On Thi Tot Nghiep Toan

7/30/2019 On Thi Tot Nghiep Toan

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N TP THITT NGHIP K THI TT NGHIP TRUNG HC PH THNGTHPT NM HC 2011-2012 Mn thi: TON THPT

Thi gian lm bi: 150 pht (khng k thi gianpht)---------------------------------------------------

s 01

I. PHN CHUNG DNH CHO TT C CC TH SINH (7,0 im)

Cu I (3,0 im): Cho hm s: 2(1 ) (4 )y x x 1)Kho st s bin thin v v th ( )C ca hm s cho.

2) Vit phng trnh tip tuyn ca th ( )C ti giao im ca ( )C vi trc honh.3) Tm m phng trnh sau y c 3 nghim phn bit: 3 26 9 4 0x x x m Cu II (3,0 im):

1)Gii phng trnh: 2 12 3.2 2 0x x

2) Tnh tch phn:

1

0

(1 ) xI x e dx

3)Tm gi tr ln nht v nh nht ca hm s: 2( 1)xy e x x trn on [0;2].Cu III (1,0 im):Cho hnh chp u S.ABCDc cnh y 2a, gc gia cnh bn v mt y bng 600. Tnh th tch ca hnh ch

II. PHN RING (3,0 im) Th sinh ch c chn mt trong hai phn di y

1. Theo chng trnh chunCu IV (2,0 im):Trong khng gian vi h to Oxyz, cho (2; 0; 1), (1; 2; 3), (0;1;2)A B C .1)Chng minh 3 imA,B,Ckhng thng hng.Vit phng trnh mt phng ( )ABC .2)Tm to hnh chiu vung gc ca gc to Oln mt phng ( )ABC .

Cu V (1,0 im):Tm s phc lin hp ca s phczbit rng: 2 6 2z z i.

2. Theo chng trnh nng coCu IVb (2,0 im): Trong khng gian vi h to Oxyzcho (2; 0; 1), (1; 2; 3), (0;1;2)A B C 1)Chng minh 3 imA,B,Ckhng thng hng.Vit phng trnh mt phng ( )ABC .2)Vit phng trnh mt cu tmB, tip xc vi ng thngAC.Cu Vb (1,0 im):Tnh mun ca s phcz= 2012( 3 )i .

s 02


Cu I (3,0 im): Cho hm s: 3 23 3y x x x 1)Kho st s bin thin v v th ( )C ca hm s cho.2) Vit phng trnh tip tuyn ca th ( )C bit tip tuyn song song vi ng thng c phng trnh y Cu II (3,0 im):

1)Gii phng trnh: 6.4 5.6 6.9 0x x x

2) Tnh tch phn:

0

(1 cos )I x xdx

3)Tm gi tr ln nht v nh nht ca hm s: 2( 3)xy e x trn on [2;2].Cu III (1,0 im):Hnh chp S.ABCc yABCl tam gic vung cn (BA =BC), cnh bn SAvung gc vi mt phng y v

di l 3a , cnh bn SBto vi y mt gc 600. Tnh din tch ton phn ca hnh chp.

II. PHN RING (3,0 im) Th sinh ch c chn mt trong hai phn di y1. Theo chng trnh chunCu IV (2,0 im): Trong khng gian vi h to Oxyz cho im (2;1;1)A v hai ng thng


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,

1 2 1 2 2 1: :

1 3 2 2 3 2

x y z x y z d d

1)Vit phng trnh mt phng ( ) i qua imAng thi vung gc vi ng thng d2)Vit phng trnh ca ng thng i qua imA, vung gc vi ng thng dng thi ct ng th

Cu Va (1,0 im):Gii phng trnh sau y trn tp s phc: 4 2( ) 2( ) 8 0z z 2. Theo chng trnh nng coCu IVb (2,0 im): Trong khng gian Oxyzcho mp(P) v mt cu (S) ln lt c phng trnh

( ) : 2 2 1 0P x y z v 2 2 2( ) : 4 6 6 17 0S x y z x y z

1) Chng minh mt cu ct mt phng.

2) Tm ta tm v bn knh ng trn giao tuyn ca mt cu v mt phng.

Cu Vb (1,0 im):Vit s phc sau di dng lng gic1

2 2z

i

s 03


Cu I (3,0 im): Cho hm s: 4 24 3y x x 1)Kho st s bin thin v v th ( )C ca hm s cho.

2) Da vo ( )C , hy bin lun s nghim ca phng trnh: 4 24 3 2 0x x m

3) Vit phng trnh tip tuyn vi ( )C ti im trn ( )C c honh bng 3 .Cu II (3,0 im):

1)Gii phng trnh: 17 2.7 9 0x x

2) Tnh tch phn:2

(1 ln )e

eI x xdx

3)Tm gi tr ln nht v nh nht ca hm s:2 2 2

1

x xy

xtrn on 1

2[ ;2]

Cu III (1,0 im):Cho hnh chp S.ABCDc yABCDl hnh vung cnh a, SAvung gc vi mt y, SXc nh tm v tnh din tch mt cu ngoi tip hnh chp S.ABCD.


