Jee 2014 Booklet6 Hwt Solutions Integral Calculus 2
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Transcript of Jee 2014 Booklet6 Hwt Solutions Integral Calculus 2
-
8/13/2019 Jee 2014 Booklet6 Hwt Solutions Integral Calculus 2
1/12
Sk`yhnhj`kr Odhssas
SNO/Kjtagrhd Ohdoudus-2 8= MWP-_fdutkfjs/Nhtmanhtkos
_fdutkfjs tf Mfna Urhotkoa Past/Nhtmanhtkos
8.
2
2
2
1 0;
2
xsao `x
`x
xofs xthj
Uut2
xthj t
hj` 2
8
2 2
xsao `x `t
82
2 2
1 0 ? ?;
`x `t t
thjofs x t
8
5 5
2
1 0 ? ? ?
`x tthj
ofs x
2. 2 2
2 5
K c x c x . g x g x `x c x c x . g x g x c x c x . g x g x `x
2
5
5 5c x c x . `x
?. Odahrdy xg x a 2 xc x x a
88 8 8
282 2 2 2
555 5 5
22
x
x x x x x x aK x a . a `x x . a a `x x . a x . a `x
8 88 2 288 8 8 82 2
5 5 5 5555
5
2 2 2 22 2
x xx x x x x xa a
x . a xa a `x x . a xa a
2 2
8 ?2 2 2
2 2 2 2
a aa a a a
0. Uut dfg x t hj`8
`x `t x
kj K8tf gat
25 225
2
858 5
82
xx a`x a `x x
Kjtagrhd Ohdoudus-2 MWP - 2
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8/13/2019 Jee 2014 Booklet6 Hwt Solutions Integral Calculus 2
3/12
Sk`yhnhj`kr Odhssas
SNO/Kjtagrhd Ohdoudus-2 25 MWP-_fdutkfjs/Nhtmanhtkos
0 08 88
2 2
a a
;. 2 ?
2 ? 2
5
K x ofs x `x
. Uut ? 2 x t tf gat?
2x `x `t
2
55 5
2 8 8 2
8 2? ? ? 2
skj t
K ofs t `t ofs t `t t
?
8. 8 1 2 8 1
2
8 8 2
2 2 8 2 8 1 2 2 2
. .
K x `x `x `x .
2. 2 2 2
5 5 5
j j jK dfg thj x `x dfg oft x `x dfg thj x `x K
K > 5
?. Dat
2
2
2 0 2
5
1 0 1 02
2 2
x
t x
t t x xc x c x x
a a
c(x) mhs axtranun wmaj 0 22 1 0 5x x x
fr 2 22 8 0 5x x x fr 2 8 8 2 2 5x x x x x fr 5 8 2x , ,
3.
1 2 2
8 8 8 8
2 5
skj ofs x ofs skj x `x skj ofs x ofs skj x `x
2 222
8 8 2
55 5
20
skj skj x ofs ofs x `x x `x x
7. 8
5
x
x
c t `t x tc t `t
@kcc. w.r.t. x tf gat 8
88
c x xc x c xx
8
22c
hj`
8
2 ?c
Pma raqukra` quh`rhtko aquhtkfj ks 28 7 2
2 5? ? ?
x x x x
2? 7 2 5x x
=. 85
8
2 8 2 8K x x x x `x
Uut 28x t hj` 2`x t `t tf gat ?,b> 8
2?U x x x hj` 3 8U x x
_f, 8 8 8
2 ? 2
5 5 5
? 3 8 8= ;U x . U x `x x x x `x x x x x 8
20 ?
5
; ; 8? ? =
2 2 2 2
xx x
0. 2
? 2
x
xg x t g t `t . . . .(k)
2
2
2 2 5g Y_aa dknkts^
2 5g Jfw `kccarajtkhtkjg aquhtkfj (k)
? 2 g x xg x x g x 2 2 2 3 2 2 g g g
?
0 2 3 22
g g
3. 88
5 5
88 8
j
ji
idkn dfg dfg x x
j j
8
5
8dfg x `x Dat 8 x i
85 8
8 5 5
8dfg i `i dfg i`i i dfg i i
=. Grhpm ks
Kjtagrhd Ohdoudus-2 MWP - 0
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8/13/2019 Jee 2014 Booklet6 Hwt Solutions Integral Calculus 2
5/12
Sk`yhnhj`kr Odhssas
SNO/Kjtagrhd Ohdoudus-2 22 MWP-_fdutkfjs/Nhtmanhtkos
Hrah fc smh`a` ragkfj 8 8
8 82 2
;. 8?
