Πανελλήνιες ἐξετάσεις στὰ Μαθηματικὰ (1999–)

(1999)

hp://gordongr.blogspot.com

(1999) http://gordongr.blogspot.com 8 2011

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I 1 1.1 1999 . . . . . . . . . . . . . . . . . . . 1.2 2000 . . . . . . . . . . . . . . . . . . . 1.3 2001 . . . . . . . . . . . . . . . . . . . 1.4 2002 . . . . . . . . . . . . . . . . . . . 1.5 2003 . . . . . . . . . . . . . . . . . . . 1.6 2003 . . . . . . . . . . 1.7 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1113 13 18 23 28 33 37 42 47 47 53 59 64 69 75 81

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2.1 1999 . . . . . . . . . . . . . . . . . . . . 2.2 2000 . . . . . . . . . . . . . . . . . . . . 2.3 2001 . . . . . . . . . . . . . . . . . . . . 2.4 2002 . . . . . . . . . . . . . . . . . . . . 2.5 2003 . . . . . . . . . . . . . . . . . . . . 2.6 2003 . . . . . . . . . . . 2.7 2004 . . . . . . . . . . . . . . . . . . . .

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87 3.1 1999 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.2 2000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.3 2001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.4 2002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.5 2003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.6 2003 . . . . . . . . . . . . . . . . . . . . . . . . . . 114 3.7 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 / 123 4.1 2000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.2 2001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 3

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4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 5 2002 . . . . . . . . . 2003 . . . . . . . . . 2003 2004 . . . . . . . . . 2004 2005 . . . . . . . . . 2005 2006 . . . . . . . . . 2006 2007 . . . . . . . . . 2007 2008 . . . . . . . . . 2008 2009 . . . . . . . . . 2009 2010 . . . . . . . . . 2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 140 145 150 156 161 166 173 180 185 190 195 200 207 212 218 225

/ 231 5.1 2000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 5.2 2001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 5.3 2002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 5.4 2002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 5.5 2003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 5.6 2003 . . . . . . . . . . . . . . . . . . . . . . . . . . 261 5.7 2003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 5.8 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 5.9 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . 279 5.10 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 5.11 2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 5.12 2005 . . . . . . . . . . . . . . . . . . . . . . . . . . 299 5.13 2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 5.14 2006 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 5.15 2006 . . . . . . . . . . . . . . . . . . . . . . . . . . 318 5.16 2006 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 5.17 2007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 5.18 2007 . . . . . . . . . . . . . . . . . . . . . . . . . . 336 5.19 2007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 5.20 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 5.21 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . 355 5.22 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 5.23 2009 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 4

5.24 2009 . . . . . . . . . . . . . . . . . . . . . . . . . . 376 5.25 2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 5.26 2010 . . . . . . . . . . . . . . . . . . . . . . . . . . 391 6 / 6.1 2000 . . . . . . . . . . . . . 6.2 2001 . . . . . . . . . . . . . 6.3 2002 . . . . . . . . . . . . . 6.4 2003 . . . . . . . . . . . . . 6.5 2004 . . . . . . . . . . . . . 6.6 2005 . . . . . . . . . . . . . 6.7 2006 . . . . . . . . . . . . . 6.8 2007 . . . . . . . . . . . . . 6.9 2008 . . . . . . . . . . . . . 6.10 2009 . . . . . . . . . . . . . 6.11 2009 . . . . . . . . . . . . 6.12 2010 . . . . . . . . . . . . . 6.13 2010 . . . . . . . . . . . . 399 399 403 407 411 415 420 424 428 432 436 441 446 450

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II7

7.1 2001 . . . . . . . . . . . . . . . . . . . 7.2 2002 . . . . . . . . . . . . . . . . . . . 7.3 2003 . . . . . . . . . . . . . . . . . . . 7.4 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

457459 459 463 467 471 477 477 482 486 490 496

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8.1 2001 . . . . . . . . . . . . . . . . . . . . 8.2 2001 . . . . . . . . . . . 8.3 2002 . . . . . . . . . . . . . . . . . . . . 8.4 2003 . . . . . . . . . . . . . . . . . . . . 8.5 2004 . . . . . . . . . . . . . . . . . . . .

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503 9.1 2001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 9.2 2002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 9.3 2003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 9.4 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 5

10 10.1 2001 . . . . . . . . . . . . . . . 10.2 2002 . . . . . . . . . . . . . . . 10.3 2003 . . . . . . . . . . . . . . . 10.4 2003 . . . . . . 10.5 2004 . . . . . . . . . . . . . . . 10.6 2004 . . . . . . 10.7 2005 . . . . . . . . . . . . . . . 10.8 2005 . . . . . . 10.9 2006 . . . . . . . . . . . . . . . 10.10 2007 . . . . . . . . . . . . . . . 10.11 2007 . . . . . . 10.12 2008 . . . . . . . . . . . . . . . 10.13 2009 . . . . . . . . . . . . . . . 10.14 2010 . . . . . . . . . . . . . . .

/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 521 526 530 535 539 545 549 555 559 565 570 576 581 587 593 593 598 603 608 613 617 621 627 632 638 643 648 654 661 661 664 668 673 677 682 686 690 694

11 / 11.1 2001 . . . . . . . . . . . . . . . . . . . 11.2 2003 . . . . . . . . . . . . . . . . . . . 11.3 2004 . . . . . . . . . . . . . . . . . . . 11.4 2004 . . . . . . . . . . 11.5 2005 . . . . . . . . . . . . . . . . . . . 11.6 2005 . . . . . . . . . . 11.7 2006 . . . . . . . . . . . . . . . . . . . 11.8 2007 . . . . . . . . . . . . . . . . . . . 11.9 2007 . . . . . . . . . . 11.10 2008 . . . . . . . . . . . . . . . . . . . 11.11 2009 . . . . . . . . . . . . . . . . . . . 11.12 2010 . . . . . . . . . . . . . . . . . . . 11.13 2010 . . . . . . . . . . 12 / 12.1 2001 . . . . . . . . . . . . . 12.2 2002 . . . . . . . . . . . . . 12.3 2004 . . . . . . . . . . . . . 12.4 2005 . . . . . . . . . . . . . 12.5 2006 . . . . . . . . . . . . . 12.6 2007 . . . . . . . . . . . . . 12.7 2008 . . . . . . . . . . . . . 12.8 2009 . . . . . . . . . . . . . 12.9 2010 . . . . . . . . . . . . . 6 . . . . . . . . . . . . . . . . . . . . . . . . . . .

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12.10 2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699 Creative Commons 3.0 , , 705

(1999) http://gordongr.blogspot.com

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. , , . 1999 , , / /. , , . , . , GordonGR . Creative Commons (, , 3.0). / : XLTEX, Kig, kmplot, Inkscape, Ubuntu Linux 10.04. http://gordongr.blogspot.com 2011

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I

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1 1.1 1999

1: . () . () () () ( ), : ) () . [ 2,5/100] . P() = ()()+(), () = 0 deg(()) < deg( ). ) () : . (). . . . . . . [ 5/100] . () = 0 deg(()) = 0. , , . 13

14

1.

) () = . = (). [ 5/100] . = P() = ( )() + () = ().

. () = 3 + + 1, . () 1 ; A. = 0, B. = 1, . = 1, . = 2, . = 2 [ 12,5/100] . =0 P(1) = 0 3 + + 1 = 0 2 + 1 = 0 ( 1) = 0 = 1,

.

2: 16 2. ) : . = 64 = 1/2, . = 64 = 1/2, . = 64 = 1/2, . = 32 = 1/2. [ 9/100] (1999) http://gordongr.blogspot.com

1.1. 1999 . ( ) = 16 = 2, = 16 = 2. = 1 1 = . 8 2

15

1 = 16 = 64, 4

. ) . [ 9/100] 1 2 1 . = = 64 = = . 8 2 2

)

. [ 7/100]

. =

1 < 1, 2 S = = = 128. 1

3: ) sin 6 + sin 4 = 2 sin 5 cos . [ 10/100] . sin 6 + sin 4 = 2 sin 6 + 4 6 4 cos = 2 sin 5 cos . 2 2


16

1.

) sin 6 + sin 4 + 4 sin 5 = 0. [o 15/100] . sin 6 + sin 4 + 4 sin 5 = 0 ()

2 sin 5 cos + 4 sin 5 = 0 2 sin 5(cos + 2) = 0 sin 5 = 0, cos 2 5 = = , . 5

4: 620 640 . : 120 . 10 . , . 48 . ) . [ 5/100] . . ( ) , . 48, = + 3 = 48 3.


1.1. 1999 ) .

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[ 5/100] . , A = 120 + S 10 = 120 + (2 + 9) 2 = 120 + 5(96 + 3) = 600 + 15.

)

. [ 5/100]

. 620 < A < 640 620 < 600 + 15 < 640 20 < 15 < 40 4 8 0. [ 9/100] . sin 2 = 2 sin cos , sin = : 1 cos 2 1 + cos 2 , cos = , 2 2

() = sin 2 (1 cos 2) 2(1 + cos ) = sin 2 cos 2 3. sin 2 cos 2 = 1 + (1) = 2, sin = 1 2 2 cos = = . = , 2 2 4 2 sin 2 cos 2 = 2 sin 2 () = sin 2 cos 2 = 2 sin 2 4 3. 4 1 2 =

) . [ 6/100] . 1 sin 2 1, 4

2 3 () 2 + 3. 2 + 3 () = 2 + 3 . (1999) http://gordongr.blogspot.com sin 2 = 1 4 2 = + 2 4 2 3 2 = + 2 4 3 = + , 8

22

1. = 2 [0, ]. 4 [ 10]

) () +

. = 2 4 2 sin 2 3 2 sin 2 + = 2 4 4 sin 2 + sin 2 + = 1 4 4 2 2 2 + 2 + 4 4 cos 4 4 =1 2 sin 2 2 2 sin cos 2 = 1 4 () + 2 cos 2 = 1 2 cos 2 = 2 5 cos 2 = cos 4 5 2 = + 2, 4 5 = + , 8 3 5 = , = . 8 8

4: . A. 10 , 6 . [ 9/100] . ( ) = 3 = 10. = 10 3 = 30. - = 30 3 . , = 30 3 . (1999) http://gordongr.blogspot.com

1.3. 2001

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B. , 3 10 . B.1. 20 . [ 8/100] . ( ) = = 30 3 = 3 10. = 30 3 3 10 = 3 10(3 3 1) = 3 10 26, - = 3 10 26 3 10( 1) = 3 10 (27 ). 20 = 3 10 7. B.2. ; [ 8/100] . = 0 3 10 (27 ) = 0 = 27,

27 .

1.3

2001

1: .1. + + + + = 0, . 0 , . [ 6,5/100] (1999) http://gordongr.blogspot.com

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1.

. , , = ()( + + + ). . , |()( + + + ) | . .2. : () . () () () , : . () = ( )() + 1 . () = ( )P() . () . () = 0. [ 6/100] . . .1. , . . 3 5 + 6 = 0 4. . 4 + 5 + 7 + 4 = 0 2. . 6 3 + 2 + 2 = 0 3. [ 6/100] . ) , ) , ) . .2. : P() = (4 + 5) + : . + 1 . 1 5 . . + . 4 [ 6,5/100] . . (1999) http://gordongr.blogspot.com

1.3. 2001

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2: 5 cos 2 14 cos 7 = 0. 3 . cos = . 5 [ 10/100] . cos 2 = 2 cos 1, 5 cos 2 14 cos 7 = 0 5(2 cos 1) 14 cos 7 = 0 10 cos 14 cos 12 = 0 5 cos 7 cos 6 = 0.

cos [1, 1], = 49 + 120 = 169 (cos ), = 3 cos = . 5 3 . , 2 sin 2, cos 2 tan 2. [ 15/100] . , 3 , 2 7 13 2 > 1, = , 10

3 4 sin = 1 cos = 1 = . 5 5 3 24 4 sin 2 = 2 sin cos = 2 = , 5 5 25 3 7 cos 2 = 2 cos 1 = 2 1 = 5 25 24 sin 2 = = . tan 2 = cos 2 7


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1.

