - sin 2x

6. cos 3x - cos 4x + cos 5x = 0

7. sin 7x - sin 3x = cos 5x

8. cos2 x - sin2 x = sin 3x + cos 4x

9.

cos

2x

-

cos

x

=

2

sin2

3x 2

10. 4 sin 3x + sin 5x - 2 sin x cos 2x = 0

11. sin 7x + cos 3x + sin x = 0

12. sin 7x sin 9x = sin 5x sin 11x

13.

sin2

x

+

sin2

3x 2

+

sin2

2x

+

sin2

9x 2

=2

14. sin4 x + cos2 x = cos 4x

15. 1 + cos x + cos 2x + cos 3x = 0

16. sin 7x cos 3x = sin 11x + cos 9x

17. cos x + cos 3x + cos 5x = 0

18. sin 3x sin 9x = sin 5x sin 7x

19. cos 7x cos 3x = cos 14x cos 10x

20. sin 11x cos 6x = sin 9x cos 4x

21. sin x cos 2x + sin 2x cos 5x = sin 3x cos 5x

22. sin2 2x + sin2 3x + sin2 4x + sin2 9x = 2

23.

cos3 x sin 3x + cos 3x sin3 x =

3 4

2.2 B?i to?n m - X?c sut

B?i 4. Chng minh rng:

1. k(k - 1)Cnk = n(n - 1)Cnk--22 2. p--kk = Cnp.Ckp B?i 5. Gii phng tr?nh:

3. Cnr = nr Cnr--11 4. Cnm+1 + Cnm-1 + 2Cnm = Cnm++21

1. A3n = 20n

2. A3x + 5A2x = 2(x + 15)

3. 3A2n - A22n + 42 = 0

4. 2Pn + 6A2n - PnA2n = 12

5.

A4n A3n+1 - Cnn-4

=

24 23

B?i 6.

11 1 6. C4x - C5x = C6x 7. C1x0++4x = C120x+-x10 8. A2x-2 +Cxx-2 = 101 9. C8x++x3 = 5A3x+6 10. (n + 1)! = 72

(n - 1)!

1. Trong lp c? 18 nam v? 12 n. Hi c? nhi?u c?ch chn 1 bn l?m lp trng?

2. Tr?n gi? c? 10 s?ch To?n, 13 s?ch L?, 17 s?ch H?a. Hi c? bao nhi?u c?ch chn 1 quyn s?ch?

3. T c?c s 1, 2, 3, 4, 5, 6, 7, 8, 9 c? bao nhi?u c?ch chn mt s hoc l? s chn hoc l? s nguy?n t?

4. Lp c? 40 hc sinh ng k? chi ?t nht mt hai trong m?n b?ng ? v? cu l?ng. C? 30 em dng k? b?ng ? v? 25 em ng k? cu l?ng. Hi c? bao nhi?u em ng k? c hai m?n th thao?

B?i 7. Lp c?c s t nhi?n t tp A = {0; 1; 2; ....; 9}. Hi lp c bao nhi?u s t nhi?n:

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GACHDUOI@

B?I TP HC K? I NM HC 2015 ? 2016

1. C? 6 ch s.

s 5 v? kh?ng c? mt ch s 0.

2. C? 6 ch s kh?c nhau.

3. C? 5 ch s kh?c nhau, trong ? c? 3 ch s chn v? 2 ch s l.

4. C? 6 ch s kh?c nhau, trong ? c? 3 ch s chn ng lin nhau v? 3 ch s l ng lin nhau.

6. C? 4 ch s kh?c nhau, trong ? c? ch s 5 v? 0 ng lin nhau.

7. C? 4 ch s m? tng c?c ch s ca n? l? s l.

8. Chn v? c? 5 ch s kh?c nhau, trong ? lu?n c? mt ch s 4.

5. C? 5 ch s kh?c nhau, trong ? c? mt ch 9. Chia ht cho 5 v? c? 4 ch s kh?c nhau.

B?i 8. Mt lp hc c? 20 hc sinh nam v? 15 hc sinh n. Gi?o vi?n ch nhim mun chn mt hc sinh nam hoc n d tri h? ca trng, hi c? bao nhi?u c?ch chn? Gi?o vi?n ch nhim mun chn mt hc sinh nam v? mt hc sinh n d l hi H?a trang, hi c? bao nhi?u c?c chn? ?p s: 35 c?ch, 300 c?ch.

B?i 9. Gieo mt ng xu c? hai mt S, N ba ln. H?y m? t kh?ng gian mu. Gi A l? bin c "mt nga xut hin ?t nht mt ln", h?y x?c nh bin c A? T?nh x?c sut ca bin c A? T?nh x?c sut mt nga xut hin ?ng hai ln?

