ÔN TẬP TOÁN 10 ĐỢT 3 (KHTN HN)

NGAN HANc cAu rrol D4r s6 to-oqT g Tfrrdng THPT Chuy€n Khoa hgc Tq nhi€n I.NQi dung 1. Phuong trinh (15 cdu). i 2. Phudng trinh vO tf (15 cau). 3. HQ phuong trlnh (15 cau). 4. Lrrong gi5,c (25 cau). II.Phin ngdn hi,ng cAu h6i ' Phrrdng trinh(cdc bei tfi 1-15) 1. Giei phrrong tr)nh 2l"l-lr+t|:2 2. Giei vd bi€n lu$n theo m m(mr-1) +2:(m*2)r 3. Giei vd bi€n lu6n theo m mn _ m _ B :fn r*I 4. Giai vd biQn ludn theo m (m - I)r2 - (2m + 1)r * m * 4 : 0 5.4ra-7r2 -5r-1:0 6. 3(z + 5)(r + 6)(r *7) :s, 7. ra- 1or3+2612- 1or*1:o 8. ra + 3r3 - r4r2 - 6r I 4:0 9. (z + 5)(r + 6)(r + 8)(z + 9) :40 10. .(z + 1)(r + 2)(r + 3)(r + 4) :72012 1,I. (r + 2)4 + (r + 8)a : 242 12. (r - Z)u + (x - 4)6 :64 13. (r2 * 3r - 4)3 + (2r2 - 5r *3)3 : (3r2 - 2r - I)3 14. (r2 - 6r - 9)' : r(r' - 4r - 9) 15. ra :24r *32 Phddng trinh v6 ti(cdc bai td 16-30) 16. 2J4r+1*Ji+2:J;r+6 l 17. \/i + \ttn +9: \6+ L + \/n +Z ls. 2\ET3 + JifiiT! :2{2r + \/{FTtor + s ts. J67 -Trl4 - ,fip r 1r: I 20. Jr - 2 + JTTZ + r/7 4 : 6 - 2r 21. \E+ 1+ JT4 + 1/@ +\@ -fi : 5 22. ft-l+JT-:tr2-6r*11 n. \Lgar+ 6r + T + rlbirTm + u : { - 2r - 12 - 2a. {r + 2 - 4t/n - 2 + 1/r + 7 - 6t/r - 2 : 3 25. +Er+L+W-r-{r+A. .: 26. (\E+r-r)(\fr4+1)-z 27. 2(r2 + 2) : 5\F + t 2s. {m-r*{i:s 29' 12*t/r+5:5 30. z3*1:2VrF1 HQ phrrong trinh(cdc bli tfi 31-45) 31. Giei vd biQn ludn theo a 32. Tim m dd tre c6 nghi€m (r,y) th6a mdn ry < 0 lr-mY:1 I lmr+u:3 JJ. l"r-(a+L)y:1 Itr- 2)r+2a:a-2 lr'*ry+y2:19 I" *, * ry:11 !{*+r)(E+1) :6 l"'*a2+r:2(r*u) I**i**:s 'lr,l r 1 r l _ [;+i+7+P:8 I" *u2 :7 Itt*a5:13+a3 34 35. 36. 37. 38. 39. I*'*1:3x-u , Ir'f1:3a-r /- )t/t}-r*1/Y-2:4 I/TO u + tlFZ: 4 13a2 44. Tim m dd tre sau c6 nghigm ("a+r*a:m*r \r'o*ra2:m 45. Tim a dd he c6 nghiQm duY nh6t Io@n+i) :E+t-ltl l"*a2:r Ldgng giSc(c6c bai tfi 46-70) 25r, 46. Cho sina: t,n . n < f .Tinh tan(a - i) 47. Cho cosr * cos!: 1,sina * siny: ]'Tinh cos(z - y)' 48. Tinh cos 18". 4e' rinh o:#r-*h, 50' chrlng minh rhng cos(a - b)cos(a + b) : cos2 a - sin2 b 40. 41. 42. 