Аналитическая геометрия. Комплексные числа: Методические указания и контрольные работы по курсу ''Высшая математика''

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. : . 1 : . . 2002 2 1. z x iy= + , x y - , i - ( , : 2 1i = ). x y z . : Re ; Imx z y z= = . z x iy= + . 1 1 1z x iy= + 2 2 2z x iy= + , , .. 1 2x x= 1 2y y= . 0 0 0 i= + . 1 1 0 i= + . 0z i y= + . z x i y= z x i y= + . , , : 1) . 1 1 1z x iy= + 2 2 2z x iy= + , ( ) ( )1 2 1 2 1 2z z x x i y y+ = + + + . 2) . 1 1 1z x iy= + 2 2 2z x iy= + , ( ) ( )1 2 1 2 1 2z z x x i y y = + . 3) . 1 1 1z x iy= + 2 2 2z x iy= + , ( ) ( )1 2 1 2 1 2 1 2 2 1z z x x y y i x y x y = + + . , , 2i 1. 4) . 1 1 1z x iy= + 2 2 2z x iy= + , 1 1 2 1 2 2 1 1 22 2 2 22 2 2 2 2z x x y y x y x yiz x y x y+ = ++ +. , , . z x iy= + ( , )M x y . , . M , z z . : y ( , )M x y x 0 3 2 2 ; argz x y z = = + = . 0z : - z , 2 k + , 0, 1, 2,...k = . arg z = , 2 2 2 2cos , sinx yx y x y = =+ + tg yx = . x y , cos ; sinx y = = . : ( )cos sinz i = + . . : 10. . ( )1 1 1 1cos sinz i = + ( )2 2 2 2cos sinz i = + , ( ) ( )( )1 2 1 2 1 2 1 2cos sinz z i = + + + , .. . 20. . ( )1 1 1 1cos sinz i = + ( )2 2 2 2cos sinz i = + , ( ) ( )( )1 1 1 2 1 22 2cos sinz iz= + , .. . n : ( )cos sinn nz n i n = + , ( )cos sinz i = + . n - ( )cos sinz i = + n , : 2 2cos sinnkk kz in n + + = + , 0, 1, ..., 1 ,k n n N= . 1. 25 22 1izi += + . , ( )25 4 2i i i i i i= = = . , ( ) ( )( ) ( )( )2 22 22 1 2 4 32 4 3 9 24 9 242 1 1 2 1 2 5 25 25 25 24i i ii i iz ii i i + + = = = = = = + + . , 9 24Re ; Im25 25z z= = . 2. 4 (3 ) (4 2 ) 2 6.(4 2 ) (2 3 ) 5 4i x i y ii x i y i + + = + + + = + . 23 4 2 (3 )(2 3 ) (4 2 ) ( 9 7 ) (12 16 )4 2 (2 3 )21 23 ,2 6 4 2(2 6 )(2 3 ) (4 2 )(5 4 )5 4 (2 3 )(14 18 ) (12 6 ) 2 44 ,3 2 6(3 )(5 4 ) (4 2 )(2 6 ) (19 7 ) ( 4 284 2 5 4xyi ii i i i ii iii ii i i ii ii i ii ii i i i ii i + = = + + = + =+ += + + = = + + + + =+ += + = + = = + + + = + ++ +)23 21 .ii== , 2 44 2(1 22 )(21 23 ) 2( 485 485 ) 970 970 1 ,21 23 (21 23 )(21 23 ) 970 97023 21 (23 21 )(21 23 ) 0 970 .21 23 (21 23 )(21 23 ) 970xyi i i i ix ii i ii i i iy ii i i += = = = = = + + = = = = = + , 1 ,x i y i= + = . 3. 1z , ( )(1 2 ) (1 )(3 4 ) 1 7i z i iz i i + + = + . , ( 5 5 ) 10i z i = , 210 2 2 (1 ) 2 2 1 .5(1 ) 1 (1 )(1 ) 1 1i i i i i iz ii i i i += = = = = + + + + 1z i= - . 32 ; arg ctg14z z ar = = = = = , z : 3 32 cos sin4 4z i = + . 1. : 5 1) ( )61 2 )i+ ; 2) ( ) ( )7 72 2i i+ + ; 3) ( )531(1 )ii+ . 2. 