🔙 Quay lại trang tải sách pdf ebook Bài tập đại số 10 nâng cao Ebooks Nhóm Zalo IGUYEN HUYOOAN (Chu bien) PHAM TH! BACH NGOC - DOAN QUYNH OANG HUNG THANG - LLTU XUAN flNH H i SO .HA XUXT BAN GIAO DUG VlfT NAM NGUYfiN HUY DOAN {Chu bien) PHAM THI BACH NGOC - DOAN QUYNH - DANG HUNG THANG - LUU XUAN TINH BAI TAP DAI s o NANGCAO (Tdi ban Idn thirndm) NHA XUAT BAN GIAO DUC VI^T NAM Ban quy^n thuoc Nha xu^'t ban Giao due Vidt Nam 01-2011/CXB/850-1235/GD Ma s6 : NB003T1 Ld I NOI DAU. Ki til nam hoc 2006 2007, ng^h Gi^o due bat ddu thuc hi^n giang day theo chucrng tiinh va sach gi^o khoa mdi Icfp 10. Di khm v6i viec d6i mod chirong trinh va sach giao khoa la ddi mdi v^ phiicfng phap day hoc va d(5i mdi c6ng tdc kilm tra danh gia k6t qua hoc tap cua hoc sinh. Di^u 66 phai duac th^ hi6n khong nhCrng trong sach giao khoa, sach giao vien mh. con trong ca sach bai tap - mOt tiii li6u kh6ng the thieu d6i vdfi giao viSn vk hoc sinh. Cu6'n Bai tap Dai so JO ndng cao nay diroc bi6n soan theo tinh thdn do. Bdii tdp Dai so 10 ndng cao g6m cac bai tap ducfc chon loc va sap x6'p m6t each h6 th6'ng, bam sat tiing chu d6 kid'n thiic trong sach giao khoa, nh^m giiip cac em hoc sinh sir dung song song vdri s^ch giao khoa, vira Cling c6 ki6'n thiic dang hoc, viJta nAng cao ki nang giai toAn. Titong tu nhu sach gi^o khoa Dai sd' 10 ndng cao, noi dung cua sach n^y g6m sau chirong : Chucrng I. Menh d^ - Tap hop Chuong II. Ham sd bac nha't va bac hai Chircmg HI. Phuong trinh v& he phuomg trinh Chucfng IV. B^t dang thirc vk bait phuong trinh Chucrng V. Th6'ng ke Chucfng VI. G6c lucmg giac va c6ng thiic lucmg giac. M6i chuong d^u ducrc md d^u bang ph^ "Nhihig kien thiJfc can nhd" P h ^ n&y t6m tat lai nhutig kiS'n thiic quan trong cua chuofng. Hoc sinh doc "Nhung kien thitc can nh&" d^ tim toi nhfing ki6'n thiic duoc van dung trong qua trinh giai bai tap. Sau khi hoc xong m6i chuong, cac em n6n tr6 lai phdn nay de' 6n tap vk ghi nhd nhirng kie'n thiic do. Tie'p theo la ph ^ "De bai" va sau do la ph ^ "Dap sd'- Huong dan Ldi giai". Cac bai tap trong phdn "De bai" duoc sap xep theo dung trinh tu cac bai hoc trong sach gido khoa. Do do hoc sinh c6 thd de dang tu lua chpn bai tap d^ lam th6m sau m6i bai hoc. Ben canh cac bai tap bam sat y^u cdu cua sach giao khoa, sach con bo sung m6t s6' bai tap vdi yeu cdu cao ban, giup hoc sinh bu6c ddu tiep can vdri nhiJng dang toan chu^n bi thi vao Dai hoc. Ngoai ra, cu6'i m6i chuong d6u c6 cdc bai tap trac nghi6m khach quan nham giup hoc sinh lam quen vol phuong phap kiem tra danh gia mdi nay. CAn chii y rang m6i cau hoi trac nghi^m khach quan, hoc sinh chi duoc Jam trong thcfi gian he't sire ban ch^ (chang ban, tir 1 de'n 2 phut). Sau khi giai bai tap, hoc sinh c6 the' tu minh ki^m tra lai ke't qua bang each d6'i chieu vdi ph^n "Ddp s6'- Hudng din - Left giai" (ngay sau phdn "De bai" cua m6i chuong). Trong phSn nay, cac tac gia chi chpn loc va nSu led giai d^y dit ciia m6t s6' it bai, eon lai ph ^ 16n cac bai d^u chi cho ddp s6' hoac dap s6' c6 \ahca. theo gpi y khi c^n thie't. Chu y rang cac hu6ng giai duoc neu trong "Huang ddn'\ tham chi trong cdc bai giai chi ti^t cung CO thI chua phai la hudng giai t6't nhSt. Cac tac gia nh ^ manh di^u nay vdi mong mu6'n : chinh hoc sinh se la nhftng ngudi dua ra nhftng Icri giai hay hon, sdng tao hon. Mac du cac tac gia da nit kinh nghidm tijt sach thf di^m va da c6' gang dl c6 duoc ban thao tO't nha't, nhung chae chin sach khdng tranh khoi con nhi^u thie'u sot. Cac tac gia ra't mong nhan dupe gop y cua ban doc g&i xa, nha't la ciia giao vien va cac em hoc sinh - nhOng ngucri true tie'p sijr dung sach. Cu6'i cung, cac tac gia to long bie't on.d^n H6i d6ng Th ^ dinh ciia BO Giao due - Dao tao da gop nhilu y kie'n quy bau, ddn Ban bidn tap sach Toan Tin, C6ng ty c6 ph ^ Dich vu xuSit ban Giao due Ha N6i - Nha xu^t ban Giao due Viet Nam da giup dd, hpp tac tich cue va c6 hieu qua trong qua trinh bien soan cu6n Bai tap Dai sd'lO ndng cao nay. CAC TAC GlA Q^huan^I MENH DE - TAP HOP A. NHONG KIEN THQC CAN NHO Menh de • Menh d^ logic (gpi tat la menh d^) la m6t eau khang dinh dung hoac mdt eau khang dinh sai. M6t menh d^ khOng the' viifa dung viita sai. • Menh dd "Kh6ng ph^i F\ ki hieu la? , dupe gpi la menh de phu dinh cua P. Menh dd P dung ne'u P sai va P sai neu P dung. • Menh dd "Ne'u P thi Q", ki hieu la/^ => Q, dupe gpi la menh dd keo theo. Menh dd k^o theo chi sai khi P diing, Q sai, • Menh dd "P ne'u va ehi ne'u Q\ ki hieu la f o g , dupe gpi la menh dd tuong duong. Menh dd nay dung khi va ehi khi P, Q ciing dung hoac cung sai. Phu dinh cua menh dd " VJC G X, P{x)" la menh dd " 3x e X, P{x) • • Phii dinh cua menh dd " 3x & X, P{x)" la menh dd " Vx e X, P{x)" Tap.hdp • Tap A dupe gpi la tap con ciia tap B, ki hieu la A c 5, ne'u mpi phan tijf cua A ddu la phdn tir ciia B. • Phep giao Ar\B -[x\x & Awkx €i B]. • Phep hpp AKJ B== [x\x & A hoact e B\. • Hieu ciia hai tap hpp A\B= {x I jc e Ava x ^ B}. • Phep l^y phkn bii : Ne'u A e £ thi OEA = E\A ^ {X\X e E\d.x ^ va cho bie't menh dd nay diing hay sai. b) Phat bie'u menh de P <:> Q va cho bi^t menh dd rtay dung hay sai. 1.5. Xet menh dd R : "Vi 120 chia he't cho 6 nen chia he't cho 9" Ne'u vie't menh dd R du6i dang "P => Q'\ hay neu noi dung cua cac menh dd P\aQ. Hoi menh dd R diing hay sai, tai sao ? 1.6. Cho hai menh dd P: "42 chia he't cho 5" ; Q: "42 chia he't cho 10", Phat bidu menh d6P =:> Q. Hoi menh dd nay diing hay sai, tai sao ? 1.7. Cho hai menh dd p.,-22003 - 1 la s6'nguyen t6'"; ^ : "16 la s6' chinh phuong" Phat bieu menh diP ^ Q,Hdi menh dd nay dung hay sai, tai sao ? 1.8. Cho hai tam giac ABC va DEF Xet cac menh dd sau P: "A = D,i = E" ; Q : "Tam giac ABC d6ng dang v6i tam giac DEF" Phat bidu menh diP => Q. Hoi menh dd nay diing hay sai, tai sao ? 1.9. Xet hai menh dd P : "7 la s6' nguyen l6" ; (2:"6! + 1 chia h^t cho 7". Phat bidu menh dd P <=> Q bang hai each. Cho bie't menh dd d6 diing hay sai. 1.10. Xet hai menh dd P : "6 la s6' nguyen t6'" ; Q:" 5\ + \ chia he't cho 6", Phat bidu menh di P <:> Q bang hai each. Cho bie't menh dd do diing hay sai. 1.11. Gpi X la tap hpp tat ca cac hoc sinh Idfp 10 of trucfng em. Xet menh dd chiia bie'n P{x) : ''x tu hoc d nha it nha't 4 giof trong mpt ngay" {x s X) Hay phat bieu cac menh dd sau bang cac cau thong thudng : a) 3x e X, P{x); h) ^x G X, Pix); c) 3x G X,P(x) ; d) Vxe X,P{x). 1.12. Xet cac cau sau day : a) Ta't ca cac hoc sinh of trucfng em ddu phai hpe luat giao thong. b) Co m6t hpc sinh Idfp 12 o trucfng em c6 dien thoai di d6ng. Hay vie't eac cau d6 du6i dang "Vx G X, P{xy hoac "3x s X, P(x)" va neu ro noi dung menh de chiia bie'n P(x) va tap hpp X. 1.13. Cho menh dd chiia hi€ti P{x) : "x = x'^" vdi x la s6' nguyen. Xac dinh tinh diing - sai ciia cac menh dd sau day : a)P(O); ' b)P(l); c)P{2)\ d)/>(-l); e) 3 A- G Z, P{x) ; g) \/x e Z, P{x). 1.14. Lap menh dd phii dinh eiia cac menh dd sau : a) Vx G R,x>x^ b) Vrt G N, «^ + 1 kh6ng chia he't cho 3. e) Vrt G N, /7^ + 1 chia het cho 4. d) 3r eQ, r^ = 3. 1.15. Xet tinh diing sai ciia cac menh dd sau va lap menh dd phii dinh eiia cac menh dd do : a) 3r G Q, 4r^ - 1 = 0. b) 3n G N, n^ + 1 chia het cho 8. c)Vx eR,x^ + x+\>0. d) V« G N*, 1 + 2 + ... + n khong ehia he't cho 11. 1.16. Cho menh dd ehiia bie'n P(x) : "x thich m6n Ngft van", trong do x \iy gia tri tren tap hpp Xcac hpc sinh ciia trudng em. a) Diing ki hieu I6gic de didn ta menh dd : "Mpi hpc sinh cua trucmg em ddu thieh m6n Ngu van." b) Neu menh dd phu dinh ciia menh dd tren bang ki hieu logic r6i didn dat menh dd phii dinh do bang cau th6ng thucmg. 1.17. Cho menh dd chiia bie'n P{x) : "x da di may bay", trong do x \&y gia tri tren tap hpp X eac eu dan eiia khu phd (hay xa) em. a) Dung ki hieu logic dd didn ta menh dd : "Co m6t ngu6i ciia khu ph6' (hay xa) em da di may bay'' b) Neu menh dd phu dinh eua menh de tren bang ki hieu I6gic r6i didn dat menh dd phii dinh bang cau th6ng thudng. §2. AP DUNG MfiNH Bt VAO SUY LUAN TOAN HOC 1.18. Phat bieu va chiing minh cac dinh If sau : a) Vn G N, n" ehia he't cho 3 => n chia he't cho 3 (gen y : Chiing minh bang phan ehiing). b) V« G N, n^ chia he't cho 6=> n chia het cho 6. 1.19. Cho eac menh dd ehiia bien P{n) : "n la s6' chan" va Q{n) : "In + 4 la s6' chan" a) Phat bidu va chimg minh dinh Ii Vn G N , P{n) => Q{n). b) Phat bieu va chiing minh dinh If dao cua dinh If tren. c) Phat bidu gpp dinh li thuan va dao bang hai each. 1.20. Cho cac menh de chiia bie'n P{n) : "n chia he't cho 5" ; Q{n) : "n ehia he't 2 2 • cho 5" va R{n): "n + 1 va n - 1 deu khOng ehia het cho 5" Sii dung thuat ngfi "didu kien e^n va dii", phat bidu va chiing minh cae dinh li dudi day : a) V/7 e N, P{n) <=> Q(n). b) V/7 G N, P{n) ^ R{n). 1.21. Cho eac s6' thuc ay,a2,—,a^^. Gpi a la trung binh e6ng ciia ehung ai + ... + a„ a = — -• n Chung minh (bang phan chiing) rang : ft nhS^t m6t trong cac s6' a^,a2,...,a„ se Idn hon hay bang a. 1.22. Sir dung thuat ngu "didu kien du" dd phat bidu cac dinh li sau : a) Ne'u hai tam giac bang nhau thi ehiing d6ng dang v6i nhau. b) Ne'u m6t hinh thang eo hai dudng cheo bang nhau thi no la hinh thang can. c) Ne'u tam giac ABC can tai A thi ducfng trung tuyen xuat phat tir dinh A cung la ducfng cao. 1.23. Sir dung thuat ngiJ "dieu kien e^n' de phat bieu eac dinh If sau : a) Ne'u mpt sd nguyen duong le dupe bieu didn thanh tong ciia hai sd ehfnh phuofng thi s5' do phai c6 dang Ak + 1 (^ e N). b) Ne'u m, n la hai s6' nguyen ducrng sao cho nr + n^ la m6t so chinh phuong thi m.n ehia het cho 12. 10 1.24. Hay phat bidu va ehiing minh dinh If dao ciia dinh If sau (ne'u eo) r6i sir dung thuat ngfl didu kien "c^n va dii" dd phat bidu g6p ca hai dinh If thuan va dao : Ne'u m, n \a hai s6 nguyen duong va m6i s6' ddu ehia he't cho 3 thi t6ng m^ + r? cung chia h^t cho 3. §3. TAP HOP VA CAC PHEP TOAN TRfiN TAP HOP 1.25. Cho A la tap hpp cac hinh binh hanh c6 bO'n goe bang nhau, B la tap hpp eac hinh chii nhat, C la tap hpp cac hinh thoi va D la tap hpp cac hinh vu6ng. Hay neu m6i quan he giiia cac tap noi tren. 1.26. Cho^ = {0;2;4;6;8|,fi={0 ; 1 ; 2 ; 3 ; 4} vaC = {0 ; 3 ; 6 ; 9|. a) Xac dinh (A u fi) u C va ^ u (B u C). Co nhan xet gi vd ke't qua ? b) Xac dinh (A n B) n C va A n (B n C). Co nhan xet gi vd ke't qua ? 