iz’u&i= dh ;kstuk 2024
d{kk & XII
fo"k; & xf.kr
vof/k & 3 ?k.Vs 15 feuV
1-
mn~ns'; gsrq vadHkkj &
Ø-l-a
1234-
2-
mn~ns';
Kku
vocks/k
Kkuksi;ksx@vfHkO;fDr
dkS'ky@ekSfydrk
;ksx
vadHkkj
izfr'kr
24
30
20
25
24
30
12
15
80
100
iz'uksa ds izdkjokj vadHkkj &
Øla123456-
3-
iw.kkZad & 80
iz'uksa dk izdkj
iz'uksa dh
la[;k
vad
izfr iz'u
dqy vad
15
1
15
7
1
10
oLrqfu"B
fjDr LFkku
vfry?kqÙkjkRed
y?kqÙkjkRed
nh?kZmÙkjh;
fuca/kkRed
;ksx
¼vadks dk½
izfr'kr
¼iz'uksa dk½
laHkkfor
le;
18.75
29.41
30
7
08.75
13.73
15
1
10
12.50
19.61
35
12
2
24
30.00
23.53
45
4
3
12
15.00
07.84
35
3
4
12
15.00
05.88
35
56
100-00
100.00
195 feuV
51
fodYi ;kstuk % [k.M ^l* ,oa ^n* esa gSa A
fo"k; oLrq dk vadHkkj &
Ø-l-a
fo"k; oLrq
1 lEcU/k ,oa Qyu
izfr'kr
vadHkkj
izfr'kr
3
03.75
2
izfrykse f=dks.kferh; Qyu
4
05.00
3
vkO;wg
5
06.25
4
lkjf.kd
5
06.25
5
lkarR;rk ,oa vodyuh;rk
8
10.00
6
vodytks ds vuqi;ksx
6
07.50
7
lekdyu
12
15.00
8
ldkdyuks ds vuqi;ksx
4
05.00
9
vody lehdj.k
6
07.50
10
lfn'k chtxf.kr
7
08.75
11
f=foeh; T;kfefr
9
11.25
12
jSf[kd izkx
s zkeu
4
5.00
13
izkf;drk
7
8.75
80
100.00
;ksx
iz'u&i= CY;w fizUV
1
lEcU/k ,oa Qyu
1(1)
2
izfrykse f=dks.kferh; Qyu
1(1)
3
vkO;wg
1(1)
4
lkjf.kd
1(1)
5
lkarR;rk ,oa vodyuh;rk
1(1)
6
vodytks ds vuqi;ksx
1(1)
7
lekdyu
1(1)
8
ldkdyuks ds vuqi;ksx
1(1)
9
vody lehdj.k
1(1)
10
lfn'k chtxf.kr
1(1)
11
f=foeh; T;kfefr
12
jSf[kd izksxzkeu
13
izkf;drk
iw.kkZad & 80
;ksx
fucU/kkRed
nh?kZmÙkjkRed
fjDr LFkku
vfry?kqÙkjkRed
y?kqÙkjkRed
dkS'ky@ekSfydrk
oLrqfu"B
nh?kZmÙkjkRed
y?kqÙkjkRed
vfry?kqÙkjkRed
fjDr LFkku
oLrqfu"B
fucU/kkRed
Kkuksi;ksx@vfHkO;fDr
nh?kZmÙkjkRed
y?kqÙkjkRed
vfry?kqÙkjkRed
fjDr LFkku
oLrqfu"B
fucU/kkRed
nh?kZmÙkjkRed
vocks/k
y?kqÙkjkRed
vfry?kqÙkjkRed
Kku
fjDr LFkku
mÌs'; bdkbZ@mi bdkbZ
oLrqfu"B
Ø-la-
fo"k; %& xf.kr
fucU/kkRed
d{kk & XII
2(1)
1(2)
1(1)
4(4)
2(1)
1(2)
1(1)
3(2)
2(1)
5(3)
2(1)
5(4)
2(2)
2(1)
8(5)
1(1)
1(2)
2(1)
6(5)
1(2)
2(1)
4*(1)
12(6)
3*(1)
1(1)
2(1)
4(3)
1(1)
1(1)
6(4)
3*(1)
1(1)
1(1)
2(1)
1(2)
7(6)
4*(1
)
1(2)
3*(1)
9(4)
4*(1)
3*(1
)
1(2)
;ksx
14(14)
4(4)
6(3)
fodYiksa dh ;kstuk %& [k.M ^l* ,oa ^n* esa izR;sd esa ,d vkarfjd fodYi gSSA
1(1)
6(6)
6(3)
3(1)
2(1)
4(1)
2(2)
4(4)
8(4)
4(1)
7(4)
6(3)
4(1)
1(1)
uksV%& dks"Bd ds ckgj dh la[;k ^vadks*a dh rFkk vanj dh la[;k ^iz'uksa* ds |ksrd gSA
4(2)
3(1)
4(1)
gLrk{kj
80(51)
'ks [kkokVh fe'ku&
l= %
vadHkkj (1 + 2 = 3)
1.
eku yhft, fd leqPp; {1, 2, 3, 4} es]a R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)} }kjk ifjHkkf"kr laca/k R gSA
fuEufyf[kr esa ls lgh mrj pqfu,A
2.
(1) R LorqY; rFkk lefer gS fdarq laØked ugha gSA
(2) R LorqY; rFkk laØked gS fdarq lefer ugha gSA
(3) R lefer rFkk laØked gS fdarq LorqY; ugha gSA
(4) R ,d rqY;rk laca/k gSA
eku yhft, fd leqPp; N es]a R = {(a, b) : a = b - 2, b > 6} }kjk iznr laca/k R gSA fuEufyf[kr esa ls lgh mRrj pqfu,&
(1) 2,4 R
3.
(4) 8,7 R
(2) F(x) = cos x
(3) F(x) = ex
(4) F(x) = x2
leqPp; A= {1, 2, 3, 4} ls Lo;a rd lHkh ,dSdh Qyu dh la[;k gksxh &
(1) 3
5.
(3) 6,8 R
R ls R esa ifjHkkf"kr fuEu esa ls dkSulk Qyu ,dSdh gS&
(1) F(x) = |x|
4.
(2) 3,8 R
(2) 4
(3) 8
(4) 6
;fn A = {1, 2, 3} gks rkss ,sls lac/a k ftuesa vo;o (1, 2) rFkk (1, 3) gks vkSj LorqY; rFkk lefer gSa fdUrq laØked ugha gS] dh la[;k
gS
(1) 1
6.
8.
9.
(3) 3
(4) 4
;fn A = {1, 2, 3} gks rks vo;o (1, 2) okys rqY;rk laca/kksa dh la[;k gS&
(1) 1
7.
(2) 2
(2) 2
(3) 3
(4) 4
eku yhft, fd f : R R ] f ( x) x 4 }kjk ifjHkkf"rk gSA lgh mrj dk p;u dhft,A
(1) f ,dSdh vkPNknd gS
(2) f cgq,d vkPNknd gS
(3) f ,dSdh gS fdarq vkPNknd ugha gS
(4) f u rks ,dSdh gS vkSj u vkPNknd gS
eku yhft, fd f ( x) 3x }kjk ifjHkkf"kr Qyu f : R R gSa lgh mrj pqfu,&
(1) f ,dSdh vkPNknd gS
(2) f cgq,d vkPNknd gS
(3) f ,dSdh gS fdarq vkPNknd ugha gS
(4) f u rks ,dSdh gS vkSj u vkPNknd gS
fuEu esa ls dkSulk Qyu ,dSdh gS\
(1) f ( x) x 2 }kjk iznr f : N N Qyu gSA
(2) f ( x) x 2 }kjk iznr f : Z Z Qyu gSA
(3) f ( x) x 2 }kjk iznr f : R R Qyu gSA
(4) buesa ls dksbZ ugha
-: Answer :1-2, 2-3, 3-3, 4-3, 5-1, 6-2, 7-4, 8-1, 9-1
(1)
'ks [kkokVh fe'ku&
10.
l= %
fl) dhft, fd okLrfod la[;kvksa ds leqPp; R esa R a, b : a b 2 ] }kjk ifjHkkf"kr laca/k R ] u rks LorqY; gS] u lefer
gS vkSj u gh laØed gSA
Ans. fn;k gS] A R okLrfod la[;kvksa dk leqPp;
rFkk R (a, b) : a b 2
1 1
LorqY; laca/k ds fy,] ge tkurs gS fd
2 2
2
lR; ugha gSA
1 1
, R vr% R ] LorqY; laca/k ugha gSA
2 2
lefer laca/k ds fy,] ge tkurs gSa fd 1 32 1,3 R ysfdu 3 ( 1) 2
3, 1 R vr% R lefer lac/a k ugha gSA
laØed laca/k ds fy,] ge tkurs gS fd 2 32 2,3 R rFkk 3 12
3,1 R
11.
ysfdu 2 12 2,1 R vr% R ,d laØed laca/k ugha gSA
fl) dhft;s fd okLrfod la[;kvksa ds leqPp; R esa lca/k R a, b : a b }kjk ifjHkkf"kr R LorqY; rFkk laØked gS fdUrq
lefer ugha gS\
Ans. R a , b : a b
(i) LorqY; %
a a lR; gS
a a a R
R LorqY;
gS
(ii) R lefer ugha gS D;ksafd ;fn a, b ls de gS] rks b, a ls de ugha gks ldrk gSA
(iii) a, b, c R ds fy, a b, b c a c vr% R laØked laca/k gSA
R ] LorqY;] laØked gS ijUrq lefer ugha gSA
12.
f ( x ) 4 x 3 }kjk iznr Qyu f : R R ij fopkj dhft,A fl) dhft, fd f O;qRØe.kh; gSA f dk izfryks Qyu Kkr
dhft,A
Ans. Qyu f : R R esa] f ( x) 4 x 3, x R
}kjk ifjHkkf"kr Qyu gSA
eku yhft, x, y R bl izdkj gS fd f ( x) f ( y )
4x 3 4 y 3
x y
f ,dSdh Qyu gSA
eku yhft, izR;sd okLrfod la[;k y R ds fy, R esa] x R bl izdkj fo|eku gS fd
f ( x) y 4 x 3 y x
y 3
4
(2)
'ks [kkokVh fe'ku&
l= %
izR;d y R ds fy, x
y 3
R bl izdkj gS fd
4
y 3 y 3
f ( x) f
4
3 y
4 4
f vkPNknd Qyu gSA vr% f ,dSdh vkPNknd Qyu gSA
x3
vr% f 1 fo|eku gSA eku yhft, g : R R es]a g ( x)
4
13.
}kjk ifjHkkf"kr gSA
fl) dhft, fd f ( x) [ x] }kjk iznr egre iw.kkZad Qyu f : R R ] u rks ,dSdh gS vkSj u vkPNknd gS] tgka [ x], x ls de
;k mlds cjkcj egre iw.kkZad dks fu:fir djrk gSA
Ans. Qyu f : R R es]a f ( x) [ x], x R
}kjk ifjHkkf"kr Qyu gS] tgka [ x], x ls de ;k mlds cjkcj egre iw.kkZad Qyu gSA
pwafd
f (1,2) [1,2] 1
f (1,9) [1,9] 1
f (1,2) f (1,9) 1 ysfdu 1.2 1.2
f ,dSdh Qyu ugha gSA
iqu% 0.7 R ds fy, R esa dksbZ x R bl izdkj ugha gS fd f ( x) 0.7 vFkkZr~ [ x] 0.7
f vkPNknd Qyu ugha gSA
vr% egre iw.kkZad Qyu u rks ,dSdh gS u gh vkPNknd gSA
14.
fl) dhft, fd f ( x) x 2 }kjk ifjHkkf"rk Qyu f : R R ] u rks ,dSdh gS vkSj u vkPNknd gSA
Ans. D;ksafd f (1) 1 f (1) ] blfy, f ,dSdh ugha gSA iqu% lgizkar R dk vo;o &2] izkr R ds fdlh Hkh vo;o x dk izfrfcac
ugha gS ¼D;ksa\½A vr% f vkPNknd ugha gSA
15.
leqPp; {1, 2, 3, .....n} ls Lo;a rd ds leLr vkPNknd Qyuks dh la[;k Kkr dhft,A
Ans. leqPp; {1, 2, 3, ......n} ls Lo;a rd ds leLr vkPNknd Qyuksa dh la[;k 1] 2] 3]---------n ds dqy Øep;ksa dh la[;k ds cjkcj
gksrh gSA vFkkZr~ nPn = n!
16.
fl) dhft, fd leqPp; A x Z : 0 x 12] esa fn, x, fuEufyf[kr laca/kksa R esa ls izR;sd ,d rqY;rk laca/k gSA
(i) R a, b :| a b |, 4 dk xq.kt gS
Ans. fn;k gS] A x Z : 0 x 12
0,1,2,3,4,5,6,7,8,9,10,11,12
(i) R a, b :| a b |, 4 dk xq.kt gS
pwfa d izR;sd a A ds fy, | a a | 0 ] tks fd 4 dk xq.kt gSA vr% R LorqY; laca/k gSa vc] eku yhft,
a, b R | a b |, 4 dk xq.kt gSA
| (b a ),4 dk xq.kt gSA
| b a |,4 dk xq.kt gSA
b, a R, a , b R
(3)
'ks [kkokVh fe'ku&
l= %
vr% R lefer laca/k gSA vc] eku yhft, a, b), (b, c R ] rc | a c | rFkk | b c | 4 ds xq.kt gSA | a c | ] 4 dk xq.kt gSA
( a, c ) R
R, laØed laca/k gSA vr% R ] ,d rqY;rk laca/k gSA
17.
fl) dhft, fd leqPp; {1, 2, 3} es]a R= {(1, 2), (2, 1)} }kjk iznr laca/k R lefer gS fdarq u rks LorqY; gS vkSj u gh laØed
gSA
Ans. fn;k gS]
rFkk
A = {1, 2, 3}
R = {(1, 2), (2, 1)},
pwfa d (1, 1), (2, 2), (3, 3), R, R, LorqY; laca/k ugha gSA vc] pwafd (1,2) R rFkk (2,1) R
R lefer laca/k gSA iqu% (1,2) R rFkk (2,1) R ysfdu (1,1) R vr% R ] laØed laca/k ugha gSa blfy, R lefer laca/k
gSA ysfdu R ] LorqY; laca/k rFkk laØed laca/k ugha gSA
'ks[kkokVh fe'ku 100 dh d{kk 10 ,oa 12 ds
fofHkUu fo"k;ksa dh uohure cqdysV MkmuyksM
djus gsrq Vsyhxzke QR CODE LdSu djsa
(4)
'ks [kkokVh fe'ku&
l= %
vadHkkj (1 + 1 + 1 + 1 = 4)
1.
tan 1 3 sec 1 (2) cjkcj gS&
2.
1
2
(2)
(4)
3
1
3
(3)
(2)
1
4
(4) 1
(3) 0
2
(4) 2 3
x
(2)
1 x2
1
1
(3)
1 x2
1 x2
7
6
(2)
5
6
(3)
3
-: Answer :1-2, 2-4, 3-2, 4-4, 5-2
fjDr LFkkuksa dh iwfrZ dhft,&
1
1
2 sin 1 cos 1 ............
2
2
1 1
1 1
Ans. 2 sin cos
2
2
2
7.
x
(4)
1 x2
7
cos 1 cos
dk eku cjkcj gS&
6
(1)
6.
2
3
sin tan 1 x , | x | 1 cjkcj gksrk gS&
(1)
5.
3
tan 1 3 cot 1 3 dk eku gS&
(1)
4.
(3)
1
sin sin 1 dk eku gS&
2
3
(1)
3.
(2)
(1)
6
3
3
3
2
3
1
sin 1 .............
2
Ans. sin 1 ( x) sin 1 x , sin 1 x dk eq[; eku ,
2 2
1
1
sin 1 sin 1
2
6
2
(5)
(4)
6
'ks [kkokVh fe'ku&
8.
l= %
1
cos 1
.........
2
Ans. cos 1 ( x) cos 1 x , cos 1 x dk eq[;eku 0,
1
1
cos 1
cos 1
2
2
9.
4
3
4
1
1
;fn sin sin 5 cos x 1 rc x .......... ...
1
1 1
cos 1 x 1
Ans. sin sin
5
1
sin 1 cos 1 x sin 1 1
5
1
sin 1 cos 1 x sin 1
5
2
2
cos 1 x
2
sin 1
cos 1 x cos 1
x
10.
1
5
1
5
1
1
sin x cos x 2
1
5
3
sin 1 sin ...................
5
Ans. sin 1 x dk eq[;eku ,
2 2
3
sin 1 sin
5
11.
3
1
sin sin
5
2
1
sin sin
5
2
5
1
tan 1 2 cos 2 sin 1 .............
2
1
1 1
1
tan 2 cos 2
Ans. tan 2 cos 2 sin
2
6
1
tan 1 2 cos tan 1 2
3
2
tan 1 (1)
4
1
sin
6 2
1
cos 3 2
tan 4 1
(6)
'ks [kkokVh fe'ku&
12.
l= %
cot tan 1 a cot 1 a ....................
1
1
Ans. cot tan a cot a cot
2
1
1
tan x cot x 2
=0
13.
1
1
tan 1 (1) cos 1 sin 1 ................
2
2
1
1 1
1 1
Ans. tan (1) cos sin
2
2
1
1
tan 1 (1) cos1 sin 1
2
2
14.
sin 1 ( x) sin 1 x
1
1
cos ( x) cos x
4
3 6
4
tan 1
2 3 8 2 9 3
3 6
12
12
4
2
7
tan 1
dk eku ------------------------- gSA
11
24
7
2
2
1 7
1 11
24
tan
tan
tan
Ans.
2 7
11
24
1
11 24
1
1
1
1 x y
tan x tan y tan
1 xy
48 77
tan 1 264
1 14
264
125
tan 1 264 tan 1
264 14
264
125
264 tan 1 125 tan 1 1
250
250
2
264
(7)
'ks [kkokVh fe'ku&
l= %
vadHkkj (1 + 2 + 2 = 5)
1.
3 × 3 dksfV ds ,sls vkO;wgksa dh dqy la[;k fdruh gksxh ftudh izR;sd izfof"V 0 ;k 1 gS\
(1) 27
2.
(2) 18
(3) 81
(4) 512
eku fyft, fd x, y, z, w rFkk p Øe'k% 2×n, 3×k, 2×p, n×3 rFkk p × k dksfV;ksa ds vkO;wg gSA ;fn n = p gks rks vkO;wg 7x – 5z
dh dksfV gSA
(1) p×2
3.
cos
;fn A sin
(1)
4.
(2) 2×n
(2)
6
3
3
2
(3)
(4)
(3) AB = 0, BA = I
(4) AB = BA = I
(3) 1 2 0
(4) 1 2 0
vkO;wg A rFkk B ,d nwljs ds O;qRØe gksaxs dsoy ;fn &
;fn A
(2) AB = BA = 0
bl izdkj gS fd A2 I rks
(1) 1 2 0
(2) 1 2 0
-: Answer :1-4, 2-2, 3-2, 4-4, 5-3
6.
(4) p×n
sin
rFkk A A' I ] gks rks dk eku gksxk\
cos
(1) AB = BA
5.