3: ( ) = log 125 = log 5. . . [ 8/100] . = + 2 = 2 = log 125 2 log 5 = 3 log 5 2 log 5 = log 5 = .

. = + + + . [ 8/100] . = + + + = ( + + + ) ( + + + ) 29 20 = (2 + 28) (2 + 19) 2 2 = (29 20) + (29 14 10 19) = 9 + 216 = (9 + 216) log 5 = 225 log 5.

. ( ) = = , . = + + + + + . [ 9/100] . ( ) = = log 5 = + 2 = = = = 2.


1.3. 2001

27

2001 , . , 1000 ( ), . ( ) = = 2, = 2 = 4 = 2 + 1. ( 1000, 2001 = 2 + 1 = 1000.) , : S = ( ) 1 4 1 4 1 = ln 5 = ln 5 . 1 41 3

4: Q() ( ), . 300. 000 , 6 . ln Q() = + , 0 , , : . Q() = 3 4 , 0. [ 10/100] 1 3 . , Q(0) = 3 Q = . 2 2 ln Q(0) = 0 + = ln 3 1 1 ln Q = + 2 2 , ln Q() = ln 4 + ln 3 ln Q() = ln(4 3) Q() = 3 4 . 3 + ln 3 = ln 2 2 + ln 3 = ln 3 ln 2 2 = ln 2 = ln 4. 2


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1.

. 1/16 . [ 8/100] . Q() = 1 Q(0) 16 . . 1/9 . [ 7/100] . 1 Q() Q(0) 9 ln 3 . ln 4 3 4 4 1 3 1 3 3 4 = 4 = 4 = 2, 3 16

1 3 ln 4 ln 3 2 ln 3 2 ln 4 ln 3 . ln 4 ln 4 ln

1.4 2002 1: . P() = . = P(). [ 9/100] (1999) http://gordongr.blogspot.com

1.4. 2002

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. , (), () P() = ( ) () + (), () deg(()) < deg( ). , P() = 0 = () = 0. = P(). , deg(()) < 1, deg(()) = 0, . = P() = ( ) () + = . . , . . = ln = , > 0. . > 0 1, , > 0 : log ( ) = log log . . tan 2 = . sin 2 = 2 tan . 1 + tan 2 1 cos 2 . 2 tan + tan . 1 tan tan [ 10/100] . ) , ) , ) , ) , ) . . : . ; . ; [ 6/100] . , , , . (1999) http://gordongr.blogspot.com

. tan( ) =

30

1.

2: , = cos 2, = cos , = 1, 0 < < . 2 . , , . [ 7/100] . cos 2 = 2 cos 1, 2 = + , , , . . . [ 8/100] . = = 1 cos = sin . . . [ 10/100] . S = = = = = 5 2 + (5 1) 2 5 (2 cos 2 + 4 sin ) 2 5 (4 cos 1 + 4 sin ) 2 5 (4 1) 2 15 . 2

3: P() = ( + ) + + 1. (1999) http://gordongr.blogspot.com

1.4. 2002 1 . P = 7 P(1) = 23, = 6 = 5. 2

31

[ 8/100] . 1 P = 7 2 P(1) = 23 ( + ) + 1 = 23 2 2 + 1 = 23 + = 11.()

1 1 1 ( + ) + 1 = 7 8 4 2 2( + ) + 4 + 8 = 56 3 + 6 = 48 (1.3)

(1.4)

(1.4) 3 (1.3) 3 = 15 = 5 (1.3) 5 = 11 = 6. . P(), = 6 = 5, 2 + 1 P() . [ 8/100] . , P() = 6 + 11 5 + 1. Horner : 6 11 5 1 3 7 6 6 14 12 7 1 P() = (6 + 14 12) + 7 2 P() = (2 1) (3 + 7 6) + 7.


32

1.

. P() > 7 = 6 = 5. [ 9/100] . 1 P() > 7 (2 1) (3 + 7 6) > 0 2 1 < 0 < , 2 = 27 < 0, 3 + 7 6 < 0. 1 , . 2

4: () = ln 1 . + 5

. (). [ 5/100] . 1 >0 + 5 1 > 0, , + 5 > 0 > 1 ( ) > 1 > > 0.

D = = (0, +). + . () = 2 ln 2. [ 10/100] . () = 2 ln 2 ln 1 = ln 4 + 5 1 =4 + 5 1 = 4 + 5 4 21 = 0 = 3 () = 7 () = ln 7.


1.5. 2003 . () > 0.

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[ 10/100] . () < 0 1 ln > ln 1 + 5 1 >1 + 5 1 > + 5 6 > 0 ( ) 3 + 2 6 > 0 ( 3) + 2( 3) > 0 ( + 2)( 3) > 0 < 2 () > 3 () > ln 3.

(ln 3, +).

1.5

2003

1: . - = + ( 1). [ 7/100] . : : = 1 = . : , = + ( 1). + 1, + = + ( + 1 1) = + . : + = + = + ( 1) + = + . , . (1999) http://gordongr.blogspot.com

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1.

. . = , : . = , . = , . = . [ 3/100] . . . . S ( ) 1 , : . S = 1 , 1 . S = 1 1 , . S = . 1 1 [ 3/100] . . . : 2 : 2 tan 2 tan tan . tan 2 = . , . tan 2 = , . tan 2 = 1 tan 1 + tan 1 tan [ 3/100] . . . : . . . , , , = . . 1, () = . [ 9/100] . ) , ) , ) (0, +). (1999) http://gordongr.blogspot.com

1.5. 2003

35

2: : cos (sin 2 + 4 sin ) = (cos 2 + 4 cos + 1) sin [ 12/100] cos 2 + 4 cos + 1 = 0. [ 13/100] . cos 2 = 2 cos 1 sin 2 = 2 sin cos , : cos (sin 2 + 4 sin ) = = = = cos 2 + 4 cos + 1 = 0 2 cos + 4 cos = 0 2 cos (cos + 2) = 0 cos = 0 cos = 2 cos = 0 = + , . 2 cos (2 sin cos + 4 sin ) cos (2 cos + 4) sin (2 cos + 4 cos ) sin (cos 2 + 4 cos + 1) sin .

3: , = 11 + 2, P() = 3 + 3. . ( ) = 9 = 2. [ 9/100] . + = 11 + 2( + 1) + 11 2 = 2, . , ( ) = 11 2 1 = 9 = 2. (1999) http://gordongr.blogspot.com

36

1.

. S = + + , , , . [ 7/100] . S = ( + + ) ( + + ) 21 11 = (2 + 20) (2 + 10) 2 2 = (21 11) + (10 21 5 11) = 10(9) + (210 55)2 = 220.

. P() = 0 . [ 9/100] . P() = ( 3) ( 3) = ( 3)( 1) = ( + 1)( 1)( 3) = 1, = 1 = 3, = = 2 ( ).

4: () = ln( 2 + 3) () = ln 3 + ln( 1). . () (). [ 6/100] . 2 +3 > 0 ( 1)( 2) > 0 < 1 > 2 < 0 > ln 2, D = (, 0) (ln 2, +). 1 > 0 > > 0, D = (0, +). (1999) http://gordongr.blogspot.com

1.6. 2003 . () = ().

37

[ 10/100] . D D = (ln 2, +) () = () ln( 2 + 3) = ln 3 + ln( 1) ln( 2 + 3) = ln 3( 1) 2 + 3 = 3 3 5 + 6 = 0 = 2 = 3 = ln 2 (.) = ln 3 (.) = ln 3.

. () > 2(). [ 9/100] . D D = (ln 2, +) () > 2() ln( 2 + 3) > 2 ln 3 + 2 ln( 1) ln( 2 + 3) > ln 3 ( 1) 2 + 3 > 9( 1) 2 + 3 > 9 18 + 9 4 8 + 3 = 0 1 3 < < 2 2 1 3 ln < < ln . 2 2

4 3 ln < ln = ln 2, 2 2 (ln 2, +) . .

1.6

2003

1: 1. , , , = . (1999) http://gordongr.blogspot.com

38

1. [ 9/100]

. , , = = = = .

2. : > 0 1, , , > 0 , : . log ( + ) = log + log , . log = log log , [ 6/100] . ) , ) , ) . 3. : cos 2 : . cos 2 = cos sin , . cos 2 = 1 2 cos , . cos 2 = 2 sin 1. [ 4/100] . . 4. : . () = + + + + 0, (). . , . [ 6/100] . ) , ) . (1999) http://gordongr.blogspot.com

. log = log .

1.6. 2003

39

2: () = ( + 1) cos(), . . () 3 4, 1 = 2 = . 2 [ 13/100] . max = + 1, + 1 = 3 = 2. 2 2 2 1 = = , = 4 = . 2 1 3 . = 2 = , () = . 2 2 [ 12/100] . = 2, = 1 2 () = 3 cos 2 () = 3 2 3 3 cos = 2 2 1 cos = 2 2 cos = cos 2 3 = + 2 2 3 2 = + 4, . 3

3: () , + 1, 0 2.


40

1.

. () = + . [ 14/100] . P() +1 1, () = + , , . , P() = ( + 1)( + ) + 0 = + + + . 0 , P(0) = 0 0 + 0 + 0 + = 0 2, + + + = 2 = 1. P() = + .

. () 2 + () 2 + () > 2. [ 11/100] . = P() 2 = + 2, : () 2 + () 2 + () > 2 + + > 0 ( + + 1) > 0 > 0, , + + 1 > 0 + 2 > 0 1 + 1 > 0 ( 1)( + + 1) + 1 > 0 ( 1)( + + 2) > 0 1 > 0, , + + 2 > 0 > 1.


1.6. 2003

41

4: 1. log(3 2 1), log(4 2 1), log(8 2 2) , ; [ 13/100] . log(3 2 1), log(4 2 1), log(8 2 2) 2 log(4 2 1) = log(3 2 1) + log(8 2 2) log (4 2 1) = log (3 2 1) (8 2 2) (4 2 1) = (3 2 1) (8 2 2) 16 2 8 2 + 1 = 24 2 14 2 + 2 8 2 6 2 + 1 = 0 1 1 62 1 1 2 = 2 = , = 36 32 = 4, (2 ), = = , 2 4 16 2 4 2 = 2 2 = 2 = 1, = 2.

2. = log 2, . [ 12/100] . = log(8 2 2) log(4 2 1) = log = + 3 = log 2 3 log 2 = 4 log 2. 2(4 2 1) = log 2. 4 2 1


42

1.

1.7 2004 1: . > 0 1, > 0 , : log = log . [ 9/100] . log = log = . = = . ( ) = = = . log = log . . . ) , log log = . log ) ( ) + S = . 2 ) () () (), () () , () (). ) , , , = . [ 4/100] . ) , ) , ) , ) . . , . . = > 0, .

. = > 0 > 0 1. . log = > 0 1 . [ 6] . ) , ) , ) . (1999) http://gordongr.blogspot.com

1.7. 2004

43

. sin cos . sin cos 0, , , , 2

[ 6/100] . : sin cos 0, , , , 2

2: . sin 2 3 cos = 0. [ 13/100] . : sin 2 3 cos = 0 2 sin cos 3 cos = 0 (2 sin 3) cos = 0 3 cos = 0 2 sin = sin cos = cos 3 = 2 + = 2 + = 2 , , 3 3 2 = 2 + = 2 + = 2 , , . 3 3 sin =


44

1.