43. (r , y -26 )ui;- 5 I" - a2 :24 I"'*2ry*r:u2 \r(r + a) :2 (r'+g2+22:L {rt+ as+23:I ["n*ya+za:1 IrO*r+L:7a \*'a'*ry+r: (r@+u*1) :3 tt"* v)2+r:1 51. Chrlng minh rHng cos2 e * sin2 E * sin(r * g) sin(r - a) I 52. Gie st cos r cosa cos z f 0.Chr1ng minh rbng sin(z - g), sin(y - z), sin(z - r) - n *ar"*g - "*g"oa" - "oar"or" -' 53. Cho sinr f O.Chrlng minh rB,ng sin 5r *" : 2cos 4r *2cos2r * 1 54. Chrlng minh rb,ng ;,."- . t*a 'U*z zlr sinz *sin3r*sinz - sin(r* a * z): 4sin Tsinf sin, 55. Chrlng minh rXng cot r - tanr - 2tan2r - 4tan4r : 8cot 8r bb. I rnn sin 350 sin 65' sin 80" A_ sin 200 * sin 50' * sin 70' 57. Tinh 58. Cho tano:2.Tinh A : 4sin2 10o * 4 sin2 50o cos 20" + cos 80o sin 2a * sin 4a A 1 * cos 2a * cos4a 59. Chrlng minh rli,ng sin2(a - 0 :sin2 a - sin2 P + zsinasinBcos(a - B) 60' ch(rng minh rang 3 + 4cos 2a I cos4a ,4 ffi:col'-0 61. Chrlng minh rb,ng " ,inartcos'r:]*f;"or+, 62. Chirng minh ri.ng ., sin3 acos3a * sin3acos3 a : 9 sin4a 4 63. Cho cos r * cos E * cos z :O'Chrlng minh rbng cos 3z * cos 39 * cos 3z : 12 cos r cos1 cos z 64. Chrlng minh rXng t'a, 'tr 2n 24n 1T sin,, +sin,, +."+sin,, :cott 4 r 65. Tinh 21 41 6n 8n lOn iZtr L4tr ,9 :cosfr +cos r, +co,s15 *cos r, *cos 1b +cos r, *cos- 66. Tinh P : cc 21 4tr 8zr rsrcosg"otg 67. Tinh ^7t ^2r .8tr A*cos';*cos";*cos"- .,,, 5'- 15 15 rii 68. Cho sin(2a + B) :7sinp.C$dng. minh rbng Stan(o + 0):4tan61 69. Tinh A: tango - tan 27o * tan63o + tan810 ?0. Chrlng minh rXng 1r Zn Bn 1 cos;-cos;*cos;:; l(lz NGAN nANc cAu Hor niivn Hoc ro odr g (Di,ng cho hqc si,nh Kh6i 10', Tradng THPT chuy€n KHTN) I. NOi dung 1. Vector, riitjm vd rlrrdrrg ttr6'ng (10 bbi). 2. Drrdng trdn (10 bb,i). 3. Ellipse (10 bbi). II. Ngdn hhng cdu h6i 1. Vector, di6m vb dttdng thlng Cdu 1l Trong h€ tqa dO Descartes cho vecto , d 1*o, A"),1 @r,ga). Dit d n-l : ragb - rbAa. a) C-hrrng minh ring r) a A (:U ri)?n7':-Tnd + . + iii) kA A l.b : (kD@ n b.) (k,l lh c6c s6 thuc bat ky) iv) ? n 17 + ?1:-t n.T +d n.? (vector.? uat r.y;. b) Chrlng minh rbns 1" n 71 : l?ll?llsin(?,711. c) Cho tam gi5.c ABC.Chrrng minh r},ng aB nfr : nd n BA : CA n CA. d) Cho tam giSc ABC.Chrlng minh rli.ng Sepc:irUB nTd. cAu 2i cho A(2,5), B(1, 1), C(3,3) a) Tim toa dO D sao cho tfi gi6c ABC D lb hinh binh hd,nh. b) Tinh diQn tfch hinh binh hiuth ABCD. cau g./fifi M' d6ixrlng M qua d trong c6,c trrrdng hgp sau a) d : r - 