1. Theo chng trnh chunCu IV (2,0 im): Trong khng gian vi h to ( , , , )O i j k , cho 2 3 2OI i j k v mt phng

phng trnh: 2 2 9 0x y z 1)Vit phng trnh mt cu ( )S c tm l imIv tip xc vi mt phng ( )P .2)Vit phng trnh mp( )Q song song vi mp( )P ng thi tip xc vi mt cu ( )S Cu V (1,0 im):Tnh din tch hnh phng gii hn bi cc ng sau y:

3 24 3 1y x x x v 2 1y x

2. Theo chng trnh nng coCu IVb (2,0 im): Trong khng gian vi h trc to Oxyz, cho im A(1;2;7) v ng thng dc

trnh: 2 11 2 1

x y z

1) Hy tm to ca hnh chiu vung gc ca imAtrn ng thng d.2) Vit phng trnh mt cu tmAtip xc ving thng d.

Cu Vb (1,0 im):Gii h pt 4 4 4log log 1 log 9

20 0

x y

x y

s 04


Cu I (3,0 im): Cho hm s: 2 11xy x


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1)Kho st s bin thin v v th ( )C ca hm s cho.2) Vit phng trnh tip tuyn vi th ( )C bit tip tuyn c h s gc bng 4.Cu II (3,0 im):

1)Gii phng trnh: 2 22 4

log log (4 ) 5 0x x

2) Tnh tch phn: 30

sin cos

cos

x xI dx

x

3)Tm cc gi tr ca tham s m hm s sau y t cc tiu ti im0

2x

3 2 23 ( 1) 2y x mx m x

Cu III (1,0 im):Cho hnh chp S.ABC c y l tam gic vung ti B, BAC= 300, SA = AC = a v SAvung gc vi m

(ABC).Tnh VS.ABCv khong cch tAn mt phng (SBC).II. PHN RING (3,0 im) Th sinh ch c chn mt trong hai phn di y1. Theo chng trnh chun

Cu IV (2,0 im):Trong khng gian vi h to ( , , , )O i j k , cho 3 2OM i k , mt cu ( )S c phn2 2 2( 1) ( 2) ( 3) 9x y z

1)Xc nh to tmIv bn knh ca mt cu ( )S . Chng minh rng imMnm trn mt cu, t vit trnh mt phng ( ) tip xc vi mt cu tiM.

2)Vit phng trnh ng thng di qua tmIca mt cu, song song vi mt phng ( ) , ng thi vung

ng thng1 6 2

:3 1 1

x y z.

Cu V (1,0 im):Gii phng trnh sau y trn tp s phc: 2 2 5 0z z 2. Theo chng trnh nng coCu IVb (2,0 im): Trong khng gian vi h to Oxyz, cho t dinABCDc to cc nh lA(1;1;1) ,

B(1;2;1) , C(1;1;2) ,D(2;2;1)1)Vit phng trnh ng vung gc chung caAB v CD.2)Vit phng trnh mt cu (S) ngoi tip t dinABCD.Cu Vb (1,0 im):Tnh din tch hnh phng gii hn bi cc ng sau y

lny x , trc honh vx = e

s 05


Cu I (3,0 im): Cho hm s: 2 2(4 )y x x 1)Kho st s bin thin v v th ( )C ca hm s cho.2) Tm iu kin ca tham s b phng trnh sau y c 4 nghim phn bit:

4 24 log 0x x b

3) Tm to ca imA thuc ( )C bit tip tuyn tiA song song vi : 16 2011d y x Cu II (3,0 im):1)Gii phng trnh:

2 2

log ( 3) log ( 1) 3x x

2) Tnh tch phn: 2

3

sin

1 2 cos

xI dx

x

3)Tm gi tr ln nht, gi tr nh nht ca hm s: 4 3x xy e e x trn on [1;2]

Cu III (1,0 im):Cho t dinSABCc ba cnh SA, SB, SCi mt vung gc vi nhau, SB =SC= 2cm, SA = 4cm. Xc nh

tnh bn knh ca mt cu ngoi tip t din, t tnh din tch ca mt cu .II. PHN RING (3,0 im) Th sinh ch c chn mt trong hai phn di y

1. Theo chng trnh chunCu IV (2,0 im): Trong khng gian Oxyz, cho im ( 3;2; 3)A v hai ng thng


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1

1 2 3:

1 1 1

x y zd v

2

3 1 5:

1 2 3

x y zd

1) Chng minh rng1

d v2

d ct nhau.

2) Vit phng trnh mt phng (P) cha1

d v2

d . Tnh khong cch tAn mp(P).

Cu V (1,0 im):Tnh din tch hnh phng gii hn bi cc ng sau y:2 1y x x v 4 1y x x

2. Theo chng trnh nng coCu IVb (2,0 im): Trong khng gian vi h ta Oxyz, cho hai ng thng

1

1 2 3:

1 1 1

x y zd v

2

1 6:

1 2 3

x y zd

1)Chng minh rng1

d v2

d cho nhau.

2)Vitphng trnh mp(P) cha1

d v song song vi2

d . Tnh khong cch gia1

d v2

d

Cu Vb (1,0 im):Tnh din tch hnh phng gii hn bi cc ng sau y:

2y x , 4x y v trc honh

s 06

I. PHN CHUNG DNH CHO TT C CC TH SINH (7,0 im)Cu I (3,0 im): Cho hm s: 3 2 22 ( 1) ( 4) 1y x m x m x m 1)Kho st s bin thin v v th ( )C ca hm s khi m = 2.