. 1 cos 2 = tan 2 sin + sin 2 2 . [ 12/100] . 1 cos 2 1 1 + 2 sin = 2 sin + sin 2 2 sin + 2 sin cos 2 sin = 2 sin (1 + cos ) sin = 1 + cos 2 sin cos = 2 cos sin = cos = tan . 2

3: () = 8 + (5 1) + 8 3 6, . . () 1 . [ 9/100] . 1 = ( 1)( + 1), , Horner . : 1 8 5 1 8 1 7 5 8 1 7 5 8 5 3 6 1 5 2 6

P() = ( 1)( 7 + (5 8) + 5) + 2 6. :


1.7. 2004 1 7 1 1 8 5 8 8 5 5 5 0 1

45

7 + (5 8) + 5 = ( + 1)( + 8 + 5). , P() = ( 1)( + 8 + 5) + 2 6.

. , . [ 4/100] . 2 6 = 0 = 3. . = 3, () = 0, () . [ 12/100] . = 3 P() = ( 1)( 8 + 15) = ( 1)( + 1)( 3)( 5). , P() = 0 1, 1, 3, 5 P() < 0 (1, 2) (3, 5).

4: . () = 1 1 2 + 3 1. 5 5 [ 13/100] . 1 1 2 + 3 1 0, 5 5 (1999) http://gordongr.blogspot.com

46

1.

2 + 3 1 0 1 = > 0. = 1 1 2, 5 1 2 1 1 2 5 1 1 5 2 log 1 log 5 log 0 log 2. 1 2

D = [ log 2, 0]. . () = 5 . : () + ( + 1) + ( + 2) + + ( + 49) = 125(5 1) . 4 [ 12/100] . 5 S, S 50 = 1 = 5. 5 1 S= = . 1 4 , 5 1 125(5 1 = 4 4 5 = 125 5 = 5 = 3. 5


2 2.1 1999

1: . A . .

) , .

[ 5/100] .

= = .

) .

[ 4/100] 47

48 .

2.

= //, A B = , , // A = . = E, . B ) = . [ 3,5/100] . () : = = = = = . A + A + A A

. ) A. = 3, = 6 = 10, :

. 3, . 6,

. 4, . 5, . 7 [ 6,5/100]

. () 10 6 = = = 5 A 3 . (1999) http://gordongr.blogspot.com

2.1. 1999

49

) A. = 4, = 6 = 8, :

. = 1 = 5, . = 5 = 1, . = 3 = 3, . = 2 = 4, . = 4 = 2.

[ 6/100] . , () 8 8 + 4 12 6 = = = = = 2. A 4 4 4 = 2 .

2: , . = 9, = 4 = 5, :


50

2.

) . [ 15/100] . , , = (A + A) = ( + ) (A + 9) = 4(5 + 4) + 9 36 = 0.

= 81 + 4 36 = 225 , = = 3. ) : . 9, . 5, . 4, . 3, . 6. [ 10/100] . , = = 4(5 + 4) = 6, . 9 15 12 < 0, = 3 > 0, 2

3: . :


2.1. 1999 ) 1.

51

[ 15/100] . () = () = 1 1 1 A A sin A = A AB sin A, 2 2 2 1 1 1 AK AB sin A = A AB sin A 2 2 2

() = (), 1. ) , .

[ 10/100] . () = () + () () = () + (), () () = ().

4: ) 402 . ( ) 25 . , .

[ 8/100] (1999) http://gordongr.blogspot.com

52

2.

. = () = = (402) = 3200. = 25 625 = . 4 4

4 = 3200 4 625 = 3200 625. 4

) , , . , ; [ 5/100] . (, R) R , . = 25 + 2R + 25 = 2R + 50 = + = 2 (402) = 80. 2R + 50 = 80 R = 15. ) (); [ 5/100] . (O, R) = 15 = 225. 4 = 3200 625 225 = 3200 400.


2.2. 2000

53

) , , ; [ 7/100] . , (, ) . , = + . = = 202 = = 202 25, 2 = (202) + (202 25) = (202) (202 25)(202) + (202 25) = 25402 25. = 5402 25.

2.2

2000

1: 1. , ,

+ = 2 +

. 2 [ 10/100]

. , , > . < 90 < . < 90, = + 2 . > 90, = + + 2 . , = , . (1999) http://gordongr.blogspot.com

54

2.

2. < :

= . [ 2,5/100] . = 2 . . 1 2. 1. = 8, = 6 = 5. :

. 7 . 4 . 10

. 9 . 11 [ 6,5/100]

. . 2. = 4, = 7, = 5, . :

. 4

. 8 . 8/3 . 5 . 3 [ 6/100]

. .

2o: //, < , A = = 90, = 4, = 3, = 5. : (1999) http://gordongr.blogspot.com

2.2. 2000 ) .

55

[ 9/100] . . , = = 3. , = + , = 5 3 = 4.

) . [ 9] . () = + 4+8 = 3 = 18. 2 2

)

.

[ 7/100] . () = () + () = + = 12. 2 2

3: (, R) = 15. :


56

2.

) R . [ 6/100] . (, R), 15 = 53. = = R3, R = 3

) (O, R). [ 6/100] . E(O, R) = R = 75. ) .

[ 6/100] . + R 3R R = 4 2 = R 4

=

3 R 33 75 3 . () = = = R3 = 2 2 2 4


2.2. 2000

57

) . [ 7/100] . E = E(O, R) () = 75 33 75 75(4 33) = . 4 4

4o: (O, R) . , , , . = , . : ) = . [ 6/100] . , , = . , , = = 90, , = = R, . = , = .


58

2.

) = = R . [ 6/100] . , , = . = ( + )( ) = (R + )(R ) = R . () = = ( )( + ) = R . = = R . ) + = 2(R + ). [ 5/100] . ,

+ = 2 + = 2( + ) = 2(R + ). 2

)

R + + = 2 . R O [ 8/100]

. () () 2 R + + = = + = + . R O

, = = = = 2 , () ( )( + ) R , ()

R + +1= + . = R O


2.3. 2001

59

2.3

2001

1: 1. , , , . [ 6,5/100] . A = 90 . , :

= = 90 A = , = 90 B = 90 B . , , ,

AB B A = = = . A A

2. , : (A = 90) . . . . 1. + 2. 3. 4. 5. 6. [ 6/100] . -4, -5, -2. (1999) http://gordongr.blogspot.com

60

2.

. 1 2: (A = 90) , = 1 = 3. 1. : . 2 . 3 . 2 . 32 [ 6,5/100] . . 2. : . 3 . 3 . 2 . 5 [ 6/100] . .

2: = 6, = 12 = 8. . . [ 7/100] . = 144 > 100 = + , > 90. . . [ 9/100] . + = 2 + , 2 = 2 + 2 2 64 + 2 36 144 = = 14. 4 4


2.3. 2001

61

. . [ 9/100] . = 2 , .

=

7 = . 2 6

3: O, O, O O = O = 150. O, O, O , , = 2, = 4 = 6. . () .

[ 12/100] . O = 360 O O = 60, () = 1 1 sin O = 2 6 sin 60 = 33. 2 2 (A) . (OB) [ 13/100] . () = () = 1 1 sin O = 2 4 sin 150 2 2 1 1 sin O = 4 6 sin 150 2 2 (A) 1 = . (OB) 3 (1999) http://gordongr.blogspot.com

.

62

2.

4: = 2R. , , = 2R. . .

. = 22R . [ 5/100] . . ,

= = (R + 2R) R = 8R = 22R. (1999) http://gordongr.blogspot.com

2.3. 2001 . = .

63

[ 10/100] .

A = E = 90 = (). ,

= = , = .

. R. [ 5/100] . () = 4R 3R = = 32R. 22R

. R. [ 5/100] . . + = (). = R 2 = = (32R) (4R) = 2R , = R2, () = 1 = 22R . 2 R 42 , = 22R = R. 2 2 (1999) http://gordongr.blogspot.com

64

2.

2.4 2002 1: 1. . [ 10/100] . , > . , . K, . :

() = () + () + () 1 1 = + + 2 2 1 1 1 = + ( + ) + 2 2 2 1 = ( + + + ) 2 1 = ( + ) . 2

2. , . . (, R), (, R) > 0, (, R) (, R). [ 1/100] .

< + A < 90. (1999) http://gordongr.blogspot.com

2.4. 2002

65 [ 1/100]

1 . E = sin . 2 [ 1/100] . (, R), R = . 180 [ 1/100] . : + = 2 + . 2

[ 1/100] . ) , ) , ) , ) , ) . . . (, R) = R, . [ 6/100]

. , , , , , , A , = = = = = A A = R. , A OA+ ( = 1, , 6) A A+ = R, A A+ = OA = OA+ . , A OA+ = 60, A = 6 60 = 360, A + = 60. A A . A A . (1999) http://gordongr.blogspot.com

66

2.

, A OA+ A A+ , . , , . , , = R. . = R3 , . 2 [ 4/100] . + 3R R3 = R = = . 2 4 2

2: , , = . 2 = 2

. = + . [ 15/100]

. + = 2 + 2 = + = + = + = + . 2 2 + 2 . 2

2


2.4. 2002 . A.

67

[ 10/100] . < + < 90. . .. = + 2 . , 2 = . A = 60.

3: , (, R) , 30. . , .

: . = R. [ 5/100] . , A , . B = 30, = = R. 2


68 .

2. (PB) = 4. (PA) [ 10]

. B = = 30, = 90 = 90 (90 30) = 30 = = 90. () = = 2. = = () = 4.

. =

2R3 . 3 [ 10/100]

1 . = 30, = = R. 2 1 2R3 = + = = R = . 4 3

4: , 7 cm, , = = = = 3 cm.


2.5. 2003 . .

69

[ 5/100] . B, = EB + ZB = 3 + 4 = 5. () = = 25 . . (, ) = 1 cm.

[ 12/100] 1 1 . () = EB EZ = 4 3 = 6 . , () = = 2 2 + + 4+3+5 = = 6. 6 = 6 = 1 cm. 2 2 . (, R) , (, R) (, ). [ 8/100] 5 . R , R = . 2 R 25 E(K, R) = = . E(, ) 4

2.5

2003

1: . (, R). . (, R) . [ 4/100] . , . ( ) ( ), ( ) ( ). (, R). (1999) http://gordongr.blogspot.com

70

2.

. = R2, . [ 4/100] . , ,

= + = 2R = R2. , = R2. . = R2 , . 2 [ 4/100] . R R2 + = R = = . 4 2 2

. , . . . [ 2/100] (1999) http://gordongr.blogspot.com

2.5. 2003

71

. . [ 2/100] . (, R) : (O, R) = R + O . [ 2/100] . . [ 2/100] . ) , ) , ) , ) . . ; [ 5/100] . .

2: = = 1 = 3. :

. A. [ 9/100] . , A. = = 1 1 3 1 = = . 2 4 2

= 2. = 60, = 120. (1999) http://gordongr.blogspot.com

72

2.

. .

[ 9/100] . () = 1 1 3 1 = 3 = . 2 2 2 4

. = . [ 7/100] . + = 2 + 7 = . 2 2

3: , , , + = 3 . :


2.5. 2003 . .

73

[ 12/100] . : + = 2 + 5 5 = = . 2 4 2

. =

3 . 2 [ 13/100]

. , , : = : = ( + ) 5 5 5 + 2 2 10 1 5 1 + = 2 5 3 . = 2 = 5 5 ME = = . 2 2 2 10


74

2.

4: , . , , , , ,

1 = = = , 3 .

: . .

[ 9/100] . () = 1 sin A 2 1 2 = sin 60 2 3 3 1 2 3 = 2 3 3 2 3 . = 18

. .

[ 7/100] . ( ), ,


2.6. 2003 = = = = = = 60 = = = 2 . 3 3 . , 18 3

75

() = ,

1 3 sin = = . 2 4

() = () 3 () 3 3 3 4 18 3 = . 12 =

. .

