c Tren nijfa mat phing bd OM c6 chi^a diem A, ve nuTa du-dng tron diTcJng kinh MF; nufa diTdng tr6n nay cat tiep tuyen tai E cua O d K.. Trung, Nam mOn ToAn _ Nguyin DiJfc Ta'n d Goi P v
Trang 1NGUYEN DlICT/iN(Chubien) NGUYEN ANHHOANG-NGUYEN DOANVU NGUYEN DlIC HOA - DO QUANG THANH - NGUYEN TH| TRINH
Trang 2LUYEN GIAI DE TRl/dC KY THI VAO L6P 10
BA MIEN BAC - TRUNG - NAM MON TOAN
NGUYIN D C T C T A N Chiu trach nhi^m xua't ban
N G U Y E N T H I T H A N H H l / O N G
B i c n tap Sura ban in Trinh bay Bia
Q U O C N H A N
H O A N G N H L T T
C o n g ty K H A N G V I E T
C o n g ty K H A N G V I ^ T
NHA XUAT BAN TONG HOP TP HO CHf MINH
NHA SACH TONG H0P
62 Nguyen Thj Minh Khai, Q l
D T : 38225340 - 38296764 - 38247225
Fax: 84.8.38222726
E m a i l : tonghop@nxbhcm.com.vn Website: www.nxbhcm.com.vn/ www.fidltour.com
Tong phat hdnh
C O N G TY T N H H M T V D|CH V y V A N H O A K H A N G V I | T
Oja Chi: 71 Dinh Tien Hoang - P.Oa Kao - Q.1 - TP.HCM
Quyet dinh xuat W n so: 1 5 2 / Q D - T H T P H C M - 2 0 1 3 do NXB Tor
Thanh Pho' Ho Chi M i n h cap ngay 07/03/201 3
In xong va npp ILAJ chieu O u v II nAm 201 3
tdi N 6 I D A U
Quyen sach Luyen giai de trUdc ky thi vdo Idp 10 ba mien Bac, Trung, Nam
mon Todn nham gop vao tu sach cua ban doc mot tai Heu toan thie't thiTc va bo
ich giup cac em hoc sinh Idp 9 on luyen va nang cao k i e n thtfc todn, chuan h i tot trong cac k i thi tuyen sinh vao Idp 10
Quyen sach gom 2 phan :
P h i n 1 : Cac de thi toan
• A D e thi tuyen sinh vao Idp 10 THPT \
• B D e thi tuyen sinh vao idp 10 chuyen
Bao gom de thi va hiTdng dan giai diTdc tuyen chon tif cac de thi tuyen sinh vao Iclip 10 cf mot so dia phu^dng (tCf nam 2010 den 2013)
P h ^ n 2 : Cac de toan on luyen
• A D e toan on luyen thi vao Idp 10 THPT
• B D e toan on luyen thi vao Idp 10 chuyen
Bao g o m 20 de toan va hiTdng dan giai do chiing toi bien soan v d i nhieu dang toan khac nhau nh^m trd giup cac e m hoc sinh ciing co, boi di/dng nang cao kien thufc toan
Chung toi da co gang t i m Idi giai m d i , hay va ngan gon cho cac bai loan va sau Idi giai cijfa m o i bai toan d c u co nhan xet va binh luan, v d i mong muon giup cdc em hoc sinh nam dUdc phUdng phap giai dang toan do, t i m k i e m bai toan tU'dng tif, b ^ i toan m d i , bai toan tong quat nhSm khdi day tiem nang tim t o i sang tao trong hoc toan 5 hoc sinh
Mac dil chung t o i da co gang rat nhieu trong bien soan, song chac han quyen sdch van con thicu sot Raft mong nhan diTdc cac y k i e n dong gop tijf ban doc de cdc Ian in sau sach dU'dc hoan thien hdn ;
X i n tran trong cam dn
Trang 3P H A N I C A C D E T H I T O A N
A Dfe THI TUYEN SINH v A O L6F 10 THPT
m so 1: De thi tuyen sinh vao Idp 10 THPT, Tp Ho Chi Minh nSm 2012 - 2013 3
D6 so 2: De thi tuyen sinh vao Idp 10 THPT, Tp Ha Npi nam 2012 - 2013 8
De s6'3: De thi tuyen sinh vao Idp 10 THPT, Tinh Dong Nai nam 2012 - 2013 13
De so 4: De thi tuyen sinh vao Idp 10 THPT, Tp Da Nang nam 2012 - 2013 16
so 5: De thi tuyen sinh vao Idp 10 THPT, Tinh Thi^a Thien Hue nam hoc
2012- 2013 19
B6 SO 6: De thi tuyen sinh vao Idp 10 THPT, Tp Can Thd nSm 2012 - 2013 24
De so 7: De thi tuyen sinh vao Idp 10 THPT, Tinh Hai Phong nam 2012 - 2013 28
D6 so 8: De thi tuyen sinh vao Idp 10 THPT, Tinh Nghe An nam 2012 - 2013 32
De so 9: De thi tuyen sinh vao Idp 10 THPT, Tinh Quang Ninh nam 2012- 2013 37
D6 so 10: De thi tuyen sinh vao Idp 10 THPT, Tinh Thanh Hda nSm 2012 - 2013 40
De s6'll: De thi tuyen sinh vao idp 10 THPT, Tinh Yen Bai nam 2012- 2013 44
D6 so 12: De thi tuyen sinh vao Idp 10 THPT, Tinh Ha Nam nam 2012- 2013 48
D6 so' 13: De thi tuyen sinh vao Idp 10 THPT, Tinh VTnh Phuc nam 2012 - 2013 51
De so 14: De thi tuyen sinh vao Idp 10 THPT, Tinh Dak Lak nam 2012 - 2013 54
De so 15: De thi tuyen sinh vao Idp 10 THPT, Tinh Tuyen Quang nam 2012 - 2013 58
D6 so 16: De thi tuyen sinh vao Idp 10 THPT, Tp Ho Chi Minh nam 2011 - 2012 61
De so 17: De thi tuyen sinh Idp 10 THPT, tinh Quang Nam nam 2011 - 2012 66
De so 18: De thi tuyen sinh vao Idp 10 THPT tinh Daklak nam 2011 - 2012 69
De so 19: De thi tuyen sinh vao Idp 10 THPT, tinh Ninh Thuan nam 2011 - 2012 72
D6 so 20: Dc thi tuyen sinh vao Idp 10 THPT, tinh Ha Tinh nam 2011 - 2012 75
De so 21: De thi tuyen sinh idp 10 THPT, tinh Thanh Hda nam 2011 - 2012 79
De so 22: Dc thi tuyen sinh vao Idp 10 THPT, tinh Kicn Giang nam 2011 - 2012 82
De so 23: De thi tuyen sinh vao Idp 10 THPT, tinh Khanh Hoa nam 2011 - 2012 85
D6 so 24: De thi tuyen sinh Idp 10 THPT, tinh Binh Dmh nam 2011 - 2012 88
D^ so 25: De thi tuyen sinh Idp 10 THPT, tinh Quang Ngai nam 2011 - 2012 92
D^' s6'26: De thi tuyen sinh vao Idp 10 THPT, Tp Da Nang nam 2011 - 2012 96
f)c so 27: Dc thi tuyen sinh vao idp 10 THPT, Tp Ha Noi nam 2011 - 2012 99
D^ so 28: De thi tuyen sinh vao Idp 10 THPT, tinh Quang Tri nam 2011 - 2012 103
De so 29: Dc thi tuyen sinh vao Idp 10 THPT tinh Nghp An nam 2011 - 2012 106
De s6'30: De thi tuyen sinh Idp 10 THPT, tinh Ninh Binh nam 2011 - 2012 110
D6 SO 31: De thi tuyen sinh Idp 10 THPT, tinh Hai DiTdng nam 2011 - 2012 114
D6 so 32: Dc thi tuyen sinh Idp 10 THPT, tinh Lang Sdn nam 2011 - 2012 118
Dd so'33: D^ thi tuycn sinh Idp 10 THPT, Tp.HCM, nam 2010 - 2011 121
Dc so 34: De thi tuycn sinh vao Idp 10 THPT, tinh Bac Lieu nam 2010 - 2011 127
De so 35: De thi tuycn sinh vao Idp 10 THPT, tinh Ba Ria - Vung Tau nam hoc
2010- 2011 129
B Dfe THI TUY^N SINH VAO L 6 P lO CHUYftN
De so' 36: De thi tuyen sinh vao Idp 10 chuyen Toan, Tp Ho Chi Minh nSm hoc
2012- 2013 132
D^ s6' 37: De thi tuyen sinh vao Idp 10 chuyen TriTdng Dai Hoc SuT Pham, Tp Ho Chi
Minh nam hoc 2012 - 2013 135
D^ so' 38: De thi tuyen sinh vao Idp 10 chuyen Toan Trifdng Dai Hoc Sif Pham, Tp Ho
Chi Minh nam hoc 2012 - 2013 141
D^' so'39: De thi tuyen sinh vao Idp 10 chuyen TriTdng phd thong Nang khie'u, DHQG
Tp.Ho Chi Minh nam hoc 2012 - 2013 146
so'40: De thi tuyen sinh vao Idp 10 chuyen TriTdng phd thong Nang khie'u, DHQG
Tp.Ho Chi Minh nam hoc 2012 - 2013 151
B6 S6'41: DC thi tuyen sinh vao Idp 10 Chuyen TriTdng THPT Dai Hoc Sir Pham Ha Noi
nam hoc 2012 - 2013 156
De s(")'42: Dc thi tuyen sinh vao Idp 10 Chuyen TriTdng THPT Dai Hoc SuT Pham Ha Noi
nam hoc 2012-2013 161
Bi so 43: De thi tuyen sinh vao Idp 10 Chuyen TriTcfng THPT Chuyen Khoa Hoc TiT
Nhien, Dai Hoc Qudc Gia Ha Npi nam hoc 2012 - 2013 165
D^ so' 44: De thi tuycn sinh vao Idp 10 Chuyen Trrfdng THPT Chuyen Khoa Hoc TiT
Nhien, Dai Hoc Qudc Gia Ha Npi nam hoc 2012 - 2013 170
so'45: De thi tuyen sinh vao Idp 10 chuyen, Tinh Dong Nai nam hoc
D6 SO' 50: De thi tuyen sinh vao Idp 10 THPT Chuyen Toan, Trifdng THPT, Tinh Hai
Du'dng nam hoc 2012 - 2013 196
Dl so' 51: Dc thi tuyen sinh vao Idp 10 THPT Chuyen, Tinh Hoa Binh nam hoc
2012- 2013 200
DC ,s6' 52: De thi tuyen sinh vao Idp 10 THPT Chuyen Todn THPT, Chuyen Phan Bpi
Chau, Tinh Nghe An nam hoc 2012 - 2013 204
t)6 s6'53: De thi tuyen sinh vao Idp 10 THPT Chuyen Lam Sdn, Tinh Thanh Hda nam
hoc 2012- 2013 209
D^ s6' 54: De thi tuye'n sinh vao Idp 10 THPT Chuyen Toan, Tinh Ba Ria - Vung Tau
nam hoc 2012- 2013 213
Dd s6' 55: Dc thi tuyen sinh vao Idp 10 THPT Chuyen Toan, TriTdng THPT Chuyen
Phan Boi Chau, Tinh Ba Rja - Vung Tau nam hoc 2012 - 2013 218
Trang 4s6' 56: De thi tuyen sinh vao Idp 10 chuyen, tru'dng Dai hoc Su pham Tp Ho Chi
Minh nam hoc 2011 - 2012 223
De so' 57: De thi tuyen sinh vao Idp 10 chuyen toan, trifdng Dai hoc Sir pham Tp Ho
Chi Minh nam hoc 2011 - 2012 227
De so'58: De thi tuyen sinh vao Idp 10 chuyen toan, THPT chuyen, Tp Ho Chi Minh
nam hoc 2011 - 2012 232
D6 so'59: De thi tuyen sinh vao Idp 10 chuyen, tru-dng THPT chuyen, Dai hoc Sir pham
Ha Noi nam hoc 2011 - 2012 237
D6 so' 60: De thi tuyen sinh vao Idp 10 chuyen toan, triTdng THPT chuyen Dai hoc Sir
pham Ha Noi nam hoc 2011 - 2012 242
Dc s6' 61: De thi tuyen sinh vao Idp 10 chuyen toan, THPT tinh Binh Dinh nam hoc
2011 - 2012 245
De so'62: De thi tuyen sinh vao Idp 10 chuyen, trirdng pho thong nang khie'u, Dai hoc
Quoc gia, Tp.HCM nam hoc 2011 - 2012 250
so'63: De thi tuyen sinh vao Idp 10 chuyen toan, trU'dng pho thong nang khie'u, Dai
hoc Quoc gia, Tp.HCM ndm hoc 2011 - 2012 255
D(1 so'64: De thi tuyen sinh vao Idp 10 chuyen, triTdng THPT chuyen KHTN, DHKHTN,
HiQG Ha Noi nam hoc 2011 - 2012 260
l.e d' 65: De thi tuyen sinh vao Idp 10 chuyen toan, triTdng THPT chuyen KHTN,
DHKHTN, DHQG Ha Npi nam hoc 2011 - 2012 264
D6 so 66: De thi tuyen sinh vao Idp 10 chuyen toan, THPT chuyen, Tp Ho Chi Minh
s6'69: De thi tuyen sinh vao Idp 10 chuyen, trirdng pho thong nang khie'u, Dai hoc
Quoc gia, Tp.HCM nam hoc 2010 - 2011 284
D^' s6' 70: De thi tuyen sinh vao Idp 10 chuyen, triTdng pho thong nang khie'u, Dai hoc
Quoc gia, Tp.HCM nam hoc 2010 - 2011 289
P H A N II C A C DE T O A N O N L U Y | N
A De toan on luyen thi vao Idp 10 THPT 295
B De toan on luyen thi vao Idp 10 chuyen 315
i';;" - i t ' / /
Cty TNHH MTV DWH Khang Viet
a) Ve do thj (P) cija ham so y = ^x^ va di/dng thang (D): y = ~ + 2 tren
cting mot he triic toa do
b) Tim toa do cac giao diem cua (P) va (D) d cau tren b^ng phep tinh
Bai 3: (1,5 diem) Thu gon cac bieu thiJc sau :
a) A = ^ + — vdi x + Vx x - 1 x - V ^ l o t - , X > 0; X ^ 1 , b) B = (2 - V3)726TT5^ - (2 + ^/3)^/26 - 15^3
