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Hinh hoc-affine

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1. H THNG BI TP TRC NGHIM CHNG I HNH HC AFFINE Cu hi 1. C th xem trng s phc C l mt khng gian affine thc c s chiu l: a. 1 chiub. 2 chiuc. 3 chiud. n chiuCu hi 2. Trong khng gian affine An , cho v l hai siu phng song song phn bit, l mphng khng cha trong . Nu ct th: a. cng ct c. trng b. song song d. cng ct hoc song song Cu hi 3. Cho v l hai siu phng phn bit v ct nhau. Nu siu phng song song vi , = v = th: a. v song song vi nhau c. v cho nhaub. v ct nhau d. Khng th kt lun v thiu d kin.Cu hi 4. Cho l mt mphng, A l mt im khng thuc . C bao nhiu lphng vi l m qua A v song song vi . a. C duy nht mt lphng c. C v s lphngb. C hai lphng d. Khng c lphng noCu hi 5. Cho v l hai ci phng c s chiu ln lt l p v q , v song song suy ra: a. Chng ct nhau cp r c. C 2 iu saib. Cho nhau cp r vi r = min(p, q) d. C 2 iu ngCu hi 6. Trong khng gian affine An cho mt siu phng v mt mphng (1 m < n. S trng hp xy ra khi xt v tr tng i ca v l: a. 2b. 3c. 4d. 5Cu hi 7. Mt mc tiu affine c th c: a. 1 c s nn duy nht c. V s c s nnb. 2 c s nn d. Khng c c s nn Cu hi 8. Trong khng gian affine A3 , cho hai mc tiu {O; , , }(1) v e1 e2 e3 + , + 2 , }(2). Cng thc i mc tiu t (1) sang (2) vi O (1, 2, 3) {O ; e1 e2 e2 e3 e3 l:1 2. x1 = x1 + x2 + 1 a. x2 = x1 + 2x3 + 2 x = x3 + 3 3 x1 = 2x1 + x2 + 1 c. x2 = x1 + 2x2 + 2 x3 = x1 + 2x2 + 3 x1 b. x2 x 3 x1 d. x2 x3= x1 + 2x2 + 1 = x1 + x3 + 2 = x3 + 3 = x1 + x2 + 3 = x1 + 2x2 + 2 = x2 + x3 + 3Cu hi 9. Chn cu ng: a. Nu l phng qua im P th M, N M N = P N P M . b. H {A0 , A1 , ..., Am } ph thuc affine khi v ch khi {A0 A1 , A0 A2 , ..., A0 Am } iu l c l c. H con ca mt h c lp suy ra c lp, ph thuc suy ra ph thuc. d. Trong khng gian afin An im c lp c nhiu nht n im.Cu hi 10. Chn cu ng: a. : V V V, ( , ) l khng gian affine. u v u v b. Khng gian affine v khng gian vect cng chiu ch khc nhau 1 im c nh. c. Mi khng gian vect l 1 khng gian affine. d. Tt c iu ng.Cu hi 11. H m + 1 im {A0 , A1 , ..., Am } ca khng gian affine ph thuc affine: a. Nu {A0 A1 , A0 A2 , ..., A0 Am } c lp tuyn tnh. b. Khi v ch khi {A0 A1 , A0 A2 , ..., A0 Am } c lp affine. c. Nu {A0 A1 , A0 A2 , ..., A0 Am1 } ph thuc affine. d. Khi v ch khi {A0 A1 , A0 A2 , ..., A0 Am } ph thuc tuyn tnh. Cu hi 12. Nu X l tp hp hu hn, X = {P1 , P2 , ..., Pm } th tng P1 + P2 + ... + Pm (xem cc Pi ,i = 1, n l khng phng ) l