[ 9/100] . (O, R), 3 = R3 = R3 R = . 3

= R =

3 = . 3 3

2.6

2003

1o: . , . [ 11/100] (1999) http://gordongr.blogspot.com

76

2.

. , A = 90, . : A = B = 90

B , , . ,

= = = = .

. , . . :

> + A < 1. [ 2/100] . , . [ 2/100] . , , , E= 4 .

[ 2/100] . , . [ 2/100] . ) , ) , ) , ) . (1999) http://gordongr.blogspot.com

2.6. 2003

77

. . . R . R . R R 1. 180 2. 2R 3. R R 4. . 360 [ 6/100] . -3, -4, -2.

2: = 23, = 1 = 30.

. = 7. [ 13/100] . . = 30, = 2, = 3. , , ,

= + = = 3. , < 90

= + 2 = 23) + 1 2 1 3 = 7, AB = 7.


78

2.

. = . [ 12/100] .

+ = 2 + 12 + 1 = 2 + = 13 7 13 = . 7 7 2

2

3: (, R) , OA = R13. . = 2, : . , R. [ 13/100] . , = OA R = 13R R = 12R (,) . = + . , . , : = ( )( + ) = 2 R13 RR13 + R = 2 R13) R = 2 12R = 2 = R6.


2.6. 2003

79

.

(O) . () [ 12/100]

. . , . :

(O) = ()

OK = = 2. OK

4: (, R) . , , .


80

2.

. = R3. [ 8/100] R . 2 R = 90, = + , MB = R = 2 3R . 2 3R . M = 2 . . = = = MB + = 3R.

. , R, . [ 8/100] . AMB = 90 = + R 3R = + = R = R. M = 90 4 4 9R 3R = + = + = 3R = R3. 4 4 , ,

() =

1 1 R 3 = R R3 = . 2 2 2

, , = R, = R3 () = R 3 . 2 () = () + () = R 3.

. , R, . [ 9/100] (1999) http://gordongr.blogspot.com

2.7. 2004

81

AM R/2 1 . = = , = 30, AB R 2 = 60. . , , , = 120. = R 120 R = . 360 3

T , = R,

() =

1 1 R3 3R = R = . 2 2 2 4

R 3R 3 . = () = = R 3 4 3 4

2.7

2004

1: . N . [ 11/100] . = = = = . , , = , = . = , () = ().


82 ,

2.

() = () + () 1 1 = + 2 2 1 = 2 2 = .

. . . . . 1. = 2. = 3. = (+) 4. = P 5. E = . [ 6/100] . -3, -1, -4. . , . .

.

= + 2. . - R : + = R . 2 . , . . , . [ 8/100] . ) , ) , ) , ) . (1999) http://gordongr.blogspot.com

2.7. 2004

83

2: R. = 150, : . . [ 10/100] . = 180 360 , 360 = 12.

150 180 =

. . . [ 8/100] . = 360 , = 360 = 30. 12

. R. [ 7/100] 1 1 . E = P = , 2 2 = 1 1 12 R R = 3R . 2 2

3: , = 2, = 1 + 2 2 () = . 4


84

2.

. = 3. [ 9/100] . .

= + 2 = , () = 2 (1 + 2)2 = , 4 2

+ 7 + 22 2 + 2

.

1 (1 + 2)2 1 () = = = 2. 2 2 2 ,

+ =

+ 7 + 22 2 + 2

+ 2 = 4 = 3.

. R . [ 8/100] . R , () = 6 E= . 4R 2


2.7. 2004

85

. . [ 8/100] . . = + 2 (2 2)3 . 3

4: (A = 90) = R = R3. (, R) (, R3).

N : . R. [ 4/100] . , = + = R + 3R = 4R = 2R.


86

2.

. B . [ 4/100] . = 2, = 30 = 60.

. R. [ 8/100] . : = = R, = = R3 . , = , () = (). ,

() = () + () = 2 () = = R R3 = R 3.

. R. [ 9/100] . , (, ) (, ). = R 120 R = 360 3

(R3) 60 R = . 360 2 , = E = E () + () = + () R R + R 3 = 3 2 5 = R 3. 6


3 3.1 1999

1: . = ( , ) = ( , ) O.

) ( ) . [ 3/100] . = + .

) , , , , = 1. [ 5,5/100] 87

3. 88 . = , = , = 0 + = 0 = 1 = 1 = 1.

) , , + cos = . + + [ 4/100] . | = || | cos cos = . || | |

|| = + , | = + = + . | + cos = . + +

. ) = (, 1), = (4, ) 0. ; . = 1, . = 3, . = 2, . = 2, . = 3. . [ 6,5/100] . = 0 4 + ( 1) = 0 ( + 3) = 0 = 3, . (1999) http://gordongr.blogspot.com

3.1. 1999

89

) = (1, 3), = (2, 23), = (3, 1). . 1. 2. 3. . /2 . /6 . /4 . 2/3 . 3/4 . /3

. [ 6/100] . 1-, 2-, 3-.

2: o = 2 + 2 = 6 + 7, . : ) 3 . [ 9/100] . : = (3, ) |3 | |6 + 6 |6 + 7 |1 = 1.

, 3 . ) 2 10 2. [ 8/100] . 2 = 12 + 14 2 2 = 10 + 12 = 10( + 1) + 2, 2. (1999) http://gordongr.blogspot.com

90

3.

) 7, + 2 7. [ 8/100] . = 7, . + 2 = 8 + 7 = 7(8 + 1) = 7.

3: (8, 0) (0, 4) . ) . [ 9/100] . , = 1 = 2 + + = (4, 2). , 2 2

, 1 = . 2 ) . [ 9/100] . = + . = 2 = 2 + 4 = 10. = 2 + 10. 1 = 2

) . MA + MB = 2OM . [ 7/100] (1999) http://gordongr.blogspot.com

3.1. 1999 . (, 2 + 10), MA + MB

91

= ( ) + ( ) + ( ) + ( ) = (8 ) + (2 10) + + (2 6) = 10 80 + 200 = 2 + (4 40 + 100) = 2 + (2 10) = 2OM .

4: 1999 . = 1, 2, 3, , 1999 : ( 1) + = 2( + 1). : ) . [ 9/100] . + (1 + 2) 2 + 1 2 = 0 + + + + = 0 = (1 + 2), = 2 = 1 2. + 4 = (1 + 2) + (2) 4(1 2) = 8 + 12 3 17 > 0 1. , , = 8 + 12 3 , . 2 1 + 2 , 2

) ( ). ; [ 8/100] (1999) http://gordongr.blogspot.com

92

3.

. ( 1) + = 0 , + 1 = 0

. 2 = 0 = 0 = 1. (1, 0). ) + 1 = 0 . [ 8/100] . . ( 1) + = 2( + 1) + 1 = 0 ( 1) + = 0 + 1 = 0

(), .

3.2 2000 1o: .1. ( , ) . [ 2/100] . ( ) + ( ) = . .2. + + A + B + = 0 ; ; [ 4,5/100] . + + A + B + = 0 , A B + 4 > 0. A , 2 2 + 4 = . 2 (1999) http://gordongr.blogspot.com

3.2. 2000

93

.3. C + = ( , ) + = . [ 6/100] . = + . = + = . C + ( + ) = ( + 1) + 2 + = 0. (3.2) (3.2) , . , =0 (.)

(3.1)

4 4( + 1)( ) = 0 + + = 0 ( ) + = 0 ( ) + 2 + + 2 + ( + ) = 0 = . + = .

(3.1) =

= + = + + = .

.1. : + = 10 (1, 3). : A. + 3 = 10, B. 5 = 8, . 3 = 10, . 3 + 2 = 3, 1 E. + = 5. 2 [ 4/100]

. . (1999) http://gordongr.blogspot.com

94

3.

B.2. . . . + 6 + 4 3 = 0 . + ( + 1) = 4 1. (0, 1), = 2 2. (3, 2), = 1 3. (3, 2), = 4 [ 4/100] . -3, -1. .3. , . . (1, 1) + = 2. . + = 4 = 2 . . + + = 0, , . [ 4,5/100] . ) , ) , ) .

2: = 6+ 0 < 6 . : ) 2 3 = 6 + 1 = 6 + 5, . [ 10/100] . , : (i) = 0, = 6, 2 3, . (ii) = 1, = 6 + 1 . (iii) = 2, = 6 + 2 = 2(3 + 1), 2, . (1999) http://gordongr.blogspot.com

3.2. 2000

95

(iv) = 3, = 6 + 3 = 3(2 + 1), 3, . (v) = 4, = 6 + 4 = 2(3 + 2), 2, . (vi) = 5, = 6 + 5 .

) () : = 3 + 1, . [ 10/100] . () : (ii) = 1, = 6 + 1 = (6 + 1) = 3(12 + 4) + 1 = 3 + 1 = 12 + 4 . (vi) = 5, = 6 + 5 = (6 + 5) = 3(12 + 20 + 8) + 1 = 3 + 1 = 12 + 20 + 8 .

) () 3. [ 5/100] . , () , (ii) (vi), = 3 + 1 = 3 + 1, , , = 3( ) = 3, = .

3: , 2 + 3 = (4, 2) 3 = (7, 8). (1999) http://gordongr.blogspot.com

96

3. [ 7/100]

) = (1, 2) = (2, 2). . 3 = (3, 6) = (1, 2) (1, 2) 3 = (7, 8) = (2, 2).

) , + 2 + 3 . [ 8/100] . || = (1) + 2 = 5, | = 2 + (2) = 8 = | 1 2 + 2(2) = 6. + + 3 2 ( + + 3 = 0 )(2 ) 2|| + 3| + (2 + 3) = 0 | 10 + 24 6(2 + 3) 0 3 = . 4

) = (3, 1) , . [ 10/100] . = + // . = (, 2), . , = ( , ) = 0 + 2 = 0 = 2 . + 2 = 3 = + (3, 1) = ( + 2 , 2 + ) . 2 + = 1 1 7 = = , 5 5 14 1 2 14 7 = . = , = , . 5 5 5 5 5 (1999) http://gordongr.blogspot.com

3.2. 2000

97

4: , ( 1) + (+1)3 = 0, , . ) . [ 8/100] . ( ) ( + 1) + ( + 3) = 0, , + 1 = 0 + 3 = 0, (1, 2). ) (2, 2), (1, 5) (1, 3). , . [ 4,5/100] . : (2, 2) ( 1) 2 + ( + 1) 2 3 = 0 = 1, = 2. (1, 5) ( 1) (1) + ( + 1) 5 3 = 0 = 1, = 1. 1 (1, 3) ( 1) 1 + ( + 1) 3 3 = 0 = , 3 + 2 5 = 0. ) . [ 6/100] . , (1) + 2 (1) + 2 5 5 (K, ) < (, ). . (K, ) = (1999) http://gordongr.blogspot.com | 2 + 2 2 5| = 3 (, ) = | (1) + 2 5 5| = 6

98

3.

) . [ 6,5/100] . 1 2 1 1 2 1 1 1 1 0 3 () = | 1 5 1 | = | 0 3 0 | = | | = 3. 2 2 2 2 1 1 3 1 2 1 0

3.3 2001 1o: .1. , , . : . |, | . [ 4/100] . | |, |( + ). [ 4/100] . ) | ( )[ = ] ( )[() = ] |. ) | | ( )[ = ] ( )[ = ] ( + )[( + ) = + ] | + .


3.3. 2001

99

.2. : , 0. : . (, ) < (, ) . (, ) = (, ) . (, ) > (, ) . (, ) = (, ) + 1 (, ) , . [ 4,5/100] . . .1. : 7|( + 5) 7|(40 ) : . 7|( + ), . 7|( + + 1), . 7|( + + 2), . 7|( + 3). [ 4/100] . . .2. 72 112. [ 4,5/100] . ( ): 112 72 40 32 = = = = 1 72 + 40, 1 40 + 32, 1 32 + 8, 4 8 + 0.