2) Vit phng trnh tip tuyn ca ( )C ti giao im ca ( )C vi trc tung.3) Tm cc gi tr ca tham s m hm s t cc tiu tix = 0.Cu II (3,0 im):1)Gii phng trnh:

2 0,52 log ( 2) log (2 1) 0x x

2) Tnh tch phn:21

0

( 1)x

x

eI dx

e

3)Cho hm s

2

2.x

y x e . Chng minh rng, 2(1 )xy x y Cu III (1,0 im):Cho hnh chp S.ABCDc yABCDl hnh ch nht c AB = a,BC= 2a. Hai mt bn (SAB) v (SAD) vu

vi y, cnh SChp vi y mt gc 600. Tnh th tch khi chp S.ABCD.II. PHN RING (3,0 im) Th sinh ch c chn mt trong hai phn di y1. Theo chng trnh chunCu IV (2,0 im): Trong khng gian Oxyz, cho (0;1;2), ( 2; 1; 2), (2; 3; 3), ( 1;2; 4)A B C D 1) Chng minh rngABCl tam gic vung. Tnh din tch ca tam gicABC.2) Vit phng trnh mt phng (ABC). Tnh th tch t dinABCD.

Cu V (1,0 im):Gii phng trnh sau y trn tp s phc: 22 2 5 0 2. Theo chng trnh nng coCu IVb (2,0 im): Trong khng gian Oxyz, cho (0;1;2), ( 2; 1; 2), (2; 3; 3)A B C 1) Chng minh rngABCl tam gic vung. Tnh din tch ca tam gicABC.

2) Vit phng trnh ng thng i qua im Bng thi vung gc vi mt phng (ABC). Xc nhimD trn sao cho t dinABCDc th tch bng 14.

Cu Vb (1,0 im):Gii phng trnh sau y trn tp s phc:2

4 8z z i

s 07


Cu I (3,0 im): Cho hm s: 3 21 2 33

y x x x


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1)Kho st s bin thin v v th ( )C ca hm s cho.2) Vit phng trnh tip tuyn ca ( )C ti im trn ( )C c honh bng 4. V tip tuyn ny ln cng h

vi th ( )C Cu II (3,0 im):

1)Gii phng trnh: 1 29 3 18 0x x

2) Tnh tch phn:21

lne x xI dx

x

3)Tm gi tr ln nht v nh nht ca hm s: 5 4 3( ) 5 5 1f x x x x trn on [1;2]Cu III (1,0 im):

Cho hnh chp u S.ABCDc cnh y 2a, gc gia mt bn v mt y bng 600

. Tnh th tch ca hnh chII. PHN RING (3,0 im) Th sinh ch c chn mt trong hai phn di y

1. Theo chng trnh chunCu IV (2,0 im):Trong khng gian vi h to Oxyz, cho (2;1; 1), ( 4; 1;3), (1; 2;3)A B C .1)Vit phng trnh ng thng ABv phng trnh mt phng (P) i qua im Cng thi vung gc v

thngAB.2)Tm to hnh chiu vung gc ca im Cln ng thngAB. Vit phng trnh mt cu tm Ctip

ng thngAB.

Cu V (1,0 im):Tm s phc lin hp ca s phczbit rng: 3 9 2 11z iz i .

2. Theo chng trnh nng coCu IVb (2,0 im): Trong khng gian vi h to Oxyzcho (2;1; 1), ( 4; 1;3), (1; 2;3)A B C 1)Vit phng trnh ng thngABv tnh khong cch t im Cn ng thngAB2)Vit phng trnh mt cu ( )S tm C, tip xc vi ng thngAB. Tm to tip im ca ng thng

mt cu ( )S .

Cu Vb (1,0 im):Tnh mun ca s phcz= 2011( 3 )i .

s 08


Cu I (3,0 im): Cho hm s:1

xy

x

1)Kho st s bin thin v v th ( )C ca hm s.2) Vit phng trnh tip tuyn vi ( )C ti cc giao im ca ( )C vi : y x3) Tm cc gi tr ca tham s k ng thng d: y kxct ( )C ti 2 im phn bit.

Cu II (3,0 im):

1)Gii bt phng trnh:

2

22

2 19 3.3

x x

x x

2) Tm nguyn hm ( )F x ca hm s ( ) 2 lnf x x x , bit (1) 1F

3)Tm gi tr ln nht, gi tr nh nht ca hm s: 3 24 3 5y x x x trn on [ 2;1]

Cu III (1,0 im):Cho hnh chp S.ABCc y l tam gic vung ti B, cnh SAvung gc vi y. Gi D, E ln lt l h

vung gc caA ln SB, SC. Bit rngAB = 3,BC= 2 v SA = 6.Tnh th tch khi chp S.ADE.