phng c: a. S chiu ln nht. c. S chiu b nht i qua cc im ny.b. S chiu ln nht i qua cc im ny. d. S chiu b nht.Cu hi 13. Hai mt phng song song "theo ngha PTTH" l 2phng song song. Chng cng l hai phng cho nhau cp: a. 0b. 1c. 2d. (n 1), n Z Cu hi 14. Trong khng gian affine A2 , cho mc tiu {O; , }. i vi mc e1 e2 tiu ny cho cc im A(1, 1), B(3, 2), C(1, 1), M (1, 7). Ta ca M i vi mc tiu {A; B, C} l:a. M (6, 7)b. M (5, 7)c. M (7, 6)2d. M (7, 5) 3. Cu hi 15. Trong khng gian affine A2 . Cho bn im A, B, C, D khng cng thuc mt mt phng v bn im P, Q, R, S to thnh cc t s n (ABP ), (BCQ), (CDR), (DAS) iu kiu cn v bn im P,Q,R,S cng thuc mt mt phng l : b. (ABP ).(BCQ).(CDR).(DAS) = 1 d. (BAP ).(BCQ).(CDR).(ADS) = 1a. (ABP ).(BCQ).(DCR).(DAS) = 1 c. (BAP ).(CBQ) = (CDR).(DAS)Cu hi 16. Trong khng gian affine A3 , nh x f c biu thc ta i vi mc tiu cho trc: x1 = 3x1 + 3x2 + 3x3 + 1 x = x1 x2 + x3 1 2 x3 = 2x1 + 2x2 + 2x3 + 3Tm nh v to nh ca im M (1, 2, 1) a. f (M ) = (12, 11, 11); f 1 (M ) = (7, 5, 3) c. f (M ) = (12, 11, 11); f 1 (M ) = (7, 5, 3)b. f (M ) = (12, 1, 11); f 1 (M ) = (7, 5, 3) d. f (M ) = (12, 11, 1); f 1 (M ) = (7, 5, 3)Cu hi 17. Trong khng gian affine A4 , phng trnh ca ci phng c s chiu b nht cha M1 (1, 1, 3, 2), M3 (1, 2, 0, 1), M2 (2, 0, 0, 0), v c phng cha (3, 3, 1, 0), b (1, 1, 1, 0) a a. x1 x2 + x4 + 2 = 0. c. x1 x2 + x4 2 = 0.b. x1 x2 x4 + 2 = 0. d. x1 + x2 x4 + 2 = 0.Cu hi 18. Cho hai ng thng d1 v d2 .Trong d1 qua A(1, 0, 2, 1) c phng (1, 2, 1, 3), d2 qua B(0, 1, 1, 1) c phng b (2, 3, 2, 4). Phng a trnh ci phng c s chiu b nht cha hai ng thng l: a. 3x1 4x2 x3 + 2x4 1 = 0 c. 3x1 4x2 + x3 2x4 1 = 0b. 3x1 + 4x2 + x3 2x4 1 = 0 d. 3x1 + 4x2 x3 2x4 1 = 0Cu hi 19. Cho ba mphng P, Q, R ln lt song song trong Am ln lt ct hai ng thng d1 v d2 ti P1 , Q1 , R1 v P2 , Q2 , R2 , trong (P QR) = p. Biu thc lin h gia Q1 Q2 v P1 P2 , R1 R2 l: a. Q1 Q2 = (1 p)P1 P2 + pR1 R2 c. Q1 Q2 = pP1 P2 + (p 1)R1 R2 b. Q1 Q2 = (1 + p)P1 P2 + pR1 R2 d. Q1 Q2 = (1 p)P1 P2 + (p + 1)R1 R2Cu hi 20. Cho hai im phn bit P v Q. Tp hp nhng im sao cho M P = k M Q l tp li nu:a. k > 0b. k < 0c. k = 0d. Tt c u saiCu hi 21. Trong khng gian affine A4 , vi cc mc tiu affine cho trc. Giao im ca ng thng AB vi cc siu phng ta vi A(4, 3, 1, 2), B(1, 2, 1, 5) l: 3 4. 3 5 7 22 11 7 11 3 17 , , ), (11, 0, 5, 11), ( , , 0, ), ( , , , 0). 5 5 5 2 2 2 3 3 3 11 2 17 3 5 7 22 11 7 b. (0, , , ), (11, 0, 5, 11), ( , , 0, ), ( , , , 0). 