αλγεβρα β λυκείου θέματα & λύσεις (Mathematica) 2 12 2014

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ΤΡΑΠΕΖΑ ΘΕΜΑΤΩΝ Άλγεβρας Β΄ τάξης Γενικού Λυκείου 2o Θέμα Εκφωνήσεις ‐ Λύσεις των θεμάτων Έκδοση 2 η (2/12/2014)

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Transcript of αλγεβρα β λυκείου θέματα & λύσεις (Mathematica) 2 12 2014

  1. 1. 2o 2 (2/12/2014)
  2. 2. 2 mathematica.gr mathematica http://www.mathematica.gr/forum/viewtopic.php?f=147&t=46866 : ,, ,,, ,,, ,,, ,,, Antonis_A,gGa,Grosrouvre,emag57 mathematica.gr
  3. 3. 3 2 GI_V_ALG_2_16950 ) ,. (10) ) (),, . (15) : ) 3x 4y 12 6x 8y 2 + = + = 2 , 3x 4y 12 0x 0y 22 + = + = . ) 3 3 1 y x 3 , y x 4 4 4 = + = + . , .
  4. 4. 4 GI_V_ALG_2_16954 :8x 2y 7(1)+ = ) (1) (10) ) ,, . (15) : ) ...8x 2y 11+ = ) ,.
  5. 5. 5 GI_V_ALG_2_16957 ,,27, . ) ; . (13) ) 5. . (12) : x y .,: x y 27+ = (1)x>y(2) ) x,y. ,,(1) ,(2) .,(x,y)=(18,9)(x,y) (17,10)= ... ) ,x y 5 = (3)(x y> ) (1),(3),2x 32 x 16= = . x 16= ,(1):16 y 27 y 11+ = = ,1611. ,.
  6. 6. 6 GI_V_ALG_2_16960 ) , (),() . (12) ) . (13) : ) (x'x) y x = + ,(=, x'x). () y x = + ,(0,2) (2,0),: (0,2)2 0 2= + = (2,0)0 2 2 1= + = = ():y x 2= + ()x'x(4,0)45 x'x, 45 1 = = . (): y x = + (4,0) :0 4 4= + = . () y x 4= : y x 4 y x 4 y 1 y x 2 x 2 x 4 x 3 = = = = + + = = . (3,1).
  7. 7. 7 GI_V_ALG_2_16962 ) : 17 , , 6 4 10 (12) ) 1 2 3 x x 2 < < < ,: 1 2 x , x 2 2 (13) : ) 17 10 : 17 7 7 3 10 10 10 10 = + = = ( 7 3 10 10 + = ) 3 , , 6 4 10 0, 2 , f(x) x= . : 3 3 6 4 10 6 4 10 < < > > ) 1 1 x x 2 = () 2 2 x x 2 = () 1 2 3 x x 2 < < < 1 2 3 x ,x , 2 . f(x) x= . : 1 2 1 2 1 2 x x x x x x 2 2 < < < : ( )1 1 2 1 2 3 3 x x x x 2 2 < < < > > > 1 2 3 x x 2 2 2 2 2 > > > 1 2 x x 2 2 2 > > > 1 2 x , x 2 2 , 2 , g(x) x= . 1 2 1 2 x x x x 2 2 2 2 > < .
  8. 8. 8 GI_V_ALG_2_16965 2 f(x) x 4x 5= + ,x R ) f 2 f(x) (x 2) 1= + . (12 ) ) , f , 2 y x= . (13 ) : ) 2 2 2 f(x) x 4x 5 x 4x 4 1 (x 2) 1= + = + + = + ) f 2 f(x) (x 2) 1= + 2 y x= 2 1 . .
  9. 9. 9 GI_V_ALG_2_16968 ) x 4 = 34x 3 0+ = ; . (10 ) ) f(x) 4x= y 1= . (15 ) : ) 34 3 3( 1) 3( 1 1) 0 4 + = + = + = , x 4 = 34x 3 0+ = ) f(x) y= . k 4x 1 4x 2k x ,k 2 4 = = = + = + GI_V_ALG_2_17647 : x 2y 8 (1) x y (2) = + = ,, . ) ,, (2,3). (13) ) ,, . (12) : ) (2)(2,3))."" ,..x 2= , 1, 0, 2= = = (1)(2,3) ,:2 2( 3) 8 8 8 = = , . ) (1).( 1, 8, 0= = = ), .
