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MATEMATIKA TEKNIK 1.BILANGAN
2. MATRIK dan DETERMINASI
3. FUNGSI
4. LIMIT
5. DERET
6. DIFERENSIAL danAPLIKASINYA
7. INTEGRASI danAPLIKASINYA
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I. BILANGAN 1. Bian!an Pada "ian!an a#i $1% 2% 3% .........& da'a( diada)an*'+,a#i-*'+,a#i da#a, ai(/0
a. P+n/aan dan '+n!/,an!an
" P+,)aian dan '+"a!ian . P+n!)a( dan +na,i) a)a,
i)a a dan " "ian!an a#i a)a a " dan a 8 " d%di ana dan d /!a
+,/'a)an "ian!an a#i.
S)+a i'/nan "ian!an
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S)+a i'/nan "ian!an
Bian!anK*'+)#
Bian!an Na(a
$ R+a &
Bian!an Kaa
$ Ia!in+, &
Bian!anRa#i*na
Bian!anI,a#i*na
Bian!an B/a( Bian!anP+aan
Bian!an B/a(N+!a(i9
Bian!an B/a(P*#i(i9 $ A#i &
N*
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• Bian!an R+a
:i'/nan "ian!an ,+a (+,di,i da,i0
a. Bian!an Ra#i*na% ai(/ "ian!an-"ian!an "/a('*#i(i9% "/a( n+!a(i9% n*
dan '+aan a;" $di ana a dan " "ian!an "/a( &". Bian!an i,,a#i*na% ai(/ "ian!an-"ian!an an! (ida)da'a( dina(a)an
#+"a!ai "+n(/) '+aan.
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. Si9a( a#*#ia(i9
Un(/) n1% n2% n3 = N "+,a)/0
• $ n1 n2 & n3 n1 $ n2 n3 &
•
$ n1
8 n2
& n3
n1
$n2
8 n3
&
d. Si9a( *d//#
i)a n = N% a)a n > > n n
Si9a( *d//# '+n/aan $ > "ian!an )+#a(/an /n(/)
'+n/aan > = N & Un(/) n 8 1 1 8 n
Si9a( *d//# '+,)aian $ 1 adaa "ian!an )+#a(/an/n(/) '+,)aian 1 = N
+. Si9a( di#(,i"/(i9
Un(/) n1% n2% n3 = N "+,a)/0
• $n1 n2& n3 n1 n3 n2 n1 #i9a( di#(,i"/(i9 )i,i
• n1 $n2 n3& n1 n2 n1 n3 #i9a( di#(,i"/(i9 )i,i
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9. Si9a( in?+#;)+"ai)an /n(/) '+n/aan
• i)a a = I% (+,da'a( "ian!an @ a #+in!!a a $ -a & >
• i)a a = I dan a >% (+,da'a( "ian!an$)+"ai)an;in?+,# da,i a& #+d+i)ian
#+in!!a0 a. 1. Bian!an di(/i# /!a a-1.
• i)a a% " = I dan " > a)a di#+"/( a#i "a!ida,i a *+ "
a di#+"/( '+"ian!% " di#+"/( '+n+"/(% )aa/"/)an #/a(/ "ian!an
"/a( a)a di#+"/( '+aan.
•Bian!an 8 di#+"/( 'an!)a( n da,i "ian!an a "ia "+,a)/8 a.a.a n "/a% di(/i# /!a 8 an.
•Bian!an 8 di#+"/( a)a, n da,i "ian!an a "ia "+,a)/ a 8n.
a 8n di(/i# /!a
2
a
1
a
1
ba
b
a 1=
b
a
na x =
2442 === x
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2. P+,(ida)#aaan
a di)a(a)an +"i "+#a, da,i'ada " di(/i# a " i)a "+"i )+i da,i'ada a a(a/ " C a% adi a " i)a dan ana" C a.
Tanda % C% $+"i "+#a, a(a/ #aa&% dan $+"i )+ia(a/ #aa& di#+"/( )+(ida)#aaan.