Bai 4: (1,5 diem) Cho phiTcJng trinh: x^ - 2mx +m - 2 = 0 (x la an so)
a) ChtJng minh rhng phtTdng trinh luon c6 nghiem phan biet vdi moi m
b) Goi X,, X2 la cac nghiem ciia phiTdng trinh ,
Tim m de bieu thiJc M = — dat gia tri nho nha't
X| + X2 - 6X|X2
Bai 5: (3,5 diem) Cho du-dng Iron (O) c6 tam O va diem M n^m ngoai diTdng
tron (O) Dirdng thang MO c^t (O) tai E va F (ME < MF) Ve cat tuyen MAB
va tiep tuyen MC cija (O) (C la tiep diem, A nam giffa hai diem M va B A
va C nKm khac phia doi vdi dirdng thing MO)
a) ChiJng minh r5ng: MA.MB = ME MF ' b) Goi H la binh chieu vuong goc ciia diem C len difdng thing MO ChiJng minh tiJ giac AHOB noi tiep
c) Tren nijfa mat phing bd OM c6 chi^a diem A, ve nuTa du-dng tron diTcJng kinh MF; nufa diTdng tr6n nay cat tiep tuyen tai E cua (O) d K Goi S la giao diem cua hai dudng thang CO va KF Chtfng minh r^ng diTdng thing MS vuong
goc vdi dufcfng thing KC
Chi?ng minh r^ng diTcfng thing MS vuong g6c vdi di/c(ng thing KC
Trang 5Luygn giai 66 truOc kl thi v^o Idp 10 ba mign BSc Trung, Nam mOn ToAn _ Nguyin DiJfc Ta'n
d) Goi P va Q Ian Itfdt la tam diTdng tron ngoai tiep cac tam giac EFS va ABS
va T la trung diem cua KS ChiJug minh ba diem P, Q, T thang hang
Hl/dNG DAN GIAI Bai 1
He phiTdng trinh c6 nghiem (x; y) la (2; -1)
c) Dat y = x^(y > 0) PhiTdng trinh trd thanh: y^ + y - 12 = 0
2) Ban doc hay giai he phi/dng trinh bKng phiTdng phdp the
3) PhiTdng trinh dang ax" + bx^ + c = 0, dat y = x^ ta diTdc phiTdng trinh bac hai
ay^ + by + c = 0 Giai tim y roi tim x
Vay (P) va (D) c^t nhau tai hai diem phan biet A(2; 1) va B(-4; 4)
Nhan xet: De tim tpa do giao diem ciia (D) va (P) bang phep tinh ta lap
phiTdng trinh hoanh do giao diem cua (D) va (P): -x^ = — + 2
4 2 Nghiem cua phu^dng trinh la hoanh dp cua cac giao diem *'
Trang 6Luygn giSi dg truflc ki thi vao I6p 10 ba mign BSc, Trung, Nam man ToAn _ Nguygn Pile Tgn
b) Theo he thiJc Vi-et ta c6: x, + XT = 2m, X | X 2 = m - 2
- 2 4 - 2 4
Do do M =
X | + X 2 - 6 x , X 2 ( X 1 + X 2 ) - 2 X , X 2 - 6 X | X 2
- 2 4 - 2 4 - 6 ( X | + X j ) ^ - 8 x , X 2 ( 2 m ) ^ - 8 ( m - 2 ) m^ - 2m + 4
b) X e t A M C A va A M B C cd
C M A (chung)
M C A = M B C (He qua gdc tao b d i tia tiep tuyen
M C 1 OC ( V i M C la tiep tuyen cua (O))
A M C O vuong tai C; C H la diTdng cao => MC^ = M H M O
c) Ta cd M K F = 9 0 " (Gdc noi tiep ch^n nuTa difdng tron, A M K F vuong tai K,
K E la dirdng cao => MK^ = M E M F
Ta cd MC^ = M A M B = M E M F = MK^ ^ M C = M K
X e t A K M S ( M K S = 9 0 " ) va ACMS ( M C S = 9 0 " ) cd: •
M K = M C , MS (canh chung)
Do dd A K M S = ACMS (canh huycn - canh gdc vuong) ^j^,,,
=> MS la dudng trung trifc cua KC
Vay M S I KC
d) • Goi I la giao d i e m cua MS va K C fij? /((.ff ; ;;
T a c d SIK = 9 0 " i -f>^-'- • ^ • ' ' '
Trang 7Luyjn giai 66 truflc k1 thi vAo I6p 10 ba mign BJc, Trung, Nam mOn ToAn _ NguySn Pile TSn
AISK vuong tai I, IT la diTdng trung tuyen => TS = TI
AMSC vuong tai C, CI la diTdng cao => MC^ = MI.MS
Ta CO MI.MS = MC^ = MA.MB
Xet AMAI va AMSB c6 AMI (chung), — = ^ (vi MI.MS = MA.MB)
MB MS
Do do AMAI ^ AMSB (c.g.c) MIA = MBS => Tu" giac ABSI noi tiep
Ta c6n CO MI.MS = ME.MF (= MA.MB) /! A ME MI
Xet AMEI va AMSF c6 EMI (chung), = (MI.MS = ME.MF) MS MF
Do do AMEI AMSF (c.g.c) => MEI = MSF => TuT giac EFSI noi tiep
Hai dirdng tron (ABSI) va (EFSI) cat nhau tai S va I c6 tarn Ian lifdt la Q, P
=> PQ la du-dng trung triTc cua doan thang SI Ma T thuoc diTdng trung tri/c
cua doan thang SI (Vi TS = TI) nen T e PQ
Vay P, Q, T th^ng hang
Nhan xet
Cau a), b) quen thupc, cau c) ne'u nhan ra
MC^ = MA.MB = ME.MF = MK' => MC = MF giijp den vdi Idi giai Cau d)
kho, chi ve cac tam P, Q khong nen ve cac du'dng tron (P), (Q) se rac roi tren
hinh ve De dang thay PQ la du'dng trung triTc cua doan thang SI, tim each
chiJng minh TS = TI
KY THI TUYEN SINH VAO LdP 10 THPT, TP.HA NOI
3) Vdi cdc cua bieu thiJc A v^ B n6i tren, hay tim cdc gia tri cua x nguyen de
gia tri cua bieu thuTc B(A - I) la so nguyen
Bai 2 (2,0 diem) Giai b^i toan sau bang each lap phiTcJng tnnh hoSc he phiTcfng tiinh:
12 Hai ngifdi cilng lam chung mot cong viec trong — gid thi xong Neu moi
ngirdi lam mot minh thi ngiTdi thuf nhat hoan thanh cong viec trong it hcfn
ngirdi thu" hai la 2 gict Hoi neu lam mot minh thi moi ngiTdi phai lam trong
bao nhieu thcfi gian de xong cong vi^c?
Cty TNHH MTV DWH Khang Vigt
Bai 3 (1,5 diem)
1) Giai he phu'Png trinh X y
X y 2) Cho phi/Png trinh: x^ - (4m - l)x + 3m^ - 2m = 0 (an x) Tim m de phiTdng
trinh c6 hai ngiem phan biet Xi, X2 thoa man dieu kien: + xj =1
Bai 4 (3,5 diem)
Cho dircfng tron (O; R) c6 du-cfng kinh AB Ban kinh CO vuong g6c vdi AB,
M la mot diem bat ki tren cung nho AC (M khac A, C): BM cat AC tai H
Gpi K la hinh chie'u cua H tren AB
1) Chu-ng minh CBKH la tuT giac noi tiep 2) Chu-ng minh ACM = ACK
3) Tren doan thang BM lay diem E sao cho BE = AM Chitng minh tam gidc
ECM la tam giac vuong can tai C
4) Gpi d la tiep tuyen cua (O) tai diem A; cho P la diem nam tren d sao cho hai diem P, C nam trong cdng mot nu-a mat phang bd AB va "^^'^^ = R
MA ChiJng minh du-dng thang PB di qua trung diem cua doan thang HK
Bai 5 (0,5 diem) Vdi x, y la cac so du-dng thoa man dieu kien x > 2y, tim gia tri
„ 2 2
nho nha't cua bieu ihuTc: M = i -
xy
Hl/dfNG D A N GIAI ' Bail, '
N /36 + 2 6 + 2
X + 16 _ >/x - 4) + ( N / ^ + 4)
, V x + 4 V x - 4 j " V x + 2 ( 7 x + 4 ) ( V x + 4 ) _ X - 4 V 2 + 4 V x + 1 6 V x + 2 x + 16Vx + 2 V x + 2
Trang 8Luygn dS trudc ki thi vao I6p 10 ba mjgn B&c Trung Mam mSn Toan , Mguygn Difc Tan
Do vay x = 14; 15; 17; 18 la cac gia t r i nguyen cua x can t i m
N h a n xet: D a y la cac bai toan de, quen thuoc
B a i 2 G o i thdi gian ngiTdi thuT nhat l a m mot m i n h xong cong viec la x (gicJ)
12 ( D i e u k i e n x > — )
T h d i gian ngiTcfi thiJ hai l a m mot m i n h xong cong viec la x + 2 ( g i d )
1
" T r o n g 1 g i d , ngiTdi thiJ nhat l a m diTdc: 1 : x = - (cong v i e c )
T r o n g 1 g i d , ngiTdi thiJ hai l a m diTdc: 1 : (x + 2) = (cong v i e c )
X ~i~ ^
(4
H T r o n g 1 g i d , hai ngiTdi l a m chung diTdc:
- + — i — (cong vice) hay 1 : — = — (cong viec)
h V a y thdi gian ngiTdi thiJ nhat lam mot minh xong cong viec la 4 g i d
T h d i gian ngiTdi thi? hai l a m mot m i n h xong cong viec la: 4 + 2 = 6 ( g i d )
N h a n xet: D a y la dang b a i : G i a i bai toan bang each lap phtfdng trinh, b a i
tocin ve cong v i c e , rat qucn thuoc v d i m o i hoc sinh
B a i 3
f l O 1) X y <=> <
Trang 9Luyjn giai 6i frUSc ki thi vSo lOp 10 ba mi6n Ba Nam mfln ToAn _ Nguyjn DCfc Ta'n
Tir (1), (2), va (3) CO KH = 2NK Do do N la trung diem cua HK ,
Vay du'cing lhang PB di qua Irung diem ciia doan thang HK
Nhan xet: Day la bai loan rat quen ihuoc doi vdi moi hoc sinh Idp 9
Vay gia tri nho nhaft cua bieu thiifc M la ^
Nhan xet: Tijr dieu kien rang buoc x > 2y, cho ta dif doan rang M dat gia tri
nho nhat khi x = 2y
Tir do, giup dieu chinh he so thich hdp "x^" va "4y^" roi van dung ba't dang
thiJc Co-si cho hai so du'dng de giai nhiT tren
x^ + y^ 4x^ + 4y^
Thao tac "bien" — thanh — giup c6 Idi giai dep
xy 4xy Tir Idi giai nay cung cho ta Idi giai khdng van dung ba't d^ng thuTc Co-si cho
hai so du'dng nhu" sau:
x^ + y^ 4x^ + 4y^ 3x^ + (x^ - 4xy + 4y^) + 4xy
y = 3x2 ^j^ (p) y = 2x - 3 c6 do thi la (d); y = kx -1- n c6 do thi la (d,) vdi k va n la nhiJng so thifc
1) V e d o thi (P)
2) Tim k va n biet (d,) di qua diem T ( l ; 2) va (d,) // (d)
Cfiu 4 (1,5 diem) Mot thuTa dat hinh chff nhat c6 chu vi bang 198m, dien tich bang 2430m^ Tinh chieu dai va chieu rong cua thuTa da't hinh chff nhat da cho
Cfiu 5 (3,5 diem) Cho hinh vuong ABCD Lay diem E thuoc canh BC, vdi E khong trung B va E khong trung C Ve EF vuong goc vdi AE, vdi F thuoc
CD Dirdng thdng AF c^t diTdng thing BC tai G Ve diTdng thing a di qua diem A va vuong gdc vdi AE, di/dng thing a c i t diTdng thing DE tai diem H
i\ • u AE CD 1) Chffngminh =
AF DE 2) Chffng minh rang tff giac AEGH la ti? giac noi tiep diTdng tron
3) Goi b la tiep tuye'n cua du'dng tron ngoai tie'p tam giac A H E tai E, bie't b cat du'dng trung trffc cua doan thang EG tai diem K Chffng minh rang KG la tiep tuye'n cua du'dng tron ngoai tiep tam giac AHE
Hl/OfNG D A N G I A I
C f i u l
l ) 7 x ^ - 8 x - 9 = 0 A' = 16 + 63 = 79; VA^ = V79
PhiTdng trinh c6 hai nghiem phan biet x, = ^ , X j = - — z - ^
Trang 10N h a n xet: D a y cQng la b a i toan de, thi sinh co the sijf dung A = l + 4 = 5 > 0
de chiirng to phu'dng trinh co hai n g h i e m phan biet
N h a n xet: D a y la b a i toan do thj ham so ciing raft quen thupc
Cau 4 NiJfa chu vi ciia ihiira difl la: 198 : 2 = 99 (m)
99
G p i chieu rpng ciia thijfa d a l la: x(m) ( D i c u k i e n x < — )
si OX • t
C h i e u d a i cua thiJa dat la 99 - x ( m )
D i e n tich ciia thiira dat la x(99 - x ) ( m ' ) hay 2 4 3 0 n r Ta co phu^dng trinh x(99 - X) = 2430 X' - 99x + 2430 = 0
N h a n xet: D a y la b a i toan giai b a i loan bang each lap phu^Png trinh, loai
toan hinh hpc ra't de va quen thupc
Trang 11Luyjn giSi di tri/flc ki thi vao Iflp 10 ba miSn B^c Trung, Nam mOn To^n _ Nguygn Difc Ta'n
Ma OEK = 90" (b la ticp tuycn cua (O)) ncn OGK = 90"
