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vu TUAN (Chu bien) - DOAN MINH CUONG - TRAN VAN HAO 0 6 MANH HUNG - PHAM PHU - NGUYI N TIEN TAI
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NHA XUAT BAN GIAO DUC VIET NAM
VUTUA N (Chu bien)
DOAN MINH CUONG - TRA N VAN HAO - D 6 MANH HUNG PHAM PHU - NGUYfiN TIEN TAI
BAITAP
DAI y 10
(Tdi bdn ldn thu ndm)
NHA XUAT BAN GIAO DUC VIET NAiVI
Ban quy6n thupc Nha xua't ban Giao due Viet Nam
01 - 2011/CXB/814 - 1235/GD Ma so : CB003T1
Ld 1 NOI DAU
Cling voi Sach giao khoa (SGK) Dai so 10, Sach bai tap la tai lieu giao khoa chfnh thiic cho viec hoc va day mon Dai so 10 Trung hoc pho thong.
Sach da dugfc mot Hoi dong chuyen mon cua Bo Giao due va Dao tao thdm dinh.
Sach bai tap Dai so 10 co ca'u true nhu sau
Mdi chuong gom :
1. Phan Kien thdc edn nhd nhac lai nhirng khai niem, menh de, eong thiic phai nhdf de van dung giai cac loai bai tap.
2. Phan Bdi tap mdu gioi thieu mot so loai bai tap hay gap hoac can liru y luyen tap.
3. Vhin Bdi tap bao g6m de bai cac loai bai tap (tu luan, trdc nghiem, tinh toan bang may tfnh bo tiii).
4. Phan Ldi gidi - Hudng ddn - Ddp sd giiip ngudi doc kiem tra, doi chie'u ket qua bai tap tu giai,
De viec hoc co ket qua cao hpc sinh khong nen xem Ibi giai, bu6ng dan trudc khi tu giai.
De viee lam bai tap giiip ndm vimg kie'n thiie dupc hpc va bie't each van dung vao giai cac loai toan, ngu6i hpc nen nghien ngSm de hieu ro
If do, nguyen nhan lam cho minh khong thanh cong (nhu chua thupc cong thiic, may moc trong tu duy, thieu sang tao trong viec dat an phu,...).
Sach bai tap Dai sd 10 bien soan lin nay khdng giai cae bai tap da cho trong SGK. Sach eung ca'p them mdt h6 thd'ng bai tap dupc bidn soan cdng phu va cd phuang phap su pham.
Cae bai tap neu trong sach trai hau he't cac loai bai tap chinh va di ttr d6 de'n khd, tiir don gian de'n phiic tap.
Cac tac gia mong rang cudn sach gdp phdn tfch cue vao hieu qua hpe tap eua ngudi hpc va giang day cua eae thdy cd giao.
Chiing tdi sSn sang tie'p thu cac y kie'n ddng gdp ctia ddc gia de sach td't hon va chan thanh cam on.
CAC TAC GIA
huang I. MENH OE. TAP HOP
§1. M$NH D £
A. KIEN THCTC CAN NHO
1. Mdi menh de phai hoac diing hoac sai.
Mdt mdnh de khdng th^ vvra diing, viira sai.
2. Vdi mdi gia tri cua bie'n thudc mdt tap hpp nao dd, mdnh de ehiia bid'n trd thanh mdt menh de.
3. Phu dinh P cua mdnh de P la diing khi P sai va la sai khi P diing. 4. Menh de "P => Q sai khi P diing va Q sai (trong mpi tnrdng hpp khac P => Q ddu diing).
5. Mdnh di dap cua mdnh d6 P ^> QlaQ => P.
6. Ta ndi hai mdnh de P va Q la hai menh de tuong duong nd'u hai menh d^ P => 2 va Q => F deu diing.
7. Kf hieu V dpc la vdi mpi. Kf hieu 3 dpc la tdn tai ft nha't mdt (hay ed ft nha't mdt).
B. BAI TAP MAU
BAI 1-
Xet xem trong cac cau sau, cau nao la mdnh de, cau nao la menh dd ehtia bid'n ?
a)7+x = 3 ; - b) 7 + 5 = 3.
Giai
a) cau "7 -H X = 3" la mdt mdnh de chiia bid'n. Vdi mdi gia tri cua x thude tap so thuc ta dupe mdt menh de.
b) cau "7 -H 5 = 3" la mdt mdnh de. Dd la mdt mdnh de sai.
BAI 2
Vdi mdi cau
sau,
tim hai
gia tri thue
cua
x de duoc
mdt
menh de diing va
mpt menh de
sai.
a) 3.Y^ + 2x
- 1 --
= 0;
b)
Gidi
4.V
+ 3 < 2x -
- 1 .
a) Vdi x = 1 ta dupc 3.1' -i- 2.1 - 1 = 0 la menh de sai ;
Vdi A = -1 ta dupc 3.(-l)^ + 2(-l) - 1 = 0 la mdnh dd diing. b) Vdi .V = - 3 ta dupe 4.(-3) -i- 3 < 2.(-3) - 1 la menh dd dting ; Vdi X = 0 ta dupc 4.0 + 3 < 2.0 - 1 la menh de sai.
BAI 3
Gia su ABC la mdt tam giac da cho. Lap mdnh di F ^> Q va menh de dao eua nd, rdi xet tfnh diing sai eiia ehiing vdi
a) P : "Gde A bang 90°" , Q : "fiC^ = AB^ + AC^" ; h)P:"A = B \ Q: "Tam giac ABC can".
Gidi
Vdi tam giac ABC da cho, ta cd
a) {P ^ Q) : "Neu gde A bang 90° thi BC^ = AB^ + AC^" la mdnh de diing.
{Q^P): "Ne'u BC^ = AB^ + AC^ thi A = 90° " la mdnh dd diing. b) (P => G) : "Nd'u A = B thi tam giac ABC can" la menh de dung. (Q=> P): "Ne'u tam giac ABC can thi A^B".
(Q => P ) la mdnh dd sai trong trudng hpp tam giac ABC ed A = C nhung A^B.
BAI 4-
Phat bieu thanh ldi cac mdnh dd sau. Xet tfnh diing sai va lap mdnh di phu dinh ciia chiing
a) 3x e R : x^ = -1 ; b) V.v &R:x'- +x + 2^ 0.
Gidi
a) Cd mdt sd thue ma binh phuong cua nd bang -1 . Mdnh de nay sai. Phil dinh cua nd la "Binh phuong eua mpi sd thuc deu khac -1 "
(Vx G R:-.v^^-l).
Menh de nay diing.
b) Vdi mpi sd thirc x deu ed x^ -i- x -h 2 ;^ 0.
Menh de nay diing vi phuong trinh x' -i- x -i- 2 = 0 vd nghiem (A = 1 - 4.2 < 0). Phil dinh ciia nd la "Cd ft nhdt mdt sd thue xma x +x-i-2 = 0" (3x e R : x^ -H X -h 2 = 0).
Mdnh d^ nay sai.
C. BAI TAP
1. Trong cac eSu sau, eau nao la mdt mdnh di, cau nao la mdt mdnh de chiia bid'n ?
a) 1 + 1 = 3 ; b)4 + x<3 ;
c) — cd phai la mdt so nguydn khdng ? d) Vs la mdt sd vd ti. 2. Xet tfnh diing sai eiia mdi mdnh de sau va phat bieu phu dinh eiia nd
a) V3 + V2 = ^ ^ ^ ; h) {yfl - Mf > S ; V3-V2
c) (>/3 -I- V12) la mdt sd huu ti;
x2- 4
d) X = 2 la mdt nghidm ciia phuong trinh —•.—— = 0.
3. Tim hai gia tri thuc cua x di tir mdi cau sau ta dupc mdt mdnh de diing va mdt mdnh de sai.
1 7
a) X < -X ; b) X < - ; c) x = 7x ; d) x < 0. X
4. Phat bidu phu dinh eiia cae mdnh de sau va xet tfnh diing sai eua chiing. a) P : "15 khdng chia hd't cho 3" ;
h)Q : "V2 > 1".
5. Lap mdnh dd P => 2 va xet tfnh diing sai eiia nd, vdi
a)P : "2<3" , Q : "-4<-6 " ;
b)P:" 4 = l", 2 : "3 = 0".
6. Vdi mdi so thue x, xet cac menh de P : "x la mdt so hmi ti", Q : "x' la mdt so huu ti".
a) Phat bieu mdnh dd P =^ 2 va xet tfnh diing sai eua nd ;
b) Phat bidu mdnh de dao cua menh dd tren ;
e) Chi ra mdt gia tri cua x ma menh de dao sai.
7. Vdi mdi sd thuc x, xet cac mdnh dd P : "x^ = 1", Q : "x =1" . a) Phat bidu mdnh de P => 2 va mdnh dd dao cua nd ;
b) Xet tfnh diing sai ciia menh de 2 =^ P ;
e) Chi ra mdt gia tri eiia x ma mdnh dd P => 2 sai.
8. Vdi mdi sd thuc x, xet cac mdnh de P : "x la mdt sd nguyen", 2 : -^ + 2 la mdt so nguyen".
a) Phat bieu mdnh de P => 2 va menh dd dao ciia nd ;
b) Xet tfnh diing sai eiia ca hai menh de tren.
9. Cho tam giac ABC. Xet cac mdnh dd P : "AB = AC", Q : "Tam giac ABC can".
a) Phat bieu menh d^ P => 2 va xet tfnh diing sai cua nd ;
b) Phat bieu mdnh de dao cua mdnh de tren.
10. Cho tam giac ABC. Phit bieu menh de dao eua cac mdnh de sau va xet tfnh diing sai cua ehiing.
a) Nd'u AB - BC = CA thi ABC la mdt tam giac ddu ;
8
b) Nd'u AB > BC thi C > A ;
e) Neu A = 90° thi ABC la mdt tam giac vudng.
11. Sir dung khai nidm "dieu kien edn", hoac "didu kidn du", hoac "dieu kidn cdn va du" (nd'u cd the) hay phat bieu cae mdnh dd trong bai tap 10.
12. Cho tii giac ABCD. Phat bidu mdt didu kidn can va dii de a) ABCD la mdt hinh binh hanh ;
b) ABCD la mdt hinh ehu nhat;
c) ABCD la mdt hinh thoi.
13. Cho da thiic /(x) = ax^ + hx + c. Xet mdnh dd "Nd'u a + Z? + c = 0 thi /(x) cd mpt nghiem bang 1". Hay phat bieu mdnh dd dao ciia menh dd trdn. Ndu mdt diin kidn cdn va dii de /(x) ed mdt nghidm bang 1.
14. Diing kf hidu V hoac 3 dl vid't cac mdnh de sau
a) Cd mdt sd nguyen khdng chia bet cho chfnh nd ;
b) Mpi sd (thue) cdng vdi 0 ddu bang chfnh nd ;
c) Cd mdt so huu ti nhd hon nghich dao ciia nd ;
d) Mpi so tu nhien deu ldn ban sd dd'i ciia nd.
15. Phat bieu thanh ldi cac menh di sau va xet tfnh dung sai ciia chiing. a) Vx G R : x^ < 0 ; b) 3x e R : x" < 0 ;
2 2
c) Vx e R : ^- ^ = x + 1 ; d) 3x e R : ^ ^ = x + 1 ; x - 1 x- 1
e)VxeR:x ^ + x+l>0 ; g)3xGR:x ^ + x-hl>0 . 16. Lap mdnh de phii dinh cua mdi mdnh de sau va xet tfnh dung sai cua nd. a) Vx e R :x.l =x ;
b) Vx 6 R : x.x = 1 ;
c) Vn e Z : « < n .
17. Lap mdnh di phu dinh ciia mdi mdnh dd sau va xet tfnh diing sai ciia nd. a) Mpi hinh vudng deu la hinh thoi ;
b) Cd mdt tam giac can khdng phai la tam giac ddu.
§2. TAP HOP
A. KIEN THOC CAN NH 6
1. AcP^(Vx, x G A=>x e P)
2. A=Po(Vx, x e A^XGP) .
B. BA! TAP MAU
BAIl
Lidt kd cac phdn tir cua mdi tap hop sau
a) Tap hpp A cac so chfnh phuong khdng vupt qua 100.
b)TaphppP= {rt e N|«(«+ 1)<20}.
Gidi
a)A= (0, 1.4,9, 16,25,36,49,64,81, 100} ;
b) 5=10,1,2,3,4} .
BAI 2
Tim mdt tfnh chat dac trung xac dinh cac phdn tir cua mdi tap hpp sau a)A = {0, 3, 8, 15,24,35} ; b) P = {-1 + Vs ;-1 - Vs}.
Gidi
a) Nhan xet rdng mdi sd thudc tap A cdng thdm 1 deu la mdt chfnh phuong. Tur dd ta ed the vid't
A = {/7^- 1 In G N, 1 5) .
a) Dimg kf hieu doan, khoang, nira khoang de vid't lai eae tap hpp b) Bieu didn cac tap hpp A, B, C, D trdn true so.
Gidi
a)A = h3;2] ; fi = (0;7]; C = (-^ ;-1) ; D = [5,+c»).