II. PHN RING (3,0 im) Th sinh ch c chn mt trong hai phn di y1. Theo chng trnh chunCu IV (2,0 im): Trong khng gian Oxyz, cho hnh hp .ABCD A B C D c to cc nh:

(1;1;1), (2; 1;3), (5;2;0), ( 1;3;1)A B D A

1)Xc nh to cc nh Cv B ca hnh hp.Chng minh rng, yABCD ca hnh hp l mt hnh ch

2). Vit phng trnh mt y (ABCD), t tnh th tch ca hnh hp .ABCD A B C D


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Cu V (1,0 im):Cho hnh phng (H) gii hn bi cc ng:1

1yx

, trc honh v x= 2. Tnh th

th trn xoay khi quay hnh (H) quanh trc Ox.2. Theo chng trnh nng coCu IVb (2,0 im):Trong khng gian Oxyz, cho hnh hp .ABCD A B C D c to cc nh:

(1;1;1), (2; 1;3), (5;2;0), ( 1;3;1)A B D A

1)Xc nh to cc nh Cv B ca hnh hp. Chng minh,ABCDl hnh ch nht.2)Vit phng trnh mt cu i qua cc nhA,B,D vA ca hnh hp v tnh th tch ca mt cu .

Cu Vb (1,0 im):Gii phng trnh sau y trn tp s phc: 2 (1 5 ) 6 2 0z i z i

s 09I. PHN CHUNG DNH CHO TT C CC TH SINH (7,0 im)

Cu I (3,0 im): Cho hm s: 3 23 1y x x c th l ( )C 1)Kho st s bin thin v v th ( )C ca hm s.2) Da vo th ( )C , hy tm iu kin ca tham s k phng trnh sau y c 3 nghim ph

3 23 0x x k Cu II (3,0 im):1)Gii bt phng trnh:

2 22 log ( 1) log (5 ) 1x x

2) Tnh tch phn:1

0

( )xI x x e dx

3)Tm gi tr ln nht, gi tr nh nht ca hm s: 3 22 3 12 2y x x x trn [ 1;2] Cu III (1,0 im):Cho hnh lng tr tam gic u .ABC A B C c tt c cc cnh u bng a. Tnh din tch ca mt cu ngoi

lng tr theo a.II. PHN RING (3,0 im) Th sinh ch c chn mt trong hai phn di y1. Theo chng trnh chunCu IVa (2,0 im): Trong khng gian vi h ta Oxyz, cho hai ng thng:

1

2 2

( ) : 3

x t

d y

z t

v2

2 1( ) :

1 1 2

x y zd

1)Chng minh rng hai ng thng1 2

( ),( )d d vung gc nhau nhng khng ct nhau.

2)Vit phng trnh mtphng (P) cha d1ng thi song song d2. T , xc nh khong cch gng thng d1 v d2 cho.

Cu V (1,0 im):Tm mun ca s phc: 31 4 (1 )z i i .2. Theo chng trnh nng coCu IVb (2,0 im):Trong khng gian vi h ta Oxyz, cho hai ng thng:

1

2 2

( ) : 3

x t

d y

z t

v2

2 1( ) :

1 1 2

x y zd

1)Chng minh rng hai ng thng1 2

( ),( )d d vung gc nhau nhng khng ct nhau.

2)Vit phng trnh ng vung gc chung ca1 2

( ),( )d d .

Cu Vb (1,0 im): Tm nghim ca phng trnh sau y trn tp s phc: 2z z , trong z l s phc lin hp ca s phcz.

s 10

I. PHN CHUNG DNH CHO TT C CC THSINH (7,0 im)Cu I (3,0 im): Cho hm s: 3 3 1y x x c th l ( )C

1)Kho st s bin thin v v th ( )C ca hm s.


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2) Vi t phng trnh ti p tuy n ca th hm s ti giao i m ca th vi trc tung. V ti p tuy n lmt h trc to vi th ( )C .

Cu II (3,0 im):

1)Gii phng trnh: 23 3

2 log log (3 ) 14 0x x

2) Tnh tch phn:1

0(2 1) xI x e dx

3)Tm gi tr ln nht vgi tr nh nht ca hm s 4 3 22y x x x trn on [1;1]Cu III (1,0 im):Cho hnh chp u S.ABCDc cnh y bng a, gc gia cnh bn v mt y bng 600. Tnh din tch xung

th tch ca hnh nn c nh Sv y l ng trn ngoi tip y hnh chp cho. II. PHN RING (3,0 im) Th sinh ch c chn mt trong hai phn di y1. Theo chng trnh chunCu IV (2,0 im): Trong khng gian vi h ta Oxyz, cho hai im ( 5;0;1), (7;4; 5)A B v m

( ) : 2 2 0P x y z

1)Vit phng trnh mt cu ( )S c ng knhAB. Tnh khong cch t tmIca mt cu n mt phng ( P2) Vit phng trnh ng thng di qua tmIca mt cu ( )S ng thi vung gc vi mt phng ( )P . T

giao im ca dv ( )P .

Cu V (1,0 im):Tm mun ca s phc:1

2 3 32

z i i

2. Theo chng trnh nng coCu IVb (2,0 im):Trong khng gian vi h to Oxyz, cho im (0;6;4)A v ng thng dc phng2 1

1 2 1

x y z

1) Hy tm to hnh chiu vung gc ca imAtrn ng thng d.2) Vit phng trnh mt cu ( )S c tm l imAv tip xc vi ng thng d.Cu Vb (1,0 im):Gii phng trnh sau y trn tp s phc

2 (3 4 ) ( 1 5 ) 0x i x i

s 11

I. PHN CHUNG DNH CHO TT C CC TH SINH (7,0 im)Cu I (3,0 im): Cho hm s: 4 2( 1) 2 1y x m x m (1)1)Kho st s bin thin v v th ( )C ca hm s khi m = 1.