KG ± OG va G thuoc dudng tron (O) (Vi tur giac AEH noi tiep difdng
tron (O)) Do do KG la tiep tuye'n cua difdng tron (O)
TiJc la KG la tiep tuyen cua du'dng tron ngoai tiep tarn giac AHE
Nhan xet: Bai todn hinh hoc cung quen thupc, de chtfng minh KG la tiep
tuye'n cua dufcJng tron (O), ne'u da den diTdc KG 1 OG can chiJng minh them
G thuoc diTdng tron (O)
2) Giai he phu'dng trinh:
D 6 S 6 4
f,, DE THI TUYEN SINK VAO LdP 10 THPT, TP.OA NANG
NAM HOC 2012 - 2013 Bai 1 (2,0 diem)
1) Giai phu'dng trinh: (x + l)(x + 2) = 0
2x + y -1
" x - 2 y = 7
Bai 2 (1,0 diem)
Rut gon bieu ihiJc A = (VlO - 7 2 ) ^ 3 + N/5
Bai 3 (1,5 diem) Bie't rang du'cfng
cong trong hinh ve ben la mot
parabol y = ax^
1) Tim he so a ' " ' ^ "
2) Goi M va N la cac giao
diem cua du'dng thang \
y = X + 4 vdi parabol
Tim toa do cua cAc diem M va N
Bai 4 (2,0 diem) Cho phiTdng trinh x^ - 2x - 3m^ = 0, vdi m la tham so
1) Giai phifdng trinh khi m = 1
2) Tim tat ca cac gia tri cua m de phiTdng trinh c6 hai nghiem Xi, X2 khdc 0
va thoa dieu kien ^ ^ =
-Bai 5 (3,5 diem) Cho hai du'dng tron (O) va (O') tie'p xuc ngoai tai A Ke tie'p
tuyen chung ngoai BC, B e (O), C e (O') DiTcfng thang BO c^t (O) tai diem
thtf hai la D
1) ChiJng minh r^ng tur gidc CO'OB 1^ mot hinh thang vuong
' Cty TNHH MTV DWH Khang Vi$t
2) Chitng minh rang ba diem A, C, D th^ng hang
3) Tir D ke ticp tuyen DE vdi diTcJng tron (O') (E la tiep diem) Chu-ng minh rang DB = DE
He phu'dng trinh c6 nghiem (x; y) duy nhat la (1; -3)
Nhan xet: Day la bai toan qua de Ba: 2 A = (VlO - V2)V3 + V5 = (V5 - l)^^^/3 + Vs
= ( V F - l ) V ^ T 2 V f = (V5-l)^(V5 + l f
= (>/5 - l)(V5 + l) = 5 - 1 = 4
Nhan xet: Bai nay qua de
Bai 3 1) parabol y = ax^ di qua diem (2; 2) o 2 = a.2^ o a = — o a =
Trang 12Luy$n giai dg trudc kl thi v^o Idp 10 ba mjgn BJc. Trung, Nam mOn ToAn _ Nguygn 0(!c IJn
Nhan xet: Day la bai loan ve ham so va do thi, dang toan nay cung ra'l quen
Nhan xet: Day la bai loan ve phiTdng trinh bac hai va uTng diing cua he thiJc
Vi-et, phat hicn X2 > 0 > X | giup c6 di/dc Idi giai cua bai toan
B a i 5 a ) B C la tiep tuyen chung
ngoai cua (O) (O') Do do BC 1
OB, BC 1 O'C =i> O'C // OB
TiJ giac CO'OB la hinh thang Ma
OBC = 90" (BC 1 OB) Do do tuf
giac CO'OB la hinh thang vuong
2) (O) va (O') tiep xuc ngoai tai A (gt)
= > 0 , O', A thang hang
Ta C O DOA = A O ^ (so le Irong va OB // O'C)
Cty TNHH MTV D W H Khang Vi$t
AOAD = C O OA ^ OD (= R) => AO AD can tai O
Ma 0 ' A = 0 ' C ( = r ) = > A O ' A C c a n l a i O '
Do vay AOAD ^ AO'AC => OAD = 0 \ ^ C SJ* '
Ta C O OAD + OAC = O ^ + OAC = 180" => Hai tia A D , AC doi nhau
Vay ba diem A, C, D ihiing hang
3) Ta C O B A D ^ 90" (goc noi tiep ch^n niJa diTdng tron)
ABDC vuong tai B, BA la du-dng cao => DB^ = DA.DC Xet ADEA va ADCE c6 EDA (chung), DEA = DCE (He qua goc tao bdi tia
tiep tuyen va day cung) Do do ADEA ^ ADCE (g.g) , ^ ,
^ ^ = ^ = ^ D E ^ = D A D C
DC DE
Ta C O DB^ = DE^ (= DA.DC) Vay DB = DE
Nhan xet: Day la bai toan hinh hoc de v^ quen thuoc
D 6 SO 5
DE THI TUYEN SINH VAO L(3P 10 THPT, TJNH THL/A THIEN HUE
NAM HOC 2012 - 2013 Bai 1 (2,0 diem)
a) Cho bid'u thiJc: C = ^ ^ ^ ^ + - (V5 + 3) ChiJng to C = V3
b) Giai phu^c^ng trinh: 3Vx - 2 - Vx^ - 4 = 0
Bai 2 (2,0 diem) Cho ham so y = x^ c6 do thj (P) va dKdng lhang (d) di qua
diem M( 1; 2) c6 he so goc k^O
a) Chijrng minh r3ng vdi moi gia tri k ^ 0 Du-clng thang (d) luon cat (P) tai hai
diem phan biet A va B
b) Goi X A va X B la hoanh 6o cua hai diem A va B Chij-ng minh rang
X A + X u - X A X B - 2 = 0
Bai 3 (2, 0 diem) < -
a) Mot xe lura di tir ga A den ga B Sau do 1 gid 40 phut, mot xe lu-a khac di tiT
ga B den ga A vdi van toe I6n hiln van toe cua xe lura thi? nha'l la 5km/h hai
xe lura gap nhau tai mot ga each ga B 300km Tim van toe cua moi xe, biet
r^ng quang diTcJng s^t tiT ga A den ga B dai 645km
•2(x + y) = 5(x - y)
20 20
b) Giai he phi/dng trinh: •
X + y x - y ;, ,^ , , , , ,
Trang 13vf,-Luy^n giai iSi truflc k1 thi v^o Iflp 10 ba miSn Bjlc, Trung, Nam mOn Toan _ NguySn Dijfc Tin
Bai 4 (3,0 d i e m ) C h o nufa diTcIng tron (O) diTdng kinh BC Lay d i e m A tren tia
doi cua tia CB Ke tiep tuyen AF vdi nuTa dir5ng lr6n (O) (F 1^ tie'p d i e m ) , tia
' AF c^t tia tiep tuyen Bx cua nuTa diTdng tron (O) tai D (tia tiep tuye'n Bx n a m
trong niJa mat phang bd BC chtfa niJa diTcJng tron (O)) Goi H la giao diem
cua BF vdi DO; K la giao diem ihuT hai cua DC vdi nuTa diTdng tr5n (O)
a) ChuTng minh rang: AO.AB = AF.AD
b) Churng minh tu" giac KHOC noi tiep
, BD DM c) Ke OM 1 BC (M thuoc doan thSng AD) Chtfng mmh
Bai 5 (1,0 diem) Cho hinh chu' nhat OABC,
COB = 30*' Goi CH la diTdng cao cua tam
giac CO, CH = 20cm Khi hinh chu" nhat OABC
quay quanh mot vong quanh canh OC co dinh
ta diTdc mot hinh tru, khi do tam giac OHC tao
lhanh hinh (H) Tinh the tich cua phan hinh try
nam ben ngoai hinh (H) (Cho n « 3,1416)
Hl/CfNG DAN GIAI Bai 1
X > 2
(X - 2)(x - 7) = 0
X - 2 > 0 9(x - 2) = x^ - 4
X > 2
x ^ - 2 x - 7 x + 14 = 0
x - 2 = 0
x - 7 = 0 X =X = 7 2
Vay nghiem cua phu'dng trinh la Xi = 2; Xa = 7
Nhan xet: Day la bai toan dc, cau b con c6 the giai nhiT sau:
= kx + 2 - k o x' - kx - 2 + k = 0 (*)
A = (-k}~ - 4(-2 + k) = k^ + 8 - 4k
= (k^ - 4k + 4) + 4 = (k - 2)^ + 4 > 0, vdi moi k
Do vay v(':ri moi gia tri k ;^ 0, (d) luon cat (P) tai hai diem phan biet A va B
b) Ta CO XA, XIJ la cac nghiem cua phiTcfng trmh (*) Theo he thiJc Vi-et ta co XA + X B = k, X A - X H = -2 + k
Do do X A + X B - X A X R - 2 = k - ( - 2 + k ) - 2 = k + 2 - k - 2 = 0
Nlian xet: Day la bai loan vc ham so va do thi kc't hdp vdi phu'dng trinh bac
hai mot an va he ihu'c Vi-ct, can ghi nhd rang: (d) cat (P) lai hai diem phan biet <=> Phifdng trinh hoiinh do giao diem co hai nghiem phan biet
Bai 3 Doi 1 gid 40 phut = ^ gid
Quang du-dng {ii ga A den cho hai xc luTa gSp nhau tcii ga each B 300km la:
645 - 300 = 345 (km) Goi van to'c cua xe lu^a thiJ nha't la x (km/gid) (Dieu kien x > 0) Van toe cua xc liJa thu' hai la x + 5 (km/gi(<) ^• ThcJi gian xe luTa thdr nhat da di la: 345 : x = 345
Thdi gian xe lufa thu" hai da di la: 300 : (x + 5) =
Trang 14Luy$n gi§i dg trutSc kl thi vio lOp 10 ba mjgn BSc Trung, Nam mOn ToAn _ Nguygn Difc Ta'n
A ' = 121 + 1035 = 1156, ^/A' = 34
X| = ^ 45 (thich hdp), x j = ^ ^ ^ "^"^ = - 2 3 (khong ihich hdp)
Vay van toe xc luTa thu" nha't (xe luTa khdi hanh tu" A) la 45 km/gid
V a n toe xe liJa thu- hai (xc lu'a khdi hanh tu" B) la:
= 7
70
<=> <
x + y 2.10 = 5(x + y)
X + y = 10
X = 7 (thich hdp) 2x = 14
X + y = 10 [x + y = 10 [y 3
Vay nghiem (x; y) cua he phu-cfng trinh la (7; 3)
N h a n x e t :
a) Giai bai toan bang each lap phiTcfng trinh loai toan chuyen dong, dang bai
toan nay cung rat thifdng xuift hicn trong cac de k i e m tra hoc k i hay thi vao
b) D B , D F lii cac tiep tuyen cua du'dng tron (O)
=i> DB = D F va DO la tia phan giac goc B D F
A D B F can tai D, DO la dUcfng phan giac ?, r
D O cung la du-ttng cao => B H 1 OD
A B O D vuong tai B, B H la diTdng cao => D B ' = D H D O "
-M a t khac B K C = 9 0 " (goc noi tiep chan nufa difdng tron) ABDC vuong tiii B, B K la difdng cao => DB^ = DK.DC
Ta CO D H D O - D K D C (= DB^)
D K D H Xet A D K H va A D O C co K D H (ehung), = (vi D H D O = D K D C )
D O DC
Do do A D K H ^ ADOC (c.g.c) => D H K = DCO Vay tu" giac K H O C noi tiep
Trang 15_ Nguyen uac lan
ACOB vuong tai C => CB = OCtanCOB = 40tan30" = ^ ^ y ^ (cm)
A K O H vuong tai K => K H = OHsin K O H = 2 0 V 3 i = 1 0 ^ (cm)
The tich hinh tru la: V, = n B C ^ O C = 7t
The tich h i n h ( H ) la:
N h a n x e t : Chia khoa bai loan la nhan ra khi quay hinh chOr nhat O A B C mot
vong quang canh C O thi tam giac O H C tao thanh hinh ( H ) g o m 2 hinh non up
vac nhau, cung ban kinh la H K , chicu cao la O K va OC; hinh chi? nhat
O A B C t c i o thanh hinh tru c6 ban kinh day la CB, chieu cao la OC
D E 8 0 6
DE THI TUYEN SINH VAO LdP 10 THPT, TP.CAN THd
NAM HOC 2012 - 2013 Cau 1 (2,0 diem)
Giai he phu'dng trinh, cac phu'rtng trinh sau day:
(vdi a > 0, a ;^ 1)
1) Rut gpn bieu thiJc K
2) T i m a de K = V2012
Cau3 (1,5 diem)
Cho phifdng trinh (an so x): x^ - 4x - m^ + 3 = 0 (*)
Cty TNHH MTV DVVH Khang Vijt
1) Chiang minh phu'ctng trinh (*) luon c6 hai nghiem phan bict v d i m o i
2) T i m gia tri ciia m de phiftJng trinh (*) c6 hai nghiem X|, X2 thoa X2 = - 5 X |
C j i u 4 ( l , 5 d i e m ) , i h X ^ ; ; i rns,,).,;;^ ,
M o t 6 to dii" dinh di tif A dc'n B each nhau 120km trong mot thdi gian quy dinh .Sail khi di difOc 1 g i d Ihl 6 to bi chan bdi xe cifii hoa 10 phut Do do de den B dung han xe phai tang van toe Ihem 6km/li Tinh van toe liic dau eiia 6 i6 Cau 5 (3,5 diem) Cho di/dng Iron (O), diem A d ngoai du'dng l i o n ve hai lie'p luyen A B va A C ( B , C lii cac iiep diem) O A cat BC l a i H
1) ChiJng minh lu" giac A B ( )C noi tiep r 2) ChiVng Miiiih BC vuong gdc \ d i O A va BA.BE = AIE.BO ,^
3) Goi I la triing diem ciia BE, difdng lhang qua I va vuong goc O I cat cae lia A B ,
AC theo thi'r liT lai D \ F Chifng minh I'lX) - iTcO va A D O F can lai O
4) ChiVng minh F la ining diem ciia A C
IMJ^NC; D A N G I A I Cau 1
|x + y = 43 [ 3 x - 2 y = 19 [ 3 x - 2 y = 19
fx - 2 1 ^x = 21 [ x - 2 1 3.2! - 2y - 19 [ - 2 y - - 4 4 ! y = 22 Vay nghiem ciia he phifdng Iririli la (x; y) = (21; 22)