1) 11ii+ ; 2) ( )31 3 3 5i i+ + + ; 3) ( )2 33 21 2 (1 )(3 2 ) (2 )i ii i+ + + ; 4) 1232i + ; 5) ( )52 2i+ ; 6) 31 22 2i + . 3. 1) 1 32 2i + ; 2) 11ii+ ; 3) ( )34 3i + ; 4) 5( 1 )i + ; 5) 1 3i ; 6) 21 32 2i + . 4. 1) (2 ) (2 ) 6(3 2 ) (3 2 ) 8i x i yi x i y+ + = + + = ; 2) 2 102 203 (1 ) 30x yi zx y zixi yi i z+ = + = + + = . 5. 1) 2 (2 ) ( 1 7 ) 0x i x i+ + + + = ; 2) 2 (3 2 ) (5 5 ) 0x i x i + = ; 3) 2(2 ) (5 ) (2 2 ) 0i x i x i+ + = . 2. 0Ax By Cz D+ + + = ( 1) , , , . . (1) , . , , 6 , . , , . , . , , , , . a, b , , : 1x y za b c+ + = . ( 2 ) , , (cos ,cos , cos ), : cos cos cos 0.x y z p + + = (3) (1) , : 2 2 21 .MA B C= + + (4) ' (1). (1) : 2 2 22 2 2 2 2 2 2 2 2cos ; cos ; cosDpA B CA B CA B C A B C A B C = + + = = = + + + + + +m. ( 5 ) (', ',z') (1) / / /2 2 2Ax By Cz DA B C+ + +=+ + (6) , , / / /cos cos cosx y z p = + + , ( 6/ ) 7 . . , , , 1 1 1 100Ax By Cz DA x B y C z D+ + + = + + + = ( 7 ) : 1 1 12 2 2 2 2 21 1 1cos AA BB CCA B C A B C+ += + + + + . ( 8 ) (7): 1 1 1A B CA B C= = . ( 9 ) : 1 1 1 0AA BB CC+ + = . ( 10 ) ( ) ( )1 1 1 2 2 2; ; , ; ;x y z x y z ( )3 3 3; ;x y z , : 1 1 12 2 23 2 311011x y zx y zx y zx y x= . ( 11) ( ) ( )1 1 1 2 2 2; ; , ; ;x y z x y z , ( )3 3 3; ;x y z ( )4 4 4; ;x y z , 1 1 12 2 23 3 34 4 411011x y zx y zx y zx y x= . ( 12) , , , : 1 1 12 2 23 3 34 4 4111 0161x y zx y zVx y zx y x= = . ( 13 ) , (V >0). 8 1. 4 3 0x y z + = : (-1; 6; 3), B(3; -2; -5), (0; 4; 1), D(2; 0; 5), E(2; 7; 0), F (0; 1; 0)? 2. , Ax + By +Cz + D == 0 , , - , . 3. : 1. 3 5 1 0x y + = ; 2. 2 3 7 0x y z+ = ; 3. 9 2 0y = ; 4. 8 3 0y z = ; 5. 5 0x y+ = . 4. : 1. ( )xz (2; 5;3) ; 2. z ( 3;1; 2) ; 3. x (4;0; 2) (5;1;7) . 5. , : 1. 2 3 12 0x y z + = ; 2. 5 3 15 0x y z+ = ; 3. 1 0x y z + = ; 4. 4 6 0x z + = ; 5. 5 2 0x y z + = ; 6. 7 0x = . 6. 5 2 10 0x y+ = . 7. 3 2 18 0x y z+ = . , , , , , . 8. (7; 5;1)P , . 9. , , . , , : 6; 29 ; 5AB BC CA= = = . 10. : 1) 2 9 6 22 0x y z + = 2) 10 2 11 60 0x y z+ + = 3) 6 6 7 33 0x y z + = 11. 15 10 6 190 0x y z + = . 9 12. , 6 , : : : 1:3: 2a b c= . 13. , 2 2 9 0x y z + = . 14. : 11 , 55 , 10a b c= = = . , . 15. 2 5 0x y z + = ( )yz . 16. , 6 2 9 121 0x y z+ + = . 17. , , (3: 6;2)P , . 18. : 1) (3;1; 1) 22 4 20 45 0x y z+ = . 2) ( )4;3; 2 3 5 1 0x y z + + = . 3) 12;0;2 4 4 2 17 0x y z + + = . 19. ( ) ( ) ( ) ( )0;6;4 , 3;5;3 , 2;11; 5 , 1; 1;4A B C D . 20. ( )1;3; 2A ( )7; 4;4B . B , AB . 21. : 1) 4 5 3 1 0x y z + = 4 9 0x y z + = ; 2) 3 2 15 0x y z + + = 5 9 3 1 0x y z+ = ; 3) 6 2 4 17 0x y z+ + = 9 3 6 4 0x y z+ = . 