  10. 10. 10 GI_V_ALG_2_17650 xcm , ycm, 38cm: 2cm 4cm, . ) . (10) ) x,y . (15) : ) xy 2x 2y 38+ = . ,(x 2)(y 4)+ . : 2x 2y 38 (x 2)(y 4) xy + = + = ) : 2x 2y 38 x y 19 x y 19 (x 2)(y 4) xy xy 4x 2y 8 xy 4x 2y 8 y 19 x y 19 x y 14 2x 19 x 4 3x 15 x 5 + = + = + = + = + = + = = = = + = = = x 5,y 14= = : ,, ,.
  11. 11. 11 GI_V_ALG_2_17651 parking10 , 830 2.700. ) . (13) ) . (12 ) : ) x,y . x y 830+ = (1) 2x ,4y . 2x 4y 2700+ = x y 1350+ = (2) ) ( )x y 830 x y 830 x 310 x 2y 1350 y 520 y 520 + = + = = + = = = 310 520 . GI_V_ALG_2_17652 :( ) 2 1+ = ) 0= 0= . (13) ) . (12) : ) ( ) 2 2 2 1 2 1+ = + + = 1 2 1 2 0 + = = 0 = 0= ) 0 2= = ( ) 2 2 1 = + = + , . = , 0 2 2 2 = = = , . 2 = + , ( 2 = , ).
  12. 12. 12 GI_V_ALG_2_17656 1 f(x) 2x,x 2 = ) ; f ; (9 ) ) f . (10) ) 1 . . (6 ) : ) f 1 2 1 2 . 2x 2 ,x . f T = . ) ) 1 , 1 2 . . f 1 1 , 2 2 . :(). ,. 3 7 , 4 4 .
  13. 13. 13 GI_V_ALG_2_17659 ) 2 y x 1 x y 1 = + = (15) ) (). (10) : ) 2 2 2 x(x 1) 0y x 1 y x 1 x 1 x 1 y x 1x y 1 y x 1 y x 1 x 0 x 1 0 x 0 x 1 x 0 y 1 y x 1 y x 1 x 1 y 2 = = + = + + = + = + = = + = + = = = = = = = + = + = = (x,y) (0,1) (x,y) (1,2)= = ) 2 y x 1= + y x 1= + ,
  14. 14. 14 GI_V_ALG_2_17663 0 x 2 < < (2x 1)(5x 4) 0+ = ,: ) 4 x 5 = . (10) ) x. (15) : ) 0 x 2 < < (1),x 0> . (2x 1)(5x 4) 0+ = 2x 1 0 + = 5x 4 0 = 1 x 2 = 4 x 5 = . x 0> , 4 x 5 = . ) 2 2 2 2 2 24 16 9 x x 1 x 1 x 1 x 5 25 25 + = + = + = = . 3 x 5 = 3 x 5 = . x 0> ,x 0 x 2 < < , 3 x 5 = . 3 x 35x 4x 4 5 = = = 1 1 4 x 3x 3 4 = = = .
  15. 15. 15 GI_V_ALG_2_17664 , 0 0 ,: 0 135+ = : ) ( ) 1+ = (10) ) 1 + + = (15) : ) 0 0 0 0 ( ) 135 (180 45 ) 45 1+ = = = = ) , ( ) ( ) 1 1 1 1 1 1 + + = = + = + = + + + = GI_V_ALG_2_17681 f(x) 2x 1,x= + ) f (10) ) x [0,2] ; (15) : ) Hg(x) 2x= 2 2, f(x) g(x) 1= + 2 1 1 + = 2 1 3+ = . ) f(x) 3 2x 1 3 2x 2 x 1 2 = + = = = = x [0,2] , x 2 =
  16. 16. 16 GI_V_ALG_2_17683 : ( 1)x 2y 3 4x ( 1)y 6 + + = + = . ) 3= ,.. (8) ) 3= ,. (8) ) 0= , . (9) : ) 3= 2x 2y 3 3 y x , x 4x 4y 6 2 + = = + = (). 1 x 2 = y 2= 1 (x,y) ,2 2 = . ) 3= 4x 2y 3 4x 2y 6 + = + = (). ) 0= x 2y 3 x 2y 3 4x y 6 8x 2y 12 + = + = = = . : 9x 9 x 1= = 1: 1 2y 3 y 2 + = =
  17. 17. 17 GI_V_ALG_2_17688 ( ) 2 2x f x x 1 = + ) ( )f x 1 . (8) ) 1;. (8) ) . (9) : ) 2 x 1 0+ > , ( )f x . : ( ) 2 2 2 2x f x 1 1 2x x 1 0 x 2x 1 x 1 + + + ( ) 2 x 1 0 x x 1= ) ( ) ( ) ( )f x 1 f x f 1 , ( )f x 1, x=1. ) ( )f x ,x x ( ) ( ) ( ) ( )2 2 2 x 2x f x f x x 1x 1 = = = + + , ( )f x .