A B
- 3 0 2 0 7
trdn ;
C D - 1 0 0 5
BAI 2
Xac dinh mdi tap hpp a) (-5 ; 3) n (0 ; 7) ; c)R\(0;+oo) ;
so sau va bieu didn trdn true sd b) (-1 ; 5) u (3 ; 7);
d) (-o); 3) n (-2 ; +«).
Gidi
a) (-5 ; 3) n (0 ; 7) = (0 ; 3). ^'/////////////////,(^ i—i ymmwMWM lm~ b) (-1 ; 5) u (3 ; 7) = (-1 ; 7). •'''•'''''\ Q ' ' ' ' 5 ' f'''"^
C) R \ (0 ; +00) = (-00 ; 0]. ],y/////////////////////,,y//,y/;t. 0
d) (-GO ; 3) n (-2 ; +QO) = (-2 ; 3). m'mmmj \ ^ ^ ^ yf/f/MJH///M - 2 0 3 •
15
C. BAI TAP
28. Xac dinh mdi tap hpp sd sau va bieu didn nd trdn true sd a) (-3 ; 3) u (-1 ; 0) ; b) (-1 ; 3) u [0 ; 5] ;
c) (-co ; 0) n (0 ; 1) ; d) (-2 ; 2] n [1 ; 3).
29. Xac dinh mdi tap hpp sd sau va bieu didn nd trdn true so a)(-3;3)\(0;5) ; b) (-5 ; 5)\(- 3 ; 3) ;
c)R\[0;l] ; d) (-2 ; 3)\(- 3 ; 3).
30. Xac dinh tap hpp A r^B, voi
a)A = [l;5];f i = (-3;2)u(3;7) ;
b) A = (-5 ; 0) u (3 ; 5) ; fi = (-1 ; 2) u (4 , 6).
31. Xac dinh tfnh diing, sai cua mdi menh de sau
a) [-3 ; 0] n (0 ; 5) = {0} ; b) (-QO ; 2) U (2 ; +oo) = (-oo ; +oo) c) (-1 ; 3) n (2 ; 5) = (2 ; 3) ; d) (1 ; 2) u (2 ; 5) = (1 ; 5). 32. Cho a, b, c, d la nhiing sd thuc vi a < b < c < d. Xac dinh eae tap hpp so sau a) {a;b)n{c;d); b) {a ; c] n [b ; d) ;
c) {a;d)\{b;c); d) {b;d)\{a; c).
§5. S6 GAN DUNG. SAI s 6
A. KIEN THUC CAN NH 6
Cho a la so gdn diing ciia a.
1. A^ = \d - a\ dupc gpi la sai sd tuydt ddi cua sd gdn diing a.
2. Nd'u A^ < (i thi d dupc gpi la dp chfnh xac ciia sd gdn diing a va quy udc vid't gpn la d = a ± d.
3. Cach vid't sd quy trdn cua sd gdn diing can cii vao dp chfnh xac cho trudc. Cho sd gdn diing a vdi dp chfnh xac d (tiic la a = a ± d). Khi dupc ydu cdu quy trdn sd a ma khdng ndi rd quy trdn dd'n hang nao thi ta quy trdn a din hang cao nhdt ma d nhd hon mdt.don vi eua hang dd.
16 '
B. BAI TAP MAU
BAI 1.
Cho so d =37 975 421 ± 150. Hay vie't sd quy trdn ciia sd 37 975 421.
Gidi
Vi dp chfnh xac dd'n hang tram ndn ta quy trdn sd 37 975 421 de'n hanj nghin. Vay sd quy trdn la 37 975 000.
BAI 2.
Bid't sd gdn dung a = 173,4592 cd sai sd tuyet dd'i khdng vupt qua 0,01. Vid't sd quy trdn ciia a.
Gidi
Vi sai sd tuydt dd'i khdng vupt qua —— nen so quy trdn ciia a la 173,5.
C. BAI TAP
33. Cho bid't V3 = 1,7320508... . Vid't gdn dung v3 theo quy tdc lam trdn de'n hai, ba, bdn chu sd thap phan cd udc lupng sai sd tuydt dd'i trong mdi trudng hpp.
34. Theo thd'ng ke, dan sd Viet Nam nam 2002 la 79715 675 ngudi. Gia sir sai sd tuydt dd'i ciia sd lieu thd'ng kd nay nhd hon 10 000 ngudi. Hay vid't sd quy trdn cua so trdn.
35. Dp eao ciia mdt ngpn nui la h = 1372,5 m ± 0,1 m. Hay vie't sd quy trdn ciia sd 1 372,5.
36. Thuc hidn cac phep tfnh sau tren may tfnh bd tui.
a) Vl3 X (0,12) lam trdn kd't qua de'n 4 chir sd thap phan.
b) ^/5 : >/7 lam trdn kd't qua dd'n 6 chii sd thap phan.
2BTDS10(C)-A 17
BAI TAP ON TAP CHUONG I
37. Cho A, B la hai tap hpp va menh d^ P : "A la mdt tap hpp con ciia B". a) Viet P dudi dang mdt menh de keo theo.
b) Lap mdnh dd dao cua P.
38. Dung kf hieu V va 3 de vie't mdnh de sau rdi lap mdnh de phii dinh va xet tfnh dting sai eiia cac mdnh de dd.
a) Mpi sd thue cdng vdi sd dd'i ciia nd deu bdng 0.
b) Mpi sd thuc khac 0 nhan vdi nghich dao cua nd ddu bang 1. c) Cd mdt sd thuc bang sd dd'i ciia nd.
39. Cho A, B la hai tap hpp, x G A va x g B. Xet xem trong cac menh di sau, mdnh dd nao dting.
a)x G AnB; h)x G Au B ;
c)xG A\fi ; d)xG fi\A.
40. Cho A, fi la hai tap hpp. Hay xac dinh cac tap hpp sau
a) (A n fi) u A ; h){AvjB)n.B;
c){A\B)uB; d){A\B)n{B\A).
41. Cho A, B la hai tap hpp khac rdng phan bidt. Xet xem trong cac mdnh de sau, menh de nao diing.
a)Aczfi\A; h)A(zAuB;
c)AnBc:AyjB; d)A\fi(=A.
42. Cho a, b, c la nhimg sd thtrc vaa 2 ) : "Nd'u x la mdt sd hiru ti thi x^ cung la mdt sd hiiu ti". Menh de diing.
b) Mdnh dd dao la "Nd'u x~ la mdt so hiiu ti thi x la mdt sd hihi ti". e) Chang ban, vdi x = v2 mdnh dd nay sai.
7. a) (P =i> 2) : "Nd'u x^ = 1 thix = 1". Mdnh dd dao la "nd'ux = 1 thi x^ = 1". b) Mdnh de dao "Nd'u x = 1 thi x^= 1" la diing.
c) Vdi X = -1 thi mdnh de (P =:?> 2 ) sai.
8. a) (P => 2) : "Nd'u x la mdt sd nguydn thi x -i- 2 la mdt sd nguydn". ( 2 => P) : "Nd'u X -I- 2 la mdt sd nguydn thi x la mdt sd nguydn".
Ca hai mdnh dd nay deu diing vi tdng, hidu ciia hai sd nguydn la mpt sd nguyen.
9. a) (P =^ 2 ) : "Nd'u AB =AC thi tam giac ABC can". Mdnh de nay diing. b) Menh di dao la "Ne'u tam giac ABC can thi AB =AC'.
Nd'u tam giac ABC can ma ed BA = BC ^ AB thi mdnh de dao sai.
10. a) "Nd'u ABC la mdt tam giac ddu thi AB = BC = CA", ca hai mdnh dd ddu diing.
b) "Ne'u C >A thi AB > BC. Ca hai mdnh dd ddu diing.
c) "Ne'u ABC la mdt tam giac vudng thi A = 90° ".
Nd'u tam giac ABC vudng tai B (hoac C) thi mdnh dd dao sai. 11. a) Didu kidn cdn va dii dd tam giac ABC ddu la AB = BC = CA. h) Dieu kien cdn va du de Afi > BC la C > A.
e) Dieu kidn dii de tam giac ABC vudng la A = 90°.
12. a) Ttr giac ABCD la mdt hinh binh hanh khi va ehi khi AB // CD va AB=CD.
20
b) Tii giac ABCD la mdt hinh chii nhat khi va chi khi nd la mdt hinh binh hanh va ed mdt gdc vudng.
c) Tir giac ABCD la mdt hinh thoi khi va chi khi nd la mdt hinh binh hanh va cd hai dudng cheo vudng gdc vdi nhau.
13. Mdnh de dao la "Nd'u /(x) ed mdt nghidm bdng 1 thi a -t- Z? + c = 0". "Didu kidn cdn va du dd /(x) = ax -\- bx -\- c cd mdt nghiem bang 1 la a + b-^ c = 0".
14. a) 3n G Z : n'i n ;
b)VxGR:x-h O = x ;
e) 3x G Q : X < - ;
X
d) Vn G N : n > -n.
15. a) Binh phuong ciia mpi sd thue deu nhd hon hoac bdng 0 (mdnh de sai). b) Cd mdt sd thuc ma binh phuong ciia nd nhd hon hoac bang 0 (mdnh di diing).
x^- l
e) Vdi moi sd thue x, — = x -i-1 (mdnh de sai);
x - 1
x^- l
d) Cd mdt sd thuc x, ma = x + 1 (menh dd diing) ;
x - 1
e) Vdi mpi sd thuc x, x^ + x -I-1 > 0 (mdnh di diing);
g) Cd mdt sd thue x, ma x^ -h x + 1 > 0 (mdnh de diing).
16. a) 3x G R : X. 1 ;^x. Mdnh dd sai.
b) 3x G R : X. X ^ 1. Mdnh de diing.
e) 3n G Z : n > n . Mdnh de diing.
17. a) Cd ft nhdt mdt hinh vudng khdng phai la hinh thoi. Mdnh dl sai. b) Mpi tam giac can la tam giac deu. Mdnh de sai.
18. a) Sai; b) Dting ; c) Sai;
d) Sai; e) Sai; g) Diing.
21
19. a)A =
n{n -I- 1)
n e N, 1 < n < 5 k
b) B
n 2 _
nGN , 2 X G A hay fi c A.
23. A = {1,2,3,6,9 , 18} ;
B= {1,2,3,5,6 , 10, 15,30) ;
Ar\B= {1,2,3,6} ;
A u fi = {1, 2, 3, 5, 6, 9, 10, 15, 18, 30} ;
A\fi = {9, 18} ;fi\A = {5, 10, 15, 30}.
24. Ar\B= {3(2/t- l)\kG Z}.
-)?
25. a)AnA=A ; h) A u A = A ;
c)A\A= 0; d)An 0 = 0 ;
e)Au0=A ; g)A\ 0 = A;
h)0\A = 0 .
26. a) fi c A ; b) A c fi ; c) fi c A ; d)Acfi ; e)A ^B; g) A n fi = 0 . 27. a) CjjQ la tap cac sd vd ti.
b) Cf^2N la tap cae so tu nhidn le.
28. a) (-3 ; 3) u (-1 ; 0) = (-3 ; 3) ; 1^3 /////'//////'//{—I—I—I—I—I—)'////////////» 0 3
b) (-1 ; 3) u [0 ; 5] = (-1 ; 5] ; ////>'/, - 1 0 5
c) (-co ; 0) n (0 ; 1) = 0 ; ///////////////////////////////////////////////////////////>
d) (-2 ; 2] n [1 ; 3) = [1 ; 2]. //////////////////////////,c >////////////////////////> 1 2
29.a)(-3;3)\(0;5 ) = (-3;0] ; ////////////////////////'(—I I T:'/////'/'///'//'//'//A -3 0
b) (-5 ; 5)\(-3 ; 3) = (-5 ; -3] u [3 ; 5); /MWM'M—i ]///////////////,[ i—)mm^ ^ -5- 3 3 5
e)M\[0 ; l] = (-oo;0)u(l ;+oo); }m^
d) (-2 ; 3) \ (-3 ; 3) = 0 . ////////////////////;y/////////////////////////////////////>
3t). a) A n fi = [1 ; 2) u (3 ; 5] ; b) A n fi = (-1 ; 0) u (4 ; 5). 31. a) Sai; b) Sai; c) Dung ; d) Sai. 32. a){a;b)n(c;d) = 0; h) (a ; c] n[b ; d) = [b ; c] ;
c) {a;d)\{b;c) = {a ; b]u [c ; d) ; d) {b ; d)\{a ; c) = [c ; d). 33. Nd'u la'y S bang 1,73 thi vi 1,73 < Vs = 1,7320508... < 1,74 ndn ta cd |V3-L73|<|1,73-1,74| = 0,01.
vay sai sd tuydt dd'i trong trudng hpp nay khdng vupt qua 0,01. Tuomg tu, nd'u ldy V3 bang 1,732 thi sai sd tuydt dd'i khdng vupt qua 0,001. Ne'u la'y S bdng 1,7321 thi sai sd tuydt dd'i khdng vupt qua 0,0001. 34. Dan sd Vidt Nam nam 2002 la 79 720 000 ngudi.
23
35. 1373 m :
36. a) 0,0062; b) 0,646310.
37. a) P : Vx (x G A z:^ x G B).
b) Menh dd dao la Vx (x G fi =^ x G A) hay "B la mdt tap eon cua A". 38. a) Vx G R : X + (-x) = 0 (dung).
