Kee Kymata
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213
0 ..)2..!.2&1# 3& 020/.3& 020..2 .2 12 )2# . 212 $02 /!. . ( . = I [ 2 #1$ .2 " ÿÿ 2 12 )2# . 212 $02 /!. . ( . = I [ 2 #1$ .2 " ÿÿÿ . )!.ùú2$! 12 W $ # 2.$*220" 9$ = − \ . 9% = \ . 212 $. /þ/.3 !31" 02. *2& 2. . 2+10& #020 * 2. )!.+û2 #0 .12 * 1 #0 .1 0 / 25.
.2 "!. * 0 *"0 .12 * 1 #/.//02..2202.20*# 1 P 2 0/"* .1#$ )22." +] 02.$*22. # = . 2 " \ = FP V ..# 1020 i) 2 "* .2 " ii) 2 0! / .2 # 1#$ )22.2"2. 2&1"0 )" ! #2 #0 .12 * 1 # .!%0202 0 1&12 #* .2 " þ2 #* .2 "!102.12 .!$22 # . [ ′[ .2$! 12 W = .2 / 02.)2 \ = 9 > UDG y = 0,05 500t – 0,5x) (SI)] >ù . L PLL ⋅ − V V
26. . #!102.121x = 0 2 # . [ ′[ .!$0212 t = 0, .020 0. .! 2. 2&1 00 1&1 \ = W 6, .!.) 0 * ./.P //02..2202.20*# 12 # . 02.$*22. # = V ..# 10202 "* .2 "2 #.! ** .2 " .!%0202 0 1&12 #* .2 " .$! 12 ..!$10 . 02. .1 0 2 # . [ ′[ #!102. 121x = 500 m; / $! 12 W V .!020.2 1 0 i) 2 . !# 12 #\.)211 !! ."2 # ii) 2 2.$*222 # iii) 2 02$# 12 # [ P >ù . P \ = W − 6, V / L LL iii) 0] V 27. þ0 1&1!. *.! ** .2 "0 . \ = W − [ 6, .!020 .2 2.$*22.2 #* .2 " 2 .)12.1/* 1 0& 2. . .$! 12 $ # 02. *2 #"/.3 !31" /.3 !31"0 )"1 0 # 02. *2& $! + 12 + W = V . W = V P >ù . P UDG@ V
214
ü ü ùÿûü ÿ þüü ù
28. .2.!$0 .020 0212 t . .! 2. 2&1 00 1&1 \ = W 6, .!.) 0 * ././/02..2202.20*# 12 # . P [ ′[ 02.$*22. # = V ..!0202 0! / 21#$ )22..2 "* .2 " .!%0202 0 1&12 #* .2 " .!%0202"0 1+10" #/ # 2 2.$*22.2. 2&1".2 02$# 110 1# !21 02 $!) . .1 0 #!102.121 [ = P /..!.121020!.323132"2. 2&1".2./3 !.1 0.2 # .2[101# !21 021# 202. [2$! 12 W V >ù . V+]P \ = W − [ (SI) 9 = 1# W − (SI), . = − W − (SI) / 3 ±x (SI)] 29. þ 2 0/"* ././/02..2202.20*# 12 # . [ ′[ 10!. ) 0 "0 .12) 1 .$0 "* .2 " Pþ0 1&1/ 10&"2"" . !102. 12 .!$ 2 # . 0 . \ = \ W . $! 12 W /* # 1 0. 2. . !1 2. & 12 . $ # . 212 $. 310" . 3 = 3 = ... /0 020)22 * ././/02. 0.20*# 1.)2 1 0 ! "2 1 0 .# 10202 .)12.1 >ù P@ 30. ò . 2 0/"* ././/02.! "2 .! 2.20*# 12 # . [ ′[ .$0 2.0 "$.!.2!12 \ = P = P = +] .!%0202 0 1&12 # * .2 "100 ..)2".!.2&0!2+10" .21[ .W 0 .\ .V > 0. 21[ m.W 0 .\ .9! 7 >ù . \ = W + [ 6, y = 0,05 2 4 t + 1,25x + (SI)] 8 31. û 02.2 .! )* . 00 1&1 \ = W − [ + (SI). ..# 1020 i) 2 2.$*22.2 #* .2 " ii) 2 122.$*22.2. 2&1"0 )"1 0 #2 #0 .12 * 1 # .!0202 2.$*22.2"2. 2&1"212 t = 0, 0 )"1 0 #2 #0 .12 * 1 # #!102.121x = 25m. .!0202 2.$*22.2"2. 2&1"212 t = 0,1s, 0 )"1 0 #2 #0 .12 * 1 # #!102.121x = 50m. P P P P ii) 9 = 9 = − 9 = ] >ù . i) V V V V
ù ù
215
32. ò . 2 0/" * . /.//02. ! " 2 02 .20*# 1 2 # . [ ′[ . $0 2 " \ = P "* .2 " P.1#$ )22. +]þ2 #* .2 " !102. 12 .!$ 2 2 # . +. W 12 1 [ . !# 1 0 . y = 0,1 m. . .!020 i) 2 # .2).! ) N = . ii) 2 0! / .2 # 1#$ )22.2 #* .2 " iii)2 2.$*22.// 1"2 #* .2 " .! 1/ !10202 .!$312 #* .2 ". .!%0202 0 1&12 #* .2 " UDG P iii) >ù . L m–1 ii) V , V V , \ = W − [ + (SI)] 33. .!.&" 2 0/+ # 2& !102.12 .!$2 0 *"$ !/" 0 # #"þ0 1&1/ 10&"2 #1 0 #20 . \ = ⋅ − W 6, .2 .!.) 0 * ././/02..2202.20*# 1 "* .2 "0 . P . )10 .2.$*22.// 1"2 #* .2 " .!%0202 0 1&12 #* .2 " )10 .. !# 1.)211 !! ."2 #2$! 12 W = V 0 )"1 0 #2"$ !/"2 .$0.)2 2.)12.1 [ = P P >ù . \ = ⋅ − W − [ 6, ⋅ − P ] V 34.