1.27. Cho A - {0 ; 2 ; 4 ; 6 ; 8 ; 10}, S = {0 ; 1 ; 2 ; 3 ; 4 ; 5 ; 6| va C=|4;5;6;7;8;9 ; 10}. Hay tim a) A n (B n C) ; b) A u (B u C) ; c) A n (B w C) ; d) (A o B) n C ; €){Ar\B')vjC. 1.28. Ve bidu d6 Ven thd hien cac phep toan sau cua eac tap A, B va C : a) A n (B u C); b)(A \B)KJ{A\C)VJ{B\ C). 1.29. Co thd noi gi vd eac tap A va B neu eac ding thd'e tap hpp sau la diing : a)Aw B = A; \y) Ar^B = A\ C)A\ B = A ; d)A\B = B\A. 1.30. Lieu CO thd ke't luan A-B dupe kh6ng ne'u A, B va C la cac tap thoa man a)A^ C = BwC ; \>) Ar\C = Br\C 1.31. Vdi m6i tap A c6 m6t s6' hihi han ph^ tir, kf hieu lAt la sd ph^ tii ciia tap A. sap xe'p cac s6' sau day theo thu: tu tang d^n : a) lAl, lAw BI, lAnBl ; b) 1A\BI, \A\ + IBI, lA^Sl. 11 132. Cho tapA={xGR| 2 2} thanh hpp cac nvra khoang. 1.34. Chimg minh rang V6 la sd v6 ti. 1.35. Cho A = {x e R | ^> 2 } vaB = U G ]R| Lc - II < U-Hay tim I jv - 2 I A^ B va An B. 1.36. ChoA=^ {;c G R | U - II < 3} vaS = |X e R | lx +21 > 5). Hay timA n B . §4. s6 GAN DUNG VA SAI S6 —, — Qung ae xap xi vz. ,.^ ^.,,,.. ^^.p 17 99 . , /- 1.37. Trong hai so —-, —- diing de xap xi V2. a) Chijmg to rang — xa'p xi V2 t6t hon. 99 r •> -5 b) Chimg minh rang sai sd tuyet ddi cua — so vdfi V2 nho hon 7,3.10 355 1.38. Cae nha toan hpc da xap xi sd n boi sd —— Hay danh gia sai sd tuyet ddi bie't 3,14159265 thi lA w BI = lAl + IBI. b) Chiing minh rang B^u (A \B) = Au B vaB n {A\B) = 0. c) Chung minh ring A-{Ar\B) u (A \ B). d) Tijr dd suy ra edng thiic sau \A u BI = lAl ^ IBI - lA n B\. 1.48. Cho A = {;t e RIU - ll>3 } va6 = U G Rl lx + 2l<5}.TimAnB . 1.49. Ngudi la goi m6t sd hOu ti r cd dang r = — la sd hiJii ti nhi phan. 2" Bie't rang trong mdi khoang tuy y ddu cd ft nha't mdt sd huu ti nhi phan. Qidng minh rang trong mdi khoang ba't ki ddu cd ft nheit 100 sd huu ti nhi phan. 13 M6t each t6ng quat ehung minh rang : Cho m6t sd nguyen ducfng M Idn tuy y. Khi do, trong mdi khoang tuy y ddu ed ft nh^t M s6 hiiu ti nhi phan. 1.50. Gia sir;c la mdt gia tri gdadung ciia v5 . Xet sd a = x + 2 . Chiing minh rang \a'j5\<\x-yf5\. tire la ne'u la'y a la gia tri g^n diing ciia v5 thi ta dupe dd ehfnh xac cao hon la la'y x. Gldl THifiU MOT S 6 CAU HOI TRAC NGHlfiM KHACH QUAN 1.51. Trong cdc menh dd dudi day menh dd nao ddng, menh dd nao sai ? a) V;c G R,x>x^. b) Vrt e N, n^ + 1 khdng chia h^t cho 3. c) Vn e N, «^ + 1 chia h^t cho 4. d)3rG = 3. Ddung Qtiung n^^ung n*Jung Dsai Dsai Dsai Dsai Trong cdc bdi tit 1.52 din bdi 1.54 hay chon phuang an tra ldi diing trong cac phuang an da cho. 1.52. Cho cac cau sau : a) Hai Phdng la mdt thanh phd d Midn Nam. b) Sdng Hdng chay qua thii dd Ha N6i. e) Hay tra ldi cau hoi nay ! d) 2 + 37 = 39 ; e) 5 + 40 = 70 ; g) Ban cd rdi tdi nay khdng ? h) A: + 2 = 11 ; Sd cau la menh di trong cae eau tren la (A)l : (B)2; (C)3; (D) 4 ; (E) 5. 1.53. Cho menh dd chiia bie'n P{x) : "jc + 15 < x^" \dix la sd thue. Menh dd diing la menh dd : (A) P{0); (B) B(3); (C) B(4); (D) Pi5). 14 1.54. Cho menh dd " Vx G R, x^ + x + 1 > 0". Menh dd phii dinh eiia menh dd tren la : (A) Vx G R, ;c^ +X + 1 <0 ; > (B) Vx G R, x^ +;c+ 1< 0 ; (C) Khong ton tai X G R ma x^ + x + 1 > 0 ; (D) 3x G R, x^ + x + 1< 0. 1.55. Trong cae menh de sau day menh dd nao khdng la dinh If: (A) V/i G N, n^\2 =^ n':2 ; (B) VM e N, n^: 3 => « : 3 ; (C) Vrt e M, «^ ; 6=^ rt ; 6 ; (D) \/n e N, n^': 9^ n : 9. 1.56. Trong eac menh dd sau day menh dd nao la mfnh dd diing. (A) Vx G R, X > - 2 => x^ > 4 ; (B) Vx G R, x > 2 => x^ > 4 ; (C) Vx G R, x^ > 4 => X > 2 ; (D) Vx G R, x^ > 4 => x > -2 . Trong cdc bdi tiJC 1.57 den 1.63, hay chon phuang an tra ldi diing trong cdc phuang an dd cho. 1.57. Trong cac sd dudi day, gia tri g^n diing ciia V65 - v63 vdi sai sd tuyet ddi be nha't la : (A) 0,12 ; (B) 0,13 ; (C) 0,14 ; (D) 0,15. 1.58. Cho tap A = {-1; 0 ; 1 ; 2}. Khi dd ta cung eo : (A) A = [-1 ; 3) n N ; (B) A = [-1 ; 3) n Z ; (C) A = [-1 ; 3) n N* ; (D) A = [-1 ; 3) n Q. 1.59. Cho doan M = [-4 ; 7] va tap A' = (-oo ; -2) ^ (3 ; +oo). Khi dd M n A^ la (A) [-4 ; -2) w (3 ; 7] ; (B) [-4 ; 2) L; (3 ; 7); (C) (-00 ; 2] u (3 ; +^); (D) (-oo ; -2) w [3 ; +oo). 1.60. Cho hai tap hpp A = {XG R |X+3< 4 + 2X} ; S = {xG R 1 5x-3<4x - 1}. Ta't ca cdc sd tu nhien thudc ca hai tap A va B la ' (A) 0 va 1 ; (B) 1 ; (C) 0 ; (D) Khdng cd sd nko. 15 1.61. Cho cac nira khoang A = (^co ; -2] ; B = [3 ; +oo) va khoang C = (0 ; 4) Khi do tap (A u B) n Cla (A) {XGRI3 3} ; (C) |x e EI3 3}. 1.62. Cho cac khoang A (-2 ; 2) ; B = (-1 ; +co) va C = -oo ; - . Khi dd giao V LJ Ac\Br\C\a (A) X G R I -1 < X < i ; (B) Ix G R 1 -2 < X < 1} ; (C) X G E I -1 < X < 11 ; (D) Ix G R I -1 < X < i| . 1.63. Cho sd thuc a < 0. Didu kien eSn va dii de hai khoang (-co ; 9a) va 4 V — ; + 00 CO giao khae tap rong la a I (A)-|<«<0 ; (B) -|)-\ 1.15. a) Menh dd dung vi vdi / = -- thi 4r'^ - 1 = 0. Menh dd phii dinh la "Vr G Q, 4r^ -1?^0 " b) Menh dd sai. Ta ehung td menh dd phii dinh "\/n e N, «^ + 1 khdng ehia he't cho 8" la diing. That vay, ne'u n la sd chan thi n^ + 1 la sd le nen khdng ehia het cho 8. Neu n la sd \e,n = 2k+\{ke N) thi n^+\= 4k{k + 1) + 2 ehia 8 du 2 ( vi k{k + 1) la sd chan). e) Menh dd diing. Menh dd phu dinh "3x G R, x^ + x + 1 < 0" d) Menh dd sai. Ta ehiing to menh dd phii dinh "3n GN,l+ 2 + ---+ n chia he't cho 11" la diing. That vay vdi n = 11 thi 1 + 2 + ••• + 11 = 66 chia he't cho 11. 1.16. a) VXGX,B(X) . b) 3x G X,P{x), nghia la "Cd mdt ban hpc sinh ciia trudng em khdng Ihi'eh mdn Ngii van". 1.17. a)"3xG A',B(x)" b) Menh dd phu dinh : "Vx e X,P{x)" nghia la : "Mpi ngudi trong khu phd (hay xa) em ddu chua di may bay" 1.18. a) "Ne'u n \a sd tu nhien sao cho n ehia he't cho 3 thi n cung ehia hdt cho 3", Ta chiing minh bang phan chiitig. Gia su tdn tai « G N de n ehia het cho 3 nhung n khdng chia hdt cho 3. Ne'u « = 3A: + 1 (/: G N) thi n^ = 3k{3k + 2) + 1 khdng chia het cho 3. Neu n = 3k-i {k e N)tlu n^ = 3k{3k - 2) + 1 khdng chia he't cho 3. b) "Ne'u n la sd tu nhien sao cho n^ chia hd^t cho 6 thi n cung chia he't cho 6". That vay ndu n^ ehia he't cho 6 thi n^ la sd chan, do dd n la sd chan, tiic la n ehia he't cho 2. Vi n^ chia he't cho 6 nen nd chia hdt cho 3. Theo cau a) didu nay keo theo n chia he't cho 3. Vi n chia he't cho 2 va 3 nen n chia he't cho 6. 18 2-BTDS10,NC - B 1.19. a) Phat bidu : " Vdi mpi sd tu nhien «, ne'u n chan thi 7n + 4 la sd chan." Chiing minh. Ne'u n chan thi In chSn. Suy ra 7n + 4 chan vi tong hai sd chan la sd chan. b) Dinh If dao : "\fn G N , Qin) => P{n)" tiic la "Vdi mpi sd tu nhien n, ne'u 7« + 4 la sd chan thi n eh^n" ChUng minh. N^u In + 4 = m chan thi In = m - 4 chin. Vay In chan nen n chan. c) Phat bidu gdp hai dinh If thuan va dao nhu sau : "Vdi mpi sd tu nhien n, n chan khi va ehi khi 7n + 4 chan" hoac "Vdi mpi sd tu nhien n, n chan ne'u va chi ne'u 7/7 + 4 chan". 1.20. a) Phat bidu nhu sau : "Didu kien edn va dii dd sd tu nhien n chia he't cho 5 la rt chia het cho 5" Chi/tng minh. Ne'u n = 5k [k e N) thi n^ = 25k^ chia he't cho 5. Ngupe lai, gia sir /z = 5jt + r vdi r = 0, 1, 2, 3, 4. Khi dd n^ = 25^^ + lOkr + r^ chia he't cho 5 nen /• phai ehia he't cho 5. Thii vao vdi r = 0, 1, 2, 3, 4, ta tha'y chi cd vdi r = 0 thi r mdi ehia he't cho 5. Do do n = 5k t\tc la n ehia het cho 5. b) Phat bidu nhu sau : "Didu kien cdn va dii dd sd tu nhien n ehia he't cho 5 la ca /7^ - 1 va /7^ + 1 ddu khdng chia het cho 5" Chimg minh. Ne'u n ehia he't cho 5 thi n^ - 1 chia 5 du 4 va /7^ + ! chia 5 du I. Dao lai, gia sir /? - \ va n + 1 ddu khdng ehia he't cho 5. Gpi /• la sd du khi chia n cho 5 (r == 0, 1, 2, 3, 4). Ta c6 n = 5k ^ r {k ^ N). Vi n^ = 25/t^ + lOkr + r^ nen suy ra ca r^ - 1 va r^ + 1 ddu khdng chia he't cho 5. Vdi r = 1 thi r^ - 1 = 0 chia hdt cho 5. Vdi r = 2 thi r^ + 1 = 5 chia he't cho 5. Vdi /• = 3 thi r^ + 1 = 10 ehia het cho 5. Vdi /• = 4 thi /-^ - 1 = 15 chia het cho 5. Vay ehi cd the r = 0 tiic \an-5k hay n chia he't cho 5. 1.21. Chiimg minh bang phan ehung nhu sau : Gia su trai lai ta't ca eac sd a^,a2....,a,^ ddu nho hon a. Khi dd ai + ^2 + • • • + a„ < na suy ra a = -^ ^ < a. Mau thuln. 19 1.22. a) Didu kien dii dd hai tam giac ddng dang la ehiing bang nhau. b) Bi mdt hinh thang la hinh thang can, didu kien dii la hai dudng cheo ciia nd bang nhau. c) Didu kien dii dd dudng trung tuye'n xua't phat tijf A eua tam giac ABC vudng gde vdi BC la tam giac do can tai A. 1.23. a) De mdt sd nguyen duong le bidu didn thanh tdng cua hai sd ehfnh phuong didu kien edn la sd dd ed dang 4^+1 . b) Cho m, n la hai sd nguyen ducfng. Didu kien e^n di m + n la sd ehfnh phuong la tfeh mn chia he't cho 12. 1.24. Dinh If dao : "Ne'u m, n \a hai sd nguyen duong \k m + n ehia he't cho 3 thi cam van ddu chia he't cho 3" Chifng minh. Ne'u mdt sd khdng chia he't cho 3 va sd kia ehia he't cho 3 thi rd rang t6ng binh phuong hai sd do khdng chia h^t cho 3. Gia sis m va n ddu khdng chia he't cho 3. Ne'u m = 3k + 1 hoac m = 3k + 2 ta ddu cd /M^ehia 3 du 1. Thanh thir m^ + n^ chia 3 du 2. Vay n^u m^ + n^ chia he't cho 3 thi ehi ed thd xay ra kha nang cam van ddu chia he't cho 3. vay : Didu kien c^n va dii dd /n^ + n^ ehia he't cho 3 (/n, /z G N*) la ca m va n ddu ehia hdt cho 3. 1.25. TaedA = B;D (z B ^ A ; D czC ; D = BnC. 1.26. a) A uB={0;l;2;3;4;6;8},(AwB) u C = {0 ; 1 ; 2 ; 3 ; 4 ; 6 ; 8 ; 9}. BL;C={0;1;2;3;4;6;9},A W (BuC)={0;l;2;3;4;6;8;9} . Tacd (AwB)^ C = Au (BuC). b)An6 = {0;2;4},(AnB) n C= {0}. B nC = {0;3},A n (B nC) = {0}. Ta cd (A nB) n C = A n (B r\C). Chu y : Cd thd chiing minh dupe rang cac ding thiic tren ludn diing y6i, A, B, C la ba tap hpp ba't ki. 1.27. a) A n (B nC)= {4;6} ; b)A ^ (B uC)= {0;1;2;3;4;5;6;7;8;9 ; 10}. 20 c)A n (B yuC) = A. d)A uB= (0 ; 1 ;2;3;4;5;6;8 ; 10}. vay (A wB) n C = |4 ; 5 ; 6 ; 8 ; 10}. e)A nB={0;2;4;6} . Vay (A nB) L^C={0;2;4;5;6;7;8;9 ; 10}, 1.28. a) b) Phctn gach cheo la hinh bi^u diin A n (B u C) Hinh l.l Ph^n gach cheo la hinh bidu difin (AsB) u (A\C) u (S\C) Hinh 1.2 1.29. a) Ne'u A^ B = A thi 6 la tap con ciia A vi theo dinh nghia ta ludn cd B(^A^B. Di kiem tra rang didu ngupe lai ciing dung. Vay AwB = A neu va chi ne'u B la tap con eiia A. b) Ne'u A n B = A thi A la lap con cua B vi theo dinh nghia ta ludn ed AnB cB . c) Neu A \B = A thi hai tap A va B phai khong giao nhau. That vay, neu ton tai X e A va X e B thi do A =^ A \ B nen x G A\B. Suy ra x khdng thupc B (mau thutn). Ngupe lai bang each ve bidu dd Ven dd tha'y ne'u AnB = 0 thi A \B = A cung diing. Vay A \ 6 ^ A nduva chi ndu An B =0 . d) Neu A \ B = B \ A thi A = B. That vay ne'u A ^ B thi phai cd mot phdn tir ciia tap nay nhung khong thupc tap kia, chang han x e A va x^ B suy rax G A\B nen x G B\ A do ddx e B vax^ A (mau thuan). Dd kiem tra rang didu ngUpe lai ciing diing. Vay A\ B = B\A n^u va ehi neu A =B. .21 1.30. a) Khong. Chang han A - {1 ; 2 ; 3 ; 4}, B = {1 ; 2}, C = {3 ; 4 ; 5}.A =^ B nhung A u C^B u C= {1 ;2;3;4;5}. b) Khdng. Chang han A - {1 : 2 ; 3 ; 4}, B = {3 ; 4}, C = {3 ; 4 ; 5}. Ta c6 A ^ B nhung A n C = B n C= {3;4} . 