Phil dinh la 3x e R : x + (-x) ^ 0 (sai).
b) VxG R\{0 } : x. - = 1 (dung).
X
Phil dinh la 3x G R {0} : x. - -/ 1 (sai).
v
e) 3x G R : x = - x (diing).
Phil dinh la Vx G R : x ^ -x (sai).
39. b) ; e).
40. a) (A n fi) u A = A ; h) {A u B) r^ B = B ; C){A\B)UB = AUB; d){A\B)r^{B\A) = 0. 41. b);e);d).
42. a) 0 ; h) {a ; c)\{b} ;
c) {a-b]; d) {a ; b).
43. a) (-2 ; 3] ; b) (-15 ; 14) ;
e)(0;5) ; d) (-co ;-1 ] u [1 ;+oo). 44. a) (-00 ; 0] u [1 ; 2] u [3 ; +oo) ; b) (-oo ; 4] u [5 ; -^-oo) ; e)(-2 ; l)u(3;7) ; d) (-1 ; 1] u [4 ; 5). 45. a) c /(x,) > /(X2).
4. Xet ehilu bid'n thien eiia mdt ham sd la tim cac khoang ddng bid'n va cac khoang nghich bid'n cua nd. Ke't qua dupc tdng kd't trong mdt bang gpi la bang bid'n thien.
5. Ham sd y = f(x) vdi tap xac dinh D gpi la ham sd chdn nd'u Vx G D thi -X G D va /(-x ) = /(x).
D6 thi eiia ham sd chdn nhan true tung lam true dd'i xiing.
6. Ham sd y = /(x) vdi tap xac dinh D gpi la ham sd le nd'u yxsD thi -X G D va /(-x ) = -fix).
D 6 thi cua ham sd le nhan gd'e toa dd lam tam dd'i xiing.
25
B. BAI TAP MAU
BAIL
Theo thdng bao eua Ngan hang TMCP Phuong Nam ta cd bang dudi day vd lai sudt tidn giri tid't kidm kieu bae thang vdi sd tidn giri tii 50 tridu VND trd ldn dupc dp dung tir 20-12-2005.
Ki ban (so thang) Lai suat (% /thang)
3
0,715
6
0,745
12
0,785
18
0,815
24
0,825
Coi lai sudt y la ham sd ciia ki ban x ( kf hieu y = /(x)).
a) Hay tim tap xac dinh ciia ham sd nay.
b) Tim cae gia tri /(3) ; /(18).
e) Hieu thd' nao vl gia tri a.f(6), nd'u sd tien giri laa(a> 50 tridu) VND ?
Gidi
a)TacdD = {3;6 ; 12; 18;24} .
b) /(3) =0,715; /(18) =0,815.
c) Theo bie'u lai sudt, nd'u giri vao quy tid't kidm la a vdi ki han 6 thang thi mdi thang se ed tiln lai la a.0,745% VND.
BAI 2.
Tim tap xac dinh cua eae ham sd sau
3x- 2
a) fix) 4x^ + 3x - 7 b) fix) = ^ ^ + ^/37^ .
Gidi
Uai ham sd trdn diu dupe cho bdng cdng thtic. Theo quy udc ta cd
a) fix) la mdt phan thiie ndn mdu thiie 4x^ -i- 3x - 7 ^ 0, tiic la 7
(x - l)(4x -^1)^0 hay x ^ 1 va x ^ ---. Vay tap xae dinh eiia ham sd dacholaD = R \ jl ;- -
26
b) Ham sd chi xac dinh vdi nhflng gia tri cua x thoa man dilu kien X *'h
13x - 5 > 0 -I
hay |x^ 3
Do dd tap xac dinh ciia ham sd da eho la
D = h u (3 ; -h oo).
BAI 3.
Xet tfnh ddng bid'n va nghich bid'n ciia cac ham sd sau trdn khoang dupc ehi ra.
a) fix) = -2x^ - 7 trdn khoang (-4 ; 0) va trdn khoang (3 ; 10); b) fix) = x-1 trtn khoang (-oo ; 7) va trdn khoang (7 ; -i-ao).
Gidi
a) V Xj, X2 G R va Xj ?t X2 , ta ed
fix,) - fix2) = -2xf - 7 - (-2x| - 7)
= 2ixl-xf)
= -2(Xi -X2)(Xi -1-X2).
V XJ , X2 e (-4 ; 0) va x, < X2, ta cd x, - X2 < 0 va Xj -1- X2 < 0. Tiir(*), suyra /(x,) - /(xj) < 0 hay fix,) < fix^). Vay ham sd ddng bid'n trdn khoang (-4 ; 0).
V Xj, X2 e (3 ; 10) va Xj < X2, ta cd Xj - X2 < 0 va Xj -1- X2 > 0. Tir (*), suy ra fix,) - fix^) > 0 hay fix,) > fix^). vay ham sd nghich bil^n trdn khoang (3 ; 10).
b) V Xj, X2 G R \ {7} va Xj ?t X2, ta cd
1 X2 _ 7(x2 — X,) /(X,)-/(X2 ) = X,-l XJ -1 ix,- 7)(X2 - 7)
(*)
(*)
27
Vx,,X2 G (-00 ; 7) va Xl <.V2, tir (*) ta cd /(A:, ) -/(X2 ) > 0 hay /(x,) > /(X2) (vi x. - .V| > 0, .Vl - 7 < 0 va .V2 - 7 < 0).
Vay ham sd nghjch bid'n trdn khoang (-00 ; 7).
Vx,,X2 G (7 ; +00) va Xi /(-^2) • vay ham sd ciing nghich bid'n trdn khoang (7 ; +co).
BAI 4
Xet tfnh chdn le cua cac ham sd sau
,.2 , ')
a) fix) = V2x + 3 ; b) /(x) = ^- ^ ; c) fix) = -x'-l.
Gidi
a) Dd tha'y tap xac dinh cua ham sd laD = ; -I- 00 va 2 G D,
nhtmg - 2 g D. Vay ham sd da cho khdng la ham sd chan cQng khdng la ham so le.
b) Tap xac dinh ciia ham sd la D = M\{0}. Nd'u x G D thi x ;t 0, do dd -X ^ 0 vi-x e D. Ngoai ra, V x T^ 0,
fi-x) =
Vay fix) la ham sd le.
i-x)' + 2 x^ +2
X -fix).
c) Ta ed tap xac dinh D = R ndn thoa man dilu kien x e D thi -x G D.
Nhung /(-I) = -2 ^ /(I) = 0 va /(-I) ^ -/(I) . Vay ham sd fix) khdng la ham sd chdn va cQng khdng la ham sd le.
C. BAI TAP
1. Bieu dd sau (h.3) bieu thi san lupng vit, ga va ngan lai qua 5 nam cua mdt trang trai. Coi y = fix), y =gix) va y =hix) tuang iing la cac ham so bilu thi su phu thudc sd vit, sd ga va sd ngan lai vao thdi gian x. Qua bieu dd, hay a) Tim tap xdc dinh cua mdi ham sd da neu ;
b) Tim cac gia tri /(2002), ^(1999), /z(2000) va ndu y nghia cua ehiing ; c) Tfnh hidu /2(2002) - hil999) va ndu y nghia eiia nd.
28
Tram t nghin con
7-
6-
5-
4-
3-
2-
1-
t
1 1 San luong vi 1 1 San luong ga an
I H San luong n^ 5,6
4.5
3,1
0,05
ai
3,8
0,3
6.0
4.0
1,0
^
6,1
4,0
1,5
6,2
4,3
2,1 1 1 1
I
99
?
99<
mo{
)
///n/i i
200 1
2002 N£
2. Tim tap xac dinh cua cac ham sd
a) J = -X + 7x - 3 ; b) y 7 + x X- + 2x - 5
e) y = ^Ax + I - V-2x + 1 ;
d) }'= ,
Vx + 9
2x + l
^''^"(2 x + l)(x-3)'
3. Cho ham sd
y = fix) =
X- + 8x - 20
2x- 3
x - 1 vdi X < 0
-x" -\- 2x vdi X > 0.
Tfnh gia tri eua ham sd dd tai x = 5 ; x = - 2 ; x = 0 : x = 2. 4. Cho ham sd
V = six) =
I V-3x + 8 vdi X < 2 ^ x -I- 7 vdi X > 2.
Tfnb cac gia tri g(-3) ; g(2) ; g(l) ; g(9).
29
5. Xet tfnh ddng bid'n, nghich bid'n cua ham sd tren cac khoang tuong iing a) V = -2x + 3 trdn R.
b) v = x^ -I- 1 Ox -I- 9 trdn (-5 ; +oo) ;
c) y^ -y—^ trdn (-3 ; -2) va (2 ; 3).
6. Xet tfnh chdn, le ciia cac ham sd
a) >• = - 2 ; h) y^ 3x- - 1 ;
c) J = -x^ -h 3x - 2 ;
§2. HA M S6y = ax + b
d) y = -x" + x^ + 1
A. KIEN THUC CAN NHO
1. Ham sd bac nha't y = ax -i- b, {a ^ 0) Tap xdc dinh D = R ;
Bang bie'n thidn
a>0
a < 0
.Y y
—00
—00 -
+ 0 0
:r- +00
X y
—00 + 0 0
+ 0 0
• *- —0 0
Do thi la mdt dudng thdng khdng song song va khdng triing vdi cae tnic toa dp.
De ve dudng thdng j = ax + d ehi edn xac dinh hai did'm khac nhau cua nd. 2. Ham sd hdng y = b
Tap xac dinh D = M ;
Ham sd hdng la ham sd chdn ;
D6 thi la mdt dudng thdng song song hoac trung vdi true hoanh va cat true tung tai diem cd toa dp (0 ; b).
3. Ham sd y = |x|
Tap xac dinh D = R ;
Ham sdj = |x| la ham sd chdn ;
Ham sd ddng bid'n trdn khoang (0 ; +oo) va nghich bid'n trdn khoang (-QO ; O). 30
B. BAI TAP MAU
|BAI 1
thdng di qua hai diem
Viet
phuong
trinh
dang y = ax -vh cua dudng
M(-
1 ; 3) va
A^(l
; 2), ve dudng thdng dd. Gidi
Vi dudng thdng cd phuong trinh dang y = ax -^b ndn ta edn xac dinh cac he sd a vi b. Dudng thdng dd di qua M(-l ; 3) va A/^(l ; 2), tiie la toa dp eiia M vi N thoa man phuong trinh y = ax -\- b. Ta c6
|3 = - a + d \2b^5
<=> s -i 1
\2 = a + b '^|2 a = - l
a = --
Vay dudng thdng da cho cd phuong
trinh la >" = -—x + —, va dd thi
cua nd duPc ve trdn hinh 4. Hinh 4 BAI 2
Hay vid't phuong trinh dudng thdng y =
ax -\- b ling vdi mdi hinh sau
a) y.
b)
y'
I
A
B/
/-2
n 0
1-
X
\ M
1 \ W
0 3 5 \ X 2 2 \
Hinh 5
Hinh 6
31
Gidi
a) Dudng thdng trdn hinh 5 di qua hai diem A(0 ; 3) va fi(-2 ; 0). Vi phuong trinh eiia dudng thdng cd dang y = ax -\- b ndn ta ed
[3 = 6
\0 = -2a + b < ^ <
b=3
3
a = —•
Vay dudng thdng cd phuong trinh la j = — x + 3. b) Tuong tu, vdi hinh 6, ta ed
l = la + b <
0 = ^a + b
a = -1 -f
Vay phuong trinh dudng thdng la y = - x +
BAI 3
Ve dd thi cua cae ham sd sau
a) V = -2x + 3 ; b)>' = fx + 2 vdi X > 2
1 vdi X < 2 :)y = -^l2.
Gidi
a) Ta tha'y cac diem A(0 ; 3) va fi[ - ; 0 ) thudc dd thi. Vay dd thi eiia ham sd la dudng thdng AB tren hinh 7.
yt
I
O
y=l A -H h
o
- ^
v =
A'
-i2
Hinh 9
32
Hinh 7
Hinh 8
b) D6 thi ciia ham sd gdm hai nira dudng thdng (h.8).
Trong nira khoang (-00 ; 2] ham sd cho bdi cdng thiic y = 1 nen cd dd thi la nira dudng thdng At.
Trong khoang (2 ; +00) ham sd cho bdi cdng thiic y = x -\-2 ndn ed dd thi la nira dudng thdng Bs khdng ke diem (2 ; 4).
c) Ham sd v = -%/2 la ham hdng, dd thi dupc ve d hinh 9.
BAI 4
Ve dd thi ciia ham sd
a) y = Ixl + 2x ; b) J = |3x - 2I.
Gidi
a) Do |x| = nen ed thd vid't
X vdi X > 0 -X vdi X < 0,
V = X + 2x = 3x vdi X > 0
X vdi X < 0.
Tir dd ta thay ham so ddng bid'n trdn toan bp true so.
Dd thi ham so da eho dupe ve trdn hinh 10. b) Ta ed
2 3x - 2 vdi X > —
Hinh 10
|3x - 2| = -
2
-3x + 2 vdi X < —•
Bang bie'n thien 2
X
_v
—00 — +00 +00~-^_^___^ ^^^,,-^+0 0
D6 thi ham sd da cho duoc ve ti
•en
hinh ] 1.