5 5 5 2 2 2 3 3 3 3 5 7 22 11 7 11 3 17 c. (0, , , ), (11, 0, 5, 11), ( , , 0, ), ( , , , 0). 5 5 5 2 2 2 3 3 3 11 3 17 3 5 7 22 11 7 d. (0, , , ), (11, 0, 5, 11), ( , , 0, ), ( , , , 0). 5 5 5 2 2 2 3 3 3a. (0,Cu hi 22. Trong khng gian affine An , h m + 1 im c lp {P0 , P1 , ..., Pm } cn gi l mn hnh vi cc nh P0 P1 ...Pm . Mi h con r + 1 gi r mt bn, h con ca cc im cn li (m (r + 1) mt bn gi l mt i din ca r mt bn . G l trng tm ca mn hnh, G1 G2 ca hai mt bn i din t s n [G1 G2 G] rm r1 r+m d. [G1 G2 G] = r1r+m r+1 rm c. [G1 G2 G] = r+1a. [G1 G2 G] =b. [G1 G2 G] =Cu hi 23. Cho A, B, C thng hng v (ABC) = k , Ai , Bi , Ci l hnh chiu ca A, B, C xung trc Ox theo phng Ox1 . Trong ai , bi , ci ln lc l ta ca A, B, C ta c : a. ai ci = k(ai bi ) c. ai ci = k(ai bi )b. ai ci = k(ai ci ) d. ai ci = k(ai ci )Cu hi 24. Trong khng affine An , X l tp hp hu hn im, X = {P0 , P1 , ..., Pm }, dim{P0 P1 + ... + Pm } = rank{P0 P1 , P0 P2 , ..., P0 Pm }, nu h im {P0 , P1 , ..., Pm } c lp th: a. {P0 P1 , P0 P2 , ..., P0 Pm } c lp affine. b. dim{P0 + P1 + ... + Pm } = m. c. {P0 P1 , P0 P2 , ..., P0 Pm } ph thuc affine d. dim{P0 + P1 + ... + Pm } = m + 1. . Cu hi 25. Cho hai ci phng v trong khng gian affine An c gi l cho nhau cp r nu: a. = v dim( ) = 0. ) = r. c. = v dim( b. l mt rphng. d. l mt (r 1)phng.Cu hi 26. Cho hai ci phng v trong khng gian affine An c gi l ct nhau cp r nu : a. l mt rphng c. l (r + 1)phngb. d. hoc v Cu hi 27. Qua mt im A cho trc c . . . . . . . . . . . . song song vi mphng cho trc : 4 5. b. (m 1)phng. d. Mt v ch mt mphnga. Mt mphng. c. Mt v ch mt (m 1)phngCu hi 28. Mi h im trong khng gian affine An thc (hoc phc ) u tn ti duy nht mt: a. Trng tmb. Trng tm trongc. Tm t cd. PhngCu hi 29. Trong khng gian affine A4 , cho hai phng c phng trnh : x1 + x2 x3 + 2x4 + 1 = 0 2x1 x2 + x3 + x4 1 = 0Vi im M (1, 2, 3, 1), phng trnh siu phng i qua v im M l : a. x1 + x2 x3 + x4 + 6 = 0 b. 9x1 + 3x2 6x3 + 15x4 + 6 = 0 c. 9x1 + 3x2 + 6x3 + 15x4 + 6 = 0 d. 9x1 + 6x2 6x3 + 15x4 + 6 = 0 (Dng cho cu 30,31,32) Trong An cho h n + 1 im c lp P0 , P1 , ..., Pn . Vi im M An , k hiu ta t c ca M i vi mc tiu ta t c {P0 , P1 , ..., Pn } l M (1 , 2 , ..., n ) cn ta affine ca M i vi mc tiu affine (P0 ; P0 P1 , ..., P0 Pn ) l M (x1 , x2 , ..., xn ). Cu hi 30. Ta t c ca cc im P0 , P1 , ..., Pn v trng tm G ca h im {P0 , P1 , ..., Pn } l : 1 n1 na.G( , ......., )b. G(n, ....., n)c.1 1 G( , ......., ) n nd. G(n, ....., n)Cu hi 