•'5) X- - 12x + 36 = 0 <r> ( X - 6 ) ' = 0 o X = 6 Vay phiftJng trinh CO nghiem kep X = 6 4) B K X D x> 2 0 1 1
Ta CO x/x - 2011 + 74x - 8044 - 3 o N/X - 201 1 + ^ 4 ( x - 2011) = 1
o Vx - 2011 ^- 27x - 2T)TT - 3 o v/x - 20 iT = 1
Trang 16Luy$n giai d6 trudc kl thi vko I6p 10 ba mign B&c Trung, Nam mOn To^n _ NguySn DCfc Ta'n
1 > 0
<=> \ « X = 2012
X - 2 0 1 1 = r
Vay phuWng tnnh CO nghicm la X = 2012 y .' '.•'^^i^vv'.i';} : j
' Nhan xet: Day la cac bai loan d e , quen ihuoc ' '
Vay phiTOng trinh (*) luon c6 hai nghicm phan bict vdi moi m
2) Theo he thiJc Vi-ct, ta co:
X| + X2 = 4, X1X2 = -m"* + 3
Ma X2 = - 5 x i (gt) Do do - 5 x i + X| = 4 - 4 x i = 4 <=> X| = - 1
Ta CO X2 = -5xi = 5 Do vay -1.5 = - m ' + 3<=>m' = 8<=>m = ±2>y2
Vay m = 2%y2 hoac m = - 2 ^ 2
Nhan xet: V i tong hai nghicm la hang so (X| + X2 = 4) do vay nen kel hcfp
vdi he thiJc nghicm cho trufdtc (X2 = -5xi) de tim du'dc Xi, X2 roi tiT do nhanh
chong CO di/dc cac gia tn m can tim
C a u 4 1 0 p h u t = - g i d
6
Goi van toe luc dau cua 6 to la x (km/h) (Dicu kien x > 0)
Thdi gian 6 l6 diT dinh di tiT A den B la: 120 : x = — (h)
X
Sau 1 gi5 6 to di du'dc la x l = x (km)
Quang du-dng con lai dai la: 120 - x (km)
Van toe cua 6 to sau khi tang la: x + 6 (km/h)
Thcli gian 6 to di quang du'dng con lai \h:
Cty TNHH MTV DWH Khang Vi$t
( 1 2 0 - x ) : ( x + 6 ) = (h)
X + 6 Theo d a u b a i , ta c6 phu'Ong Irinh:
= 48 (nhan); X2 = - 2 1 - 6 9
= -90 (loai)
1 • - ^ Vay van toe luc dau cua 6 to la 48 km/h '
Nhan xet: Giai bai loan bang each lap phUdng Irinh, loai loan chuyen dong,
dang loan nay quen thuoc doi vdi hoe sinh \6p 9 Can chu y d6n vi phai
thong nha't
Cau 5 1) A B , AC la cac tiep tuyen cua
difcfng tron (O) (gt)
^ AB 1 OB, AC 1 OC
=> ABO = ACO = 9 0 "
=> Tu" giac ABOC noi tiep 2) A B , AC la cac tiep tuyen cua du'dng tron (O) (gt) =^ AB = AC, AO la tia phan giac cua goe BAC (tinh cha't cua hai tiep tuyen eat nhau) AABC can lai
A, AO la diTdng phan giac nen AO eung la du'dng cao cua lam giac ABC
=^ B C l OA
Xet AEAB va AEBO eo AEB = BEO (= 90")
Do do AEAB ^ AEBO (g.g)
Vay BA.BE = AE.BO
BAE = EBO (cung phu vdi goe ABE)
BA AE
BO BE
3) Ta CO OID = OBD = 90" => Tvt giac OIBD noi licp =>
Ma OB = OC (= R) =^ AOBC can lai O :=> BCO = IBO
IDO = I B O
Do vay DIO = BCO (= I B O )
Trang 17Luy$n giai 6i truflc k1 thi vgio lap 10 ba miSn BJc Trung, Nam mOn loin _ Nguygn DCic TSn
Mat khac OIF + OCF = 90" + 90" = 180"
Ttf giac lOCF noi tiep => BCO = IFO
Vay tarn giac DOF can tai O A
4) ADOF can tai O c6 01 la di/ctng cao ( 0 1 1 D F ) nen 01 cung la difcJng trung tuyen
Ti? giac BDEF co I la trung diem cua BE (gt) va I la trung diem cua DF
(chuTng minh trcn)
Do do BDEF la hinh binh hanh => EF // BD
Xet AABC CO E la trung diem cua BC (AABC can tai A diTdng cao AE cung
la dirdng trung tuyen) va EF // BD
Vay F la trung diem cua AC
Nhan xet: Day la bai loan de, quen thuoc
D E S67
D E T H I T U Y E N S I N H V A O L(3P 1 0 T H P T , T J N H H A I P H O N G
N A M H O C 2 0 1 2 - 2 0 1 3 Phan I Trac nghi^m (2,0 diem)
Hay chon chi 1 chuT cai diJng trifdc cau tra 15i dung
Cau 1 Dieu kien xac dinh cua bicu thuTc Vx - 1 la:
Cty TNHH MTV DWH Khang Vigt
Cfiu 6 Tam giac MNP vuong tai M biet M N = 3a, MP = 3^/3a Khi do tanP
Cfiu 8 Cho hinh chi? nhat c6 AB = 4cm, BC = 3cm Quay hinh chi? nhat do xung
quanh AB ta difcJc mot hinh tru The tich cua hinh tru do bkng:
b) Tim cac gia tri cua m de phiTdng trinh (1) c6 it nhat mot nghiem khong diTdng 3) Tim hai so biet tdng cua chiing b3ng 8 va so thu" nhat gap 3 Ian so thi? hai
Bai 3 (3,0 diem) Cho tam gidc ABC c6 ba goc nhon va AB = AC DiTdng tron
tarn O dirdng kinh AB = 2R c^t cdc canh BC, AC Ian lirdt tai I , K Tiep tuyen ciia dirdng trdn (O) tai B c^t A I tai D, H la giao diem cua A I va BK
a) Chu-ng minh tiJ giac IHKC noi tiep
b) Chu-ng minh BC la tia phan giac cua D B H va tiJ giac BDCH la hinh thoi c) Tinh dien Uch hinh thoi BDCH theo R trong tnrdng hdp tam giac ABC deu
Trang 18Luy^n giSi (SJ truflc k1 thi v^o I6p 10 ba m\in BJc, Trung, Nam mOn Toan _ NguySn Bijfe TSn
Vay phifdng trinh (1) luon c6 nghiem vcti moi m
b) De phifdng trinh (1) c6 it nha't mot nghiem khong diTcfng khi va chi khi
• PhiTdng trinh (1) c6 hai nghiem trdi da'u o P < 0 o - m - 1 < 0 o m > - 1
• PhiTdng trinh (1) c6 mot nghiem b^ng 0 1
3) Gpi so thi? nha't la x, so thi? hai la 8 - x
V i so thuf nha't gap 3 Ian so thiJ hai, nen ta c6 phiTdng trinh x = 3(8 - x)
So thiJ nha't la 6
So thiJ hai la 8 - 6 = 2
Nhan xet: Cau 1 va 3 de, cau 2 tim cac gia tri cua m de (1) co it nha't mot
nghiem khong diTOng ta tim m de co 1 nghiem bing 0, co hai nghiem am
Cty TNHH MTV^jwnjviiang VjQt
Bai 3 a) Ta co AIB = 90" (Goc noi tiep chan nijfa diTcJng iron)
• B K _ L A C iron) r:> BK _L AC => HKC = 90"
Ti? giac IHKC co HIC + HKC = I8O"
Do do tiJ giac IHKC npi tie'p b) Ta CO AABC can tai A (gt), A I la diTdng cao nen A I la tia phan giac cua goc BAG =^ B A I = l A C
Ma B A I = D B I (He qua tao bdi tia tie'p tuyen va day cung)
lAC = K B I (Hai goc npi tie'p cilng ch^n cung KI)
Do do D B I = K B I Vay BC la tia phan giac cua goc DBH ABHD co B I dUcfng phan giac va la du^dng cao nen tarn giac BHD can tai B => B I la di/dng trung tuyen => I H = ID
TiJ giac BDCH co I la trung diem cua BC, HD Nen tiJ giac BDCH la hinh binh hanh Ma HD 1 BC " * Vay tiJ giac BHCD la hinh thoi
c) AABC deu => AC = BC = AB = 2R, A B I = 60" , AIAB vuong tai I ^ A I = ABsinABI = 2Rsin60" = 2 R — = V3R " '
2
H la triTc tam cua tam giAc ABC (AI 1 BC, BK 1 AC) ma AABC deu
Trang 19LuyQn giai ai truflc kl thi vito Iflp 10 ba m\in BJc, Trung, Nam mOn Toan _ Nguygn Ditc Tan
1) Dung cac phcp bicn ddi tiTdng diTdng dc c6 Kli giai
2) Van dung cau 1) dc tim gia trj nho nhat cua A, lilf gia thic't 2x + 3y < 2 cho ta
dir doan A dat gid nho nhat khi 2x = 3y = 1 Do vay, ap dung bat d^ng thufc
Co-si cho hai so diTdng ta c6 2x + 3y > 2 72x.3y <=> 2x -i- 3y > 2 ,j6xy Day
cung la mot nut th^l cua bai toan
a) T i m dieu kicn xac de A xac djnh va rut gon A
b) T i m tat ca cac gia tri x de A > ^
Cty TNHH MTV DWH Khang Vigt c) T i m tat ca cac gia Iri cua x de B = —A dat gia tri nguyen
C&u 2. (1,5 diem) Quang dtfdng A B dai 156km M o t ngiTdi di xe may tCr A, mot
ngi/cti di xe dap tu" B Hai xe xuat p h a t ciing mot luc va sau 3 gicJ gap nhau Biet rang van toe cua ngU'di di xe may nhanh hcfn van toe cua ngtfdi di xe
dap l a 28km/h Tinh van toe cua moi xe
CSu 3 (2 diem) Cho phiTdng trinh: x^ - 2(m - l) x -i- m^ - 6 = 0 (m la tham so)
a ) Giai phU'dng t r i n h khi m = 3.
f-b) T i m m de phu'dng trinh c6 hai nghiem X|, X2 thoa man x^ + X j = 16
CSu 4 (4 diem) Cho diem M nam ngoai diTcfng tron tam O V e tiep tuyen M A , M B
vcfi dirdng tron (A, B l a cac tiep diem) Ve cat tuyen M C D khong di qua tam O (C n^m giOra M va D), O M c^t A B va (O) Ian liTdt tai H va I ChuTng minh
a) Tvt giac M A O B noi tiep
Trang 20Luy?n giai dg trmSc kl thi vao Idp 10 ba m'tin BSc, Trung, Nam mOn Toan _ NguySn Dijfc Tin
("au a), b) de, cau c) nham giup phan loai hoc sinh, chia khoa b a i toan la
nhan ra B > 0 va vi 3>/x + 6 > 6 nen B < — o B < 2 - de c6 difdc B = 1
6 3 hoSc B = 2 Tif do tim diTdc gia trj ciia x de B la so nguyen
CSu 2 Goi van toe cua ngtfdi di xe dap la x(km/h) (Dieu k i e n x > 0)
V a n toe cua ngU'di di xe may la x + 28 (km/h)
>• Trong 3 gid xe may di du'dc quang du'dng la:
3(x + 28) = 3x + 84 (km/h)
Trong 3 gid xe dap di dUdc quang du'dng la: 3x(km/h)
Theo dau bai ta c6 phu'cfng trinh: ^
3x + 84 + 3 x = 1 5 6
o 6x + 84 = 256
6x = 72
<=> X = 12 (thich hdp)
Vay van toe cua ngurdi di xe dap la 12 (km/gi6)
V a n toe cua ngU'di di xe may la: ' "
1 2 + 28 = 40 (km/h)
N h a n x e t : Giai bai toan b^ng each lap phu'cfng trinh, loai toan chuyen dong
rat quen thuoc do'i vdi hoc sinh Idp 8, 9
C&u 3 a) m = 3, phiTdng trinh trd thanh x^ = 4x + 3 = 0
C 6 a + b + c = l + (-4) + 3 = 0
Cty TNHH MTV DVVH Khang Vi?t
PhiTdng trinh c6 hai nghiem phan biet X| = 1, X2 = - = 3
b) A' = (m - 1) - (m - 6) = m - 2m + 1 - m^ + 6 = - 2 m + 7 ^ : , ,:
Phu'dng trinh CO hai nghiem X,, X2
<:> A' > 0 « 2 m + 7 > 0 < = > m < Theo he thuTc Vi-et ta c6: Xi + X2 = 2(m - 1) = 2m - 2; X1X2 = m^ - 6
Vay m = 0 la gia tri bai toin ve phu'dng trinh bac hai mot an va van dung he
thtfc Vi-et, bai toan nay eung de va quen thuoc
Cfiu 4 a) M A , M B Ih cic tiep tuyen cua
diTdng tron (O) (gt)
c) M A , M B la cac tiep tuyen cua diTdng tr6n (O) => M A = M B , M O la tia phan giac goc A M B , A M A B can tai M , M O la dudng phan giac nen M O la diTdng cao cua tam giac M A B => M O 1 A B
A M A O vuong tai A eo A H la diTdng cao => O H O M = 0 A ^ M H M O = MA^
Trang 21Luy§n gici 6n Join _ NguySn Bute Ta'n
HI' CH H I ' H I
Vay CI la tia phan giac cua goc MCH
Nhan xet: Cac cau a), b), c) de va quen
thuoc Cau d) la cau kho, day la phan M
kho nhat cua do thi nhim giiip phan
loai, phai that sir Unh hoat mdi nhan ra
ding thiJc MC M I CO diTcJc tir AMCH ^ AMOD, A I la diTdng phan gidc
CH H I cua tam giac A M H va MA.OD = AH.MO
Co mot Icfi giai rat hay, neu phat hien siT xuat hien ciia diem E la diem doi
xiJng cua diem I qua O
Cdc ban c6 phat hien diTcJc Idi giai sau khong?