Hinh 11
3.B1BS10{C)-A 33
C. BAI TAP
7. Ve dd thi ciia cac ham sd sau va xet tfnh chdn le eiia chiing
2 4
a) y = --X + 2 ; h) y = -x -I ;
c) y = 3x ;
8. Ve dd thi ham sd
d) y = 5 ; 5) >- = V2-I .
2x - 1 vdi X > 1
— x + 1 VOl X < 1.
9. Viet phuong trinh dudng thdng song song vdi dudng thang y = 3x -2 va di qua diem
a) M(2 ; 3); b) A/(-l ; 2).
10. Xdc dinh cae hd sd a va fc de dd thi eiia ham so y = ax -\- b di qua eae diem sau a) A(^ | ; - 2 j va fi(0 ; 1) ; b) M(-l ; -2) va Ni99 ; -2) ;
c)fi(4;2)va2(l;l) .
11. Vid't phuong trinh dudng thing y = ax -\- b iing vdi hinh sau a) b) c)
Hinh 12 Hinh 13 Hinh 14 34 3.fflDS10(C) - B
12. Cho ham sd y = |-x - 3| + |2x + l| + |x + ij. Xet xem diem nao trong cac diem sau day thudc dd thi ciia nd.
a) Ai-l ; 3) ; b) fi(0 ; 6) ;
c)C(5;-2); d)D(l;10).
13. Lap bang bid'n thidn va ve dd thi cua mdi ham sd
3 a)y = \2x - 3\ ;
h) y -X + 1
c) y = 1-2x1 - 2x.
§3. HAM S 6 BAC HAI
A. KIEN THUC CAN NHO
1. Ham sd bae hai y = ax -\- bx -\- c, {a ^ 0) cd tap xac dinh D = 2. D6 thi eua ham sd bac hai y = ax + i>x + c la mdt dudng parabol cd dinh la diem / b -A] . .,.-., . J . u ' b -—— ; —— , CO true dot ximg la duong thang x = --—•
2a 4a J ' 2a
Parabol nay quay be ldm ldn tren nd'u a > 0 (h. 15), xud'ng dudi nd'u a < 0 (h. 16).
a<0
Hinh 16
3. Dl ve dudng parabol y = ax + bx -\- c, ia^O)ta thuc hien cac budc sau
Xdc dinh toa dd dinh l\ --r- ; -r-
\ 2a 4a
35
Ve true dd'i xiing d la dudng thing x = - 2a
Xic dinh giao diem ciia parabol vdi eae true toa dd (nd'u cd). Xac dinh thdm mdt sd diem thudc dd thi. Chang ban, diem ddi xiing vdi giao didm eiia dd thi vdi true tung qua true dd'i xiing cua parabol.
Dua vao kd't qua trdn, ve parabol.
4. Bang bid'n thidn
a>0 a<0
X y
—00 + 0 0
b
2a
-A
^ 4a-^
+ 0 0 + 0 0
X y
—00
b
2a
-A
^4a ^
+ 0 0 —00
—00 ^
B. BAI TAP MAU
BAIl
Lap bang bie'n thidn va ve dd thi cae ham so
a) y = -x + 2x - 2 ; h)y = 2x + 6x + 3.
Gidi
a) Ham so da eho la ham sd bac hai vdi a- -I ; b = 2vic = -2. Ta cd - ^ = 1 ; A = b^ - 4ac = -4 ; - ^ = -1 . 2a 4a
Vi a < 0, ta cd bang bid'n thidn
—00 + 0 0
• —00
36
Ham sd ddng bid'n trdn khoang (-00 ; 1) va nghich bid'n trdn khoang (1 ; +co).
Parabol y = -x^ + 2x - 2 cd dinh la /(I ; -1), true dd'i xiing la dudng thdng d : X = I ; giao diem ciia dd thi vdi true tung la diim A (0 ; -2). Diim dd'i xiing vdi A qua d la A'(2 ; -2) thude dd thi. Dd thi ciia ham so duoc ve trdn hinh 17.
Hinh 17
b) Dd'i vdi ham sd da cho ta cd Vi a > 0, ta cd bang bid'n thidn
2a
3_ . __A_ 2 ' 4a
3_ '2
X y
— 00
+ 00 ...___^
3
2
-^^ ^ 3 ^ ^ 2
+ 00
^_^+0 0
Parabol y = 2x + 6x + 3 cd true dd'i
xiing la dudng thdng d : x = -— ; dinh
3 3 ; giao diem vdi true tung 2 ' 2^
(0 ; 3) va eae giao diim vdi true hoanh la
^M^;0^ va - 3 + V3 ;0 Do thi
ciia ham sd' duoc ve trdn hinh 18. Hinh 18 37
RATI
Xac dinh ham sd bac hai 2
J = 2x + Z7X + c, bid't ring dd thi
cua nd
a) Cd true dd'i xiing la dudng thang x = 1 va cdt true tung 1 .ai diim (0 4);
b)Cddinhla/(-l ;-2);
c) Di qua hai diim i4(0 ;
-l)vaS(4;0);
d) Cd hoanh dd dinh la 2
va di qua diem M(l ; -2). Gidi
De xac dinh ham sd ta phai xac dinh cac hd so 6 va c tii cae dilu kidn da cho.
a) Ta cd -- - = 1 <» 6 = - 2a = - 4 ; 4 = 2.0 + &.0 + c <» c = 4. 2a Ham sd edn tim la j = 2x - 4x + 4.
b) Ta cd - ^ = -l=>b = 2a = 4; • 2a 4a = -2
, ., 4ac-b^ ^ do do ;; = -2 . 4a (1)
Thay a = 2vab = 4 vao (1) ta dupc —- — = -2 <=> c = 0. 8 ^
' * 2
Ham sd can tim lay = 2x + 4x.
c) Vi parabol di qua A(0 ; -1) va fi(4 ; 0) ndn ta ed he phuong trinh
[-1 = 2.0 + Z7.0 + c Jc = -1
lo = 2.16 + 4.6 + c "^ 132 + 4.6 + c = 0 ^ ^
7 ^1
Ham sd can tim la v = 2x x - 1. 4
c = - l b = - 31
d) Ta cd - — = 2 => 6 = -8 ; parabol di qua diem M(l, -2) ndn -2 = 2.1 + 6.1 + c <=>-2 = 2 - 8 + c =^ c = 4.
Ham sd can tim la j = 2x - 8x + 4.
38
BAI 3
Vid't phuong trinh cua parabol y = ax^ + bx + c ling vdi mdi hinh sau
Hinh 19 Hinh 20 Hinh 21
Gidi
a) Trdn hinh 19, ta tha'y diem 7(3 ; 4) la dinh ciia parabol va diim (-1 ; 0) thudc dd thi. Ta cd
2a = 3 (1) ; 4a = 4 (2) va 0 = a - 6 + c (3). Tii (1) suy ra 6 = -6a. Thay bieu thiie eiia 6 vao (2) ta dupc
4ac - 36a^
Ta = 4 <» c - 9a = 4 ivia^Q).
Thay b = -6a va c = 4 + 9a vao (3) ta dupc
a + 6a + 4 + 9a = 0<=> a 4'
Tir dd 6 = -6a = I- va c = 4 + 9a = 4 - ^ = ~
2 4 4
1 2 3 7
Vay j = -^A- +2^+4 "
b) Trdn hinh 20, ta thdy dd thi cit true tung tai diem ed toa dp (0 ; 2) ndn suy ra c = 2. Vi true dd'i xiing ciia dd thi la dudng thing x = 1 ndn b
2a = 1, ngoai ra vi dd thi di qua diim M(3 ; 4) nen ta cd
39
c = 2
\b = -2a <» •
c = 2
c = 2
6 = -2a <»
" = 3
-I
6 = -2 a <:> •
4 = 9a + 36 + c 2 . 4
4 = 9a - 6a + 2 2.
3a = 2
c = 2.
vay y = ^x- -^A- +
e) Theo hinh 21, parabol cd dinh 7(2 ; 0) va di qua diem (0 ; 2). vay c = 2 ; - — = 2 ^ 6 = -4a ; 2a
A = 6^ - 4ac =^ 16a^ - 8a = 0 ^ a = - (vi a ^ 0) va 6 = -4a = -2 . 1 2 Tiif dd J = —X - 2x + 2.
C. BAI TAP
14. Xae dinh true dd'i xiing, toa dp dinh, eae giao diim vdi true tung va true hoanh cua parabol
a) y = 2x -x-2; c) J = -i^x^ + 2x - 1;
b)}' = -2 x - x + 2 ;
1 9
d) y =T-^ -2 x + 6.
15. Lap bang bid'n thien va ve dd thi eiia ham sd bac hai
a) J = 2x^ + 4x - 6 ; b) ); = - 3x^ - 6x + 4 ;
2
c) J = 2^^ + 2x + 1 ; d) y = - 2x - 2.
16. Xac dinh ham sd bac hai y = ax - 4x + c, bid't ring dd thi ciia nd a) Di qua hai diem A(l ; -2) va fi(2 ; 3) ;
b)Cddinhla/(-2;-l) ;
e) Cd hoanh dd dinh la - 3 va di qua diim fi(-2 ; 1) ;
d) Cd true ddi xiing la dudng thing x = 2 va cit true hoanh tai diim M(3 ; 0). 40
17. Vid't phuomg trinh cua parabol j = ax^ + 6x + c iing vdi mdi dd thi dudi day. yn
Hinh 22 Hinh 23
18. Mdt chide ang-ten chao parabol cd chilu cao h = 0,5 m va dudng kfnh d = 4 m. d mat cit qua true ta dupc mdt parabol dang y = ax (h.24). Hay xae dinh hd sd a.
1 2 ~
19. Mdt chid'c cdng hinh parabol dang y = -^x ed ehilu rdng d - S m. Hay
tfnh chilu cao h ciia cdng (h.25).
y^
0
/ h '
^ \ X 8m \
Hinh 24 Hinh 25
BAI TAP ON TAP CHUONG II
20. Hai ham s6y = x + 4viy- x^ -1 6
x - 4 cd chung mdt tap xac dinh hay khdng ?
21. Cho ham sd y = fix) nghich bid'n trdn khoang (a ; 6), khi dd ham sd y = -fix) cd ehilu bid'n thidn nhu thd' nao trdn khoang {a ; b) 1
41
22. Tim giao diem eua parabol 3; = 2x + 3x - 2 vdri cac dudng thing a) }' = 2x + 1 ; b) J = X - 4 ; c) y =-x - 4 bing each giai phuong trinh va bing dd thi.
Hudng ddn. Di xae dinh hoanh 'dp giao diim cua hai dd thi cd phuong trinh tuong ling la y =fix) va y = g(x) ta phai giai phucfng trinh/(x) = gix).
23. Lap bang bid'n thidn va ve dd thi ciia ham sd j = x^ - 2UI + 1.
2 2 8
24. Ve dd thi ciia ham sd y = —x - - x + 2
L6| GIAI - HUONG DAN - DAP SO
1. a) Tap xae dinh ciia ca ba ham sd y =fix), y - gix) viy- hix) la Z) ={ 1998 ; 1999 ; 2000 ; 2001 ; 2002}.
b) /(2002)= 620000 (con) ; g(1999)= 380000 (con) ; /2(2000)= 100 000 (eon). Nam 2002 san lupng eiia trang trai la 620 000 eon vit ; nam 1999 san lupng la 380 000 con ga ; nam 2000 trang trai cd san lupng la 100000 con ngan lai.
c) /i(2002) - /2(1999) = 210 000 - 30 000 = 180 000 (con). San lupng ngan lai cua trang trai nam 2002 tang 180 000 con so vdi nam 1999.
2. a) D = R ;
b) D = M\ {-1 - Ve ; - 1 + V6} vi x^ + 2x - 5 = 0 <:> c) Ham sd xdc dinh vdi cac gid tri cua x thoa man
X = -l-V 6 X = -1 + V6.
4x + 1 > 0 va -2x + 1 > 0 hay X > - - va X < 4- •^ 4 2
Vay tap xdc dinh ciia ham sd da cho la D = l_ J_ •4 ' 2.
d) Ham sd xac dinh vdi cac gia tri eua x thoa man
Jjc + 9 > 0 fx > -9
[x^ + 8x - 20 ^ 0 [x^ -10 va x ^ 2.
vay tap xae dinh cua ham sd da cho la D = [-9 ; +oo) \ {2}.
e)D = R\{-i;3}.
3. /(5)= -5^ + 2.5 = -25 + 10 = -15 (vi 5 > 0);
/(-2)= ^l2^!i ^ = I (vi -2 < 0) ; /(O) = 3 ; /(2) = 0.
4. gi-3)= V-3.(-3) + 8 = Vl7 (vi -3 < 2); g(2) = 3 ; gH) = Vs ; g(9) = 4. 5. a) V Xj, X2 e R ; ta ed
/(x,) - /(X2) = -2xi + 3 - (-2x2 + 3) = -2(^1 - X2).