.2 "!. * 0 *"0 .12 * 1 #/.//02.! "202.20*# 1 2 # . [ ′[ 0!1 2 0/" * . 0 2 " \ = FP " * .2 " P FP.2.$*22. # = $! 12 W 2 * .3 012 1 0 V .!$2 # . [ ′[ +.2 1 0 / 02.)22$! 12 W /!$02..)2 11 !! ."2 # * 0 .22 .! 23 ! . .!%0202 0 1&12". !# 1"2 #1 0 #101# !21 02 $!) .!%0202 0 1&12". !# 1"0 )"1 0 ù2 #0 .12 * 1 #2 $01# 202. [ cP101# !21 02 $!) .1$0/10202 !32 #0 .12 * 1 #/0 2 #1 0 #2"$! "12T 3T 5T " 0, , T, , 2T, . 2 2 2 >ù . \ = W + \ = W − \ FP W V ]
216
ü ü ùÿûü ÿ þüü ù
35. ! !. * 0 *"0 .12 * 1 #.!$0212 t = 0, .020 0. 0&2.! 2. 2&1 00 1&1 \ = W W10V\10FP )20/.//0P 2..2 "2 # .2[* . 02.$*22. # = V .)1 0 .2 "* .2 " )20.!$0 .2. . 2+ 02. .1 0 2 #0 .12 * 1 #2 .$0.) 2 .)12.1[ P .!%0202 0 1&12". !# 1"2 #1 0 #. .# 10202 2 2"2$! 12 W1 = 5,625 s. .0 .2 2"31"2 #1 0 #2 .!. &$! 12 / )1 .$0.)2 1 0 .1 0 2 2 /.$! 12 W1 = 5,625 s) $031 3 = + .2 .3 !/.//02.2 * . 0..!.121020!.32 02. 2"31"2 #1 0 #101# !21 02 $!) >ù . P V \ 0 = W − W 10 V \ 10 FP , 10 cm, 3 = + / P .)2 ! "2 @ 36. ,0&! * 01 0..!.&"# 2& 2" .". !# 1.)21 1 !! ."/ 02..)2 0 1&1 \ = W + 3 W10V\10FP $! 12 W !102.12 1202. !# 1.)21 1 !! ."2"õ2. 0! 0.)211 !! ."2".2!23 !2 * . #.!02..).#2 $0/./ 010.)12.1G FP ..!0202& . 3 .!%0202 0 1&12 #* .2 " #/.//02..2 "2 # . [ ′[ 2 #0 .12 * 1 # 0.!$212"".! "2023 ! .!%0202"0 1+10" #/ # 2 2.$*22.2"2. 2&1".2 02$# 1 101# !21 02 $!) . . )! 2 #0 .12 * 1 # #!102.121 x = 10 cm. /..!.121020!.323132"2. 2&1".2./3 !.1 0.2 # .2[101# !21 02 .)12.12 #"[.)2 2$! 12 W V 0.1$0/10202 12 )2# 2 #* .2 "2$! 12 W V [ >ù . \ = W + − W10V x\10FP FP FP ), . = − W − W10V.10 )] 9 = 1# W − W10VV10 V V
ù ù
217
37. 2. 1$ .2. 3. 2. /* !.3" .y (cm) (ÿ !.1210" . 0!1 2 0/" * . t=0 2 2 /.//02..2 "!. * 0 *" 0 .12 * 1 # ! " 2 02.20*# 12 # . [ ′[ x (cm) 5 10 15 20 ,0&! * 0&".!$[ 2 .!2 # !. * 0 .12 * 1 # 0 1 2" ! 3 !0" #.!$ # !.3".- - 2 !.1210" , . ,, .!020 . 2 " * .2 " . 2 0! / 2 # y (cm) (ÿÿ * .2 " x=0 2 2.$*22.// 1"2 #* .2 " 2 2 0 1&12 #* .2 " / 2 . !# 1 .) 2 1 1 !! 0,4 0,8 1,2 1,6 t (s) ." 2 2.$*22. 2. 2&1" . 2 02$# 1 0 )" ! # 2 # 0 .12 * 1 #2 !102.121[ FP -2 2$! 12 W V 0.1$0/10202 12 )2# 2 #* .2 ".22$! 12 W V [ FP \ = W − W10V\.[10FP >ù . FPV V FP / ] V 38. þ 2 0/" 0!1 * . 2 #" \ = P /3 rad) .//02..2 "!. * 0 *"0 .12 * 10 1 #.2202.20*# 12 # . [ ′[ þ 0 1&1/ 10&"2""2 #!102.12 .!$ 2 # . [ ′[ 0 . \ = \ &W 2 1$ . / 02.!.3.!12.12"31"2 #* .2 " 10 1# !21 0 2 .)12.1 [ .) 2 2 x (m) 2 $! 12 W V ..!0202 0! / 2 #* .2 ".2 "* .2 " )10 .2.$*22.// 1"2 #* .2 " .!%0202 0 1&12 #* .2 " /.!020.2$! 12 W V..2 1 0 ù2 #0 .12 * 1 #2 .$0.)2 .)12.1[ P i) 2 . !# 12 #.)211 !! ."2 # ii) 2 2.$*222 # iii) 2 02$# 12 # 0.1$0/10202 12 )2# 2 #* .2 "2$! 12 W V P >ù . VP \ = W − [ (SI) V P iii) 0] / i) 0 ii) − V
218
ü ü ùÿûü ÿ þüü ù
û9
S
.2 "!. *0 .12 * 1 #/.//02. * .0 1&12 # #0 . \ = W − [ (SI). #t1 ..!0202 0! / 2 #* .2 ".2 "* .2 " #t2 )10 .2.$*22.// 1"2 #* .2 " ût − NJ * ù # )22.2 # 1 #0 . ! = .!0202 2.12 #* .2 " .P 22// 12 #* .2 "/0 .!.2!02..+ 0.0 !0." = .
)/0 : ü . , = ) # : = ∑ P L 9 = 9 !û9 . û9 = 6 ⋅ # ⋅ ûW . 6 ⋅ ûW W P >ù . VP I = 2 2 2 !#y 02 = 8 ⋅ 10 −6 2 ] V m
39.