Ve dirdng kinh IE cua dtfdng tron (O) OD = OE (= R)
=> A ODE can tai O => HOD = 20ED
Ma MCH = HOD (Tu- giac CDOH npi tiep)
Nen MCH = 26ED Mat khdc MCI = OED (Ttf gidc CDEI noi tiep)
Vay CI la tia phan giac cua goc MCH
oiy iNnn ivii v uvvn ^^a^g vi?t
[x + 2y = 4 ' cau 2 (2,0 diem) Cho phiTdng trinh (an x): x^ - ax - 2 = 0 (*) " '
1) Giai phu'dng trinh (*) vdi a = 1
2) ChuTng minh rang phu'cfng trinh (*) c6 hai nghiem phan biet vdi mpi gia tri cua a
3) Gpi Xi, X2 la hai nghiem cua phu'Png trinh (*) Tim gia tri cua a de bieu thiJc
N = xf + (X| + 2)(X2 + 2) + X2 CO gia tri nho nhat
cau 3 (2,0 diem) Giai bai toan bkng each lap phu'cfng trinh, he phiTdng trinh
Quang dU'cJng song AB dai 78km Mot chiec thuyen may di tijr A ve phia B
Sau do 1 gicJ, mot chiec ca no di tijf B ve phia A Thuyen va ca no gap nhau tai C each B 36km Tinh thdi gian cua thuyen, thdi gian ciia ca no da di tuf luc khdi hanh den khi gap nhau, biet van toe cua ca no Idn hcfn van toe cua thuyen la 4km/h
cau 4 (3,5 diem) Cho tam giic ABC vuong tai A, tren canh AC lay diem D
(D ^ A, D C) Dirdng tron (O), diTdng kinh DC cil BC tai E (E ^ C)
1) ChiJng minh ti? giac ABED npi tiep
2) Dufdng thing BD c^t dirdng tron (O) tai diem thiJ hai I Chtfng minh ED la tia phan giac cua goc AEI
3) Gia sur tan ABC = V2 Tim vi tri cua D tren AC de EA la tiep tuyen cua du-dng tron diTdng kinh DC
Cfiu 5 (0,5 diem) Giai phiTdng trinh: 7 + 27^ - x = (2 + sf^)y/l - x
Trang 22LuyQn giai OJ triOc kl thi vAo Ii3p 10 ba mign 6^0, Trung, Nam mOn ToAn _ Nguygn Bijfc Ta'n
Nhan xet: Bao toan nay de va rat quen thuoc
CSu 2 1) Khi a = 1 PhiTdng trinh trd thanh x^ - x - 2 = 0
a, c trai dau nen phu'cfng trinh c6 hai nghiem phan biet vdi moi a
3) Theo he thuTc Vi-et, ta c6: x, + X2 = a, X1X2 = - 2
Vay gia tri nho nha't cua N la 5 khi va chi khi a = - 1
Nhan xet: Day la bai toan ve phu'cfng trinh bac hai va he thufc Vi-et, bai toan
nay cung quen thuoc va de, can liTu y x^ + X2 = (x, + X2)^ - 2x,X2
CSu 3 Goi thcJi gian ciia ca no di ttr luc khcli hanh den khi gSp nhau la x(gicJ)
(Dieu kien x > 0)
Thdi gian cua thuyen mdi di tiJf luc khdi hanh khi gap nhau la x + 1 (gid)
Quang diTcfng tiJf A den C dai: 78 - 36 = 42 (km)
V^ntdc cua thuyen may la:42: (x-i-l) =
Van toe cua ca no 1^:
42
X + 1 (km/gicJ)
36 : X = — (kmAi)
Theo dau bai ta c6 phu'cfng trinh:
Cty TNHH MTV DWH Khang Vigt
CSu 4
1) Ta CO DEC = 90" (Goc noi tiep ch^n nuTa du-dng tron) A DEC = BAD (= 90")
Do do ti? giac ABED noi tiep
2) Die = 90" (Goc noi tiep ch^n nufa
dirdng tron) Ta c6 BAC = BIC = 90"
Ti? giac ABCI noi tiep
=> ABI = A C I Ma DEI = DCI (hai goc noi tiep cilng ch^n cung DI) Mat khac ABD = AED (TiJ gidc ABCD noi tiep) Do do AED = DEI
Vay ED la tia phan giac cua goc AEI
3) Gia sijr EA la tiep tuyen cua diTdng tron duTdng ki'nh DC
Ta CO AED = ACE (He qua goc tao bdi tiep tuyen va day cung)
Ma ABD = A E D Dodo ABD = ACE Nen ADB = ABC
Ta CO tanADC = tanABC = V2 tan ADB = Nen ^ = V2
ABV2
di^dng tron diTdng kinh DC
NhSn xet: Bai toan hinh hoc nky cQng la dang toan quen thupc, cSu 3 se c6
IfJi giai neu phat hi$n ADB = ABC c6 di/dc tiJf ABD = ACE
5 Digu kien 0 < X < 7
Trang 23LuyOn giii 66 tn;., IM vio Iflp 10 ba mign iguySn DiJc TSn
Vay phifdng trinh da cho c6 hai nghiem X] = 3,5; X2 = 3
Nhan xet: Day la bai todn phiTdng trinh chiJa can, bai toan kho nha't cua de
thi nh^m de phan loai hoc sinh, dat y = V7 - x de c6 {y - Vx j ( y - 2) = 0
giup CO diTdc Idi giai cua bai toan
[x + y = 2
Bai 2 (2,0 diem) Cho bieu thu-c: A = 1 1 a ^ + l
2 + 2Va 2 - 2Va 1 - a
1) Tim dieu kien xac dinh va rut gon bieu thiJc A
2) Tim gia tri ciia a; bie't ^ < 'J
Bai 3 (2,0 diem)
1) Cho difdng thing (d): y = ax + b Tim a; b de diTdng th^ng (d) di qua diem
A ( - l ; 3) va song song vdi diTdng thing (d'): y = 5x + 3
CtyTNHH MTV DWH Khang Vijt
2) Cho phu'dng trinh: ax^ + 3(a + l)x + 2a + 4 = 0 (x la an so) Tim a de phiTdng trinh da cho c6 hai nghiem phan biet Xi; X2 thoa man xf + X j = 4 '
Bai 4 (3,0 diem) Cho tam giac deu ABC c6 diTdng cao A H Tren canh BC lay
diem M ba't ky ( M khong trung B; C; H) TiT M ke MP; M Q Ian liTdt vuong
g6c vdi cac canh A B ; AC (P thuoc AB; Q thuoc AC)
1) Chi?ng minh: Tvt giac APMQ noi tiep du'Sng tron
2) Gpi O la tam du'dng tron ngoai tiep tu* giac APMQ ChiJng minh OH 1 P Q 3) Chtfng minh r^ng: MP + MQ = A H
Bai 5 (1,0 diem) Cho hai so thtfc a; b thay ddi, thoa man dieu kien a + b > 1 va
a > 0 Tim gia tri nho nha't cua bieu thiJc A = ^ ^ + b^ ,
4a V'",,""'' '
H U d N G D A N G I A I
Bai 1 1 a) x - 1 = 0 <=> x = 0 + 1 <=> x = 1
Phu'dng trinh da cho c6 nghiem la x = 1
b) Phifdng trinh x ^ - 3 x + 2 = 0 c 6 a + b + c = l + (-3) + 2 = 0 nen c6 hai
Vay nghiem (x; y) cua he phiTdng trinh la (3; - 1 )
Nhan xet: Bai toan nay qua de doi vdi moi hoc sinh Idp 9, thi sinh diT thi
ch^c c h i n CO tron 2 diem 6 bai toan nay
Bai 2 1 A xac dinh <=> <
2(1 + V a ) ( l - V a ) ( l + a) _ 1 + a - Va - aVa + 1 + a + Va + aVa - 2a^ - 2 ,,i ; -
2 ( l - a ) ( l + a)
= 2a - a ^ _ 2a(l - a) _ a j • ; 2(1 - a)(l + a) ~ 2(1 - a)(l + a) ~ 1 + a '
Trang 24Luygn giai 6i tru6c kl thi vAo I6p 10 ba miSn B^c, Truiig, r^jm mfln lo&n _ Nguygn Dire Ta'n
Nhan xet: Day la bai toan de
Vay phifdng trinh c6 hai nghiem phan biet vdi a ;^ 0
Cty TNHH M T V D W H Khang Vi^t
<=> 9(a + I)^ - (4a^ + 8a) - 4a^ = 0
<^ 9a^ + 18a + 9 - 4a^ - 8a - 4a^ = 0
<=>a^+10a + 9 = 0 < = > a ^ + a + 9a + 9 = 0
o a(a + 1) + 9(a + 1) = 0 o (a + l)(a + 9) = 0 « a = - 1
a = - 9 v ; «
a = - 1 (thich hdp), a = - 9 (thich hdp) Vay a = - 1 hoSc a = - 9
N h § n x e t : 1) Day la bai toan ve do thj h^m so' bac nhat rat quen thuoc va de v i i -2) Day la bai toan ve phifdng trinh bac hai mot an he thtfc Vi-et Lufu y rang
fa ;^ 0 phiTdng trinh hai nghiem phan bi$t <=>
A = b'^ - 4ac > 0 Bai 4 1) Ta c6: M P 1 A B (gt) => M P A = 90"
Va M Q 1 A C (gt) M Q A = 90" TiJ giac A P M Q cd M P A + M Q A = 90", M P A = 90"
Do do trJ giac A P M Q noi tiep diTdng tron dirdng kinh A M , tarn la trung diem cua A M 2) Ta cd O la tarn diTdng tron ngoai tiep tiJ gi^c A P M Q Mat khac A H M = 90"
=> M thuoc du-dng tron difdng kinh O M
Do vay H e diTdng tron Ma B A H = H A C (AABC deu, A H la diTdng cao nen cung 1^
dirdng phan giac)
=> HP = H Q Vay PO I H Q
3) SMAB + SMAC = SABC = > - M P A B + - M Q A C = - A H B C
2 2 2
=> ^ M P B C + ^ M Q B C = ^ A H B C (AABC deu => A B = BC = A B )
D o d d M P + M Q = A H Nhfin xet: Day la bai toan hinh hoc qua de vk rS't quen thuoc
Bai 5 Ta c6 a > 0, a + b = 1 Do do A = ^ ^ - ^ + b^ = 2a + + b^
4a , , j 4a
•nr' 1
Trang 25Luy^n giSi ai truOc k1 thi vao Idp 10 ba miSn Bjc, Trung, Nam mOn Join _ Nguyln Dufc Tin
Vay gia tri nho nhat c u a A la ^ ,
Nhan x e t : D a y la bai toan ciTc tri dai so, cac bien c6 dieu kien rang bupc
(a > 0 , a + b > 1), bai toan nay kho nhat trong de thi nay nh^m giup phan
loai hpc sinh, bang diT doan a = b = ^ thi A dat gia trj nho nhat, chung ta
kheo leo viet A thanh
a) Tinh gia tri cua y khi x = 1
b) V e d o thi cua ham so (1)
2) Giai phiTdng trinh: 4x^ - 7x + 3 = 0
C a u 2. (2,0 diem) Cho bieu thtfc M = — ^ = + = ^ ^ - ^
3 - V x 3 + V x x - 9 1) T i m dieu kien cua x dc bieu thtfc M c6 nghla Rut gpn bieu thtfc M
2) T i m cac gia tri cua x de M > 1
Cfiu 3. (2,0 diem) M o t dpi the* mo phai khai thac 260 tan than trong mot thcfi gian
nha't dinh Trcn ihiTc tc, moi ngay dpi deu khai thac vu'dt dinh mtfc 3 tan, do do
hp da khai thac du'dc 261 tan than va xong trufctc thdi han mot ngay
Hoi theo ke hoach moi ngay dpi thd phai khai thac bao nhicu ta'n than?