Ta tha'y nd'u Xj > X2 thi -2(x, - X2) < 0, tiie la
/(Xl ) -/(X2 ) < 0 <^ /(Xl X /(X2 ).
vay ham sd da eho nghich bid'n trdn R.
b) V Xl, X2 e R ; ta cd
fix,) - fixj) = xl + lOxi + 9 - xf - 10x2 - 9
= (Xi - X2)(Xi + X2) + 10(Xi - X2)
= (Xi -X2)(Xi +X2+IO). (*)
Vxi, X2 e (-5 ; +00) va Xi < X2 ta cd Xi - X2 < 0 va Xi + X2 + 10 > 0 vi Xl > - 5 ; X2 > -5=>Xi + X2 > -10 .
vay tir (*) suy ra fix,) - /(xj) < 0 <:^ fix,) < /(x2).
Ham sd ddng bie'n trdn khoang (-5 ; +00).
c) V Xl, X2 e (-3 ; -2) va x, < X2 ta ed Xi - X2 < 0; Xi + 1 < -2 + 1 < 0 ; X2 + 1 < - 2 + 1 < 0=>(xi + 1)(X2 + 1) > 0. Vay
Do dd hdm sd ddng bid'n trdn khoang (-3 ; -2).
43
V X,. .Vj G (2 ; 3) vax, < .is , tuong tu ta cung cd/(xi) < /(X2). Vay ham sd ddng bie'n trdn khoang (2 ; 3).
a) Tap xac dinh D = R va Vx e D cd -x e D va fi-x) = -2 = /(x). Ham sd la ham sd chin.
b) Tap xac dinh D = R ; Vx e D cd -x G D va/(-x) = 3 (-x)^ - 1 = = 3x - 1 = fix). Vay ham sd da cho la ham sd chan.
c) Tap xac dinh D = R, nhung/(I) =- 1 + 3 - 2 = 0 cdn/(-I) =-l-3- 2 = - 6
ndn /(-I) ^ /(I) va /(-I) ^ -/(I).
Vay ham sd da cho khdng la ham so chin eiing khdng la ham so le. d) Tap xac dinh Z) = R \ {0} ndn nd'u x ^ 0 va x e D thi -x e D. Ngoai ra,
fi-x) = -i-xf + i-xf + 1 -x' + x^ + 1 -X^+x2+ l - X - X X
Vay ham so da cho la ham sd le.
fix).
a) Dd thi la hinh 26. Ham sd khdng la ham sd chin, khdng la ham sd le. b) Dd thi la hinh 27. Ham sd khdng la ham so chdn, khdng la ham sd le. c) D6 thi la hinh 28. Ham sd la ham sd le.
Hinh 26 Hinh 27
d) Do thi la hinh 29. Ham sd la ham sd chin. e) Dd thi la hinh 30. Ham sd la ham sd chdn. yi.
v = 5
o
O
Hinh 28
^ - I
y =j2-\
' H -
Hinh 29
Hinh 30
Hinh 31
D6 thi ham sd dupc ve trdn hinh 31. Diem (1 ; 1) thudc dd thi, diim 1 ; khdng thude dd thi.
9. Hudng ddn. Cic dudng thdng deu cd phuong trinh dang y = ax + b. Cic dudng thing song song vdi nhau diu cd ciing mdt he sd a. Do dd cac phuong trinh eiia cac dudng thing song song vdi dudng thdng y = 3x - 2 deu ed he sda = 3.
a) Phuong trinh can tim cd dang y =• 3x + 6. Vi dudng thdng di qua diem M(2 ; 3), ndn ta cd 3 = 3.2 + 6 <» 6 = -3 .
vay phuong trinh cua dudng thdng dd la y = 3x - 3.
b) y = 3x + 5.
10. Hudng ddn. De xdc dinh cae he sd a va 6 ta dua vao toa dp cae diim ma dd thi di qua, lap he phuong trinh cd hai an a va 6.
[2 ^ 2
a) Vi dd thi di qua A — ; - 2 I ndn ta cd phuong trinh a.— + 6 = -2 . Tuong tu, dua vao toa dp ciia fi(0 ; 1) ta cd 0 + 6 = 1.
vay, ta cd he phuong trinh
2« u
- - + 6 = -2
3 <=> [0 = a + 6 [6 = 3.
Dudng thdng cd phuong trinh la y = -3x + 3.
h) y = -4x; c) y = x - 2.
12. Hudng ddn. De xet xem mdt diim vdi toa dd eho trudc ed thudc dd thi cua ham sd y = fix) hay khdng ta chi edn tfnh gia tri eiia ham sd tai hoanh dd cua diim da cho. Nd'u gia tri cua ham sd tai dd bdng tung dp cua diim dang xet thi diim dd thudc dd thi, cdn nd'u ngupc lai thi diim dang xet khdng thudc dd thi.
a) Vdi diem Ai-l ; 3). Ta ed
|-(-l)-3 | + |2.(-l) + l| + |-l + l|=2 + 1 + 0 = 3, bdng tung dd eua diem A, do dd diim A thudc dd thi.
b) Diim B khdng thudc dd thi ;
c) Diem C khdng thudc dd thi;
d) Diim D khdng thudc dd thi.
13. a) Ta ed the vid't
2x - 3 vdi X > —
y =
2
-2x + 3 von X < —• 2
Tii dd ed bang bid'n thidn va dd thi ciia ham sd J = |2x - 3| (h.32)
y=-2x+3 >' = 2x:-3
X
y
46
—00
+00 ^^ ^
3
2
" ^ ^ 0 -—•
+00 +00
b) Bang bid'n thidn v^ d6 thi ciia
ham so y = --X+ 1 (h.33)
X —00 4_
3 +00
+00 .+00 Hinh 33
c) Ta cd thi vid't y = I-4x vdi X < 0
I 0 vdi X > 0
va dd thi ciia ham sd y = |-2xi - 2x
dupc ve tren hinh 34.
14. a) CJ day a = 2 ; 6 = - 1 ; c = -2 . Ta cd A = (-1)^- 4.2.(-2) = 17.
•) 1 ,
True ddi xiing la dudng thang x = — ; dinh / 1 _1Z
4 ' 8 ; giao vdi true tung
tai diim (0 ; -2).
Dl tim giao diim vdi true hoanh ta giai phuong trinh
2x -x- 2 = 0<»Xi 2 =
l±^/^7
Vay cac giao diem vdi true hodnh la 1 +Vl7
^ 4 ;0 va ^>-^;0 ^
V y
1 , r 1 17
b) True ddi xiing x - -— ; dinh / ; giao VOl true tung tai 4' 8
diim (0; 2); giao vdi true hoanh tai cac diim
1 +Vn ;0 va Vl7-1
c) True-dd'i xiing x = 2 ; dinh 7(2 ; 1) ; giao vdi true tung tai diim (0 ; -1) ; giao vdi true hoanh tai cae diem (2 + %/2 ; 0) va (2 - V2 ; 0). d) True dd'i xiing x = 5 ; dinh 7(5 ; 1) ; giao vdi true tung tai diem (0 ; 6).
Parabol khdng cat true hoanh | A = ---<0
47
15. a) Ham sd bac hai da cho cd a = 2; 6 = 4; c = -6 ;
Vay 2a 1 ; A = 6^-4a r = 64 ; - A
4a
Vi a > 0 , ta cd bang bie'n thidn
-00 +00 + 00 -8 + 00
Ham sd nghich bid'n trdn khoang (-co ; -1) va ddng bid'n trdn khoang (-1 ; -i-oo).
De ve dd thi ta cd true dd'i xiing la dudng iiidng x = -1 ; dinh /(-I ; -8) ; giao vdi true tung tai didm (0 ; -6) ; giao vdi true hoanh tai cdc diim (-3 ; 0) va(l;0).
Dd thi cua ham sd v = 2.v~ + 4x - 6 duoc ve trdn hinh 35
\-3 i
sD \ '; . > \
> ^ \ •
^ \ • • + \ •
^x\ : • (S \ :
]l
0 X
w \ : • '*' \ '.
' - 6 • - 8
Hinh 35
b) Bang bid'n thien
Hinh 36 +00
-00' —oo
48
Ham sd ddng bid'n trdn khoang (-00 ; -1) va nghich bid'n trdn khoang (-1 ; +<»).
Dinh parabol 7(-l ; 7). D6 thi ciia ham sd y = -3x^ - 6x + 4 dupe ve tren hinh 36.
c) Bang bie'n thidn
—00 +00
+00 -1 +00
Ham sd nghich bid'n trdn khoang (-QO ; -2) va ddng bid'n trdn khoang (-2 ; +«)).
Dinh parabol /(-2 ; -1).
1 2 Dd thi ham sd y = — x + 2x + 1 dupe ve trdn hinh 37.
Hinh 37 Hinh 38
d) Bang bid'n thidn
-00 +00
-2
-00
—00
a-BTDS10(C)-A 49
Ham sd ddng bie'n trdn nira khoang (-00 ; 0] va nghich bid'n trdn nira khoang [0 ; +00 ) la ham sd chan.
Dinh parabol 7(0 ; -2) ; dd thi di qua diem (1 ; -4) va diim (-1 ; -4). Dd thi ham sd y = -2x - 2 dupc ve trdn hinh 38.
16. Cae ham sd bac hai cdn xdc dinh diu cd 6 = -4.
f-2 = a - 4 + c fa + c = 2 fa = 3
a) Ta cd <^ -^ < <=> <^
3 = 4a- 8 + c 4a + c = ll c =-1 .
Vay ham sd edn tim lay = 3x - 4x - 1
h)y 2 . r- X 22 ^ 13 ,, 2>-. . x-4x-5 ; c) y = —-x^ -4x- — \ d)y = x -4x + 3.
17. a) Dua trdn dd thi (h.22) ta thdy parabol ed dinh 7(-3 ; 0) va di qua diim
(0 ; -4). Nhu vay c = - 4 ; --— = - 3 <» 6 = 6a. Thay c = - 4 va 6 = 6a vao 2a
bilu thiie A = 6 - 4ac va 6 = -—•
36a +16a = 0 a = - - (vi a ^ 0) 4 2 8
vay phuong trinh cua parabol la y = -—x - — x - 4.
u^ 4 2 8 5
b)>' = -^x +9X-- .
18. He sda = —•
8
19. Chilu cao eua cdng /j = 8 m.
20. Khdng.
21. Ham sd y - -fix) ddng bid'n tren khoang (a ; 6). 22. a) Xet phuong trinh 2x^ + 3x - 2 = 2x + 1 c^ 2x^ + x - 3 = 0 o
X, = 1
3^
'2
Vay parabol da cho va dudng thing y = 2x + 1 cd hai giao diem la (1 ; 3) vd I -^-,-2
50 )IUS10(C).
b) Xet phuong trinh 2x^ + 3x - 2 = x - 4
c^ 2x^ + 2x + 2 = 0
<»x +X+ 1 =0. (*)
Phuong trinh (*) ed bidt thirc A = 1 - 4 = -3 < 0, do dd phuong trinh vd nghiem.
Vay parabol da cho va dudng thang
y - X - 4 khdng cd giao diim.
e) Xet phucfng trinh
2x^ + 3x-2 = -x-4«>2x^ + 4x + 2 = 0
=> x^ + 2x +1 = 0 => X = -1 . Vay
parabol da cho va dudng thdng
y = -X - 4 tid'p xiie nhau tai diim
ed toa dd (-1 ; -3).
Dd thi duoc ve trdn hinh 39. Hinh 39 23. Tap xae dinh eiia ham so la D = R. Ngodi ra
9 I I 9 I I ~ fi-x) - (-x) - 2|-x| + l = x -2|x| + l = fix). Ham sd la ham sd chan.
D6 thi cua nd nhan true tung lam true dd'i xiing. Dl xet chidu bie'n thidn va ve dd thi ciia nd chi edn xet chilu bid'n thien va ve dd thi cua nd trdn nira khoang [0 ; +oo), rdi ldy dd'i xiing qua
Oy. Vdi X > 0, cd fix) = x^ - 2x + 1.
Bang bid'n thidn
X
fix)
0 1 +00 1^^^^^^ +00
Dd thi ciia ham sd da cho dupc ve d
hinh 40. Hinh 40 51
24. Vi |/(x)| = |/(x), nd'u/(x)>0
l-/(x), ne'u/(x)<0
ndn dl ve dd thi eiia ham sd y = /(x)| ta ve dd thi eua ham sd y = fix), sau dd giii nguydn phdn dd thi d phfa tren true hoanh va lay dd'i xiing phdn dd thi ndm phfa dudi true hoanh qua true hodnh.
2 9 8 Trong trudng hpp nay, ta ve dd thi eua ham so y = ^x - — x + 2, sau dd
giii nguydn phdn dd thi iing vdi cac nira khoang (-co ;1] va [3 ; +00). Lay dd'i xirng phdn dd thi iing vdi khoang (1 ; 3) qua true hoanh.
2 2 8
-x^ -- X + 2 3 3 dupc ve tren hinh 41 (dudng
Hinh 41
52
Dd thi cua ham sd y = liin net)
huang HI. PHl/ONG TRINH. HE PHl/ONG TRJNH
§1. DAI CLfONG V^ PHtfONG TRINH
A. KIEN THljrC CAN NH 6
1. Phuong trinh dn x la mdt mdnh dl chu^a bid'n dang fix) =g(x), trong dd fix) va gix) la cdc bilu thiic ciia x.
2. Dieu kien xdc dinh cua phuong trinh (gpi tdt la dilu kidn cua phuong trinh) la nhiing dilu kidn cua in x dl cdc bilu thiic trong phuong trinh diu cd nghia.