22. : 1) ( )2;7;3 4 5 1 0x y z + = ; 2) : 2 5 3 0x y z + + = 3 7 0x y z+ = ; 3) ( )0;0;1L ( )3;0;0N 3 ( )xy . 23. : 11 2 10 15 0x y z = 11 2 10 45 0x y z = . 24. 3 6 2 14 0x y z + = . 10 25. : ( ) ( ) ( )0;0;2 , 3;0;5 , 1;1;0A B C ( )4;1;2D . . 26. , . 27. , : 1) ( ) ( ) ( )3; 1; 0 , 0; 7; 2 , 1; 0; 5 ( )4; 1; 5 ; 2) ( ) ( ) ( )1; 1; 1 , 0; 2; 4 , 1; 3; 3 ( )4; 0; 3 . 28. : 1) 4 2 3 0 , 3 5 0 , 3 12 62 7 0x y z x y z x y z + = + + = + + = ; 2) 5 8 0 , 2 3 1 0 , 2 3 2 9 0x y z x y z x y z+ = + + = + = ; 3) 2 5 4 0 , 5 2 13 23 0 , 3 5 0x y z x y z x z + = + + = + = . 29. 4 3 1 0x y z + = 5 2 0x y z+ + = : 1) ; 2) ( )1;1;1 ; 3) y ; 4) 2 5 3 0x y z + = . 3. ; : 1 1 1 100Ax By Cz DA x B y C z D+ + + = + + + = . ( 14 ) , (14). , ; , . , (14) : x mz ay nz b= + = + . ( 15 ) . ( )1 1 1; ;x y z ( )2 2 2; ;x y z , , : 11 1 1 12 1 2 1 2 1x x y y z zx x y y z z = = . ( 16 ) : 2 1 2 1 2 1; ;x x m y y n z z p = = = , : 1 1 1x x y y z zm n p = = ; ( 17 ) ;m n p : : cos : cos : cosm n p = , ( 18 ) : 2 2 22 2 22 2 2coscoscosmm n pnm n ppm n p= + + = + + = + + . ( 19 ) (19) , . . x a y b z cm n p = = 1 1 11 1 1x a y b z zm n p = = ( 20 ) : 1 1 12 2 2 2 2 21 1 1cos mm nn ppm n p m n p+ += + + + + . ( 21 ) : 1 1 1m n pm n p= = . ( 22 ) : 1 1 1 0mm nn pp+ + = . ( 23 ) , (17), (14) . : (14) , , (15), z (15) : 12 1x a y b zm n = = ; 0c = 1p = . : (14) ,a b c - , ,m n p , 1 1 1 1 1 1: : : :B C C A A Bm n pB C C A A B= . ( 24 ) ,m n p , , , . 0x a y b z cn p = = 0x a = y b z cn p = ; x . 0 0x a y b z cp = = 0x a = 0y b = ; z . ( ), ,a b c cos , cos , cos , : cos cos cosx a y b z c= = . ( 25 ) (17) 2 2 21m n p+ +. (25) ( ); ;x y z ( ), ,a b c . (17) . (20) 1 1 11 1 10a a b b c cm n pm n p = . ( 26 ) 13 1. : 1) 1 1 100Ax By CzA x B y C z+ + = + + = ; 2) 1 100Ax CzA x C z+ = + = ; 3) 1 100Ax DA x D+ = + = ; 4) 1 100Ax By Cz DB y D+ + + = + = . 2. D 3 2 6 04 0x y zx y z D + = + + = z ? 3. B D 2 9 03 0x y zx By z D + = + + + = ( )xy ? 4. 1 1 1 100Ax By Cz DA x B y C z D+ + + = + + + = : 1) x ; 2) y ; 3) z 4) ( )yz ; 5) ( )xz ; 6) ? 5. , 5 8 3 9 02 4 1 0x y zx y z+ + = + = 14 . 6. 3 2 5 01 0x y zx y z+ + = + = ? 7. 4 2 5 03 2 0x y zx y z + = + + = 2 3 0x y z+ + = . 8. , ( , , )a b c . 9. , : ( ) ( ) ( ) ( )0; 0;2 , 4; 0; 5 ; 5; 3; 0 ; 1; 4; 2A B C D . 10. , : ( )3;0;1 , ( ) 40;2;4 ; 1; ; 33. 11. ( )1 1, ,0x y ( )2 2;0;x z . . 12. 1) 1 5 24 3 12x y z += = ; 2) 7 312 9 20x y z += = . 13. , ( )1; 5;3A , 600, 450 1200. 14. , : 1 2 53 6 2x y z + = = 3 12 9 6x y z += = . 15. , : ( ) ( ) ( ) ( )3; 1;0 , 0; 7;3 , 2;1; 1 , 3;2;6A B C D . 