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8/3/2019 Sol.Paper-2
1/4
WJ]O QHJSZIUQDZ QUNGD"L ZSNGD ! >
WJ]OQHJS FJ] IUQD UDQU SNSD]"=
>) s / s
>2 x / | nie s
s
>2 x
|
(glvdi,
Neelig mjuh wd gdu
=s 2 =x s 2 x
Ijw qtmu}naulig mjuh wd gdu
s
=2 =|
s
>2 | s 2
|
>
Uhd}dfj}d x 2|
>
x| 2 >
=) x 2 | {
x / | / { 2 3
=x / { 2 3
{ 2 =x
==
==
|{
x{
2
==
==
xx5
xx5
2
4
1
1) x= >lq fnauj} jf f(x, 2 x5 / sx1 / =x= 1x / ~
(x >, nie (x / >, n}d n`qj fnauj} f(x,
M| fnauj} uhdj}db
f(>, 2 3
> / s / = 1 / ~ 2 3
s / ~ 2 3 )))(l,
f( >, 2 3
> s / = / 1 / ~ 2 3
s / ~ 2 6 )))(ll,
Nee (l, nie (ll,
=~ 2 6
~ 2
1 s 2 1)
5) 1
=
=
>
5
1
)))))))))) i
>i
2 i
>
4) 7 1x/> / (1x/5,2 4x/1 4x/=
j} 1x/> _ 7 / 11R 2 4x/= _4>R
j} 1x/> _=3R 2 4x/= _5R
j} 1x/> _=3R 2 4x/> _=3R
j}
>x
4
1
2 >
j}
>x
4
1
2
3
4
1
x / > 2 3 x 2
>
6) (>3>,= 2 hnvd 1 iji"{d}j elgluq
(>3>3>,= 2 hnvd 4 iji"{d}j elgluq
Qj (>3>3>3>3>,= 2 hnvd < iji"{d}j elgluq
7) Wd hnvd n / m / a 2 6 )))(l,
(n / m / a,= 2 16
_Ji q~tn}lig mjuh qledq jf (l,R
n= / m= / a= / =(nm / ma / na, 2 16
n= / m= / a= / = 8 2 16
_ nm / ma / na 2 8R
n= / m= / a= 2 (16 >6, 2 =3)
Ijw wd hnvd
n1 / m1 / a1 1nma 2 (n / m / a, (n= / m= / a=
nm ma na,
2 (n / m / a, _(n= / m= / a=, (nm / ma / na,R
2 6 _(=3 8,R 2 (6 >=, 2 7=)
^tdq) > = 1 5 4 6 7 8 < >3 >> >= >1 >5 >4 >6 >7 >8 >< =3
Niq) E A E A N A N A M M N M A M M A E E E N
^tdq) => == =1 =5 =4 =6 =7 =8 =< 13 1> 1= 11 15 14 16 17 18 1< 53
Niq) N N N N N M M E N M E N M A N A A A E M^tdq) 5> 5= 51 55 54 56 57 58 5< 43 4> 4= 41 45 44 46 47 48 4< 63
Niq) M M M N A M N E E M N M N N N M M N A A
^tdq) 6> 6= 61 65 64 66 67 68 6< 73 7> 7= 71 75 74 76 77 78 7< 83
Niq) N N A M N A A A M E M E N A M M M A A E
^tdq) 8> 8= 81 85 84 86 87 88 8<
8/3/2019 Sol.Paper-2
2/4
WJ]O QHJSZIUQDZ QUNGD"L ZSNGD ! =
8) ==
11
mnmn
mn
2 n / m
8)71 / 5)=7 2 >1)
R
>3)n|mx
,R|mxm=xn(n|,|m|n=xn(mx_ ============
2,n|mx(
,n|mx(nmx|=
,n|mx(
,n|mx(|m
,n|mx(
,n|mx(xn ====
2 n= x= / m=|= / =nmx|2 (nx / m|,=
>>) Glvdi 1=7>6>6 2
15 1 x
1
55812
1
51 x
1 558 2 1 x65
M| ajbsnl}lqlji 65x 2 558 x 2 7
>=) =6 2 45
>6)=< / =6)3 / 33=6)3
/ 3333=6)3
2 4)5 / )45 / )345 / )3345 2 4) _>/ 5 / 5= / 51 / 55R56> _> / 5 / >6 / 65 / =46R56> 15> 15> lq elvlqlm`d m| >>
>5) I 2 1=1~ / >=4
>=4~1=1
Qliad 1=1~ lq elvlqlm`d m| >=4 m| >< wd gdu >> nq }d"bnlied}
>4) i= 2 x(i / >,= 2 x / |i= / > / =i 2 x / |=i 2 x / | i= >2 x / | x > 2 | >(x / =,= 2 i= / 5 / 5i
2 x / 5 / = =i
2 x / 5 / =(|
>,2 x / 5 / =| =2 x / =| / =
>6) [d}jdq li uhd s}jetau jf > = 1 5 )))))))))))))
5< 43
2 >= (= 4 >3 >= >4 =3 == =4 13
1= 14 53 5= 54 43,
>7) x 2 ==
x=
2 = / =
x5 2 5 / = / 5 = 2 6 / 5 =
5x
>2
=56
>
=56
=56
2
5
=56
5x
52 6 5 =
x5 / 5x
52 6 / 5 = / 6 5 = 2 >=)
>8) _1(>=1 / i,R= Qj lu lq n itbmd} whlah lq nbt`uls`d jf 54>47 / i n`qj nq bt`uls`djf 54>47 lq =1) i 2 5 Qj uhnu lu lq elvlqlm`d m| =>8{>15|>6< whdi elvlede m| 484 glvdq}dbnlied} 2 485)
=3) (16=15, (114>=, (45==1,
2 ))))))))) 6 )))))))) > ())))))) 5, ())))))) 4,2 ))))))))) 6 )))))))))) 3qj tilu elglu 2 6 Niq)
7>) (M, >= / > (=,= /> (1,= />))))))))))
2 = 4 >3 >7)))) (4,= / > 2 =6
7=) (4 =, > (< =, = (>6 =, 1
(=< =,
5 (45 =,
4 >31 (E,
ij) lq bt`uls`lde m| = nie uhdi qtmu}naudef}jb
> = 1 5 4))))))) }dqsdulvd`|
75) Fl}qu duud} jf dvd}| ud}b lq bjvde uh}dd qudsq
mnaown}e li dnah idxu ud}b) Qdajie itbmd}
jf dvd}| ud}b lq bjvde uwj qudsq mnaown}e
nie uhl}e `duud} lq bjvde uwj qudsq fj}wn}e)
Hdiad uhd idxu ahjlad wjt`e md U>>D)
8/3/2019 Sol.Paper-2
3/4
WJ]O QHJSZIUQDZ QUNGD"L ZSNGD ! 1
74) (M, Li flgt}d (N, (4 165, 2 44>)Li flgt}d (M, (78,2 6==)
76) (M, A`dn}`|
Li flgt}d N R 6 1 5 = 2 >8 8 2 >3
Li flgt}d MR
< 4
4 1 2 54
>4 2 13 Li flgt}d A R 6 4 = 4 2 13 >3 2 =3
77) (M, Uhd snuud}i jf uhd qd}ldq lq aaa mmm
nnn*aaa mmm nnn*a)
78) (A, bij * ijs~ * js~}q * s~}qu
El}dauljiq ;(7
8/3/2019 Sol.Paper-2
4/4
WJ]O QHJSZIUQDZ QUNGD"L ZSNGD ! 5
>, 2 =3
Qj Qtil` lq =>qu f}jb uhd mjuujb)