3. Nd'u fixo) = ^(XQ) thi XQ dupe gpi la nghidm cua phuong trinh fix) = gix). 4. Gidi mdt phuong trinh la tim tap tdt ca cae nghidm eua nd. 5. Hai phuong trinh fix) = gix) (1) va /i(x) = g,ix) (2) dupc gpi la
tuong duong nd'u chiing cd tap nghidm bing nhau (cd thi rdng). Kfhidu(l)<»(2).
6. Nd'u thuc hidn cae phep bid'n ddi sau day trdn mdt phucmg trinh ma khdng lam thay ddi dilu kidn xac dinh cua nd thi ta dupc mdt phuong trinh mdi tuong duong.
a) Cdng hay trii hai vl vdi eung mdt so hay eung mdt bilu thiic.
b) Nhan hoac chia hai vd' vdi ciing mdt sd' khac 0 hoac vdi cimg mdt bilu thiic ludn cd gia tri khac 0.
7. Nd'u mdi nghidm eua phuong trinh (1) eung la nghidm eua phuong trinh (2) thi ta ndi phuong trinh (2) la phuong trinh he qud ciia phuong trinh (1).
Kfhieu(l)=^(2).
Ching ban, vdi so nguydn duong n tuy y ta cd
fix)^gix)^[fix)]"^[gix)]".
8. Phuong trinh hd qua cd thi cd nghiem ngoqi lai, khdng phai la nghiem ciia phuong trinh ban ddu. Mud'n loai nghidm ngoai lai ta phai thir lai vao phuong trinh ban ddu.
53
9. Ngoai cdc phuong trinh mdt dn cdn cd cac phuong trinh nhieu dn. Nghidm ciia mdt phuong trinh hai dn x, y la mdt cap sd thue (xg ; yo) thoa man phuong trinh dd, edn nghidm cua mdt phuong tnnh ba dn x, y, z la mdt bd ba sd thuc (XQ ; JQ '•< ^o) '^boa man phuong trinh dd.
10. Trong mdt phuong trinh (mdt hoac nhieu in), ngoai cac chii ddng vai trd an so cdn cd thi cd cac chii khac dupc xem nhu nhirng hing sd va dupe gpi la tham sd.
Giai va bidn luan phuong trinh ehiia tham so la xet xem khi nao phuang trinh dd vd nghidm, khi ndo cd nghidm tuy theo cdc gia tri cua tham sd va tim cdc nghidm dd.
B. BAI TAP MA U
Gidi
a) Bilu thiic d ve trai cd nghia khi x ^^ 2 va x ?^ -2. Bieu thiic 6 ve' phai ed nghia khi x< 3.
Dilu kidn cua phucmg trinh la
x<3,x?t2vax ^ -2.
b) Bilu thiie d vl trdi cd nghia khi x > 2, cdn ve' phai cd nghia khi x < 1. Dilu kidn eiia phuong trinh la x < 1 va x > 2.
Ta tha'y khdng ed gia tri nao eua x thoa man ca hai dilu kidn nay.
Chu y. Khi khdng ed gia tri ndo eua x thoa man dilu kidn cua phucmg trinh thi phuong trinh da cho vd nghidm.
BAI 2
Chiing td cdc phuong trinh sau vd nghidm
= 3 + V4 - X.
54
a) "!-""" - Vx 3 ; b) Vx - 4 - X V-x + 2
Gidi
a) Dieu kidn eiia phuong trinh la x < 2 va x > 3. Khdng ed gid tri ndo eua x thoa man dieu kidn nay. Vay phuong trinh vd nghidm.
b) Dilu kidn cua phuong trinh la x > 4 va x < 4, tiic la x = 4. Gid tri nay khdng thoa man phuong trinh da cho (vl trai bdng -4, ve phai bdng 3).
Vay phuong trinh vd nghidm.
BAI 3
Cho phucmg tnnh
(x + 1)^ = 0
va phuong tnnh chiia tham sd a
(1)
2
ax -
- (2a + l)x + a = 0.
(2)
Tim gia tri
eua
trinh (2).
a sao
cho phuong Gidi
trinh (1) tuomg duomg vdi
phuong
Dieu kien cdn. Gia sir cac phuong trinh (1) va (2) tuomg duomg. The thi, nghidm x = -1 eua phucmg trinh (1) eiing phai la nghidm ciia phuomg trinh (2). Vay
a.(-l)2 - (2a + l).(-l) + a = 0 hay a = -| -
Dieu kien du. Nd'u a = -— thi
4
(2) «-ix2-lx-l.= 0
<=> x^ + 2x + 1 = 0
« (x + 1)2 = 0 «» (1).
Phuong trinh (1) va (2) tuong duong.
Ket ludn. Phuomg tnnh (1) va (2) tuomg duong khi va chi khi a = --• 55
BAI 4
Giai cae phucmg trinh
a) Vx + 1 + X = 3 + Vx + 1 ; b) Vx-5- x = 2 + Vx-5 .
Gidi
a) Dieu kidn eua phuong trinh la x > -1 . Ta cd
Vx + 1 + X- = 3 + Vx + 1
<::>X= 3 + Vx + 1 - Vx + 1
=>x = 3.
Gia tri X = 3 thoa man dilu kidn x > -1 va nghidm diing phuomg trinh. vay nghidm cua phuomg trinh da cho la x = 3.
b) Dilu kidn cua phuomg tnnh la x > 5. Ta cd
Vx - 5 - X = 2 + Vx - 5
<» -X = 2 + Vx - 5 - Vx - 5
=> X = -2 .
Gia tri x = -2 khdng thoa man dilu kidn x > 5.
vay phuong trinh da cho vd nghidm.
BAT 5
Giai cac phuong tnnh
, 2x + 1 x + 2
yjx -3 \lx -3 b) 2^' = ^ . Vx + 1 Vx + 1
Gidi
a) Dilu kidn ciia phuong trinh la x > 3.
Vdi dilu kidn dd, ta cd
2x + I x + 2 2x + 1 I x + 2 « • .7733 = 4=^.47^ Vx- 3 Vx- 3 Vx- 3 Vx- 3 ^2x+l= x + 2=:>x=l.
56
Gia tri X = 1 khdng thoa man dilu kidn x > 3 ndn bi loai.
vay phuong trinh da eho vd nghidm.
b) Dilu kidn eua phuong trinh la x > -1 . Vdi dilu kien dd, ta ed
2x^ 8 2x^ j r 8 /
V «> , -vx + 1 = , -Vx + 1 x + 1 Vx + 1 Vx + 1 Vx + 1
=::>2x2 = 8=>x2 = 4=>x = 2 hoac x = -2.
Gia tri x = -2 khdng thoa man dilu kidn cua phuong trinh ndn bi loai. Gia tri X = 2 thoa man dilu kidn va nghidm diing phuong trinh.
Vay phuong trinh da cho cd nghidm la x = 2.
C. BAITAP
1. Vid't dilu kidn cua eae phuomg trinh sau
a) V2x + 1 = - ; b) / =3x^ + x+l; V2x2 + 1
. X 2 ,^ 2x + 3 I 7
c) , = , ; d) — = Vx +1 .
Vx- 1 Vx + 3 x- 4
2. Xac dinh tham sd m dl cac cap phuomg trinh sau tuong duong
mx
a) X + 2 = 0 va —=^ + 3m - 1 = 0 ;
x + 3
b) x^ - 9 = 0 va 2x2 + (m - ^^^ _ ^^^ + 1) = 0.
3. Giai cac phuong tnnh
a) Vx + 1 + X = Vx + 1 + 2 ; b) x - V3 - x = Vx- 3 + 3 ; c) x^ - V2-X = 3 + Vx- 4 ; d) x^ + V-x - 1 = 4 + V-x - 1.
57
4. Giai cae phuong trinh
a) c)
3x- + 1
Vx- 1 Vx- 1 ' 3x^ - X - 2 = V3x-2 ;
, , x'^ + 3x + 4 I b) , — = Vx + 4
Vx + 4
d) 2x + 3 + x^+ 3
V3X-2 """ ' "' ' ' x- 1 x- 1
5. Xdc dinh m de mdi cap phucmg trinh sau tuong ducmg
a) 3x - 2 = 0 va (m + 3)x - m + 4 = 0 ;
b) X + 2 = 0 va mix^ + 3x + 2) + m^x + 2 = 0.
§2. PHl/ONG TRINH QUY V^ PHl/ONG TRINH BA C NHXT , BACHAI
A. KIEN THOC CAN NH 6
1. Gidi vd bien ludn phuong trinh
ax + 6 = 0. (1)
He sd
a;^0
a = 0
6 ^ 0 6 = 0
Kd't luan
Phuong trinh (1) cd nghidm duy nhdt x = —•
Phuong trinh (1) vd nghidm.
Phuong trinh (1) nghidm diing vdi mpi x.
Khi a ^ 0 phuong trinh (1) dupc gpi liphuon'g trinh bdc nhdt mdt dn. 58
2. Gidi vd bien ludn phuong trinh bdc hai
ax + 6x + c = 0, (a^O). (2)
Bidt thiic A = 6^ - 4ac A > 0
A = 0
A < 0
3. Dinh li Vi-et
Ke't luan
Phucmg trinh (2) ed hai nghidm
-6 +VA
"^••2- 2a •
Phuong trinh (2) cd nghidm kep x = -—-• 2a
Phucmg trinh (2) vd nghidm.
6 c
Nd'u phuong trinh (2) cd hai nghiem Xi, X2 thi Xi + X2 = — , XiX2 = — •
Ngupc lai, nd'u hai so M va v cd tdng M + v = 5 va tfch uv = P thi M va v la cae nghidm eiia phucmg trinh x - Sx + fi = 0.
4. Phuong trinh trung phuong ax + 6x + c = 0, (a T^: 0) ed thi dua vl phuomg trinh bae hai bang each dat r = x , (r > 0).
5. Cd thi khii da'u gia tri tuydt dd'i trong phuong trinh ehda dn trong dd'u gid tri tuyet ddi nhd sir dung dinh nghia
, , \a nd'u a > 0
-a nd'u a<0 . \a\ =
Dac biet, dd'i vdi phucmg trinh |/(x)| = |g(x)|, ta cd
\fix)\=\gix)\^[fix)f = [gix)f
hoac
|/(x)| = \gix)\ O fix) = gix)
fix) = -gix).
6. Khi giai phuong trinh ehda dn dudi dd'u cdn thdc bdc hai ta thudng binh phuomg hai v l dl khii da'u can thiic va dua tdi mdt phuomg trinh hd qua.
59
B. BAI TAP MAU
BAIl
Giai vd bidn luan cac
phuong
trinh
sau theo tham so m
2
a) m (x + 1) -
L = (2 -
m)x;
^^ (2/77 - l)x +
2
x - 2 2
— = m +
1.
Gidi
a) 777 (X + 1) - 1 = (2 - 777)X
2 2
(W + m - 2)X =1-77 7
<=> (777 - 1)(777 + 2)X = -(777 - l)(/77 + 1).
Nd'u 777 5^ 1 va 777 5!i - 2 thi phuong trinh ed nghiem duy nha't x = — •
Ne'u 777 = 1 thi mpi sd thuc x di u la nghidm eua phuomg trinh. Nd'u 777 = - 2 thi phuong trinh vd nghidm.
b) Dilu kien ciia phuong trinh la x ?t 2. Khi dd ta cd
(2/77 - l)x + 2 , _ ,, - ^
•^^ '-- = 777 + 1 => (2/77 - 1)X + 2 = (777 + l)(x - 2)
^ (777 - 2)X = -2(777 + 2). (3)
Vdi 777 ^ 2 phucmg trinh (3) cd nghidm duy nha't x = -• "777-2
Nghidm nay thoa man dilu kien ciia phucmg trinh da cho khi va Chi khi
-2(777 + 2) ^
777-2
hay -2m -4^2m-4<:^m^0.
Vdi 777 = 2 phuong trinh (3) trd thanh 0.x = -8 , phuong tnnh ndy vd nghidm, do dd phuong trinh da cho vd nghidm.
Kdt ludn. Khi 777 = 2 hoac 777 = 0 phucmg trinh vd nghidm.
Khi 777 ^ 2 va 777 ;t 0 phuong trinh ed nghidm duy nhdt la x = —i . 60
BAI 2—-
Cho phuong trinh bae hai
2 9
X + (2/7! - 3)X + 777" - 2/7? = 0.
a) Xdc dinh m de phucmg trinh cd hai nghidm phan bidt;
b) Vdi gia tri nao cua 777 thi phuong trinh ed hai nghidm va tfch cua chiing bing 8 ? Tim cac nghidm trong trudng hpp dd.
Gidi
a) Phuomg trinh cd hai nghidm phan bidt khi bidt thiic A > 0. Ta cd A = (2/77 - 3)2 - 4(7772 _ 2^) = _ 4 ^ + 9
- 1
A>0<»-4/77 + 9>0<=>77J<
9
Vay khi hi < — phuong trinh cd hai nghidm phan bidt.