1.31. a) \A n B\, \A\, lA u BI ; b)lA\BUA wBl, lAl + lBl. 1.32.A = (2 ; 3) u(-3;-2) . 1.33.A-[2,+oo)LJ ( - 00 ;-2] . 1.34. Chutig minh bang phan chiing. Gia su v6 = — la mot sd hiiu ti trong dd a, h la hai sd nguyen duong va (a, b) = \. Suy ra 6/? = a Vay a chia he't cho 2 va chia het cho 3. Suy ra a chia he't cho 2 va chia he't cho 3 tiic la a chia het cho 6. Dat a = dk {k e N*). Thay vao ta dupe 6b^ = 36k^ hay b ~ 6k Lf luan tuong tu nhu tren ta suy ra b chia het cho 6. vay a va b CO ude ehung la 6. Didu nay mau thuln vdi gia thie't a, b khdng ed udc ehung Idn hon 1. 1.35.TacdA = {x G MlO 1,414213. u o do u < 99 DoddO< - ^ - V2 < 1,414286- 1,41421*3 « 0,000073. 22 1.38. Ta cd (su dung may tfnh bo tiii) : 355 113 3,14159292... < 3,14159293. 355 Do vay 0 < — - n < 3,14159293 - 3,14159265 ^ 0,00000028. Vay sai sd tuyet ddi nho hon 2,8.10"^ 1.39, 1.40. Ta ed AL^ = BLLD = 2, do dd AL =^2 . Lai ed BD = 3, suy ra dien tfeh ciia hinh ehunhatla3V2 =3.1,41421356... «4,24264... «4,24. Chii sd 3 (hang phan tram) la chu" sd chae do 0,00312 < 0,005. Do dd C cd 3 chu" sd chae (d hang don vj, hang phdn chuc va hang phdn tram). ..2 Hinh 1.3 1.41. 1.42. 1.43. 1.44. 1.45 . 1 ^' = rrx (1 - x) . Sai sd tuong ddi la S^ - —~ a _ *^ 1 - x^ I + v " 1 - ; a) Ne'u mdt ngudi la ki su thi ngudi do ed lay nghd. b) Ne'u mdt ngudi khdng cd lay nghd thi ngudi dd khong cd thu nhap cao. c) Ne'u mot ngudi la fci su thi ngudi a'y cd thu nhap cao. Menh de phii diiih la "3n e N, n^ + /7 + 1 khdng la sd nguyen td" Menh dd phii dinh dung. Vf du vdi /? = 4 thi /i~ + /7 + 1 =2 1 chia he't cho 3 nen la,hpp sd. Dinh li dao : "Ne'u hai sd duong a, b cd trung binh edng va trung binh nhan bang nhau thi chiing bang nhau.' Chitng minh. Gia sir a, b la hai sd duong sao cho a + b = ^ab Khi dd a + b-2yfab =^0 c> (Ja - Sf=0 ::^ a = h. vay didu kien c^n va dii de hai sd duong bang nhau la trung binh epng va trung binh nhan cua ehung bang nhau. a) Gia sii ea bdn gde ddu nhpn. Khi dd tdng cua bdn gde cua tu" giac se nho hon360" (mau thuan). Tuong tu gia sii ca bdn gde ddu tii. Khi dd t6ng ciia bon gde ciia lii giiic se Idn hon 360° (mau thuan). 23 b) Gia siix + y + xy = -1 . Suy rax + y + xy + 1 = (x + l)(y + 1) = 0. vay phai ed hoac x = -1 hoac y = -1 (mau thu^n). 1.46. a) Menh dd sai. b) Menh dd diing. e) Menh dd sai. d) Menh dd diing (vi v6im= I thi n chia he't cho m vdi mpi n). e) Menh di diing (vi v6i n = 0 thi n ehia h^t cho m vdi mpi m). 1.47. a) Hien nhien. b) De tha'y bang each ve so d6 Ven. c) Dd tha'y bang each ve so dd Ven. d) Ta cd lA uBl = IB! + lA \Bl, (do cau a) va b)). (1) Lai cd A = (A \B) u(A n B) ( do c)) thanh thir UI = IA\BI + IA n BI. vay IA\BI = IAI-IA n BI. (2) Thay (2) vao (l)ta dupe IAWBI = IAI + I6l-IA n BI. 1.48. Tae6A = (4;+oo)w (-oo ;-2); B = (-7 ; 3). Vay A n B = (-7 ;-2). 1.49. Gia sir (a ; b) la mdt khoang ba't ki. Ta chia (a ; b) lam 100 khoang eon rdi nhau. Theo nhan xet tren trong mdi khoang con dd ddu cd chiia mdt sd hixu ti nhi phan. Cac sd hiJu ti nhi phan nay khae nhau do cac khoang con khong giao nhau. Vay (a ; b) chiia ft nha't 100 sd hiiu ti nhi phan. Md rOng : Ta chia khoang (a ; b) lam M khoang eon rdi nhau. Theo nhan xet tren trong mdi khoang con dd ddu cd chiia mot sd hiiu ti nhi phan. Cac sd hUu ti nhi phan nay khae nhau do eac khoang con khdng giao nhau. Vay {a ; h) chiia ft nha't M sd huu ti nhi phan. 1.50. Dat M - X - V5 va V = fl - V5 Ta cd rz 2x + 5 - xVs - 2V5 (2 - %/5)(x - VS) (2 - V5)M -=^~^5- J ^ - 772 " - ;, + 2 • 24 vay a-yf5\=\v\= \u\^^~^ <^^~^ \u\<\u\ = x-yf5 • I x + 2 2 1.51. Cau a) la menh dd sai. cau b) la menh dd diing. That vay neu /7 = 3)t thi /7^ + 1 = 9A:^ + 1 chia 3 du 1. Ne'u /7 = 3jt + 1 thi /i^ + 1 = 9k^ + 6Jt + 2 chia 3 du 2. Neu n =3k + 2thin^ +\ = 9k^ + 12/: +5 chia 3 du 2. cau e) la menh dd sai. That vay ne'u n = 2k thi n^ + \ = 4k^ + \ chia 4 du 1. N^u n = 2k+lthi n^ + \ =4k^ +4^ + 2 chia 4 du 2. Cau d) la menh dd sai do V3 la sd v6 ti. 1.52. Phuong an (D). (Cae cau a), b), d), e) la cac menh dd). 1.53. Phuong an (D). 1.54. Phuong an (D). 1.55. Cae menh dd (A), (B) va (C) la menh dd diing Menh de (D) la sai vi vdi n = 3 thi 3^ = 9 ehia h^t cho 9 nhung 3 khdng chia he't cho 9. Do dd menh dd (D) khdng phai la dinh If. Vay ta chpn phuong an (D). 1.56. (A) la menh dd sai. That vay vdi x = 0 thi 0 > -2 nhung 0 < 4. (B) la menh dd diing. (C) la menh dd sai. That vay vdi x = -3 thi (-3)^ = 9 > 4 nhung -3 < 2. (D) la menh dd sai vi chang han, khi x = -3 thi (-3)^ > 4 nhung -3<-2 . Do dd ta chpn phuong an (B). 1.57. Sii dung may tfnh cho ta V65 - >/63 «0,I25003815... Do dd ta chpn phuong an (B). 1.58. Phuong an (B). 1.59. Phuong an (A). 1.60. Phuong an (A). I 1.61. Phuong an (C). 1.62. Phuong an (D). 1.63. Phuong an (A). 25 uang II *' HAM SO A. NHQNG KIEN THLfC CAN NHO Ham so' Trong bang sau day, y ~ f(x) \a mot ham sd vdi tap xac dinh S), K la mdt khoang (niia khoang hay doan) nam trong 3). Tinh chat cua ham so The hien qua do thi yo = fi^) (vdixothudcy^). Diem (XQ ; yp) thude do thi cua ham sd. y 1 1 Ham sd/d6ng bie'n tren K : VXi,X2 eKlX^ /(Xi)/(xi)>/i;x2). O Tren K, dd thi eua ham sd/d i xudng (theo chidu tang ciia ddi so'). 26 Ham sd/khdng ddi tren K : y = m{m\a hang sd). y =f{x) la ham sd chan : Vx € 3): -X G ® va/(-x) =/(x). y =f(x) la ham sd le : Vx e ^"^: -X G y^' va/(-x) = -fix). yk o b X W thi eua ham sd/nlm tren dudng thang song song (hoac triing) vdi Ox. Dd thi cua ham so fed true ddi xiing la true tung Oy. B6 thj eua ham sd/ed tam ddi xung la gde toa dp O. 27 Ham so bac nhat • Ham sd cho bdi bidu thiic y = ax + b (a^^tQ). Tap xac dinh : • Bang bien thien : X y = ax + b {a>0) - 0 0 +00 ^ ^ ^ +00 —00 ^"^'''''^ X y = ax + b ia<0) - 0 0 +00 ~-*' -0 0 • D6 thi ciia ham s6 y = ax + b {a ^ 0) la dudng thing cd he sd gde bang a, eat Ox tai (—; 0) va cat Oy tai (0 ; b). a • Ne'u (di) va (J2) la hai dudng thing phan biet ed he sd gde la Oj va ^2 thi; (Ji) 7/(^2) <^ i?i =?^ «2 ' d^ X dn <^ a^.a2 - -1 . Ham so bac hai • Ham sd cho bdi bieu thiic y^ax + fex + c (« ^ 0). Tap xac djnh : M • Bang bie'n thien : . ^ 2 y = ax •>t-bx + c - 0 0 b 2a + 0 0 f^'^ X 2 L. y~ax + bx + c —00 / h 2a A "4^ \ + 0 0 (fl>0 ) " \ \ A/ 4a («<0) / —00 \ - 0 0 » D6 thi ciia ham sd y ~ ax •¥ bx + c {a ^ H) \a. parabol cd dinh la didm ' A.__ A 'l . b . /2fl ' 4a, ; ed true ddi xiing la dudng thang x =-—', hudng be 16m len tren khi o > 0 va xudng dudi khi o < 0. 4 28 Pheptinhtiendothj Cho ham sd y =f{x) cd d6 thi (G) ; p va ^ la hai sd khdng am. • Khi tinh tie'n (G) len tren q don vi, ta dupe d6 thi cua ham sd y -f{x) + q. • Khi tinh tie'n (G) xudng dudi q don vi, ta dupe dd thi ciia ham sd y =^x) - q. • Khi tinh tie'n (G) sang trai p dan vi, ta dupe dd thi ciia ham sd y ~f{x + p). • Khi tjnh tie'n (G) sang phai p don vi, ta dupe dd thj ciia ham sd y -fix - p). B. DE BAI §1. DAICl/ONGVfiHAMSd Khai niem ham so 2.1. Dudng tron tam O ban kfnh r khdng phai la d6 thi ciia mdt ham sd. Nhung ntfa dudng tron g6m cae didm ed lung dd khdng am ciia dudng tron tam 0 ban kfnh r (h. 2.1) la dd thi eiia mdt ham sd. Hay vi^t bidu thiic xac dinh ham sd dd va cho bie't tap xac dinh cua nd, bie't rang dudng tron tam O ban kfnh r \a tap hpp cic didm cd toa d6 2 2 2 (x ; y) thoa man he thu'c x + y = r 2.2. Tim tap xac dinh cua cdc ham sd sau : a)y = x - 1 x^- l V2X+T 2x^ - X - 1 3x + 4 Hinh 2.1 Nita dir6ng tron ban kfnh r = 2 c)y = (X - 2)Vx + 4 ' 29 2.3. Cho ham sd f{x) = x + 1 }[xV\ ne'u X > 0, ne'u - 1 < X <0. I x- 1 a) Tim tap xac dinh cua ham sdy =/(x). b)Tfnh/(0),/(2),/(-3),/(-l). 2.4. Cho ham sd fix) =x^+ylx~l. a) Tim tap xac dinh cua ham sd. b) Dung bang sd hoac may tfnh bo tiii, tfnh gia tri gdn dung eiia /(4), fi-Ji) .fin) chinh xac de'n hang ph^n tram. Sir bien thien cua ham so 2.5. Hay lap bang bie'n thien ciia ham sd cd dd thi la niia dudng tron cho tren hinh 2.1. 2.6. Dd thi eiia mdt ham sd xac dinh tren R dupe cho tren hinh 2.2. Dua vao dd thi, hay lap bang bie'n thien cua ham sd dd. Hay cho bie't gia tri Idn nha't hay nho nha't ciia ham sd (ne'u cd). 2.7. Bang each xet ti sd Z^^zWXfi)^ j^-y X2 -X i neu su bie'n thien eiia cac ham sd sau (khdng yeu e^u lap bang bie'n thien cua nd) tren cac khoang da cho : a) y = x + 4x + 1 tren mdi khoang (-oo ; -2) va (-2 ; +oo); b) y = -x^ + 2x + 5 tren mdi khoang (-oo ; 1) va (1 ; +oo); e) y = X x + 1 tren mdi khoang (-oo ; -1) va (-1 ; +co); tren mdi khoang (-oo ; 2) va (2 ; +co). 30 d) y = 2x + 3 -x + 2 Ham so' chin va ham so le 2.8. Cd hay khdng mOt ham sd xac dinh tren E viia la ham sd ehSn vira la ham sd le ? 2.9. Cho hai ham sd y = fix) vay = g(x) xac dinh tren K. Dat Six) = fix) + g(x) vaP(x) =/(x)^(x). Chiing minh rang : a) Ne'u y = fix) vay = g{x) la nhflng ham sd chan thi y = S{x) va y = Pix) Cling la nhiing ham sd chan. b) Ne'u y =fix) va y = gix) la nhiing ham sd le thi y = Six) la ham sd le va y = Pix) la ham sd chan, c) Ne'u y = fix) la ham sd chan, y = gix) la ham sd sd le thi y = Pix) la ham sd le. 2.10. Xet tfnh chan le ciia eac ham sd sau : a) y = 3x'* + 3x^ - 2 ; b) y = 2x^ - 5x ; e) y = X tx| ; d) y = Vl + x + Vl - x ; e) y = V1 + X - vl - X. Tjnh tie'n do thj song song vdi true toa do 2.11. Trong mat phing toa dd, cho cae didm A(-l ; 3), Bi 2 ; -5), da ; b). Hay tinh toa dd cae didm ed dupe khi tinh tid^n cac didm da cho : a) Len tren 5 don vi; b) Xudng dudi 3 don vi ; c) Sang phai 1 don vi; d) Sang trai 4 don vi. 2.12. Ham sd y = 4x - 3 ed dd thi la dudng thing id). a) Gpi idi) la dudng thing ed dupe khi tinh tie'n id) len tren 4 don vi. Hoi (rfj) la dd thi ciia ham sd nao ? b) Gpi id2) la dudng thing cd dupe khi tinh tid^n id) sang trai 1 don vi. Hoi id2) la d6 thi cua ham sd nao ? e) Em cd nhan xet gi vd hai k^t qua tren ? 