(3) n×3
5 2
3 6
x y
x rFkk y dk eku Kkr dhft,] ;fn x y
rFkk
gSA
0 9
0 1
5 2 3 6
Ans. ;gka ij x y x y
0 9 0 1
8 8
;k x x y y 0 8
8 8
1 8 8
;k 2x 0 8 ;k x 2 0 8
4 4
;k x 0 4
5 2 3
6
lkFk gh x y x y 0 9 0 1
5 3 2 6
;k x x y y 0 0 9 1
(8)
'ks [kkokVh fe'ku&
l= %
2 4
1 2 4
;k 2 y 0 10 ;k y 2 0 10
1 2
5
;k y 0
7.
fuEufyf[kr lehdj.k ls x ,oa y dk eku Kkr dhft,A
x
2
7
5
y 3
3 4
1 2
x
Ans. fn;k x;k gS 2
7
2 x
10
7 6
15 14
5 3 4
y 3 1 2
3 4
2
;k 14 2 y 6 1
2 x 3
10 4
7 6
15 14
7 6
15 14
7
6
;k 14 1 2 y 6 2 15 14
2 x 3
;k 15
¼laxr vo;oksa dh rqyuk djus ij½
;k 2 x 3 7
;k 2 x 7 3
rFkk 2 y 4 14
rFkk 2 y 14 4
;k 2 x 4
rFkk 2 y 18
;k x
4
2
;k x 2
8.
7 6
2 y 4 15 14
6
rFkk y
18
2
rFkk y 9
cos x sin x 0
;fn F ( x) sin x cos x 0 gS rks fl) dhft, F ( x).F ( y ) F ( x y )
0
0
1
Ans. F ( y ) ds fy, x ds LFkku ij y j[ks rFkk blh izdkj F ( x y ) ds fy, x ds LFkku ij x y j[ksA
cos x sin x 0 cos y sin y 0
cka;k Hkkx F ( x).F ( y ) sin x cos x 0 sin y cos y 0
0
0
1 0
0
1
cos x cos y sin x sin y cos x sin y sin x cos y 0
;k F ( x).F ( y ) sin x cos y cos x sin y sin x sin y cos x cos y 0
0
0
1
cos( A B ) cos A cos B sin A sin B
sin( A B ) sin A cos B cos A sin B
(9)
'ks [kkokVh fe'ku&
l= %
cos( x y sin( x y ) 0
;k F ( x).F ( y ) sin( x y ) cos( x y ) 0
0
0
1
;k F ( x).F ( y ) F ( x y )
9.
2 0 1
;fn A 2 1 3 gS rks A 2 5 A 6 I dk eku Kkr dhft,A
1 1 0
2 0 1 2 0 1
A A A 2 1 3 2 1 3
Ans.
1 1 0 1 1 0
2
4 0 1 0 0 1 2 0 0
A 4 2 3 0 1 3 2 3 0
2 2 0 0 1 0 1 3 0
2
5 1 2
A 9 2 5
0 1 2
2
5 1 2
2
A 5 A 6 I 9 2 5 5 2
0 1 2
1
2
5
A 5 A 6 I 9
0
2
1
2
10
10
1 2 5
2
5
1
1 0 0
3 6 0 1 0
0 0 1
1 0
0
1
5 6 0 0
15 0 6 0
5 0 0 0 6
0
5
5 10 6 1 0 0 2 5 0
A 5 A 6 I 9 10 0 2 5 6 5 15 0
0 5 0 1 5 0 2 0 6
2
1 1 3
A 5 A 6 I 1 1 10
4
5 4
2
(10)
'ks [kkokVh fe'ku&
10.
l= %
3 2
1 0
;fn A 4 2 rFkk I 0 1 ,oa A 2 k . A 2 I gks rks k dk eku Kkr dhft,A
3 2 3 2
2
Ans. A A.A
.
4 2 4 2
98
6 4
2
;k A 12 8 8 4
1 2
2
;k A 4 4
A2 k . A 2 I
1 2
3 2 1 0
k
2
4 4
4 2 0 1
1 2 3k
4 4 4k
2k 2 0
2k 0 2
2k
1 2 3k 2
2k 2
4 4 4k
leku vkO;wg ds xq.k/keZ }kjk leku vkO;wg ds laxr vo;oksa dks leku j[kus ij
3k 2 1 k 1
2 k 2 k 1
4k 4 k 1
4 2 k 2 k 1
11.
vr% k 1
2
;fn A 4 rFkk B 1 3 6 gS rks lR;kfir dhft, ( AB )' B' A' gSA
5
tgka A' ] vkO;wg A dk ifjorZ vkO;w gSA
2
A 4
Ans. ;gka
,oa B 1 3 6
5
2
2 6 12
blfy, AB 4 1 3 6 4 12 24
5
5 15 30
4
5
2
vr% ( AB )' 6 12 15
12 24 30
(11)
'ks [kkokVh fe'ku&
l= %
1
vc A' 2 4 5 ,oa B' 3
6
4
5
1
2
blfy, B' A' 3 2 4 5 6 12 15
6
12 24 30
vr% ( AB )' B' A'
12.
1 0
2 3
B
rFkk
gS rks ( A 2 B )' Kkr dhft,A
2
1 2
;fn A' 1
1 0
2 3
B
Ans. fn;k x;k gS& A'
rFkk
1 2
1 2
1 1
2
vr% B ' 0
( A 2 B )' A' (2 B)'
( A 2 B )' A'2 B '
2 3 1 1
( A 2B)'
2
1 2 0 2
2 3 2 2
( A 2 B )'
1 2 0 4
2 2 3 2
( A 2 B )'
1 0 2 4
4 5
( A 2 B )'
1 6
13.
;fn A rFkk B leku dksfV ds lefer vkO;wg gS rks n'kkZb, fd AB lefer gS] ;fn vkSj dsoy ;fn A rFkk B Øefofues; gS] vFkkZr~
AB = BA gSA
Ans. fn;k gS fd A rFkk B nksuksa lefer vkO;wg gS] blfy, A' = A rFkk B' = B gSA
eku yhft, fd AB lefer gS rks = (AB)' = AB
fdUrq (AB)' = B'A' = BA
vr% AB = BA
foykser& ;fn AB = BA gS rks ge fl) djsxsa fd AB lefer gSA
vc (AB)' = B'A'
(AB)' = BA
( A ,oa B lefer gS)
vr% AB lefer gSA
(12)
'ks [kkokVh fe'ku&
14.
l= %
1 2 0 0
1 2 12 0 1 2 0 gSA
x ds fdl eku ds fy,
1 0 2 x
1 2 0 0
1 2 12 0 1 2 0
Ans. fn;k x;k gS&
1 0 2 x
0
;k 1 4 1 2 0 0 0 2 22 0
x
0
;k 6 2 42 0
x
;k 0 4 4 x 0
4 4 x 0 4 x 4
vr% x
4
1
4
vr% x 1
'ks[kkokVh fe'ku 100 dh d{kk 10 ,oa 12 ds
fofHkUu fo"k;ksa dh uohure cqdysV MkmuyksM
djus gsrq Vsyhxzke QR CODE LdSu djsa
(13)
'ks [kkokVh fe'ku&
l= %
vadHkkj (1 + 1 + 1 + 2 = 5)
1.
x
x 1
dk eku gksxk\
x 1
x
(1) 0
2.
(2) 1
x
2
6
(2) –6
(4) 0
(3) –1
(4) + 1
(3) |A|n+1
(4) |A|n–1
rhu lajs[k fcUnqvksa ls cus f=Hkqt dk {ks=Qy gksxk &
(2) 1
;fn n dksfV dk ,d oxZ vkO;wg A gks rks |adj(A)| gksxk&
(1) |A|
5.
(3) + 6
2
(1) 0
4.
(4) 3
;fn 18 x 18 6 gks rks x cjkcj gS&
(1) 6
3.
(3) –1
(2) |A|n
;fn A dksfV nks dk O;qRØe.kh; vkO;wg gks] rks del (A–1) gksxk&
1
(2) del ( A)
(1) del(A)
(3) 1
-: Answer:1-2, 2-3, 3-1, 4-4, 5-2
6.
3 x
3 2
;fn x 1 4 1 rks x dk eku Kkr dhft,A
Ans. fn;k gS fd
3 x
x 1
3 2
4 1
;k 3 x 2 3 8
;k x 2 8
vr% x 2 2
7.
1 2
;fn A 4 2 rks fn[kkb, | 2 A | 4 | A |
1 2
Ans. fn;k x;k gS A
4 2
| A | 2 8 6
1 2 2 4
2 A 2
4 2 8 4
(14)
(4) 0
'ks [kkokVh fe'ku&
l= %
| 2 A | 2 4 8 4 8 32 24
vr% | 2 A | 24 4(6) 4 | A |
vr% | 2 A | 4 | A |
8.
,d f=Hkqt dk {ks=Qy Kkr dhft, ftlds 'kh"kZ (3, 8), (-4, 2) ,oa (5, 1) gSA
Ans. f=Hkqt dk {ks=Qy
3 8 1
1
1
4 2 1 3(2 1) 8(4 5) 1(4 10)
2
2
5 1 1
1
2
(3 72 14)
9.
61
oxZ bdkbZ
2
2 3
x
3
;fn 4 5 2 x 5 rks x dk eku Kkr dhft,A
Ans. fn;k x;k gS
2 3
4 5
x
3
2x 5
vFkkZr~ 10 12 5 x 6 x
vFkkZr~ 2 x
vFkkZr~ x 2
10.
2 3
vkO;wg A 1 4 dk lg[kaMt Kkr dhft,A
A11 4, A12 1
Ans. ge tkurs gS fd
A21 3, A22 2
A
A
4
11
21
vr% adjA A A 1
22
12
11.
3
2
1 2
ds lHkh vo;oksa ds milkjf.kd o lg[kaM Kkr dhft,A
3
lkjf.kd 4
Ans. vo;o aij dk milkjf.kd M ij gSA
;gka a11 1 ] blfy, M 11 a11 dk milkfj.kd 3
M 12 vo;o a12 dk milkjf.kd 4
M 21 vo;o a21 dk milkjf.kd 2
M 22 vo;o a 22 dk milkjf.kd 1
(15)
'ks [kkokVh fe'ku&
l= %
vc aij dk lg[kaM Aij gSA blfy,
A11 (1)11| M 11 (1) 2 (3) 3
A12 (1)12| M12 (1)3 (4) 4
A21 (1) 2 1| M 21 (1) 3 (2) 2
A22 (1) 2 2| M 22 (1) 4 (1) 1
12.
lkjf.kd
1 2 4
1 3 0
4
1 0
dk eku Kkr dhft,A
Ans. /;ku nhft, fd rhljs LraHk esa nks izfof"V;ka 'kwU; gSa blfy, rhljs LraHk C3 ds vuqfn'k izlj.k djus ij gesa izkIr gksrk gS fd
4
1 2
4
1
0
1 2
4 1
0
1
2
1 3
4(1 12) 0 0
52
13.
lkjf.kdska dk iz;ksx djds A(1,3) vkSj B(0, 0) dks tksM+us okyh js[kk dk lehdj.k Kkr dhft, vkSj k dk eku Kkr dhft,A ;fn
,d fcUnq D(k, 0) bl izdkj gS fd ABD dk {ks=Qy 3 oxZ bdkbZ gSA
Ans. eku yhft,] AB ij dksbZ fcUnq P(x, y) gS rc
ABP dk {ks=Qy = 0
0 0 1
1
blfy, 2 1 3 1 0
x y 1
blls izkIr gS
1
( y 3 x) 0
2
;k y 3 x
tks vHkh"V js[kk dk lehdj.k gSA
fdUrq ABD dk {ks=Qy 3 oxZ bdkbZ fn;k gS vr%
1 3 1
1
0 0 1 3
2
k 0 1
gesa izkIr gS
3k
3
2
vr% k 2
(16)
'ks [kkokVh fe'ku&
14.
l= %
2 3
iznf'kZr dhft, fd vkO;wg A 1 2
lehdj.k A2 4 A I 0 ] tgka I ,2 2 dksfV dk ,d rRled vkO;wg gS vkSj 0,2 2 dksfV dk ,d 'kwU; vkO;wg gSa bldh
lgk;rk ls A1 Kkr dhft,A
Ans. ge tkurs gS fd
2 3 2 3 7 12
A 2 A. A
1 2 1 2 4 7
7 12 8 12 1 0
0 0
2
vr% A 4 A I 4 7 4 8 0 1 0 0 0
vc A2 4 A I 0
| A |
2 3
1 2
4 3 1 0
A 2 4 A I 0 dks A1 ls xq.kk djus ij
A 1.( A 2 4 A I ) A 1 .0
( A 1 . A) A 4 A 1 A A1 0
A 4 I A 1 0
A 1 4 I A
4 0 2 3 2 3
A 1
0 4 1 2 1 2
2
1
vr% A 1
15.
3
2
3 7
6 8
;fn A 2 5 vkSj B 7 9 gS rks lR;kfir dhft, fd ( AB ) 1 B 1 A1 gSA
3 7 6 8 67 87
Ans. gr tkurs gS fd AB
2 5 7 9 47 61
67 87
D;ksafd | AB | 47 61 4087 4089 2 0
( AB ) 1 dk vfLrRo gS vkSj bls fuEufyf[kr izdkj ls O;Dr fd;k tkrk gSA
61 87
61
87
2
1
1
2
( AB ) 1
.adj ( AB)
47
67
AB
2 47 67
2
2
(17)
'ks [kkokVh fe'ku&
l= %
3 7
vkSj | A | 2 5 15 14 1 0
6 8
vkSj | B | 7 9 54 56 2 0
blfy, A1 ,oa B 1 nksuksa dk vfLrRo gS vkSj fuEu izdkj ls fy[kk tk ldrk gSA
A1
1
adj ( A)
( A)
1 5 7 5 7
A 1
1 2 3 2 3
1 9
1
1
,oa B | B | adj ( B ) 2 7
1 9
1
1
vr% B . A 2 7
8
6
8 5 7
1 45 16 63 24
6 2 3
2 35 12 49 18
1 61 87
B 1 A 1
2 47 67
vr% ( AB ) 1 B 1. A1
16.
1 1 2 2 0 1
vkO;wgksa ds xq.kuQy 0 2 3 9 2 3 dk iz;ksx djrs gq, fuEufyf[kr lehdj.k fudk; dks gy dhft,A
3 2 4 6 1 2
x y 2z 1
2 y 3z 1
3x 2 y 4 z 2
Ans. fn;k x;k xq.kuQy
1 1 2 2 0 1 2 9 12 0 2 2 1 3 4 1 0 0
0 2 3 9 2 3 0 18 18 0 4 3 0 6 6 0 1 0
3 2 4 6 1 2 6 18 24 0 4 4 3 6 8 0 0 1
1 1 2
vr% 0 2 3
3 2 4
1
2 0 1
9 2 3
6 1 2
(18)
'ks [kkokVh fe'ku&
l= %
vc fn, x, lehdj.k fudk; dks vkO;wg ds :i esa fuEufyf[kr izdkj ls fy[k tk ldrk gSA
1 1 2 x 1
0 2 3 y 1
3 2 4 z 2
1
x 1 1 2 1 2 0 1 1
;k y 0 2 3 1 9 2 3 1
z 3 2 4 2 6 1 2 2
x 2 0 2 0
;k y 9 2 6 5
z 6 6 4 3
vr% x 0, y 5, z 3
'ks[kkokVh fe'ku 100 dh d{kk 10 ,oa 12 ds
fofHkUu fo"k;ksa dh uohure cqdysV MkmuyksM
djus gsrq Vsyhxzke QR CODE LdSu djsa
(19)
'ks [kkokVh fe'ku&
l= %
vadHkkj (1 + 1 + 2 + 2 + 2 = 8)
1.
2.
Qyu f ( x) [ x] vlarr gS&
(1) izR;sd okLrfod la[;k ij
(2) izR;sd ifjes; la[;k ij
(3) izR;sd iw.kkZad ij
(4) izR;sd vifjes; la[;k ij
;fn y cos x rks
dk eku gksxk &
dx
dy
(1)
3.
sin
x
(2)
2 x
(3)
x
sin
x
2 x
(3) x 6 5 log x
(2) x 2 5 6 log x
cos x 2
3
(2)
;fn y x, log e x rks
(1)
2 cos x
3
(3)
cos x 2
3
sin
x
x
(4) x 2 6 5 log x
1
1 x
(2)
1
x
(3) log e 1 x
1-3, 2-1, 3-1, 4-3, 5-2
d
e
dx
d
Ans. dx e
x
x
------------------ gSA
1
2 e
x
(4)
cos x 2
3
d2y
dk eku gksxk &
dx 2
-: Answer :-
6.
(4)
Qyu 3 y sin x 2 x dk x ds lkis{k vodyu gS&
(1)
5.
x
x 3 log x dk x ds lkis{k f}rh; dksfV dk vodyt gS&
(1) x 5 6 log x
4.
sin
e
x
1
2 x
e
x
e
4 x
x
4 xe
x
7.
d x
a ---------------------- gSA
da
Ans.
d x
a xa x 1
da
8.
d2y
----------------- gSA
;fn y tan x gS rks
dx 2
1
dy
1
d2y
1
,
Ans. dx 1 x 2 dx 2
1 x2
2
2x
2x
1 x
2 2
(20)
(4) 1 log e x
'ks [kkokVh fe'ku&
9.
l= %
Qyu f ( x) | ( x 3 |, x R, x ds eku ---------- ij vodyuh; ugha gSA
Ans. x 3 ij f ( x) | ( x 3) | vodyuh; Qyu ugha gksrk gSA
10.
| x |
, if x 0
f ( x) x
0, if x 0
| x |
, if x 0
f
(
x
)
x
Ans. ;gka]
0, if x 0
f ( x) lim
vc] LHL xlim
0
x 0
x0h
|x|
x
j[kus ij] x 0 h 0
|0h|
h
lim
1
h 0 0 h
h 0 h
lim
rFkk
RHL lim f ( x) lim
x0h
x 0
j[kus ij x 0 h 0 lim
h 0
x 0
|x|
x
|0h|
h
lim 1
x
0
0h
h
LHL RHL
vr% f ( x), x 0 ij vlrr Qyu gSA
11.
f ds lrr~rk dh tkap dhft,] tgka f fuEufyf[kr izdkj ls ifjHkkf"kr gS
sin x cos x, if x 0
f ( x)
1,
if x 0
sin x cos x, if x 0
Ans. ;gka] f ( x )
1,
if x 0
vc]
x0h
LHL lim f ( x) lim (sin x cos x)
x 0
x 0
j[kus ij] x 0 h 0
limsin 0 h cos0 h lim sinh cosh 0 1 1
h 0
rFkk
x0h
h 0
RHL lim f ( x) lim (sin x cos x)
x 0
x 0
j[kus ij] x 0 h 0
limsin 0 h cos0 h limsinh cosh 0 1 1
h 0
h 0
LHL RHL f (0)
vr% f ( x), x 0 ij lrr Qyu gSA
ge tkurs gSa fd x 0 ds fy, f ( x) sin x cos x lrr Qyu gS rFkk x 0 ds fy, f ( x) sin x cos x lrr~ Qyu gSA
vr% f ( x), x ds izR;sd eku ds fy, lrr gSA
(21)
'ks [kkokVh fe'ku&
l= %
fuEufyf[kr iz'uksa esa k ds ekuksa dks Kkr dhft,] rkfd iznr Qyu fufnZ"V fcUnq ij lrr gksA ¼iz-la- 12&13½
12.
k cos x
2 x , ; fn x 2
f ( x)
3,
; fn x
2
k cos x
2 x , ; fn x 2
f ( x)
Ans. ;gka]
3,
; fn x
2
vc]
x
LHL lim f ( x) lim
x
2
x
2
x
h j[kus ij] x
x
2
2
k cos
2x
h0
k cos h
2
lim k sinh
lim
h 0
h 0 2h
2 h
2
sin x
1
lim
x
0
x
k sinh k
k
lim
1
h 0 2
h
2
2
rFkk
x
RHL lim f ( x) lim
x
2
2
x
h j[kus ij] x
k cos x
2x
2
2
h0
k cos h
2
lim k sinh lim lim k sinh
lim
h 0
2h
h 0 2
h
h 0 2 h
2 h
2
sin x
1
lim
x
0
x
k
k
1
2
2
f 3
2
iqu%]
pwfa d f ( x), x
2
ij lrr Qyu gSA
k
LHL RHL f 3
2 2
k 6
(22)
'ks [kkokVh fe'ku&
13.
l= %
kx 1, ; fn x 5
f ( x)
}kjk ifjHkkf"kr Qyu x 5 ij
3x 5, ; fn x 5
kx 1, ; fn x 5
Ans. ;gka] f ( x)
3x 5, ; fn x 5
f ( x) lim (kx 1)
vc] LHL xlim
5
x 5
x 5h
j[kus ij] x 5 h 0
limk 5 h 1 lim5k kh 1 5k 1
rFkk
RHL lim f ( x) lim 3 x 5
h 0
x 5h
h 0
x 5
x 5
j[kus ij] x 5 h 0
lim35 h 5 lim10 3h 10
iqu%]
f (5) 5k 1
h 0
h0
f ( x) kx 1
f ( x) x 5 ij lrr~ Qyu gSA
LHL RHL f (5)
LHL RHL f (5) 5k 1 10 k
14.