, (112, 72) = (72, 40) = (40, 32) = 8 (5, 4) = 8. (1999) http://gordongr.blogspot.com

100

3.

.3. 72 112 72 112. [ 4/100] . : 8 = = = = 40 32 40 (72 40) = 72 + 2 40 72 + 2(112 72) 3 72 + 2 112.

2: , || = 1, | = 2 , = | = 2 + 3 = 2 : , . . . [ 5/100] 1 | . = || | cos , = 1 2 cos = 1 2 = 1. 3 2 . ||, || . [ 8/100] . || = |2 + 3 | = |2 + 3 | = 4|| + 12 + 9| | = 4 + 12 + 36 = 213 (1999) http://gordongr.blogspot.com . 3

3.3. 2001 || = | 2 | = | 2 | = || 4 + 4| | = 1 + 16 4 = 13.

101

. . [ 7/100] . = (2 + 3 ( 2 ) |) = 2|| 6| | = 2 1 24 = 23.

. . [ 5/100] . = , = || || cos = 21313 cos = 26 cos . cos = 23 . 26

3: + 6 + 9 = 0. . . [ 7/100] . ( + 3) = 0 ( + 3 )( + 3 + ) = 0 = + 3 = ( + 3) = + 3 = ( + 3). (1999) http://gordongr.blogspot.com

102

3.

. . [ 7/100] . , , = 1 = 1 1, .

. (, ) > 0 > 0 , = (3, ) = (16, 4) . [ 6/100] . : = 1 = 1 = 3 = 1 3 4 = 1 = 4. (3, 4). 16 = 1 = 3 = 1 3 4 = 1 = 1 = 4. (3, 4) , 16 . . , . [ 5/100] . = 2 > 0. , 8 4 = 2 3 = . 3 = 16 . 3

4: . + + 6 + 8 = 0, , . , , . [Mo 7/100] (1999) http://gordongr.blogspot.com

3.3. 2001

103

. + + + + = 0, = 6, = 8 = 0. + 4 = 36 + 64 > 0, (3, 4) = 3 + 16 . , , = 0, = 0 . . , 3 + 2 = 0. . , + + 6 + 8 = 0 , . [ 6/100] . = 3 = 4, + 2 = 0. + 2 = 0, = = 0 . . , , , + + 2 = 0, OA OB = 0. [ 6/100] . : + 6 + + 8 = 0 + 6 + ( + 2) + 8( 2) = 0 + (3 4 + 2) + 2 8 = 0

(3.3)

+ = (3 4 + 2) = 2 8. , : OA OB = 0 + = 0 + ( 2)( 2) = 0 2 + 2( + ) + 4 = 0 16 + 4 2(3 4 + 2) + 4 = 0 2(3 + 4 2) = 0 3 + 4 2 = 0.


104

3.

3 + 2 = 0 = 1 2 = . 3 . , () . [ 6/100] . , , (3.3) 4 6 = 0. = 16 + 24 = 40 , = 4 210 = 2 10. 2

, = 4 10.

0 0 1 1 (OAB) = | 2 + 10 4 10 1 | 2 2 10 4 + 10 1 = 1 2 + 10 4 10 | | 2 2 10 4 + 10 1 = 2 + 10 4 + 10 2 10 4 10 2 = 210.

3.4 2002 1o: ; . , [ 4/100] . . (1999) http://gordongr.blogspot.com

3.4. 2002

105

. . [ 9/100] . = ( , ), = ( , ) . OAB, (0, 0) ( , ), B( , ), OA = , O = , . |A| = |OA| + |OA| 2|OA| |O| cos . , |A| = ( ) + ( ) = + 2 + + 2 , |OA| = + , |O| = + . + 2 + + 2 = + + + 2|| | cos | || | cos = + | = + .

. , . . , . , . det(, , ) // (, = 1. ) . , , [, ] = (, ) [, ] , (, ) , . (1999) http://gordongr.blogspot.com

106

3.

. + + A + B + = 0 + 4 > 0 A B K , . 2 2 [ 8/100] . ) , ) , ) , ) . . ( , ). , . ) + = ) + = 1 ) = 2 )

1. = ( + ) 2. + = + = 1 4. + = 1 5. = 6. = 1. [ 4/100]

=1

. -2, -3, -1, -5.

2: . . [ 5/100] . = 2 + 1, = 2 + 1, , . = (2 + 1)(2 + 1) = 4 + 2 + 2 + 1 = 2(2 + + ) + 1 = 2 + 1 = 2 + + . . . , . ( + 1) 2 [ 10/100] . , ( + 1) , . 2 2 , () , + 1 + 1 ( + 1) . , . 2 2 (1999) http://gordongr.blogspot.com

3.4. 2002 . , .

107 ( + 1) 2 [ 10/100]

. = 2 + 1 , : ( + 1) (2 + 1)(4 + 4 + 2) = = (2 + 1) [2( + ) + 1] 2 2 .

3: = 4. : . . [ 6/100] . = 2 = 2. (1, 0) = 1. . 2 . 2 [ 10/100] . + = , , . 1+0= = (O, ) = 2 2 =

| 0 + 0 + | + 2 = 2 + = = .

=

2 2

= = = 0 + + 1 = 0. = = = 0 +1 = 0. (1999) http://gordongr.blogspot.com

108

3.

. = 1. [ 9/100] . ( , ) , = 2( + ) 2 + 2 = 0, 2 = . = 1 1, 2 = 1 = 2. , , = 4 4 = 4 = 1. (1, 2) + 1 = 0.

4: + 2 cos 2 sin 1 = 0, 0 < 2. . , . [Mo 9/100] . + + + + = 0, = 2 cos , = 2 sin , = 1. + 4 = 4 cos + 4 sin + 4 = 4 + 4 = 8 > 0, K R = + 4 8 = = 2. 2 2 A B , = (cos , sin ) 2 2

. = , 2 (1, 2). [ 9/100] . , = , + ( 1) = 2, (1, 2) 2

1 + (2 1)( 1) = 2 = + 3.


3.5. 2003

109

. (0, 0) = 1. [ 7/100] . ( , ) + = cos + sin = 1. (0, 0) = 1.

3.5

2003

1o: 0 . , , = . [ 7/100] . = , , : = = = = ( + ) + + 0, .

. , . . ( ) = . [ 2/100] . + = ( , ) + = . (1999) http://gordongr.blogspot.com

110

3. [ 2/100]

. (, 0), (, 0) 2 =1 + = . [ 2/100] . , AB AB = OA OB. [ 2/100] . ) , ) , ) , ) . . . , 0, ; [ 5/100] . . ; [ 5/100] . . | ( )[ = ].

. .

2: . : (1999) http://gordongr.blogspot.com

3.5. 2003

111

. = 8 = 2 + 1 . [ 12/100] . = 2, , = (2) = 8 = 8 = . = 2 + 1, , = (2 + 1) = 8 + 12 + 6 + 1 = 2 + 1 = 4 + 6 + 3. . ( + 1) . [ 13/100] . = 2, , ( + 1) = 2(4 + 1) = 2, , = (4 + 1). = 2 + 1, , ( + 1) = (2 + 1)(4 + 4 + 2) = 2 = (2 + 1)(2 + 2 + 1).

3: , (2 1, 3 + 2), (1, 2) (2, 3) 2. . , . [ 8/100] . = 2 1 = 3 + 2 3 7 = + . 2 2 (1999) http://gordongr.blogspot.com 2 + 1 = 2 3 + 1 2 = 2 3 7 3 = + . 2 2 =

112

3.

. = 1, : .

[ 8/100] . , (1, 5) . [ 9/100] . ) = 1 (1, 5), 1 5 1 1 5 1 1 1 1 3 0 3 | 1 2 1 | = | 0 3 0 | = | | = . 1 2 2 2 2 2 2 3 1 1 2 0

() =

) 1 1 2 | 1 2 1 = 0 | = 0 + 1 = 0. 2 3 2 3 1 (1, 5) (A, ) = |1 5 + 1| 1 + 1 = 3 2

9 ( 1) + ( 5) = . 2

4: : = 2 4 + 2 = 3 , > 0. . 0, 0, 3 . 2 [ 8/100] (1999) http://gordongr.blogspot.com 3 2

3.5. 2003 .

113

+

=1

3 3 = . 4 2 3 3 < , = = , = . , 4 2 3 3 0, . 0, 2 2 = . K , , . 2 2 [ 8/100] . . = 4 + 2 = 3 ( ) + 2 3 . 2 2

= 4 4(3 ) = 16 ( ), = 2 4 3 < 0, = 2 = > 0, . 2

= , = = , K , 2 2 , . 2 . K , 2 . [ 9/100] (1999) http://gordongr.blogspot.com

114

3.

. = + + = 0 2 2 = 1. 3 4 + 2 = 3 + = 0 2 2 = 1. = 1, .

3.6 2003 1o: . = ( , ) . || = + . [ 9/100] . , = . . , = = . , = + || = + .

. , . . ( = 0. ), [ 2/100] (1999) http://gordongr.blogspot.com

3.6. 2003

115

. , . | |, = . [ 2/100] . + B + = 0 = (, ). [ 2/100] . , 0 = 2 2 = 2. [ 2/100] . ) , ) , ) , ) . . . . . + = 1, > > 0 . = 1, > 0, > 0 . = 2, > 0 . + = , > 0 1. 2. 3. 4. 5. . [ 8/100] . -5, -3, -4, -1.

2: (14, 5) (2, 1). (1999) http://gordongr.blogspot.com

116

3.

. 2 4 = 0. [ 13/100] . = + , = 1 = 1 = 2 2 4 = 0. 2 . , (4, 0) (0, 2) . [ 12/100] . = 0 = 4 (4, 0). 1 = 0 = 0 2 = 2 (0, 2). 2 1 = 2 1 2 + = 2. 2

3: 5 2, : . + 2003 5. [ 12/100] . , = 5 + 2 = 5 + 2. + 2003 = = = = = (5 + 2) + ( = 5 + 2) 2003 25( + ) + 20( + ) + 8 2003 25( + ) + 20( + ) 1995 55( + ) + 4( + ) 399 5.


3.6. 2003 . 8 + 9 5.

117

[ 13/100] . 8 + 9 = = = = 8(5 + 2) + 9(5 + 2) 5(8 + 9) + 34 5(8 + 9) + 30 + 4 5(8 + 9 + 6) + 4,

4.

4: + 4 + 2 + 3 = 0 (2, 1). . (2, 1) = 2. [ 6/100] . 4 + 4 + + 2 + 1 + 3 5 = 0 ( 2) + ( + 1) = 2 (2, 1) = 2. . (2, 1). [ 10/100] . (2 2) + (1 + 1) = 4 0, . ( , ) ( 2)( 2) + ( + 1)( + 1) = 2. M ( 2)(2 2) + ( + 1)(1 + 1) = 2 = 0 ( , ) ( 2) + (0 1) = 2 ( 2) = 1 2 = 1 = 1 = 3.

= 1 (1, 0) ( 2) + + 1 = 2 + + 1 = 0. = 3 B(3, 0) 2++1 = 0 + 3 = 0. (1999) http://gordongr.blogspot.com

118

3.

. , , . [ 9/100] . 2 1 1 1 1 1 1 1 (MAB) = | 1 0 1 | = | | = 2 = 1. 3 1 2 2 2 3 0 1

3.7 2004 1o: . , . : = 1. [ 10/100] . = ( , ) = ( , ). , , 0, = = . , = 0 + = 0 = 1 = 1.

. . ; [ 5/100] . , , , . (1999) http://gordongr.blogspot.com

3.7. 2004

119

. , . . 0 0, + + = 0 . [ 2/100] . = 2, = . 2 [ 2/100] . , , , , 0. | |, |( + ). [ 2/100] . , , , :

(AB) =

1 det(AB, A). 2 [ 2/100]

. . [ 2/100] . ) , ) , ) , ) , ) .