  18. 18. 18 GI_V_ALG_2_17692 ) : ( ) x x 0 2 + + + = (10) ) [ )x 0,2 : x x 2 = + (15) : ) : x x 2 2 + = ( x x 2 2 + + = ) x x 2 = () x x 2 + = . ( ) x x+ = () ( ) x x x x 0 2 + + + = = ) x x x x 2 = + = (()) 2x 0 x 0 = = x x 2 2 2 = = , . [ )x 0,2 , 2 0 2 2 0 4 4 2 + < + < 1 3 0 4 1 4 1 4 3 4 4 + < < < , , 0= x 2 = 2 0 2 2 0 4 4 2 < < 1 5 0 4 1 4 1 4 5 4 4 < < < , 1= 3 x 2 2 2 = =
  19. 19. 19 GI_V_ALG_2_17693 ) : 17 , , 6 4 10 (12) ) 1 2 3 x x 2 < < < ,: 1 2 x , x 2 2 (13) : ) 17 10 , : 17 7 7 3 10 10 10 10 = + = = ( 7 3 10 10 + = ) 3 , , 6 4 10 0, 2 ,y=x . : 3 3 6 4 10 6 4 10 < < > > ) 1 1 x x 2 = () 2 2 x x 2 = () 1 2 3 x x 2 < < < 1 2 3 x ,x , 2 . y=x. : 1 2 1 2 1 2 x x x x x x 2 2 < < <
  20. 20. 20 ( )1 1 2 1 2 3 3 x x x x 2 2 < < < > > > 2 1 2 3 x x 2 2 2 2 2 + > > > 1 2 x x 2 2 2 > > > 1 2 x , x 2 2 , 2 , y=x. : 1 2 1 2 x x x x 2 2 2 2 > < .
  21. 21. 21 GI_V_ALG_2_17698 fC f . : ) 1 2 3f(x ),f(x ),f(x ). (10) ) ;. (10) ) f 2x ;. (13) : ) y y : 1 3 2f(x ) f(x ) f(x )< < ) . , 1 2 3x x x< < 1 2 3f(x ) f(x ) f(x )< < , 1 2 3x x x< < 1 2 3f(x ) f(x ) f(x )> > () 1 3 2f(x ) f(x ) f(x )< < . ) ( )2 2x ,f(x ) 2f(x ). 2x .
  22. 22. 22 GI_V_ALG_2_17699 3 5 = , (). ) . (10) ) . (15) : ) 2 2 2 2 2 23 9 16 1 1 1 5 25 25 + = + = + = = 4 5 = 4 5 = . , 0> 4 5 = . ) , = . 3 ( ) 5 = = = 4 ( ) 5 = = = . 2 = , 3 (2 ) ( ) 5 = = = = 4 (2 ) ( ) 5 = = = = . : = +
  23. 23. 23 GI_V_ALG_2_17703 : 1( ):2x y 1 = 1( ):( 1)x y 6 = . ) . (8) ) 3= . (8) ) ; . (9) : ) ToD 2 1 3= + = . D 0 3 0 3= = = 1( ):2x y 1 = 1( ):2x y 6 = . ) ) ,D 0= 3= (). .