2a *2 : 0 vd, M(I,4). b) d : 2r * y -2 : 0 vh, M(6,5). c) d : 4r - 14A - 29 :0 vd, M(1, 2). Cau a./Cho tam gitrc ABC c6 trung didm mOt canh Ib M(1,2). Bi6t hai trung tuy€n xu6t ph6t tt hai dinh c6 phuong tr)nh lAn ltrOt Ih d1 : rly-3 : 0 vb d2 : 2r-y*4 : 0. Vi6t phrrong trinh c5,c canh cta tam gi6,c ABC. Cdu 5. Cho hinh chrl nh6,t ABCD c6 1(6;2) lb giao cii6m ctra hai drrdrrg ch6o AC vit" BD. Diiim M(1;1) thuQc drrdng thd.ng AB. Trung ditim ,O cira cqnh CD nXm tr6n drtdng thd,ng r + A - 5 :0. Vi6t phrrong trinh cpnh AB. Cdu 6. Cho tam gi6c ABC c6 C(-3,1). Phan giac AD c6 phrrong trinh r*3y *I2 : 0, drrdng aao AH c6 phuong trinh r * 7A +'32: 0. Lap phrrong trinh c6c cpnh tam gi6c. Cdu 7. Cho tam giftc ABC c5 A(2,0),8(4,I),C(1,2). a) Vi6i phrrong trinh ph6,n gi6c trong g5c A. b) Tirn tsa dO didm D thuQc BC )d, chdn drrdrrg ph6,n giSc ngodi g5c A. Cdu 8. Cho didm A(a,b), a> 0,b > 0. Vi6t phrrong trinh dudng thhng d di qua.4 khOng di qua g6c O cft tia Or,Oy tai M,.ly' sao cho MO + Ol/ nh6 nh6t. CAu 9. Cho hai hinh vuOng ABCD vd AB'C'D'ctng hu6ng. Chrlng minh rX,ng cdc dudng thd,ng BBt,CCt vb" DDt ddng quy. Cdu 10. Cho A(a,0) vd, B(0, b), a,b > 0. M di chuy6n tr€n d,oan OA, l/ di chuy6n tr€n do4n OB sao cho AM : ON. Chrrng minh rXng trung tr',tc MN luOn di qua ditim c6 dinh vb, tim tqa dO di6m d6. 2. Dddng trbn CAu 11. Ld,p phrtdng tr)nh dttdng trdn (C) trong c6c tnldng hop sau a) Dudng trbn (C) di qua A(1, 3), B(5,6), C(7,0). b) Dudng trdn (C) t6m I(-4,2), ti6p xirc v6i dudng thd,ng d: 3r*4g - 16:0. c) Drrdng trdn (C) ti6p xric d: r t2A -5:0 tai,4(3,1) vd di qua 8(6,4). d) Drrdng trdn (C) di qua A(I,2),8(3,1) c5 tdm nb,m tr€n d,: 7r *3y -l1 : 0. e) Drrirng trbn (C) di qua A(4,2),8(5, 1) ti6p xfic dudng thd,ng d, : r - 29 * 5 : 0. f) Drrirngtrdn (C) c6 b6nkinh R: \/t ti6pxfc vdid: r-A - 6:0 tai A(4,-2). g) Dudngtrbn (C) ddngt6,mvdi (f) , 12+y2 -4r-6y- 17:0vbti6pxircvdi 3r-49*7:0. Cdu 12. Cho d: 3z* 4y-3:0 vA,drrdngtrdn (C) t 12 +A2 -r-7y:0 a) Tim tqa dO giao didm cria d ve, (C). b) Lap phuong trinh ti6p tuy6n cria (C) tai chc giao didm d5. Cdu 13. LQp phrrong trinh ti6p tuy6n d cria (C) bi6t ' a) (C) : 12 +A2 - 2r - Bg *1 : 0 vd d song song vtli dttdng