Si9a(-#i9a(0
• a > i)a dan ana i)a a '*#i(i9
• a C > i)a dan ana i)a a n+!a(i9
• a > i)a dan ana i)a @a C >
• a C > i)a dan ana i)a @a >
• i)a a C " dan " C % a)a a C
• i)a a C "% a)a a C " /n(/) #+(ia' • i)a a C " dan C d% a)a a C " d
• i)a a C " dan '*#i(i9% a)a a C "
• i)a a C " dan n+!a(i9% a)a a "
• i)a > C a C " dan > C C d% a)a a C "d
(anda C "*+ di!an(i d+n!an (anda dan (anda
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3. In(+,?a $S+an!&
• In(+,?a "/)a
In(+,?a (+,"/)a In(+,?a (+,"/)a da,i a )+ " di(/i# $a% "&%a)a i'/nan "a!ian da,i R adaa0 $a% "& 8 a C 8
C "H% da'a( di!a"a,)an0
• In(+,?a (/(/' "/)a $#+(+n!a (+,"/)a& In(+,?a (/(/'"/)a da,i a )+ " di(/i# a%"&% a)a i'/nan "a!ian da,i
R adaa0 a% "& 8 a 8 C "H% da'a( di!a"a,)an0
• In(+,?a (/(/'
In(+,?a (+,(/(/' da,i a )+ " di(/i# a% "J% a)a i'/nan"a!ian da,i R adaa0
a% "J 8 a 8 "H% da'a( di!a"a,)an0
• In(+,?a "/)a (/(/' $in(+,?a #+(+n!a (+,(/(/'&
In(+,?a #+(+n!a (+,(/(/' da,i a )+ " di(/i# $a% "J% a)a
i'/nan da,i R adaa0 $a% "J 8 a C 8 "H% da'a(di!a"a,)an0
a "
a "
a "
a "
I ( i ( ( ) ( i
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In(+,?a-in(+,?a (a) (+,in!!a0
• A 8 8 C aH 8 - C 8 C aH -% aH
• B 8 8 aH 8 a C 8 C H a% H
• < 8 8 aH 8 - C 8 aH -% aH
• D 8 8 aH 8 a 8 C H a% H
• E 8 = R H 8 - H - % H
aa-
a
- a
a
-
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S*a La(ian;PR I1. T+n(/)an "+#a,an a,!a 8 #+"a!ai "ian!an
"/a( "ia0
a&. 84 256 "&. 27 83
2. T+n(/)an '+,(ida)#aaan i)a0
a&. 4 a C " di)ai)an d+n!an 2 dan -2"&. -2 8 5 d+n!an +n!!an(i (anda d+n!an
3. T+n(/)an a,!a /(a) da,i0
a&. 2 $ 6 @ &
"&. 2 8 1>>
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5. Bian!an +
Bian!an + "ia#a di!/na)an daa R//#-,//# I/Fi#i)a dan I/ T+)ni)
! = fakulteit/factorail/fakultas 5 ! = 5.4...".= "#
Bila$%a$ e = & '"(("((45) *iasa +itulis +e$%a$ 5 +esi,al&-aitu & '"((
+a$ +ise*ut Bila$%a$ Tra$se$+e$tal
• ika $ a+ala *ila$%a$ asli ,aka $ faktorial 0 $! 1 = $ 0$ 2 "1 0$3 1.........."
Berlaku ula $ ! = $ 0$ 3 "1 !
o$to6 " ! = "7 ! = ."7 ! = .."7 4 ! = 4..." = 4
•
....597182818284,2!
1...............
!3
1
!2
1
!1
11)1(
)1
1(
1
0
=++++++=+=
+=
→
∞→
nk mil
h
mil e
k
h
h
h
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6. P+,/(a#i dan K*"ina#i
• i)a da,i i'/nan n /n#/, $*"+)& an! "+,ainan% a)a"ana)na #/#/nan $a,a '+n!/,/(an& /n#/, i(/ di#+"/(
'+,/(a#i i'/nanO (+,#+"/(. i)a da,i i'/nan n /n#/, an! "+,ainan di"/a(#/#/nan an! (+,di,i da,i
/n#/, $ ) C n &% a)a "ana) #/#/nan i(/ di#+"/("ana)na '+,/(a#i
(+,#+"/( d+n!an a"ian )% di,//#)an0
di ana0 n "ana)na /n#/, $*"+)& dan ) "ana)na an!!*(a
daa )+*'*).