9
5
n'kkZb, fd f ( x ) | cos x | }kjk ifjHkkf"kr Qyu ,d lrr~ Qyu gSA
Ans. eku yhft, g ( x ) cos x rFkk h( x) | x |
ge tkurs gS fd izR;sd x R ds fy, g ( x ) cos x lrr~ Qyu gSA
iqu% izR;sd x R ds fy, ekikad Qyu h( x) | x | lrr~ Qyu gksrk gSa vr% la;kstd Qyu]
izR;sd x R ds fy,] (hog )( x) h( g ( x)) h(cos x) | cos x | lrr Qyu gksxkA
vr% f ( x ) | cos x | ] izR;sd x R ds fy, lrr~ Qyu gSA
15.
1 x2
,0 x 1
y cos 1
2
1 x
(i)
dy
?
dx
Ans. tan 1 x vFkkZr~ x tan j[kus ij]
2
2
1 1 x
1 1 tan
y
cos
cos
1 x2
1 tan 2
1 tan 2
1 tan 2 cos 2
y cos 1 cos 2 2 2 tan 1 x
x
ds lkis{k vodyu djus ij]
dy
d
2
2
tan 1 x
dx
dx
1 x2
1
d
1
tan x
1 x2
dx
(23)
'ks [kkokVh fe'ku&
l= %
2
1 1 x
,0 x 1
(ii) y sin
2
1
x
Ans. x tan tan 1 x j[kus ij]
2
1 1 tan
1
y
sin
sin cos2
2
1
tan
1
y sin sin 2 2
y
x
2
2 y
2 tan 1 x
2
ds lkis{k vodyu djus ij]
dy d
d
2
tan 1 x
dx dx 2
dx
dy
2
0
dx
1 x2
18.
dy
2
dx 1 x 2
y sin tan 1 e x
dy
?
dx
Ans. eku yhft, y sin tan 1 e x
x
ds lkis{k vodyu djus ij]
dy d
sin tan 1 e x
dx dx
dxd tan e
1 1e
cos tan 1 e x
cos tan 1 e x
cos tan 1 e x
1
d
1
dx tan x 1 x 2
1
1
1 e
x 2
2 x
x
¼J`a[kyk fu;e ls½
d x
e
dx
e x cos tan 1 e x
1 e2 x
(24)
'ks [kkokVh fe'ku&
17.
l= %
y log cos e x
dy
?
dx
Ans. eku yhft, y log cos e x
ds lkis{k vodyu djus ij]
x
cos1e dxd cose
dy d
log cos e x
dx dx
x
x
¼J`a[kyk fu;e ls½
tan e .e e tan e
1
d x
sin e x
e
x
dx
cos e
x
18.
x
x
x
2x
y cos 1
1 x 1
2
1 x
dy
?
dx
Ans. x tan tan 1 x j[kus ij]
2 tan
y cos 1
2
1 tan
2 tan
sin 2
1 tan 2
y cos 1 sin 2
1
y cos cos 2 2
y
x
19.
2
2 y
2
sin 2 cos 2 2
2 tan 1 x
ds lkis{k vodyu djus ij]
1
dy
2
0
dx
1 x2
dy
2
dx 1 x 2
yx xy
tan x
dy
?
dx
Ans. fn;k gS] y x x y
nksuksa rjQ dk y?kqx.kd ysus ij]
log y x log y x x log y y log x
x
1
d
1
dx tan x 1 x 2
ds lkis{k vodyu djus ij]
(25)
'ks [kkokVh fe'ku&
l= %
d
x log y d y log x
dx
dx
1 dy
1
dy
x y dx log y y x log x dx
x dy
dy y
y dx log x dx x log y
x
dy y
y log x dx x log y
x y log x dy y x log y
dy y y x log y
y
x
dx x x y log x
dx
20.
x cos , y b cos
dy
?
dx
Ans. fn;k gS] x a cos rFkk y b cos
ds lkis{k vodyu djus ij]
dy
dy
a sin rFkk
b sin
d
d
21.
dy
dy d dy d b sin b
dx dx d
a sin a
d
dy dy / dt
dx dx / dt
x sin t , y cos 2t
dy
?
dx
Ans. x sin t rFkk y cos 2t
t ds lkis{k vodyu djus ij]
dx
dy
cos t rFkk
sin 2t 2
dt
dt
dy
dy dx dy dt
dx dx dt dx
dt
2 sin 2t 22 sin t cos t
4 sin t
cos t
cos t
dy dy / dt
dx dx / dt
sin 2 2 sin cos
(26)
'ks [kkokVh fe'ku&
22.
l= %
d2 y
?
dx2
y log log x
Ans. eku yhft, y log log x
x
ds lkis{k vodyu djus ij]
dy
1 d
log x 1 . 1 1
dx log x dx
log x x x log x
d2y d 1 d
1
iqu% x ds lkis{k vodyu djus ij] dx 2 dx x log x dx x log x
1 x log x
2
23.
1
x log x
2
d
x log x
dx
d
d
x dx log x log x dx ( x)
1
1 log x
x. log x.1
x log x x
x log x 2
1
2
;fn y 5 cos x 3 sin x gS] rks fl) dhft, fd
d2y
y0
dx 2
Ans. fn;k gS] y 5 cos x 3 sin x
x ds
lkis{k vodyu djus ij]
dy
5sin x 3cos x
dx
iqu% x ds lkis{k vodyu djus ij]
d2y d
5 sin x 3 cos x 5 cos x 3 sin x 5 cos x 3 sin x y
dx 2 dx
d2y
y0
dx 2
24.
;fn y 500e 7 x 600e 7 x gS] rks n'kkZb, fd
d2y
49 y
dx 2
Ans. fn;k gS] y 500e 7 x 600e 7 x
x
ds lkis{k vodyu djus ij]
dy
d
d
500e 7 x 7 x 600e 7 x 7 x
dx
dx
dx
500e 7 x .7 600e 7 x .(7)
iqu% x ds lkis{k vodyu djus ij]
(27)
'ks [kkokVh fe'ku&
l= %
d2y
7 500 e 7 x .7 7 600e 7 x 7
2
dx
49 500e 7 x 600e 7 x
25.
y log x
log x
d2y
49 y
dx 2
ds nksuksa rjQ dk y?kqx.kd ysdj vodyu dhft,A
Ans. eku yhft, y log x log x
nksuks rjQ dk y?kqx.kd ysus ij]
log y log log x
log x
log m
log y log x log log x
x ds
n
n log m
lkis{k vodyu djus ij]
1 dy
d
d
log x loglog x log log( x) log( x)
y dx
dx
dx
(log x)
26.
1 1
1 1
log(log x) 1 loglog x
log x x
x x
dy y
1 log(log x)
dx x
(log x) log x
1 log(log x) log x log x 1 log(log x)
x
x
x
ax 1, ; fn x 3
a vkSj b ds mu ekuksa dks Kkr dhft, ftuds fy, f ( x )
}kjk ifjHkkf"kr Qyu x 3 ij larr gSA
bx 3, ; fn x 0
Ans. fn;k gS% Qyu x 3 ij larr gSaA blfy, ck,a i{k dh lhek nk,a i{k dh lhek f (3)
f ( x) lim f ( x) f (3)
xlim
3
x 3
ax 1 lim bx 3 3a 1
xlim
3
x 3
3a 1 3b 3 3a 1
3a 3b 2 a b
2
3
(28)
'ks [kkokVh fe'ku&
l= %
vadHkkj (1 + 1 + 2 + 2 = 6)
1.
,d 10 m f=T;k ds csyukdkj Vadh esa 314 m3/h dh nj ls xsga w Hkjk tkrk gSa Hkjs x, xsgwa dh xgjkbZ dh o`f) nj gS&
(1) 1 m/h
2.
(4) 0.5 m/h
(2) 2,0
(3) 2,
(4) 0,2
,d o`r dh f=T;k r = 6 cm ij r ds lkis{k {ks=Qy esa ifjorZu dh nj gS&
(1) 10
4.
(3) 1.1 m/s
fuEufyf[kr esa ls fdl varjky esa y = x2 e–x o/kZeku gS\
(1) ,
3.
(2) 0.1 m/h
(2) 12
(3) 8
(4) 11
,d mRikn dh x bdkb;ksa ds foØ; ls izkIr dqy vk; #Ik;ksa esa R(x)= 3x2 + 36x + 5 ls iznr gSA tc x = 15 gS rks lhekar vk;
gS&
(1) 116
5.
(3) 90
(4) 126
(3) cos 3x
(4) tan x
fuEufyf[kr esa dkSu ls Qyu 0, 2 esa gzkleku gS\
(1) cos x
6.
(2) 96
(2) sin x
fuEufyf[kr varjkyksa esa ls fdl varjky esa f ( x) x 400 sin x 1 }kjk iznr Qyu f gzkleku gS\
(1) 0,1
(2) ,
2
(3) 0,
2
(4) buesa ls dksbZ ugha
-: Answer :1-1, 2-4, 3-2, 4-4, 5-1, 6-4
7.
;fn f ( x) | x 1 | 3 rks f (x) dk vf/kdre eku ------------------------------- gSA
Ans. fn;k x;k Qyu g ( x) | x 1 | 3
ge tkurs gS fd izR;sd x R ds fy,] | x 1 | 0
izR;sd x R ds fy,] | x 1 | 0
izR;sd x R ds fy,] | x 1 | 3 3
g dk mPpre eku rHkh Kkr fd;k tk ldrk
tc | x 1 | 0 ;k x 1
vFkkZr | x 1 | 0 x 1
g dk mPpre eku g (1) | 1 1 | 3 3
8.
,d o`r dh f=T;k 0-7 lseh@lsd.M dh nj ls o`f) gks jgh gS] rc o`r dh ifjf/k esa ifjorZu dh nj -------------- gksxhA
Ans. eku yhft, le; t ij o`r dh f=T;k r gS vkSj bldh ifjf/k c gSA
rc c 2r
(29)
'ks [kkokVh fe'ku&
l= %
dc
dr
2
¼ t ds lkis{k vodyu djus ij] J`a[kyk fu;e }kjk½
dt
dt
ifjf/k dh o`f) dh nj]
tgka]
dr
f=T;k dh o`f) dh nj gSA
dt
dr
0.7 lseh@ls
dt
dc
dr
2 (0.7) lseh@ls 14 lseh@ls ¼ 0.7 lseh@ls] fn;k gS½
dt
dt
vr%
9.
ifjf/k dh o`f) dh nj 14 lseh@ls gSA
varjky 1,5 esa f ( x) 2 x 3 15 x 2 36 x 1 }kjk iznr Qyu ds fujis{k mPpre vkSj fujis{k fuEure ekuksa dks Kkr dhft,A
Ans. gesa Kkr gS
f ( x) 2 x 3 15 x 2 36 x 1
;k f ' ( x) 6 x 2 30 x 36 6( x 3)( x 2)
/;ku nhft, f ' ( x) 0 ls x 2 vkSj x 3 izkIr gksrs gSaA
vc ge bu fcUnqvksa vkSj varjky 1,5 ds vaR; fcUnqvksa vFkkZr x 1, x 2, x 3 vkSj x 5 ij f ds eku dk ifjdyu djsx
a sA
vc
f (1) 2(13 ) 15(12 ) 36(1) 1 24
f (2) 2(23 ) 15(2 2 ) 36(2) 1 29
f (3) 2(33 ) 15(32 ) 36(3) 1 28
f (5) 2(53 ) 15(5 2 ) 36(5) 1 56
bl izdkj] ge fu"d"kZ ij igqaprs gS fd varjky 1,5 ij Qyu f ds fy, x 5 ij fujis{k mPpre eku 56 vkSj x 1 ij fujis{
fuEeure eku 24 gSA
10.
varjky Kkr dhft, ftuesa f ( x) sin x cos x,0 x 2 }kjk iznr Qyu f o/kZeku ;k gzkleku gSA
Ans. Kkr gS fd
f ( x) sin x cos x,0 x 2
;k f ' ( x) cos x sin x
5
vc f ' ( x) 0 ls sin x cos x ftlls gesa x ,
izkIr gksrs gSAa D;ksafd 0 x 2
4 4
fcUnq x
4
vkSj x
3
5
5
varjky 0,2 dks rhu vla;qDr varjkyksa] uker% 0, 4 , 4 , 4 vkSj 4 ,2 esa foHkDr djrs
4
gSaA
(30)
'ks [kkokVh fe'ku&
l= %
5
/;ku nhft, fd f ' ( x) 0 ;fn x 0, 4 4 ,2
5
vr% varjkyksa 0, 4 vkSj 4 ,2 esa Qyu f o/kZeku gSA
5
vkSj f ' ( x) 0 ] ;fn x 4 , 4
5
vr% f varjky 4 , 4 esa gzkleku gSA
11.
fn[kkb, fd iznr Qyu f ]
f ( x ) x 3 3 x 2 4 x, x R
R ijo/kZeku Qyu gSA
Ans. /;ku nhft, fd
f ( x) 3x 2 6 x 4
3 x2 2x 1 1
2
3 x 1 1 0 ] lHkh x R ds fy,
blfy, Qyu f , R ij o/kZeku gSA
12.
fl) dhft, fd iznr Qyu f ( x) cos x
(a ) (0, ) esa gzkleku gS
(b) ( ,2 ) esa o/kZeku gS
(c) (0,2 ) esa u rks o/kZeku vkSj u gh gzkleku gSA
Ans. /;ku nhft, fd f ' ( x) sin x
(a) pwfa d izR;sd x 0, ds fy, sin x 0 ] ge ikrs gSa fd f ' ( x) 0 vkSj blfy, 0, esa f gzkleku gSA
(b) pwfa d izR;sd x ,2 ds fy, sin x 0 ] ge ikrs gS fd f ' ( x) 0 vkSj blfy, ,2 esa f o/kZeku gSA
(c) mijksDr (a) vkSj (b) ls Li"V gS fd 0,2 esa f u rks o/kZeku gS vkSj u gh gzkleku gSA
13.
os varjky Kkr dhft, ftuesa f ( x) 4 x 3 6 x 2 72 x 30 }kjk iznr Qyu f (a) o/kZeku (b) gzkleku gSaA
Ans. ;gka
f ( x) 4 x 3 6 x 2 72 x 30
;k f ' ( x) 12 x 2 12 x 72
12 x 2 x 6
12 x 3x 2
blfy, f ' ( x) 0 ls x 2,3 izkIr gksrs gSAa x 2 vkSj x 3 okLrfod js[kk dks rhu vla;Dq r varjkyks]a uker% ,2, 2,3
vkSj 3, esa foHkDr djrk gS ¼fp=kuqlkj½
(31)
'ks [kkokVh fe'ku&
l= %
varjkyksa ,2 vkSj 3, esa f ' ( x) /kukRed gS tcfd varjky 2,3 esa f ' ( x) _.kkRed gSA QyLo:i Qyu f varjkyksa
,2 vkSj 3, esa o/kZeku gS tcfd varjky 2,3 esa Qyu gzkleku gSa rFkkfi f , R ij u rks o/kZeku gS vksj u gh gzkleku
gSA
14.
,slh nks la[;k,a Kkr dhft,] ftudk ;ksx 24 gS vkSj ftudk xq.kuQy mPpre gksA
Ans. lc ls igys ge nks la[;k x vkSj 24 x ekurs gS fQj ge y x24 x ekurs gSa vc vodyu dk vuqiz;ksx djds nksuksa la[;k
Kkr djrs gSA
eku yhft, igyh la[;k x gS] rc nwljh la[;k 24 x gSA
¼ nks la[;kvksa dk ;ksx 24 gS½
;fn y ds lkis{k dk xq.kuQy dks n'kkZrk gS rks
y x24 x 24 x x 2
x ds
lkis{k vodyu djus ij]
dy
d2y
24 2 x vkSj
2
dx
dx 2
vc
dy
0 j[kus ij
dx
24 2 x 0
x 12
x 12 ij]
d2y
2 0
dx 2
blfy, la[;kvksa dk xq.kuQy mPpre gksxk tc la[;k x 12 gksxh vkSj 24 12 12 blfy, la[;k, 12 vkSj 12 gSA
15.
fl) dhft, fd Qyu f ( x) log | cos x | 0, 2 esa fujUrj gzkleku vkSj 2 , esa fujUrj o/kZeku gSA
f ( x) log(cos x)
Ans. fn;k gS]
f '( x)
1
( sin x) tan x ¼ x ds lkis{k voyu djus ij½
cos x
varjky 0, 2 es]a tan x 0
¼ tan x izFke prqFkkZa'k esa gS½
tan x 0
¼ tan x izFke prqFkkZa'k esa gS½
0, esa ] f ' ( x) 0
2
vr% 0, 2 esa f fujUrj gzkleku gSA
(32)
'ks [kkokVh fe'ku&
l= %
vc varjky 2 , esa tan x 0
¼ tan x f}rh; prqFkkZa'k esa gS½
tan x 0
, esa f ' ( x) 0
2
blfy,] 2 , esa f fujUrj o/kZeku gSA
16.
fl) dhft, fd R esa fn;k x;k Qyu f ( x) x 3 3 x 2 3 x 100 o/kZeku gSA
Ans. fn;k gS]
f ( x) x 3 3 x 2 3 x 100
¼ x ds lkis{k vodyu djus ij½
f ' ( x) 3 x 2 6 x 3
f ' ( x) 3 x 2 2 x 1 3 x 1
2
2
x R esa x 1 0 ] D;ksafd ,d iw.kZ oxZ _.kkRed ugha gks ldrk gSA
vr% R ij f ' ( x) 0 blfy,] fn;k x;k Qyu f o/kZeku Qyu gSA
17.
fl) dhft, fd Qyu f ( x) log sin x, 0, 2 esa fujUrj o/kZeku vkSj 2 , esa fujUrj gzkleku gSA
f ( x ) log(sin x)
Ans. fn;k gS]
f ' ( x)
1
(cos x) cot x
sin x
¼ x ds lkis{k vodyu djus ij½
varjky 0, 2 es]a f ( x) cot x 0
D;ksafd cot x izFke prqFkkZa'k esa /kukRed gksrk gSA
0, esa f fujUrj o/kZeku gSA
2
varjky 2 , esa] f ' ( x) cot x 0
D;ksafd cot x f}rh; prqFkkZa'k esa _.kkRed gSA
, ij f fujUrj gzkleku gSA
2
18.