9
b) Phuomg trinh cd hai nghidm khi 777 < —• Theo dinh If Vi-et ta cd
7772 - 2/77 = 8 <» 7772 - 2/77 - 8 = 0 o m = -2
m = 4.
Vdi 777 = 4 > — phucmg trinh vd nghidm.
2
Vdi 772 = -2 phucmg trinh tro thdnh x - 7x + 8 = 0 va cd hai nghidm 7±vr7
^1.2=^ —
Vay vdi /77 = - 2 phuomg trinh da eho cd hai nghiem va tfch eiia ehiing bdng 8. rr • w' A'^^ 7 ± Vl7 Hat nghiem do la X12 = r
RAT 3
2
2 777 - 3)X + 777 = 0.
Cho phucmg trinh mx + (
a) Xac dinh m Ae phuong
trinh cd nghiem kep
va tim nghidm
kep dd.
b) Vdi gia tri nao ciia m 13 „
•Vl+X2=^ ?
thi phuong trinh ed
liai nghiem Xi,
X2 thoa man 61
Gidi
a) Phuong trinh cd nghiem kep khi 777 ^ 0 va A = 0. Ta cd A = (/?7 - 3) - 4/72 = m - 10/77 + 9.
Phuong trinh trung phuong m -IO/77 + 9 = 0 cd bdn nghidm m = ±1 va 772 = ±3.
Vdi 772 = 1 hoac 772 = - 3 phuong trinh da cho ed nghidm kep x = 1. Vdi m = -l hoac m = 3 phucmg trinh da cho cd nghidm kep x = -1 . b) Dilu kidn de phucmg trinh cd hai nghiem la
/72 9t 0 va A = 772'^ - 10/772 + 9 > 0.
Theo dinh If Vi-et ta cd
Theo dl bai ta phai cd
Xl +X 9 = — = ' a
3-/772 ^3
m
3-/72' 777
« 4/72^ +13/72-12 = 0 m = —4
<=>
3
/72 = — • 4
Vdi 772 = - 4 thi A = 105 ;
3 ^. , 945
Voi /72 = — thi A = ——•
4 256
Ca hai gia tri tim dupc cua 777 diu thoa man dilu kien A > 0.
3 , , 13
Vay khi 772 = -4 hoac /72 = — thi tdng hai nghidm cua phuong trinh la -—• BAI 4
= |3x - 2 ;
= x2 + 2x - 4. .
62
Giai cdc phucmg trinh a) |2x - 3| = X - 5 ;
| 3 x -1 | I I e) ' ^J=\x 3 ; x + 2 ' '
sau
b) 2x + 5
d) 4x + 1
Gidi
a) Cdch 7. Khi X > - ta cd |2x - 3| = 2x - 3.
2 ' '
Liic dd phuong trinh trd thanh 2x - 3 = x - 5, suy ra x = -2. 3 Gia tri x = -2 khdng thoa man didu kidn x > — ndn bi loai.
3 Khi X < - ta cd |2x - 3| = -2x + 3.
g
Liic dd phuong trinh trd thdnh -2x + 3 = x - 5, suy ra x = — •
8 3
Gia tri X = — khdng thoa man dieu kidn x < — ndn bi loai. Ke't ludn. Phuomg trinh da eho vd nghidm.
Cdch 2. Binh phuong hai vd' phuong trinh da cho ta dupc phucmg trinh he qua. Ta cd
|2x - 3|= X - 5 ^ (2x - 3)2 = (x - 5)2
<» 4x2 _ j2x + 9 = x2 - lOx + 25
<:*3x2-2x- 16 = 0.
g
Phuong trinh cud'i ed hai nghidm Xi = -2, X2 = -•
g
Thii lai, ta thdy ca hai gid tri Xi = -2, X2 = :r- deu khdng phai la nghidm ciia phuong trinh da eho ndn bi loai.
Vay phuong trinh da cho vd nghidm.
b) Binh phuong hai vl phucmg trinh da cho ta dupc phuong trinh tuong duong |2x + 5| = |3x - 2| c^ (2x + 5)2 = (3x - 2)2
<=> 4x2 ^ 20x + 25 = 9x2 _ i2x + 4
o5x2-32x-2 1 =0.
63
3 Phuomg trinh eudi cd hai nghidm Xi = 7, X2 =
3
vay phuong trinh da cho cd hai nghidm Xi = 7 va X2 = --r
c) Dilu kidn cua phuong trinh la x 9^ -2.
Ta chia khoang de khir dau gia tri tuydt dd'i.
Vdi X > 3 thi |x - 3| = x - 3 va |3x - l| = 3x - 1.
Vdi dilu kien dd phuong trinh da cho trd thanh
3x- l
x + 2 = x-3 . (1)
Tacd (1) <»3x- 1 =x2-x- 6
<:> x2 - 4x - 5 = 0
<=> Xl = -1 , X2 = 5.
Vi X > 3 ndn chi cd X2 = 5 la nghidm cua phuong trinh.
Vdi - < X < 3 thi |x - 3| = -X + 3 va |3x - l| = 3x - 1.
Khi dd phucmg trinh da cho trd thanh
3x- l
x + 2 •X + 3. (2)
Ta ed (2) «• 3x - 1 = -x2 + x + 6
o x2 + 2x - 7 = 0
o X3 =- 1 + 2V2, X4 =-1-2V2 .
Vi - < X < 3 ndn chi ed gid tri X3 = -1 + 2V2 la nghidm cua phuong trinh.
Vdi X < - thi |x - 3| = -x + 3 vd |3x - l| = -3x + 1.
Khi dd phuong trinh da cho trd thanh
-3x + l
. + 2 =-"' • (3)
64
Tacd (3) <^ -3x+ 1 =-v2 + x + 6
<=i> x2 - 4A- - 5 = 0 <=> X5 = -1 , xg = 5.
Vi X < - nen chi ed gid tri .v^ = -1 la nghiem cua phuong trinh. 3
Ket ludn. Phuong trinh da cho cd ba nghidm x = 5. x = 2V2 - 1, x = -1 . d) Vdi X > -— thi |4x + l| = 4x + 1. Khi dd phuong trinh da cho trd thanh
4x+ 1 =.v2 + 2x-4 . (*)
Tacd (*) c^x2-2x- 5 = 0
<^x, 1 + V6, X2 = 1 - Ve .
Vi X > -— nen chi cd x, = 1 + Vd la nghidm ciia phuong trinh.
Vdi X < -— thi |4x + l| = -4 x - 1. Khi dd phuong trinh da cho trd thanh
-4x- 1 =x2 + 2x-4 . (**) Tacd (**) x2 + 6x- 3 = 0
<^ A-3 = - 3 + 2V3 , X4 =- 3 - 2V3 .
Vi X < - - ndn chi ed gid tri X4 =- 3 - 2V3 la nghiem ciia phuong trinh.
Vay phuong trinh da cho ed hai nghiem x = 1 + Vd va x = - 3 - 2V3 .
Giai cac phuong trinh sau
a) V4.V - 9 = 2x - 5 ; b) Vx2 - 7 x + 10 =3x -- 1 .
BTDS10(C)-A 6 5
Gidi
9
a) Dieu kien eua phuong trinh la x > — • Ta cd.
V4x- 9 = 2x - 5 ^ 4x - 9 = (2x - 5)2
=^ 4x - 9 = 4x2 - 2Q^ ^ 25
^2(2x2 - 12x+ 17) = 0.
, . ,. 6 + V2 6-V2
Phuong trinh cudi co hat nghiem Xi = —-— , X2 = — r
6-V2 Gia tri X2 = —r — khdng thoa man dilu kien cua phuomg trinh ndn bi loai.
Thay Xi = — - — vao phuong ttinh ban ddu ta thay gia tri cua hai vd' bdng nhau.
6 +V2
Vay nghidm eiia phuong trinh da cho la x = —
2
b) Dilu kidn eua phuong trinh lax -7x+10>0 .
Ta ed
Vx2 - 7x + 10 = 3x - 1 => x2 - 7x + 10 = (3x - 1)2
=> x2 - 7x + 10 = 9x2 - 6x + 1
^ 8x2 + X - 9 = 0.
9 Phuong trinh eudi ed hai nghiem Xi = 1, X2 = -—•
o
9 . > , Ca hai gia tri 1 va -— ddu thoa man didu kidn cua phuong trinh da cho. 8 Thir lai ta thay phuong trinh da cho chi ed mdt nghiem x = 1. vay nghidm eua phuong trinh da cho la x = 1.
BAI 6
Giai va bidn luan cac phuomg trinh sau theo tham so 777
a) |4x - 3m\ = 2x -\- m ; b) |3x - /77|= |2x + //z + l| ;
, (777 + 3)X + 2(3/77 + 1) C) -^ ^ = (2/72 - l)x + 2.
x + 1
6 6 5.EnDS10(C)-B
Gidi
a) Ta xet hai trudng hpp
3/72
Vdi X > -— phuong trinh da cho trd thdnh
4 x - 3/72 = 2X + 772 <=> 2X = 4/72 < ^ X = 2/77.
3/72
Ta cd 2/77 > —— <=> 772 > 0.
4
Vay vdi /?2 > 0 thi phuong trinh cd nghidm x = 2m.
3/72
Vdi X < —- phuong trinh da cho trd thanh
- 4 x + 3/72 = 2x + 772 <^ 6X = 2/72 <» X = 772
^ , 777 3/77 3/72 772 „ 5/72
Tac o — <——<^—; — — >0< ^ -—- > 0 <=> 772 > 0. 3 4 4 3 12
772
Vay vdi m>0 phuong trinh cd nghiem x = —
Ket ludn. Vdi 772 > 0 phuong trinh cd nghidm x = 2mvix = m
Vdi 772 = 0 phuomg trinh cd nghidm x = 0.
Vdi /72 < 0 phucmg trinh vd nghidm.
b) Ta cd
|3x - m\ = |2x + 772 + l| <»
Ta tha'y
(1) <^X = 2/72 + 1,
(2)<^5x = -l o x =
3x-/72 = 2x + /72 + l
3x - 772 = -2 x -772-1 .
'5'
1 f\ 'X
(1) (2)
Hai nghidm nay triing nhau khi 2/72 + 1 = - - <» 2/72 =--<=> m = --• 67
3
Kit ludn. Vdi mi^ - - phuong trinh cd hai nghiem phan biet
.V = 2/72 + 1 va X = — . 5
3 1 Vdi 772 = -— phuong trinh cd nghidm kep x =
Ghi ehu. Vi hai ve cua phuong trinh la nhiing bilu thiic khdng am ndn ta cung ed the binh phuong hai ve dl dupe mdt phucmg trinh tuong duong.
c) Dilu kidn ciia phuong trinh la x ^t -1 . Khi dd ta cd
jm + 3)x + 2(3/72 + 1)
; = (2/72 - 1)X + 2
X + 1
«> (772 + 3)X + 2(3/72 + 1) = [(2/72 - l)x + 2](x + 1)
<» (772 + 3)x + 2(3/72 + 1) = (2/w - l)x2 + (2//2 + l)x +2
«>(2/72- l)x2 + (777- 2)X- 6/72 = 0. (*) Vdi 772 = — phuong trinh (*) trd thdnh
-|x- 3 =0<^ x = -2 .
Gia tri x = - 2 thoa man dilu kien ciia phuong trinh da cho.
Vdi m* - phuong trinh (*) la mdt phuomg trinh bac hai cd bidt thiic
A = (/72 - 2)2 + 24/72(2/72 - 1)
= 49/722 - 28/7? + 4
= (7/72 - 2)2 > 0.
2
Khi m^ -- phuong trinh (*) cd hai nghidm phan bidt
^1,2 = 2 - 772 ± (7/72 - 2)
2(2/72 - 1)
68
Tadatx, = ^,X 2 = -2 .
3/72 1
Gia tri ^ - 1 khi va chi khi 3m ^ -2m + 1 hay m^ --• • 2/72 - 1 5
2
Khi /?? = — phuong trinh (*) ed nghidm kep x = -2 .
Ket ludn
Khi m= — hoac m= -- phuong trinh cd mdt nghidm x = -2 .
2
Khi 777 = — phucmg trinh cd nghiem kep x = -2 .
1 1 2 Khi m^ — , mj^--vim^ — phuong trinh ed hai nghiem phan bidt
3/72 . ^
"'=2;;^^^"^^-2 -
C. BAI TAP
6. Giai va bidn luan theo tham sd m cac phuong trinh sau
.y (fjj — 2)x + 3
a) mim - 6)x + /?2 = -8 x -\- m - 2 ; b) ; = 2/77 - 1 ; x + 1
, (2/77 + 1)X - 772 ,, (3/72 - 2)x - 5 „ c) ; = X + /72 ; d) = -i. x - 1 X - 772 7. Cho phucmg trinh
im + 2)x2 + (2/72 + l)x + 2 - 0.
a) Xdc dinh m de phuong trinh cd hai nghidm trai ddu vd tdng hai nghidm bing -3 .
b) Vdi gid tri nao cua m thi phuong trinh cd nghiem kep ? Tim nghidm kep dd.