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". Mi#a)an : A% B% < H
B+,a'a "ana) #/#/nan i'/nan (+,#+"/( da'a( di"/a(Q
aa"0 "ana)na /n#/, 3% "ana)na an!!*(a daa)+*'*) 3
n 3 dan ) 3
adi0 ada 6 #/#/nan
• i)a da,i i'/nan n "/a /n#/, an! "+,ainan a)andi#/#/n d+n!an a#in!-a#in! #/#/nan an! (+,di,i da,i )/n#/, $ ) n & (an'a +'+,a(i)an /,/(anna% a)a"ana) #/#/nan i(/ di#+"/( "ana)na )*"ina#ii'/nan (+,#+"/( d+n!an a"i)an ) di,//#)an0
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aa"an0 n % ) 6
ada 4 ,+!/
i)a #+)+*'*) (+,#+"/( (+,di,i da,i 6 *,an! a)a"ana) ,+!/ da'a( di#/#/n0
1 ,+!/
) #/)/ )+ $i @ 1&% i#0 #/)/ )+ 4 ) 4 -1 3% a)a)*+#i+n "in*i/O
#/)/ )+ i dina(a)an d+n!an0
#/)/ )+ 4 adaa0
846
504
1.2.3
7.8.9
!3!.6
1.2.3.4.5.6.7.8.9
!)69(!6
!9
!)(!
!====
−=
−=
k nk
n
C k
n
1!6
!6
!)66(!6
!6
!)(!
!==
−=
−=
k nk
n
C k
n
)1().....2)(1(!!)(!
!
−−−−=⇒−=
k nnnnnk nk
nC n
k
33
!3
)2)(1(
!)(!
!ba
nnnba
k nk
nU
nk k nn
k
−− −−=−
=
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aa"0 a. "ana)na /n#/, 3% "ana)na an!!*(a daa)+*'*) 3
$
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7. Bin*i/ N+(*n
Bin*i/ N+(*n adaa '+n++#aian "in*i/ $#/)/d/a& d+n!an ,//#0
) #/)/ )+ $i @ 1&
#/)/ )+...1% ...........2%..............d#(............n% di ana adan "% #+,(a n adaa
"ian!an a#i
K*+#i+n "in*i/O da'a( di,//#)an d+n!an0
S/)/ )+...$S& % diana ) S @ 1% a)a0
Pandan! "+n(/)-"+n(/) di "aa ini0
• $ a " &n
• $ a " &2 a2 2 a " "2
• $ a " &3 a3 3 a2 " 3 a "2 "3
nn
n
nn
n
n
n
n
n
nbC baC baC aC ba ++++=+ −
−− 1
11
10
......)(
C k
n
)!(!
!
k nk
nC k
n −=
k k nn
k
bak nk
nU
−
−
=
!)(!
!
n
k
U
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Mi#a)an /n(/) #/)/ )+ 4 da,i n 5% a)a i 4 dan ) 4@ 1 3% )*+#i+n #/)/ )+ 4 adaa0
K*+#i+n-)*+#i+n "in*i/ (+,#+"/( dida'a( da,i "+n(/)#+!i(i!a 'a#aO d+n!an 'an!)a( "ian!an $ n & "/a('*#i(i9 dan )+i.
1
1 11 2 1
1 3 3 1
1 4 6 4 1
1 5 1> 1> 5 1
Un(/) 'an!)a( "ian!an '+aan a/'/n "/a( n+!a(i9a(a/ '*#i(i9 di!/na)an "+n(/)0
102
20
!2!.3
!3.4.5
)!35(!3
!5
!)(!
!===
−=
−=
k nk
nC
k
n
nk k n
nnnnnn
bbak
k nnnn
bannnn
bannn
bann
ban
aba
++−−−−
++
−−−+
−−+
−++=+
−
−−−−
...............!
)1(..().........2)(1(........................
!4
)3)(2)(1(
!3
)2)(1(
!2
)1(
!1)(
4433221
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Un(/) $ a " &1;2 .............. $'a)ai a,!a '+nd+)a(an&
a3 "2 1> a2 "3 5 a "4 "5
Ma)a )*+#i+n #/)/ )+ 4 adaa0 ) 4 @ 1 3
". T+n(/)an #/)/ )+ 4 da,i $3a @ 2"&
aa"0 n ) 4 @ 1 3
3535
3388
3
108864).8.(243.56
)2()3()38(!3
!8
!)(!
!
baba
babak nk
nU
k k n
−=−=
−−
=−
=
−−
nk k n
n
bbak
k nnnn
bannnn
bababaaba
++−−−−++
−−−+
−−+
−++=+
−
−−−−
...............!
)1(..().........2)(1(........................
!4
)3)(2)(1(
!3
)2)(1(
!2
)1(
!1)(
443321
21
21
2221
21
121
2
1
21
21
21
21
32335 10!)35(1.2.3
1.2.3.4.5
!)(!