40. .1 0.# 2& .!013.!* .2. 012.0!1$* W. .!020 . 2.12 #* .2 "10.)12.1P.)2 0 !0. #.0!1010$! )/12 .V.)00/03 0.0 ./ * 20 cm 2 , #!102.101 0 ù2 .$0.)2 .)12.1P.0 .0212/0*# 1// 1"2 #* .2 " : 0)1.)12.1.)2 2.12 #* .2 "0 .1 0 ⋅ − P : >ù . ⋅ − - P@ P 41. .1 0.12.0!"1$* " 3 0 013.!* .2. .ù 2.12 #* .2 "10.)12.1 U = P .)2 0 . , .!02010 , )1.)12.1 U .)2 2.12 #* .2 "0 . , = * û 02.)2 2.12 #* .2 "1¶ .1 0 2 #0 .12 * 1 #0 .. 2 # 202!.+ # 2 # 2 #" 2. 2&1" 2& !& 2 # 0 .12 * 1 # 1¶ .#2) 2 1 0 û ./1$*0 , ; = N\ ; ) #N12.0! i) ù 10.)12.1 U = P .)2 2 2 "2. 2&1"2& !& 2 #0 .-
12 * 1 #0 . \ = ⋅ − P .!0202 2 "2"2. 2&1"2& !&
2 #0 .12 * 1 #12 .)12.1 U 2 #0!&2 .2 " . ii) )1 2 "!0 . 02. 01$*"2""+1202 2 "2"2. 2&1"2& !& 2 #0 .12 * 1 #12 .)12.1 U ./ .1.120 >ù . P L − P ii) 300% ]
42. 0 . 1 0 2 12 0 0*0! 0202. 03 0. 0! * # !0 0 32 # 0 12.0!)!# )12.) 0"2 02)û #!02.21 .03. 0.).! )* .2 0&! * 00!1 2 "2. 2&1"2""0 .12.0!)1 0 \ .!.2! * 0)2.2 " .".2 ."// 1"[2 #* .2 "1$ .2 -
ù ù
219
2././ $³)!´2. .. *2 # .)12.1G P ..!0202 0! / 2 #* .2 ".2 "* .2 " )10 .2.$*22.// 1"2 #* .2 " * ,0&! * 0)203 0.2 # 0! *0 .1)2! 1 .)2.22// 12 # * .2 "/0 $ # 0.+ 0.0 !0." i) ./. 1020.22 2 "2. 2&1"2& !& 2 # 0! *0 .22+ 02. .+". .!# ) .120.)2 ii) ù . 1210" /* !& 2" 03 0." 2 # 0! * .) 2 .!.&" 2& 03. 0.+ # 2& 0 .. 212 $. U U ./0 020)2.2. 22. \ U 2&1"2& !& .#2+ .1$*01$1 = \ U
)/0 .2 0!+2 .(ii): þ00 ) 0 0 !0..)2 0 ./..) #"2 #"* #"þ0 !0. 2. 2&1"0 .. 2 #202!.+ #2 # 2 #"2. 2&1"+./* * #" 0.2 0" U U .1$*0 ( = N ⋅ \ ⋅ U = N ⋅ \ ⋅ U ) #N12.0! P i þ0 !0. #0 0.2. 02.10 V * #"2& & 2. 0 .. .2& .2 & 2 #"þ $. 0 !0. # .!$012 * ..2. 02.10 0. *20! . 0! * 0. 2 01 .2. ..0 22&12 # 2 #"2"2. 2&1".+". .!# ) .120.)2 @ >ù . VP
43. û* .! * .2./ # 2 #"./."1#$ )22."/.// 2.12 / 0 .12) 1 12 /./0*# 1 0. 2023 !./ 2! 2.$*22."þ /0*# 1// 1"2& # 2& 0 . ." [ ′[ ,0&! * 0.!$2 # .2 1 0 212 . .!* 10" #! . * 2. .) 2. * .2. # 3 # 1¶ .#2) $ # 2 /. 0 1&1 \ = \ &W û ./ . W 0 .\ .9!.. ..)2.* .2.1212".!$"2 # . ..!0202 0 1&12 #121 #* .2 " #! *20.)21# 2& /* .#2+ # 2& .! 1/ !10202"10"2& /01 + .2"10"2& + .!0202 .)12.1 02. * i) /* /./ $+ /01 + ii) /* /./ $+ + iii) 0 )"/01 *.2"02 "2 # ." 7 / . 1$0/1020 2 12 )2# 2 # 121 # * .2 " 2" $! " 12 " W = . 7 W = [ >ù . \ = \ 1# W 10"/01 + [ û = N + N ∈ = 7 ii) iii) ] 10" + [ = N N ∈ = i)
220
ü ü ùÿûü ÿ þüü ù
44. ù+. ../01 )121 #* .2 " . # !.3".!.1210" . 2 #"101# !21 02 $!) . !# 1" .) 2 1 1 !! ." 10 1# !21 0 A ú û 2 $!) ú2 / . )1$ ./ 02.2 12 )2# 121 #* .2 " t=0 1020 2& .2$! 12 W = . . 1$0/1020 2 12 )2# 2 # * .2 " 2" $! " T T 3T , T. 12 " , , 4 2 4 0 0" !3" 0 3. 02. 0 !0. 2. 2&1" 2& !& 2 # .2 " 2" .!. &$! "12 " 45. ¶ .121 * ./* )!.2 #0 .12 * 1 #.$ # .)2 / /01 )û. 1210" . . 212 $. . .0 . 02. *2 #"/.3 !31" 0 .2 2 "2"2. 2&1" #020 * . 2 2 ".0 )".)2.* .2. #/ #! * 2 121 * .0 . \ >ù . UDG \ ] 46. þ0 1&1.! ** .2 "2 /.//02..2202.20*# 12 # . [ ′[ !0 .!.30 02 !3 \ = \ N[ − &W +.* . #/.//02..22 .! 2 .20*# 12 # . [ ′[ 0 1&1$02 !3 \ = \ N[ + &W ) # N = û* * .2.2. .1# # ./ #! * 121 * .10$ !/ 0 # #"$ # 0 1+10" \ = \ N[ − &W . \ = \ N[ + &W . ../0 020)20 1&12 #121 #* .2 "$02 !3 y = 2 y 0 kx1# &t. .!0202"10"2& /01 + .2& + .# 10202 .)12.1 02. * i) /* /./ $+ /01 + ii) /* /./ $+ + iii) 0 )"/01 *.2"02 "2 # ." >ù [ û = Q ii) iii) ] [ = Q + Q ∈ = L 47. +!. ) 0 " 0 .12) 1 020 02. .2 2 /0*# 1 2 # . [ ′[ û* 1 0.2 #ùú.$ # 02. *2 #".)12.1 FP . .!$ # . 2. . 2+ 2. .2.)!#3. 02 /.1#$ )22. +]. 2 " \ = FP .2 "2 #0 .12 * 1 #/.// 2.2./* 2 0/0!1.* .2. #.! 2. )&2"2. 2&1"2& 1 0& ù.ú "* .2 "0 . FP ,0&! * 0.!$2 # . [ ′[ 2 1 22".)12.1"ùú 02 ù.!120!.2 ú /0 ü1"0&! * 0.!$2 #$!) #2$! 12 .22 .2.* .2. 1# . 2+ 2.12 2.0 ..2 1 0 2\ .9!
ù ù
221
. . !%020 2 0 1&1 2 # 121 # * .2 " # ! *20 .) 2 1# 2& /* # 2&
.!0202"10".2 .! )2& /01 + . + #1$ .2 2. 21 * ..2 "2 #0 .12 * 1 #1$ .202. ) 02. *2& 1 0& ùúù!120!2 #ù./0 2 #ú2.* .2./.// 2..22 /..20*# 1 .!%0202 0 1&12". !# 1"\101# !21 02 $!) .2/0*20!! "2./0 . 022 1 0 2 [ >ù . \ = 1# W [\10FPW10V x û = (2k + 1) cm, − 5 ≤ k ≤ 4 /01 [ . = N FP − ≤ N ≤ 0" \ = W \10FPW10V@ 48. .$ !/020 02. 2&1 00 1&1 \ = 1#
[ W [\10FPW10V
.)1 0 .2 12 2 "2"2. 2&1" . !020 2 0! / 2 " * .2 " . 2 2.$*22. // 1" 2 # 2!$ 2 " * .2 "
.!%0202"0 1+10"2& 2!0$) 2& # 2& .)21# 2& & ! #%02 121 * . ,0&! * 0.!$2 # .2 1 0 2.2 212 W 0 .\ .9! P >ù . FP VFP V x x y1 = 4 2 5t − , y 2 = 4 2 5t + (x, y1, y210FPW10V @ 12 12 49.