31 _2 2.13. Gia sii ham sd y = — cd dd thj la (//). a) Ne'u tjnh tie'n (//) xudng dudi 3 don vi thi ta dupe d6 thi cua ham sd nao ? b) Ne'u tinh tie'n (//) sang phai 2 don vi thi ta dupe dd thi ciia ham sd nao ? e) Ne'u tinh tie'n (//) len tren 1 don vi r6i sang trai 4 don vi thi ta dupe dd thi eua ham sd nao ? §2. HAM S6 BAC NHXT 2.14. Ve dd thj ciia mdi ham sd sau : 1 J3 a)y = 2x-3; b) y = -- x +1 ; c) y = — x + 2. 2.15. Trong mdi trudng hpp sau, tim eac gia tri ciia k sao cho dd thi ciia ham sd y = -2x + kix+\) a) Di qua gd'c toa dd O ; b) Di qua didm M(-2 ; 3); c) Song song vdi dudng thing y-yj2x. 2.16. Tim cac cap dudng thing song song trong cae dudng thing eho sau day : a) 3y - 6x + 1 = 0 ; b) y = -0,5 x - 4 ; c)y = 3+ | ; d)2y + x = 6; e) 2x-y = 1 ; f)y = 0,5x+ 1. 2.17. Ve dd thi cua mdi ham sd sau vk lap bang bie'n thien ciia nd : a) y = I 3x + 5| ; b)y = -2|x-l| ; c)y = -ii2 x + 3|+| . 2.18. Trong mdi trudng hpp sau, xac dinh avkb sao cho dudng thing y = ax + b a) Cat dudng thing y = 2x + 5 tai didm cd hoanh dd bing -2 va cat dudng thing y = ~3x + 4 tai didm cd lung dd bang -2 ; 32 b) Song song vdi dudng thing y= - x v^ di qua giao didm cua hai dudng thing y = -x^ + 1 vay = 3x + 5. 2.19. a) Cho didm Aix^ ; y^. Hay xac dinh toa dp ciia didm B, bie't ring B dd'i xu^g vdi A qua true hoanh. b) Cbiing minh ring hai dudng thing y = x - 2 va y = 2 - x ddi xiing vdi nhau qua true hoanh. c) Tim bidu thiic xac dinh ham sd y = fix), biet ring d6 thi eiia nd la dudng thing dd'i xiing vdi dudng thing y = -2x + 3 qua true hoanh. 2.20. a) Cho didm AixQ ; yo). Hay xac dinh toa dd cua didm B, bie't ring B ddi xiing vdi A qua true tung. b) Chiing minh ring hai dudng thing y = 3x + 1 va y = -3x + 1 ddi xiing vdi nhau qua true tung. c) Tim bidu thiic xac dinh ham sd y = fix), bie't ring dd thi cua nd la dudfng thing ddi xiing vdi dudng thing y = 0,5x - 2 qua true tung. 2.21. Mdt tia sang ehie'u xien mot gde 45° de'n didm O tren bd mat eua mdt cha't long thi bi khue xa nhu hinh 2.3. Ta lap he toa dp Oxy nhu da thd hien tren hinh ve. a) Hay tim ham sd y - fix) ed dd thi triing vdi dudng di eiia tia sang ndi tren. b) Lap bang bie'n thien ciia ham s6y-fix). 2.22. a) Tim didm A sao cho dudng thing y == 2mx + 1 - m ludn di qua A, du m la'y ba't cii gia tri nao. b) Tim didm B sao eho dudng thing y = mx ^ 3 - x ludn di qua B, dii m la'y ba't cii gia tri nao. 2.23. Trong mdi trudng hpp sau, tim eac gia tri ciia m sao eho a) Ba dudng thing y = 2x,y = -3-xva y = mx + 5 phan biet va ddng quy. b) Ba dudng thing y = -5(x +l), y = mx + 3va y = 3x + m phan biet va ddng quy. 3-BTDS10,NC-A 33 §3.HAMS6BACHAI 2 2 2.24. Cho ham s6 y = - x a) Khao sat su bie'n thien va ve d6 thi iP) ciia ham sd da eho. b) Ne'u tinh tie'n iP) len tren 2 don vi thi ta dupe dd thi ciia ham sd nao ? c) Ne'u tinh tie'n iP) xudng dudi 3 don vi thi ta dupe dd thi eiia ham sd nao ? yl3 2 2-25. Cho ham sd y = ——x a) Khao sat su bien thien va ve dd thi iP) eiia ham sd da cho. b) Ne'u tinh tien (f) sang phai 1,5 don vi thi ta dupe dd thi eiia ham sd nao ? c) Ne'u tinh tie'n (/*) sang trai 2 don vi thi ta dupe dd thi eua ham sd nao ? 2 2.26. Cho ham sd y = 2x cd dd thi la parabol (f). Phai tinh tie'n (/*) nhu thd nao de dupe dd thi eua ham sd a) y = 2x^ + 7 ; b) y = 2x^ - 5 ; c)y = 2(x + 3)^ ; d)y = 2(x-4) ^ e) y = 2(x - 2)^ + 5 ; /) y = 2x^ - 6x +1 ? 2.27. Khong ve dd thi, tim toa dp dinh, phuong trinh true ddi xiitig eiia mdi parabol sau day. Tim gia tri nhd nha't hay Idn nha't eiia mdi ham sd tuong ling. a)y = 2(x + 3)^-5 ; b)y = -(2x-l) ^ + 4 ; e)y = - V2x^ + 4x 2.28. Khao sat su bie'n thien va ve dd thi ciia cae ham sd sau : a) y = x^ + X + 1 ; b) y = -2x^ + x - 2 ; c) y = -3? + 2x -1 ; d) y = ix ^ - X + 2. 2.29. Cho ham sdy = -x + 4x - 3. a) Khao sat su bie'n thien va ve dd thi ciia ham sd da cho. b) Dua vao dd thi, hay neu cac khoang tren dd ham sd chi nhan gia tri duong. c) Dua vao dd thi, hay neu cac khoang tren dd ham sd chi nhan gia tri am. 34 a-BTDsio.Nc-e 2.30. Cung yeu cdu nhu bai 2.29 dd'i vdi cac ham sd sau : ^ 2 3 a)y = x -X+- ; b) y - -2x^ + 3x - -^ ; c) >- = 0,5x'' - 3x. 2.31, 2.32. y!,t^ Ve do thi ciia mdi ham sd sau rdi lap bang bien thien cua nd : a) y = ^x^ + 2x - 6 2 b) y = |-0,5x' + 3x-2,5 |. Ve do thi ciia mdi ham sd sau rdi lap bang bie'n thien eua nd : f-2x + 1 ne'u x> 0 a) fix) 0= . . [x^ + 4x + 1 ne'u X < 0 ; b)/(x) = -x^- 2 ne'u X < 1 2x - 2x - 3 ne'u x > 1. , Ve dd thi ciia ham sd y = -x + 5x + 6. Hay sir dung dd thi dd bien luan 2.33 theo tham sd m sd didm ehung eua parabol y - -x + 5x + 6 va dudng thing y = m. , Mdt parabo! cd dinh la didm /(-2 ; -2) va di qua gd'c toa dp. 2.34 a) Hay eho bie't phuong trinh true ddi xutig cua parabol, bie't ring nd song song vdi true tung. b) Tim didm ddi xiing vdi gde toa dp qua true ddi xiing trong eau a). c) Tim ham sd cd dd thi la parabol da eho. 2.35 a) Kf hieu (P) la parabol y = ax + bx + c ia ^ 0). Chiing minh ring ne'u mdt dudng thing song song vdi true hoanh, cat iP) tai hai didm phan biet A va B thi trung didm C ciia doan thing AB thude true dd'i xiing ciia parabol (P). b) Mpt dudng thing song song vdi true hoanh cat dd thi (P) ciia mdt ham sd bac hai tai hai didm M(-3 ; 3) va A'(l ; 3). Hay cho bie't phuong trinh true ddi xiing eua parabol (P). 35 •J 1 2.36. Ham sd bac hai fix) -ax + bx + c c6 gia tri nho nh^t bang — khi x = ^^ va nhan gia tri bang 1 khi x = 1. a) Xac dinh eac he sd a, b va c. Khao sat su bie'n thien va ve dd thi (?) eiia ham sd nhan dupe. b) Xet dudng thing y = mx, kf hieu bdi id). Khi id) cit (?) tai hai didm A va B phan biet, hay xac dinh toa dd trung didm cua doan thing AB. BAI TAP 6 N TAP CHl/ONG II 2.37. Chiing minh ring y = 0 la ham sd duy nha't x^c dinh tren R va cd d6 thi nhan true hoanh lam true ddi xiing. Hudng din. Til dinh nghia ham sd ta ed nhan x6t ring mdi dudng thing song song vdi true tung thi cat dd thi ciia ham sd tai khdng qua mdt didm. 2.38. Gia sir y =fix) la ham sd xac dinh tren tap dd'i xiing 5 (nghia la ne'u x G 5 thi -X e S). Chiing minh ring : a) Ham sd Fix) = - |/(x) +/(-x)] la ham sd chSn xdc dinh tren 5. b) Ham sd Gix) = i |/(x) -/(-x)] la ham sd le xac dinh tren 5. 2.39. Gpi A va S la hai didm thude dd thi ciia ham sd/(x) = (m - l)x + 2 va cd hoanh dd lin lupt la -1 va 3. a) Xac dinh toa dd ciia hai didm A va B. b) Vdi didu kien nao ciia m thi didm A nim d phfa tren true hoanh ? e) Vdi didu kien nao cua m thi didm B nim d phfa tren true hoanh ? d) Vdi didu kien nao eiia m thi hai didm A va S ciing nim d phfa tren true hoanh ? Tii dd hay tra ldi cau hoi ; Vdi didu kien nao eiia m thi fix) > 0 vdi mpi X thude doan [-1 ; 3] ? 2.40. Cho ham sd y = -3x^ cd dd thi la parabol (?). a) Ne'u tinh tie'n (?) sang phai 1 don vi rdi tinh tie'n parabol viira nhan dupe xudng dudi 3 don vi thi ta dupe dd thi eiia ham sd nao ? b) Ne'u tinh tie'n (?) sang trai 2 don vi rdi tinh tie'n parabol viira nhan dupe len tren 2 don vi thi ta dupe d6 thi eiia ham sd nao ? 36 2.41. Tim ham sd bac hai cd dd thi la parabol (?), bidt rang dudng thing y = -2,5 cd mdt didm ehung duy nha't vdi (?) va dudng thing y = 2 cit (?) tai hai didm cd hoanh dd la -1 va 5. Ve parabol (?) ciing cac dudng thing y = -2,5 va y = 2 tren ciing mdt mat phing toa dd. Gidl THifiU MC T S 6 CAU HO I TRAC NGHlfeM KHAC H QUAN Trong cdc bdi tO: 2.42 de'n 2.49, hay chon phuang an tra Icfi dung trong cdc phuang dn dd cho. 2.42. Tim didm thude dd thi eiia ham sd y = - x - 2 trong eac didm cd toa dp la (A) (15;-7); (B) (66 ; 20) ; (C)(V2-1;V3); (D) (3 ; 1). 2.43. Ham sd cd dd thi triing vdi dudng thing y = x + 1 la ham sd (A)y=(VxTl) ; (B)y=i—-^ ; (C) y = x(x + 1) - x^ + 1 ; (D) y - ^^^^^^ 2.44. Dudng thing song song vdi dudfng thing y= V2 x la (A) y = 1 -V2x ; (B) y - -|=x - 3 ; (C)y+V2x=2 ; (D)y--|. x = 5. 2.45. Mudn cd parabol y = 2(x + 3) , ta tinh tie'n parabol y = 2x (A) Sang trai 3 don vi; (B) Sang phai 3 don vi ; (C) Len tren 3 don vi ; (D) Xudng dudi 3 don vi. 2 2 2.46. Mudn ed parabol y = 2(x + 3) - 1 , ta tinh tie'n parabol y = 2x (A) Sang trai 3 don vi r6i sang phai 1 don vi ; (B) Sang phai 3 don vi rdi xudng dudi 1 don vi; (C) Len tren 1 don vj rdi sang phai 3 don vi ; (D) Xudng dudi 1 don vi rdi sang trai 3 don vi. 37 2.47. True ddi xiing eiia parabol y = -2x + 5x + 3 la dudng thing (A)x= - ; (B)x = - - (C)x= - ; 2.48. Ham sd y = 2x^ + 4x ~ 1 (D)x = - - (A) Dong bie'n tren khoang (-^ ; -2) va nghich bie'n uen khoang (-2 ; + oo); (B) Nghich bie'n tren khoang (-oo ; -2) va ddng bie'n tren khoang (-2 ; + oo); (C) Ddng bie'n tren khoang (-co ; -1) va nghich bie'n tren khoang (-1 ; + co); (D) Nghich bie'n tren khoang (-oo ; -1) va ddng bie'n tren khoang (-1 ; + co). 2.49. Ham sd y - -x - 3x + 5 cd 3 (A) Gia tri Idn nha't khi x = - ; (B) Gia tri Idn nha't khi x = - - (C) Gia tri nho nha't khix= - \ (D) Gia tri nho nha't khi x = -•- Trong mdi bdi tif bdi 2.50 den bdi 2.52, hay ghep mdi thanh phan cua cdt trdi vai mot thanh phan thich hap d cot phai deduac khdng dinh dung. 2.50. 2 ; 2) la a) Didm ( dinh ciia parabol . l)y = 2x' + 2x+ 1. b) Didm r 1 P la dinh cua parabol 2)y = x'-x+l . I 2'2 , 2.51. Xet parabol {P) : y = ax^ + bx + c a) Chic chin (?) cd dinh nim d phfa dudi true hoanh b) Chic chin (?) ed dinh nim d phfa tren true hoanh 38 3)y = -0,25x' + x+ 1. 1) ne'u a 0 va c < 0 3) ne'u <3 < 0 va c> 0 4) ne'u (3 > 0 va t > 0 2.52. Xet parabol (?) : j = ax" + /jx + c vdi a < 0, t^ = \? ~ 4ac. a) Chic chin (?) cit true hoanh tai hai didm ed hoanh dp duong b) Chic chin (?) cit true hoanh tai hai didm ed hoanh dd am l)ne'u A>0 , & 0, 6 > 0 va (•> 0 3) neu A>0 , ^<0vac> 0 4) ne'u A>0 , &>Ovac< 0 C. DAP SO - HUONG DAN - LOi GIAI 2.1. y=ylr ~ x^ , xac dinh tren doan [-/•; r]. Chu y. Ham sd y = -Vr^ - x"^ cd dd thi la nia dudng tron gdm eac didm thupc dudng tron dang xet va ed tung dp khong duong (cGng ed tap xac djnh la [~r ; r]). 2.2. a)R\{-l ; 1}. b)(-|;+oo)\{l). e)(-4;+0D)\{2} . 2.3. 2.4. 