15 16. 2 3 3 9 02 3 0x y zx y z = + + = . 17. 5 6 2 21 03 0x y zx z + + = + = . 18. 3 4 2 02 2 0x y zx y z = + = 4 6 2 03 2 0x y zy z+ = + = . 19. ( )2; 5;3 : 1) z ; 2) 1 2 34 6 9x y z += = ; 3) 2 3 1 05 4 7 0x y zx y z + = = . 20. ( )xz , 2 1 53 2 1x y z + = = . 21. , 1) 1 7 52 1 4x y z = = 6 13 2 1x y z += = ; 2) 4 1 02 3 0x zx y+ = + = 3 4 02 8 0x y zy z+ + = + = . 22. , ( )2;3;1A 1 22 1 3x y z+ = = . 23. 5 2 14 3 2x y z += = . 16 24. ( )4;0; 1A , : 1 3 52 4 3x y z + = = 2 15 1 2x y z += = . 25. , 3 52 3 1x y z+ = = 10 75 4 1x y z += = , , 2 1 38 7 1x y z+ = = . 26. 7 3 91 2 1x y z = = 3 1 17 2 3x y z = = . 4. x a y b z cm n p = = ( 27 ) 0Ax By Cz D+ + + = , ( 28 ) . , , (27). , ,x m a y n b z p c = + = + = + ; (28) . (27) (28) 2 2 2 2 2 2sin Am Bn CpA B C m n p+ += + + + + . ( 29 ) (27) (28) 0Am Bn Cp+ + = . ( 30 ) 17 A B Cm n p= = . ( 31 ) , (27) (28), : 00Aa Bb Cc DAm Bn Cp+ + + = + + = . 1. 12 9 14 3 1x y z = = 3 5 2 0x y z+ = / 2. : 1) 1 32 4 3x y z+ = 3 3 2 5 0x y z + = ; 2) 13 1 48 2 3x y z = 3 4 1 0x y z + = ; 3) 7 4 55 1 4x y z = 3 2 5 0x y z + = . 3. , 2 3 0x y z+ = 3 5 11 5 2x y z += = 5 3 42 4 6x y z += =. 4. A 3 5 1 0Ax y z+ + = 1 24 3 1x y z += = ? 5. A B 6 7 0Ax By z+ + = 2 5 12 4 3x y z + += = ? 6. (3; 2;4) 5 3 7 1 0x y z+ + = . 18 7. , 2 3 14 5 2x y z+ = =. 8. ( )4; 3; 1A 2 3 0x y z+ = . 9. , : 1) 1 3 22 1 5x y z + += = 4 3 3 0x y z+ + = ; 2) 1 24 7 3x y z = = 5 8 2 1 0x y z = ; 3) 2 13 4 1x y z+ += = 3 2 15 0x y z + = . 10. , ( )3; 1; 2 4 35 2 1x y z += = . 11. 2 3 15 1 2x y z += = , 4 3 7 0x y z+ + = . 12. 4 14 3 2x y z += = 3 8 0x y z + + . 13. , 3 1 25 2 4x y z + = = 8 1 63 1 2x y z = = , , . 14. , ( )3; 2; 5 4 3 13 0x y z+ + = 2 11 0x y z + = . 15. , 2 17 3 5x y z+ = = 1 3 27 3 5x y z += = . 16. , ( )4; 3; 1P : 6 2 3x y z= = 1 3 45 4 2x y z+ = = . 19 17. , 3 4 22 1 2x y z + = = 5 2 14 7 2x y z+ = = . 18. 5 23 1 4x y z+ = = , 15 0x y z+ + = . 19. 7 5 14 3 6x y z = = 2 7 1 0x y z+ + = ? 20. ( )1; 0; 7P 3 2 15 0x y z + = , 1 34 2 1x y z = = . 21. ( )7; 9; 7P 2 14 3 2x y z = = . 22. 7 31 2 1x y z+ = = , ( )3; 2; 6 . 23. , ( )4; 3; 10P 1 2 32 4 5x y z = = . 24. 2 13 4 2x y z += = 7 1 33 4 2x y z = = . 25. ( ) ( ) ( )4; 1; 2 , 2; 0; 0 , 2; 3; 5A B C . , B . 26. A , 25. 27. AB 25 , . 28. , . , . 29. , , , . 20 1. . . . .: , 1994. 206. 2. . ., . . . .: , 1964. 304. 3. . . . .: , 1966. 336. 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 : ..

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