.2 "!. * 0 *"0 .12 * 1 #/.// 2./* 0!1. 2 0/* .2.2. .0!!3 2..)2"0 1+10" [ [ \ = W − + . \ = W + , ) #2.[\1, y2 02!+ 2.10FP. $!) "W10V ..!0202 0! / 2 "* .2 ".2 2.$*22.// 1"2& # 2& . !%020 2 0 1&1 2 # 121 # * .2 " # ! *20 .) 2 1# 2& /* .#2+ # 2& .0 .2.$*22.0 )"1 0 #2 #0 .12 * 1 # 01# 202. [ FP 2$! 12 W = V P [ >ù . V , 6 cm, \ = 1# W [\10FPW10V 9 @ V
50. +!. ) 0 "0 .12) 1 020 02..22/0*# 12 # . [ ′[ ù! ) * ././/02..2 "2 #0 .12 * 1 #.2202.20*# 1.0!!30x 2..)2 0 1&1 y1 = 4 t − ) #2.[\1 02!+ 2.10FP.2 W10V 10
222
ü ü ùÿûü ÿ þüü ù
. . !%020 2 0 1&1 2 # 2 0/ *" * .2 " / # 2 #" 2 )2. 1# 0 02 ! * 0 / #!0.2 "2 #0 .12 * 1 #121 * ..0&!1020)2121x = 0 212 t = 0 0 .y .V > 0. . !%020 2 0 1&1 2 # 121 # * .2 " # / #!0 1# 2& /* ! * 0 & # 2& .!%0202"0 1+10"101# !21 02 $!) .2 2.$*22..2 02$# 10 )"1 0 #2 #0 .12 * 1 #2 !102.121 01# 202. x = 2,5 cm. [ [ >ù . \ = W + (x, y210FPW10V \ = 1# W [\10FPW10V FP FP V = 4 2 1# t 910 W10V . = −4 2 2 t .10 W10V @ V V .2 "!. * 0 *" 1 #2 020 02..22/0*# 1 [ ′[ / #!02.121 0!1 * .2 0!!302..)2 0 1&1 [ \ = 1# W [\10FPW10V . .!0202 2 "2 0! / 2 "* .2 " . 2 2.$*22. // 1" 2& /* 2!0$) 2& # 2& 2& & 1# / *!102 121 * .
.!%0202"0 1+10"2& 2!0$) 2& # 2& 2. . 021# 2 #"/ #! * 2 121 * . )1 0 . 2 2 " 2. 2&1" /* 1 0& ù ú 2 # 0 .12 * 1 # 2. . !1 2.12"10" [ = − FP . [ = + FP . 212 $. /.!0202 .! )Q2& + 2 #121 #* .2 " #1$ .2 2. 02. * 2& 1 0& ù.ú
002. # 0.2 .21#$ )22.2& 1# . ) 2& # 2& )20/ #!02..2 "2 #0 .12 * 1 # . 121 * .û.12+ # 0)2 02. * 2& 1 0& ù.ú2 #0 .12 * 1 #1$ .2 2.Q± 0"û0/ #)2 2.212.12& 1 0& ù.ú/0 02. 0 i) .!020 0 .2 "* .2 ". .0! / "2& # 2& #/ #! * 2 121 * . ii) .!%0202 0 1&12 # #121 #* .2 " P >ù . FPVFP V x x y1 = 3 2 5t − , y 2 = 3 2 5t + (x, y1, y210FPW10V 20 20 [ cm, s ii) \ = 1# / 0" 0 i) W [\10FPW10V @
51.
52. 0!. ) 0 "0 .12) 1 0 # #"/.// 2./* .! * .2.2. ./ #! * 121 * ..0!!3 2..)2"0 1+10" # .2)".! )" \ = \ &W − N[ + 3 . \ = \ &W + N[ ) # N = ./0 020)2
ù ù
223
& . N
.#.!02.!$31 3 . 02"10"2& /01 + .2& + 0 +/0 . 02 .)12.1 02. */./ $+ /01 + /./ $+ + 3 >ù 10"/01 + x û = 0 + (2n + 1) , n ∈ Z 4 4 3 10" + x = 0 + n , n ∈ Z ] 4 2 .2.$*22.// 1"2& # 2& 0 . # =
+ÿ ÿ+ ü ÿù !& 02!"0 1+10" [ = ⇔ [ = + N [ = − + N 1# [ = 1# ⇔ [ = ± + N [ = ⇔ [ = N 1# [ = ⇔ [ = + N [ = ± ⇔ [ = + N 1# [ = ± ⇔ [ = N !& 02! .! .! 1 .2 "./.3 !" 1# . − = 1# .1# + . 1# . + = 1# .1# − . 02.1$ .21 .! 1 2& 10 ) 0 . .+ . − . + = 1# 02.1$ .21 & 10.! 1 .2. .1# = . + + . −
224
ü ü ùÿûü ÿ þüü ù
ù ù ü!&210" . "0 " /. +. ... 2102012".!.2&0!&210" . "0 ".!0 .!%020 12 3* .. 210& 2 .! )2"0!+21"./0 .).#2) 2 ! . #. 212 $0121&12. 21 .!.2!1 !1 * 02.1* .#.2 2.$*22.2 #* .2 ".V .2 2.$*22.2. 2&1"2& !& 2 #0 .12 * 1 # 1.
.22// 10 )"* .2 "1¶ .0 .12) 1 . 02.3!02.* 02.3!02.0 !0..* ) .2.1 0.2 #0 .12 * 1 #$ # 2 /.312 /.$! 12 / 02.3!02.0 !0.. ! 0 !1 2.$*22.
2. õ2. .0! /)* .. 0 1 // 1" .2.$*222 # 012.0! 1#$ )222 # 012.0! 2 "* .2 "/0 02. 02. / 02. 2.2 "* .2 ".1#$ )222 # 3.
.2 "!. * 0 *"0 .12 * 1 #/.//02.0!1 .! )* . $&!".+ 00"0 !0." . . )!.2 #0 .12 * 1 #2. . 2+ 2..22/0*# 1// 1"2 #* .2 " $ .2 2.³)!´.³ /0"´ þ 122.$*22.2. 2&1"2& !& 2 #0 .12 * 1 #. !02..)2 1$1 9 = . /þ2.$*22.// 1"2 #* .2 ". !02..)2 0 1&1 # = &\ .
4.
.2 "!. * 0 *"0 .12 * 1 #/.// 2./. .! * .2.$&!".+ 00"0 !0." . . )!.2 #0 .12 * 1 #2. . 2+ 2.02.12/0*# 1// 1"2"/.2.!.$" þ 122.$*22.2. 2&1"2& !& 2 #0 .12 * 1 #. !02..)2 0 1&1 9 = . $ .2 2.³# + .2.´.³.!.+ .2.´ /þ2.$*22.// 1"2 #* .2 ". !02..)2 0 1&1 # = &\ .
198
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7.
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199
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200
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201
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208
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11. . # 0 2.1* .2 " ¶ .1 0 0 *".1)2! # 1 #!102.12.0!"1$* " ..!0.! * .2.$+! #./0 020)2 2.12 #* .2 " 02. 02.. 212!)3&". . 02 202!& 2".)12.1".)2 #.!02 * . 12. # 0121 * . .1 0.2 #0 .12 * 1 # 2./01 . . 0" 13. 0 !. ) 0 " 0 .12) 1 / #!02. 121 * . )1 0 . .)12.1 02. * ./* /./ $+ /01 + /* /./ $+ + 0 )"/01 *.)2 1120!2 # . 14.
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209
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17.
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ü ü ùÿûü ÿ þüü ù
210
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19.