a
dk og U;wure eku Kkr dhft, ftlds fy, varjky (1,2) esa f ( x) x2 ax 1 ls iznr Qyu fujUrj o/kZeku gSA
lcls igys] ge f ' ( x) Kkr djrs gS] rc 1 x 2 ds fy,] f ' ( x) 0 j[kdj a dk U;wure eku Kkr dhft,A
Ans. fn;k gS]
f ( x) x 2 ax 1
f ' ( x) 2 x a
varjky ¼1] 2½ es]a 1 x 2 2 2 x 4
2 a 2 x a 4 a
vr% f (x) fujarj o/kZeku Qyu gS] rc 2 a 0
[ 2 x a ( 2 a ) ds fy,] f ' ( x) 0 ]
( 2 a ) 0 a 2 blfy,] a dk U;wure eku a 2
(33)
'ks [kkokVh fe'ku&
19.
l= %
varjky Kkr dhft, ftuesa fuEufyf[kr Qyu f fujarj o/kZeku ;k gzkleku gSA
(i) f ( x) x 2 2 x 5
Ans. (i) eku yhft, fd
(ii) f ( x) 10 6 x 2 x 2
f ( x) x 2 2 x 5
f ' ( x) 2 x 2 0
¼ x ds lkis{k vodyu djus ij½
f ' ( x) 0 j[kus ij]
2 x 2, x 1
(34)
'ks [kkokVh fe'ku&
20.
l= %
,d xqCckjk tks lnSo xksykdkj jgrk gS] dh f=T;k ifjorZu'khy gSA f=T;k ds lkis{k vk;ru ds ifjorZu dh nj Kkr dhft, tc
f=T;k 10 lseh gSA
Ans. eku yhft, fd xksykdkj xqCckjs dh f=T;k r vkSj vk;ru V gSA
4
rc] r 10 lseh vkSj V r 3
3
2
f=T;k r ds lkis{k vk;ru ds ifjorZu dh nj dr 3 3r
dV
4
¼ r ds lkis{k vodyu djus ij½
¼ r 10 lseh½
vr% xqCckjs dk vk;ru 400 lseh @lseh dh nj ls c<+ jgk gSA
,d oLrq dh x bdkb;ksa ds mRiknu ls lacaf/kr dqy ykxr c(x) ¼#Ik;s es½a c( x) 0.007 x 3 0.003x 2 15 x 4000 ls iznr
gSA
lhekar ykxr Kkr dhft,] tcfd 17 bdkb;ksa dk mRiknu fd;k x;k gSA
4r 2 4 (10) 2 400
3
21.
lhekar ykxr ifj.kke ds lkis{k dqy ykxr ds ifjorZu dh nj gSA
Ans. lhekar ykxr
dc
dt
dc
0.007(3 x 2 ) 0.003(2 x) 15 0.021x 2 0.006x 15
dt
tc x 17 ] lhekar ykxr 0.021(17) 2 0.006(17) 15
0.021(289) 0.006(17) 2 15 6.069 0.102 15 20.967
vr% tc 17 bdkb;ksa dk mRiknu fd;k x;k gS] rks lhekar ykxr 20-967 #i;s gSA
22.
fdlh mRikn dh x bdkb;ksa ds foØ; ls izkIr dqy vk; R(x) #i;ksa esa R( x) 13x 2 26 x 15 ls iznr gSA lhekar vk; Kkr
dhft, tc x 7 gSA
lhekar vk; bdkb;ksa dh la[;k ds foØ; ds lkis{k dqy vk; ds ifjorZu dh nj gSA
Ans. lhekar vk;
dR d
13x 2 26 x 15 13 2 x 26 26 x 26
dx dx
tc x 7 ] rks lhekar vk; 26(7) 26 182 26 208
vr% lhekar vk; 208 #i;s gSA
23.
fl) dhft, f ( x) sin x ls iznr Qyu
(i) 0, esa fujarj o/kZeku gSA
2
(ii) , esa fujarj gzkleku
2
(iii) 0, esa u rks o/kZeku vkSj u gh gzkleku gSA
Ans. fn;k x;k Qyu f ( x) sin x gSA
x
ds lkis{k vodyu djus ij]
f ' ( x) cos x
(35)
'ks [kkokVh fe'ku&
l= %
(i) vr% izR;sd x 0, ds fy, cos x 0
2
¼ cos x izFke prqFkkZa'k esa /kukRed gS½
f ' ( x) 0
blfy,] 0, 2 esa fujarj o/kZeku gSA
(ii) vr%] izR;sd x , ds fy, cos x 0
2
¼ cos x f}rh; prqFkkZa'k esa /kukRed gS½
f ' ( x) 0
blfy,] f , 2 ,
(iii) tc x 0, ge ns[krs gS fd 0, esa f ' ( x) 0 vkSj , esa f ' ( x) 0
2
2
blfy, f ' ( x), (0, ) esa /kukRed vkSj _.kkRed gSA
blfy, (0, ) esa f (x) u rks o/kZeku gS vkSj u gh gzkleku gSA
24.
,d vk;r dh yEckbZ x,5 lseh@feuV dh nj ls ?kV jgk gS vkSj pkSM+kbZ y,4 lseh@feuV dh nj ls c<+ jgh gSA tc x 8 lseh
vkSj y 6 lseh gS] rc vk;r ds (a) ifjeki (b) {ks=Qy ds ifjorZu dh nj Kkr dhft,A
Ans. eku yhft, fd fdlh le; l ij] vk;r dh yEckbZ] pkSMk+ bZ] ifjeki vkSj {ks=Qy Øe'k% x, y, P vkSj A gS] rc P 2( x y )
vkSj A xy
------1
vk;r dk ifjeki = 2¼yEckbZ $ pkSM+kbZ½ vkSj {ks=Qy = yEckbZ × pkSM+kbZ
;g fn;k gS fd
vkSj
dx
5 lseh@feuV
dt
¼ ve fpUg n'kkZrk gS fd yEckbZ ?kV jgh gS½
dy
4 lseh@feuV
dt
(a) vc] P 2( x y )
t ds lkis{k vodyu djus ij]
ifjeki ds ifjorZu dh nj dt 2 dt dt
dP
dx
dy
2 5 4 lseh@feuV 2 lseh@feuV
dy
dx
4
5 &
dt
dt
vr%] vk;r dk ifjeki 2 lseh@feuV dh nj ls ?kV ¼ ve fpUg½ jgk gSA
(b) ;gka] vk;r dk {ks=Qy A xy
t ds lkis{k vodyu djus ij]
{ks=Qy ds ifjorZu dh nj
dA
dy
dx
x y
dt
dt
dt
(36)
'ks [kkokVh fe'ku&
l= %
dy
dx
4
5 &
dt
dt
dA
8 4 6 5
dt
2
32 30 2 lseh @feuV
vr%] vk;r dk {ks=Qy 2 lseh2@feuV dh nj ls c<+ jgh gSA
uksV& ;fn ifjorZu dh nj c<+ jgh gS] rks ge ¼ ve fpUg½ ysrs gS vkSj ;fn ifjorZu dh nj ?kV jgh gS rks ge ¼ ve ÇpUg½ ysrs
gSA
25.
,d ifjorZu'khy ?ku dh dksj 3 lseh@ls dh nj ls c<+ jgh gSA ?ku dk vk;ru fdl nj ls c<+ jgk gS] tcfd dksj 10 lseh yEch
gS\
Ans. eku yhft, ?ku dh dksj dh yEckbZ x gS] vkSj vk;ru V gS] rc V x 3
le; ds lkis{k vk;ru ds ifjorZu dh nj]
dV d 3
dx
( x ) 3x 2
dt dt
dt
¼J`a[kyk fu;e ls½
;g fn;k x;k gS fd ?ki dh dksj 3 lseh@ls dh nj ls c<+ jgh gSA
blfy,] tc x 10 lseh]
dx
3 lseh@ls
dt
dV
3x 2 (3) 9 x 2 lseh3@ls
dt
dV
9 (10) 2 900
dt
vr% tc dksj dh yEckbZ 10 lseh gS] rks ?ku dk vk;ru 900 lseh3@ls dh nj ls c<+ jgk gSA
26.
fuEufyf[kr fn, x, Qyuksa ds mPpre ;k fuEure eku] ;fn dksbZ gks rks] Kkr dhft,A
f ( x) | x 2 | 1
Ans. fn;k x;k Qyu f ( x) | x 2 | 1
ge tkurs gS fd izR;sd x R ds fy,] | x 2 | 0
blfy,] izR;sd x R ds fy,] f ( x) | x 2 | 1 1
f dk U;wure eku rHkh Kkr fd;k tk ldrk gS tc | x 2 | 0
vFkkZr | x 2 | 0 x 2
f dk U;wure eku f (2) | 2 2 | 1 0 1 1
blfy,] f (x) dk U;wure eku &1 gS ysfdu x 2 ij dksbZ mPpre eku ugha gSA
(37)
'ks [kkokVh fe'ku&
l= %
vadHkkj (1 + 1 + 1 + 2 + 3 + 4 = 12)
cgq fddYih; iz 'u %&
1.
log
10
x dx dk eku gS\
(1) x log x - x+c
2.
(2)
a
x
(1) a log a c
log x x dx
sin x
(1)
6.
o
(4)
3 32 1
x
x c
2
2
ax
c
(2)
log a
(3)
a x 1
c
x 1
(4) x a x 1 c
(2) 0
(3) x + c
(4) c
180
cos x o
(3)
180
cos x c
(4)
180
cos x o c
d
3
[f (x)] 4x 3 4 rFkk f (2) 0 rc f (x) gS\
dx
x
dx
1 129
x3
8
x x log x
1
tan
1
0
(1) 1
2
9.
1
2 32
x 2x 2 c
3
(2)
3
(2) x
(x
2
(1) 0
3
1 129
x4
8
4
(3) x
1 129
x3
8
(4) None of these
dk eku gS\
(1) 1 log x c
8.
2 32 1 2
x x c
3
2
dx cjkcj gS\
180
cos x o c
4
(1) x
7.
(3)
(2)
dk eku gS\
(1) 1
5.
(4) log10 e x log x x c
da dk eku gS\
x
4.
(3) log e 10 x log x x c
1
x
dk izfrvodyu gS\
x
1 13
(1) x 2 x c
3
3.
1
x
2x 1
2
1 x x
(2) log10 1 log x c
(3) log e 1 log x c
(4) log e log x c
(2) 0
(3) &1
(4)
dx gS\
x cos x tan 5 x 1)dx
4
gS\
(2) 2
(3)
4
(38)
(4)
'ks [kkokVh fe'ku&
1
10.
dx
1 x
l= %
dk eku gS\
2
0
(1) 0
11.
e
x
(2)
cos
(1)
13.
(3)
4
(4)
sec x(1 tan x)dx gS\
(1) e x cos x c
12.
2
2
(2) e x sin x c
(3) e x sec x c
(4) e x tan x c
1
(3) x sin 4x c
8
(4)
2x dx gS\
x 1
sin 4x c
2 8
cos ec 2
sec 2
x
2
(1) cot
x
2 dx
(2)
x 1
sin 4x c
2 4
x 1
sin 4x c
2 8
dk eku gS
x
xc
2
(2)
1
x
cot x c
2
2
(3) 2 cot
x
xc
2
(4) 2 cot
x
xc
2
b
14. f (a b x) f (x) rc
x f (x) dx
gS\
a
b
ab
f (a b c)dx
(1)
2 a
b
ab
f (x)dx
(2)
2 a
b
ab
f (x)dx
(3)
2 a
fjDr LFkku dh iwfrZ dhft,&
dx
1.
sin 2 2x cos2 2x
2.
1 sin
3.
4.
5
5.
sec x(sec x tan x)dx
1
3
5
dk eku -------------- gS
x cos 4 x dx dk eku ------------------ gSA
x 5 dx dk eku ---------------------------- gSA
log5 x
dx dk eku----------------------- gSA
dk eku --------------- gSA
(39)
b
ab
f (x)dx
(4)
4 a
'ks [kkokVh fe'ku&
6.
7.
x
1 x
2
l= %
1 x t
dx dk eku---------------- gSA
sin x
1 cos x
2
dx dk eku ---------------- gSA
;fn f ,d fo"ke Qyu gS rc
2 x
I 2
5
8.
1
f (x)dx ..................
dt
t
I 2 log t c
5
I 2log 1 x c
tan 1 x
6 1 x 2 dx ¾--------------------- gSA
I
2
10.
1
x dx
dk eku----------------------- gSA
1
&% mŸkjekyk %&
1-
1
tan 2 x cot 2 x C ;k log cos ec 4 x cot 4 x C
2
2-
0
3-
3
8
4-
567-
gksxkA
1
9.
dx dt
8
x3
2.
2
dx
x
2
I log 2
3.
e
x
dx
dk eku gS\
e x
gy%&
I
I
dx
e
x
ekuk ex t e x dx dt
dt
t 1
I
vfry?kq RkjkRed iz 'u %&
¾ tan 1 (t) c
1
e x
e x dx
e 2x 1
10- log 2
1.
c
I 2 log 2 0
1
c
1 cos x
9-
2
I 2 log x 1
C
2
32
2
dx dk eku Kkr djks\
x
Sol. I 1
1
log 1 x 2 c
2
0
2
log 1 x
I 2 log 2 log(1)
1
C
x
tan x sec x c
8-
1
2
dx
dk gy Kkr djks\
x x
Sol. I
dx
x x
I
dx
x 1 x
2
¾ tan 1 (ex ) c
4.
I
2cos x 3sin x
dx dk ?ku gSa\
6cos x 4sin x
gy%& I
2cos x 3sin x
dx
6cos x 4sin x
ekuk 3cos x 2sin x t
(40)
'ks [kkokVh fe'ku&
l= %
3sin x 2cos x dx dt
8.
dt
2t
I
x
2 dx
I
gy%&
2 x
2 cos
2
1
I log 3cos x 2sin x c
2
tan 2
I cos(sin 1 x)dx dk eku gS\
x
2
¾ 2 tan x c
I cos cos 1 1 x 2 dx
1 x
2
dx
x
dx
2
x
sec2 1 dx
2
gy%& ekuk sin 1 x
x sin rc cos 1 x 2
I
1 cos x
dx
1 cos x
2sin 2
1
I log(t) c
2
5.
I
4
9.
tan x dx
0
x
1
I
1 x 2 sin 1 x c
2
2
gy%& ¾ log sec x 04
6.
¾ log sec 4 log sec 0
x
x
I sin 2 cos 2 dx dk eku gS\
2
2
0
log
2 x
2 x
gy%& I cos 9 sin 2 dx
0
0
I sin x 0
y?kqmŸkjh; iz'u %&
1.
I sin sin o 0
7.
x
1 cos dx
4
gy%&
dx
2x x 2
Sol. I
I
x
2cos 2 dx
8
I
x
2 cos dx
8
I
x
2 8sin c
8
8 2 sin
2 log(1)
1
log 2
2
I cos x dx
dk eku Kkr djsa\
dx
2x x 2
dx
x 2x 1 1
2
dx
(x 1) 2 1
dx
1 (x 1) 2
I sin 1 (x 1) c
x
c
8
(41)
¼iw.kZ oxZ cukus ij½
'ks [kkokVh fe'ku&
2.
a
log a (sin 2 x)
l= %
dx dk eku Kkr djksa\
Sol. I a log a (sin
2
x)
dx
0
I sin x dx
2
¼lw= e
log e x
vr% I ¾ 1
x½
I ¾ tan 1 t
1
1
1
I ¾ tan 0 tan 1
1
1
I 1 dx cos 2x dx
2
2
I ¾ 0
4
x 1 sin 2x
I
c
2 2
2
I
x sin 2x
I
c
2
4
3.
e
1
e2x 1 dx
5.
dk eku Kkr djks\
ex ex e x dx
2sin x 2x cos c
ex e x dx
6.
ekuk e + e = t
x
I
-x
e x e x dx dt
vr% I
dt
t
I log e x e x c
sin x
2
0 1 cos 2
Sol. ekuk I ¾
x
ekuk cos x = t
- sin x dx = dt
x 1
2
x 1 1 ex dx
2
x 1
lq= & e x f (x) f 1 (x) dx e x f (x) c
vr%& I
dx dk eku gS\
sin x
2
0 1 cos 2
xe x dx
1
1
ex
dx
2
x 1 x 1
I log t c
4.
(cos x cos ) (cos x cos )
dx
cos x cos
2 cos x cos dx
e x e x
I¾
cos 2x cos 2
dx
cos x cos
2
e x e x e x
I¾
4
(2cos 2 x 1) (2cos 2 1)
dx
cos x cos
e 2x 1
e2x 1 dx
Sol. I ¾
dt
1 t2
0
1 cos 2x
I
dx
2
2x
rc t = 0
2
x=
1
x
dx
7.
ex
c
1 x
I x e x dx
2
0
gy%& ekuk x 2 t 2x dx dt
vc x = 0 ij t = 0 rFkk x = 1 ij t =
;fn x = 0 rc t = 1
1
I
(42)
1
et dt
2 0
'ks [kkokVh fe'ku&
l= %
0
1
I e t
1
2
1
vr% I ¾
1
e 1 c
2
4
8.
sin
2
1
A
B
(1 t) (2 t) (1 t) (2 t)
x dx
1 ¾ A(2 + t) + B(1 + t)
t = -1 ij A 1
gy%& sin 2 x ,d le Qyu gSA
t = -2 ij B 1
4
1
vr%& I 2 sin 2 x x
vr% I ¾
0
4
1
0
I ¾ (log 2 - log 3) - (log 1 - log 2)
I ¾ 2log 2 - log 3
1
2
1
1
sin 2x 4
I x
2 0
sin
2 0 0
I
4
2
(1 t) (2 t) dt
I ¾ log(1 t) log(2 t) 0
1 cos 2x
I 2
dx
2
0
4
0
vkaf'kd fHkUuksa esa fo;ksftr djus ij
4
¾ I
dt
(1 t) (2 t)
I ¾ log 4 - log 3
4
I log 3
2.
dx
9 8x x 2
Sol. I
dk eku Kkr djks\
dx
9 8x x 2
iw.kZ oxZ cukus ij
nh?kZ mŸkjh; iz ' u%&
2
1.
cos x
dx dk eku Kkr djks\
(1
sin
x)(2
sin
x)
0
I¾
I¾
I=
I¾
I
2
cos x dx
(1 sin x)(2 sin x)
0
Sol. ekuk I
ekuk sinx = t
vr% cos x dx = dt
;fn x = 0 rc t = 0
x
2
dx
x 2 8x 9
dx
2
[x 8x 9 16 16]
dx
(x 4)2 25
dx
(5) 2 (x 4) 2
1 x 4
vr% I sin 5 c
rc t = 1
(43)
'ks [kkokVh fe'ku&
3.
l= %
cos 2x dx
(cos x sin x)2
Sol. ekuk I ¾
I¾
I¾
1 1
t 2 2
I
log
C
1
1 1
t
2
2
2 2
dk eku Kkr djks\
1
cos 2x dx
(cos x sin x)2
t 1
I log
C
t
(cos2 x sin2 x)dx
(cos x sinx)2
(cos x sin x) (cos x sin x)
(cos x sin x) 2
ex 1
I log x C
e
dx
uksV%& bl loky dks vkaf'kd fHkUu }kjk Hkh gy dj ldrs
gSA
(cos x sin x)dx
I ¾ (cos x sin x)
ekuk cosx + sin x = t
5.
lekdyu
(-sin x + cos x) dx = dt
gy%& I
dt
vr% I ¾
t
I log(cos x sin x) c
e
dx
1 dk eku Kkr djksa\
I
1 x2
I 1 x 2 sin 1 x x c
3
x
6.