C a u 4 (3,0 diem) Cho nuTa di/dng Iron tarn O, diTcJng kinh A B = 12cm Tren nuTa
mat phang bd A B chuTa nuTa diTcfng tr6n (O) ve cdc tia tiep tuyen A x , By M
la mot d i e m thupc nCfa diTdng iron (O), M khong trung vdi A v a B A M cj(t
g y tai D , B M cMt A x tai C, E la trung diem cua doan thang B D
b) Bang gia tri
Trang 26Nhan xet: Bai todn nay cung qua de
Cfiu 3 Gpi so tan than theo ke" hoach moi ngay dpi thd phai khai thac la x (tan)
(Dieu kien x > 0)
So tan than theo thifc te moi ngay dpi thP khai thac du'cfc la: x + 3 (tan)
Vay theo ke hoach moi ngay dpi thd phai khai thac 26 tan than
Nhan xet: Bai toan bang cdch lap phiTdng trinh la dang bai toan de va quen
thupc
CSu 4 a) Ta c6 AMB = 90" (Goc npi tiep chan nufa diTdng tron)
Xet AABC va ABDA c6 BAG = DBA (=90"),
ABC = BDA (cung phu vdi goc MDB)
ABDM vuong tai M , ME la du'cfng trung
tuye'n (E la trung diem cua BD)
=> EM = EB = ED
Xet AOME va AOBE c6: OM = OB (= R), EM = EB, OE (canh chung)
46
Cty TNHH MTV DWH Khang Vi?t
Do do AOME = AOBE (c-c-c) => OME = OBE = 90"
Vi EM 1 OM, M e (O) Do vay OM la tiep tuyen cua dtfdng tron (O)
3) Ta CO FA, FM la cac tiep tuyen cua (O) => AF = MF TiTdng tiT EM = BE
Ax_L AB, By 1 AB => Ax//By ;' Ti? giac AFEB la hinh thang:
_ (AF + BE).AB _ (MF + EM).AB _ EF.AB
S A F E B ^
-Qua A vc dU'dng thang song song vdi EE cat By d K
Ttf giac AFEK la hinh binh hanh ncn EF = AK Ma AB 1 BK nen AK > A B
Nhan xet: Day la bai toan hinh hpc rat quen thupc
Cfiu 5 Dieu kien x > 34; y > 21; z > 4
da cho thanh A ' + B ' + C ' = 0 tu-c la
tinh diTdc gia tri cua bieu thuTc T i :
X : ••••
Trang 27i '-J
b) Giai he phifdng trinh:
Luyjn giJi dS trudc ki thi vAo Idp 10 ba mign BSc, Trung Nam mOn ToAn _ NguySn Dufc la'n
CSu 3 (2 diem) Trong mat phang toa dp Oxy cho Parabol (P) c6 phiTdng trinh
y = x^ va dirdng thang (d) c6 phiTdng trinh y = 2mx - 2m + 3 (m la tham so)
a) Tim toa dp cac diem thupc (P) bie't tung dp cua chung b;ing 2
b) ChiJng minh rang (P) va (d) cat nhau tai hai diem phan biet vdi mpi m
Gpi yi, y2 la cac tung dp giao diem cua (P) va (d), tim m de yi + y2 < 9
Cfiu 4 (3,5 diem) Cho du'dng Iron tam O, du'dng kinh AB Tren tiep tuyen cua
dirdng tron (O) tai A lay diem M (M khac A) TiT M ve tiep tuyen thiir hai
MC vdi (O) (C la tiep diem) Ke CH vuong goc vdi AB (H e AB), MB cat
(O) tai diem thi? hai la K va cat CH tai N ChiJng minh rang:
a) Tu" giac AKNH la tiJ giac npi tiep
c) Goc KAC bang goc OMB ^ < :
d) N la trung diem cua CH
CSu 5. ( 1 diem) Cho ba so thiTc a, b, c thoa man: a > 1 ; b > 4; c > 9
Tim gia tn Idn nhat cua bieu thu'c: P =
Bai nay qua de, chia khoa cua cau b la Vs - i V ^ = ^^V6 - V 2 k h a
thuoc do'i vdi nhicu hoc sinh
quen
CSu 2 a) Phu-dng trinh x ^ - 5 x + 4 = 0 c 6 a + b + c = l + (-5) + 4 = 0
Do do phu"dng trinh c6 hai nghiem X| = 1 ; X2 = - =4
a b) 3 XX + - y = 2y = 1 5
Nhan xet: Day cung la bai toan de do'i vdi mpi hoc sinh Idp 9
Cau 3. a) Ta CO 2 = x^ <=> X = ± ^
Vay cac diem thupc (P) c6 tung dp bang 2 la (sjl;!^; (-N/2;2)
b) PhiTdng trinh hoanh dp giao diem cua (P) va (d) x^ = 2mx - 2m + 3 <=> x^ - 2mx + 2m - 3 = 0 (*) A' = m^ - 2m + 3 = (m^ - 2m + 1 ) + 2 = (m - 1 ) ^ + 2 > 0, vdi mpi m
Do do (*) luon CO hai nghiem phan biet vdi mpi m
Vay (P) va (d) c^t nhau tai hai diem phan biet vdi mpi m
Gpi giao diem cua (P) va (d) la (x,, y,) va (x., y2); x,, X2 la nghiem cua (*) Theo he thu'c Vi-et ta c6 X| + X2 = -2m, X1X2 = 2m - 3 va y, = x f , y 2 = X j
Nh§n xet: Day la bai toan ve do thi ham so va ket hdp van dung he thu'c
Vi - et cung la bai todn quen thupc do'i vdi hpc sinh Idp 9
cau 4 a) AKB = 90" (Goc npi tie'i; -han nuTa dirdng tron)
AHN = 90" (CH 1 AB) Tu'giac AKNH co AKB + AHN - I8O"
Do do tiJ giac AKNH npi tiep
4 0
Trang 28b) MA la tiep tuyen cua difdng tr6n (O)
= > M A ± AB
AAMB vuong lai A, AK la dudng cao
Vay AM^ = MK.MB
c) MA, MC la cac tiep tuyen cua (O) (gt)
=> M A = MC, MO \k tia phan giac g6c AMC,
OM la tia phan giac goc AOC AMAC can tai
M, MO la dirdng phan giac nen MO la diTdng
cao, dU'cJng trung tuyen cua tam giac MAC
ACB = 90" (Goc npi tiep ch^n nijra diTdng tron)
Vay N la trung diem cua CH
Nhan xet: Bai toan hinh hoc nay la bai toan quen thuoc
2) Tim a de he phi/dng trinh c6 nghiem duy nhat
CSu 3 (2,0 diem) Mot hinh chff nhat c6 chieu rpng bang mot nufa chieu dai Biet rkng neu giam moi chieu di 2m thi dien tich hinh chff nhat da cho giam
di mot nuTa Tinh chieu dai hinh chi? nhat da cho
Cfiu 4 (3,0 diem) Cho diTdng tron (O; R) (diem O co dinh, gia tri R khong ddi) diem M n^m ben ngoai (O) Ke hai tiep tuyen MB, MA, (B, C la cac tiep
<Jiem) cua (O) va tia Mx n i m giffa hai tia MO va MC Qua B ke diTdng thang song song vdi Mx, du-dng thing nay cat (O) tai diem thi? hai la A Ve diTdng kinh BB' cua (O) Qua O ke dirdng thing vuong goc vdi BB', diTcfng thing nay c^t MC va B'C Ian liTdt tai K va E ChiJng minh r^ng
' ^ B6n diem M , B, O C cdng n^m tren mot diTdng tron
E>oan thing ME = R
Khi die'm M di dpng ma OM = 2R thi diem K di dpng tren mot diTdng tron cd
^inh, Chi ro tam va ban kinh cua difdng tron do
" 5 (1,0 diem) Cho a, b, c 1^ cAc so diTPng thoa man a + b -i- c = 4 ChiJng
"linhr^ng: V 7 + Vb^ + V?>2>/^
Trang 29Luygn giai dl truOc k1thi vao Iflp 10 ba mign B^c Trung, Nam mOn To^n _ NguySn DiJc Tafn
HUdNG DAN GIAI
y = -5 + ax
Vi a^ + 6 > 0, vdi moi a nen (1) c6 nghiem vdi moi a Do vay he phuTdng tni
CO nghiem (x; y) vdi moi a
Nhan xet: Day la bai todn ve he phu'dng trinh bac nhat hai an, b^i todn ^
do'i vdi moi hoc sinh Idp 9
Cfiu 3 Goi chieu dai cua hinh chi? nhat da cho la x(m) (Dieu kien x > 4)
Chieu rong cua hinh chu* nhat la - (m)
Dien tich ciia hinh chff nhat da cho 1^: x - = — (m^)
2 2
Cty TNHH MTV OWH Khang Vigt
Chieu dai cua hinh chfr nhat sau khi giam la:
Chieu rpng ciaa hinh chiJ nhat sau khi giam la:
Dien tich cua hinh chiJ nhat mdi la:
x - 2(m)
| - 2 ( m ) ( x - 2 ) - - 2 U Theo dau bai, ta c6 phu'dng trinh
X| = 6 + 2\/5 (thoa man dieu kien x > 4)
X2 = 6 - 2^y5 (khong thoa man dieu kien x > 4) Vay chieu dai cua hinh chu' nhat da cho la 6 + 2^5 (m)
Nhan xet: Giai bai toan bang each lap phu'dng trinh, loai toan c6 noi dung
hinh hoc la dang bai toan de, quen thupc do'i vdi moi hoc sinh
CSu 4 1) MB, MC la cac ticp tuyen
ciia dirdng Iron (O) (gt)
= > M B 1 0 B , M C 1 0 C
=i> MBO = 90" , MCO = 90" M
-Do do M, B, O, C cung Ihupc difdng tr6n dudng kinh OM
2) MB, MC la cac tiep tuyen cua du'dng tron (O) (gt)
=> OM la tia phan giac goc BOC (Tinh cha't cua hai tiep tuyen c^t nhau)
=> MOB = - B O C Ma B B ^ - -BOC (He qua gdc noi tiep) 2 2
; f;
Dodo MOB = BB'C Xet ABOM va AOB'E c6: MBO = EOB' (= 90"), OB = OB' = (= R) MOB = CB'E Do do ABOM = AOB'E (g.c.g) => OM = B'E
Matkhacco MOB = OWE ^ OM//B'E
Tiif giac MOB'E cd OM // B'E, OM = B'E => Tu" giac MOB'E la hinh binh hanh=>ME = OB' =R
Trang 30Luy$n giai lii frijflc ki thi vao I6p 10 ba mign BJc Trung, Nam mOn To^n _ Nguyjn Difc Ta'n
OB R 1 3) AOBM vuong tai B => sinBMO = = = - => B M O = 30"
OM 2R 2
Ta CO OMC = OMB = 30", K O M = OMB (so le trong va M B // OE)
Nen iCOM = 30" • , : ^.cn^,, ? ^ ,
OMC + COM = 90" (ACMO vuong tai M) Do d6 COM = 60"
• , O K 2 OK 3
Ma O c o d i n h
2 /3R
Vay K di dong tren du'dng tron tarn O, ban kinh — - —
Nhan xet: Cac cau 1), 2) raft quen thuoc Cau 3) khong kho khan l^m, chung
ta nhan ra KOC = 30" Do vay c6 diTdc OK =
3
CSu 5 V i a, b, c > 0 thoa man a + b + c = 4 Nen 4 a, 4 > b, 4 > c
D o d 6 V ? + Vb^ + t / ? > V ? + ^ + V 7 = a + b + c = 4
Vay V 7 + Vb^ + V ? > 7 ^ - 2 V 2
Nhan xet: Tuf a, b, c > 0 thoa man a + b + c = 4 de c6 4 > a, 4 > b, 4 > c den
vdi Idi giai tren that hay, s^ng tao Cac ban hay tim them cac each giai khac
nffa cho bai toan nay
1) Hai 6 to tit A den B dai hdn 200km Biet van toe xe thuT nha't nhanh hdn van
toe xe thi? hai Ik lOkm/h nen xe thu* nhat den B sdm hcfn xe thiJ hai 1 gicJ
Tinh van toe moi xe
2) Rut gon bieu thtfc: A - 1 - ^ = (x + V x ) ; vdi x > 0
Cty TNHH MTV DWH Khang Vi^t
C&u 3 (1,5 diem) Cho phiTdng trinh: - 2(m + 2)x + + 4m + 3 = 0
1) ChiJng minh r i n g : PhiTOng trinh tren luon c6 hai nghiem phan biet X | , X2 vdi
mpi gia tri cua m
2) Tim gia tri cua m de bieu thiJc A = xf + X j dat gia tri nho nha't
CSu 4 (3,5 diem) Cho tarn giac ABC c6 ba goc nhon npi tie'p diTcfng tron tam O (AB < A C ) Hai tie'p tuyen tai B va C cat nhau tai M A M cat di/dng tron (O) tai diem thiJ hai D E la trung diem doan A D EC c^t diTdng tron (O) tai diem thu" hai F ChiJug minh r^ng:
Phurpng trinh c6 hai nghiem phan biet X | = - ; ' ' i = " J
2) Do thi ham so y = ax + b di qua A(2; 5) v^ B ( - 2 ; chiTa ro so)
Ja = 2
b - 1
2a + b = 5 2b = 2
Nhain xet: Day la bai toan de, quen thuoc
2 1) Gpi van toe xe thtf hai la x (km/h) (Dieu kien x > 0) Van toe xe thiJ nha't la x + 10 (km/h)
Trang 31Luy§n (; rrung, Nam mOn Join _ Nguygn Pile TSn
2 0 0
Thdi fiian xc thi? nhal di tCf A den B la: 200 : (x + 10) = ^ (h)
Thdi gian xe thiJ hai di lit A den B la: 200 : x = 200 (h)
Xe thijr nhat den B sdm hdn xc thuT hai 1 gid, nen ta c6 phiTdng Irmh