?
8 5
88 8
j
r
rdfg U dfg dfg x `x
j j
?
? ?
?
?8 8
8
x
dfg x `x x dfg x `x
x
8 8
8 8? ?
?5 55
5
?8 8 ?
8
`xdfg x `x x dfg x x
x
2 ? ?dfg
85.
2 2?
5 5
8 ? 8 2 0
8 2 8 2
skj x skj x skj x skj x`x
skj x skj x
YQskjg ?( ? ? 0 )skj x skj x skj x ^
2
5
8 8 2 8skj x skj x
8. Dat 2 2
5 5
c t `t H, tc t `t B
22c x B x H
2 2 2 2
2
5 5 5 5
2c x `x B `x x H `x H . . . .(k)
hj`
2 2 2
?
5 5 5
2
xc x `x B x `x x H `x B . . . .(kk)
_fdva (k) hj` (kk) tf gat B
_kjoa H > B 0
H B
Kjtagrhd Ohdoudus-2 MWP - 1
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8/13/2019 Jee 2014 Booklet6 Hwt Solutions Integral Calculus 2
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Sk`yhnhj`kr Odhssas
SNO/Kjtagrhd Ohdoudus-2 2? MWP-_fdutkfjs/Nhtmanhtkos
7.
8 8 8
2 2 2
5 5 5
8 8
88 8 8
xx x xx axa a a`x `x `x
xx x x
8
5
88 2
xa a
x
=. 2 2
2 0 2 0
2 5
2skj x skj x skj x ofs x `x skj x skj x ofs x `x
Dat skj x t
88
1 ?2 0
55
2 22
1 ?
t tt t `t
2 ? 12 2 01 ? 81 81
8.
0 02 2
5 58 8
ofs x `x ofs x `x
skj x skj x
0 2
2
5
22
ofs x `xofs x
?.
_mh`a` Hrah
88 ?
2 8
2
5 5
22 2 2 ?8
xx `x thj x
x
8 ? 2
22 ? ?
1. _mh`a` hrah >
0
2
2
2 0x `x0
22
2
0 02 0
2 2
x xdfg x x
2 2 2 ? 2 0 2 ? 2 2dfg dfg
2 0 ? 2 2 ? = ? 0 2 ?dfg dfg
7. 0 0 00
? 2
55 5 5
2
?x `x x x `x x x `x
83
?K
0
5
K x `x Orktkohd pfkjts hra wmara x K
2x K
8 0
5 8
5 8 ?K `x `x
0
5
83 7?
? ?x `x
Kjtagrhd Ohdoudus-2 MWP - 3
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8/13/2019 Jee 2014 Booklet6 Hwt Solutions Integral Calculus 2
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Sk`yhnhj`kr Odhssas
SNO/Kjtagrhd Ohdoudus-2 20 MWP-_fdutkfjs/Nhtmanhtkos
;. c(x) ks
Ckj` hrah fc smh`a` ragkfj wmkom ks hrah
2 1 3
5 2 1 3
8 8 2ofs x `x `x skj x `x
85. b o
h o
c x o `x
x o t
`x `t
b o b
h o h
c x o `x c t `t
8.
8 2 2 2
8 2
8 82
8 8
x x`x
x x
8 2 2 2
5
8 82 2
8 8
x x`x
x x
8 2
5
8 82
8 8
x x`x
x x
8 2 8 2
22
5 5
0 02 2
88
x x`x `x
xx
2
0 8Hs ks va kj 5
28
xx ,
x
8 2 8 28 2
2
2 25
5 5
0 22 0 0 8
8 8
x x`x `x j x
x x
?0 8
0
j j
?0
0j
00
?
j
2.2
0y hx
20x hy
Hrah
00 2 ? 2 ?
55
2 00
0 ? 82
hhx h x x
hx `xh h
2 2?2 30
? 82
h h
2?2 83
?h
283
?
h
Kjtagrhd Ohdoudus-2 MWP - 7
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8/13/2019 Jee 2014 Booklet6 Hwt Solutions Integral Calculus 2
8/12
Sk`yhnhj`kr Odhssas
SNO/Kjtagrhd Ohdoudus-2 21 MWP-_fdutkfjs/Nhtmanhtkos
1.