th8,ng 5r *L2y-6 : 0. b) (C), 12 + A2 - 6r *2E :0 vil d vu6ng g6c vrti drrdng th&ng 3r - y* 6 : 0. Cdu 14. Cho dudng trdn (C) : x2 +A2 - 4r -2A:0 vd A(3, -2). ViCt phrrong trinh ti6p tuy6n cria (C) ke td / vh tim toa dd c5.c ti€p diiim. Cdu 15. Cho c6.c dudng trbn (C): 12*y2:r (C^), 12 + a2 - 2(*+ 1)r )- 4my- 5 : 0 a) Tim tQ,p hqp tdm cria (C-) b) Chrrng minh c5 hai drrdng trbn cria hs (C*) ti6p xirc (C). Vi6t phuong trinh ti6p tuy6n chung cria hai drrdng trdn nA,y. C6u 16. a) Cho ,4(3,3) vd E|(6,4) tim c6c didm M trln trqc holinh sao cho TfrB : 45". =" t; Cho drrdng thEng d : Ar * By : Az I82. Ditim p di chuy6n tr€n d. Tr6n tia Of i6y e sao "Uo Op.Oe: 1. Chfng minh rXng Q luon nBm trcn drrdng trdn c6 dinh. Hay vi6t phrr<rng tr)nh drrdrrg trbn iI6. cAulT.Chodudngtrbn(C):12+a2:R2vddidmM("'96)ongob'idrrdngtrdn' K6 ii6p tuy6n MT1,"MT2 tdi (C)' vli Tr,?2 ld ti6p di6m' a) Vi6t- phttdng trinh rludng th&ng "1?2' U) Cia su M di cluydn treri clrrdng thd,ng d c6 dintr. Chrlng rnirrh rX,ng 7r?z ludn di qua rn6t ditim c5 dinh. Cdu 18. Cho A(a,0) vb, .B(-a,0). Chrlng minh rb,ng di6m M th6a man ffi : n udi k > 0, k + I th\ lu6n nB.m tr€n mOt dudng trdn c6 dinh. vi6t phrrong trinh drrdng trdn d6. Cdu 19. Tr6n mat ph&ng tqa dO c,ho A(m,O) vh, B(0, n) sao cho m,n > 0 vb'm*n: a ttrung dtii. Drrang tror, ft, OB) c6t dudng trdn (B' OA) tai P,Q' Chring minh rbng rrO riu" di qua ai6- "6 dinh khi A, B thav d6t" Cdu 20. 1l6n md,t phfr,ng tqa dO cho '4(b, h), F(0, h), B(c,h),C(a,0)' D(-a' 0)' a) Chrlng minh rbng trl gi5,c ABCD ld hinh thang vit" FC : FD' b) AC JAt nn @i E. C6,c drrdng trbn ngopiti6p tam gi6' (FAD) vd (FBC) c6t nhau tai G kh5c F' Chrlng minh rllng E, F,G th&ng hhng' 3. Ellipse Cdu 21. Cho drrdng trdn (O) chrla drrdng trbn (O'). Cho drrdng trdn (1) ti6p xirc trong ,.Oi (O) vd, ti6p xric "ngoir,i (O'). Chfng *intt t6,m I luOn nb,m tren mOt Ellipse c6 dinh' cdu22. cho hinh thang ABCD,d6,y ldn AB c6 dinh, d6y nh6 CD.: b kh6ng'd6i' C-l ai.rr";j" ;;" a,,,o An + BC : k > 0 khdng adl. Cnrtng minh rB'ng giao didm 1 cria hai drrdng c.heo iC,rJfO f"O" n},m tr€n mq1 Eilipse c6 ainn khi C, D thav doi' Cdu 23. Cho A, B, C thl,ng ha,ng tren drrdng thfrng d theo thf tU d6' Drrdng