Van toe xe thiJ nha't la: 40 + 10 = 50 (km/h)
• 1) Giiii bai toan bang each lap phi/dng trinh loai toan chuyen dpng rat quen
ihupe vdi hoc sinh
2) Riit gon bieii ihifc chuTa can dang ddn giiin
cau 3 1) A' = (m + 2f - (m^ + 4m + 3) = m^ + 4m + 4 - m' - 4m - 3
= 1 > 0, vdi moi m Vay phiTdng trinh c6 hai nghiem phan biet vdi moi m
2) Theo he thu'e Vi-el ta c6
1) Can tim each chi^ng minh A' (hoSc A) di/dng vdi moi m
2) Theo he ihufc Vi-et ta c6 x, + Xj = 2m + 4, x,X2 = n r + 4m + 3
+ 3)
Cty TNHH MTV DVVH Khang Vi§t
Do do A = xf + X2 = ( X | + Xjf - 2 X | X 2 = 2 ( m + 2 ) ^ + 2 > 2
Tuf do tim di/dc gid tri nhd nhat cua A
C&u 4 1) E la irung diem day cung A D va A D ^ 2R '
=:> OE J_ AD Ta c6 M B la tiep tuyen cua (O)
=> MB 1 OB.Vi O B M - OEM (= 90") Tur giac OEBM noi tiep
2) Xet AMBD va AMAB c6 B M D (chung),
MBD ^ M A B (He qua gdc tao bdi tia tiep tuyen
va day cung) Do do AMBD AMAB (g.g)
MB _ M D
M A ~ M B
V a y M B ^ ^ M A M D 3) MB, MC la cac tiep tuyen ciJa difdng trdn (O) (gt)
OM la tia phan giac cua goe BOC => MOC = ^ B O C
1 Mat khac BFC = - B O C (He qua gdc npi tiep) Vay BFC = MOC
4) Ta CO MEO + MCO = 180" => Ti? giac MEOC noi tiep
^ MEC = M O C Ma BFC = MOC (cau 3) Do do MEC = BFC
Ta CO MEC va BFC la hai gdc dong vi Vay BF // A M •
Nhan xet: Day la bai toan hinh hoc quen thuoc va de Cau 4) la cau kho nha't cua bai toan nay, tiir giac MEOC la chia khda giiip cd Idi giai
Cfiu 5 Ap dting bat dang IhiJc Co-si cho hai so diTdng, ta cd
X
trong vice van dung bat d^ng ihuTc Co-si cho hai so dtfdng, giup cd difdc Idi giai b^i toan
Trang 32Luy5n giai dg truOc kl thi vao I6p 10 ba mign Bjc, Trung, Nam mOn Jo&n _ NguySn D(lc Ta'n ^
b) Giai he phiTdng irinh: 3y + 4x = 10
e) Giai phiTdng trinh: Vx^ - 6x + 9 = x - 2011
CSu 2 (2,5 diem) Mot ea no chay xuoi dong lit A den B roi chay ngiTdc dong tir
B den A het ta't ca 4 gid Tinh van toe ca no khi niTdc yen iSng, biet rSng
quang song dai 30km va van toe dong niTdc la 4km/gi(l
CSu 3 (2,5 diem) Tren difcJng tron (O) lay hai diem M, N sao cho M, O, N
khong thang hang Hai tiep tuycn tai M, N vdi difdng tron (O) cat nhau tai A
Tir O ke dirdng vuong goc vdi OM cat AN tai S TiT A kc di/dng vuong goc
vdi AM cat ON tai I Chiang minh:
a) SO = SA
b) Tam giac OIA can
CSu 4 (2,0 diem)
a) Tim nghiem nguyen cua phi/dng trinh: x^ + 2y^ + 2xy + 3y - 4 = 0
b) Cho tam giac ABC vuong tai A Goi I la giao diem cac difdng phan giac
trong Biet AB = 5cm, IC = 6cm Tinh BC
, , , Hiring din giai
CSu 1 a) A' = 9 - 9 = 0 Phifdng trinh co nghiem kcp x = = 3
Vay he phu^dng trinh c6 nghiem (x; y) la
NhSn xet: Day la bai toan de, quen thuoc
CSu 2 Goi van toe cua ca no khi nifdc yen lang la x(km/gicf) (Dieu kicn x > 4)
Van to'c cua ca no khi xuoi dong la x + 4 (km/gid) va khi ngifdc dong la x - 4
Wi a - b + c = l - ( - 1 5 ) + (-16) = 0
Do vay X, = -1 (khong thich hdp); X2 = 16 (thich hdp)
Vay van toe cua ca no khi nu^dc yen lang la 16km/gi5
NhSn xet: Giai bai toan bang each lap phu'dng trinh loai toan chuyen dong,
dang toan nay ra't quen thuoc doi vdi moi hoc sinh Idp 9
CSu 3 a) AM, AN la cac tiep tuyen cua
difcfng tron (O) (gt) AO la tia phan giac cua goc MAN va OA la tia phan giac cua goc MON Ta c6 AM 1 OM (AM la tiep tuyen cua difdng tron (O)),
SO 1 OM (gt)
=> AM // SO => MAO = AOS
Mk MAO = AOS (AO la tia phan giac
goc MAN) Suy ra AOS = OAS
=i> ASOA can tai S Vay SO = SA b) Ta CO lA 1 AM (gt), OM 1 AM =^ lA // OM lAO = AOM
Ma AOM - AOI (OA la tia phan giac cua goc MON) | Nen lAO = AOI Do do tam giac OIA can tai I
NhSn xet: Day la bai toan hinh hoc quen thuoc, de Mau chol cua bai toan
la van dung tinh cha't cua hai tiep tuye'n c^t nhau
CSu 4 a) x ' + 2y'= 2xy + 3y - 4 = 0
o (x^ + 2xy + y^) + (y^ + 3y - 4) = 0 o (x + y)^ + (y _ i)(y + 4) = o (*)
Vi (x + y)2 > 0 Do do (y - l)(y + 4) < 0
y - l > 0 , y + 4 < 0
y - l < 0 , y + 4 > 0 <=> y > 1, y < -4 y < 1, y > -4
<=> -4 < y < 1 Ma y e Z Do do y e {-4; - 3 ; -2; -1; 0; 1}
Trang 33Luygn giai dfi trUOc kl thi vao lap 10 ba mign BJc Trung, Nam mOn Toan _ Nguygn Dire TSn
Thay cac gia tri ciia y e {-4; - 3 ; - 2 ; - 1 ; 0; 1} ta c6 (x; y) = (4; - 4 ) ; (1; - 3 ) ;
( 5 ; - 3 ) ; ( 2 ; 0 ) ; ( - 2 ; 0 ) ; ( - 1 ; 1)
Vay nghicm nguycn (x; y) cua phiTdng trinh la:
(4; - 4 ) ; ( 1 ; - 3 ) ; (5; - 3 ) ; (2; 0); (-2; 0); ( - 1 ; 1)
b) Ve CH 1 B I tai H, CH c^t AB 6 D ABCD
CO BH la diTdng cao, difdng phan gi^c
=^ ABCD can tai B ^ ^* ^
^ B D = BC D a t B C = x
Ta CO ABC + ACB - 90" (AABC vuong tai A)
HIC - IBC + ICB - - ABC + ^ A C B = 45"
AHIC vuong tai H ^ HC = ICsin HIC = 6sin45" = 6 ^ = 3V2
N e n I H = N C = 3V2 va D C = 2 H C = 6 N/2
Ta CO A D = BD - AB = X - 5
AABC vuong tai A A B ' + A C ' = B C ' nen A C ' = - 25
AACD vuong tai A => AD^ + AC^ = CD^
Do do (X - 5)' + x^ - 25 = [Ssflf o x^ - 5x - 36 = 0
<=> (x - 9)(x + 4) = 0 o x = 9 (thich hdp)
X = - 4 (khong thich hdp)
Vay BC = 9cm
Nhan xet: Day la bai toan kho nhkm de phan loai hoc sinh Bai 4 (1) tiJf
+ 2y^ + 2xy + 3y - 4 = 0 giup den (x + y)^ + (y - l ) ( y + 4) = 0 Can
n h a n r a C y - l ) ( y + 4 ) < 0 v a ket hdp y e Z de c6 y e { - 4 ; - 3 ; - 2 ; - 1 ; 0; 1}
CAu 4 (2) V i CO BIC = 135" do vay ve CH 1 BI tai H, goi D Ik giao diem cua
CH va A B Lam xuat hien cac diem H, D d tren that hay
c) x'* + 5 x ^ - 3 6 = 0 ^ d) 3 x ' - x V 3 + V 3 - 3 = 0
B a i 2 : ( l , 5 d i e ' m ) a) Ve do thi (P) cua ham so y = -x^ va diTdng thing (D): y = - 2 x - 3 tren cung mot he true toa do
b) Tim toa do cac giao diem cua (?) va (D) d cau tren b^ng phep tinh
Bai 3: (1,5 diem) Thu gpn cac bieu thu-c sau : f'"^' sntrtirfq viV
a) Chiang minh rang phiJdng trinh luon luon c6 nghiem vdi mpi m
b) Goi X|, X2 la cdc nghiem cua phu'dng trinh
Tim m de bieu thiJc A = xf + xf - X1X2 dat gia tri nho nha't
Bai 5: (3,5 diem) Cho dirdng tr6n (O) c6 tam O, diTcJng kinh BC Lay mot diem A
tren dirdng tr5n (O) sao cho AB > AC Tir A, ve A H vuong goc vdi B (H thuoc BC) TiT H, ve HE vuong goc vdi AB va HF vuong goc vdi AC (E thupc AB, F thupc AC)
a) Chufng minh rhng AEHF la hinh chff nhat va OA vuong goc vdi EF
b) Dirdng thang EF cat diTdng tron (O) tai P va Q (E nkm giffa P va F) , ,
Chufng minh : AP^ = AE.AB Suy ra APH la tam giac can
c) Goi D la giao diem cua PQ va BC ; K la giao diem cua A D v^ di/dng tron (O) (K khdc A) Chufng minh r^ng AEFK la mot tuf giac npi tiep
Gpi I \k giao diem ciia KF va BC Chu-ng minh IH^ = IC.ID
Trang 34Luy§n giai dg trifdc ki thi vao \0p 10 ba miSn BJc Trur;, r
a b) De nhan ra nen giai he phu'dng trinh bang phtfdng phap cong dai so
c) PhiTdng trinh trung phiTdng ax'' + bx^ + c = 0 (a 0) Dat x^ = y ta diTdc
phiTdng trinh bac h a i : ay'^ + by + c = 0
d) Phu'dng trinh dang ax^ + bx + c = 0 (a ;^ 0) c6 a - b + c = 0 thi phu'dng trinh c6
c mot nghiem X| = - 1 , con nghiem kia X2 = —
• D o thi ham so y = - 2 x - 3 la diTdng thang (D) di qua (0 ; - 3 ) va ( - 1 ; -1)
• Do thi ham so':
!'i ! id <y> situ
Cty TNHH MTV DWH Khang Vin
b) Phu'dng trinh hoanh do giao diem cua (D) va (P) : -x^ = - 2 x - 3 <=> x^ - 2x - 3 = 0
V a y toa do cac giao diem cua (P) va (D) la ( - 1 ; - 1 ) ; (3 ; - 9 )
Nhfin xet : T i m toa do cac giao diem cua (D) va (P) bang phep tinh, triTdc
tien lap phiTdng trinh hoanh do giao diem cua (D) va (P) : - x ^ = - 2 x - 3 (*) Cac nghiem cua (*) la hoanh dp cua cac giao diem
Trang 35Luy$n giai ai truSc k1 thi v^o I6p 10 ba miSn B&c, Trung, Nam mOn Jo&n _ Nguygn Dure J&n
a) Lfu tien 1, true cac can thiJc cf mau
b) N h a n ra rkng (Vx + l)(Vx - 4 ) = x - 3 V x - 4 Quy dong mau cac phan
thiJc, roi thiTc hien phep tinh
Tit do de tim diTdc gia tri nho nhat cua A
B a i 5: a) B A C = 90" (gdc noi tiep ch^n nuTa diTdng tron)
H E A = 90" ( H E 1 AB), H F A = 90" (HF 1 AC) => TO giac A E H F la hinh chiJr nhat
Gpi J la giao d i e m cua A H va EF TO giac A E H F noi tiep diTdng tron tam J
JA = JF => A J A F can tai J => JFA = JAF
~X7T7 imnn iin i v u v v i i rvriaiiy vi?i
A O A D cd A H , D E la hai diTdng cao ( D E 1 OA, A H 1 OD) c^t nhau tai J
J la triTc tam cua tam giac O A D => OJ 1 A K => OJ di qua trung diem cua A K
N e n OJ la du'dng trung triTc cua doan thang A K =:> JK = JA
=> K thupc dirdng tron (J)
Ta cd A , K , F, H , E cung thupc mot dtfdng tron (J)
Do vay tii' giac A E F K noi tiep
M a t khac I H F = H A F (cung phu v d i A H F ) ,
H K F = H A F ( A , K , F, H , E cung thupc mot diTdng tron) I H F = H K F
Ti? giac A E H F cd B A C = 90", H E A = 90", H F A = 90" Chu y rkng:
O A C = O C A , JFA = J A F (J la giao d i e m cua A H va EF)
65
Trang 36Luy$n giai dg truOc k1 thi vao Idp 10 ba mign 6^0. Trung, NarrTmOn Toan _ Nguy6n Ddc W\
OA ± EF (la bai loan quen thuoc d Idp 8)
b) AP' = A H ' (= AE.AB) => AP = AH
Dc chiirng minh hai doan thang AP, AH bang nhau nhieu khi phai nghi den
chi^ng minh AP' = A H ' ne'u Irong hinh ve bai loan c6 xua't hi^n lam giac
vuong, cac lam giac dong dang a :| ,i ,'
c) De Iha'y A, E, H, F cilng thuoc diTdng Iron (J ; JA) jj ''Rff/iar-'jil ,,.)