31. Lin h gia ta t c (0 , 1 , ..., m ) v ta affine (x1 , x2 , ..., xn ) ca cng im M i vi hai mc tiu chn l : a.x1 = 1 , ..., xn = n ; 0 = 1 (x1 + x2 + ... + xn ). b. x1 = 1 , ..., xn = n ; 0 = 1 + (x1 + x2 + ... + xn ). c. x1 = 1 , ..., xn = n ; 0 = 1 (x1 x2 ... xn ). d.x1 = 1 , ..., xn = n ; 0 = 1 + (x1 x2 ... xn ). Cu hi 32. Vi j < k . Gi l phng tng ca cc im M, P0 , P1 , ..., Pm , m khng c im Pj , Pk . Gi s im M c ta (0 , 1 , ..., m ), i = 0. T s n [Pj Pk M ] l: a.x jb.x jc.j xd. j xCu hi 33. iu kin cn v h (m + 1) im ca khng gian affine c lp l :5 6. m i=0 m i OMi = 0 vb.m i OMi = 0 va. i=0 mc. i=0 md. i OMi = 0 v i OMi = 0 vi = 0 0 = ....... = m = 0. i=0 mi = 0. i=0 mi = 0 0 = ....... = m = 0. i=0 mi=0i = 0 0 = ....... = m = 0. i=0Cu hi 34. Trong khng gian affine A3 , xt 4 im khng ng phng A, B, C, D. Qu tch tm t c ca h im c trng s {(A, m + 1), (B, 2m 3), (C, 4 3m), (D, 1)} trong m l mt s thc l: 4 1 2 a. ng thng i qua M (1, , ) vi vect ch phng = ( , 1, 0). u3 2 3 4 1 = ( 2 , 1, 0). b. ng thng i qua M (1, , ) vi vect ch phng u 3 2 3 4 1 = ( 2 , 1, 0). c. ng thng i qua M (1, , ) vi vect ch phng u 3 2 3 4 1 = ( 2 , 1, 0). d. ng thng i qua M (1, , ) vi vect ch phng u 3 2 3Cu hi 35. Phng trnh tham s ca phng c phng trnh tng qut : x1 + x2 2x3 + 3x4 = 1 x + 2x2 x3 + 2x4 = 1 1 x1 x2 4x3 + 5x4 = 3 x1 x1 = 3t1 4t2 1 x2 x2 = t1 + t2 + 2 b. a. x3 x3 = t1 x4 = t2 x 4 l: x1 x1 = 3t1 + 4t2 1 x2 x2 = t1 t2 d. c. x3 x3 = t1 x4 = t2 x4= 3t1 4t2 + 1 = t1 + t2 = t1 = t2 = 3t1 4t2 1 = t1 + t2 + 2 = t1 = t2Cu hi 36. Trong khng gian affine An thc, im G thuc an thng th G l tm t c ca h im c trng s: a. {(A, t), (B, 1 + t)}b. {(A, t 1), (B, t)}c. {(A, t), (B, 1 t)}d. Mt kt qu khCu hi 37. Trong khng gian affine A4 , phng trnh tng qut ca ci phng c s chiu b nht cha im M (1, 0, 2, 2) v c phng (2, 1, 4, 4), b (0, 0, 7, 7) a l a. c.x1 + 2x2 1 = 0 2x1 + x3 x4 = 0 x1 + 2x2 1 = 0 x3 x4 = 0b. d. 6x1 2x2 + 1 = 0 x2 x4 = 0 x1 2x2 + 1 = 0 x3 x4 = 0 7. Cu hi 38. Chn p n ng : a. Trong khng gian affine An lun lun tn ti duy nht h m 1 im c lp vi 0 m n + 1, mi h im nhiu hn m 1 im u khng c lp. b. Trong khng gian affine An lun c nhng h m im c lp 0 m n + 1, mi h im nhiu hn m + 1 u khng c lp. c. Trong khng gian affine An lun lun c nhng h im m 1 im c lp vi 0 m n + 1, mi h im nhiu hn n im u ph thuc. d. Trong khng gian affine An lun lun c nhng h im m im c lp vi 0 m n + 1, mi h im nhiu hn n + 1 im u ph thuc. Cu hi 39. "Nu l mphng