!bababa
k nk
nU
k k nn
k
=−
=−
=
−−
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S*a La(ian;PR III1. T+n(/)an )*+#i+n a7"5 da,i "+n(/) $a "&12
2. :i(/n! )*+#i+n 86 da,i "+n(/) $8 @ 2&11
3. T+n(/)an )*+#in da,i 86 'ada "+n(/) $28 @ 3&
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. Ind/)#i L+n!)a'Ind/)#i +n!)a' di!/na)an /n(/) +"/)(i)an #/a(/,//# "+,a)/ /n(/) #+(ia' "ian!an a#i n $ n 1% 2%3% .........&
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". $ a " &2 a2 2 a " "2
B/)(i0
Mi#a)an a 2 dan " 3 $ 2 3 &2 22 2.2.3 32
T+,"/)(i "+,a)/0 a. 1 3 5 .................. $2n @ 1& n2
". $ a " &2 a2 2 a " "2
. M+na,i a,!a P+nd+)a(anDa,i $ 1 8 &n% i)a n )+i +nd+)a(i n* $ n > &% a)a$ 1 8 &n 1 n8
i)a $ 1 8 &a 1 a 8
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In!a( n $'+aan& Bia V8 )+i di"andin!)an 8% a)a #/)/ )+ 3 d#( da'a(dia"ai)an% #+in!!a0
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1>.Bian!an K*'+)#:i'/nan "ian!an-"ian!an an! "+,"+n(/) a i"% diana a dan " "ian!an ,+a% #+dan! a(a/ i2 - 1% di#+"/(i'/nan "ian!an )*'+)#.
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i)a 82 @ 68 34 > 8 3 5i a(a/ 8 3 @ 5i
Daa '+,na(aan 8 3 5 i% diana0
• 3 di#+"/( "a!ian ,+a da,i 8
• 5 di#+"/( "a!ian iain+, da,i 8
Ga"/n!an )+d/ana di#+"/( "ian!an )*'+)# Bian!an )*'+)# $"a!ian ,+a& $"a!ian iain+,&
W'+,a#i 'ada "ian!an )*'+)#
a. P+n/aan dan '+n!/,an!an "ian!an)*'+)# Ba!ian ,+a dan "a!ian iain+, a,/# di!a,a' #+a,a
(+,'i#a
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i2& $1-2i& $6 - 7i 2>& $1 @ 2 i&
$ 26 @ 7 i& $1 @ 2 i& 26 @ 7 i @ 52 i
14 i2
26 @ 5 i @ 14 12 @ 5 i
P+,)aian 2 9a)(*, "ian!an )*'+)# a#i '+,i(/n!an'ada //na +,/'a)an "ian!an )*'+)# /!a.
S+)a,an! i(/n!0 $ 5 i& $ 5 @ i& 25 4> i @ 4> i @ 64i2
25 64
:a#ina #+'+n/na ,+a% )+na'aQQQQQQ
K+d/a 9a)(*, id+n(i)% )+/ai (anda '/# $& dan in/#$-&
Pa#an!an "ian!an )*'+)# #+aa ini di#+"/(
"ian!an )*'+)# )*n/!a(O:a#i )ai d/a "ian an )*n / a( #+a/ +,/ a)an
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. P+"a!ian "ian!an )*'+)#• M+"a!i "ian!an )*'+)# d+n!an "ian!an ,+a
Mi#a)an0
• M+"a!i "ian!an )*'+)# d+n!an "ian!an )*'+)#
Mi#a)an0
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11. Bian!an )*'+)# #+a,a !,a#Da'a( dina(a)an #+a,a dia!,aa(i#% ai(/ d+n!an#+"/a !a,i# da,i (i(i) a#a $ > & )+ (i(i) (+,(+n(/ an!
+na(a)an '/# $ & a(a/ in/# $ - & 'ada #)aa an!di(+n(/)an. Ga,i# an! +na(a)an "+#a, $+a('anan!na& dan a,a $+a( +n/n/))an )+'aa ana)'ana& di#+"/( ?+)(*,O.
• i)a $3& 8 $-1& $-3& 9a)(*, $-1& +n+"a")an ?+)(*,"+,"ai) a,a 1>>.
3-3
-3 -2 -1 > 12 3
-3 -2 -1 > 12 3
1>>-3 3
Fa)(*, $ 1& #+(a,a d+n!an i2 ai(/ i 8 i adi +n!ai)an
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Fa)(*, $-1& #+(a,a d+n!an i2 ai(/ i 8 i% adi +n!ai)and+n!an i #+)ai a)an +/(a, ?+)(*, #+"+#a, >> daaa,a '*#i(i9 '+n!/)/,an #/d/(% ai(/ daa a,a"+,aanan d+n!an a,/ a.