2.5. a) [-1 ; +00). b)/(0) = -1 ; /(2) = I ;/(-l) = 0 ; /(-3) khdng xac dinh. a) [1 ;+«)). b)/(4)= 16+ V3 =« 17,73; /(72)^2,64 ; fin) ^ 11,33. X y= yJ4-X^ - 2 0 -^-" ^ 0 ^ _ ^ 2 ___ ^ 2 ^""^"•^—-^ 0 39 2.6. —00 +C0 -1 1,6 2,4 +00 +00 -4,4 Ham sd cd gia tri nhd nha't bing -4,4 khi x = 2, nhung khdng cd gia tri Idn nha't. 2.7. ,)/(^2)-/U|),^^^,^^4 . Xj Xj Tren khoang (-oo ; -2), ta cd X2 + x^ + 4 < 0 nen him sd nghich bie'n. Tren khoang (-2 ; +GO), ta ed X2 + XJ + 4 > 0 nen ham sd ddng bid^n. X2 - Xi Tren khoang (-co ; 1), ta cd -X2 - Xj + 2 > 0 nen ham sd ddng bie'n. Tren khoang (1 ; +co), ta cd -X2 - Xj + 2 < 0 nen ham sd nghich bie'n. c) Vdi hai sd phan biet Xj va X2 thude tap xac dinh eua ham sd', ta cd : fiX2)-fix,) = Xf Xi X2 + 1 Xi + 1 (Xi + 1)(X2 + 1) ' /(X2)-/(X^) ^ 1 X2 - Xj (Xi + 1)(X2 + 1) ' Dodo : - Neu Xi < - 1 va X2 < -I thi (X] + 1)(X2 + 1) > 0 va suy ra ham sd ddng bie'n tren khoang (-co ; -1). - Ne'u xi > - 1 va X2 > - 1 thi (xj + 1)(X2 + 1) > 0 va suy ra ham sd cung ddng bien tren khoang (-1 ; +00). 40 1 (Xi + 1)(X2 + 1) 1 (Xi + 1)(X2 + 1) >o, >0, d) -^^^2) /Ui) ^ Z . Tiif dd suy ra ham sd da eho X2 - X| (-X2 + 2)(-Xi + 2) ddng bie'n tren mdi khoang (-00 ; 2) va (2 ; +co). 2.8. De tha'y ham sd' y = 0 la ham sd xac dinh tren R, viia la ham sd chan, viia la ham sd le. Gia sir ham sd y =/(x) la mot ham sd bat ki ed tfnh eha't nhu the'. Khi dd vdi moi x thude M, ta cd : fi-x) -fix) (vi/i a ham sd chan); /(-;c) = -/(x) (vi/lahamsdle). TCr dd suy ra vdi moi x thude M, xay rafix) = -fix), nghia \afix) = 0. Vay y = 0 la ham sd duy nha't xac dinh tren R, vixa la ham sd chin, viia la ham sd le. 2.9. a) De dang suy ra tir gia thie't va dinh nghia ham sd chan. b) Vdi X tuy y thude M, ta cd : /(~x) = -fix) va gi-x) = -g(x) (vi / va g la nhitng ham sd le); do dd 5(-x) -/(-x ) + gi-x) = -fix) - gix) = -\fix) + gix)] = -Six), ?(-x) =/(-x)^(-x) = [-fix)][-gix)] =fix)gix) = ?(x). vay y = iS(x) la ham sd le va y = ?(x) la ham sd chin. c) Vdi X tuy y thuoc E, ta cd :/(-x) =/(x) va gi-x) = -gix) (vi/la ham sd chin va g la ham sd le); do dd ?(-x) =fi-x)gi-x) =/(x)[-^(x)] = ~fix)gix) = -Pix). Vay y = ?(x) la ham sd le. 2.10. a) Ham sd chan (tdng ciia ba ham sd chin). b) Ham sd le (long ciia hai ham sd le). c) Ham sd le (tich cua ham sd le y = x va ham sd chan y = |x|). d) Tap xac dinh eua ham sd/(x) = Vl + x + Vl - x la doan [-1 ; 1]. Vdi moix thude doan [-1 ; 1], ta ed : 41 fi-x) = Vl -X + Vl + x =/(x). vay y =/(x) la ham sd chin. e) Tap xac dinh eda ham sd gix) = Vl + x - Vl - x la doan [-1 ; 1]. Vdi moi X thude doan [-1 ; 1], ta ed : ^(-x) = VT ^ - Vl + x = -gix). vay y = gix) la ham sd le. 2.11. a) Khi tinh tien len tren 5 don vi, ta dugfe : Ai-l ; 3) h^ A^i-\ ; 8); B(2 ; -5) h^ ?i(2 ; 0) ; da ; b) \-^ Cyia ;b + 5). b) Khi tinh tie'n xudng dudi 3 don vi, ta dugfe : A(-l ; 3) h^ ^2(^1 ; 0); ^(2 ; -5) h^ ?2(2 ; -8) ; Cia ; b) h^ C2{a •,b-3). c) Khi tinh tidn sang phai 1 don vi, ta duoc : A(-l ; 3) h^ A^iO ; 3); 5(2 ; -5) h^ ^3(3 ; -5) ; da ; b) h^ 03(0 + 1 ; &). d) Khi tinh tie'n sang trai 4 don vi, ta duoc : Ai-l; 3) h^ A4(-5 ; 3); B(2 ;-5) h^ S4(-2; -5) ; C(a ; b) \-^ C^ia-4\b). 2.12. a) (Ji) la dd thi eiia ham sdy = (4x - 3) + 4 hay y = 4x+ 1. b) (^2) '^ ^^ thi eiia ham sd y = 4(x + 1) - 3 hay y = 4x + 1. e) Dudng thing y = 4x + 1 cd thd ed duoc bing each tjnh tie'n dudng thing y - 4x ~ 3 theo hai each nhu trong a) va b). -2 2.13. a) Tinh tie'n (//) xudng dudi 3 don vi, ta duoe dd thi ham sd y = 3, -3x - 2 hayy=— - — - 2 b) Tinh tie'n (//) sang phai 2 don vi, ta duoc dd thi ham sd y = x - 2 c) Tinh tie'n (//) len tren 1 don vi rdi sang trai 4 don vi, ta duoc do thi ... . - 2 _ . x + 2 ham so y = + 1, hay y - 7 -^ x + 4 ^ x + 4 42 2.14. a), b) Hpc sinh tu giai. c) D6 thi hinh 2.4. Giao didm vdi true tung : (0 ; 2). Giao didm vdi true hoanh : 2A5.a)k = 0. b)^ = 1. C)k^2+yl2 2.16. cac cap dudng thing song song la a) va e) ; b) va d); c) va/). 2.17. a) D6 thi hinh 2.5.a). Hinh 2.4 a) c) r-2x + 2 khi X > 1 b) Ham sd cd the vie't dang y=< DO thi hinh 2.5.b). 2x - 2 khi X < 1 - x + 1 khi X > -— e) Ham sd ed the vie't dang y = • . Dd thi hinh 2.5.c). x + 4 khix< - — 2 (Hpc sinh tu lap bang bie'n thien). 2.18. a) Tren dudng thing y = 2x + 5, didm ed hoanh do bing - 2 la Ai-2 ; 1). Tren dudng thing y ^ -3x + 4, diem cd tung dp bing -2 la 5(2 ; -2). Vay dudng thing cin tim di qua hai didm A va B. Tit dd, a va b phai thoa man he 43 f~2a + fo = 1 [2a + b=^ -2. <. 3, 1 Suyraa-- - ,b = --- h) Giao didm M eiia hai dudng thing y = - —x + 1 va y = 3x + 5 cd toa dd 1 la nghiem ciia he phuong trinh y=--x+ l y = 3x + 5. He nay ed nghiem (x ; y) =-- ; — ]. Vay dudng thing cin tim song 1 /' 8 11 ^ song vdi dudng thing y = - x va di qua didm Ml --; — I. Tut dd suy ra 1 . , 15 a = — \ab -—- 2 7 2.19. a) B(xo ; -yo). b) Mudn chiing minh hai dudng thing (c/,) va ((^2) dd'i xiing vdi nhau qua true hoanh, ta ehiing minh ring ne'u A(xo ; yo) la mdt didm tuy y thude idi) thi didm ddi xiing vdi A qua true hoanh, tiic la diem BixQ ; -yo) thude (J2) va ngupe lai. That vay, gpi id^) la dudng thing y = x - 2, ((^2) la dudng thing y - 2 - x, ta cd ^(^0 ; yo) e (^1) '^yo = xo-2 <^ -yo = 2 - xo <^ ?(xo ; -yo) e id2). TCr dd suy ra dpem. e) Tuong tu nhu cau tren, ta dd dang chiing minh dupe ring dd thi eiia hai ham sd y =/(x) va y = -fix) ddi xiing vdi nhau qua true hoanh. Do dd, dudng thing ddi xung vdi dudng thing y = -2x + 3 qua true hoanh la dd thi eiia ham sd y ~ -(-2x + 3), tde la ham sd y = 2x - 3. 2.20.a)?(-Xo;yo). b) Chiing minh tuong tubal 2.19.b). c) y ^ -0,5x -2. Gai y. Trudc het chdng minh ring dd thi ciia hai ham sd y =fix) va y =/(-x) ddi xiing vdi nhau qua true tung. 44 -^11 \ J-, ^ \~x khi X < 0 2.21. a)/(x)='^ [-2x khi X > 0. b) Hpe sinh tu lap bang bie'n thien. 2.22. a) Gia sii didm A cin tim ed toa d6 (XQ ; yo). Khi dd, vi A thude dudng thing y = 2mx + 1 - m vdi mpi m nen ding thu'c yo = 2w XQ + 1 - m, hay (2 Xo - l)m + 1 - yo = 0 xay ra vdi mpi m. Didu dd ehi cd thd xay ra khi ta cd ddng thdi 2xo -1= 0 va 1 - yo = 0, nghia la XQ = — va yo = 1. Vay toa dd eiia ^ la (—; 1). Ngupe lai, de thiy gia tri ciia ham sdy = 2mx + 1 - m tai x = —ludn bing 1 vdi mpi m, chiing to dd thi eiia nd ludn di qua didm Ai—; 1) vdi mpi m. b) B(0 ; -3). Goi y. Cach giai tuong tu eau a). 2.23. a) Hai dudng thing y = 2x va y = - 3 - x cit nhau tai M(-l ; -2). Dudng thing thd" ba y = mx + 5 cung di qua didm M khi va chi khi - 2 = m(-l) + 5, tiic la m = 7. Thii lai ta tha^y m thoa man didu kien cua diu bai. b) Hai dudng thing y = -5(x + 1) va y = 3x + m cit nhau tai N ^ m + 5 5m - 15 Dudng thing y = mx + 3 cung di qua A^ khi va chi khi 5m - 15 f m + 5^ ^ •=m\ — + 3. 8 Giai phuong trinh tren ddi vdi in m, ta dupe m = -13vam = 3. -V6im = -13, ba dudng thing y = -5(x + 1), y = -13x + 3 va y = 3x - 13 ddng quy lai didm N^H ; -10). - Vdi m = 3, hai dudng thing y = mx + 3 va y = 3x + m trung nhau va triing vdi dudng thing y = 3x + 3. Do dd trudng hpp nay bi loai. Ket ludn. m = -13 . 2.24. a) Hpe sinh tu giai. b) y = -x ^ + 2. c) y = -x ^ - 3. 45 2.25. a) Hpc sinh tu giai. c)y = -^i x + 2f 2.26. a) Tinh tie'n (?) len tren 7 don vi. b)y - -^(x-1,5) ' b) Tinh tie'n (?) xudng dudi 5 don vi. c) Tinh tie'n sang trai 3 don vi. d) Tinh tie'n sang phai 4 don vi. e) Tinh tie'n sang phai 2 don vi rdi tinh tie'n tie'p len tren 5 don vi. /) Tjnh tie'n sang phai 1,5 don vi rdi tinh tie'n tie'p xudng dudi 3,5 don vi. 2.27. Ke't qua dupe neu trong bang sau Parabol y = 2(x + 3)^-5 y = -(2x-l)^ + 4 = -4(x-^)^ + 4 y = -^^x^ + 4x= -V2(x - V2)^ + 2%^ Dinh (-3;-5) 4; 4) (V2;2^ ) True ddi xiing x = -3 1 x=V2 Gia tri nhd nhat -5 Gia tri Idn nhit 4 2V2 2.28. a) Ta cd the vie't ham sd y = x + x + 1 dudi dang y=\^+\ 3 ^-4' Ttr dd suy ra dd thi ciia nd la mpt parabol hudng bd iQm len tren va ed dinh tai i 1 2 ' 4 ; ham sd da eho nghich bie'n tren khoang (-c» ; --) , ddng bie'n tren 4t Hinh 2.6 khoang (-—; +QO) va cd gia tri nho nha't bing — khi x = -— Hpc sinh tu lap bang bie'n thien. 46 Dd ve dd thi eiia ham sd nay, ta lap bang mdt vai gia tri ciia nd nhu sau X y - 2 3 - 1 1 1 2 3 4 0 1 1 3 Dd thi ciia ham sd ed dang nhu hinh 2.6. b) Dua ham sd da cho vd dang y = - 2 ^ " 4 15 TiJf dd suy ra ham sd ddng bie'n tren khoang (-oo ; —), nghich bie'n tren khoang ( —; +oo) va cd gia tri Idn nhit bing —— khi x =—. Hpc sinh tu lap bang bien thien va a 4 ' ve dd thi ciia nd. c) Hpc sinh tU giai. d) Hpc sinh tu giai. 2.29. a) Ham sd y = -x^ + 4x -3 cd thd vie't dupe dudi dang y = -(x-2)2+l . Tii do suy ra ham sd ddng bie'n tren khoang (-co ; 2), nghich bie'n tren khoang (2 ; +oo). Bang bie'n thien : '00 +00 y= -X + 4x - 3 - 0 0 —00 Ham sd cd gia tri Idn nha't bing 1 khi x = 2. Q6 thi eua nd la mdt parabol di qua eac didm (0 ; -3), (1 ; 0), (2 ; 1), (3 ; 0), (4 ; -3) (h.2.7). Tir dd thi ta tha'y : b) Ham sd ehi nhan gia tri duong ne'u x e (1 ; 3). c) Ham sd ehi nhan gia tri am ndu X e (-00 ; 1) u( 3 ; +oo). Hinh 2.7 47 2.30. a) Hpc sinh tu khao sat su bien thien cua ham sd. Ham sd cd dd thi nhu hinh 2.8a. Ham sd nhan gia tri duong vdi mpi x e R. b) Hpe sinh tu khao sat su bie'n thien cua ham sd. Ham sd cd dd thi nhu hinh 2.8b. 3 3 Ham sd nhan gia tri am vdi mpi x ^— (khi x = — . ham sd nhan gia tri bing 0). c) Hpe sinh tu khao sat su bie'n thien eiia ham sd. Ham sd ed dd thi nhu hinh 2.8e. a) b) Hinh 2.8 c) Ham sd nhan gia tri am ndu x e (0 ; 6) va nhan gia tri duong ndu X G (-00 ; 0) LJ (6 ; +oo). 2.31. a) y = -x^ +2x- 6 Q6 thi (h. 2.9a). Bang bie'n thien —00 -6 -2 +00 +00 ,+00 48 b)y = |-0,5x^ + 3x-2,5| Dd thi (h. 2.9b). y \ \ .2 0 -2 1 a) Hinh 2.9 Bang bie'n thien •2,5 / \ / ' ^ \ / \ / ' \ / A h X • / -> 1^ ' \ X / \ 1 / \ / ^ ' '' \ / •.. 1 ^ \ I ^ — ' ' \ b) —00 +00 +00 +00 2.32. a) y = -2x + 1 ne'u x> 0 x^ + 4x + 1 ndu X < 0. Dd thi (h. 2.10a). Hpc sinh tu lap bang bien thien. b)y = -x^- 2 ne'u X < 1 2x^ -2x- 3 ne'u x > I. Dd thi (h. 2.10b). Hpc sinh tu lap bang bie'n thien. 4-BTOSlO.NC-A 49 a) 2.33. Hoc sinh tu ve dd thi. b) Hinh 2.10 Do parabol hudng bd 16m xudng dudi va ed dinh tai didm p^" ^ i2-j nen : - Ndu m> 12— thi dudng thing va parabol khdng cd diem ehung. - Neu m= 12— thi dudng thing va parabol cd mot diem ehung. - Neu m < 12— thi dudng thing va parabol cd hai diem ehung phan biet. 2.34. a) Phuong trinh true ddi xiing la x = -2 . b) Diem ddi xdmg vdi 0(0 ; 0) qua true x = - 2 la diem M(-4 ; 0). 2 , c) Ta phai tim a (a ^ 0), b va. c sao cho ham s6 y = ax + bx + c c6 do thi la parabol dinh /(-2 ; -2) va di qua didm O. Tii gia thie't ta ed cac he thde sau : b „ A b^ -4ac ^ . ^ - - - = -2 ; --—= = - 2 va c = 0. 