2: = (1, 2) = (2, 3). . = 5 3 . [ 8/100] . = 5 3 = 5(1, 2) 3(2, 3) = (1, 1). , || = (1) + 1 = 2.


120

3.

. . [ 8/100] . , tan = 1 3 7 = 1 = = . 1 4 4 3 . 4

, =

. , = ( , ) . [ 9/100] . = 0 + 2 = 0 + = 0 = 0 = 1.

3: = 12 5, . . . [ 7/100] . = 2(63)+1 = 2+1, = 63, . . 4. [ 8/100] . = 4 3 8 + 3 = 4(3 2) + 3 = 4 + 3, = 3 2. , 3. (1999) http://gordongr.blogspot.com

3.7. 2004

121

. = ( + 15)( 1) 64. [ 10/100] . A = = = = = = ( + 15)( 1) (144 120 + 25 + 15)(144 120 + 25 1) (144 120 + 40)(144 120 + 24) 8 (18 15 + 5) 8 (18 15 + 3) 64 (18 15 + 5)(18 15 + 3) 64.

4: 3+4+6 = 0 3+4+16 = 0. . . [ 7/100] . , 3 . (2, 0) , 4 . , ( , ) = (, ) = |3 (2) + 4 0 + 16| 3 + 4 = 10 = 2. 5

. . [ 8/100] . , , 3 , 3 + 4 + = 0, . 4 , , 3 (B, ) = (B, ) = 3 + 4 | + 6| = | + 16| + 6 = 16 = 11. |3 ( ) + 4 0 + 6| |3 ( ) + 4 0 + 16|

3 + 4


122

3.

, 3 + 4 + 11 = 0. . = 43. [ 10/100] . . = 0 3 +4 +6 = 0, = 2 = 0. = (2, 0). , . , = R, = = 23 2 = (, ) = , = + R = (23) + 2 R = 4. ( + 2) + = 16. |3 (2) + 4 0 + 16| 3 + 4 = 2.


4 / 4.1 2000

1: . ) F() = () + (). , , F () = () + (). [ 8/100] . F DF () = F( + ) F() ( + ) () ( + ) () = + .

, , , . F F () = lim () = lim

( + ) () ( + ) () + lim = () + ().

A. ) () , () 0, . : (), ()(), () [o 4,5/100] 123

124

4. /

. () ()()

= (), = ()() + () (),

() ()() () () = . () ()

. ) . () . + . + cos . sin . 4 ( ) 1. 1 sin 2. 3 8 3. 2 + 3 4. sin cos 5. 2 6. 3 4 7. sin + cos [ 8/100] . -5, -1, -7, -6. : () = , 0 : + , : A. , B. . , E. [ 4,5/100] . . B. )

2: A. . (1999) http://gordongr.blogspot.com

4.1. 2000 1 10 2 3 50 % N 35 1 100 10 1 4 9 10

125

[ 16/100] . : = 10, = 1 = = 35 10 = 25. = = 10 = 0, 2 % = 20%. 50 25 = 0, 5 % = 50%. 50

= 1 0, 2 0, 5 = 0, 3 % = 30%. = 0, 3 50 = 15, N = = 50. : = 2 25 = 50, = 3 15 = 45, = 4 25 = 100, = 9 15 = 135, = 105, = 245. , : 1 2 3 10 25 15 50 % N 0,2 20 10 0,5 50 35 0,3 30 50 1 100 10 50 45 1 4 9 10 100 135

B. . [ 4/100] 105 = = 2, 1. . = 50 25 26 , 2+2 = 35, = = 2. 2 (1999) http://gordongr.blogspot.com

126

4. /

. = 0, 49. : 1 = = . =

[ 5/100] . : 1 1 105 = = 0, 494. = = 245 = 50 50

3: 120 , 24 , 20 12 . . : . ; [M 8/100] . : : . : . P(A) = 24 1 20 1 12 1 = , P(B) = = P(A B) = = . 120 5 120 6 120 10 1 1 1 4 + = . 5 6 10 15

P(A B) = P(A) + P(B) P(A B) =

. ; [M 8/100] (1999) http://gordongr.blogspot.com

4.1. 2000

127

1 1 1 . P(A B) = P(A) P(A B) = = P( ) = 5 10 10 1 1 1 P() P(A B) = = . 6 10 15 P(A B) ( ) = P(A B) + P( ) = 1 1 1 + = . 10 15 6

. ; [M 9/100] . P(A B ) = P(A B) = 1 P(A B) = 1 4 11 = . 15 15

4: 100 . : [) % 0 5 10 5 10 15 10 15 12 15 20 15 20 25 18 25 30 18 30 35 12 . /- 15 ; [ 5/100] . , = 100, = %, . 15 + + + = 63 .

. 35 : ) /- 12, 5 ; . (1999) http://gordongr.blogspot.com

128

4. / [M 10/100]

) /- , ; . [ 10/100] . ) 12, 5 /- 22, 5 , + + = 39 . ) , /- 30 , = 12 . /- .

4.2 2001 1: .1. : ( ) = () ( ). [ 8,5/100] . = AB , (AB)(AB ) = A (AB)(AB ) = . P(A B) + P(A B ) = P(A)P(A B) = P(A) P(A B).

.2. , (=,,) : . ( ) 1 (), [ 2/100] . () (). [ 2/100] . ) =, ) . (1999) http://gordongr.blogspot.com

4.2. 2001

129

.1. , : . . () + () < 1. . () = ( ) 2() = (). [ 4/100] . ) , ) . .2. 5 1 : , () = () = ( ) 4 12 . 1 4 . 5 12 . 2 3 1 . . 6 [ 2,5/100] . . .3. , : () = 1 1 1 , P(B) = ( ) = . 3 4 5 ) P(A B) ) P(B A) ) P(A B) 1. 2. 3. 4. 5. [ 6/100] . -2, -5, -3.

2: () = cos + sin . (1999) http://gordongr.blogspot.com

130

4. /

A. () + () = 0. [ 8/100] . , () = sin + cos , () = cos sin . , () + () = 0.

. (0, 1). [ 8/100] . (0) = 1 (0) = 1 = 1 + . . : 2 = 2. 2 2 [ 9/100] . 2 = 2 (1) + 2 1 = 2 = 4. 2 2

3: , 80 . . () .. F [45, 55) 0, 2 [55, 65) 0, 5 [65, 75) [75, 85) (1999) http://gordongr.blogspot.com

4.2. 2001

131

. , . [ 8] . F = 0, 2 = 0, 2 F = F + 0, 5 = 0, 2 + = 0, 3. , = 2 0, 2 = 0, 4, = 1 (0, 2 + 0, 3 + 0, 4) = 0, 1. , F = F + = 0, 5 + 0, 4 = 0, 9 F = 1. . . [ 9/100] . = 50, = 60, = 70 = 80. = + + + = = 64.

. 80 , . . 65 . [ 4/100] . 55 75 . [ 4/100] . , = : = 16, = 24, = 32, = 8. , () (), || = 80, || = + = 40 || = + = 76. , P(A) = P(B) = |A| 40 = = 0, 5 || 80 |B| 56 = = 0, 7. || 80


132

4. /

4: , , 50% 12 , 16% 10 . .

. .

[ 6/100]

. , = 12 = 10, = 2.


4.2. 2001

133


134

4. /

. . [ 6/100] . CV = . . 4. 000, /- 14 16 ; [ 6/100] 99, 7 95 . [14, 16] = [+2, +3] = 2, 35% 2 2, 35 , 4000 = 47 . 100 . , , . CV. [ 7/100] . = + 5, = 1, , 4000 = + 5 = 17 = = 2. 2 CV = = 17 2 1 5 CV = = . 17 6 6 17 1 1 = > . 6 10

4.3 2002 1: . , , , , , , . (1999) http://gordongr.blogspot.com

4.3. 2002

135

. , , = 1, 2, , ; [ 3/100] . . . , = 1, 2, , ; [ 3/100] . , = . . : i) 0 1, = 1, 2, , . ii) + + + = 1. [ 4/100] . = > 0 , > 0 = = 1. , + + ++ = = = 1.

+ + =

.1. , ( ) = () + (). [ 8/100] . , , |A B| = |A| + |B|. P(A B) = |A B| |A| + |B| |A| |B| = = + = P(A) + P(B). || || || ||


136

4. /

.2. . . [ 5/100] . P(A) = |A| . ||

.2.. : i) P(), ii) (). [ 2/100] . P() = || || 0 = 1 P() = = = 0. || || ||

2: () = 2 . + 1

. . [ 4/100] . +1 0 1. D = {1} = ( 1) (1, +).

. lim (). [ 4/100] . , lim () = (3) = . . [ 7/100] . () = 2 (2) ( + 1) 2( + 1) 2 = = . + 1 ( + 1) ( + 1) 6 3 23 = = . 3+1 4 2


4.3. 2002

137

. = 2 + 5. [ 10/100] . ( , ) , 2 ( ) = . ( + 1) = 2 + 5 2, 2 = 2 ( + 1) = 1 = 2 = 0. ( + 1) = 2 D = (2) = 4 = 2 + 8. = 0 D = (0) = 0 = 2.

3: 10 , : 8, 10, 13, 13, 15, 16, 18, 14, 14, 9. . , . [ 6/100] . = 8 + 10 + 13 + 13 + 15 + 16 + 18 + 14 + 14 + 9 = 13. 10

, 8, 9, 10, 13, 13, 14, 14, 15, 16, 18, = 13 + 14 = 13, 5. 2 , 13 14, 2.

. , . [ 6/100] (1999) http://gordongr.blogspot.com

138

4. /

. R = 18 8 = 10. = (8 13) + (10 13) + + (9 13) = 3. 10

CV = 3 = . 13

. 10%, . [ 13/100] . = 0, 9 , = 1, , 10 = 0, 9 = 0, 9 . 0, 9 = = CV CV = = 0, 9 .

4: , () + () 2( ). () = P(A B) P(A B) , . . P(A B) P(A B). [ 5/100] . P(A B) = P(A) + P(B) P(A B) 2P(A B) P(A B) = P(A B).


4.3. 2002 . ()

139 P(A) + P(B) . 2 [ 13/100]

. , () = 3 P(A B) 3 P(A B) . () 0 , : , , P(A) + P(B) 2 P(A) + P(B) , 2 P(A B) P(A B) 0 P(A B) + P(A B) P(A B) + P(A B) 0 P(A B) P(A B) 2 P(A B) P(A B) 0 P(A B) + P(A B) , P(A B) P(A B) 2 P(A) + P(B) . 2

P(A) + P(B) , + 2

P(A) + P(B) , +, 2

=

P(A) + P(B) . 2

. , , (P(A)) = (P(B)). [ 7/100] . , , P(A B) = 0 P(A B) = P(A) + P(B), (P(A)) = P(A) P(A B) + P(A) 0 = P(B) + P(A) (P()) = P(B) P(A B) + P(B) 0 = P(A) + P(B) . . P(A) = P(B), , . :-) (1999) http://gordongr.blogspot.com

140

4. /

4.4 2003 1: . () = () = 1. [ 8/100] . ( + ) () + () = = = 1. 0 , . () = lim () = lim 1 = 1.

. ; [ 6/100] . D, , , < () < () D, , , < () > ().

. . [ 6/100] . , , , . (1999) http://gordongr.blogspot.com

4.4. 2003

141

. , . . . . . . ((())) = (()) () , . . , = . . . [ 5/100] . ) , ) , ) , ) , ) .

2: /- , 55% , 40% /- 30% . . : . . [ 5/100] . : : . P(A) = 0, 55, P(B) = 0, 4 P(A B) = 0, 3. P(A B) = P(A) + P(B) P(A B) = 0, 55 + 0, 4 0, 3 = 0, 65. . . [ 5/100] . A B = A B , (A B) (A B ) = A (A B) (A B ) = , P(A B) = P(A) P(A B) = 0, 55 0, 3 = 0, 25.