2) Vit phng trnh tip tuyn ca ( )C ti im trn ( )C c honh bng - 3 .3) Tm cc gi tr ca tham s m hm s (1) c 3 im cc tr.Cu II (3,0 im):1)Gii phng trnh: 2 0,5log ( 3) log ( 1) 3x x

2) Tnh tch phn:21

0( )xI x x e dx

3)Cho hm s 4 2x xy e e . Chng minh rng, 13 12y y y

Cu III (1,0 im):Cho khi chp S.ABCc SAvung gc vi mt y (ABC), tam gicABCvung cn tiB, SA= a, SB hp

mt gc 300.Tnh th tch ca khi chp S.ABC.


1. Theo chng trnh chunCu IV (2,0 im): Trong khng gian Oxyz, cho ng thng dv mt phng (P) ln lt c pt

3 2

: 1 ,( ) : 3 2 6 0

x t

d y t P x y z

z t


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1)Tm to i mAgiao i m ca ng th ng dv mp(P). Vi tphng trnh mt ph ng (Q) i qua i mthi vung gc vi ng thng d.

2)Vit phng trnh mt cu ( )S tm (2;1;1)I , tip xc vi mp(P). Vit phng trnh mt phng tip din cu ( )S bit n song song vi mp(P).

Cu V (1,0 im):Tm phn thc v phn o ca s phcz i

z i, trong 1 2z i

2. Theo chng trnh nng coCu IVb (2,0 im):Trong khng gian Oxyz, cho ng thng dv mt phng (P) ln lt c pt

3 1: ,( ) : 3 2 6 0

2 1 1

x y zd P x y z

1)Chng minh rng ng thng dct mt phng (P) nhng khng vung gc vi (P). Tm to im Aim ca ng thng dv mp(P).

2)Tm phng trnh hnh chiu ca ng thng dln mp(P).

Cu Vb (1,0 im):Gii phng trnh sau y trn tp s phc: 2 4 4 0iz z i

s 12



2 42

xy x


2) Tnh din tch hnh phng gii hn bi th ( )C v trc honh.3) Tm m phng trnh sau y c ng 2 nghim phn bit: 4 22 2 0x x m Cu II (3,0 im):

1)Gii phng trnh: 2 2 22 2 3 0x x

2) Tm nguyn hm ( )F x ca 21

( ) 3 4 xf x x ex

bit rng (1) 4F e

3) Vit phng trnh tip tuyn ca th hm s 3 1y x x , bit tip tuyn song song vi n2 1y x .

Cu III (1,0 im):

Cho hnh chp tam gic u c cnh y bng 6 , ng cao h= 2. Hy tnh din tch ca mt cu ngoi tichp .


1. Theo chng trnh chunCu IV (2,0 im): Trong khng gian Oxyz, cho ( 1;2; 1), (2;1; 1), (3;0;1)A B C 1)Vit phng trnh mt cu i qua 4 im O,A,B,Cv xc nh to tmIca n.

2)Tm to imMsao cho 3 2AM MC . Vit phng trnh ng thngBM.

Cu V (1,0 im): Tnh1 2

x x , bit1 2,x x l hai nghim phc ca phng trnh sau y:

23 2 3 2 0x x

2. Theo chng trnh nng coCu IVb (2,0 im):Trong khng gian vi h to Oxyzcho ng thng dv mt phng (P) ln lt c

trnh d:

1 2

2

1

x t

y t

z

, (P): 2 2 1 0x y z .

1) Vitphng trnh mt cu c tm thuc d, bn knh bng 3 v tip xc (P).2) Vitphng trnh ng thng i qua imM(0;1;0), nm trong mp(P) v vung gc vi ng thng d

Cu Vb (1,0 im): Gi1 2

;z z l hai nghim ca phng trnh 2 1 0z z trn tp s phc. Hy x

1 2

1 1

A z z .


82/87

s 13


Cu I (3,0 im): Cho hm s: 2 2( 2) 1y x 1)Kho st s bin thin v v th ( )C ca hm s.

2) Da vo th (C) bin lun s nghim phng trnh: 4 24x x m.Cu II (3,0 im):

1)Gii phng trnh:2 2

log ( 5) log 2 3x x

2) Tnh tch phn:3ln 2

0

1x

x

eI dx

e

3)Tm gi tr ln nht, gi tr nh nht ca hm s:3 2

1

xy

xtrn on[1;4]

Cu III (1,0 im):Cho hnh lngtr .ABC A B C c yABCl tam gic u cnh bng a. Hnh chiu vung gc ca A xu

phng (ABC) l trung im caAB. Mt bn ( )AA C C to vi y mt gc bng 45 . Tnh th tch ca khi ny.