trdn (1) ;6" ;; trruy Arii ludn ti6p ii. a tai ,4. Ti6p tuy6n kh5'c d cira (1) k6 tt B, C cdt nhau Lai K. a) chrlng minh rhng K lu6n nhm tr6n mot Ellipse (E) c6 clintr khi (1) thay dtii' Uj Cito a(-S,0), B(-3,0), C(3,0) vi6t phuong trinh ctra (E)' cdu 24. X6c dinh do ddi hai truc, ti6u crl, tdm sai, tqa d0 ii€u di6m, toa do c6c dinh, phrrong,tr)nh dudng chud,n ctra c6c Ellipse sau ,r' a' a) . +7:r. LlI t ,, .iz u' 1 b) 25+T:no Cdu 25. Tim t6,m sai cria Ellipse bi6i a) M6i ti€u didm nhin truc nh6 du6i g6c vudng. b) Kho6,ng c6c girla hai dinh tr6n hai truc bb,ng ti6u crJ. *2 Cdu 26. Cho Ellipse (E) : ? * r': 1. Tim c6c diiim tren (E) sao cho a) C6 b6n kinh qua ti6u didm nAy bB,ng 3 Idn bdn kinh qua ti6u di6m kia. b) Nhin hai tiOu didm drrdi g6c 120'. c) Nhin hai ti€u di6m dudi g6c vu.6ng. Cd:u 27. Vi6t phuong trinh chfnh t6c cria Ellipse (E) bi6t a) Ellipse (E) qua M(-2\/8,2), dO dir,i trsc nh6 lb 6. b) Ellipse (E) qua M(4,-rt),N(2\/r,3). c) Ellipse (E) qua B(2,-Z),c6 t6,m ,ui ":25. d) Ellipse (E) qua CGrt,2) vb dudng chud,n r: *4. CAu 28. Cho Ellipse (,8) : 4 *t: 1td,m O.A,B thuQc (E) thay d6i sao cho ,w \u) a2 t bz - OA L OB. Chrrng minh rXng .4,B lu6n ti6p xric dudng trdn co dinh. Cdu 29. Cho Ellipse (,8) : t * t: 1 t6,m O, mOt ti6u di6m lb F c6 hob,nh dO 'v \u ) . a2 , bZ - r vv uvo.A^r sv duong. M di chuydn tr€n (E) lu6n c6 tung dO dudng. D6.t p:ifrF a) Tinh MF theo a,b vb" 9. b) MF c|,t (E) tai di6m thrl hai Mt. Chtrngminh ri,ng h . afu Unu,rg adi khl M di chuyiSn. CAu 30. Cho Ellipse (E) : 4**:1t6.m O, ti6u didm Jr1, Fz.M di chuyi5n tron (B) a) Chrlng minh rH,ng M n.M F2 + O M2 IuOn kh6ng ddi. b) Chrlng minh rB,ng (MFt - 1v7Fr)' - 4MO2ludn kh6ng d6i. . r l _ [;+i+7+P:8 I" *u2 :7 Itt*a5: 13+ a3 34 35 . 36 . 37 . 38 . 39 . I*'*1:3x-u , Ir'f1:3a-r /- )t/t}-r*1/Y-2:4 I/TO u + tlFZ: 4 13a2 44. Tim m dd tre sau c6 nghigm ("a+r*a:m*r 'o*ra2:m 45 :64 13. (r2 * 3r - 4 )3 + (2r2 - 5r *3) 3 : (3r2 - 2r - I )3 14. (r2 - 6r - 9)' : r(r' - 4r - 9) 15. ra :24r *32 Phddng trinh v6 ti(cdc bai td 16 -30 ) 16 rang 3 + 4cos 2a I cos4a ,4 ffi:col'-0 61. Chrlng minh rb,ng " ,inartcos'r:]*f;"or+, 62. Chirng minh ri.ng ., sin3 acos3a * sin3acos3 a : 9 sin4a 4 63. Cho