1
6
dt
t(t 1)
dx
cot x dk eku gksxkA
3
gy%& ekuk
dt
t t
2
3
dt
I
1 1
4 4
6
dt
2
dx dt rc x sin t
I t cos t sin t c
ekuk e t e dx dt
x
t2 t
dx
I t cos t 1 cos t dt
e x dx
I x x
e e 1
I
1 x2
dx dks gy dhft,&
vr% t sin t dt
dx
gy%& I ex 1
I
x sin 1 x
1
x
I
1 x2
ekuk sin 1 x t
I ¾ log t + c
4.
x sin 1 x
1 1
t
2 2
I
6
dx
1 cot x
sin x
sin x cot x
dx
&&&&& lehdj.k ¼1½
dx
&&&&& lehdj.k ¼2½
P3 xq.k/keZ yxkus ij
2
3
I
6
(44)
cos x
cos x sin x
'ks [kkokVh fe'ku&
l= %
lehdj.k ¼1½ $ lehdj.k ¼2½ ls &
2
I¾
2I (1)dx
4 log 1
0
6
2I x
3
6
I¾
2I
3 6
6
I¾
12
I¾
2I
I
I¾
1
4 log 1
1
0
| x 1| dx
Kkr dhft,&
I¾
0
4
gy%& | x 1| dx
0
4
0
1
I
I
4
1
1
4 5
2
2
2.
fl) dhft,
4
0
log 2 log(1 tan x) dx
Sol. ekuk I ¾
4 log(1
0
I¾
4 log(1 tan x)dx
0
log 2
8
x dx
0 a 2 cos2 x b2 sin 2 x dx
I¾
tan x)dx
log 2
8
dk eku Kkr djks\
x dx
dx &&&&&& (1)
a cos x b 2 sin 2 x
2
2
( x)dx
0 a 2 cos2 ( x) b2 sin 2 ( x)
I ¾ 0
tan x)dx &&&&&& ¼1½
I ¾ 0
xq.k/keZ P4 yxkus ij
4 log 1
0
4 log 2dx
0
xq.k/keZ P4 yxkus ij
4 log(1
0
2
dx
tan x
Sol. I 0
fuEcU/kkRed iz 'u %&
1.
4 log
1
0
2I ¾ log 2 0
4
x 2
x2
I x x
0 2
1
2
1 16
1
I 1 0 4 1
2
2 2
tan x 1 tan x
dx
1 tan x
I ¾ log 2 x 4 I lehdj.k (1) ls
0
vr%& f (x) (x 1)dx (x 1)dx
1
1
4 log
0
(x 1) x
| x 1|
(x 1) x
1
tan x
dx
tan x
4
7.
tan x
4
dx
1 tan tan x
4
tan
dx
x dx
2 2
2
a cos x b sin x
a cos x b2 sin 2 x
2
dx
a cos x b sin2 x
2
2I ¾ 0
tan x dx
4
2
2
dx
a cos x b sin2 x
2
2
2a
a
0
0
f (x)dx 2 f (x)dx
;fn F(2a - x) = F (x)
(45)
'ks [kkokVh fe'ku&
l= %
1
a cos x b 2 sin 2 x
vr% ;gk¡ F(x) ¾
2
I¾
2
a 2 cos 2 ( x) b 2 sin 2 ( x)
1
a cos b 2 sin 2 x
F( x)
2
2
0
2
0
I
0
I¾
0
dx
I
1 sin x
sin x
dx
1 sin x
2I ¾ 0
sec x dx
a b 2 tan 2 x
2I ¾
0
sin x(1 sin x)dx
(1 sin x)(1 sin x)
(sin x
1
I¾
b a
1 t
tan a
0
2I ¾ 0
a2 t2
1
tan tan 1 0
ab
I¾
0
ab 2
sin 2 x
cos 2 x
dx
2
2I ¾ sec x tan x tan x dx
0
I¾
sin 2 x)dx
1 sin 2 x
sin x
dt
2I ¾ 0
x rc t
2
0
2
2I ¾ sec x tan x sec x 1 dx
0
2I ¾ sec x tan x x 0
2I ¾ sec tan sec 0 tan 0 0
2I ¾ 1 0 1 0 0
2
2ab
Sol. ekuk I ¾
x sin x dx
dx
1 sin x 0 1 sin x
2
vr% I
b
dx
sin x
2
b sec2 x dx = dt
lekdyu
1 sin x
I ¾ 0
ekuk b tan x = t ;fn x = 0 rc t = 0
3.
( x)sin x
dx
sin x
(cos2x dk va'k o gj esa Hkkx yxkus ij½
I¾
1 sin( x)
I ¾ 0
dx
2
2
a cos x b 2 sin x
dx
2
2
a cos b 2 sin 2 x
I
yxkus ij
( x)sin( x)
I¾
2
¾ F(x)
vr% 2I ¾ 02
x sin x
P4 xq.k/keZ
1
F ( x) ¾
0 1 sin x dx &&&&& (1)
2I ¾ 2
0
2I ¾ 2 2
x tan x dx
dk eku Kkr djks\
sec x tan x
I
x tan x dx
0 sec x tan x
sin x
cos x
I ¾ 0 1
sin x
cos x cos x
x
2
2
2
4.
log sin x dx
dk eku Kkr djks\
0
2
gy%& ekuk I logsin x dx &&&&&&& lehdj.k ¼1½
0
(46)
'ks [kkokVh fe'ku&
l= %
R 1 xq.k/keZ yxkus ij
5.
2
x sin x
1 cos
2
0
I log cos x dx &&&&&&&& lehdj.k ¼2½
0
lehdj.k 1 $ lehdj.k 2 ls &
x sin x
dx
1
cos 2 x
0
x ds fu"dklu fu;e ls &
2I log sin x cos x dx
I
0
sin 2x
2I log
2
0
1
2
2
0
0
I
2I log sin 2x dx (log 2)dx
I tan 1 (1) tan 1 1
2
2
I
2 4 4
vc ¾ I1 log(sin 2x)dx
0
dt
2
I
x 0 ij t 0
x
ij t
2
1
I1 (log sin t)dt
20
2a
a
0
0
6.
2
2
2
4
I
I
f (x)dx 2 f (x)dx
I
;fn f (2a x) f (x)
1
2
dt
2 1 1 t2
1
I tan 1
1
2
2I I1 log 2
2
P6 xq.k/keZ
sin x
dx
2 0 1 cos 2 x
ekuk cos x t rFkk x 0 ij t 1
sin x dx dt
x ij t 1
dx
ekuk 2x t dx
dx dk eku Kkr djksa\
gy%& ekuk I
2
2
x
2
dx
sin 3 x sin(x )
dx
sin x [sin x cos cos x sin
3
dx
sin 4 x [cos cot x sin
cos ec 2 x dx
vr% I1 2 log sin t dt
I
pj t dks x esa fy[kus ij
ekuk %&
0
cos cot x sin
cos cot x sin 1
2
I1 logsin x dx I
sin cos ec2 x dx dt
0
vr% 2I I log 2
2
1
I log 2 log
2
2
2
(47)
I
1
dt
sin
t
I
1
2 t c
sin
'ks [kkokVh fe'ku&
I
l= %
2
cos cot x sin c
sin
2
cos x
I
cos
sin c
sin
sin x
I
7.
2 sin x cos cos x sin
c
sin
sin x
8.
x dx
1 sin x
dks gy dhft,&
0
x dx
1
sin x
0
gy%& ekuk I
x dk fu"dklu djus ij
1 sin x
2 0 1 sin 2 x
2 sin(x )
I
c
sin
sin x
I
(2sin 2)cos d
5 cos2 4sin dk eku Kkr djks\
1 sin x
I
dx
2 0 cos 2 x
gy%& ekuk I
(2sin 2)cos d
5 cos 2 4sin
I
sec2 x tan x sec x dx
20
I
tan x sec 0
2
cos d dt
I
I
(3t 2)
t 4t 4
tan x sec tan sec
2
I
I
(3t 2)
dt
(t 2) 2
(0 1) (0 1)
2
¾ I
I
(2sin 2)cos d
5 1 sin 2 4sin
ekuk sin t
2
vkaf'kd fHkUu esa fu;ksftr djus ij
3t 2
A
B
¾ (t 2) 2 (t 2) (t 2) 2
¾ (3t 2) A(t 2) B
t = 0 ij -2 = - 2 A + B
t = 2 ij 4 = B
vr% A = 3
dt
dt
vr % I 3 (t 2) 4 (t 2)2
I 3 log l(t 2)
4
c
(t 2)
I 3log(sin 2)
4
C
(sin 2)
(48)
'ks [kkokVh fe'ku&
l= %
vadHkkj (1 + 1 + 2 + 2 = 6)
cgq fodYih; iz' u
1.
oØ y = x2 rFkk js[kk y = 4 ds chp dk izFke prqFkkZa'k esa {ks=Qy gS\
(1)
2.
32
3
(2)
4.
(2) 3
(1) 2
(2)
nh?kZo`Ÿk
x 2 y2
2 dk x-v{k ds Åij dk {ks=Qy gS\
2
1
(2)
(2)
(3) 12
(4)
3
2
(3)
2
2
(4) 4
(3) 4
(4)
9
4
(3)
9
3
(4)
9
2
vUrjky [a, b] esa nks oØ y f (x) rFkk y g(x) gS rFkk f (x) g(x) rFkk x a, x b ds e/; bu oØks ls f/kjk {ks=Qy gS\
b
(1)
b
f (x) g(x) dx
(2)
a
7.
8
3
oØ y 2 4x, y v{k js[kk y 3 ls f/kjk {ks=Qy gS\
(1) 2
6.
(4)
o`r x 2 y2 2 dk x v{k ds Åij dk {ks=Qy gS\
(1) 2
5.
16
3
(3)
x y
1 rFkk funsZ'kh v{kksa ds chp dk {ks=Qy gS\
2 3
js[kk
(1) 6
3.
64
3
g(x) f (x) dx
b
(3)
a
f (x) g(x) dx
b
(4)
a
f (x)
g(x) dx
a
x = 0 ls x rd oØ y sin x ds chp {ks=Qy gS\
(1) 1
(2) 2
(3) 3
(4) 4
8.
(0, 4)
y=x
B
C
A
(4, 0)
2
2
x + y = 16
bl fp= esa Nka;kfdr Hkkx dk {ks=Qy gS\
4
(1)
2
4
4
16 x 2 dx
(2)
0
16 x 2 dx
(3)
2 2
(49)
4
16 x 2 dx
(4)
2
16 x 2 dx
'ks [kkokVh fe'ku&
l= %
9.
fuEu esa ls fdl Hkkx dk {ks=Qy vkafdd :i ls _.kkRed vk;sxkA
(1) I
(2) II
(3) III
(4) None
fjDr LFkku dh iwfrZ djsa&
1.
oØ y x 3 , x v{k ,ao dksfV;ks x 2, x 0 rd {ks=Qy -------------- gSA
2.
oØ y x 2 rFkk js[kk y 4 ds chp dk {ks=Qy gSA
3.
x 2 y2
2 dk x v{k ds Åij dk {ks=Qy------------- gSA
nh?kZo`Ÿk
3
2
&% mŸkjekyk %&
(1) 4 oxZ bdkbZ
(2)
32
oxZ bdkbZ
3
(3) 2 6 oxZ bdkbZ
vfry?kq R jkRed iz 'u
1. oØ y 2 x js[kkvksa x 1, x 4 vkSj x&v{k ds chp izFke prqFkkZa'k dk {ks=Qy Kkr djksaA
4
4
1
1
gy%& {ks=Qy ¾ ydx xdx
4
2
2 3
x
¾ 3
1
2 32 32
¾ 3 4 1
¾
2
14
7
oxZ bdkbZ
3
3
(50)
'ks [kkokVh fe'ku&
2.
l= %
ijoy; y2 4ax ,oa blds ukfHkyEc ds chp dk {ks=Qy
gS\
y
1
x
x
(a, 0)
3
y1
vfHk"B {ks=Qy ¾ 2x ydx
0
x=a
¿ ijoy; y 2 4x, y v{k ds lkis{k lefer gSAÀ
gy%&
3
¾ 2x 2 xdx
ijoy; dh ukfHk ¼a] 0½ gksrh gSA
0
a
3
2 32
4
x
¾
3 0
vfHk"B {ks=Qy ¾ 2 ydx
0
a
2 2 a xdx
¾
0
¾ 8 3 oxZ bdkbZ
a
2 32
¾ 4 a 3 x
0
¾
3.
8 a
8a
a a 0
3
3
8
[3 3 0]
3
y?kq rjkRed iz 'u
2
oxZ bdkbZ
y | x 3 | dk xzkQ [khph,%&
ijoy; y2 ¾ 16x rFkk js[kk x = 4 ls f?kjk {ks=Qy Kkr
djks\
Sol. fp= cukus ij
1.
Y
gy%& y | x 3 |
2
A
X'
y = 16x
c x=4
B
x=4
Y'
vHkh"B {ks=Qy ¾ OABO dk {ks=Qy
¾ 2 × OACO dk {ks=Qy
4
4.
¾ 2 y dx
oØ y2 4x ,ao js[kk x 3 ds chp dk {ks=Qy Kkr djks\
gy%&
0
(51)
X
'ks [kkokVh fe'ku&
l= %
4
¾ 2 4 x dx
0
4
2 3
¾ 8 x 2
3
0
16
¾ 3 4 2 0
3
16
128
8 ¾
¾
oxZ bdkbZ
3
3
2.
vfHk"B {ks=Qy ¾
x = 0 ls x 2 ds e/; oØ y = cos x ls f?kjk {ks=Qy
Sol.
1
1
(3x x)dx
2
3
2
1
3x 2
3 3x 2
2x
2x
¾ 2
1 2
2
Y
3
¾
O
(3x 2)dx
¿ izFke {ks=Qy x v{k ds uhps gS vr% ekaikd yxkuk gSAÀ
Kkr djks\
X'
2
3
1 25 13
oxZ bdkbZ
6 6
3
X
Y'
oØ y = cos x dk x = 0, ls x 2 ds chp {ks=Qy
Nk;kafdr gS &
vfHk"B {ks=
¾ 4 0 2 y dx
¾ 4 0 2 cos x dx
¾ 4 sin x 0 2
¾ 4 sin 2 sin 0
3.
¾ 4 oxZ bdkbZ
js[kk y 3x 2, x v{k ,ao dksfV;ksa x = - 1 ,oa x = 1 ds
chp {ks=Qy Kkr djksaA
(1) o
(2) o
(3) o
(4) o
gy%&
'ks[kkokVh fe'ku 100 dh d{kk 10 ,oa 12 ds
fofHkUu fo"k;ksa dh uohure cqdysV MkmuyksM
djus gsrq Vsyhxzke QR CODE LdSu djsa
(52)
'ks [kkokVh fe'ku&
l= %
vadHkkj (1 + 1 + 1 + 3 = 6)
3
1.
d 2 y dy 2
dy
vody lehdj.k 2 dx sin dx 1 0 dh ?kkr gS&
dx
(1) 3
(2) 2
(3) 1
(4) ifjHkkf"kr ugha gSA
(3) 0
(4) ifjHkkf"kr ugha gSA
Ans. 4
2.
vody lehdj.k 2 x 2
(1) 2
d2y
dy
3 y 0 dh dksfV gS&
2
dx
dx
(2) 1
Ans. 1
3.
pkj dksfV okys fdlh vody lehdj.k ds O;kid gy esa mifLFkr LosPN vpjksa dh la[;k gS&
(1) 0
(2) 2
(3) 3
(4) 4
Ans. 4
4.
rhu dksfV okys fdlh vody lehdj.k ds fof'k"V gy esa mifLFkr LosPN vpjksa dh la[;k gS&
(1) 3
(2) 2
(3) 1
(4) 0
Ans. 4
5.
fuEufyf[kr vody lehdj.kksa esa ls fdl lehdj.k dk O;kid gy y c1e x c2 e x gS&
d2y
y0
(1)
dx 2
d2y
y0
(2)
dx 2
d2y
1 0
(3)
dx 2
d2y
1 0
(4)
dx 2
Ans. 2
4
6.
2
d s
ds
3s 2 0 fuEufyf[kr lehdj.k dh ?kkr rFkk dksfV Kkr dhft,A
dt
dt
Ans. bl vody lehdj.k esa mifLFkr mPpre dksfV vodyu
d 2s
d 2s
gS
A
blfy,
bldh
dks
f
V
2
gS
A
rFkk
dh vf/kdre ?kkrkad
dt 2
dt 2
1 gS vr% bl lehdj.k dh ?kkr 1 gSA
2
7.
d2y
dy
2 cos 0 fuEufyf[kr lehdj.k dh ?kkr rFkk dksfV Kkr dhft,A
dx
dx
d2y
Ans.
vodyu dh dksfV 2 rFkk bl lehdj.k dk cka;k i{k vodyuksa esa cgqin ugha gS] blfy, bldh ?kkr ifjHkkf"kr ugha gSA
dx 2
(53)
'ks [kkokVh fe'ku&
8.
l= %
d2y
y 0 dk gy gSA
lR;kfir dhft, fd Qyu y a cos x b sin x ftlesa a, b R ] vody lehdj.k
dx 2
Ans. fn;k gqvk Qyu gS y a cos x b sin x -----1
lehdj.k 1 ds nksuksa i{kksa dks x, ds lkis{k mÙkjksrj vodyu djus ij ge ns[krs gS&
dy
a sin x b cos x
dx
d2y
a cos x b sin x
dx 2
d2y
,oa y dk eku fn, gq, vody lehdj.k esa izfrLFkkfir djus ij izkIr djrs gSA
dx 2
cka;k i{k a cos x b sin x a cos x b sin x 0 nk;ka i{k
9.
dy
x 1
vody lehdj.k dx 2 y , y 2 dk O;kid gy Kkr dhft,A
dy x 1
Ans. fn;k x;k gS fd dx 2 y , y 2 ---------------1
lehdj.k 1 ds pjksa dks i`Fkd djus ij ge izkIr djrs gS &
2 y dy x 1dx ---------------2
lehdj.k 2 ds nksuksa i{kksa dk lekdyu djus ij
(2 y)dy ( x 1)dx
vFkok 2 y
y2 x2
x c1
2
2
vFkok
x 2 y 2 2 x 4 y 2c1 0
x2 y 2 2x 4 y c 0
c 2c1 lehdj.k dk O;kid gy gSA
10.
y log ydx xdy 0 vody lehdj.k dk O;kid gy Kkr dhft,A
Ans. fn;k gS y log ydx xdy 0
y log ydx xdy
dx
dy
x y log y
dx
x
dy
y log y
;gka log y t j[kus ij&
(54)
'ks [kkokVh fe'ku&
l= %
1
dx
1
dy dt
.dt
y
x
t
log | x | log | t | c1
log | x | log | log y | c1
log | x | log | log y | c1
log
x
c1
log y
x
e c1 A ¼ekuk½
log y
x A log y
log y
1
x cx
A
y e cx ¼vfHk"V gy½
11.
dy
sin 1 x vody lehdj.k dk O;kid gy Kkr dhft,A
dx
Ans. fn;k gS
dy
sin 1 x ;k dy sin 1 dx
dx
dy sin
1
xdx
y (sin 1 x). | dx
sin 1 x dks igyk Qyu ekudj [k.M'k% lekdyu djus ij
1
y sin 1 x x
y x sin 1 x
1
2
1 x2
xdx
2x
1 x2
dx
vc 1 x 2 t j[kus ij
2 xdx dt
y x sin 1 x
1
2
dt
t
1
1
1 t 2
x sin 1 x
2 1 1
2
y x sin 1 x t c
y x sin 1 x 1 x 2 c
(55)
'ks [kkokVh fe'ku&
12.
l= %
xdy ydx x 2 y 2 dx le?kkrh; vody lehdj.k dk gy Kkr dhft,A
Ans. iz'ukuqlkj &
xdy ydx x 2 y 2 dx
xdy y x 2 y 2 dx
2
2
dy y x y
f ( x, y )
dx
x
;k
f ( x , y )
y 2 x 2 2 y 2
x
y x 2 y 2
º f ( x, y )
x
f ( x, y ),0 ?kkr dk le?krh; Qyu gSA
vc y vx j[kus ij
dy
dv
vx
dx
dx
v x
dv vx x 2 v 2 x 2
dx
x
x v 1 v 2
x
v 1 v2
x
dv
1 v 2 ;k
dx
1
1 v2
dv
dx
x
nksuksa i{kksa dk ledkyu djus ij &
dv
1 v
2
dx
x
log | v 1 v 2 | log | x | log C
v
y
j[kus ij
x
log
1 y2
y
log | x | log C
x
x2
;k log
y 1 y2
log | x | log C
x
(56)
'ks [kkokVh fe'ku&
l= %
y x2 y2
x2
C
y x 2 y 2 Cx 2
13.
x
dy
y
y x sin 0 le?kkrh; vody lehdj.k dk gy Kkr dhft,A
dx
x
Ans. iz'ukuqlkj x
dy
dx
dy
y
y x sin 0
dx
x
y
x f ( x, y )
y x sin
x
y x sin
f ( x , y )
x
y y x sin y
x
x
x
º f ( x, y )
f ( x, y ) le?kkrh; Qyu gS ftdh ?kkr 'kwU; gSA
y vx j[kus ij]
dy
dv
vx
dx
dx
;k v x dv
dx
vx x sin
x
vx
x v sin v
nksuksa i{kksa dk lekdyu djus ij &
2 log v
v v log v dv
dx
x
1 log v 1
v(1 log v)dv log | x | log C
log | v | log | log v 1 | log | x | log C
log | log v 1 | log v log x log x log | Cvx |
log v 1 Cvx
log v 1 Cvx
log
y
y
1 C. .x
x
x
y
y
, log 1 Cy
x
x
y
x
vfHk"V gy Cy log 1
(57)
'ks [kkokVh fe'ku&
14.
l= %
2 xy y 2 2 x 2
dy
0, y 2 ;fn x 1 le?kkrh; vody lehdj.k dk gy Kkr dhft,A
dx
2
2
Ans. vody lehdj.k 2 xy y 2 x
2x2
dy
0
dx
dy
2 xy y 2
dx
dy 2 xy y 2
--------------------------1
dx
2x2
vc y vx j[kus ij &
dy
xdv
v
dx
dx
vx
x
dv 2 x.vx v 2 x 2
dx
2x2
dv v 2
1 dx
v 2 dv
dx 2
2 x
nksuksa i{kksa dk lekdyu djus ij&
v
2
1 dx
dv .