  24. 24. 24 GI_V_ALG_2_17704 ( )f x 32x, x= ) ,f. (12) ) f . (13) : ) 2 T 2 = = 3, 2x 1 2x 2 x , 2 = = + = + 32x 1 2x 2 x , = = = ) x 0 4 2 3 4 2x 0 2 3 2 2 2x 1 0 1 0 1 f(x)=32x 3 0 3 0 3 [ ]0,
  25. 25. 25 GI_V_ALG_2_17709 1 2 1:2x y 5, : 2x 3y 9, :3x 2y 7+ = + = + = ) i) 1 2, ii) 1 3, (12) ) ) 2 3, 1 (13) : i) 2x y 5 2x 3y 9 + = + = . :4y 4 y 1= = 1:2x 1 5 x 3 = = A(3, 1) . ii) 2x y 5 4x 2y 10 3x 2y 7 3x 2y 7 + = = + = + = . : x 3 x 3 = = 1:6 y 5 y 1+ = = , A(3, 1) . ) A(3, 1) , 2 3, 1
  26. 26. 26 GI_V_ALG_2_17717 25. 14 16,374. ) xyo, . (12) ) ; (13) : ) 16 x 14 y 374 x y 25 + = + = ) ( )16 x 14 y 374 16 25 y 14 y 374 (1) x y 25 x 25 y (2) + = + = + = = . (1) :y 13= . (2):x 25 13 12= = : ,: :16 x 16 12 192 = = . :14 y 14 13 182 = = . GI_V_ALG_2_17725 f(x) ( 3x) 3x 2 = + . ) f(x) 23x= (10) ) f . (13) : ) ( 3x) 3x = 3x 3x 2 = ,,: f(x) 23x= . ) 2, 2 2 T 3 =
  27. 27. 27 GI_V_ALG_2_17732 f : (2,3)(4,5). ) f. (13) ) fxx2, f(0) 0> (12) : ) f(2,3)(4,5), f(2)=3f(4)=5.f,2f(4)3>5,. f,. ) fxx2,f(2)=0. 2 ,3 4 ) 2= 4= 1 2 = .(7) ) 2= 1 2 = , i. f(x) 3= (10) ii. f [ ]0,8 . (8) : ) f (x) , 1= + 0= > :maxf(x) 1 3 1 1 3= + = + + = 1 3 2+ = = 4= 2 1 T 4 4 4 2 2 = = = = ) 2= 1 2 = f(x) 3 x 2 = i. f(x) 3 3 x 3 x 1 x 2 2 2 2 x 2 x 2 2 2 2 2 = = = = = + = + x 4 x 4 1, Z = + = + . ii. [ ]0,8 ,: ( ) x 0 1 2 3 4 5 6 7 8 f x 0 3 0 3 0 3 0 3 0 f .
  28. 46. 8 GI_V_ALG_4_17838 52 28 21 0+ + = . . 4 5 = . 2 < < ,: i. 7 2 25 = 24 2 25 = . ii. : [ ] 2 2 13 2 2 12 1822 25 2 2 + + = + + : ) 52 28 21 0+ + = (1). 2 2 2 1= (2). (1): ( )2 2 2 2 5 2 1 28 21 0 10 5 28 21 0 10 28 16 0 5 14 8 0 + + = + + = + + = + + = : y= [ ]y 1,1 . : 2 5y 14y 8 0+ + = 5, 14, 8= = = 2 2 4 14 458 196 160 36 0= = = = > . : 1,2 14 36 14 6 y 2 25 10 = = = , : 1 14 6 8 4 y 10 10 5 + = = = 2 14 6 20 y 2 1 10 10 = = = < ,. : 4 y 5 = = ) 2 2 2 < < < < 2 34.