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y t=0
y0
ù
(1)
ú
x
- y0
(2) t1
x
+
(3)
t2
û x .2 " !. * 0 *" 0 .12 * 1 #2 020 02.12/0*# 1 2 # . [ ′[ /.//02. 0!1 .! )* .2 0!!30[ 2..)2 0 1&1 \ = W − [\10FPW10V 0 ..)2..!.2&1# 3& 020/.3& 020..2 . * ././/02.! "2 .! 2.20*# 12 # . [ ′[ . 2 "2 #* .2 "0 .FP.0! /)"2 #V P "* .2 "0 .FP.2.$*22.// 1" . V / þ 2.$*22. 2" 2. 2&1" 2 # 1 0 # 2 # 0 .12 * 1 # 0 1# 202. cm . [ = FP 2$! 12 W V0 . V = 40 s
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ù ù
211
þ2.$*22.2. 2&1"2 #1 0 #ú2$! 12 W V0 . V = −40 /þ2.$*22.// 1"2 #* .2 "0 . # =
cm . s
P . V
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. 2 2. 2&1" /* 1 0& 2 # 0 .12 * 1 # 2. . .$ # .) 2 \ U . 1210" U U . 212 $.1# / 2. 02. *2 #" 021$1 = . \ U 10" U U . 212 $.1# / 2. 021$1
/ 2 "2. 2&1"2& 1 0& 2 #0 .12 * 1 #0 .. 0 !22 .)2 .)12.12 #".)2 .!.&"2& # 2& 22. û* .! * .2. /.// 2. .2 " !. * 0 *" 0 .12 * 1 # 12/0*# 12 # . [ ′[ 0. 2020".20#* 10".0!!3 2..)2"0 1+10" \ = \ N[ − &W . \ = \ N[ + &W ) # N = 0 ..)2..!.2&1# 3& 020/.3& 020..2 .þ0 1&12 #121 #* .2 " #/ #!01# 2& /# # 2& 0 . \ = \ N[ ⋅ 1# &W 10"2& + 0 . x K = (2n + 1) , n ∈ Z . 4 10"2& /01 + 0 . x û = (2n + 1) , n ∈ Z . 2
/ þ 2.$*22. 2. 2&1" 2& + 12" 10" x k = (4n + 1) , n ∈ ý / 02. .) 4 2 0 1&1 V = −2&y 0 &t
23.
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y +
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2
3
t2 t3
x
ü ü ùÿûü ÿ þüü ù
212
7 0 +2 12 )2# .
. 12 )2# . 212 $012$! 12 W =
7 $! 12 W 2 1 0 ù2 #0 .12 * 1 # 02.! "202.20*# 10 +2 1 0 ú 02.! "2 .! 2.20*# 1 þ/.3 !31" 02. *2& 2. . 2+10& #020 * 2.1 0.+.û0 .1 0 / 7 / 1 0 ù2 #0 .12 * 1 #2$! 12 W + ûW ) # ûW < 0! / "2"2. 2&1" #020 * 2.1 0.2 #0 .12 * 1 # $0 )&2"2. 2&1"2 # ) /# . 0 !0. 212 $012$! 12 W =
24. 2 1$ . ÿ / 2./* 12 )2#. . 121 #0!1 #* .2 "2 / #!02.10!. ) 0 "0 .12) 1 .0!!302..)2 0 1&1 [ W \ = \ 1# y +
(1)
(2)
(I)
2y0 t1 ú
W = W +
ù
x
û
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ü = f (x) (III)
ù
x
ù ù
213
0 ..)2..!.2&1# 3& 020/.3& 020..2 .2 12 )2# . 212 $02 /!. . ( . = I [ 2 #1$ .2 " ÿÿ 2 12 )2# . 212 $02 /!. . ( . = I [ 2 #1$ .2 " ÿÿÿ . )!.ùú2$! 12 W $ # 2.$*220" 9$ = − \ . 9% = \ . 212 $. /þ/.3 !31" 02. *2& 2. . 2+10& #020 * 2. )!.+û2 #0 .12 * 1 #0 .1 0 / 25.
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26. . #!102.121x = 0 2 # . [ ′[ .!$0212 t = 0, .020 0. .! 2. 2&1 00 1&1 \ = W 6, .!.) 0 * ./.P //02..2202.20*# 12 # . 02.$*22. # = V ..# 10202 "* .2 "2 #.! ** .2 " .!%0202 0 1&12 #* .2 " .$! 12 ..!$10 . 02. .1 0 2 # . [ ′[ #!102. 121x = 500 m; / $! 12 W V .!020.2 1 0 i) 2 . !# 12 #\.)211 !! ."2 # ii) 2 2.$*222 # iii) 2 02$# 12 # [ P >ù . P \ = W − 6, V / L LL iii) 0] V 27. þ0 1&1!. *.! ** .2 "0 . \ = W − [ 6, .!020 .2 2.$*22.2 #* .2 " 2 .)12.1/* 1 0& 2. . .$! 12 $ # 02. *2 #"/.3 !31" /.3 !31"0 )"1 0 # 02. *2& $! + 12 + W = V . W = V P >ù . P UDG@ V
214
ü ü ùÿûü ÿ þüü ù
28. .2.!$0 .020 0212 t . .! 2. 2&1 00 1&1 \ = W 6, .!.) 0 * ././/02..2202.20*# 12 # . P [ ′[ 02.$*22. # = V ..!0202 0! / 21#$ )22..2 "* .2 " .!%0202 0 1&12 #* .2 " .!%0202"0 1+10" #/ # 2 2.$*22.2. 2&1".2 02$# 110 1# !21 02 $!) . .1 0 #!102.121 [ = P /..!.121020!.323132"2. 2&1".2./3 !.1 0.2 # .2[101# !21 021# 202. [2$! 12 W V >ù . V+]P \ = W − [ (SI) 9 = 1# W − (SI), . = − W − (SI) / 3 ±x (SI)] 29. þ 2 0/"* ././/02..2202.20*# 12 # . [ ′[ 10!. ) 0 "0 .12) 1 .$0 "* .2 " Pþ0 1&1/ 10&"2"" . !102. 12 .!$ 2 # . 0 . \ = \ W . $! 12 W /* # 1 0. 2. . !1 2. & 12 . $ # . 212 $. 310" . 3 = 3 = ... /0 020)22 * ././/02. 0.20*# 1.)2 1 0 ! "2 1 0 .# 10202 .)12.1 >ù P@ 30. ò . 2 0/"* ././/02.! "2 .! 2.20*# 12 # . [ ′[ .$0 2.0 "$.!.2!12 \ = P = P = +] .!%0202 0 1&12 # * .2 "100 ..)2".!.2&0!2+10" .21[ .W 0 .\ .V > 0. 21[ m.W 0 .\ .9! 7 >ù . \ = W + [ 6, y = 0,05 2 4 t + 1,25x + (SI)] 8 31. û 02.2 .! )* . 00 1&1 \ = W − [ + (SI). ..# 1020 i) 2 2.$*22.2 #* .2 " ii) 2 122.$*22.2. 2&1"0 )"1 0 #2 #0 .12 * 1 # .!0202 2.$*22.2"2. 2&1"212 t = 0, 0 )"1 0 #2 #0 .12 * 1 # #!102.121x = 25m. .!0202 2.$*22.2"2. 2&1"212 t = 0,1s, 0 )"1 0 #2 #0 .12 * 1 # #!102.121x = 50m. P P P P ii) 9 = 9 = − 9 = ] >ù . i) V V V V
ù ù
215
32. ò . 2 0/" * . /.//02. ! " 2 02 .20*# 1 2 # . [ ′[ . $0 2 " \ = P "* .2 " P.1#$ )22. +]þ2 #* .2 " !102. 12 .!$ 2 2 # . +. W 12 1 [ . !# 1 0 . y = 0,1 m. . .!020 i) 2 # .2).! ) N = . ii) 2 0! / .2 # 1#$ )22.2 #* .2 " iii)2 2.$*22.// 1"2 #* .2 " .! 1/ !10202 .!$312 #* .2 ". .!%0202 0 1&12 #* .2 " UDG P iii) >ù . L m–1 ii) V , V V , \ = W − [ + (SI)] 33. .!.&" 2 0/+ # 2& !102.12 .!$2 0 *"$ !/" 0 # #"þ0 1&1/ 10&"2 #1 0 #20 . \ = ⋅ − W 6, .2 .!.) 0 * ././/02..2202.20*# 1 "* .2 "0 . P . )10 .2.$*22.// 1"2 #* .2 " .!%0202 0 1&12 #* .2 " )10 .. !# 1.)211 !! ."2 #2$! 12 W = V 0 )"1 0 #2"$ !/"2 .$0.)2 2.)12.1 [ = P P >ù . \ = ⋅ − W − [ 6, ⋅ − P ] V 34.