2 x
v 2 1
1
log | x | C
2 1 2
1 1
log | x | C
v 2
iqu% v
y
j[kus ij &
x
x 1
log | x | C ------------------------- 2
y 2
vc x 1, y 2 j[kus ij
1 1
log1 C
2 2
C
C
1
2
dk eku lehdj.k 2 esa j[kus ij
x 1
1
log | x |
y 2
2
x 1
1 log | x |
y 2
2x
vr% vfHk"V gy y 1 log | x |
(58)
'ks [kkokVh fe'ku&
15.
vody lehdj.k tan 1 y x dy 1 y 2 dx dk gy Kkr dhft,A
Ans. fn;k gqvk vody lehdj.k fuEufyf[kr :i ls fy[kk tk ldrk gSA
dx
x
tan 1 y
----------------------1
dy 1 y 2 1 y 2
1
dx
tan 1 y
p
p
x
Q
Q
1
lehdj.k 1] dy 1
1 ] ds :i dk jSf[kd vody lehdj.k gSA ;gka
,oa 1 1 y 2
1 y2
1
I .F . e
1 y 2 dy
e tan
1
y
blfy, vody lehdj.k dk gy gS&
xe tan
1
y
tan 1 y tan 1 y
e
dy C ------------------2
2
1 y
tan 1 y tan 1 y
e
I
dy
2
1 y
1
tan 1 y t j[kus ij ge ikrs gSa fd 1 y 2 dy dt
vr% I te t dt, I te t 1.et et , I te t et (t 1)
I e tan
1
y
(tan 1 y 1)
lehdj.k 2 esa I dk eku izfrLFkkfir djus ij&
x.e tan y e tan y tan 1 y 1 C ikrs gSA
1
1
1
x tan 1 y 1 Ce tan y ¼vfHk"V gy½
16.
dy
3 y cot x sin 2 x ; y 2 ;fn x
rks vody lehdj.k dk fof'k"V gy Kkr dhft,A
dx
2
dy
3 y cot x sin 2 x
dx
Ans. vody lehdj.k
;g
dy
py Q ds :i dk vody lehdj.k gS&
dx
vr% p 3 cot x
I .F . e
p .dx
e
Q sin 2 x
3 cot xdx
3
e 3 log sin x e log(sin x )
1
cos ec 3 x
3
sin x
I .F . cos ec 3 x
vody lehdj.k dk gy&
y.I .F . Q.I .F .dx C
y. cos ec 3 x sin 2 x. cos ec 3 xdx C
2 sin x. cos x. cos ec 3 .dx C
(59)
l= %
'ks [kkokVh fe'ku&
l= %
2 cot x. cos ec x dx C
y. cos ec 3 x 2 cos ec x C
y 2 sin 2 x C sin x
x
2
rFkk y 2 j[kus ij C 4
vr% y 2 sin 2 x 4 sin 3 x
17.
fcUnq (0,0) ls xqtjus okys ,d ,sls oØ dk lehdj.k Kkr dhft, ftldk vody lehdj.k y ' eT sin x gSA
Ans. y ' eT sin x
dy
eT sin x
dx
dy eT sin xdx
dy
eT sin xdx
y eT sin xdx C
ekuk fd I e x sin xdx
e x dks igyk Qyu ekudj [k.M'k% lekdyu djus ij
I e x ( cos x) e x ( cos x)dx
e x cos x e x cos xdx
e
x
cos xdx dk [k.M'k% lekdyu djus ij&
I e x cos x e x sin x e x sin xdx
e x cos x e x sin x I
2 I e x cox e x sin x
1
I e x ( cos x sin x)
2
I dk eku lehdj. 1 esa j[kus ij
1
y e x ( cos x sin x) C
2
x 0, y 0 j[kus ij
1
o 1 C
2
C
1
2
1
2
blfy, lehdj.k dk vfHk"V gy gksxk& y e x (sin x cos x)
(60)
1
2
'ks [kkokVh fe'ku&
18.
l= %
fdlh cSad esa ewy/ku dh o`f) r% okf"kZd dh nj ls gksrh gSa ;fn 100 #i;s 10 o"kksZa esa nqxqus gks tkrs gSa rks r dk eku Kkr dhft,A
(loge2 = 0.6931)
Ans. ekuk fdlh le; t ij ewy/ku p gS ,oa nj r gSA
vr%
dP Pr
dt 100
dP Pr
dP rdt
dt 100
P 100
nksuksa i{kksa dk lekdyu djus ij
dP
r
P 100 dt
log P
r
t log C
100
log P log C
r
t
100
r
log
t
P
r
P
t e 100
C 100
C
t
P Ce 100
t
-----------------------1
tc t 0, P 100 rks lehdj.k ls
100 Ce º C 100
lehj.k 1 esa C dk eku
P
j[kus ij
r
t
100
100e
rFkk tc t 10, P 200 rks lehdj.k 1 ls
r
200 100e 100
t 10
r
2 e 10
r
log 2
10
r 10 log 2 10 0.6931
r 6.931 %
'ks[kkokVh fe'ku 100 dh d{kk 10 ,oa 12 ds
fofHkUu fo"k;ksa dh uohure cqdysV MkmuyksM
djus gsrq Vsyhxzke QR CODE LdSu djsa
(61)
'ks [kkokVh fe'ku&
1.
l= %
vadHkkj (1 + 1 + 1 + 1 + 1 + 2 = 7)
lfn'k a iˆ 2 ˆj 3kˆ ds fnd~ dkslkbu gS&
(1)
1 2 3
,
,
7 7 7
(2)
1
14
,
2
14
,
3
14
(3) 1 , 2 , 3
(4) buesa ls dksbZ ugha
Ans. 2
2.
;fn a 2iˆ ˆj 2kˆ rFkk b iˆ ˆj 2kˆ gks rks a b ds vuqfn'k ek=d lfn'k gksxk &
(1)
1 ˆ ˆ
(i j )
2
(2)
1 ˆ ˆ
(i k )
2
(3)
1 ˆ ˆ ˆ
(i j k )
2
(4) buesa ls dksbZ ugha
Ans. 2
3.
;fn nks lfn'kksa a vkSj b ds chp dk dks.k gS rks a.b 0 gksxk ;fn&
(1) 0
(2) 0
2
2
(3) 0
(4) 0 2
Ans. 2
4.
;fn A(1, 1,2), B (2, 3, 5) rFkk C(1, 5, 5) ABC ds 'kh"kZ gks rks {ks=Qy gksxk&
(1)
61 oxZ bdkbZ
(2)
1
61 oxZ bdkbZ
2
(3) 61 2 oxZ bdkbZ
(4) buesa ls dksbZ ugha
Ans. 2
5.
lfn'k a iˆ 2 ˆj kˆ dk lfn'k b 4iˆ 4 ˆj 7 kˆ ij iz{ksi gksxk &
(1) 19
(2) 19 9
(3)
19
9
(4)
19
9
Ans. 4
6.
lfn'k a iˆ ˆj 2kˆ ds fnd~ vuqikr yhf[k, vkSj bldh lgk;rk ls fnd~&dkslkbu Kkr dhft,A
Ans. lfn'k r aiˆ bˆj ckˆ rc a, b, c lfn'k r ds fnd~&vuqikr gksrs gS vr% a 1, b 1, c 2
;fn l , m, n fn, gq, lfn'k ds fnd~&dkslkbu gS rks
| r | 12 12 2 2 6
a
1
l
|r |
6
b
1
m
|r |
6
c
2
n
|r |
6
1
vr% fnd~&dkslkbu
6
,
2
gSA
6 6
1
,
(62)
'ks [kkokVh fe'ku&
7.
l= %
fn, gq, lfn'kksa a 2iˆ ˆj 2kˆ vkSj b iˆ ˆj kˆ ] ds fy, lfn'k a b ds vuqnf'k ek=d lfn'k Kkr dhft,A
Ans. lfn'k a 2iˆ ˆj 2 kˆ
vkSj b iˆ ˆj kˆ
a b 2iˆ ˆj 2kˆ ( iˆ ˆj kˆ )
( 2 1)iˆ ( 1 1) ˆj ( 2 1) kˆ
iˆ 0. ˆj kˆ
iˆ kˆ
vc | a b || iˆ kˆ | 12 2 2 2
1
(a b ) ds vuqfn'k ek=d lfn'k | a b | (a b )
iˆ kˆ
2
iˆ
2
kˆ
2
1 ˆ 1 ˆ
i
k gSA
2
2
8.
lfn'k a iˆ 3 ˆj 2kˆ dk lfn'k b iˆ 2 ˆj kˆ ij iz{ksi Kkr dhft,A
Ans. lfn'k a dk b ij iz{ksi
vr% ek=d lfn'k
1
(a.b )
|b |
6
1
( 2.1 3.2 2.1)
5
6
6
6 3
nks lfn'kksa a vkSj ds b ds ifjek.k Kkr dhft,A ;fn buds ifjek.k leku gS vkSj buds chp dk dks.k 60º gS rFkk budk vfn'k
9.
1
| b | 12 2 2 12 6
(10)
10
1
gSA
2
xq.kuQy
Ans. iz'ukuqlkj]
| a || b |
rFkk a.b
a
1
2
rFkk b ds chp dk dks.k 60 º
ekuk | a || b | k
(63)
'ks [kkokVh fe'ku&
l= %
1
a.b
vc cos 2 1 2
| a || b | k .k 2k
cos 60º
1
2k 2
1
1
2
2 2k
k2 1
k 1
ijUrq k | a || b |
k 1 | a || b |
vr% | a | 1
rFkk | b | 1
10.
lekUrj prqHkZqt dk {ks=Qy Kkr dhft, ftldh lyaXu Hkqtk,a a 3iˆ ˆj 4kˆ vkSj b iˆ ˆj kˆ }kjk nh xbZ gSA
Ans. fdlh lekUrj prqHkqZt dh layXu Hkqtk,a a vkSj b gS rks mldk {ks=Qy | a b | }kjk izkIr gksrk gSA
rks
iˆ
a b 3
ˆj
1
kˆ
2 5iˆ ˆj 4kˆ
1 1 1
a b 25 1 16 42
11. ;fn a iˆ 7 ˆj 7 kˆ vkSj b 3iˆ 2 ˆj 2kˆ rks | a b | Kkr dhft,A
Ans. a iˆ 7 ˆj 7 kˆ
b iˆ 7 ˆj 7 kˆ
rks
ˆj kˆ
iˆ
a b 1 7 1
3 2 2
( 14 14)iˆ ( 2 21) ˆj ( 2 21) kˆ
0iˆ 19 ˆj 19 kˆ
a b | 19iˆ 19 kˆ |
19 2 192
361 361
722
19 2
(64)
'ks [kkokVh fe'ku&
12.
l= %
eku yhft, a iˆ 5 ˆj kˆ ] b 3iˆ 2 ˆj 7 kˆ ] c 2iˆ ˆj 4kˆ rks ,d ,slk lfn'k d Kkr dhft, tks a vkSj b gS ij yEc
gks vkSj cd 15
Ans. iz'ukuqlkj
a iˆ 5 ˆj kˆ
b 3iˆ 2 ˆj 7 kˆ
c 2iˆ ˆj 4 kˆ
a rFkk b ij yEc lfn'k a b
rc
iˆ ˆj kˆ
a b 1 4 2
3 2 7
( 28 4)iˆ (7 6) ˆj ( 2 12) kˆ
32iˆ ˆj 14 kˆ
ekuk
d (a b )
d (32iˆ ˆj 14 kˆ )
ijUrq cd 15
( 2iˆ ˆj 4kˆ ) (32iˆ ˆj 14 kˆ ) 15
2 32 1 ( 1) 4 ( 14) 15
64 1 ( 56) 15
9 15
15 5
9 3
5
d (32iˆ ˆj 14kˆ)
3
160 ˆ 5 ˆ 70 ˆ
i j k
3
3
3
1
(160iˆ 5 ˆj 70kˆ)
3
13. lfn'kksa a 2iˆ 3 ˆj 4kˆ vkSj
Ans. ekuk a rFkk b dk ifj.kkeh
rc c a b
b iˆ 2 ˆj kˆ ds ifj.kkeh ds lekUrj ,d ,slk lfn'k Kkr dhft, ftldk ifjek.k 5 bdkbZ gSA
c gSA
2iˆ 3 ˆj kˆ (iˆ 2 ˆj kˆ)
3iˆ ˆj
(65)
'ks [kkokVh fe'ku&
c
cˆ
vr%
l= %
1
ds vuqfn'k ek=d lfn'k | c | (c ) cˆ
3iˆ ˆj
c
10
3 ˆ
1 ˆ
i
j
10
10
ds vuqnf'k og lfn'k ftldk ifjek.k 5 bdkbZ gks
3 ˆ
1
5
i
10
10
ˆj
15 ˆ
5 10 ˆ
i
j
10
10 10
3
5
10 iˆ
10 ˆj
2 10
14.
3
1
10 iˆ
10 ˆj
2
2
lfn'k (a b ) vkSj (a b ) esa ls izR;sd ds yEcor ek=d lfn'k Kkr dhft, tgka a iˆ ˆj kˆ ] b iˆ 2 ˆj 3kˆ gSA
Ans. iz'ukuqlkj]
a b 2iˆ 3 ˆj 4kˆ a b ˆj 2 kˆ
ˆj
iˆ
kˆ
iˆ
(a b ) (a b ) 1 1 1 2 1 3 2
1 1 1 2 1 3
ˆj
3
kˆ
4
0 1 2
( a b ) ( a b ) 2iˆ 4 ˆj 2kˆ c ¼ekuk½
| c | 2 2 4 2 2 2 4 16 4
24 2 6
vr% vfHk"V ek=d lfn'k
c
1 ˆ 2 ˆ 1 ˆ
i
j
k gSA
|c |
6
6
6
15.
n'kkZb;s fd ox, oy, oz v{kksa ds lkFk cjkcj >qds gq;s lfn'k dh fnd~&dkslkbu dksT;k,a
Ans. ekuk ,d lfn'k OP , ox , oy rFkk oz ds lkFk cjkcj dks.k cukrk gS rks
OP dh fnd~ dksT;k,sa cos , cos , cos
l
cos 2 cos 2 cos 2 1
cos 2
2
1
3
vr% OP dh fnd~&dkslkbu
1
3
,
1
3
,
1
3
m2 n2 1
gSA
(66)
1
3
,
1
3
,
1
3
gSA
'ks [kkokVh fe'ku&
16.
Ans.
l= %
iˆ. ˆj kˆ ˆj.iˆ kˆ kˆ.iˆ ˆj
iˆ. ˆj kˆ ˆj. iˆ kˆ kˆ. iˆ ˆj dk eku D;k gSA
iˆ. iˆ ˆj ˆj kˆ. kˆ
| iˆ |2 | ˆj |2 | kˆ |2
11 1
1
fcUnqvksa A(1, 2, 2) vkSj B(2, 3, 1) dks feykus okyk lfn'k AB D;k gksxkA
Ans. A(1, 2, 2) vkSj B(2, 3, 1)
17.
PQ x
AB 2 1iˆ 3 2 ˆj 1 2 kˆ
2
x1 iˆ y 2 y1 ˆj z 2 z1 kˆ
ˆ
AB iˆ ˆj k
18.
;fn nks lfn'k a rFkk b bl izdkj gS fd | a | 2, | b | 3 vkSj a.b 4 rks | a b | Kkr dhft,A
Ans. | a b |2 a b . a b
a.a a.b b .a b .b
| a |2 2a.b | b |2
(2) 2 2 4 (3) 2
489
5
| a b |2 5
| a b | 5
19.
;fn ,d f=Hkqt dh nks Hkqtk;sa lfn'k iˆ 2 ˆj 2kˆ rFkk 3iˆ 2 ˆj kˆ ls fu:fir gks] rks f=Hkqt dk {ks=Qy Kkr djksA
Ans. ekuk a iˆ 2 ˆj 2kˆ
iˆ
a b 1
ˆj
2
b 3iˆ 2 ˆj kˆ
kˆ
2 2 4iˆ 1 6 ˆj 2 6kˆ
3 2 1
6iˆ 5 ˆj 8kˆ
1
2
vr% f=Hkqt dk {ks=Qy | a b |
1
125
2
1
5 5 oxZ bdkbZ
2
1
36 25 64
2
(67)
'ks [kkokVh fe'ku&
l= %
20.
fjDr LFkkuksa dh iwfrZ dhft,&
(i)
fnd~dkslkbu ds lekuqikrh la[;k,sa lr , mr , br lfn'k r ds ---------- dgykrs gSA
Ans. fnd~vuqikr
(ii)
nks lfn'k a rFkk b leku lfn'k dgykrs gS ;fn muds ------------- gSA
Ans. ifj.kke ,oa fn'kk leku
(iii) lfn'k a iˆ ˆj 2kˆ ds fnd~dkslkbu ---------- gksx
a sA
Ans.
(iv)
1
6
,
1
6
,
2
6
;fn a rFkk b nks 'kqU;rj lfn'k gSA rc a.b 0 ;fn vkSj dsoy ;fn ----------- gSA
Ans. a rFkk b ijLij yEcor gSA
'ks[kkokVh fe'ku 100 dh d{kk 10 ,oa 12 ds
fofHkUu fo"k;ksa dh uohure cqdysV MkmuyksM
djus gsrq Vsyhxzke QR CODE LdSu djsa
(68)
'ks [kkokVh fe'ku&
l= %
vadHkkj (1 + 1 + 3 + 4 = 9)
1.