142

4. /

. . [ 7/100] . B A = A B, (A B) (A B) = B (A B) (A B) = , P(B A) = P(B) P(A B) = 0, 4 0, 3 = 0, 1.

. . [ 8/100] . P(A B) = P(A B ) = 1 P(A B ) = 1 0, 25 = 0, 75.

3: () = . 1

. : : . . (1, 1) . {1, 1} . (1, +). [ 5/100] . 1 0 (). 1.

. () < 0 . [ 7/100] . () = + 1 < 0, D. ( 1)


4.4. 2003 . lim [( + 1) ()]

143

[ 6/100] . ( + 1)() = ( + 1) lim [( + 1) ()] = lim

= . ( 1)( + 1) 1

1 = . 1 2

. (0, (0)) . [ 7/100] . C (0, 1) 0 + 1 (0) = = 1. 0 < (0 1) tan = 1 tan = tan 3 3 = . 4 4

4: / , , ( ) ( ) . 1 7 8 14 9 6 5 4 3 12 4 5 (1999) http://gordongr.blogspot.com

144

4. /

. . [ 6/100] . = = 7 + 14 + 6 + 4 + 12 + 5 = 8. 5 1+8+9+5+3+4 =5 6

, , 1, 3, 4, 5, 8, 9 4+5 = 4, 5. 2 , , 4, 5, 6, 7, 12, 14 6+7 = 6, 5. 2 . . [ 5/100] . (1 5) + (8 5) + (3 5) + (5 5) + (3 5) + (4 5) 23 = = 6 3 (7 8) + (14 8) + (6 8) + (4 8) + (12 8) + (5 8) 41 = = . 6 3 , CV = = 41 23 = = 0, 554 CV = = 0, 462. 83 53 CV < CV . . 20% 5 , ; [ 8/100] . = 1, 2, = 1, 2 = 1, 2 5 = 6. , = , + 5 = + 5 = 13. , (1999) http://gordongr.blogspot.com

4.5. 2003

145

. . [ 6/100] . = 1, 2, , + 5 = = 0, 462. , = 1, 2 = 1, 2 0, 554 = 0, 665. = , 0, 665 = = 0, 11 6

, CV = CV =

0, 462 = = 0, 03. CV > CV , 13 .

4.5

2003

1: . , : ( ) = 1 (). [ 9/100] . A = A = , P(A) + P(A ) = P() P(A ) = 1 P(A).

. : , ( + ) () . lim , 0 . . lim ( ) ( ) , , 0.

( + ) ( ) . lim , , 0 . . lim ( + ) + () , , 0. [ 5/100] . . (1999) http://gordongr.blogspot.com

146

4. /

. : . , . , . , . . [ 5/100] . . . . [ 6/100] . , , > 0 CV = = , < 0

2: () = 1. . . [ 5/100] . 1 0 1 || 1 1 1. D = (, 1] [1, +). (1999) http://gordongr.blogspot.com

4.5. 2003 . , = 3, 32 . 4

147

[ 10/100] . 1, () =

( 1) 2 1

=

1

.

, 3 (3) = 3 3 1 = 3 8 = 3 22 = 32 32 = . 22 4

. () =

() 3 , 2, lim (). 2 [ 10/100]

. (2) = 3, () 3. lim () = (2) =

2 2 1

=

2 3

=

23 . 3

3: 30 , 1 30. . 5. , , . (), P(B) [ 6/100] (1999) http://gordongr.blogspot.com

148

4. /

. = {1, 2, , 30} A = {2, 4, , 30} B = {5, 10, 15, 20, 25, 30}. , , P(A) = P(B) = |A| 15 = = 0, 5 || 30 6 |B| = = 0, 2. || 30

. ( ) [ 6/100] . A B = {10, 20, 30} P(A B) = 3 1 = . 30 3

P(A B) = P(A) + P(B) P(A B) = 0, 5 + 0, 2 0, 1 = 0, 6.

. ( ) [ 6/100] . (A B) (A B ) = (A B) (A B ) = , P(A B ) = P(A) P(A B) = 0, 5 0, 1 = 0, 4, P(AB ) = P(A)+P(B )P(AB ) = P(A)+1P(B)P(AB ) = 0, 5+10, 20, 4 = 0, 9.

. P(A ) ( ) [ 7/100] . P(A B) = P() P(A B) = 0, 2 0, 1 = 0, 1 (A B) (A B ) = , P(A ) ( ) = P(A B) + P(A B ) = 0, 1 + 0, 4 = 0, 5.


4.5. 2003

149

4: . 50% 65 Kg, 47, 5% 65 Kg 75 Kg. . , . [ 6/100] . = 65 + 2 = 75, = 5. = = 65. . . [ 6/100] . CV = . . , 55 Kg 70 Kg. [ 6/100] . [55, 70] = [ 2, + ] = [ 2, ] [, + ] 47, 5 + 34% = 81, 5% . . 55 Kg 60 Kg 27. . [ 7/100] . [ 2, ] 13, 5%. , 27 100 13, 5 = 27 = = 200. 100 13, 5 (1999) http://gordongr.blogspot.com 5 1 1 = = < 65 13 10

150

4. /

4.6 2004 1: . () = 0. [ 8/100] ( + ) () = = 0. 0 , , () = lim () = 0, . . () = . . [ 5/100] (1999) http://gordongr.blogspot.com

4.6. 2004

151

. D, , ,

lim () = ( ).

. , . . . . 95% ( , + ), . . , . [ 6/100] . ) , ) , ) . . . . . ) , ) ) . [ 6/100] . -4, -2, -1. 1) 2) 3) ( ) 4)

2: () = 4 + 3 3 .


152

4. /

. . [ 10/100] . [ 0 3 0] 0 3, D = [0, 3) (3, +). B. (). [ 15/100] . D, () = lim () = lim( 1) lim + 3 = 43.

( 4 + 3) + 3 3 + 3

=

( 1)( 3) + 3 3

= ( 1) + 3

3: 200 , 5 45 . :

(.) [ , ) [5, 15) [15, 25) [25, 35) [35, 45) ( )

(.) ( %) 60

. (.) ( )

. . (F %) 68

180 200 100


4.6. 2004

153

. . [ 10/100] . : = 10, = 20, = 30 = 40. : = 200, N = 200, = 20, % = 10. F = 100. % = 60 = 30, = = 60, F % = % = 30. % = F % F % = 68 30 = 38, = 38 = 76, N = N + = 60 + 76 = 136. = = 180 136 = 44 % = 44 = 22 F % = F % + % = 68 + 22 = 90. , :

(.) (.) . (.) [ , ) ( ) ( %) ( ) ( ) ( %) ( ) [5, 15) 10 60 30 60 [15, 25) 20 76 38 136 [25, 35) 30 44 22 180 [35, 45) 40 20 10 200 200 100

. ( , %) . [ 5/100] (1999) http://gordongr.blogspot.com

154

4. /

.

. . [ 5/100] . = 10 60 + 20 76 + 30 44 + 40 20 = = 21, 7. 200

. 25 . [ 5/100] . 25 + = 44 + 20 = 64, 64. 000.

4: 5 () = 2 + + 10. 2 P(A) P(B) , . (1999) http://gordongr.blogspot.com

4.6. 2004 . P(A) = 1 1 P() = . 2 3

155

[ 9/100] . () = 6 5 + 1. , 1 1 , = 25 24 = 1 = , = . 3 2 () 0 : 1 1 , , +, , 3 2 1 1 , , + 3 2 1 1 1 1 , , , , 3 2 3 2 1 1 547 , = 3 3 54 1 1 83 , = . 2 2 8 P(A) = 1 1 P(B) = . 2 3 1 1 3 2

2 . P(A), P(B) P(A B) = , 3 : i. P(A B) ii. P(A B) iii. P[(A B) ] iv. P[(A B) ( )]. [ 16/100] . (i) P(A B) = P(A) + P(B) P(A B), P(A B) = P(A) + P(B) P(A B) = 1 1 2 1 + = . 2 3 3 6


156

4. /

(ii) AB = AB , (AB)(AB ) = A (AB)(AB ) = . P(A B) + P(A B ) = P(A) 1 1 1 P(A B) = P(A B ) = P(A) P(A B) = = . 2 6 3 1 5 (iii) P((A B) ) = 1 P(A B) = 1 = . 6 6 1 1 1 (iv) (ii), P(B A) = P(B) P(A B) = = . 3 6 6 , (A B) (B A) = , P(A B) (B A) = P(A B) + P(B A) = 1 1 1 + = . 6 6 2

4.7 2004 1: . , () (). [ 7/100] . A B, .. A X = B A X = , X = B A. P(A) + P(X) = P(B) P(A) P(B). . . ; . . [ 6/100] . ) , . ) . (1999) http://gordongr.blogspot.com

4.7. 2004

157

. , , . . lim () = lim () = , lim (() () = . . A, () ( ) . . ()() = ()()+ ()(), . . =

1 > 0.

. ( ) = 1 + (). . . [ 12/100] . ) , ) , ) , ) , ) , ) .

2: () = + 2 .

. . [ 9/100] . , () = ( + 2) + 1 = . ( )

() 0 1. : (, 1), (, 1] (1, +), [1, +) 1, (1) = .


4. / 1 . () + () = . 158 [ 8/100] . () + () = + 2 + 1 1 = , .

. (0, (0)). [ 8/100] . (0) = (0) ( 0) = + 2.

3: 25 14, 10 11. . 15 . [ 12/100] . , , . + + = 11 10 = 110 = 14. : = 14 + + + + + = 14 25 + + + + + = 14 25 110 + + + = 350 + + = 240 + + 240 = = 16. 15 15


4.7. 2004

159

. 25 5000, (CV). [ 13/100] . = 14 + + = 5000. = = = = = = ( ) + + ( ) 25 + ( + + ) 2( + + ) 25 + + + 2 25 + + + 25 5000 14 + 25 4.

= 2 CV = 2 1 = = . 14 7

4: = {1, 2, 3, 4, 5, 6} () = + 4 + 2, . P(1) = P(3) = P(5) = 2P(2) = 4P(4) = 2P(6), : . P(1), P(2), P(3), P(4), P(5), P(6). [ 8/100] . P(1) = P(3) = P(5) = 2P(2) = 4P(4) = 2P(6) = P(1) = , P(2) = , P(3) = , P(4) = , P(5) = , P(6) = . 2 4 2


160

4. /

P(1) =

P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 1 + ++ ++ =1 2 4 2 17 =1 4 4 = . 17

2 4 1 4 2 4 , P(2) = , P(3) = , P(4) = , P(5) = , P(6) = . 17 17 17 17 17 17

. , : . : . [ 8/100] . 5 . 17 B = {1, 3, 5} P(B) = P(1) + P(3) + P(5) = A = {2, 4, 6} P(A) = P(2)+P(4)+P(6) = 2 1 2 + + = 17 17 17

4 4 4 12 + + = . 17 17 17 17

. , : . [ 9/100] . , () = 2 + 4. , , = 4 16 = 4( 2)( + 2). , () 0, < 0 2 < 0 < 2 = 1. = {1}, P() = P(1) = 4 . 17


4.8. 2005

161

4.8

2005

1: . : P(A B) = P(A) + P(B) P(A B). [ 10/100] . , P(A B) = |A B| || |A| |B| |A B| = + || || || = P(A) + P(B) P(A B).

. . ; [ 3/100] . ; [ 4/100] . ) . ) . (, ). . , , . . () > 0 , . [ 2/100] (1999) http://gordongr.blogspot.com

162

4. / () () + () () () = , () [ 2/100]

)

, .

) . [ 2/100] . P(A) > P(B). [ 2/100] . ) , ) , ) , ) .