.2 "!. * 0 *"0 .12 * 1 #/.//02.! "202.20*# 1 2 # . [ ′[ 0!1 2 0/" * . 0 2 " \ = FP " * .2 " P FP.2.$*22. # = $! 12 W 2 * .3 012 1 0 V .!$2 # . [ ′[ +.2 1 0 / 02.)22$! 12 W /!$02..)2 11 !! ."2 # * 0 .22 .! 23 ! . .!%0202 0 1&12". !# 1"2 #1 0 #101# !21 02 $!) .!%0202 0 1&12". !# 1"0 )"1 0 ù2 #0 .12 * 1 #2 $01# 202. [ cP101# !21 02 $!) .1$0/10202 !32 #0 .12 * 1 #/0 2 #1 0 #2"$! "12T 3T 5T " 0, , T, , 2T, . 2 2 2 >ù . \ = W + \ = W − \ FP W V ]
216
ü ü ùÿûü ÿ þüü ù
35. ! !. * 0 *"0 .12 * 1 #.!$0212 t = 0, .020 0. 0&2.! 2. 2&1 00 1&1 \ = W W10V\10FP )20/.//0P 2..2 "2 # .2[* . 02.$*22. # = V .)1 0 .2 "* .2 " )20.!$0 .2. . 2+ 02. .1 0 2 #0 .12 * 1 #2 .$0.) 2 .)12.1[ P .!%0202 0 1&12". !# 1"2 #1 0 #. .# 10202 2 2"2$! 12 W1 = 5,625 s. .0 .2 2"31"2 #1 0 #2 .!. &$! 12 / )1 .$0.)2 1 0 .1 0 2 2 /.$! 12 W1 = 5,625 s) $031 3 = + .2 .3 !/.//02.2 * . 0..!.121020!.32 02. 2"31"2 #1 0 #101# !21 02 $!) >ù . P V \ 0 = W − W 10 V \ 10 FP , 10 cm, 3 = + / P .)2 ! "2 @ 36. ,0&! * 01 0..!.&"# 2& 2" .". !# 1.)21 1 !! ."/ 02..)2 0 1&1 \ = W + 3 W10V\10FP $! 12 W !102.12 1202. !# 1.)21 1 !! ."2"õ2. 0! 0.)211 !! ."2".2!23 !2 * . #.!02..).#2 $0/./ 010.)12.1G FP ..!0202& . 3 .!%0202 0 1&12 #* .2 " #/.//02..2 "2 # . [ ′[ 2 #0 .12 * 1 # 0.!$212"".! "2023 ! .!%0202"0 1+10" #/ # 2 2.$*22.2"2. 2&1".2 02$# 1 101# !21 02 $!) . . )! 2 #0 .12 * 1 # #!102.121 x = 10 cm. /..!.121020!.323132"2. 2&1".2./3 !.1 0.2 # .2[101# !21 02 .)12.12 #"[.)2 2$! 12 W V 0.1$0/10202 12 )2# 2 #* .2 "2$! 12 W V [ >ù . \ = W + − W10V x\10FP FP FP ), . = − W − W10V.10 )] 9 = 1# W − W10VV10 V V
ù ù
217
37. 2. 1$ .2. 3. 2. /* !.3" .y (cm) (ÿ !.1210" . 0!1 2 0/" * . t=0 2 2 /.//02..2 "!. * 0 *" 0 .12 * 1 # ! " 2 02.20*# 12 # . [ ′[ x (cm) 5 10 15 20 ,0&! * 0&".!$[ 2 .!2 # !. * 0 .12 * 1 # 0 1 2" ! 3 !0" #.!$ # !.3".- - 2 !.1210" , . ,, .!020 . 2 " * .2 " . 2 0! / 2 # y (cm) (ÿÿ * .2 " x=0 2 2.$*22.// 1"2 #* .2 " 2 2 0 1&12 #* .2 " / 2 . !# 1 .) 2 1 1 !! 0,4 0,8 1,2 1,6 t (s) ." 2 2.$*22. 2. 2&1" . 2 02$# 1 0 )" ! # 2 # 0 .12 * 1 #2 !102.121[ FP -2 2$! 12 W V 0.1$0/10202 12 )2# 2 #* .2 ".22$! 12 W V [ FP \ = W − W10V\.[10FP >ù . FPV V FP / ] V 38. þ 2 0/" 0!1 * . 2 #" \ = P /3 rad) .//02..2 "!. * 0 *"0 .12 * 10 1 #.2202.20*# 12 # . [ ′[ þ 0 1&1/ 10&"2""2 #!102.12 .!$ 2 # . [ ′[ 0 . \ = \ &W 2 1$ . / 02.!.3.!12.12"31"2 #* .2 " 10 1# !21 0 2 .)12.1 [ .) 2 2 x (m) 2 $! 12 W V ..!0202 0! / 2 #* .2 ".2 "* .2 " )10 .2.$*22.// 1"2 #* .2 " .!%0202 0 1&12 #* .2 " /.!020.2$! 12 W V..2 1 0 ù2 #0 .12 * 1 #2 .$0.)2 .)12.1[ P i) 2 . !# 12 #.)211 !! ."2 # ii) 2 2.$*222 # iii) 2 02$# 12 # 0.1$0/10202 12 )2# 2 #* .2 "2$! 12 W V P >ù . VP \ = W − [ (SI) V P iii) 0] / i) 0 ii) − V
218
ü ü ùÿûü ÿ þüü ù
û9
S
.2 "!. *0 .12 * 1 #/.//02. * .0 1&12 # #0 . \ = W − [ (SI). #t1 ..!0202 0! / 2 #* .2 ".2 "* .2 " #t2 )10 .2.$*22.// 1"2 #* .2 " ût − NJ * ù # )22.2 # 1 #0 . ! = .!0202 2.12 #* .2 " .P 22// 12 #* .2 "/0 .!.2!02..+ 0.0 !0." = .
