De Thi HSG Binh Phuoc 20132014
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S GIO DC V O TO
S GIO DC V O TOK THI CHN HC SINH GII CP TNH THPT BNH PHC
NM HC 2013 - 2014 THI CHNH THC
Mn: Ton
( thi c 01 trang)
Thi gian lm bi: 180 pht (khng k thi gian giao )
Ngy thi: 03/10/2013Cu I:(THPT:4,0 im; GDTX: 4,0 im) Cho hm s:
(1)
1. Kho st s bin thin v v th ca hm s (1).
2. Vit phng trnh tip tuyn ca , bit tip tuyn ct ng tim cn ng v tim cn ngang ln lt ti sao cho , vi .
Cu II:(THPT:5,0 im; GDTX: 6,0 im)
1. Gii h phng trnh:
2. Gii phng trnh:
Cu III:(THPT:4,0 im; GDTX:4,0 im)
1. Trong mt phng vi h trc ta , cho hnh ch nht c , im thuc vo ng thng c phng trnh: . ng thng i qua v trung im ca on c phng trnh: . Tm ta ca v , bit im c honh dng.2. Cho tam gic nhn ni tip ng trn . Gi ln lt l cc im di ng trn cung nh , sao cho thng hng. Gi , ln lt l hnh chiu vung gc ca ln cc ng thng tng ng v ln lt l hnh chiu vung gc ca ln cc ng thng . Gi l giao im ca hai ng thng v . Tm gi tr ln nht ca din tch tam gic (theo ).Cu IV:(THPT:3,0 im; GDTX:3,0 im) Cho hnh chp c y l hnh ch nht, tam gic u cnh v nm trong mt phng vung gc vi y. Gc gia mt phng v mt phng y bng .1. Tnh th tch khi chp theo .
2. Tnh khong cch gia hai ng thng v theo .
Cu V:(THPT:2,0 im; GDTX:3,0 im) Cho l ba s dng. Tm gi tr ln nht ca biu thc:
Cu VI:(THPT:2,0 im) Cho dy s c xc nh: .
Xt dy s . Tm .
------------------HT------------------
Th sinh khng c s dng ti liu. Gim th khng gii thch g thm.
Lu : i vi th sinh hc ti cc trung tm GDTX th khng lm cu VI.S GIO DC V O TO
HNG DN CHM THI CHN HC SINH GII BNH PHC
CP TNH THPT NM HC 2013 2014
(Hng dn chm c 06 trang)
MN: TON
Ngy thi: 03/10/2013I VI TH SINH THPTCuLi giiim
I1
Cho hm s: . Kho st s bin thin v v th (C) ca hm s .
2,0
TX:
0,25
( phng trnh ng TCN: y = 2
( phng trnh ng TC: x = 20,5
( Hm s nghch bin trn tng khong xc nh.
Hm s khng c cc tr.0,5
Bng bin thin:
0,25
Giao im vi trc tung: A(0; 3/2)
Giao im vi trc honh: B(3/2;0)0,25
th:
0,25
2Vit phng trnh tip tuyn ca (C), bit tip tuyn ct ng tim cn ng v tim cn ngang ln lt ti A, B sao cho , vi I(2;2).2,0
Gi
PTTT ca (C) ti M:
0,5
Do v tam gic AIB vung ti I ( IA = IB nn h s gc ca tip tuyn k = 1 hoc k = -1. v nn ta c h s gc tip tuyn k = -1.0,5
0,5
( c hai phng trnh tip tuyn:
;
0,5
II1Gii h phng trnh:
2,5
k:
0,5
Pt(2)
1,0
Pt(1)
1,25
H cho tng ng:
Vy h phng trnh c 2 nghim:
0,75
2Gii phng trnh:
2,5
k: (*)0,5
Pt tng ng:
0,75
0,75
Nghim tha mn (*)Phng trnh c 2 h nghim:
0,5
III1Trong mt phng vi h trc ta , cho hnh ch nht c , im thuc vo ng thng c phng trnh: . ng thng i qua v trung im ca on c phng trnh: . Tm ta ca v , bit im c honh dng.2,0
Gi , M l trung im AB, I l giao im ca AC v d2: 3x 4y 23 = 0.Ta c ng dng
0,5
M nn ta c:
Vy C(1;5).0,5
Ta c:
0,5
Do
0,5
2Cho tam gic nhn ni tip ng trn . Gi ln lt l cc im di ng trn cung nh , sao cho thng hng. Gi , ln lt l hnh chiu vung gc ca ln cc ng thng tng ng v ln lt l hnh chiu vung gc ca ln cc ng thng . Gi l giao im ca hai ng thng v . Tm gi tr ln nht ca din tch tam gic (theo ).
2,0
Chng minh gc
K KH vung gc vi BC (H thuc BC), ta c:
(t gic PEBD ni tip)
Suy ra:
Tng t, ta chng minh c:
Vy (do PQ l ng knh)0,5
Chng minh :Tht vy, xt hnh thang vung vung ti D v D nn , du = xy ra khi
0,5
Xt tam gic . Ta c:
Vy din tch ln nht ca tam gic bng khi
1,0
IV1Cho hnh chp c y l hnh ch nht, tam gic u cnh v nm trong mt phng vung gc vi y. Gc gia mt phng v mt phng y bng .