• i)a 3 8 i 3i% 3i 8 i 3 i2 - 3% a)a #+a,a !,a# da'a(di!a"a,)an #""0
-3 -2 -1 >1 2 3
3
3i
>>
-3 -2 -1 > 12 3
3
3i
X i-3
X i
• i)a )+d/a "a!ian ,+a dan iain+, d+n!an #/"/ XX
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• i)a )+d/a "a!ian ,+a dan iain+, d+n!an #/"/ XX1
dan YY 1 #+'+,(i "ia#a% a)a0
• i)a in!in +na(a)an 3 2 5 #+"a!ai /a d/a?+)(*,% a)a )+d/a ?+)(*, di !a"a,)an #+a,a "+,an(ai#+"a!ai "+,i)/(0
X X1
Y
Y 1
-3 3
3i
-3i
>8 i
•S)aa #/"/ X+na(a)an "ian!an
,+aXX1 adaa #/"/
"ian!an ,+a•S)aa #/"/ Y+na(a)an "ian!an
iain+, YY 1 adaa #/"/
iain+,
> 1 2 3
4 5
3 2
3 2 5
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• i)a "ian!an )*'+)# 3 2i% a)a di/a)an?+)(*, an! +na(a)an 3 d+n!an ?+)(*, an!+na(a)an 2i
P+,na(aan !,a# ini di)+na #+"a!ai dia!,a A,!andO• A(a/ d+n!an +n!!/na)an #)aa #/"/ XY di ana
#/"/ X #+"a!ai #/"/ ,+a dan #/"/ Y #+"a!ai #/"/iain+,.
> 1 2 34 5
32i32i
X
2>
>
P+,a(i)an ?+)(*, 2iadaa 2 8 i an! +/(a,
?+)(*,na >>.Z+)(*, an! +na(a)an $3 2i& adaa #+"/a?+)(*, da,i 'an!)a ?+)(*,'+,(aa $(i(i) a#a& )+//n! ?+)(*, )+ d/a
-6 -4 -2 > 24 6
4 2 3i2
X
X
Y
• P+n/aan dan '+n!/,an!an "ian!an )*'+)# #+a,a
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• P+n/aan dan '+n!/,an!an "ian!an )*'+)# #+a,a!,a#
Mi#a)an /a an(a,a 1 5 2i dan 2 2 3i
> 1 2 3 4 56 7
2 2
3i
5
4
3
2
1
1 5
2i
1
2
adi 1 2 $5 2i& $ 2 3i& 7 5i /a d/a ?+)(*, daa
dia!,a A,!and+,/'a)an dia!*na da,i aa,an !+nan! an!
di"+n(/) *+ )+d/a?+)(*, (+,#+"/(.
• Un(/) '+n!/,an!an an(a,a 5 2i dan 2 3i
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• Un(/) '+n!/,an!an an(a,a 1 5 2i dan 2 2 3i
da'a( dia)/)an d+n!an #+di)i( #ia#a(% di ana 1 - 2
1 $- 2& $- 2& ?+)(*, n+!a(i9 da,i 2 $"+,aanan
a,a&.
> 1 2 3 4 56 7
2 2
3i
5
4
3
2
1
1 5
2i
adi 1 -
2 $5 2i& - $ 2
3i& 3 - i
'+n!/,an!and/a ?+)(*, daadia!,a A,!and
+,/'a)andia!*na da,i aa,an
!+nan! an!
di"+n(/) *+ )+d/a?+)(*,(+,#+"/(% ai(/0
1 -
2 $5 2i& - $ 2 3i&
1 $-
2&
$5 2i& $ -2
- 3i&
1 @
2 3
- i
-2 - $2
3i&
TUGAS IZ
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TUGAS IZ
K+,a)an #*a di "aa ini0
1. a&. $6 5i& @ $4 @ 3i& $2 @ 7i&
"&. $3 5i& @ $5 @ 4i& @ $-2 @ 3i&2. a&. $4 @ 5i& $3 2i&
"&. $3 4i& $2 @ 5i& $1 @ 2i&
&. $2 7i& $2 @ 7i&
3.
4. Ga"a,)an dia!,a A,!and /n(/) +na(a)an
?+)(*, da,i #*a n*. 1
i
iic
i
ib
i
ia
43
)21)(32().
35
53).
2
54).
+
−+
−
+
−
−
8/18/2019 1 BILANGAN
39/41
n
8/18/2019 1 BILANGAN
40/41
n
n
8/18/2019 1 BILANGAN
41/41
n
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