II. PHN RING (3,0 im) Th sinh ch c chn mt trong hai phn di y1. Theo chng trnh chunCu IV (2,0 im): Trong khng gian Oxyz, cho hai im (0;1; 4), (1;0; 5)A B v ng thng

1 4 1:1 4 2

x y z

1)Vit phng trnh ng thngABv chng minh rngAB v cho nhau.2) Vit phng trnh mt phng (P) cha hai imA,Bngthi song song vi ng thng . Tnh kho

gia ng thng v mt phng (P).

Cu V (1,0 im):Tnh din tch hnh phng gii hn bi: 2 12 36y x x v 26y x x 2. Theo chng trnh nng coCu IVb (2,0 im): Trong khng gian Oxyz, cho hai ng thng:

1 2

13 1

: 1 :1 2 12

x tx y z

y t

z

1) Chng minh1v

2cho nhau. Vit phng trnh mp(P) cha

1v song song

2.

2)Tm imA trn1v imB trn

2sao cho di onABngn nht.

Cu Vb (1,0 im):Trn tp s phc, tm B phng trnh bc hai 2 0z Bz i c tng bnh phnghim bng 4i

s 14


Cu I (3,0 im): Cho hm s: 2 11

xyx

1)Kho st s bin thin v v th ( )C ca hm s.2) Vit phng trnh tip tuyn ca ( )C ti im trn ( )C c tung bng 5.3) Tnh din tch hnh phng gii hn bi ( )C v hai trc to .Cu II (3,0 im):

1)Gii phng trnh: 20.5 2

log ( 5) 2 log ( 5) 0x x

2) Tnh tch phn:1

01I x xdx

3)Tm gi tr ln nht, gi tr nh nht ca hm s: 2

( 2)

xy e x trn on[1;3]


83/87

Cu III (1,0 i m): Cho hnh chp S.ABCc yABCl tam gic vung tiB, cnh SAvung gc vi mt

060SCB ,BC= a, 2SA a . GiMl trung im SB.1) Chng minh rng (SAB) vung gc (SBC).2) Tnh th tch khi chpMABC

II. PHN RING (3,0im) Th sinh ch c chn mt trong hai phn di y

1. Theo chng trnh chunCu IV (2,0 im): Trong khng gian Oxyz, cho 4 im ( 1;1;1), (5;1; 1), (2;5;2), (0; 3;1)A B C D 1)Vit phng trnh mt phng (ABC). T chng minhABCDl mt t din.2)Vit phng trnh mt cu (S) c tm l im D, ng thi tip xc vi mt phng (ABC). Vit phng tr

din vi mt cu (S) song song vi mp(ABC)Cu V (1,0 im):Gii phng trnh sau y trn tp s phc: 4 25 36 0z z

2. Theo chng trnh nng caoCu IVb (2,0 im): Trong khng gian vi h ta Oxyz, cho ng thng (d) v mt phng (P) ln

phng trnh :3 1 3

2 1 1

x y zv mt phng (P): 2 5 0x y z .

1)Tm ta giao imca ng thng dv mtphng (P) .2)Tnh gc gia ng thng dv mt phng (P) .3)Vit phngtrnh hnh chiu vung gc ca ng thng dln mt phng (P).

Cu Vb (1,0 im):Gii h phng trnh sau : 2 22

4 .log 4

log 2 4

y

y

x

x

s 15



2( ) 2 33

xy f x x x


2) Vit phng trnh tip tuyn ca ( )C ti im trn ( )C c honh 0

x , vi0

( ) 6f x .

3) Tm tham s m phng trnh 3 26 9 3 0x x x m c ng 2 nghim phn bit.Cu II (3,0 im):1)Gii phng trnh: 4 4 2 42 17.2 1 0x x

2) Tnh tch phn:0

(2 1)sinI x xdx

3)Tm gi tr ln nht, gi tr nh nht ca hm s 2 4 ln(1 )y x x trn on [2;0]

Cu III (1,0 im):Cho hnh lng tr ng .ABC A B C c yABCl tam gic vung tiB,BC= a, mt ( )A BC to vi y m

0

30v tam gic A BCc din tch bng 2

3a . Tnh th tch khi lng tr .ABC A B C .


1. Theo chng trnh chunCu IV (2,0 im): Trong khng gian vi h to Oxyz, cho hai im (7;2;1), ( 5; 4; 3)A B v m

( ) : 3 2 6 38 0P x y z

1) Vit phng trnh tham s ca ng thngAB. Chng minh rng,AB ||( )P .2) Vit phng trnh mt cu ( )S c ng knhAB.3) Chng minh ( )P l tip din ca mt cu ( )S . Tm to tip im ca ( )P v ( )S

Cu V (1,0 im):Cho s phc 1 3z i. Tm s nghch o ca s phc: 2 .z z z

2. Theo chng trnh nng co


84/87

Cu IVb (2,0 im): Trong khng gian vi h to Oxyz, cho cho im (1; 3; 2)I v ng

4 4 3:

1 2 1

x y z

1)Vit phng trnh mt phng (P) i qua imIv cha ng thng .2)Tnh khong cch t imIn ng thng .3)Vit phng trnh mt cu (S) c tm l imIv ct ti hai im phn bitA,B sao cho on thng

di bng 4.