69
8. Cho phuong trinh
9x2 + 2(/722- l)x+ 1 =0.
a) Chiing td rdng vdi m>2 phucmg trinh cd hai nghidm phan bidt am. b) Xac dinh m de phuomg trinh cd hai nghidm xi, X2 ma Xi + X2 = -4. 9. Giai cdc phuong trinh
a) |x - 3| = |2x - l| ; b) |3x + 2| = x + 1 ;
5x - 2 2
c) J -^=x-2 ; d) 3x- 5 =2x +X-3 . X + 3 ' ' ' '
10. Giai cdc phuong trinh
a) V3x-4 = X - 3 ; b) Vx2 - 2x + 3 = 2x - 1 ;
e) V2x2 +3x + 7 = x + 2 ; d) V3x2 _ 4^ _ 4 =72x + 5. 11. Giai vd bidn luan theo tham so m eae phucmg trinh sau a) |3x + 2m\ -x - m ; b) |2x + /72| = |x - 2m + 2J ;
c) mx + (2/72 -l) x + /72-2 = 0; d) — = 772 - 1. 2x - 1
§3. PHl/ONG TRINH
VA Hfi PHl/ONG TRINH BAC NHX T NHI£ U XN
A. KIEN THOC CAN NH 6
1. Phuong trinh bac nhdt hai an x, y cd dang
ax + 6j = c,
trong dd a, 6, c la cae sd thuc da eho vd a, 6 khdng ddng thdi bdng 0. 70
2. Hd hai phuomg trinh bae nhdt hai dn x, y cd dang
[aiX + 6i>' = c,
[ujx + b2y = C2,
trong dd ea hai phuong trinh diu la phucmg trinh bac nhat hai an. Cd hai each giai hd phucmg trinh bac nhat hai dn quen thudc.
a) Phuong phdp the. Tix mdt phuong trinh cua he bilu thi mdt an qua dn kia rdi thay vao phucmg trinh edn lai.
b) Phuong phdp cpng. Bid'n ddi cho he sd eiia mdt an trong hai phuong trinh la hai sd dd'i nhau rdi cdng timg ve hai phucmg trinh lai.
3. Dang tam giac ciia he ba phuong trinh bac nhat ba an la
a,x + b,y + c,z = d,
bjy + CjZ = dj (1)
C-,! ^3,
hoac
a,x
a2X + 62^
= d,
= d. (2)
a3X + 63^ + C3Z = d^.
Cdch gidi. Tir phuong trinh cud'i cua he (1) tfnh dupc z, thay vao phuomg trinh thii hai tfnh dupe y rdi thay vao phucmg trinh ddu tfnh dupe x.
Tir phucmg trinh ddu cua he (2) tfnh dupc x, thay vdo phuong trinh thii hai tfnh dupc y rdi thay vao phuong trinh thii ba tfnh dupc z.
4. He ba phuong trinh bae nhdt ba dn cd dang
aiX + 6i>' + CiZ = d,
a^x + 62^ + C2Z = dj
a^x + b-^y + C3Z = d^.
Cdch gidi. Diing phuomg phap Gau-xo khu ddn dn sd de dua vl he phuong trinh dang tam giac.
71
B. BAI TAP MAU
BAI 1
Giai cae hd phuong trinh
f 3x - 4y = 2
'-4x + 5>' = -3
a)
[-5x + 3^ = 4 ; '3 2 1
b)^
7x + 3^ = 8 ;
c)-
4^-5 ^ = 2 1 4 1 ,r^5^^-3 '
d)<
Gidi
0,4x-0,3y = 0,6 -0,3x-0,2j = -L3.
a) Tix phuong trinh thii nhdt suy ra
X =
4^ + 2
Thay bieu thiic eiia x vao phuong trinh thii hai ta dupc
^4^ + 2^ , , 11 22
+ 3>' = 4=i> -—y =— ^y = -2.
Tirdd
,= it?l±l,_2.
Vay nghiem ciia he phuong trinh la (-2 ; -2).
b) Ta cd
-4x + 53; = - 3 (-28x + 35y = -21
[7x + 3v = 8 ^ I 28x + 12j = 32.
Cdng timg ve hai phuong trinh ta dupc 47_y = 11 ' = l,8
l-0,3x - 0,2>' = -1,3 ^ l-l,2 x - 0,8y = -5,2.
Cdng timg ve hai phuong trinh ta dupc -l,7 j = -3,4 suy ra j = 2. Thay y = 2 vao mdt trong hai phuong trinh cua he, ta dupe x = 3. vay nghiem cua he phucmg trinh la (3 ; 2).
BAI 2.
Tm mdt sd cd hai chii so, bid't hidu ciia hai ehii sd dd bdng 3. Nd'u vid't cac , . 4 , ehii sd theo thii tu ngupc lai thi dupc mdt sd bang — sd ban dau tni di 10.
Gidi
Gpi chii sd hang chue la x, ehii sd hang don vi la y thi sd phai tim la lOx + y. Dilu kien bai toan la x, y nguyen va 1 < x < 9, 0 < y < 9.
Sd ban ddu la lOx + y thi sd vie't theo thii tu ngupc lai la lOy + x.
Theo gia thie't sd vid't theo thii tu ngupc lai phai nhd ban sd ban ddu, cho ndn phai ed x > y. Ta ed he phuong trinh
X - y = 3
4
X + lOy = --(lOx + y) - 10.
73
Thay x = y + 3 vao phuong trinh thii hai eiia he, ta dupc
lly = 55 =>y = 5 =^x = 8.
Vay sd phai tim la 85.
BAI 3.
Giai he phucmg trinh
' 2x - 3y + 2z = 4
-4x + 2y + 5z = -6
<
2x + 5y + 3z =8.
Gidi
Nhan hai ve ciia phuong trinh ddu vdi 2 rdi cdng timg ve vdi phucmg trinh thii hai, ta dupe he phuong trinh
'2x - 3y + 2z = 4
- 4y + 9z = 2
2x + 5y + 3z = 8.
Nhan hai vl ciia phuong trinh ddu vdi -1 rdi cdng tiing vd' vdi phuomg trinh thii ba, ta dupc hd phuong trinh
'2x - 3y + 2z = 4
- 4y + 9z = 2
8y+ z = 4.
Nhu vay, ta da khu dupe in x trong hai phucmg trinh eudi. De khir dn y trong phuong trinh thii ba, ta nhan hai vl cua phuong trinh thir hai vdi 2 rdi cdng timg ve vdi phuong trinh thii ba ta dupe he phuong trinh ed dang tam giac
'2x - 3y + 2z = 4
- 4y + 9z = 2
19z = 8.
Q
Tir phuong trinh cud'i suy ra z = — • Thay gia tri nay ciia z vao phuong
17 trinh thii hai, ta dupe y = —• Cud'i eiing, thay ede gia tri cua y va z vira tim 38 171 dupc vdo phuong trinh adu ta tim dupe x = ^=7--
74
Vay nghidm eiia hd phuong trinh la
ix;y •,z) = 171 . 2Z . A
76 ' 38 ' 19
BAI 4
Giai he phuong trinh
-3x + 2y - z = -2
<
5x - 3y + 2z = 10 2x - 2y - 3z = -9.
Gidi
Nhdn xet. Dd'i vdi hd phuong trinh nay, vide khir dn x khdng don gian ldm. Tuy nhidn, nd'u ehii y dd'n hd sd eiia z d ba phuong trinh, ta thdy dd khir dn z d hai phucmg trinh eudi.
Nhan hai vl ciia phuong trinh ddu vdi 2 rdi cdng timg ve vdi phucmg trinh thii hai. Nhan hai vl cua phuong trinh ddu vdi -3 rdi cdng tiing ve vdi phuong trinh thii ba, ta dupe hd phuomg trinh
-3x + 2y - z = -2
-x+ y = 6
llx-8 y =-3 .
Dd'n day, ta thdy dd khii dn x (hoac dn y) trong phuong trinh thii ba. Ching ban, nhan hai ve ciia phuong trinh thii hai vdi 8 rdi cdng timg vl vdi phuong trinh thii ba, ta dupc
-3x + 2y - z = -2
-x+ y =6
3x = 45.
Hd phucmg trinh nay cd dang tam giac. Giai ldn lupt tit phuong trinh thii ba ldn ta dupc x = 15, y = 21, z = -1 .
Ddp sd: (x ; y ; z) = (15 ; 21 ; -1).
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BAI 5 .
Ba cd Lan, Huong va Thuy ciing theu mdt loai ao gid'ng nhau. Sd ao ciia Lan theu trong 1 gid ft hon tdng sd ao ciia Huong va Thuy theu trong 1 gid la 5 do. Tdng sd do ciia Lan theu trong 4 gid va Huong theu trong 3 gid nhidu hon sd ao cua Thuy theu trong 5 gid la 30 do. Sd do ciia Lan theu trong 2 gid cdng vdi sd do ciia Huong theu trong 5 gid va sd ao ciia Thuy theu trong 3 gid tdt ea dupc 76 ao. Hdi trong 1 gid mdi cd theu dupc may ao ?
Gidi
Gpi X. y, z lan lupt la sd do eiia Lan, Huong, Thuy thdu trong 1 gid. Dilu kidn la x, y, z nguyen duomg.
Tir gia thie't ciia bdi toan ta cd
X = ;-• + z - 5
< 4x + 3y - 5z = 30
[2x + 5y + 3z = 76
[x - y - z = -5
<» ' = 3
d) -0,2x + 0,5y = 1,7
5
3-^'
5 2 7-^' = 3^
0,3x + 0,4y = 0,9.
13. Mdt cdng ti cd 85 xe chd khach gdm hai loai, xe chd dupc 4 khach va xe chd dupc 7 khach. Dung ta't ca so xe dd, tdi da cdng ti chd mdt ldn dupc 445 khdch. Hdi cdng ti dd cd may xe mdi loai ?
14. Giai ede he phuong trinh
a)
X - 2y + r =12 2x- y + 3z =18 -3x + 3y + 2z = - 9 ;
b)
X + y + z = 7 3x - 2y + 2_- = 5 4x - V + 3z = 10.
15. Giai cac he phuong trinh sau bing may tfnh bd tiii 3 7 4
a)
r-3' = ^ 2 2 2 r^7 ' = 9'
b) 3.7x + 4,3y = -2,5 -5.1x + 2,7y= 4,8.
16. Giai cae he phuong trinh sau bdng may tfnh bd tiii
a)
3xi + 4x2 ~ ^-^?= 12 -4xi + 2x0 + 7x3 = 7 5xi + 6x2 - 4x3 =12 ;
b)
0,3x-4,7y + 2,3r = 4,9 -2,lx + 3,2y+ 4,5z = 7,6 4,2x-2,7 y + 3,7z = 5,7.
17. Mdt ehii cira hang bdn le mang 1 500 000 ddng dd'n ngan hang ddi tiln xu de tra lai cho ngudi mua. Ong ta ddi dupc ta't ca 1 450 ddng tiln xu cac loai 2000 ddng, 1000 ddng va 500 ddng. Biet ring sd tiln xu loai 1 000 ddng bdng hai ldn hieu ciia sd tidn xu loai 500 ddng vdi sd tiln xu loai 2000 ddng. Hdi mdi loai cd bao nhidu ddng tiln xu ?
18. T m gia tri ciia m de cac he phuong trinh sau vd nghidm
a) 13x + 2y = 9
I mx - 2 V = 2
77
BAI TAP ON TAP CHUONG III
19. Hay vidt dilu kidn ciia mdi phuomg trinh
a) V-3x + 2 = ; b) Vx- 2 + x = 3x2 + 1 - ^ _ 4 . x + 1
3x + 5 fz—-7 ., Vx + 4
c) , = V2x + 1 ; d) -^ = X + ^
V3x2 + 6x + 11 ^ - 9
20. Xac dinh 772 de mdi cap phucmg trinh sau tuong duong
a) 3x - 1 = 0 va ^"^ V + 2/72 - 1 = 0 ;
x - 2
b) x2 + 3x - 4 = 0 va 772x2 - 4x - m + 4 = 0.
21. Giai va bidn luan cae phucmg trinh sau theo tham sd m
a) 2m(x - 2) + 4 = (3 - m\ ; b) ^"^^^^^ = 3/72 + 2 ;
, 8/72X , . 1. 1 .X ( 2 -m)x
e) = (4/72 + l)x + 1 ; d) ^ - ^ = (772 - l)x - 1.
22. Cho phuong trinh
3x2 _^ 2(3^ .^ i)x-[-3m^ - m-i-I = 0.
a) Vdi gid tri ndo eiia 772 thi phuong trinh vd nghiem ?
b) Giai phuong trinh khi 772 = -1 .
23. Cho phuomg trinh
im + l)x2 + (3/72 - 1)X + 2/77 - 2 = 0.
Xdc dinh m de phucmg trinh cd hai nghidm Xi, X2 ma Xi + X2 = 3. Tfnh cac nghidm trong trudng hpp dd.
24. Giai eae phuong trinh
a) |3x - l| = 2x - 5 ; b) |2x + l| = |4x - 7| ;
I I I I |2x + 7| , ,
c) 5x + 2 + 3x - 4 = 4x + 5 ; d) J ^ = 3x - 1.
I I I I X -I '
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