1. Tnh th tch khi chp theo .1,5
H, M ln lt l trung im ca AB v CD
Ta c:
0,5
Gc gia (SCD) v mt y l
0,25
Ta c
0,25
0,5
22. Tnh khong cch gia hai ng thng v theo .1,5
K ng thng d i qua A v d//BD. Trong mt phng (ABCD) k ng thng ( i qua H , ( ( d v ( ct d ti J, ( ct BD ti I. trong (SHI) k HK vung gc vi SI ti K.Khi :
0,5
Ta c ng dng
EMBED Equation.DSMT4 0,5
Xt vung ti H, ta c:
Vy
0,5
VCho l ba s dung. Tm gi tr ln nht ca biu thc:
2,0
0,75
0,75
Vy
= vi
0,75
Vy gi tr ln nht ca khi
0,75
VICho dy s uc xc nh: .
Xt dy s . Tm .
2,0
Ta c .
Khi :
t . Khi ta c dy mi c xc nh bi:
0,25
Chng minh l dy tng:Xt hiu:
Do nn suy ra dy l dy tng.0,25
Chng minh (xn) khng b chn hay :Gi s (xn) b chn, do dy tng v b chn nn tn ti gii hn hu hn.
Gi s dy (xn) c gii hn hu hn, t .
T cng thc truy hi
Ly gii hn hai v, ta c: (khng tha mn)
Do dy cho khng c gii hn hu hn.0,5
Ta c:
EMBED Equation.DSMT4 M:
0,5
Do , ta c:
M nn
0,5
Ch : Nu th sinh lm cch khc m ng th vn chm im ti a.S GIO DC V O TO
HNG DN CHM THI CHN HC SINH GII
BNH PHC
CP TNH THPT NM HC 2013 2014
(Hng dn chm c 06 trang)
MN: TON
Ngy thi: 03/10/2013I VI TH SINH HC TI CC TRUNG TM GDTXCuLi giiim
I1
Cho hm s: . Kho st s bin thin v v th (C) ca hm s .
2,0
TX:
0,25
( phng trnh ng TCN: y = 2
( phng trnh ng TC: x = 20,5
( Hm s nghch bin trn tng khong xc nh.
Hm s khng c cc tr.0,5
Bng bin thin:
0,25
Giao im vi trc tung: A(0; 3/2)
Giao im vi trc honh: B(3/2;0)0,25
th:
0,25
2Vit phng trnh tip tuyn ca (C), bit tip tuyn ct ng tim cn ng v tim cn ngang ln lt ti A, B sao cho , vi I(2;2).2,0
Gi
PTTT ca (C) ti M:
0,5
Do v tam gic AIB vung ti I ( IA = IB nn h s gc ca tip tuyn k = 1 hoc k = -1. v nn ta c h s gc tip tuyn k = -1.0,5
0,5
( c hai phng trnh tip tuyn:
;
0,5
II1Gii h phng trnh:
3,5
k:
0,5
Pt(2)
1,0
Pt(1)
1,25
H cho tng ng:
Vy h phng trnh c 2 nghim:
0,75
2Gii phng trnh:
2,5
k: (*)0,5
Pt tng ng:
0,75
0,75
Nghim tha mn (*)Phng trnh c 2 h nghim:
0,5
III1Trong mt phng vi h trc ta , cho hnh ch nht c , im thuc vo ng thng c phng trnh: . ng thng i qua v trung im ca on c phng trnh: . Tm ta ca v , bit im c honh dng.2,0
Gi , M l trung im AB, I l giao im ca AC v d2: 3x 4y 23 = 0.
Ta c ng dng
0,5
M nn ta c:
Vy C(1;5).0,5
Ta c:
0,5
Do
0,5
2Cho tam gic nhn ni tip ng trn . Gi ln lt l cc im di ng trn cung nh , sao cho thng hng. Gi , ln lt l hnh chiu vung gc ca ln cc ng thng tng ng v ln lt l hnh chiu vung gc ca ln cc ng thng . Gi l giao im ca hai ng thng v . Tm gi tr ln nht ca din tch tam gic (theo ).
2,0
Chng minh gc
K KH vung gc vi BC (H thuc BC), ta c:
(t gic PEBD ni tip)
Suy ra:
Tng t, ta chng minh c:
Vy (do PQ l ng knh)0,5
Chng minh :
Tht vy, xt hnh thang vung vung ti D v D nn , du = xy ra khi
0,5
Xt tam gic . Ta c:
Vy din tch ln nht ca tam gic bng khi
1,0
IV1Cho hnh chp c y l hnh ch nht, tam gic u cnh v nm trong mt phng vung gc vi y. Gc gia mt phng v mt phng y bng .
3. Tnh th tch khi chp theo .1,5
H, M ln lt l trung im ca AB v CD
Ta c:
0,5
Gc gia (SCD) v mt y l
0,25
Ta c
0,25
0,5
24. Tnh khong cch gia hai ng thng v theo .1,5
K ng thng d i qua A v d//BD. Trong mt phng (ABCD) k ng thng ( i qua H , ( ( d v ( ct d ti J, ( ct BD ti I. trong (SHI) k HK vung gc vi SI ti K.
Khi :
0,5
Ta c ng dng
EMBED Equation.DSMT4 0,5
Xt vung ti H, ta c:
Vy
0,5
VCho l ba s dung. Tm gi tr ln nht ca biu thc:
3,0
0,75
0,75
Vy
= vi
0,75
Vy gi tr ln nht ca khi
0,75
Ch : Nu th sinh lm cch khc m ng th vn chm im ti a.
0
0
1/4
-
+
0
4
+(
1
t
f(t)
f(t)
0
0
1/4
-
+
0
4
+(
1
t
f(t)
f(t)
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