;fn nks js[kkvksa ds fndvuqikr
a1, b1 , c1
rFkk a2 , b2 , c2 gS rks os ijLij yEcor gksxh] ;fn&
(1) a1b2 a2 b1 c1c2 0
(2) a12 b12 c12 1
(3) a22 b22 c22 1
(4) a1a 2 b1b2 c1c2 0
Ans. 4
2.
;fn ,d js[kk x, y vkSj z v{k ds lkFk Øe'k% 90º ,135º ,45º dks.k cukrh gS rks bldh fnd~dkslkbu gksxh&
(1)
1
2
,
1
2
,0
(2) 0,
1
2
,
1
(3)
2
1 1
, ,0
2 2
1 1
(4) 0, ,
2 2
Ans. 2
3.
nks ijLij yEcor js[kkvksa ds fnd~vuqikr 1] 2] 3 rFkk 3] 2] gS rks dk eku gksxk&
(1)
7
3
(2) 5
(3)
5
3
(4)
7
3
Ans. 4
4.
x-v{k ds lekUrj rFkk ewy fcUnq ls tkus okyh js[kk dk lehdj.k gksxk&
(1) r ˆj
(2) r (iˆ ˆj kˆ )
(3) r iˆ
Ans. 3
5.
n'kkZb, fd cUnq (2, 3, 4), (-1, -2, 1), (5, 8, 7) laj[s k gS&
Ans. ekuk fn;s x;s fcUnq A(2, 3, 4), B(-1, -2,1) rFkk C(5, 8, 7) gS%
AB
1 22 2 32 1 4 2
32 52 32
3 25 9 43
BC
5 12 8 2 7 12
62 102 6 2
36 100 36 2 43
AC
2 52 3 32 4 7 2
32 52 32
9 25 9 43
AC AB 43 43 2 43 BC
vr% fcUnq A, B rFkk C lajs[k gSA
(69)
(4) r ( ˆj kˆ )
'ks [kkokVh fe'ku&
l= %
fcUnq 5,2,4 ls tkus okyh rFkk lfn'k 3iˆ 2iˆ 8kˆ ds lekarj js[kk dk lfn'k lehdj.k Kkr dhft,A
Ans. ges Kkr gS fd&
6.
a 5iˆ 2iˆ 4kˆ vkSj b 3iˆ 2iˆ 8kˆ
blfy,] js[kk dk lfn'k lehdj.k gS&
r 5iˆ 2iˆ 4kˆ 3iˆ 2 ˆj 8kˆ
7.
fn, x, js[kk ;qXe
r 3iˆ 2 ˆj 4 kˆ iˆ 2 ˆj 2 kˆ
vkSj
r 5 iˆ 2 ˆj 3iˆ 2 ˆj 6 kˆ
ds e/; dks.k Kkr dhft,A
Ans. eku yhft,
ekuk js[kkvksa ds chp dks.k gks rks
b1.b2
cos
| b1 || b2 |
cos
cos
iˆ 2 ˆj 2kˆ . 3iˆ 2 ˆj 6kˆ
1 4 4 9 4 36
3 4 12 19
3 7
21
19
21
cos 1
8.
fcUnq 1,2,3 ls xqtjus okyh js[kk dk lehdj.k Kkr dhft, tks lfn'k 3iˆ 2 ˆj 2kˆ ds lekUrj gSA
Ans. fLFkfr fcUnq Aa ls xqtjus okyh js[kk AP rFkk lfn'k b ds lekUrj js[kk dk lehdj.k
r a b
;gka ij a iˆ 2 ˆj 3kˆ
vkSj b 3iˆ 2 ˆj 2kˆ
vHkh"V js[kk AP dk lehdj.k
r iˆ 2 ˆj 3kˆ 3iˆ 2 ˆj 2kˆ
9.
fuEu js[kk ;qXe ds chp dks.k Kkr dhft,A
x y z
x 5 y 2 z 3
vkSj
2 2 1
4
2
8
Ans. js[kk
x y z
x 5 y 2 z 3
ds fnd~ vuqikr 2] 2] 1 gS rFkk js[kk
ds fnd~ vuqikr 4] 1] 8 gSA
2 2 1
4
1
8
a1 2, b1 2, c1 1
(70)
'ks [kkokVh fe'ku&
l= %
a 2 3, b2 c2 8
;fn nks js[kkvksa ds e/; dks.k gks] rks
a1a2 b1b2 c1c2
cos
a12
b12 c12 a22 b22 c22
2 4 2 1 1 8
2 2 2 2 12 4 2 12 8 2
8 28
4 4 1 16 1 64
18
9 81
18
2
3 9 3
2
cos 1
3
10.
P dk eku Kkr dhft, rkfd js[kk,a
1 x 7 y 14 z 3
3
2P
2
vkSj
7 7x y 5 6 z
ijLij yEc gksA
3P
1
5
Ans. nh gqbZ js[kkvksa dks ekud :i esa j[kus ij]
x 1 y z z 3
3 2P / 7
2
vkSj
x 1
y 5 z 6
3P / 7
1
5
js[kk
x 1
y2
z3
2P
ds fnd~ vuqikr 3, ,2 gS rFkk
3
2P / 7
5
7
js[kk
x 1 y 5 z 6 x 1
y 5 z 6
3P
ds fnd~ vuqikr ,1,5 gSA
3
1
5 3P / 7
1
5
7
js[kk,a yEcor gks rks &
a1a 2 b1b2 c1c2 0
3 3P 2 P 1 2 5 0
7
7
9P 2P
10
7
7
11P
10
7
P
70
11
(71)
'ks [kkokVh fe'ku&
11.
;fn js[kk,a
l= %
x 1 y 5 z 6
x 1 y 1 z 6
vkSj
ijLij yEc gSA rks k dk eku Kkr djksA
3
2k
2
3k
1
5
Ans. nh xbZ js[kk,a
x 1 y 2 z 3
x 1 y 1 z 6
esa fnd~ vuqikr 3,3k ,2 gS rFkk
ds fnd~ vuqikr 3k ,1,5 gSa
3
2k
2
3k
1
5
rc js[kk,a ijLij yEc gS rc
a1a2 b1b2 c1c2 0
3 3k 2k 1 2 5 0
9 k 2k 10 0
7 k 10 0
7 k 10
k
12.
10
7
js[kk,a ftudh lfn'k lehdj.k fuEufyf[kr gS] ds chp dh U;wure nwjh Kkr dhft,A
r 1 t iˆ t 2 ˆj 3 2t kˆ
r s 1iˆ 2 s 1 ˆj 2 s 1kˆ
Ans. iz'ukuqlkj igyh js[kk dk lehdj.k
r 1 t iˆ t 2 ˆj 3 2t kˆ
iˆ 2 ˆj 3kˆ t iˆ
r a1 b1 ls bldh
a1 iˆ 2 ˆj 3kˆ rFkk
ˆj 2 kˆ
rqyuk djus ij
b1 iˆ ˆj 2kˆ
iz'ukuqlkj nqljh js[kk
r ( s 1)iˆ ( 2 s 1) ˆj ( 2 s 1) kˆ ;k
r iˆ ˆj kˆ s iˆ 2 ˆj 2kˆ
r a2 b2 bldh rqyuk djus ij
a 2 iˆ ˆj kˆ ] b2 iˆ 2 ˆj 2kˆ
ge tkurs gS fd ,slk
r a1 b1 vkSj r2 a2 b2 ds chp
U;wure nwjh d
a2 a1 .b2 b1
b1 b2
-------------------1
;gka a2 a1 iˆ ˆj kˆ iˆ 2 ˆj 3kˆ ˆj 4kˆ
rFkk b1 b2 iˆ ˆj 2kˆ iˆ 2 ˆj 2kˆ
(72)
'ks [kkokVh fe'ku&
l= %
iˆ ˆj kˆ
1 1 2 (214)iˆ (2 2) ˆj (2 1) kˆ
2 2
1
2iˆ 4 ˆj 3kˆ
vr% | b1 b2 | 4 4 2 32 29
budk eku 1 esa j[kus ij
d
13.
ˆj 4kˆ. 2iˆ 4 ˆj 3kˆ 0 2 1 4 4 3
29
8
29
js[kkvksa
x 1 y 1 z 1
x 3 y 5 z 7
vkSj
ds chp dh U;wure nwjh Kkr dhft,A
7
6
1
1
2
1
x 1 y 1 z 1
7
6
1
Ans. nh xbZ js[kk,a
vkSj
29
x 3 y 5 z 7
1
2
1
;gka ij x1 1, y1 1, z1 1
rFkk x2 3, y2 5, z 2 7
vkSj a1 7, b1 6, c1 1
a 2 1, b2 2, c2 1
d
x2 x1
y2 y1
z 2 z1
a1
b1
c1
a2
b2
c2
b1c2 b2 c1
nks js[kkvksa ds chp dh U;wure nwjh
2
c1a2 c2 a1 a1b2 a2 b1
2
2
3 1 5 1 7 1
7
6
1
2
6 2
2
1
1
1 7 14 6
2
2
116
116 2 29
116
vr% vHkh"V U;wure nwjh 2 29 bdkbZ
uksV& bu js[kkvksa ds chp dh nwjh budks lfn'k :i esa cnydj iz'u 12 dh rjg Kkr dj ldrs gSA
(73)
'ks [kkokVh fe'ku&
14.
l= %
fcUnq (1, 2, -4) ls tkus okyh vkSj nksuksa js[kkvksa
x 8 y 19 z 10
x 15 y 29 z 5
vkSj
ij yEc js[kk dk lfn'k
3
16
7
3
8
5
lehdj.k Kkr dhft,A
Ans. ekuk vHkh"V js[kk dk lehdj.k r iˆ 2 ˆj 2kˆ b1iˆ b2 ˆj b3 kˆ
;g js[kk nksuksa nh xbZ js[kkvksa ds yEcor gS vr%
-----2
3b1 16b2 7b3 0
rFkk 3b1 8b2 5b3 0 -----3
lehdj.k 2 o 3 ls &
b3
b1
b2
80 56 15 12 24 48
b1 b2 b3
2 3
6
vr% b1 , b2 , b3 ds lekuqikrh eku lehdj.k 1 esa j[kus ij
js[kk r iˆ 2 ˆj 4kˆ 2iˆ 3 ˆj 6kˆ
15.
js[kk ;qXeksa ds chp dk dks.k Kkr dhft,&
x 2 y 1 z 3
x 2 y 4 z 5
rFkk
2
5
3
1
8
4
Ans. nh xbZ js[kkvksa ds fnd~vuqikr&
a1 2, b1 5, c1 3
a 2 1, b2 8, c2 4
nksuksa js[kkvksa ds chp dk dks.k gks rks &
cos
a1a2 b1b2 c1c2
a12
b12 c12 a22 b22 c22
2 1 5 8 3 4
4 25 9 1 64 16
2 40 12
38 81
26
9 38
26
9 38
cos 1
(74)
--------1
'ks [kkokVh fe'ku&
l= %
vad Hkkj (4)
1.
fuEufyf[kr O;ojks/kksa ds vUrxZr z 5x 3y dk vkys[kh; fof/k ls vf/kdrehdj.k dhft,
5x 2y 10, 3x 5y 15, x 0, y 0
gy %& fn, O;ojks/k
5x 2y 10 &&&&&&& O;ojks/k ¼1½
3x 5y 15 &&&&&&&O;ojks/k ¼2½
x 0, y 0 &&&&&&& O;ojks/k ¼3½
O;ojks/k ¼1½ ls lehdj.k 5x 2y 10 ;k
x y
1 rFkk ewy fcUnq ¼0] 0½ 0 10 larq"V djrk gSA
2 5
O;ojks/k ¼2½ ls lehdj.k 3x 5y 15 ;k
x y
1
5 3
rFkk ewy fcUnq ¼0] 0½] 0 15 larq"V djrk gSA
O;ojks/k ¼1½] ¼2½ o ¼3½ ds vkys[kksa ls ifjc) {ks= OABC lqlaxr {ks= gSA
dks.kh; fcUnqvksa ds laxr Z ds eku
A (2, 0) rc Z = 5(2) + 3(0) = 10
20 45
20
45 235
B , rc Z 5 3
&&& vf/kdre
19
19 19
19 19
C(0, 2) rc Z 5(0) 3(2) 6
20 45
2.
235
vr%& fcUnq 19 , 19 ij Z dk vf/kdre eku
izkIr gksrk gSA
19
fuEufyf[kr O;ojks/kksa ds vUrxZr Z = 3x + 5y dk vkys[kh; fof/k ls U;wuŸkehdj.k dhft,&
x 3y 3, x y 2, x, y 0
gy%& fn, O;ojks/k x 3y 3 &&& O;ojks/k ¼1½
x y 2 &&&& O;ojks/k ¼2½
x 0, y 0 &&&& O;ojks/k ¼3½
(75)
'ks [kkokVh fe'ku&
l= %
O;ojks/k ¼1½ ls lehdj.k x 3y 3 ;k
x y
1
3 1
O;ojks/k ¼1½ ls lehdj.k x 3y 60 ;k
rFkk ewy fcUnq ¼0] 0½] 0 3 larq"V ugha djrk gSA
O;ojks/k ¼2½ ls lehdj.k x y 2 ;k
O;ojks/k ¼2½ ls lehdj.k x y 10 ;k
x y
1 rFkk ewy
2 2
x y
1
10 10
rFkk ewy fcUnq ¼0] 0½ vlfedk x y 10, 0 10 dks
larq"V ugha djrk gSA
O;ojks/k ¼3½ ls lehdj.k x y rFkk fcUnq ¼1] 0½] vlfedk
x y 0, 1 0 dks larq"V ugha djrk gSA
vlfedk ¼O;ojks/k½ 1] 2] 3] o 4 ds vkys[kksa ls ifjc) {ks=
ABCD lqlaxr {ks= gSA
fcUnq (0,0) 0 2 larq"V ugha djrk gSA
O;ojks/k ¼1½] ¼2½ o ¼3½ ds vkys[kksa ls vifjc) {ks= vFkkZr~
lqlaxr {ks= ¼Nk;kafdr½ fn[kk;k x;k gSA
dks.kh; fcUnqvksa ds laxr Z ds eku
A (3, 0) rc Z = 3(3) + 5(0) = 9
dks.kh; fcUnqvksa ds laxr Z ds eku&
A(0, 10) ij Z = 3(0) + 9 (10) = 90
B (5, 5) ij Z = 3(5) +9(5) = 60 &&& U;wuRke
C (15, 15) ij Z = 3 (15) + 9 (15) = 180 &&vf/kdre
D (0, 20) ij Z = 3 (0) + 9 (20) = 180 &&& vf/kdre
vr%& lqlaxr {ks= fcUnq B (5, 5) ij Z dk U;wure eku 60
gSA rFkk vf/kdre eku 180 nks fcUnqvksa C (15, 15) o D (0,
20) gSA o CD ij fLFkr izR;sd fcUnq ij Hkh vf/kdre eku
180 gSA
3 1
3 1 14
B , rCk Z 3 5 7 &&&& U;wure
2 2
2 2 2
C(0, 2) rc Z 3(0) 5(2) 10
3 1
3.
x
y
1
60 20
vr%& fcUnq 2 , 2 ij Z dk U;wure eku 7 izkIr gksrk gSA
fuEufyf[kr O;ojks/kksa ds vUrxZr Z = 3x + 9y dk jSf[kd
izkx
s zkeu ds xzkQh; fof/k }kjk U;wurehdj.k o vf/kdrehdj.k
dhft,&
x 3y 60
x y 10
xy
x 0, y 0
gy %& fn, O;ojks/k
x 3y 60 &&&&& O;ojks/k ¼1½
x y 10 &&&&& O;ojks/k ¼2½
x y ;k x y 0 &&&& O;ojks/k ¼3½
x 0, y 0 &&&& O;ojks/k ¼4½
(76)
'ks [kkokVh fe'ku&
l= %
vadHkkj (1 + 1 + 2 + 3 = 7)
1.
1
2
;fn P( A) , P( B) 0 rc P( A | B) gS&
(1) 0
2.
5.
(3) A B
(2) A B
(4) P ( A) P( B )
nks iklksa dk ,d tksM+k mNkyk tkrk gS rks izR;sd ikls ij ,d le vHkkT; la[;k izkIr gksus dh izkf;drk D;k gksxh\
(1) 0
4.
(4) 1
;fn A vkSj B nks ?kVuk,a bl izdkj gS fd P( A | B) P( B | A) 0 rc
(1) A B
3.
(3) ifjHkkf"kr ugha
(2) ½
(2)
1
3
(3)
1
12
(4)
1
36
nks ?kVukvksa A vkSj B dks ijLij Lora= dgrs gS] ;fn
(1) A vkSj B ijLij viothZ gS
(2) P(A'B') = [1-P(A)][1-P(B)]
(3) P(A) = P(B)
(4) P(A) + P(B) = 1
A }kjk lR; cksyus dh izkf;drk
4
gSA ,d flDdk mNkyk tkrk gS rFkk A crkrk gS fd fpr iznf'kZr gqvkA okLrfod :i esa fpr
5
izdV gksus dh izkf;drk gS&
(1)
6.
4
5
1
5
(4)
2
5
(2) P ( A | B) P( A)
(3) P ( A | B) P( A)
(4) buesa ls dksbZ ugha
(2) B A
(3) B
(4) A
(3) P ( B | A) P ( B )
(4) P ( B | A) P ( B )
(3) P(B|A)=0
(4) P(A|B)=0
(3) 0.5
(4) 0
;fn P( A | B) P( A) ] rc fuEu esa ls dkSu lgh gSA
(1) P ( A | B) P( B)
9.
(3)
;fn A vkSj B nks ,slh ?kVuk,a gS fd P( A) 0 vkSj P( B | A) 1 gS] rks
(1) A B
8.
1
2
;fn A vkSj B ,slh ?kVuk,a gS fd A B rFkk P( B) 0 rks fuEu esa ls dkSu Bhd gS&
P ( B)
(1) P( A | B) P( A)
7.
(2)
(2) P ( A B) P( A).P( B )
;fn A vkSj B ,slh nks ?kVuk,a gS fd&
P( A) P ( B ) P( A B) P( A) ] rc
(1) P(B|A) = 1
10.
(2) P(A|B)=1
;fn P(A) = 0.5 gS rks P(A') dk eku gksxk\
(1) 0.2
(2) 0.8
-: Answer :1-3, 2-4, 3-4, 4-2, 5-1, 6-3, 7-1, 8-3, 9-2, 10-3
(77)
'ks [kkokVh fe'ku&
l= %
Hkkx&c
11.
,d ifjokj esa nks cPps gSA ;fn ;g Kkr gks fd cPpksa esa ls de ls de ,d cPpk yM+dk gS] rks nksuksa ds yM+dk gksus dh D;k izkf;drk
gS\
Ans. ekuk fd b yM+dk ,oa g yM+dh dks fu:fir djrs gSA rks ijh{k.k dk izfrn'kZ lHkf"V gS&
S = {(b, b), (b, g), (g, b), (g,g)}
ekuk fd E ,oa F Øe'k% fuEu ?kVukvksa dks n'kkZrs gS&
E : nksuksa cPPks yM+ds gSA
F : cPpksa esa ls de ls de ,d yM+dk gSA
rc E = {(b,b)}, F = {(b, b), (g, b), (b, g)}
vc E F b, b
vr% P( F )
3
1
,oa PE F
4
4
1
P E F 4 1
P E | F
blfy,
3 3
P F
4
12.