2a 4a 4a 1 1 2 Tir dd tinh dupe a=- ,b = 2,c = 0va ham sd can tim la y = - x + 2x. 50 4-BTDSlO.NC-B 9 1 2.35, a) Ta da biet true ddi xuTig ciia parabol y = ax + ^x + c la dudng thang __b_ 2^' Gia su (d) la dudng thing da cho (song song vdi true hoanh). Ta bie't ring id) la dd thi cua ham sd khong ddi y = m vdi m la mot sd nao dd. Gia thie't cho id) eit (?) tai hai didm phan biet A va B c6 nghia la phuong trinh ax + bx + c = m hay ax+bx + c-m = 0 (1) cd hai nghiem phan biet ; hon nUa, hai nghiem a'y chinh la cac hoanh dp x^ ciia didm A va xg eua diem B. Theo dinh li Vi-et, ta cd x^ + Xg = — Do dd trung diem C ciia doan thine AB ed hoanh do la Xr = — — = - — • ^ • ^ 2 2a Didu dd ehung td diem C thuoc dudng thing x = --—, tiJc la thude true 2a ddi xiing cua parabol (?). Chuy. Dudng thing id) song song vdi true hoanh nen vudng gde vdi true ddi xutig ciia (?). Do dd, khi id) cit (?) tai hai diem AvaB thi hai diem a'y ddi xiing vdi nhau qua true ddi xiing va trung diem C cua doan AB phai ihudc true ddi xiing. b) Ap dung ke't qua tren, trung diem K eua doan MN phai thude true ddi xiing ciia parabol (?). Diem K cd hoanh dd la ——— = -I . Vay true ddi xiing ciia parabol (?) cd phuong trinh la x = -1 . 2.36. a) • Vl ham sd cd gia tri nho rtha't bins — khi x = — nen --— = — va 4 2 2a 2 A b'^ - 4ac 3 ^ —- suy raa = -b va -a-\-4c = 3. 4a 4a 4 Vi ham sd ed gia tri bing 1 khi x - 1 nen/(I) = a + 6 + c = 1, suy ra c = 1 (do a - -b). Do dd « = 4c - 3 = 1 va ^ = -1 . 2 Vay ham sd can tim iay = x -x+1 . 51 • Do he sd a ^ 1 > 0 va gia tri nhd nha't ciia ham sd dat dupe tai x = — nen ham sd nghich bie'n tren khoang (-00 ; —) va dong bien tren khoang (— ; +co). Bang bien thien : —00 +00 +00 + 0 0 y = x -x+ 1 Ham sd cd dd thi nhu hinh 2.11. b) Dudng thing y = mx cit parabol (?) tai hai didm M^A ; y/i) va B(xg ; yg) ndu va ehi ne'u phuong trinh X - X + 1 = mx hay x^-( l +m)x+ 1 - 0 (1) ed hai nghiem phan biet, tiic la biet thiic A = (1 + m)^-4 = m'-\-2m-3 duong. Khi dd, hai nghiem ciia (I) chinh la x^ va x^. Theo dinh li Vi-et, ta ed Hinh 2.11 x^ + Xg = 1 + m. Tir (2) ta suy ra hoanh dp trung diem C eiia doan thing AB la _ x^ + Xg _ 1 + m Xr- = 2 2 (2) Do C la mot diem thude dudng thing id) nen tung dp y^ cua nd thoa man y^=.mxc = m(l + m) I + m m{\ + m) 52 Ke't ludn. Toa dp trung diem ciia doan thing AB la C vdi dieu kien m + 2m - 3 > 0. 2.37. Hien nhien ham sd y - 0 xac dinh vdi mpi x va cd dd thi ddi xiing qua true hoanh. Gia su ham sd y -/(x) xac dinh tren R, cd dd thi (G) nhan true hoanh lam true ddi xiing. Khi dd Vx G M : M(x ; y) e (G) <^ M'(x ; -y) e (G). Didu nay ed nghia la VxGE:y=/(x)^-^=/(x) . Suy ra y = 0 vdi mpi x. vay ham sd y = 0 la ham sd duy nha't cd dd thi ddi xiing qua true hoanh. Chu y. Ciing cd the ehiing minh ring (G) triing vdi true hoanh. That vay, ne'u trai lai thi phai ed mot diem M(xo ; yo) thude (G) va yo T^ 0. Khi dd, do tinh ddi xiing qua true hoanh, didm M'(xo ; -yo) cung thude (G). Ta ed dudng thing MM' song song vdi true tung, cit (G) tai hai didm phan biet M va M' Dd la dieu khdng the xay ra ddi vdi dd thi cua mdt ham so. 2.38. a) F(-x) = ^ |/(-x) +/(x)] - ?(x). b) Gi-x) = i {fi-x) -fix)] = - i Ifix) -fi-x)] = -Gix). 2.39. a) A(-l ; -m + 3), 5(3 ; 3m - 1). b) A nim o phia tren true hoanh khi va chi khi -m + 3 > 0, tu'c la m < 3. e) B nim cf phia tren true hoanh khi va chi khi 3m - I > 0, tu'c la m >- . d) Ca hai didm A va 6 ddu nim d phia tren true hoanh khi va ehi khi cae didu kien ndi trong cau b) va c) dong thdi dupe thoa man, nghia la - < m < 3. Khi dd, loan bd doan thing AB nim d phia tren true hoanh. Ndi each khae : (m - l)x + 2 > 0, Vx G [-1 ; 3] c^ - < m < 3. 2.40. a) y = -3(x - 1)^ - 3 ; b) y - -3(x + 2)^ + 2. 2.41. Dudng thing y = -2,5 song song vdi true hoanh. Do dudng thing nay CO mot diem ehung duy nhat vdi parabol (?) nen diem ehung iy chinh la dinh ciia parabol (?). Tir do suy ra dinh I cua parabol (?) ed Uing dp y = -2,5. 53 Dudng thing y = 2 cung song song vdi true hoanh. Do dd trung didm C ciia doan thing AB nim tren true ddi xiing ciia parabol. -1 + 5 Hoanh dd cua diem C la x = = 2. Vay true ddi xiing cua parabol la dudng thing x = 2, suy ra hoanh dp dinh I cua (?) la x = 2. Toa dp cua / la (2 ; -2,5). Til dd suy ra ne'u (?) la dd thi cua ham sd 2 fix) = ax + bx + c thi /(-I) = a-b + c = 2, -—- = 2 va 2(3 A Ja b^ - 4ac 4^ = -2,5. Tir dd suy ra a =—. b = -2 , c = -— va ham 1 2 1 • sd can tim la y = — x - 2x -— Dd thi cua ham sd nhu tre^n hin 2 2 h 2.12. • Hinh 2.12 2.42. Phuong an (B). 2.43. Phuong an (C). Chii y ring cac ham sd con lai ddu cd tap xac dinh khae R. 2.44. Phuong an (D). 2.45. Phuong an (A). Chi can chii y ring cin phai tinh tie'n sang trai. 2.46. Phuong an (D). Chu y. Tranh nhim lin ve phuong va hudng tinh tie'n. 2.47. Phuong an (C). Chii y. Tranh eac nhim lin vd da'u va nhim lin giua —- 2a . b va — a - 2.48. Phuong an (D). 2.49. Phuong an (B). 2,50.(a)<^(3);(b)^(l). 2.51. (a) ^ (2) ; (b) ^ (3). 2.52. (a)<^(4);(b)<^(l). 54 phuang III PHUONG TRINH BAC NHAT VA BAC HAI •* « A. NHONG KIEN THLJC CAN NHO Cac kie'n thUe dupe neu sau day cd bd sung mot vai ke't qua di nhan thay va dupe sic dung nhieu trong thuc hanh giai toan. 1. Cac phep bien doi tirong duong cua phuang trinh 1) Thuc hien eac phep bien ddi ddng nhat trong tCmg ve' nhung khdng lam thay ddi tap xac dinh eua phuong trinh. 2) Them vao hai ve' cua phuong trinh ciing mdt bidu thiic xac dinh vdi mpi gia tri ciia an thude tap xac dinh cua phuong trinh (trudng hpp hay dung la quy tac chuyen ve). 3) Nhan hai vd eiia phuong trinh vdi ciing mot bieu thiie xae dinh va khae 0 vdi mpi gia tri cua in thupc tap xac dinh cua phuong trinh (chii y ring chia cho mot sd tiCc Id nhan vdi nghich dao eua sd dd). 4) Binh phuong hai ve' eua mdt phuong trinh cd hai ve' ludn ciing dau khi in lay mpi gia tri thupc tap xac dinh cua phuong trinh. 2. Phep bien doi cho phuong trinh he qua Binh phuong hai vd cua mdt phuong trinh. 3. Giai va bien iuan phuong trinh dang ax + fa = 0 •£1^0: phuong trinh ed mpt nghiem duy nha't x = • a^Ovab^O : phuong trinh v6 nghiem. • (3 = /> = 0 : phuong trinh nghiem diing vdi mpi x. 55 4. Giai va bien luan phuong trinh bac hai mot an ax^ + bx + c = 0 (1) 2 2 vdi biet thde A = b ~ 4ac hay biet thiic thu gpn A' = 6' - ac (vdi b = 2b'). A < 0 (A' < 0) : (1) v6 nghiem. • A - 0 (A' - 0) : (1) cd mdt nghiem kep x = - —- 2a ' b' X = I. V a A > 0 (A' > 0): (1) cd hai nghiem phan biet x = -b±yfK( -/?' ± VA^ X = 2(2 V a 5. Djnh li Vi-et (thuan va dao) : 2 Hai sd Xi va xj la hai nghiem eiia phuong trinh bac hai ax + 6x + c = 0 khi vd chi khi ehiing thoa man hai he thiic Vi-et sau : _ b _c Xi + XT — ; XIX T — — • a a Dinh li Vi-et cd the dupe ling dung dd : - Nhim nghiem ciia phuong trinh bac hai. - Tim hai sd biet tdng va tich cua ehiing : Ne'u hai sd cd tdng bing S va tich bing P thi hai sd dd la hai nghiem ciia phuong trinh x^ -Sx-¥P = 0. (Tat nhien, didu kien tdn tai eua hai sd ndi tren la S^ - 4P > 0.) - Phan tich mpt tam thiic bac hai thanh nhan tii : Cho tam thiic bac hai fix) = ax + 6x + c. Neu phuong trinh bac hai fix) = 0 cd hai nghiem (cd thd triing nhau) xi va X2 thi tam thiic bac hai fix) cd thd phan tich dupe thanh nhan tii nhu sau: 2 ax + bx +c = aix-x{)ix-X2). - Tmh gia tri cae bidu thde ddi xiing ciia hai nghiem ciia phuong trinh bac hai : o = X] + X-1 = — — ', r = X1X9 — — j ' '^ a ^ ^ a x\+ x\^^ -2P; x\+ x\=S^ - 3PS. 56 - Xet da'u cae nghiem cua phuong trinh bac hai: Phuong trinh cd hai nghiem trai dau o P <0. Phuong trinh ed hai nghiem duong <^ A > 0, /* > 0 va 5 > 0. Phuong trinh cd hai nghiem am o A > 0, /^ > 0 va 5 < 0. 6. Giai va bien luan he hai phuong trinh bac nhat hai an iax + by = c i o ? ? (a + />^ ?^ 0 va d^ + ^'^ ^ 0). \a' X + by = C (2) a b D - = ab' - db ; a' b' D . - ^y = c c a a' b b' c c' = cb' - c'b ; = ac - ac. • D ?^ 0 : (2) cd mpt nghiem duy nha't (x ; y), trong dd x=—^, y =-j ^ • D = 0, D^ ?t 0 hoac D^ ?t 0 : (2) v6 nghiem. • D = D^ = D^ = 0 : (2) ed vd sd nghiem (x ; y) tinh theo edng thiic -by + c -ax + c ineubitO). Chiiy X = — a (ne'u a^O) hoac • X e y = Khi giai va bien luan he phuong trinh ed chiia tham sd dang \ax + by = c lO'x + b'y = c'. cd thd xay ra trudng hpp a = b = 0 (hoac d = b' = 0). Khi dd, ta sir dung cae kdt luan dd thiy sau day : , - Phuong trinh Ox + Oy = c vd nghiem ndu c ^ 0, nghiem diing vdi mpi x va vdi mpi y ndu c = 0. 57 - Trong mdt he phuong trinh, neu mpt phuong trinh eua he v6 nghiem thi he v6 nghiem. - Trong mpt he hai phuong trinh, neu mpt phuong trinh eiia he nghiem diing vdi mpi gia tri cua cae in thi tap nghiem cua he phuong trinh dd triing vdi tap nghiem cua phuong trinh con lai. , 7. Giai he phuOng trinh bac hai hai an 1) He phuong trinh trong dd ed mpt phuong trinh bac nhat : Diing phuong phap the. 2) He phuong trinh ma mdi phuong trinh trong he khdng thay ddi khi thay the' dong thdi x bdi y va y bdi x : Diing phuong phap dat an phu S = X + y ; P = xy. B. DE BAI §1. DAI CUC^ G Vt PHUONG TRINH 3.1. Tim didu kien eiia mdi phuong trinh sau rdi suy ra tap nghiem cua nd : a) X - Vx- 3 = V3-X + 3 ; b) V-x^ +4x-4 - = x^ - 4 ; e) Vx - Vl - X = V-x -2 ; d)x + 2Vx + l=:l - V-x - 1. 3.2. Tim nghiem nguyen eua mdi phuong trinh sau bing each xet didu kien xac dinh cua nd : a) sl4-x -2= y[x -x\ b) 3 Vx + 2 - V2 - x +2V2 3.3. Giai cac phuong trinh sau : a) X + Vx - Vx - 1 ; b) x^ + V2 - x = V2 - x + 9. 3.4. Trong cac phep bien ddi sau; phdp bien ddi nao cho ta phuong trinh tuong duong, phep bie'n ddi nao khdng cho ta phuong trinh tuong duong ? 7 a) Lupe bo sd hang r o ca hai ve cua phuong trinh .A. i 2,7 ^ 7 X + 1 + = x - 1 2x + X-1 ' 58 b) Luoc bo sd hang d ca hai ve' eua phuong trinh x - 2 x^ + 1 + - ^ =2x+ ^ x-2 X - 2 ' c) Thay the' (V2x-l) bdi 2x - 1 trong phuong trinh (V2x-l) =3x + 2 ; , 2 d) Chia ca hai ve' qua phuong trinh x + 3 = x +3 eho x ; x^ + 1 1 e) Nhan ca hai ve' eiia phuong trinh =2 + — vdi x. X X 3.5. Trong cac phep bidn ddi neu trong bai tap 3.4, phep bie'n ddi nao eho ta phuong trinh he qua ? 3.6. Kidm tra lai ring cac bien ddi sau day lam mat nghiem eua phuong trinh : , , 2 a) Chia ca hai ve cua phuong trinh sau cho x - 3x + 2 (x + l)(x^ - 3x + 2) = x^ - 3x + 2 ; b) Chia ca hai ve' ciia phuong trinh sau eho Vx - 1 (x + 4)Vx~l = (Vx-l) . 