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


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



5/12

Sk`yhnhj`kr Odhssas


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



6/12

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



7/12

Sk`yhnhj`kr Odhssas


;. 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



8/12

Sk`yhnhj`kr Odhssas


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 - =


9/12

Sk`yhnhj`kr Odhssas


=.(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


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

Jee 2014 Booklet6 Hwt Solutions Integral Calculus 2

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Transcript of Jee 2014 Booklet6 Hwt Solutions Integral Calculus 2