Cu Vb (1,0 im): Gi1 2,z z l hai nghim ca phng trnh: 2 2 2 2 2 0z z i . Hy lp mt ph

bc hai nhn1 2,z z lm nghim.

s 16


Cu I (3,0 im): Cho hm s: 4 21

22

y x x

1)Kho st s bin thin v v th ( )C ca hm s nu trn.

2) Dng th ( )C bin lun s nghim ca phng trnh: 4 24 2x x m.3) Tnh din tch hnh phng gii hn bi th ( )C vi trc honh.Cu II (3,0 im):1)Gii phng trnh:

22

log ( 2) 2 log 2x x

2) Tnh tch phn:2

2 2

0( 1)I x x dx

3)Tm gi tr ln nht v gi tr nh nht ca hm s: 24y x Cu III (1,0 im):Hnh chp S.ABCcBC= 2a, yABCl tam gic vung ti C, SABl tam gic vung cn ti Sv nm tro

phng vung gc vi mt y. GiIl trung im cnhAB.1) Chng minh rng, ng thng SIvung gc vi mt y ( )ABC .2) Bit mt bn (SAC) hp vi y (ABC) mt gc 600. Tnh th tch khi chp S.ABC.II. PHN RING (3,0 im) Th sinh ch c chn mt trong hai phn di y

1. Theo chng trnh chunCu IV (2,0 im): Trong khng gian vi h to Oxyz, cho hai im (3;1; 1), (2; 1; 4)A B vmt phng ( ) : 2 3 1 0P x y z

1) Vit phng trnh ng thngABv phng trnh mt cu ng knhAB.2) Vit phng trnh mt phng ( )Q cha hai imA,B, ng thi vung gc vi mp(P).

Cu V (1,0 im):Gii phng trnh sau y trn tp s phc: 3 25 2 0z z z 2. Theo chng trnh nng coCu IVb (2,0 im): Trong khng gian vi h to Oxyz, cho mt phng (Q): 2 2 2 0x y z 1)Vit phng trnh mt cu ( )S tmI(3;1;2) tip xc vi (Q). Tm to tip im.

2)Vit phng trnh mt phng (P) i qua hai im (1; 1;1), (0; 2;3)A B , ng thi to vi mt cu ( )S m

trn c bn knh bng 2.

Cu Vb (1,0 im):Trn mt phng phc, tm tp hp cc im biu din s phcztha iu kin:2 4 2z i i z

s 17


Cu I (3,0 im): Cho hm s:2 ( 3)

2

x xy


2) Vit phng trnh tip tuyn ca ( )C ti giao im ca ( )C vi trc honh.3) Tm iu kin ca k phng trnh sau y c nghim duy nht: 3 23 0x x k .


85/87

Cu II (3,0 im):

1)Gii phng trnh:22 6 6

12 2.4x x

x

2) Tnh tch phn:33

20 1

xI dx

x

3)Tm gi tr ln nht v nh nht ca hm s: 5 4 33 9y x x x trn on [ 2;1]

Cu III (1,0 im):Cho khi chp S.ABCcABCv SBCl cc tam gic u c cnh bng 2, 3SA a . Ttch khi chp S.ABCtheo a.


1. Theo chng trnh chunCu IV (2,0 im): Trong khng gian vi h to Oxyz, cho tam gicABCc to cc nh:A(1;1;2),B(0;1;1) v C(1;0;4).

1) Chng minhABCl tam gic vung. Xc nh to im D bn imA,B,C,Dl bn nh ca mt hnht.

2) GiMl im tho MB = 2MC. Vit phng trnh mt phng (P) i qua imMv vung gc vi nBC. Vit phng trnh mt cu tmA,tip xc vi mp(P).

Cu V (1,0 im):Tnh din tch hnh phng gii hn bi cc ng sau y:2 2

( 1) ,y x x y x x v 1x

2. Theo chng trnh nng coCu IVb (2,0 im): Trong khng gian vi h to Oxyz, cho im (1;2; 3)M v ng thng

d:3 1 1

2 1 2

x y z

1) Tm to hnh chiu vung gc ca imMln ng thng d. Vit phng trnh mt cu tmM, tip xc2) Vit phng trnh mp(P) i qua imM, song song vi dv cch dmt khong bng 4.

Cu Vb (1,0 im):Cho s phc 1 3z i . Hy vit dng lng gic ca s phc 5z .

s 18


Cu I (3,0 im): Cho hm s:3 2

1

xy

x

1)Kho st s bin thin v v th ( )C cahm s.2) Vit pt tip tuyn ca ( )C bit tip tuyn vung gc vi ng thng : 1 0x y 3)Tm cc gi tr ca k ( )C v : 3d y kx ct nhau ti 2 im phn bit.Cu II (3,0 im):1)Tm gi tr ln nht v nh nht ca hm s: 3 2( ) 2 3 12 1f x x x x trn on [ 1;3]

2) Tnh tch phn:1

(ln 1)e

I x dx

3)Gii phng trnh: 12 2log (2 1).log (2 2) 6x x

Cu III (1,0 im):

Cho mt hnh tr c di trc 2 7OO .ABCDl hnh vung cnh bng 8 c cc nh nm trn hai y sao cho tm ca hnh vung l trung im ca on OO . Tnh th tch ca hnh tr .