B?i 10. Mt hp ng 4 vi?n bi xanh, 3 vi?n bi v? 2 vi?n bi trng. Chn ngu nhi?n 2 vi?n bi. T?nh x?c sut chon c 2 vi?n bi c?ng m?u? T?nh x?c sut chn c 2 vi?n bi kh?c m?u?

B?i 11. Mt hp cha 4 qu cu m?u , 5 qu cu m?u xanh v? 7 qu cu m?u v?ng. Ly ngu nhi?n

c?ng l?c ra 4 qu cu t hp ?. T?nh x?c sut sao cho 4 qu cu c ly ra c? ?ng mt qu cu m?u

v? kh?ng qu? hai qu cu m?u v?ng.

?p

s:

740 1820

.

B?i 12.

1. Cho tp A = {0, 1, 2, 3, 4, 5}, t A c? th lp c bao nhi?u s t nhi?n gm 5 ch s kh?c nhau,

trong ? nht thit phi c? ch s 0 v? 3.

?p s: 384.

2. C? bao nhi?u s t nhi?n c? 5 ch s kh?c nhau m? trong mi s lu?n lu?n c? mt hai ch s chn

v? ba ch s l.

?p s: 960.

3. Gi S l? tp hp c?c s t nhi?n c? hai ch s. Chn ngu nhi?n mt s t tp hp S. T?nh x?c sut

s c chn c? ch s h?ng n v v? h?ng chc u l? ch s chn.

?p s 2/9.

B?i 13. Chn bn qu?n b?i trong mt c b?i t? l kh gm 52 qu?n b?i. T?nh x?c sut chn c 4 qu?n K? T?nh x?c sut chn c ?ng mt qu?n K? T?nh x?c sut chn c ?t nht mt qu?n K?

B?i 14. Gieo mt ng xu ri gieo mt con s?c sc. H?y m? t kh?ng gian mu? T?nh x?c sut ca bin c A: "ng xu xut hin mt sp"? T?nh x?c sut ca bin c B: "con s?c sc xut hin mt 6 chm"? T?nh x?c sut ca bin c C: "con s?c sc xut hin mt l"? Chng minh A v? B l? hai bin c c lp.

2.3 Nh thc Newton

B?i 15.

Khai

trin

c?c

nh

thc:

(2

-

x)7,

(3x

+

y)6,

(x

+

2 x

)8

,

(1

-

x)6,

2x

+

1 2

6,

x2

-

3 x

7;

B?i 16. T?m s hng cha x15 trong khai trin (3 - x3)7.

B?i 17. T?m h s ca x7 trong khai trin (2x - 1)11? T?m h s ca x25y10 trong khai trin x3 + xy 15.

B?i 18. T?m h s ca x8 trong khai trin (x2 + 2)n, bit: A3n - 8Cn2 +Cn1 = 49. B?i 19. Bit h s ca x2 trong khai trin ca (1 + 3x)n bng 90. H?y t?m n?

?p s: 280.

B?i 20. T?nh tng c?c h s ca khai trin (3x - 2)2015.

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GACHDUOI@

B?I TP HC K? I NM HC 2015 ? 2016

2.4 H?nh hc kh?ng gian

B?i 21. Cho ch?p S.ABCD c? AD kh?ng song song vi BC. Ly im I thuc SA sao cho SA = 3IA, im J bt k? thuc SC, M l? trung im SB v? G l? trng t?m tam gi?c SCD.

1. T?m giao tuyn ca (SAD) v? (SBC); 2. T?m giao im E ca AB v? (IJM); 3. T?m giao im F ca BC v? (IJM); 4. T?m giao im N ca SD v? (IJM);

5. Gi H l? giao im ca MN v? BD, chng minh H, E, F thng h?ng;

6. Chng minh IG song song (ABCD);

7. T?m giao tuyn ca (BIG) v? (ABCD).

B?i 22. Cho t din ABCD c? M l? mt im tr?n cnh AB. Mt phng () qua M ng thi song song vi hai ng thng AC v? BD.

1. T?m giao tuyn ca mt phng () vi c?c mt ca t din.

2. Thit din ca t din ct bi mt phng mt phng () l? g??

B?i 23. Cho h?nh ch?p S.ABC c? G l? trng t?m tam gi?c SAC, I l? im thuc AB sao cho AI = 2BI. Chng minh IG (SBC). X?c nh giao tuyn ca hai mt phng (SIG) v? (SBC). X?c nh thit din ca h?nh ch?p ct bi mt phng () i qua I ng thi song song vi SB v? BC?