2: :

. : [4, 8) [8, 12) [12, 16) [16, 20] N F


4.8. 2005

163 [ 11/100]

. : = 5, = 10, = 25, = 10 N = 5, N = 15, N = 40, N = 50 = 50 = 5 10 25 10 = 0, 1, = = 0, 2, = = 0, 5, = = 0, 2 50 50 50 50

F = 0, 1, F = 0, 3, F = 0, 8, F = 1. = 6, = 10, = 14, = 18. , : [4, 8) 6 5 [8, 12) 10 10 [12, 16) 14 25 [16, 20] 18 10 50 0, 1 0, 2 0, 5 0, 2 1 N 5 15 40 50 F 0, 1 0, 3 0, 8 1

. . . = 6 5 + 10 10 + 14 25 + 18 10 = = 13, 2. 50

. /- 10; . 10 + = 5 + 5 = 10. 2

3: , , , : (i) 7 , . 8 (ii) P(B), P(A B) 1 5 3 15 X = , , , = lim . 2 4 6 + 5 (1999) http://gordongr.blogspot.com

164

4. / [ 5/100]

. .

. 1, 5 3 15 3( 5) 3 = = 6 + 5 ( 1)( 5) 1

= lim

3 15 3 3 = lim = . 6 + 5 1 4

. P(B), P(A B) . [ 8/100] 5 > 1 4 1 3 . A B P(A) < P(B) < . 2 4 1 3 : P(B) = , P(A B) = . 4 2 . . : 1. . [ 6/100] 2. . [ 6/100] . P(A B) = P(A) + P(B) P(A B) P(A) = P(A B) P(B) + P(A B) = 7 3 1 5 + = . 8 4 2 8

, (A B ) (A B ) = A (A B ) (A B ) = , 1 P(A B ) + P(A B ) = P(A) P(A B ) = . 8


4.8. 2005

165

4: , 1 () = , (0, +). . (1, 1). [ 7/100] . () = 1 , (0, +). ,

(1) = (1) ( 1) = + 2.

. (, ) , , O . , . [ 10/100] 1 . M, . 1 () = 2 + , > 0. () = 21 1 () 0 1 > 0 > 0 , : (1, +), , [1, +) (0, 1), , (0, 1] = 1, (1) = 4. (1, 4). (1999) http://gordongr.blogspot.com 1 1. > 0 1 .

166

4. /

. () = 5 = 2. . [ 8/100] . Y = + 2, = 1, 2, 3, 4, 5. = + 2 = 5 + 2 = 3 = = 2.

4.9 2005 1: A.1. F(), () () F() = () + (). , , F () = () + (). [ 9/100] . F F( + ) F() ( + ) + ( + ) () () = ( + ) () ( + ) () = + . 0, , F () = F () = lim () = lim

( + ) () ( + ) () + lim = () + ().

A.2. X, > 0 , < 0; [ 4/100] . CV = CV = , ,

> 0 , < 0

, 0. ||


4.9. 2005

167

. , . . . [ 2/100] 1 . > 0, (ln ) = . [ 2/100] . , , F , N . [ 2/100] . . [ 2/100] . , () = (), || = ||. [ 2/100] . . [ 2/100] . ) , ) , ) , ) , ) , ) .

2: () = ln , , . . . [ 3/100] . > 0, D = (0, +). (1999) http://gordongr.blogspot.com

168

4. /

. . [ 5/100] . () = 2.

. , (1, 1) = 3 2. [ 10/100] . (1) = 1 ln 1 1 = 1 = 1 (1) = 2(1)1 = + 2. 1 = (1) + (1) ( 1) = ( + 2) 1. + 2 = 3 = 1. . lim ( () ). [ 7/100] 1 . () = + 2 = + 2 , lim( () ) = lim( + 2 ) = 4 + 32 = 36.

3o: , 50% 20. 81, 5% (16, 22), : 3, 2, , . (1999) http://gordongr.blogspot.com

4.9. 2005 . = 20 = 2.

169

[ 10/100]

. 50% (, ) (, +). , = 20. 81, 5% ( 2, + ), = 2.


170

4. /


4.9. 2005

171

. , ( , + ) 95% . [ 5/100] . 95% ( 2, + 2). = 2. . R , R () = ( + 4) + 9. 2 [ 10/100] . R = 6 = 12. () = 6 24 + 18. () = 12 24. () 2, : (, 2), , (, 2] (2, +), , [2, +) 2, (2) = 6.

4: = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} . , , A B = {1, 2, 3, 4, 5, 6}, + 1 2. A B = {1, 3, 4}, A B = {2, 6} = 1 (1999) http://gordongr.blogspot.com

172

4. /

. (), (), (). [ 9/100] . |A B| 6 = = 0, 6 || 10 |A B| 3 P(A B) = = = 0, 3 || 10 |A B| 2 P(A B) = = = 0, 2. || 10 P(A B) = A B = A B (A B) (A B ) = A, (A B) (A B ) = , P(A) = P(A B) + P(A B ) = 0, 5. , P(A B) = P(A) + P(B) P(A B) P(B) = 0, 4. || = 3, P() = || = 0, 3. || + 1 2 1 3 {1, 2, 3}

. , . [ 3/100] . || = 5, || = 4, A B = {1, 2, 3, 4, 5, 6} A B = {1, 3, 4}, A = {1, 2, 3, 4, 6} = {1, 3, 4, } {5, 7, 8, 9, 10}. , B = {1, 3}. , ( ) ( ) = ( ) ( ) = . P( ) + P( ) = P() P( ) = 0, 2.


4.10. 2006

173

. , . [ 3/100] . , P( ) + P( ) = P() P( ) = 0, 2. , P( ) ( )) = P( ) + P( ) = 0, 2 + 0, 2 = 0, 4.

. , 3, 5, , = { > 24}. [ 10/100] . + 3 + 5 = 3. 3 ( 3) + (3 3) + (5 3) 8 = , 3 3 , P() = 8 > 24 > 9 > 3 {4, 5, 6, 7, 8, 9, 10}. 3 || 7 = = 0, 7. || 10

4.10

2006

1o: A. . ( ()) = (), . [ 10/100] (1999) http://gordongr.blogspot.com

174

4. /

. () = ()( + ) ()() ( + ) () ( + ) () = = .

( 0) , , ()() = lim () = lim

( + ) () = ().

B.. , ; [ 3/100] . , A B = . . . ; [ 4/100] . , , A ,

lim () = ( ).

. , . . , A, () ( ) . [ 2/100] . , , , . [ 2/100] (1999) http://gordongr.blogspot.com

4.10. 2006 . 0 1 1 = .

175

[ 2/100] . . [ 2/100] . ) , ) , ) , ) .

2: , 50 . , : ( ) /- ( ) 0 +4 1 5 + 8 2 4 3 1 4 2 50 . . [ 3/100] . + + = 50 13 + 11 = 50 = 3.

. : . [ 7/100] (1999) http://gordongr.blogspot.com

176

4. / + + 5 0( + 4) + 1(5 + 8) + 2(4) + 3( 1) + 4(2) = 5 24 + 5 = 5 77 = = 1, 44. 50

. =

. . [ 7/100] . , . ( ) ( ) N 0 7 7 1 23 30 2 12 42 3 2 44 4 6 50 50 25 26 , 1+1 = = 1. 2

. 3 . [ 8/100] . : 3 , || = 2 + 6 = 8 || = 50, P(A) = 8 = 0, 16. 50


4.10. 2006

177

3o: , ( + 4) . . , . . [ 7/100] . + ( + 4) = + 9 + 16 . , () = . + 9 + 16

= 81 67 = 17, , = 9 17 2

, > 0, . D = + 9 17 9 + 17 9 17 , +. = 0, 2 2 2

1 . 19 100 , , . [ 8/100] . 1 = 10 + 16 = 0 = 2 = 8. + 9 + 6 19

= 8 8 + (8 + 4) = 152 > 100, . = 2 ( ), ( + 4) = 36 ( ), 38 . 36 9 = . 38 12 (1999) http://gordongr.blogspot.com

178

4. /

. , , ; [ 10/100] . D () = () 0 [ + 16 0 D] 0 2. , : (0, 2), , (0, 2] (2, +), , [2, +) 2, (2) = 1 . 19 + 16 + 9 + 16 (2 + 9) = . ( + 9 + 16) ( + 9 + 16)

4: , () = 2 + + 4 + 10, 0. . (1, (1)) , = 2 . [ 5/100] . [0, +) (0, +), 2 () = 4 + + . (1) = 2 + . 2 + = 0 = 2. 2 () = 2 + 2 + 4 + 10 () = 4 + 2 + . = (1) + (1) ( 1) = 14.


4.10. 2006

179

. (4) = (1) = . , 13 , 8. (i) (10, 16). [ 10/100] (ii) . > 0 , . [ 10/100] . = 14 = 2. [0, 8], [0, 3]. 1 3 1 = (100 99, 7) = 2000. 2 100 [10, 16] = [ 2, + ] = [ 2, ] [, + ] 1 1 95% + 99, 7% , 2 2 1 (95 + 99, 7) = 1630 2 100 . CV = 1 1 = > 7 20

. , = + = 14+ = = 2. 2 , CV = 14 + 2 1 6. 14 + 10


180

4. /

4.11 2006 1: A. , : P( ) = 1 (). [ 9/100] . A A = A A = P(A) + P(A ) = 1 P(A ) = 1 P(A).

.1 A; [ 3/100] . A A, () ( ). .2 ; [ 3/100] . . . , . . , . [ 2/100] . (cos ) = sin , . [ 2/100] . (CV) . (1999) http://gordongr.blogspot.com

4.11. 2006

181 [ 2/100]

. . [ 2/100] . , . P() P(A B) P(). [ 2/100] . ) , ) , ) , ) , ) .

2: () = ( + + 9) , . (2, ) = + 3 , : . = 1 = 6. [ 12/100] . (2) = (4 + 2 + 9). , , () = ( + + 9) + (2 + ) = + (2 + ) + 9 + . , = (2) ( 2) + (2) = (2) + (2) 2 (2) = (8 + 3 + 9) + (12 4 9). = + 3 , 8 + 3 + 9 = 1 12 4 9 = 3. = 1, = 6. (1999) http://gordongr.blogspot.com

182

4. /

. . [ 13/100] . () = ( 6 + 9) () = ( 4 + 3). () 0 4 + 3 0 1 3, : (, 1) (3, +), (, 1] [3, +) (1, 3), [1, 3] 1, (1) = 4 3, (3) = 0.

3o: . 0, 7 0, 1. . . . [ 15/100] . , . P(A B A B) = 0, 7 P(A B ) = 0, 1. P(A B) = 1 P(A B ) = 1 0, 1 = 0, 9 P(A B A B) = 0, 7 P(A B) P(A B) = 0, 7 P(A B) = 0, 2. . (1999) http://gordongr.blogspot.com

4.11. 2006

183

. 0, 6 : i. , ii. . [ 10/100] . (A B) (A B ) = A (A B) (A B ) = , P(A) = P(A B) + P(A B ) = 0, 2 + 0, 6 = 0, 8. , P(A B) = P(A) + P(B) P(A B)0, 9 = 0, 8 + P(B) 0, 2P(B) = 0, 3. , (A B) (A B) = B (A B) (A B) = , P(B) = P(A B) + P(A B)P(A B) = 0, 3 0, 2 0, 1.

4: 2006 : ( ) ( ) [ , ) 0, 1 [ , ) [ , ) 0, 3 [ , ) 10 1 , : (1999) http://gordongr.blogspot.com

184

4. / [ 10/100]

. 2. = 10 7 = 2, , 2 , . + [3, 5), [5, 7), [7, 9), [9, 11) 4, 6, 8, 10. . , . [ 5/100] . + + + = 10, 1 + + 0, 3 + 2 = 1

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Πανελλήνιες ἐξετάσεις στὰ Μαθηματικὰ (1999–)