II. PHN RING (3,0 im)Th sinh ch c chn mt trong hai phn di y

1. Theo chng trnh chunCu IV (2,0 im): Trong khng gian vi h to Oxyz, cho ng thng v mt phng ( ) ln

phng trnh3 2 3

:1 1 3

x y z; ( ) : 2 1 0x y z

1)Chng minh rng ng thng song song vi mt phng (). Tnh khong cch t ng thng n m().

2)Tm to giao im Aca ng thng vi mt phng ( )Oxy . Vit phng trnh mt cu tm A, tip mt phng ().


86/87

Cu V (1,0 im): Cho 2(1 2 )(2 )z i i . Tnh mun ca s phc z


Cu IVb (2,0 im): Trong khng gian vi h to Oxyz, cho imM(1; 1;1), mt phng ( ) : 2 0P y z

ng thng1

1:

1 1 4

x y z, 2

2

: 4

1

x t

y t

z

1) Tm to im M i xng vi imMqua ng thng 2.2)Vit phng trnh ng thng ct c hai ng thng 1, 2v nm trong mp(P).

Cu Vb (1,0 im):Cho hm s2 1

1

( 1)mx m xyx

. Tm m hm s chai im cc i v cc tiu n

pha so vi trc tung.

s 19


Cu I (3,0 im): Cho hm s: 4 21 3 5

4 2 4y x x

1)Kho st s bin thin v v th ( )C ca hm s.2)Vit phng trnh tip tuyn ca ( )C ti im cc tiu ca n.

3) Tm cc gi tr ca tham s m phng trnh sau y c 4 nghim phn bit:4 26 1 4 0x x m Cu II (3,0 im):1)Gii bt phng trnh: 2 22 5.6 9.9x x x

2) Tnh tch phn:2

2

0( 1) xI x e dx

3)Tm gi tr ln nht, gi tr nh nht ca hm s: 4 2( ) sin 4cos 1f x x x

Cu III (1,0 im):

Cho hnh lng tr ng ABC.A'B'C'c yABCl mt tam gic vung ti A vAC= a, 060C . ng cca mt bnBB'C'Cto vi mt phng (AA'C'C) mt gc 030 . Tnh th tch ca khi lng tr theo a.

II. PHN RING (3,0 im) Th sinh ch c chn mt trong hai phn di y1. Theo chng trnh chunCu IV (2,0 im): Trong khng gian vi h ta Oxyz, cho mt phng (P) c phng trnh

2 2 1 0x y z v im (1;3; 2)A 1)Tm ta hnh chiu caAtrn mt phng (P).2)Vit phng trnh mt cu tmAv i qua gc ta O.Cu V (1,0 im):Cho s phc z tha mn: 2(1 ) (2 ) 8 (1 2 )i i z i i z . Tm phn thc, phn o

mun ca s phcz.2. Theo chng trnh nng coCu IVb (2,0 im): Trong khng gian vi h ta Oxyz, cho ng thng (d) c phng trnh

2 11 2 3

x y z v im (1; 2;3)A

1)Tm ta hnh chiu caAtrn ng thng (d)2)Vit phng trnh cu tmA, tip xc vi ng thng d.

Cu Vb (1,0 im):Cho hm s2 3

1

x xy

x( )C . Tm trn ( )C cc im cch u hai trc to .

s 20

I. PHNCHUNG DNH CHO TT C CC TH SINH (7,0 im)

Cu I (3,0 im): Cho hm s: 3 21 1 1

23 2 6

y x x x



87/87

2)Tm cc gi tr ca tham s m phng trnh sau y c 3 nghim phn bit:3 22 3 12 1 2 0x x x m

Cu II (3,0 im):1)Gii bt phng trnh: 1 62 2 24x x

2) Tnh tch phn:2

2

1

lne

x xI dx

x

3)Vit phng trnh tip tuyn ca th hm s ti cc giao im ca n vi ng thng 2 1y x .Cu III (1,0 im):Mt hnh nn c thit din qua trc l mt tam gic vung cn c cnh gc vung bng a.

a)Tnh din tch xung quanh v din tch ton phn ca hnh nn. b)Tnh th tch ca khi nn tng ng.II. PHN RING (3,0 im) Th sinh ch c chn mt trong hai phn di y1. Theo chng trnh chun

Cu IV (2,0 im): Trong khng gian vi h ta ( , , , )O i j k , cho hnh hp .ABCD A B C D c

0, , 2 3 , 3OA OB i OC i j k AA k ,

1)Vit phng trnhmt phng ( )ABA v tnh khong cch t C n ( )ABA .

2)Tm to nh Cv vit phng trnh cnh CD ca hnh hp .ABCD A B C D

Cu V (1,0 im): Cho1 3

2 2z i . Tnh 2 1z z


Cu IV (2,0 im): Trong khng gian vi h ta ( , , , )O i j k , cho hnh hp .ABCD A B C D c

0, , 2 3 , 3OA OB i OC i j k AA k ,

1)Tm ta cc nh C,D v chng minh rng .ABCD A B C D l hnh hp ch nht.2)Vit phng trnh mt cu ngoi tip hnh hp .ABCD A B C D .

Cu Vb (1,0 im): Cho1 3

2 2z i . Tnh 2011z

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