3.7. Giai eac phuong trinh sau bang each binh phuong hai ve': a)|2x + 3| = l ; b) |2-x | = 2x - 1 ; c) V3x-2 - 1 -2x ; d)V5~2x = Vx - 1. 3.8. Tim didu kien xac dinh ciia phuong trinh hai in sau rdi suy ra tap nghiem ciiand V-x2-(y + l)^+xy = (X + l)(y + 1). §2. PHUONG TRINH BA C NHA T VA BA C HAI M6 T X N Phuang trinh bac nhat 3.9. Tim cac gia tri ciia p dd phuong trinh sau v6 nghiem (4/-2)x= l + 2p-x. 59 3.10. Tim cae gia tri ciia q di mdi phuong trinh sau cd v6 sd nghidm : a) 2qx ~ \ =x + q ; b) q^x - q^ 25x - 5. 3.11. Tim cae gia tri eiia m de mdi phuong trinh sau chi cd mdt nghiem : a) (x - m)(x - 1) = 0 ; b) m(m - l)x = m - 1. '3.12. Giai va bien luan cae phuong trinh sau theo tham sd m : a) 2mx = 2x + m + 4 ; b) m(x + m) = x + 1. Phuang trinh bac hai 3.13. Vdi mdi phuong tiinh sau, bie't mdt nghiem, hay tim tham sd m va nghiem con lai: a) (2m^ - 7m + 5)x^ + 3mx - (5m^ - 2m + 8) = 0 cd mdt nghiem la 2. b) (5m^ + 2m - 4)x^ - 2mx - (2m^ - m + 4) = 0 cd mdt nghiem la -I. 3.14. Giai va bien luan cac phuong trinh sau theo tham sd m : a) mx + 2x + 1 = 0 ; b) 2x^ - 6x + 3m - 5 = 0 ; c) (m + l)x^ - (2m + l)x + (m - 2) = 0 ; d) (m^ - 5m - 36)x^ - 2(m + 4)x + 1 = 0. 3.15. Tim cac gia tri cua tham sd m di mdi phuong trinh sau cd hai nghiem bing nhau : a) x^ - 2(m - l)x + 2m + 1 - 0 ; b) 3mx^ + (4 - 6m)x + 3 (m - 1) = 0 ; e) (m - 3)x^ - 2(3m - 4)x + 7m - 6 = 0 ; d) (m - 2)x - mx + 2m - 3 = 0. 3.16. Bien luan sd giao diem cua hai parabol sau theo tham sd m : 2 X 2 y = x + mx + 8 va y = x + x + m. Dinh li Vi-et 3.17. Vdi mdi phuong trinh sau, biet mpt nghiem, tim m va nghiem con lai : 2 a) X - mx + 21 = 0 cd mpt nghiem la 7 ; b) X - 9x + m = 0 cd mpt nghiem la -3 ; c) (m - 3)x^ - 25x + 32 = 0 ed mpt nghiem la 4. 60 3.18. Gia SIX Xi, x^ la cac nghiem cua phuong trinh 2x - 1 lx+ 13 - 0. Hay tinh : a ) X, + X2 ; \ 4 4 c) Xi - xJ ; b ) x\ + X2 ; ^)t(^^'^^)^?^-^')- 3.19. Gia sii xj, X2 la cae nghiem ciia phuong trinh x + 2mx + 4 = 0. Hay tim ta't ca cac gia tri eua m dd cd ding thUc : ( ^ \' + ^•^ 2 = 3. \^\ J 3.20. Tim tat ca cac gia tri ciia a de hieu hai nghiem cua phuong trinh sau bang 1 2x^-(a + l)x + a + 3 = 0. 3.21. Gia su Xiva X2 la cac nghiem cua phuong trinh bac hai ax + 6x + c = 0. Hay bieu didn cac bidu thde sau day qua eac he sd a,bvac a) x\ + x\ ; Xi Xo b) Xj^ +x ^ ; d) Xj - 4x|X2 + X2. 3.22. Tim tit ca cac gia tri duong cua k de cac nghiem cua phuong trinh 2x- -ik + l)x + l = k^ trai da'u nhau va ed gia tri tuyet ddi la nghich dao cua nhau. 3.23. Hay tim tat ca cac gia tri ciia k dd phuong trinh bac hai {k + 2)x^ -2kx-k = 0 ed hai nghiem ma sip xep tren true sd, ehung ddi xiing nhau qua diem x = l. 3.24. Gia sii a, b \a hai sd thoa man a> b>0. Khdng giai phuong trinh abx^ - (fl + &)x + 1 = 0, hay tinh ti sd giua tdng hai nghiem va hieu giua nghiem Idn va nghiem nho eiia phuong trinh dd. 61 3.25. Giai eac phuong trinh sau day : a) X* - 5x^ + 4 = 0 ; b) x^ - 13x^ + 36 = 0 ; e) x"^ - 8x^ - 9 = 0 ; d) x^ - 24x^ - 25 = 0. 3.26. Cac he sd a, b va c cua phuong trinh triing phuong ax + bx + c = 0 phai thoa man didu kien gi dd phuong trinh do a) V6 nghiem ? b) Cd mpt nghiem ? c) Cd hai nghiem ? d) Cd ba nghiem ? e) Cd bdn nghifm ? §3. MOT S6 PHI/ONG TRINH QUY vfi PHUONG TRINH BAC NHXT HOAC BAC HAI 3.27. Giai va bien luan cac phuong trinh sau theo tham sd a : a) = a\ b) = a-3 ; c) -— - 2. x - 1 x- 2 ax+ 3 3.28. Giai cae phuong trinh : a) Vx^ + X + 1 = 3 - X ; b) Vx^ + 6x + 9 = |2x - l| ; c) x(x + 1) + x(x + 2) = x(x + 4) 1 - x 1 + xy U- x / 14 - X 3.29. Giai cac phuong trinh : a) —-j - + ^ - 1 ; x + 1 X-2 • ,,2x- l 3x- l x- 7 x - 1 3x 5 '^^ X ~2x- 2 "~2" ' 62 3.30. Giai cae phuong trinh : 4x 5x 3 a) -z + x^ + X + 3 x^ - 5x + 3 2 ' b) x - 1 x- 2 x- 4 x- 5 x + 2 x + 3 x + 5 x + 6 3.31. Giai va bien luan cac phuong trinh sau theo tham sd m : a) I 3mx - 1 I = 5 ; b) | 3x + m | ^ I 2x - 2m | ; 3.32. Giai va bien luan cac phuong trinh sau : a) (x - 2)(x - mx + 3) =^ 0 ; ,, (x + l)(mx + 2) b) T = 0 ; X — 3m 2 mx - 1 m _ m{x + 1) x - 1 "^ x + 1 " j.^ _i 3.33. Cho tam giac ABC nhpn cd canh BC = a, dudng cao AH = h. Mdt hinh chU nhat MNPQ npi tie'p trong tam giac iM & AB • N ^ AC ; P, Q G BC) cd ehu vi bing 2p {p la dp dai eho trudc). Hay tfnh dp dai canh PQ cua hinh ehu nhat MNPQ, bien luan theo p, a, h. §4. PHUONG TRINH VA HE PHUON G TRINH BAG NHX T NHI£ U A N 3.34. Xet tap hpp cac didm ed toa d6 (x ; y) la nghiem ciia phuong trinh ax + by = c. Tim dieu kien ciia a, b, c di : a) Tap hpp didm dd la mpt dudng thing di qua gd'c toa dp ; b) Tap hpp didm dd la mpt dudng thing song song vdi true tung ; c) Tap hpp diem dd la mpt dudng thing song song vdi true hoanh ; d) Tap hpp didm dd la true tung ; e) Tap hpp didm dd la true hoanh ; g) Tap hpp dd la mpt dudng thing cit hai true Ox va Oy tai hai diem phan biet. 63 3.35. 3.36. 3.37. Giai eac phuong trinh sau va minh hoa tap nghiem tren mat phing toa dp : a)2x + 3y = 5 ; b) 0.x + 3y - 6 ; c) 2x + O.y = 4 ; d) 2x + 3y = 0. Giai va bien luan eac phuong trinh sau theo tham sd m : a) mx + (m - I)y = 5 ; b) mx + my = m + 1. Bing dinh thde, hay giai cac he phuong trinh sau : 5x - 3y = 1 ; b) 'V2x + 4y = 1 a) '3x + 2y = -7 2x + 4Sy = 5. 3.38. 3.39. Tinh nghiem gin dung cua cac he phuong trinh sau (chinh xac de'n hang phin tram) : jVSx + V3y = V2 |3x + (Vs - 2)y = 1 [V2x - V3y = V5 ; [(V^ - l)x + ^^y = V5. Giai va bien luan cae he phuong trinh theo tham sd a : a) ax + 2y - 1 b) ia 2)x + ia --4)y = 2 X + (« - l)y - a. ia + l)x + i3a + 2)y = -1 ; c) ia - l)x + (2a -3)y = a ia + \)x + 3y = 6; d) 3(x + y) _ ^ x - y 2x--y-^ .., 3.40. 3.41. Giai eac he phuong trinh : j3|x| + 5y- 9 = 0 "^\2x-\y\ = l; Giai he eac phuong trinh b) ixl - (2 = 1 y-2 x = 5 ia la tham sd). a ) • ^ + ^. 3 X y 9 10 b) + X - 2y X + 2y 3 4 X - 2y X + 2y = 3 = -1 . 64 .^ y ~ ' 3.42. Mpt ca no chay tren sdng trong 8 gid, xu6i ddng 135 km va ngupe ddng 63 km. Mpt lin khae, ea nd cung chay tren sdng trong 8 gid, xu6i ddng 108 km va ngupe ddng 84 km. Tinh van tde ddng nude chay va van toe ciia ca no (bie't ring van toe that cua ca nd va van tde ddng nude chay trong ea hai lin la bing nhau va khong ddi). 3.43. Cho hai dudng thing (df,) : (m - l)x + y = 5 va (0^2) : 2x + my = 10. a) Tim m dd hai dudng thing (Jj) va (.^2) cit nhau. b) Tim m di hai dudng thing (Jj) va (fl'2) song song. e) Tim m de hai dudng thing (rfj) va (^2) trung nhau. 3.44. Cho ba dudng thing idy) :2x + 3y = -4 ; id2):3x + y=l; id^) : 2mx + 5y = m. a) Vdi gia tri nao cua m thi (d^), id2), id^) ddng quy tai mpt didm ? b) Vdi gia tri nao eiia m thi ((^2) va id^) vudng gde vdi nhau ? 3.45. Vie't phuong trinh cua dudng thing trong mdi trudng hpp sau :, a) Cat true Ox tai didm ed hoanh dp la 5 va cit true Oy tai didm ed tung dd la - 2. b) Di qua hai didm AH ; -1) va S(3 ; 5). 3.46. Giai cac he phuong trinh bac nhat ba in : a) X + y = 25 y + z = 30 2 + X - 29 ; b) 2x + y + 3z - 2 -X + 4y - 6z = 5 5x- v + 3z - -5 . 3.47. Su dung may tinh bo tiii dd tim nghiem gan diing cua he phuong trinh sau (chinh xac de'n hang phin tram) : '4x + V2y + z = 1 [(V2 + l)x + y + V3z = -1 a) V3x + V3y + 2z = V2 b) X + V5y + 3z = V3 ; V. + V2y + V5z - V2 V3x + (V3 + l)y - z - Vs. 5BTDS10.NC-A 65 3.48. Cd ba Idp hpe sinh lOA, \0B, IOC gdm 128 em cung tham gia lao ddng trong cay. Mdi em Idp lOA trdng dupe 3 cay bach dan va 4 cay bang. Mdi em Idp 106 trdng dupe 2 cay bach dan va 5 cay bang. M6i em Idp IOC trdng dupe 6 cay bach dan. Ca ba Idp trdng dupe la 476 cay bach dan va 375 cay bang. Hoi mOi Idp cd bao nhieu hpc sinh ? 3.49. Bdi todn co. Hay giai bai toan dan gian sau : Em di chaphien Anh gvti mot tien Cam, thanh yen, quyt Khdng nhieu thi it Mua du mot tram Cam ba dong mdt Quyt mot dong nam Thanh yen tuai tdt Nam dong mot trdi Hoi mdi thii mua bao nhieu trai, bie't ring mpt tien la 60 ddng ? §5 M6T S 6 vi DU V£ H£ PHUONG TRINH BAC HAI HAI XN Giai cae he phuong trinh sau 3.50. a) c) '2x - y - 7 = 0 y^ - x^ + 2x + 2y + 4 - 0 ; 2x^ + X + y + I = 0 x^ + 12x + 2y + 10-0 . (x + y + 2)(2x + 2y - 1) - 0 3x^ - 32y2 + 5 - 0 ; b) b) '4x + 9y = 6 3x^ + 6xy - X + 3y = 0 ; '(x + 2y + l)(x + 2y + 2) = 0 xy + y^ + 3y + 1 = 0. r3(x+.y) = xy [x^ + y^ = 160 ; 66 5-BTDSlO.NC - B 3.53.a)|'"'-^'= ^ ,Jx 2 + y^=25-2x y xy + x^ =2 ; [y(x + y) = 10 ; c) 2(x + y)2+2(x-y) 2 = 5ix^ - y^) x^ +y^ = 20. BAI TAP 6 N TAP CHUONG III 3.54. Phuomg trinh dang ox + ^ = 0 (in x) v6 nghiem trong trudfng hop nao, ed vd sd nghiem trong trudng hpp nao ? Ap dung. Tim cac gia tri ciia tham sd m sao cho phuong trinh mim - 2)x = m a) Cd nghiem duy nha't; b) Vd nghiem ; e) Cd vo sd nghiem ; d) Cd nghiem. 3.55. Cho he phuong trinh (ax + by = c , (I) < (an la X va y) thoa man dieu kien db'c' ^ 0. [fl'x + b'y = C '• Chiing minh ring : a) Ne'u —1^ ?^ — thi he (I) cd nghiem duy nha't. a b ' b) Neu —r = T7 5^ -r thi he (I) v6 nghiem. a b c ' c) Neu — = -— = —- thi he (I) cd vd sd nghiem. a b c ' Ap dung. Tim eac gia tri cua tham sd a sao eho he phuong trinh Ua + l)x + 3y ^ a [x + ia- l)y = 2 CO vd sd nghiem. 67 3.56. Giai va bien luan cae phuong trinh theo tham sd m : a) (2m - l)x - 2 = m - 4x ; 2 c) m(x + 1) = m - 6 - 2x. b)mlx-l) + 1 = - (4m + 3)x; 3.57. Giai va bien luan cac phuong trinh theo tham sd m : , (2m - l)x + 2 a ^^ '- -m+ 1 x - 2 3.58. Giai cae he phuong trinh 2x - y + 3z = 4 a) \3x - 2y + 2z = 3 5x - 4y - 2 ; c) |x - y| = V2 2x- y = -1 . b) (m-l)(m + 2)x^^^^ ^ 2x + l X + y = 16 y + z = 28 z + X = 22 ; 3.59. Cho he phuong trinh (m - l)x + (m + l)y = m (3 - m)x + 3y = 2. a) Tim eac gia tri eiia m dd he phuong trinh ed nghiem. Khi dd, hay tmh theo m cae nghiem eua he. b) Tim nghiem gin dung cua he, chinh xac de'n hang phin nghin khi m - VS - 2. 3.60. Giai va bifn luan cae phuong trinh theo tham sd m : 3.61, a)t2x + mj = |2x + 2m- l |; e) (mx - 2)(2x + 4) = 0. Giai cae phuong trinh b)|mx+ l [ = |2x-m-3| ; a) 1 + x - 2 x + 3 (2 - x)(x + 3) x^ - X -1 2 2 10 50 b) L_L =2x. X - 3 3.62. Sir dung dd thi dd bien luan sd nghiem eiia cae phuong trinh sau theo tham s6 k : a) 3x^-2x = k- b)x^-3|xl-jt + 1 =0 . 