B?i 24. Cho h?nh ch?p S.ABCD c? ?y l? h?nh b?nh h?nh. Gi M, N, P l? trung im SB, BC, SD

1. T?m giao tuyn ca (SAC) v? (SBD); (SAB) v? (SCD); (SCD) v? (MNP);

2. T?m giao im ca CD v? (MNP); giao im ca AB v? (MNP);

3. T?m giao tuyn ca (SAC) v? (MNP), suy ra thit din ca h?nh ch?p v? (MNP).

B?i 25. Cho h?nh ch?p S.ABCD c? ?y l? h?nh b?nh h?nh t?m O. Gi M, N, P ln lt l? trung im ca SB, SD v? OC.

1. Chng minh MN song song vi (ABCD). X?c nh giao tuyn ca (MNP) v? (ABCD).

2. T?m giao tuyn ca mt phng (MNP) vi mt phng (SAC).

3. T?m giao im ca SA vi mt phng (MNP).

4. T?m thit din ca h?nh ch?p vi mt phng (MNP).

5.

Mt phng (MNP) ct c?c cnh SA, BC, CD ln lt ti I, J, K. T?nh

IA IS

,

JB JC

v?

KC KD

.

B?i 26. Cho h?nh ch?p S.ABCD c? ?y l? h?nh thang vi AB CD v? AB > CD. Gi M v? N ln lt l? trung im ca SB, SC.

1. Chng minh MN song song vi (ABCD), x?c nh giao tuyn ca (AMN) v? (ABCD)?

2. T?m giao tuyn ca (SAD) vi (SBC)?

3. T?m giao im ca ng thng SD vi mt phng (AMN)?

4. T?m thit din ca h?nh ch?p S.ABCD vi mt phng (AMN)?

4

GACHDUOI@

B?I TP HC K? I NM HC 2015 ? 2016

3 thi th

S GD-T NAM NH Trng THPT Xu?n Trng B

THAM KHO gachduoi@

KIM TRA CHT LNG HC K? I Nm hc 2015 ? 2016

M?N TO?N 11

(Thi gian l?m b?i: 90 ph?t)

[ S 1]

C?u 1. (2,0) T?m Tp x?c nh ca c?c h?m s sau:

sin x + cos x 1. y = 2 sin x - 2 cos x

tan 4x 2. y = tan2 x - 1

C?u 2. (2,0)

1. Trong mt phng ta Oxy, cho ng thng d : 2x - y + 2 = 0. T?m nh ca d qua ph?p quay t?m O g?c quay 270. 2. Trong mt phng ta Oxy, cho im I(2; -3) v? ng tr?n (C) : x2 + y2 - 2x + 2y - 7 = 0. T?m nh

ca ng tr?n (C) qua ph?p v t t?m I t s k = -3.

C?u 3. (5,0) Gii c?c Phng tr?nh sau:

1. sin 3x + cos 2x = 0

3. sin 2x + 4(cos x - sin x) = 4

2. 2 tan x + cot x - 3 = 0

4. sin 2x + cos 2x - 3 sin x - cos x + 1 = 0

C?u

4.

(1,0)

T?m

nghim

thuc

on

[0; 2013]

ca

PT:

5x sin(

-

) - cos( x

-

)

=

2 cos

3x

24

24

2

[ S 2]

C?u 1. (2,0) T?m Tp x?c nh ca c?c h?m s sau:

sin x + cos x 1. y = 2 sin x - 2 cos x

tan 4x 2. y = tan2 x - 1

C?u 2. (2,0)

1. Trong mt phng ta Oxy, cho ng thng d : 2x - y + 2 = 0. T?m nh ca d qua ph?p quay t?m O g?c quay 270. 2. Trong mt phng ta Oxy, cho im I(2; -3) v? ng tr?n (C) : x2 + y2 - 2x + 2y - 7 = 0. T?m nh

ca ng tr?n (C) qua ph?p v t t?m I t s k = -3.

C?u 3. (5,0) Gii c?c Phng tr?nh sau:

1. sin 3x + cos 2x = 0 2. 2 tan x + cot x - 3 = 0

3. sin 2x + 4(cos x - sin x) = 4 4. sin 2x + cos 2x - 3 sin x - cos x + 1 = 0

C?u 4. (3,0) Cho h?nh ch?p S.ABCD c? ?y l? h?nh thang ABCD vi ?y ln AD d?i gp ?i ?y nh BC. Gi I l? trung im AD. 1. X?c nh giao tuyn ca hai mt phng (SBC) v? (SAD) 2. Chng minh: IC (SAB) 3. Ly M l? mt im tr?n cnh SD. X?c nh thit din ca h?nh ch?p khi ct bi mp(IBM).

--------HT-------- CH?C C?C EM T KT QU CAO!

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