Tim each chrfng minh JK = JA
d) De nhan ra AlHF ^ A I K H , ncn c6 diTdc IF.IK = I H ^
Tim each chuftig minh IC.ID = IF.IK, nghia la can chufng minh AICK ^ AIFD
a) Giai phu'dng Irinh (1) khi m = 4
b) Tim cdc gia Iri ciJa m de phu'dng Irinh (1) c6 hai nghiem X i , Xj Ihoa man
b) Xac dinh a, b de diTdng th^ng (d): y = ax + b c^l true tung tai diem c6 lung
do b^ng -2 va c^l do thi (P) n6i tren tai diem c6 hoanh do bkng 2
Bai 4: (4 diem) Cho nuTa diTcJng lr6n (O ; R) diTcfng kinh AB Gpi C la diem
chinh giffa cua cung AB Tren tia doi cua tia CB lay diem D sao cho CD = CB
OD c^t AC tai M TiT A, ke AH vuong goc vdti OD (H thuoc OD) AH c^l DB
tai N va c^l nufa di/dng Iron (O ; R) lai E
a) ChiJng minh MCNH la tuT giic npi liep va OD song song vdi EB
b) Gpi K la giao diem cua EC va OD ChuTng minh r^ng ACKD = ACEB
Suy ra C la trung diem cua KE
c) Chtfng minh tam giac EHK vuong can v^ M N song song vdi AB
(I) Tinh theo R dien tich hinh Iron ngoai liep tiJ giac MCNH
-Vay nghiem ciia he phifdng tnnh da cho la (x ; y) = (1 ; 2 )
T a c 6 : a + b + c = l + (-4) + 3 = 0 Vay nghiem cua phiTdng tnnh la x, = 1 ; X2 = - = 3
CJ tren da giai he phi/dng Irinh bac nha'l hai an n^y bIng phiTdng phap cpng
^ai so Ban dpc hay giai b^ng phi/dng phap the
m
Trang 37Luyjn giSi truOc k1 thi vao I6p 10 ba miln BJc Trung, Nam mOn ToAn _ Nguyin Dure Ta'n
2) Day la bai loan van dung h? thiJc Vi-6t, can lUu y — + — c6 nghla
b) (d) c^t true tung tai diem c6 tung do bkng -2 nen b = -2
Diem thuoc (P) c6 hoanh do bing 2 thUung do cua diem do la - 2^ = 1
4
3 3 Diem (2 ; 1) thuoc (d) o 1 = a.2 - 2 o a = - Vay a = - ; b = -2
Nhan xet: Day la bai toan de ve do thi ham so y = ax^ (a ^ 0)
Bai 4: a) ACB = 90" (goc npi tiep ch^n niJa difdng tr6n)
MCN + MHN = 90" + 90" = 180"
nen npi tiep mot dU 'dng tron
Mat khac AEB = 90" (goc npi tiep chin
nufa dU 'dng tron)
Taco : O D l A E ( g t ) v a E B l A E ( A E B = 90°) ^
VayOD//EB , O
b) Xet ACKD va ACEB CO :
iCCD = ECB (doi dinh), CD = CB (gt), CDK = CBE (so le U-ong va OD // BE)
Do do : ACKD = ACEB (g.c.g) =^ CK = CE
Vay C la trung diem cua doan thSng KE
CtyTNHH MTV DWH Khang Vijt
AEC = ABC (hai goc noi tiep cilng ch^n cung AC)
AEHK vuong tai H c6 HEK = 45" => AEHK vuong can tai H
Ma HC la diTcJng trung tuyen cua tarn giac EHK Do do HC la diTdng phan giac cua tam giac EHK KHC= iEHK = 45" i = ?!
Ma MNC = MHC (tur giac MCNH npi tiep) nen MNC = 45"
Ta CO : MNC = ABC = 45", MNC va ABC dong vj Vay MN // AB
d) Xet AABD co AC, DO la hai diTdng trung tuycn (O, C Ian liTpl la trung diem cua AB, BD) c^t nhau tai M
=> M la trong tarn cua tarn giac ABD => DM = — DO => —
R 3 3 Tu- giac MCNH npi tiep c6 M?iN = 90" =^ MN la difdng kinh cua diTdng tron ngoai tiep tiir giac MCNH => Ban kinh cua diTcJng tron ngoai tiep ti? giac MCNH bang ^ : 2 = -
• Cic cau 1); 2); 3) de, quen thupc
• Cau 4 la cau kho nhat cua bai toan 4, ph^t hien M Ik trpng tam cua tam giac
2R
ABD de CO MN // OB, tif do c6 MN = — giup c6 difpc M giai bai toan
DE THI TUYEN SINH VAO LdP 10 THPT, TJNH DAKLAK
NAM HQC 2011 - 2012
B&i 1: (2 diem) • '
1) Giai cac phu'Png trinh sau ; ^ ^ ^ , a)9x' + 3 x - 2 = 0 b) x'+ 7x^- 18 = 0 " ^ ^ ' ! 2) Vdi gia tri nao cija m thi do thi hai ham so :
y = 12x + (7 - m) va y = 2x + (3 + m) cil nhau tai mpt diem tren true tung
Trang 38Bai 2: (2 diem)
1) Riit gon bieu thuTc : A = 1
2) Cho bieu thiJc : B =
a) Rut gon bieu thiJc B
b) T i m gia tn cua x de bieu thiJc B = 3
Bai 3: (1,5 diem)
2y - X = m + 1 2x - y = m - 2
a) Giai he phi/dng trinh (1) khi m = 1
b) Tim gia tri cua m de he phu'dng trinh (1) c6 nghiem (x ; y) sao cho bieu thuTc
P = + y^ dat gia tri nho nhat
Bai 4: (3,5 diem) Cho tam giac ABC c6 ba goc nhon va noi tiep dU'ffng tron (O)
hai dU'cfng cao BD va CE cua tam giac ABC cat nhau tai diem H DiTdng
thang BD c^t diTdng tron (O) tai diem thuT hai P, difdng thang CE cat diTdng
tron (O) tai diem thu" hai Q ChiJng minh :
a) Tuf giac BEDC noi tiep
b) HQ.HC = HP.HB
c) D E / / P Q
d) Du"dng thing OA la diTdng trung triTc cua PQ
Bai 5: (1 diem) Cho x, y, z la ba so thifc tuy y Chufng minh:
x^ + y^ + - yz - 4x - 3y > - 7
Hifdng d§n giai
B a i 1: a)9x^ + 3x - 2 = 0 Ta c6: A = 9 + 72 = 81 ^ > /A = 9
-3 + 9 1 - 3 - 9 - 2 PhU'dng trinh c6 hai nghiem phan biet: X i = = - ; X2 = = —
2.9 3 2.9 3 b) x^ + 7x' - 18 = 0. Dat x^ = y (y > 0)
Phirdng trinh trd thanh: y^ + 7y - 18 = 0, A = 49 + 72 = 121 V A = 11
yi = ""^ ^ ^ ^ = 2 ; yz = "'^ ^ ^ ^ = - 9 (loai v i - 9 < 0) ]
y = 2 nen x^ = 2 o x = ±>/2
PhiTcfng trinh c6 hai nghiem phan biet \
Nhan x e t : Day la bai toan de, quen thuoc doi vdli moi hoc sinh Idp 9
Cty TIMHH MTV < Khang Vi$t
Vay he phtfdng Irinh da cho c6 nghiem la (x ; y) = (0 ; 1)
^ay Pdat gia tri nho nhat b i n g ^ khi m = i ^
Nh§n xet : Ca hai cSu 1) ; 2) deu giai b^ng phiTcfng phap the Ban doc hay
giai b^ng phu'dng p h i p cong dai so
Trang 39Luy?n giii truflc k1 thi v^o Wp 1 ' 'n BJc, Trung, Nam mOn Toan _ Nguyen Pile lan
Bai 4: a) B D , CE la hai di/tfng cao cua tarn gidc A B C (gt)
^ B D C = B E C = 90" => TiJ giac B E D C npi tiep
c) Ta CO : PQC = PBC (hai goc noi tiep cilng chan cung CP)
D E C = D B C (tiir giac B E D C n o i tiep)
= (\-2Y+ ^y-z + T y " r - 7 >-7, vdi mpi X, y, z e R
Nhan xet : V i trong da thiJc x^ + y^ + z^ - yz - 4x - 3y c6 chiJa cdc ddn thift
x^ ; y ^ ; z^ ; - y z ; - 4 x ; - 3 y nen nghl den viec v a n dung h^ng d^ng thiJc :
( A ± B)^ t h e m chut kheo leo, giup chung ta den :
X + y + z - yz - 4x - 3y = (x - 2) +
0
+ ^ 2 ^ - y - z ) + — 4 y -— 2) - 7 DjfesO 19
DE THI TUYEN SINH VAO LdP 10 THPT, TJNH NINH THUAN
< 1 NAM HQC 2011 - 2012 Bai 1: (2 d i e m ) Cho difcJng thang (d): y = - x + 2 va parabol (P): y = x l
a) V e (d) va (P) tren cung mot he true tpa dp
b) B ^ n g do thj hay xac dinh tpa dp cac giao d i e m cua (d) va (P)
Cty TNHH MTV DWH Khang Vi$t
„ x 5 3: (2 diem) Cho bieu thiJc : P = ^ + 3(1 - N / X ) , vdi x > 0
X + 2 V x + 4 a) RUt gpn b i e u thtfc P
2P J,) T i m cac gia t n nguyen difpng cua x de bieu thufc Q = ^ — - nhan gia tri nguyen
p^i 4: (3 diem) Cho tam giac A B C c6 g6cBAC= 6O", diTdng phan giac trong
cua goc A B C la B D va diTdng phan giac trong cua goc A C B la C E c^t nhau tai I ( D G A C va E e A B )
a) Chtfng m i n h tu" giac A E I D n p i tiep du'cJc trong mot diTdng tron
•p) Ch^ng m i n h rSng : I D = I E c) ChiJug m i n h r^ng : B A B E = B D B I Bai 5: (1 d i e m ) Cho hinh vuong A B C D Qua d i e m A ve m o t diTcfng t h i n g cat canh B C t a i E va di diTdng thang C D tai F Chiang minh r^ng : x
Trang 40LuySn gi&\ tru6c k1 thi vao Iflp 10 ba miSn B&c, Trung Nam mOn ToAn _ NguySn Birc Tgn
B a i 4: a) AABC c6 BAC + ABC + ACB = I8O" A
Ma BAC = 60" (gt) nen ABC + ACB = 120"
iBC = ^ ABC (BD la tia phan giac)
ICB = - ACB (CE la tia phan giac)
Ta CO : DIC = IBC + ICB ( D I C la goc ngo^i cua tarn giac IBC)
Do do Die = - ( A B C + A C B ) = 6O" Ttf giac A E I D c6 D I C = E A D ( = 6O")
2
=> TiJ giac AEID npi tiep diTdc trong mot difcJng tron
b) AABC CO BD, CE la hai dufdng phan giac c^t nhau tai I
=> I la tarn dUtfng tron noi tiep tarn gi^c ABC
A I la tia phan giac goc BAC => E A I = I A D
'' Xet dUdng tron (AEID) c6 IAD = EAI Suy ra ID = IE Vay I D = IE
c) XetABEIva A B D A c o : EBI (chung), BEI = BDA (tuf giac AEID npi Uep)
Do do ABEI ^ ABDA (g.g) BE H I
V a y B A B E = BD.BI
BD BA
N h a n x e t : Day cung la bai toan de, quen thuoc
B a i 5: Qua A ve du"dng thang vuong goc vdi AE, du"dng thang nay c^t CD tai M
Txjt giac ABCD la hinh vuong
2
Ttf giac MAEC c6 :
MAE + MCE = 90" + 90" = I8O"
=> TiJ giac MAEC noi tiep => A E M = A C M = 4 5 " "
AMAE vuong tai A c6 A E M = 45" => AMAE vuong can tai A => AE = A M
AAMF vuong tai A, A D la dtfdng cao :
AD^ AM^ AF^
Ma A D = AB (ABCD la hinh vuong), AE = A M (chuTng minh tren)
1 1 1
Vay Y = Y +
r-AB^ AE^ AF^
N h a n x e t : Day cijng la bai toan quen thuoc, chung ta c6 the chifng minh
AE = A M bang each chtfng minh AABE = A A D M
DE THI TUYEN SINH VAO LdP 10 THPT, TJNH HA TINH
NAM HOC 2011 - 2012
B a i l : (2 diem) ' n
a) Tim m de dirdng thang y = (2m - l)x + 3 song song vdi dudng thang y = 5x - I
b) Giai he phUdng trinh ^ ^
13x - 2y = 4
\r 1 >
1 - Va I + Va jl^Va + 1 vdi a > 0 va a ;^ 1
B a i 2: (2 diem) Cho bieu thiJc : P =
a) Rut gon bieu thtfc P
^) Vdi nhffng gia tri nao cua a thi P > ^ •
fiai 3: (2 diem) '
^) Tim toa do giao diem cua do thi cac ham so : y = x^ va y = - x + 2
Xac dinh cac gia trj cua m de phiTdng trinh x ^ - x + I - m = 0 c 6 2 nghiem