)/0 : ü . , = ) # : = ∑ P L 9 = 9 !û9 . û9 = 6 ⋅ # ⋅ ûW . 6 ⋅ ûW W P >ù . VP I = 2 2 2 !#y 02 = 8 ⋅ 10 −6 2 ] V m
39.
40. .1 0.# 2& .!013.!* .2. 012.0!1$* W. .!020 . 2.12 #* .2 "10.)12.1P.)2 0 !0. #.0!1010$! )/12 .V.)00/03 0.0 ./ * 20 cm 2 , #!102.101 0 ù2 .$0.)2 .)12.1P.0 .0212/0*# 1// 1"2 #* .2 " : 0)1.)12.1.)2 2.12 #* .2 "0 .1 0 ⋅ − P : >ù . ⋅ − - P@ P 41. .1 0.12.0!"1$* " 3 0 013.!* .2. .ù 2.12 #* .2 "10.)12.1 U = P .)2 0 . , .!02010 , )1.)12.1 U .)2 2.12 #* .2 "0 . , = * û 02.)2 2.12 #* .2 "1¶ .1 0 2 #0 .12 * 1 #0 .. 2 # 202!.+ # 2 # 2 #" 2. 2&1" 2& !& 2 # 0 .12 * 1 # 1¶ .#2) 2 1 0 û ./1$*0 , ; = N\ ; ) #N12.0! i) ù 10.)12.1 U = P .)2 2 2 "2. 2&1"2& !& 2 #0 .-
12 * 1 #0 . \ = ⋅ − P .!0202 2 "2"2. 2&1"2& !&
2 #0 .12 * 1 #12 .)12.1 U 2 #0!&2 .2 " . ii) )1 2 "!0 . 02. 01$*"2""+1202 2 "2"2. 2&1"2& !& 2 #0 .12 * 1 #12 .)12.1 U ./ .1.120 >ù . P L − P ii) 300% ]
42. 0 . 1 0 2 12 0 0*0! 0202. 03 0. 0! * # !0 0 32 # 0 12.0!)!# )12.) 0"2 02)û #!02.21 .03. 0.).! )* .2 0&! * 00!1 2 "2. 2&1"2""0 .12.0!)1 0 \ .!.2! * 0)2.2 " .".2 ."// 1"[2 #* .2 "1$ .2 -
ù ù
219
2././ $³)!´2. .. *2 # .)12.1G P ..!0202 0! / 2 #* .2 ".2 "* .2 " )10 .2.$*22.// 1"2 #* .2 " * ,0&! * 0)203 0.2 # 0! *0 .1)2! 1 .)2.22// 12 # * .2 "/0 $ # 0.+ 0.0 !0." i) ./. 1020.22 2 "2. 2&1"2& !& 2 # 0! *0 .22+ 02. .+". .!# ) .120.)2 ii) ù . 1210" /* !& 2" 03 0." 2 # 0! * .) 2 .!.&" 2& 03. 0.+ # 2& 0 .. 212 $. U U ./0 020)2.2. 22. \ U 2&1"2& !& .#2+ .1$*01$1 = \ U
)/0 .2 0!+2 .(ii): þ00 ) 0 0 !0..)2 0 ./..) #"2 #"* #"þ0 !0. 2. 2&1"0 .. 2 #202!.+ #2 # 2 #"2. 2&1"+./* * #" 0.2 0" U U .1$*0 ( = N ⋅ \ ⋅ U = N ⋅ \ ⋅ U ) #N12.0! P i þ0 !0. #0 0.2. 02.10 V * #"2& & 2. 0 .. .2& .2 & 2 #"þ $. 0 !0. # .!$012 * ..2. 02.10 0. *20! . 0! * 0. 2 01 .2. ..0 22&12 # 2 #"2"2. 2&1".+". .!# ) .120.)2 @ >ù . VP
43. û* .! * .2./ # 2 #"./."1#$ )22."/.// 2.12 / 0 .12) 1 12 /./0*# 1 0. 2023 !./ 2! 2.$*22."þ /0*# 1// 1"2& # 2& 0 . ." [ ′[ ,0&! * 0.!$2 # .2 1 0 212 . .!* 10" #! . * 2. .) 2. * .2. # 3 # 1¶ .#2) $ # 2 /. 0 1&1 \ = \ &W û ./ . W 0 .\ .9!.. ..)2.* .2.1212".!$"2 # . ..!0202 0 1&12 #121 #* .2 " #! *20.)21# 2& /* .#2+ # 2& .! 1/ !10202"10"2& /01 + .2"10"2& + .!0202 .)12.1 02. * i) /* /./ $+ /01 + ii) /* /./ $+ + iii) 0 )"/01 *.2"02 "2 # ." 7 / . 1$0/1020 2 12 )2# 2 # 121 # * .2 " 2" $! " 12 " W = . 7 W = [ >ù . \ = \ 1# W 10"/01 + [ û = N + N ∈ = 7 ii) iii) ] 10" + [ = N N ∈ = i)
220
ü ü ùÿûü ÿ þüü ù
44. ù+. ../01 )121 #* .2 " . # !.3".!.1210" . 2 #"101# !21 02 $!) . !# 1" .) 2 1 1 !! ." 10 1# !21 0 A ú û 2 $!) ú2 / . )1$ ./ 02.2 12 )2# 121 #* .2 " t=0 1020 2& .2$! 12 W = . . 1$0/1020 2 12 )2# 2 # * .2 " 2" $! " T T 3T , T. 12 " , , 4 2 4 0 0" !3" 0 3. 02. 0 !0. 2. 2&1" 2& !& 2 # .2 " 2" .!. &$! "12 " 45. ¶ .121 * ./* )!.2 #0 .12 * 1 #.$ # .)2 / /01 )û. 1210" . . 212 $. . .0 . 02. *2 #"/.3 !31" 0 .2 2 "2"2. 2&1" #020 * . 2 2 ".0 )".)2.* .2. #/ #! * 2 121 * .0 . \ >ù . UDG \ ] 46. þ0 1&1.! ** .2 "2 /.//02..2202.20*# 12 # . [ ′[ !0 .!.30 02 !3 \ = \ N[ − &W +.* . #/.//02..22 .! 2 .20*# 12 # . [ ′[ 0 1&1$02 !3 \ = \ N[ + &W ) # N = û* * .2.2. .1# # ./ #! * 121 * .10$ !/ 0 # #"$ # 0 1+10" \ = \ N[ − &W . \ = \ N[ + &W . ../0 020)20 1&12 #121 #* .2 "$02 !3 y = 2 y 0 kx1# &t. .!0202"10"2& /01 + .2& + .# 10202 .)12.1 02. * i) /* /./ $+ /01 + ii) /* /./ $+ + iii) 0 )"/01 *.2"02 "2 # ." >ù [ û = Q ii) iii) ] [ = Q + Q ∈ = L 47. +!. ) 0 " 0 .12) 1 020 02. .2 2 /0*# 1 2 # . [ ′[ û* 1 0.2 #ùú.$ # 02. *2 #".)12.1 FP . .!$ # . 2. . 2+ 2. .2.)!#3. 02 /.1#$ )22. +]. 2 " \ = FP .2 "2 #0 .12 * 1 #/.// 2.2./* 2 0/0!1.* .2. #.! 2. )&2"2. 2&1"2& 1 0& ù.ú "* .2 "0 . FP ,0&! * 0.!$2 # . [ ′[ 2 1 22".)12.1"ùú 02 ù.!120!.2 ú /0 ü1"0&! * 0.!$2 #$!) #2$! 12 .22 .2.* .2. 1# . 2+ 2.12 2.0 ..2 1 0 2\ .9!