0 1 02 2
5 5 0 1 0
ofs x skj x `x ofs x skj x `x skj x ofs x `x ofs x skj x `x
3. 8 8
;;;;
5 5
8 8 8 8x x `x x x `x
8 8
;; ;; 855
5 5
8 x x `x x x `x 8
855 858
5
8
855 858 85855
x x
=.
0
8 0
8 0 8 0 0
2
0 0 8 0
8 8
8 8 8 8 8 8
thj x `xx x
t thj t `t thj t `t thj t `t t t t t t t
Dat
8t
x
K K K > 5
0.(O) Dat
2 2
5 58
/ /`x skj x
K `xoft x skj x ofs x
Hdsf
2 2
5 5
2
2 2
/ /skj / x x ofs x
K `xskj / x ofs / x ofs x skj x
2
5
2 82 0
/
K `x fr K
1.(O) Dat
2 2 2 2
5 50 ; 0 ;
x`x x `xK
ofs x skj x ofs x skj x
2
2 2 2 2
5 5
2 20 ; 0 ;
/`x `x
Kofs x skj x ofs x skj x
2 2
2 2
5 5
2 2
0 ; 0 ;
/sao x`x `t
thj x t
, wmara thj x t
28
2
5 5
2 2 ? ?
; ; 2 2 30 ;
`t. thj t
t /
3.(B) Dat C x c x c x . g x g x tmaj C x c x c x . g x g x C x
2
2
5
/
/
C x ` x
7.(H)
2 22 25 58 8 8
1 0 ;0 ;
`x`x
x xx x
8 8
5
8 8 8 8
1 2 2 ? ? 1 0 3 35
x xthj thj
Kjtagrhd Ohdoudus-2 MWP - =
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8/13/2019 Jee 2014 Booklet6 Hwt Solutions Integral Calculus 2
9/12
Sk`yhnhj`kr Odhssas
SNO/Kjtagrhd Ohdoudus-2 23 MWP-_fdutkfjs/Nhtmanhtkos
=.(H) Raqukra`
0 2
5 0
/ /
/
hrah ofs x skj x `x skj x ofs x `x
> 0 2
5 02 2 8
/ /
/skj x ofs x ofs x skj x
;.(O) 2
2 2
25
8 2 8 2 2 ? 0
x`y ` y
y t t `t x x x x
`x `x
2
2
05 2
?
` yx
`x
85.(O) 2 0
5
/
K skj x `x
. Uut 2x t tf gat 2
5
2 /
K t skj t `t
2
52 2
/tofst skjt
8.(@) Uut ?x t tf gat
82 82 82
8
82 5
8 8 882 0
? ? ?
P P
K c t `t c t `t K K
2.(@)
22 8 2 ? 282 ? 0
5 5 8 2 ?
xx `x x`x x `x x `x x `x
? 2 1 28 8 8 82 8 ; 0 ?2 ?2 ? 0 1
/ /
? 2 1 203; 8 82 ?35 ? 1
/ /. .
?.(B) Pma ourvas21y x hj` 8y |x | kjtarsaot ht (2, 8) hj` (8, 2).
Raqukra`
2 2
2
8 8
1 8hrah x `x |x |`x
2
2
2 2 8
8 8
11 1
2 2 1
x xx `x x skj
12 0
2 8
8 2
8
|x | x | t | t , wmara 8t x
585 8
2 2
52 5 2
1
2 2 2
t tt `t t `t
Raqukra`1 8
0 2hrah
.
0.(H)
2 22 2
0 20 0
83 =2 2 2 8
x xx x
c x x a xa xa a
_kjoa
2
2
? ? ?8 8
8
8
j j
j jr r
r
r jdkn dkn
jr j r
j
8
2?
5
8 2?8
x `x dfgx
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8/13/2019 Jee 2014 Booklet6 Hwt Solutions Integral Calculus 2
12/12
Sk`yhnhj`kr Odhssas
85.(@)
2
8sao sao
ofs ofs
c / xK c x `x `x
x
. Uut8
tx
tf gat
2 5 5ofs ofs
sao sao
K c t `t c x `x K K K