68 3.63. Cho ham sd y - x^ + x - 2 cd dd thj la parabol iP), ham s6 y = 3x + k cd dd thi la dudng thing id). a) Hay bien lua.n sd nghiem eiia phuong trinh x + x - 2 = 3x + /:, tCr dd suy ra sd didm ehung cua parabol iP) va dudng thing id). b) Vdi gia tri nao ciia k thi dudng thing id) cit parabol (P) tai hai didm nim d hai phia khae nhau eiia true tung ? e) Vdi gia tri nao cua k thi dudng thing id) cit parabol iP) tai hai diem phan biet Of vd ciing mpt phia eua true tung. Khi dd hai didm ay nim d phia nao cua true tung ? 3.64. Cho hai phuong trinh x^ - 5x + ^ = 0 (1) va x^-lx + 2k = 0 (2). a) Vdi gia tri nao ciia k thi phuong trinh (1) ed hai nghiem va nghiem nay gip ddi nghiem kia ? b) Vdi gia tri nao eiia k thi phuong trinh (2) cd hai nghiem X] va X2 thoa man x^ + X2 = 25 ? e) Vdi gia tri nao eua k thi ea hai phuong trinh ciing cd nghiem va mpt trong cac nghiem eua phuong trinh (2) ga'p ddi mdt trong cae nghiem eiia phuong trinh (1) ? 3.65. Giai cac he phuong trinh sau : a) b) c) d) 2x^ - xy + 3y^ = 7x + 12y - 1 ;c - y + 1 = 0 ; [(2x + 3y - 2)(x - 5y - 3) - 0 [x-3y = i; x^ + y^ + 2x(y - 3) + 2y(x - 3) + 9 = 0 2(x + y)-xy + 6 = 0; 'x^-2y^ =lx ^ ,^., ^, y^ - 2x^ =ly. \x^ + y^ - 2(a + 1) 3.66. Cho he phuong trinh [ix-^yY=4. a) Giai he phuong trinh vdi a = 2. b) Tim cac gia tri cua a di he cd nghiem duy nha't. 69 Gidl THifiU MOT S 6 GAU HOI TRAC NGHlfiM KHACH QUAN Trong cdc bdi tu: 3.67 den 3.71, hay chon phuang.an tra ldi diing trong cdc phuang dn dd cho. 1 /'3 O 3.67. Didu kien xac dinh eua phuong trinh x + • . = la V2x + 4 X 3 iA) X > -2va X ^ 0 ; (B) x > -2 , x ^t 0 va x < - ; 3 (C) X > -2 va X < — ; (D) Khdng phai cae phuong an tren. 3.68. Cap (x; y) = (1 ; 2) la nghiem eua phuong trinh (A) 3x + 2y = 7 ; (B) x - 2y - 5 ; (C) 0.x + 3y = 4 ; (D) 3x + O.y = 2. f 3x + 4y = -5 3.69. Nghiem ciia he phuong trinh la I-2x + y = -4 (A)(l;-2); (B) (\_ -7 3' 4 J' (C)f-i;-5|; (D)(-2;l). 3.70. Cho phuong trinh bac hai ox^ +/7x + c = Oed hai nghiem x^,X2 cung khae 0. Phuong trinh bac hai nhan —va — lam nghiem la : X, X2 (A) cx^ -rhx + a = 0; (B) 6x^ + ax + c = 0; (C) cx^ + £7x + /? = 0; (D) ax^ + ex + Z> = 0. --,„,„..,, , . , (m + l)x - 1 , 3.71. Tap nghiem cua phuong tnnh = 1 trong trudng hpp m ^^^ 0 la (A)5={^} ; (B)S=0 ; (C) 5 = R ; (D) KhOng phai cac phuong an tren. 70 Trong cdc bdi 3.72 va 3.73 , hay ghep mdi dong 6 cot trdi vdi mdt dong a cdt phai deduac mot khdng dinh diing. 3.72. Cho phuong trinh x^ + 2mx + m^ - 2m - 1 = 0. a) Ne'u m> — b) Ne'u m < —- c) Ne'u m = — 1) thi phuong trinh da eho v6 nghiem. 2) thi phuong tnnh da cho cd v6 sd nghiem. 3) thi phuong trinh da eho ed mpt nghiem kep. 4) thi phuong trinh da cho cd hai nghiem phan biet. 3.73. Cho he phuong trinh mx + 9y = 6 X + my = -2. 1) thi he phuong trinh da cho v6 nghiem. a) Ne'u m = 3 b) Ne'u m = - 3 c) Ne'u m^±3 2) thi he phuong trinh da cho cd mot nghiem. 3) thi he phuong trinh da cho cd v6 sd nghiem. 4) thi he phuong tiinh da eho nghiem dung vdi mpi gia tri ciia hai in. C. DAP SO - HUONG DAN - LOI GIAI 3.1. a)5 - {3} ; b)^ - {2}. c) Khdng cd sd thue nao thoa man ddng thdi hai didu kien x > 0 va -X - 2 > 0. vay phuong trinh v6 nghiem. d) Phuong trinh vd nghiem. 3.2. a) Didu kien xac dinh ciia phuong trinh la 0 < x < 4. Thii true tiep cac gia tri eiia x thupc tap {0 ; 1 ; 2 ; 3 ; 4} vao phuong trinh, ta tha'y phuong trinh ed cae nghiem x-0; x = 4vax-2 . b) Dieu kien xac dinh cua phuong trinh la - 2 < x < 2. Thu true tiep cac gia tri eua x thupc tap {-2 ; -1 ; 0 ; 1 ; 2 } vao phuong trinh, ta thay phuong trinh cd mot nghiem x = 0. 71 3.3; a) Vd nghiem. b)x = -3 . 3.4. a) Khong ; b) Co ; e) Khdng ; d) Khdng ; e) Khdng. 3.5. a), b), c) vae). 3.6. a) Ta tha'y khi x = 1 hoac x = 2 thi x^ - 3x + 2 = 0. Do dd x = 1 va x = 2 la hai trong cae nghiem ciia phuong trinh da eho. Nhung sau khi bie'n d6i, ta dupe phuong trinh x + 1 = 1 ; phuong trinh nay khdng nhan x = 1 va X = 2 lam nghiem. b) Sau khi bie'n ddi, ta dupe phuong trinh ix + 4) = (Vx - 1 )^ Phuong trinh nay khdng nhan x = 1 lam nghiem, trong khi x = 1 la nghiem eiia phuong trinh ban dau. Chu y. Hai bai loan tren eho tha'y : Neu chia ca hai ve' cua mpt phuong trinh eho mpt bieu thiic thi cd thd lam mat nghiem eiia phuong trinh. 3.7. a)x = - l vax = -2 ; b)x=l ; c) V6 nghiem ; d)x = 2. 3.8. Didu kien ciia phuong trinh la - x^ - (y + 1)^ > 0 hay x^ + (y + 1)^ < 0. Didu nay tuong duong vdi x = (y + 1) - 0, tiic la (x ; y) = (0 ; -1) (vi ndu trai lai, em hay chiing minh rang ta ludn ed x + (y + 1) > 0). Thu true tie'p X ~ 0 va y = - 1 vao phuong trinh, ta tha'y cap sd (0 ; -1) diing la nghiem ciia phuong trinh da eho). 3.9. p = ^ 3.10. a) 2^x - 1 = X + ^ o (2x-m = 0 hoac x-l= 0 c^ x~m hoac x = 1. Vay phuong trinh ehi ed mpt nghiem khi m = 1. b) m ?^ 0 va m ?^ 1. 72 3.12. a)Tacd: 2mx = 2x + m + 4<^2(m- l)x = m + 4 (1) - Vdi m - I ^ 0 hay m ?^ 1, chia hai ve' cua (1) eho 2(m - 1) ta dupe m + 4 X = 2(m - 1) - Vdi m -1 = 0 hay m = 1, phuong trinh (1) trd thanh 0.x = 5, vd nghiem. b) Phuong trinh cd nghiem duy nha't x = ~ im + 1) khi m ^ I, nghiem diing vdi mpi x khi m = \. 3.13. a) Do X = 2 la nghiem ndn thay vao phuong trinh ta dupe : 4(2m^ - 7m + 5) + 6m - (5m^ - 2m + 8) = 0 hay 3m^ - 20m +12-0 . Giai phuong trinh tren (in la m) ta ed ke't qua m e J6 ; —[. Vdi m = 6 , phuong trinh da cho trd thanh 35x^ + 18x-176 =0 OQ va cd hai nghiem la x, =2 vax2 = '^ • 2 Vdi m = —, phuong trinh da cho tro thanh 11 2 n 80 ^ 40 va CO hai nghiem la Xj =2 va Xj - "T T b) Vdi m = 1, nghiem thii hai la - ; vdi m - -- , nghiem thii hai la — 3.14. a) Ne'u m = 0 thi phuong trinh cd nghiem x=-— Ne'u m^O thi phuong trinh cd A' = 1 - m. + Ne'u 1 - m < 0 tde la m >1 thi phuong trinh da cho vd nghiem. + Ndu 1 - m = 0 tde la m = 1 thi phuong trinh da cho cd mpt nghiem kep X--1 . 73 + Ne'u 1 - m > 0 tiic la m < 1 thi phuong trinh da cho ed hai nghiem phan biet •1-Vl- m . -l + VT^ Xi = va X2 = m m m vay, vdi m e ( - 00 ; 0) u (0 ; 1) thi phuong trinh cd hai nghiem -1 - Vl- m . -1 + VT^ m Xi = va X2 = m m Vdi m = 0, phuong trinh cd nghiem x = -— - Vdi m = 1, phuong trinh cd nghiem kep x = - 1. Vdi m e (1 ; +QO), phuong trinh v6 nghiem b) Phuong trinh cd A' = 9 - 2(3m - 5) = - 6m + 19. Vdi m e "2" ' + °° ' phuong trinh v6 nghipm. 19 3 Vdi m= —r. phuong trinh cd nghiem kep x = — 6 2 ( 19'l Vdi m e ~ ^ ' "2" ' phuong trinh cd hai nghiem 3-Vl9-6 m . 3 + Vl9-6 m X = va X = 2 2 e) Vdi m = -1 , phuong trinh cd nghiem x = 3. Vdi m ^ -1 , phuong trinh ed A = (2m + 1)^ - 4(m + l)(w - 2) = 8m + 9. Do dd, vdi m G ~ ^ ' ~ o" ' pbuong trinh vd nghidm. 9 Vdi m = -•-, phuong trinh ed mpt nghiem kep x = 5. 8 ( 9 ^ Vdi m e -— ; 1 w (1 ; +oo), phuong trinh ed hai nghiem _ 2m + 1 - VSm + 9 . _ 2m + 1 + V8m + 9 """ 2(m + l) ^^''~ 2(m + l) 74 2 d) m - 5m - 36 = 0 o m = - 4 hoac m = 9 Vdi m = - 4, phuong trinh trd thanh Ox = 1 nen vd nghiem. Vdi m = 9, phuong trinh trd thanh -26x + 1 =: 0 nen cd nghiem x = — • Vdi m g {-4;9}, tacd A' = (m + 4) - (m - 5m - 36) = 13m + 52 Tir dd suy ra : Vdi m G ( -00 ; -4], phuong trinh v6 nghiem . Vdi m e ( - 4 ; 9) w (9 ; +oo), phuong trinh cd hai nghiem _ m + 4 - yji3im + 4) . _ m + 4 +Vl3(m + 4) X — v 9 a X — • " ^v „ m - 5m - 36 m - 5m - 36 Vdi m = 9, phuong trinh cd nghiem x = —— 26 3.15. a)me{0;4 } ; b) m - ^ ; 11 .. 14 ± 2V7 c)m e <|-2;- h d) m = ^ 3.16. Hoanh dd giao didm hai parabol la nghiem eua phuong trinh 2 2 X +mx + 8= x +x + m. Phuong trinh tren tuong duong vdi phuong trinh (1 - m)x = 8 - m. Tir do suy ra : Neu m = 1 thi hai dd thi khdng ed didm ehung. Neu m ^i thi hai dd thi cd mpt diem ehung. 3.17. a) Gpi nghiem thii hai la X2. Theo dinh U Vi-et, ta ed : [7 + X2 = m [7x2 =21 - Giai he tren ta dupe X2 = 3, m = 10. b)x2 - 12 ;m= -36. 32 29 75 11 13 3.18. Theo dinh li Vi-et ta cd Xj + X2 = — ; X|.X2 = — (dd tha!y hai nghiem ddu duong). Do dd : 11 13 11 473 a) Xi +X2=(Xi+X2) -3x,X2(xi+X2)= ly ] " ^-y- y = g • b) xt + X2' - [ix, + X2)' - 2x1X2 f - 2xfxl = ~ - C) Xj - X2 = (Xi - X2){Xi + X2)t(Xi + X2) - 2 X]X2]. Tacd : 2 2 I I Vl7 (X, - X2) -(X i +X2) - 4X|.X2=> k - X2 = -r - Gia sirxj X2, ta ed x^ - x^ = ~T2~"^ d ) - 269 26 Gmy. ^(I-X2 ] + ^(I-X? ) = ^ + ^-2XIX2=^L1^_2XIX2 . X2 V ^ I X-^\ ' / X2 Xi ' X,X2 3.19. m = ± V2 + Vs Gai y. Didu kien dd phuong trinh ed nghiem la : A' = m^ - 4 > 0 <:> Iwl > 2. The o din h li Vi-et, ta c d fxi + X2 = -2m X1X2 = 4 r ^ \ nen v^2y / V- ^ 4 4 -i2 (Xj + X2) -2x^X2 1 ^^2 _ L X, + X 2 2 - X, X 1-^2 2 2 X1X2 (4m^-8)^ 16 - 2 . 76 + ^•^Ij - 2 . Tacd ^ x V ^-^2 ; + ^ =3o(4m^~8) - = 80 «(m^-2) ^ = 5 <»m^-2+V5 => m = ± V2 + V5 Cac gia tri nay ddu thoa man didu kidn \m\ > 2. 3.20. a e {-3; 9} Gai y. Didu kien dd phuong trinh ed nghiem la A = (<3 + 1)^ - 8(fl + 3) > 0 <^ fl^ - 6^ - 23 > 0. (*) Gpi hai nghiem eua phuong trinh da cho la xj, X2 (gia sir X2 > X|) a + \ Theo dinh li Vi-et ta ed X, + x-^ = 2 2 a + 3 Vl - 2 ' 2 2 Do X2 - X[ = 1 nen (x2 - X|) = (xj + X2) - 4x,X2 ^ 1, suy ra ^^^~- -2ia + 3)=loa'^- 6a- 27 = 0 <^ a = 9 hoac a =-3 Rd rang ca hai gia tri nay deu thoa man (*) vi a - 6a - 23 ~ 4 > 0. 3.21. a) Xi +X2 =(xi+X2) - 2xi.X2=—-2 — = a ^ a b)X i +X2 =(Xj+X2 ) - 3 XiX2(Xi + X2) ^ a' . 1 1 X, + x-, c) — + — = -^ ^ Xj X2 XiX H2 d)xi^-4xiX2 + X2^-(xi+X2)^- 6rir2-^ ^ 6c _b^ -6ac 3.22. k = 3. Gai y. Gpi Xi, X2 la hai nghiem ciia phuong trinh . Ap dung dinh li Vi-et va theo yeu c^u bai toan ta cd X2 = va 1 _ -t + 2 Xi + Xn — X| — — Xi Xo — Xi r-p 7-k' y-^i I 11 I'k^ 7 TiJf —- — = -\tac6k =9,dok>Qntnk = 3. w " , 1 w ^ u t^ ur 5-V4T 5 + V4I Voi k = 3 nghiem cua phuong tnnh la xj = — , X2 = 7 — 3.23. Gpi X|, X2 la hai nghiem ciia phuong trinh : ik + 2)x^ -2kx - k = 0 thoa man yeu c^u bai loan. Khi dd —^-—- = 1 nen Xj + X2 = 2 Ngoai ra 2k 2k Xy + X-, =-— - nen -— - =2, dodo k = k +2. ^ ^ k + 2 k + 2 Suy ra khong tdn tai k thoa man bai toan. 3.24. Gpi xj, X2 la hai nghiem ciia phuong trinh sao cho x^ > X2 Khi do, do a > b > 0 nen Xi - X2 = V(Xi + X2)^ - 4x^X2 = / _ . i. ^ 4 \(a - b\ a ~ b a + b ab VI ab I ab Xt + Xn = a + b ab a + b Suy ra ti sd giua t6ng va hieu hai nghiem bang a — b 3.25. a)x=-±l, x = ±2;b)x-±2 , x = ±3;c) x = ±3;d) x = ±5 . 2 3.26. Dat y = x , ta ed phuong trinh bac hai ay^ + by + c = 0. (I) a) Phuong trinh triing phuong da eho vd nghiem ne'u va chi ne'u • Phuong trinh (1) vd nghiem, tiic la A = 6 - 4ac < 0, hoac • Phuong trinh (1) chi cd nghiem am, tiic la A > 0, ac > 0 va a& > 0 ->Ova--< 0 a a J b) Phuong trinh triing phuong da cho cd mpt nghiem ne'u va chi ne'u phuong trinh (1) cd mdt nghiem y = 0, nghiem kia khdng duong, tiic la c -Ovaa&>0 . 78