  29. 47. 9 i) : 2 2 2 2 2 2 4 16 9 3 1 1 1 1 5 25 25 5 + = = = = = = 2 3 5 = . 2 2 4 16 32 25 7 2 2 1 2 1 2 1 5 25 25 25 25 = = = = = 3 4 24 2 2 2 5 5 25 = = = ii) : [ ] 2 2 13 2 2 12 131 12 7 241822 25 2 2 181 25 25 25 25 25 25 17 18 17 18 25 25 + + + = = = + + + = = = +
  30. 48. 10 AL_4_17839 : (1)x+3y=3 x (+1)y=3 + , . ) 0 0(x ,y ), 0 0x y= (10) ) : i. . (6) ii. . (4) ) 4, 2, 2= = = . (5) : . : 1 3 D 1 1 = = + 2 2 ( 1)( 1) 3 1 3 4 ( 2)( 2) + = = = + x 3 3 D 3( 1) 9 3 6 3( 2) 3 1 = = + = = + y 1 3 D 3( 1) 3 3 6 3( 2) 1 3 = = = = : D 0 ( 2)( 2) 0 2 + 2 ,: yx 0 0 DD 3( 2) 3( 2) 3 3 (x ,y ) , , , D D ( 2)( 2) ( 2)( 2) ( 2) ( 2) = = = + + + + , : 0 0x y= i) 2= ,: (21)x+3y=3 x+3y=3 x 3y 3 x 3 3y x (2+1)y=3 x 3y=3 + = = + + ,:(x,y) (3 3k,k),k=
  31. 49. 11 ii) 2= ,: ( 2 1)x 3y 3 3x 3y 3 x y 1 0x 0y 2 x ( 2 1)y 3 x y 3 x y 3 x y 3 + = + = + = + = + + = = = = , ) 3= , 2= , 2= ,
  32. 50. 12 GI_V_ALG_4_17840 x 2y 1 x y + = + = , ) (10) ) 1= ( )o ox ,y , [ ) 0,2 ox = oy = (7) ) 1= ( )1 1x ,y , , 1x = 1y = . (8) : ) x 2y 1 (): x y + = + = : x y 1 2 1 2 1 1 D 2 , D 2 D 1 1 1 = = = = = = = D 0 2 0 2 ,() ( ) yx DD 1 1 x,y , , , D D 2 2 2 2 + = = = + + D 0 2 0 2= = = ,: x 2y 1 x 2y 1 () x 2y 2 x 2y 2 + = = = = ,. ) 1= () ( ) ( )o o 1 1 1 x ,y , 1,0 1 2 1 2 + = = + + [ ) 0,2 o 1 x= = o 0 y= = , = ) 1= () ( )1 1 1 1 1 1 2 x ,y , , 1 2 1 2 3 3 + = = + + ,, 1 1 x 3 = = 1 2 y 3 = = : 2 2 4 1 5 1 1 1 9 9 9 + = + = = ,.
  33. 51. 13 GI_V_ALG_4_17841 .,, t sec t h(t) 8 6 ,0 t 180 30 = + ) , . (8) ) . (3) ) , .0180sec; (4+2=6) ) : i. h(t) (3) ii. h(t)0 t 90 (5) ( ) t 0 15 30 45 60 75 90 h t
  34. 52. 14 : ) t g(t) 6 30 = :maxg(t) 6= ming(t) 6= maxh(t) 8 maxg(t) 8 6 14= + = + = minh(t) 8 ming(t) 8 6 2= + = = , 14m 2m : t t t t h(t) 14 8 6 14 6 6 1 30 30 30 30 2 = + = = = = t 2 30 2 = + t 2 t 60 15 t 60 15, 30 2 = + = + = + 1 11 0 t 180 0 60 15 180 15 60 165 4 4 + , 0= 1= 2= t 15= t 75= t 135= : t t t t h(t) 2 8 6 2 6 6 1 30 30 30 30 2 = + = = = = t 2 30 2 = t 2 t 60 15 t 60 15, 30 2 = + + = = , 1 13 0 t 180 0 60 15 180 15 60 195 4 4 , 1= 2= 3= t 45= t 105= t 165= ) :() d (AE) (BE) maxh(t) minh(t) 14 2 12m= = = = 6m ) 2 T 60 sec 30 = = 180 3 60 = .