ù ù
221
. . !%020 2 0 1&1 2 # 121 # * .2 " # ! *20 .) 2 1# 2& /* # 2&
.!0202"10".2 .! )2& /01 + . + #1$ .2 2. 21 * ..2 "2 #0 .12 * 1 #1$ .202. ) 02. *2& 1 0& ùúù!120!2 #ù./0 2 #ú2.* .2./.// 2..22 /..20*# 1 .!%0202 0 1&12". !# 1"\101# !21 02 $!) .2/0*20!! "2./0 . 022 1 0 2 [ >ù . \ = 1# W [\10FPW10V x û = (2k + 1) cm, − 5 ≤ k ≤ 4 /01 [ . = N FP − ≤ N ≤ 0" \ = W \10FPW10V@ 48. .$ !/020 02. 2&1 00 1&1 \ = 1#
[ W [\10FPW10V
.)1 0 .2 12 2 "2"2. 2&1" . !020 2 0! / 2 " * .2 " . 2 2.$*22. // 1" 2 # 2!$ 2 " * .2 "
.!%0202"0 1+10"2& 2!0$) 2& # 2& .)21# 2& & ! #%02 121 * . ,0&! * 0.!$2 # .2 1 0 2.2 212 W 0 .\ .9! P >ù . FP VFP V x x y1 = 4 2 5t − , y 2 = 4 2 5t + (x, y1, y210FPW10V @ 12 12 49.
.2 "!. * 0 *"0 .12 * 1 #/.// 2./* 0!1. 2 0/* .2.2. .0!!3 2..)2"0 1+10" [ [ \ = W − + . \ = W + , ) #2.[\1, y2 02!+ 2.10FP. $!) "W10V ..!0202 0! / 2 "* .2 ".2 2.$*22.// 1"2& # 2& . !%020 2 0 1&1 2 # 121 # * .2 " # ! *20 .) 2 1# 2& /* .#2+ # 2& .0 .2.$*22.0 )"1 0 #2 #0 .12 * 1 # 01# 202. [ FP 2$! 12 W = V P [ >ù . V , 6 cm, \ = 1# W [\10FPW10V 9 @ V
50. +!. ) 0 "0 .12) 1 020 02..22/0*# 12 # . [ ′[ ù! ) * ././/02..2 "2 #0 .12 * 1 #.2202.20*# 1.0!!30x 2..)2 0 1&1 y1 = 4 t − ) #2.[\1 02!+ 2.10FP.2 W10V 10
222
ü ü ùÿûü ÿ þüü ù
. . !%020 2 0 1&1 2 # 2 0/ *" * .2 " / # 2 #" 2 )2. 1# 0 02 ! * 0 / #!0.2 "2 #0 .12 * 1 #121 * ..0&!1020)2121x = 0 212 t = 0 0 .y .V > 0. . !%020 2 0 1&1 2 # 121 # * .2 " # / #!0 1# 2& /* ! * 0 & # 2& .!%0202"0 1+10"101# !21 02 $!) .2 2.$*22..2 02$# 10 )"1 0 #2 #0 .12 * 1 #2 !102.121 01# 202. x = 2,5 cm. [ [ >ù . \ = W + (x, y210FPW10V \ = 1# W [\10FPW10V FP FP V = 4 2 1# t 910 W10V . = −4 2 2 t .10 W10V @ V V .2 "!. * 0 *" 1 #2 020 02..22/0*# 1 [ ′[ / #!02.121 0!1 * .2 0!!302..)2 0 1&1 [ \ = 1# W [\10FPW10V . .!0202 2 "2 0! / 2 "* .2 " . 2 2.$*22. // 1" 2& /* 2!0$) 2& # 2& 2& & 1# / *!102 121 * .
.!%0202"0 1+10"2& 2!0$) 2& # 2& 2. . 021# 2 #"/ #! * 2 121 * . )1 0 . 2 2 " 2. 2&1" /* 1 0& ù ú 2 # 0 .12 * 1 # 2. . !1 2.12"10" [ = − FP . [ = + FP . 212 $. /.!0202 .! )Q2& + 2 #121 #* .2 " #1$ .2 2. 02. * 2& 1 0& ù.ú
002. # 0.2 .21#$ )22.2& 1# . ) 2& # 2& )20/ #!02..2 "2 #0 .12 * 1 # . 121 * .û.12+ # 0)2 02. * 2& 1 0& ù.ú2 #0 .12 * 1 #1$ .2 2.Q± 0"û0/ #)2 2.212.12& 1 0& ù.ú/0 02. 0 i) .!020 0 .2 "* .2 ". .0! / "2& # 2& #/ #! * 2 121 * . ii) .!%0202 0 1&12 # #121 #* .2 " P >ù . FPVFP V x x y1 = 3 2 5t − , y 2 = 3 2 5t + (x, y1, y210FPW10V 20 20 [ cm, s ii) \ = 1# / 0" 0 i) W [\10FPW10V @
51.
52. 0!. ) 0 "0 .12) 1 0 # #"/.// 2./* .! * .2.2. ./ #! * 121 * ..0!!3 2..)2"0 1+10" # .2)".! )" \ = \ &W − N[ + 3 . \ = \ &W + N[ ) # N = ./0 020)2
ù ù
223
& . N
.#.!02.!$31 3 . 02"10"2& /01 + .2& + 0 +/0 . 02 .)12.1 02. */./ $+ /01 + /./ $+ + 3 >ù 10"/01 + x û = 0 + (2n + 1) , n ∈ Z 4 4 3 10" + x = 0 + n , n ∈ Z ] 4 2 .2.$*22.// 1"2& # 2& 0 . # =
+ÿ ÿ+ ü ÿù !& 02!"0 1+10" [ = ⇔ [ = + N [ = − + N 1# [ = 1# ⇔ [ = ± + N [ = ⇔ [ = N 1# [ = ⇔ [ = + N [ = ± ⇔ [ = + N 1# [ = ± ⇔ [ = N !& 02! .! .! 1 .2 "./.3 !" 1# . − = 1# .1# + . 1# . + = 1# .1# − . 02.1$ .21 .! 1 2& 10 ) 0 . .+ . − . + = 1# 02.1$ .21 & 10.! 1 .2. .1# = . + + . −
224
ü ü ùÿûü ÿ þüü ù
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