  35. 53. 15 ) : 0 h(0) 8 6 8,h(15) 8 6 14 , 30 2 = + = = + = ( ) 3 h(30) 8 6 8,h(45) 8 6 2 2 = + = = + = ( ) 5 h(60) 8 6 2 8,h(75) 8 6 8 6 14, 2 2 = + = = + = + = ( ) ( )h(90) 8 6 3 8 6 8= + = + = : ( ) t 0 15 30 45 60 75 90 h t 8 14 8 2 8 14 8 :
  36. 54. 16 GI_V_ALG_4_17842 : ( ) ( ) 21 f x x c d, x 2 = c,d , ( )A 0,16 ( )B 4,0 ) , c,d . (10) ) c 6= d 2= , i. f . (3) ii. , f ( ) 21 g x x 2 = (6) iii. , f , f , . (6) : ) ( ) ( ) 21 f x x c d, x 2 = ( )A 0,16 ( )B 4,0 , . ( )f 0 16= ( )f 4 0= ( ) ( ) 2 2 2 21 1 f 0 16 0 c d 16 c d 16 c 2d 32 (1) 2 2 = = = = ( ) ( ) ( ) 2 2 2 21 1 f 4 0 4 c d 0 16 8c c d 0 16 8c c 2d 0 (2) 2 2 = = + = + = (1)(2): 16 8c 32 0 8c 48 c 6 + = = = (1) 36 2d 32 4 2d d 2 = = =
  37. 55. 17 ) c 6= d 2= ( ) ( ) 21 f x x 6 2, x 2 = . i. fC x x ( )y f x y 0 = = ( ) ( ) ( ) 2 2 2 x 6 2 x 8 1 f x 0 x 6 2 0 x 6 4 2 x 6 2 x 4 = = = = = = = fC x x ( )B 4,0 ( ) 8,0 fC y'y ( )f 0 16= A . ii. ( ) 21 g x x 2 = 2 y x= (0,0), . ( ) ( ) 21 f x x 6 2 2 = gC ,, 2 y y 6+ x x . iii. f x 6= ( )f 6 2= . ( ],6 f . [ )6,+ f .
  38. 56. 18 GI_V_ALG_4_17843 f ( )f(x) x k= + ,,,k . ) ,: i. f (3) ii. T f (3) ) ,,k . . (9) ) 1 3, 2 = = k 2= , 0x A ,. (10) : ) : i. maxf 5= minf 1= ii. T 4= ) , . maxf 5 k 5 (1)= + = minf 1 k 1 (2)= + = (1)(2)2k 4 k 2= = , 3= . 2 1 T 4 4 2 4 2 = = = =
  39. 57. 19 k 2= , 3= 1 2 = . ) k 2= , 3= 1 2 = 1 f(x) 3 x 2 2 = + . 1 f(x) 3 x 2 2 = + 0 7 A x , 2 , ( )0 7 f x 2 = ( )0 0 0 7 1 7 1 7 f x 3 x 2 3 x 2 2 2 2 2 2 = + = = 0 0 1 3 1 1 3 x x 2 2 2 2 = = 0 1 x 2 6 = 0 1 x 2 2 6 = + 0 1 x 2 2 6 = + 0 x 4 3 = + 0 5 x 4 , 3 = + ,, 05 x 6< < 5 4 6 5 4 6 14 12 7 1 1 3 3 < + < < + < < < 14 17 2 5 1 1 12 12 12 12 < < + < < + 5 5 5 4 6 5 4 6 10 12 13 3 3 < + < < + < < < 10 13 1 12 12 < < = 1= 0 5 17 x 4 3 3 = + =
  40. 58. 20 GI_V_ALG_4_17844 ) : 2 2 x y 1 x y 1 + = + = (12) ) (), 0 2 , 1+ = . (13) : ) ( ) 22 2 x y 1x y 1 x y 1 1 2xy 1x y 1 x y 2xy 1 + + = + = =+ = + = x y 1 x y 1 xy 0 x 0 y 0 + = + = = = = x y 1 y 1 (x,y) (0, 1) x 0 x 0 + = = = = = x y 1 x 1 (x,y) ( 1,0) y 0 y 0 + = = = = = (x,y) (0, 1)= (x,y) ( 1,0)= ) : 1+ = 2 2 + =1 2 2 1 + =1 + = x, =y= 1 x 1, 1 y 1 2 2 x y 1 x y 1 + = + = , ()(x,y) (0, 1)= (x,y) ( 1,0)= ,. : 0= =1 ,0 2 , 3 2 = , 1= =00 2 , = B,
  41. 59. 21 GI_V_ALG_4_17846 f(x) x= g(x) 2x= . ) f g ., f(x) g(x), [ ]x 0,2 (8) ) , ( )2x x 1= [ ]0,2 . (4) )