,d ikls dks nks ckj mNkyk x;k vkSj izdV gqbZ la[;kvksa dk ;ksx 6 ik;k x;kA la[;k 4 ds U;wure ,d ckj izdV gksus dh lgizfrca/k
izkf;drk Kkr dhft,A
Ans. ekuk fd E ?kVuk la[;k 4 dk U;wure ,d ckj izdV gksuk rFkk F ?kVuk nksuksa iklksa ij izdV la[;kvksa dk ;ksx 6 gksus dks n'kkZrk
gSA
rc E 4,1, 4,2, 4,3, 4,4, 4,5, 4,6 , 1,4 , 2,4, 3,4, 5,4 , 6,4
,oa F 1,5, 2,4, 3,3, 4,2 , 5,1
ge tkurs gS fd P( E )
rFkk PE F
11
5
,oa P( F )
36
36
2
36
2
P E F 36 2
vr% okafNr izkf;drk P E | F PF 5 5
36
13.
;fn fn;k x;k gS fd nksuksa iklksa dks QSd
a us ij izkIr la[;k,a fHkUu&fHkUu gSAa nksuksa la[;kvksa dk ;ksx 4 gksus dh izkf;drk Kkr dhft,A
Ans. nks iklksa dks mNkyus ij izfrn'kZ lef"V S = 6×6 = 36
ekuk fd ?kVuk A = nks la[;kvksa dk ;ksx 4 gSA
(78)
'ks [kkokVh fe'ku&
l= %
A = {(1, 3), (2, 2), (3,1)}
vr% n(A) = 3
nks iklksa dh mNky esa leku la[;k okys ifj.kke
= {(1, 1), (2, 2), (3, 3), (4, 4), (5,5), (6,6)}
?kVuk B = iklksa ij mNky ij fHkUu&fHkUu vad izkIr gksxk
= 36 - 6 = 30
n|B| = 30
A B 1,3, 3,1
rc n A B 2 ] P A B
P ( A)
2
36
3
30
, P( B)
36
36
2
P A B 36 2
1
P A | B
vr%
30 30 15
P B
36
14.
,d vufHkur ikls dks nks ckj mNkyk x;kA eku ysa A ?kVuk igyh mNky ij fo"k; vad izkIr gksuk vkSj B f}rh; mNky ij fo"k;
vad izkIr gksuk n'kkZrs gSA ?kVukvksa A vkSj B ds Lokra=; dk ijh{k.k dhft,A
Ans. ;fn lHkh 36 ekSfyd ?kVukvksa dks lelHkkO; eku ys rks
P( A)
18 1
18 1
,oa P( B)
36 2
36 2
lkFk gh P( A B) P ¼nksuksa mNkyksa eas fo"k; vad izkIr gksuk½
9 1
36 4
1 1
2 2
vc P( A).P( B) .
1
4
Li"Vr;k P A B P A.P B
vr% A vkSj B Lora= ?kVuk,a gSA
15.
fl) dhft, fd ;fn E vkSj F nks Lora= ?kVuk,a gS rks E vkSj F' Hkh Lora= gksxhA
Ans. D;ksafd E ,oa F Lora= ?kVuk,a gS blfy,
P E F P E .P F .......................1
mijksDr osu vkjs[k ls Li"V gS fd E F rFkk E F ' ijLij viothZ ?kVuk,sa gSA
(79)
'ks [kkokVh fe'ku&
l= %
lkFk gh E E F E F '
vr% P ( E ) PE F P E F '
P E F ' P ( E ) P ( E ).P ( F ) ¼lehdj.k 1 ls½
P E F ' P ( E )[1 P ( F ' )]
P E F ' P ( E ).P ( F ' )
vr% E vkSj F ' Lora= ?kVuk,a gSA
Hkkx&l
16.
,d flDds dks mNkyus ds ijh{k.k ij fopkj dhft,A ;fn flDds ij fpr izdV rks flDds dks iqu% mNkysa ijUrq ;fn flDds ij
iV izdV gks rks ,d ikls dks QsadsaA ;fn ?kVuk de ls de ,d iV izdV gksuk dk ?kfVr gksuk fn;k x;k gS rks ?kVuk ikls ij
4 cM+h la[;k izdV gksuk dh lizfrca/k izkf;drk Kkr dhft,A
Ans. ijh{k.k dk izfrn'kZ lef"V gS&
S = {(H, H), (H, T), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6)}
tgka (H, H) n'kkZrk gS fd nksuksa mNkyksa ij fpr izdV gqvk vkSj (T, i) n'kkZrk gS fd igyh mNky ij ,d iV vkSj ikls dks Qsd
a us
ij la[;k i izdV gqbZA
vr% 8 ekSf[kd ?kVukvksa (H, H), (H, T), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6) dh Øe'k%
1 1 1 1 1 1 1 1
, , , , , , ,
4 4 12 12 12 12 12 12
izkf;drk fu/kkZfjr dh tk ldrh gSA
eku ysa F ?kVuk U;wure ,d iV izdV gksuk vkSj E ?kVuk ikls ij 4 ls cM+h la[;k izdV gksuk dks n'kkZrs gSaA
rc E H , T , T ,1, T ,2 , T ,3T ,4, T ,5, T ,6
(80)
'ks [kkokVh fe'ku&
l= %
E T ,5, (T ,6) vkSj E F T ,5 , T ,6
vc P ( F ) PH , T P T ,1 PT ,2 PT ,3 PT ,4 P T ,5 PT ,6
1 1 1 1 1 1 1 3
4 12 12 12 12 12 12 4
vkSj PE F PT ,5 PT ,6
1
1 1
12 12 6
1
P E F 6 2
vr% P E | F PF 3 9
4
17.
52 iÙkksa dh vPNh rjg QsaV xbZ xM~Mh esa ls ,d ds ckn ,d rhu irs fcuk izfrLFkkfir fd, fudkys x,A igys nks iÙkksa dk ckn'kkg
vkSj rhljs dk bDdk gksus dh D;k izkf;drk gS\
Ans. eku ysa fd K ?kVuk fudkyk x;k irk ckn'kkg gS dks vkSj A ?kVuk fudkyk x;k irk bDdk gS dks O;Dr djrs gSaA Lo"Vr;k gesa
P(KKA) Kkr djuk gSA
vc P( K )
4
52
lkFk gh P( K | K ) ;g Kkr gksus ij fd igys fudkyk x;k irk ckn'kkg gS ij nwljs irs dk ckn'kkg gksus dh izkf;drk dks n'kkZrk
gSA vc xM~Mh esa (52 1) 51 iÙks gS ftuesa rhu ckn'kkg gS
blfy,
P( K | K )
3
51
varr% P A | KK rhljs fudkys x, irs dk bDdk gksus dh lizfrca/k izkf;drk gS tc fd gesa Kkr gS fd nks ckn'kkg igys gh fudkys
tk pqds gSaA vc xM~Mh esa 50 iÙks jg x, gSa
blfy, P A | KK P A | KK
4
50
izkf;drk ds xq.ku fu;e }kjk gesa izkIr gksrk gS fd
P ( KKA) P ( K ) P ( K | K ) P ( A | KK )
4 3 4
2
52 51 50 5525
(81)
'ks [kkokVh fe'ku&
18.
l= %
rhu vfHkUu fMCcs I, II vkSj III fn, x, gSa tgka izR;sd esa nks flDds gSaA fMCcs I esa nksuksa flDds lksus ds gS] fMCcs II esa nksuksa flDds
pkanh ds gSa vkSj fMCcs III esa ,d lksus vkSj ,d pkanh dk flDdk gSA ,d O;fDr ;kn`PN;k ,d fMCck pqurk gS vkSj mlesa ls ;kn`PN;k
,d flDdk fudkyrk gSA ;fn flDdk lksus dk gS] rks bl ckr dh D;k izkf;drk gS fd fMCcs esa nwljk flDdk Hkh lksus dk gh gS\
Ans. eku ys E1 , E2 vkSj E3 Øe'k% fMCcs I, II vkSj III ds p;u dks fu:fir djrs gS
rc PE1 PE2 PE3
1
3
lkFk gh eku ysa A ?kVuk *fudkyk x;k flDdk lksus dk gS* dks n'kkZrk gSA
rc P(A|E1) = P ¼fMCcs I ls lksus dk flDdk fudyuk½ =
2
=1
2
P(A|E1) = P ¼fMCcs II ls lksus dk ,d flDdk fudyuk½ = 0
P(A|E3) = P ¼fMCcs III ls lksus dk flDdk fudyuk½ =
1
2
vc fMCcs esa nwljk flDdk Hkh lksus dk gksus dh izkf;drk
= fudkyk x;k lksus dk flDdk fMCcs I ls gksus dh izkf;drk
= P(E1|A)
vc cst&izes; }kjk
P ( E1 | A)
PE1 P A | E1
P E1 P ( A | E1 ) P( E2 ) P A | E2 P E3 P A | E3
1
1
2
3
1
1
1 1 3
1 0
3
3
3 2
19.
,d O;fDr ds ckjs esa Kkr gS fd og 4 esa ls 3 ckj lR; cksyrk gSA og ,d ikls dks mNkyrk gS vkSj crykrk gS fd ml ij vkus
okyh la[;k 6 gSA bl dh izkf;drk Kkr dhft, fd ikls ij vkus okyh la[;k okLro esa 6 gSA
Ans. eku yhft, fd E O;fDr }kjk ikls dks mNky dj ;g crkus dh fd ml ij vkus okyh la[;k 6 gS dh ?kVuk gSA eku yhft, fd
S1 ikls ij la[;k 6 ugha vkus dh ?kVuk gSaA rc
P(S1) = la[;k 6 vkus dh ?kVuk dh izkf;drk
1
6
P(E|S1) = O;fDr }kjk ;g crkus ij fd ikls fd la[;k 6 vkbZ gS tcfd ikls ij vkus okyh la[;k okLro esa 6 gS] dh izkf;drk
= O;fDr }kjk lR; cksyus dh izkf;drk
3
4
P(E|S2) = O;fDr }kjk ;g crkus ij fd ikls ij la[;k 6 vkbZ gS tcfd ikls ij vkus okyh la[;k okLro esa 6 ugha gS] dh izkf;drk
= O;fDr }kjk lR; cksyus dh izkf;drk 1
3 1
4 4
(82)
'ks [kkokVh fe'ku&
l= %
vc cst izes; }kjk
P(S1|E) = O;fDr }kjk ;g crkus dh izkf;drk fd la[;k 6 izdV gqbZ gS] tc okLro esa la[;k 6 gSA
1 3
P S1 PE | S1
1 25 3
6 4
P S1 PE | S1 P S 2 PE | S 2 1 3 5 1 8 8 8
6 4 6 4
vr% vHkh"V izkf;drk
20.
3
gSA
8
A vkSj B ckjh&ckjh ls ,d ikls dks mNkyrs gS tc rd fd muesa ls dksbZ ,d ikls ij N% izkIr dj [ksy dks thr ugha ysrkA ;fn
A [ksy dks 'kq: rks muds thrus dh Øe'k% izk;fdrk Kkr dhft,A
Ans. eku yhft, S lQyrk ¼ikls ij 6 izdV gksuk½ dks vkSj F vlQyrk ¼ikls ij 6 izdV u gksuk½ dks O;Dr djrs gSaA
1
6
vr% P( S ) , P( F )
5
6
P (A ds igyh mNky esa thruk½ = P(S) =
1
6
A dks rhljh mNky dk volj rc feyrk gS tc A igyh mNky esa vkSj B nwljh mNky eas vlQy gksrs gSA blfy,
2
5 5 1 5 1
P(A dk rhljh mNky esa thruk½ = P(FFS) = P(F)P(F)P(S)=
6 6 6 6 6
4
5 1
blh izdkj P(A dk ikapoh mNky esa thruk½ = P(FFFFS) =
6 6
2
4
1 5 1 5 1
vkSj blh izdkj vU; vr% P(A thruk½ ....
6 6 6 6 6
1
6
6
1 25 11
36
P(B thruk½ = 1 - P (A thruk½ = 1
21.
6 5
11 11
,d fo'ks"k leL;k dks A vkSj B }kjk Lora= :i ls gy djus dh izkf;drk,a Øe'k%
gy djus dk iz;kl djrs gSa] rks izkf;drk Kkr dhft, fd &
(i) leL;k gy gks tkrh gSA
(ii) muesa ls rF;r% dksbZ ,d leL;k gy dj ysrk gSA
(83)
1
1
vkSj gSaA ;fn nksuksa Lora= :i ls] leL;k
2
3
'ks [kkokVh fe'ku&
l= %
Ans. A vkSj B }kjk leL;k gy djus dh izkf;drk Øe'k%
;k
2
gSA
3
(i) leL;k gy u gksus dh izkf;drk
1
1
1
1
1
vkSj
vkSj u gy djus dh izkf;drk Øe'k% 1 ;k vkSj 1
2
3
2
2
3
1 2 1
2 3 3
1
3
nksuksa dh leL;k gy gksus dh izkf;drk 1
2
3
(ii) ;fn leL;k ds gy gksus dks S vkSj u gy gksus dks F fu:fir djsa rks rF;r% ml leL;k dks gy SF + FS <ax ls gy fd;k
tk,xkA
bldh izkf;drk 2 1 3 1 2 3
1
22.
1
1
1
1 2 1 1 1 2 1 1
2 3 2 3 2 3 3 2
FkSys 1 esa 3 yky rFkk 4 dkyh xsans gSa rFkk FkSyk 2 esa 4 yky vkSj 5 dkyh xsna s gSaA ,d xsna dks FkSyk 1 ls FkSyk 2 esa LFkkukUrfjr
fd;k tkrk gS vkSj rc ,d xsan FkSys 2 ls fudkyh tkrh gSA fudkyh xbZ xsan yky jax dh gSA LFkkukUrfjr xsan dh dkyh gksus dh
izkf;drk Kkr dhft,A
Ans. FkSys 1 esa 3 yky vkSj 4 dkyh xsans gSA
rFkk FkSys 2 esa 4 yky vkSj 5 dkyh xsna sa gSA
eku yhft, ?kVuk E1 rFkk E2 FkSys 1 ls yky xsan vkSj dkyh xsan fudkyus dh gksa] rc
3
4
P E1 , P E2
7
7
?kVuk A : yky jax dh xsan fudkyuk
,d yky xsan FkSys 1 ls fudky dj 2 esa j[k nh xbZA bl izdkj FkSys 2 esa 5 yky vkSj 5 dkyh xsna sa gks xbZA
P A / E1
5
10
,d dkyh xsan FkSys 1 ls fudkydj FkSyk 2 esa j[k nhA bl izdkj nwljs FkSys esa 4 yky vkSj 6 dkyh xsans gSaA
P A / E2
4
10
cst izes; ls]
PE1 / A
PE2 P A / E2
PE1 P A / E1 PE2 P A / E2
4 4
16
7
10
3 5 4 4 15 16
7 10 7 10
16
31
(84)
'ks [kkokVh fe'ku&
l= %
1.
cgqfodYih; iz'u &
(i)
;fn dksbZ laca/k R, leqPp; N ij bl izdkj ifjHkkf"kr gS fd R = {a, b} : a - b + 2=0, b > 6} gks fuEu esa ls lR; ugha gS&
(1) (5, 7) R
(ii)
(2) (6, 8) R
(2)
cos 2
sin 2
sin 2
cos 2
1
3
(4)
2
3
dk eku Kkr djks ;fn
(3) AA1 A 1 A 0
(4) buesa ls dksbZ ugha
(3) –1
(4)
4
(2) 1
1
2
d2y
;fn y log x gks rks
dk eku gS&
dx 2
(1)
(vi)
(3)
(2) AA1 A 1 A I
(1) 0
(v)
1
3
vkO;wg A rFkk A–1 ,d nwljs ds izfrykse gS rks
(1) AA 1 A 1 A
(iv)
(4) (4, 6) R
1
sin cos 1 dk eku gS&
3
2
(1) 0
(iii)
(3) (7, 9) R
1
(2)
x
1
2x
(3)
1
2x2
(4)
1
2x2
|sin 2x| + 4 dk U;wure eku gS&
(1) 5
(2) 6
(3) 4
(2) 2e 2 x C
(3)
(4) –1
(vii) e2x dk ledkyu gS&
(1) e 2 x C
(viii) nh?kZo`r
(4)
e2x
C
2
x2 y2
1 dk x v{k ds Åij {ks=Qy gS&
9
4
(1) 36
(ix)
2
C
e2x
(2) 6
(3) 3
(4) 6
(3) 3
(4) 4
oØ y = sin x dk x = 0 ls x = rd dk {ks=Qy gS&
(1) 1
(2) 2
(85)
'ks [kkokVh fe'ku&
l= %
2
(x)
d3y
dy
vody lehdj.k dx3 cos dx 0 dh ?kkr gS&
(1) 1
(xi)
(3) 3
(4) vifjHkkf"kr
(2) 1
(3) –1
(4) 2
(2) 0, 1, 0
(3) 1, 0, 0
(4) 1, 1, 1
(2) 2
iˆ. iˆ kˆ ˆj. iˆ kˆ kˆ. iˆ ˆj dk eku gS&
(1) 0
(xii) z v{k dh fnd~ dkslkbu gS&
(1) 0, 0, 1
(xiii) js[kk r 3iˆ 2 ˆj 4kˆ iˆ 2 ˆj 2 kˆ rFkk js[kk r 2iˆ ˆj kˆ 2iˆ ˆj mkˆ ijLij yEcor gks rks m dk eku gS&
(1) 2
(2) –2
(3) 4
(4) –4
(xiv) nks ?kVuk, A vkSj B Lora= gks rks lR; gS&
(1) P A B P ( A) P ( B )
(2) P A B P ( A) P ( B )
(3) P A B P ( A) P ( B )
(4) P A B P ( A) P ( B )
( A)
(xv) ;fn P ( A) 0.80, P( B ) 0.50 rks P ( B ) Kkr djksA
(1)
5
8
(2)
8
5
(3) 1
(4) Kkr ugha dj ldrs
2.
fjDr LFkkuksa dh iwfrZ djks&
(i)
3
sin 1 sin dk eku -------------------------- gSA
5
(ii)
tan 1
(iii)
1
tan 1 cot 1 3
gks rks x dk eku --------------- gSA
2
x
(iv)
y sin cos x
(v)
o`r dh ifjf/k esa ifjorZu dh nj f=T;k ds lkis{k ------------------------ gksxk ;fn f=T;k 5 lseh gSA
(vi)
vody lehdj.k
(vii)
a 2iˆ ˆj kˆ ds vuqfn'k ek=d lfn'k ----------------------- gSA
3.
vfry?kq rjkRed iz 'u&
(i)
;fn x 1 4 1 gks rks x dk eku Kkr djksA
3 sec
1
2 dk eku -------------------- gksxkA
3 x
2
gks rks
dy
dk eku ----------------------- gSA
dx
dy
e x y dk O;kid gy --------------------- gSA
dx
3 2
(86)
'ks [kkokVh fe'ku&
l= %
(ii)
a11
a21
a12
a22
a13
a23
a31
a32
a33
(iii)
fl) dhft, Qyu f ( x) x 3 3 x 2 4 x, x R ,d fujUrj o/kZeku Qyu gSA
(iv)
(v)
Qyu f ( x) sin 2 x dk U;wure eku fdruk gksrk gSA
(vi)
ds fy, vo;o a11 vkSj a21 dk lg[k.M Kkr djksA
1 cos 3 x dx dk eku Kkr djksA
1 x
x dx dk eku Kkr djksA
dy
2 y x 2 dk lekdyu xq.kkad (I.F.) Kkr djksA
dx
(viii) a ,d ek=d lfn'k bl izdkj gS fd x a , x a 8 rks x ds ifjek.k dk eku gSA
(ix) ;fn | a.b || a b | gks rc a vkSj b ds chp dks.k Kkr djksA
(x) a 2iˆ 3 ˆj 2kˆ ij b iˆ 2 ˆj kˆ dk iz{ksi Kkr djksA
(vii) jSf[kd lehdj.k x
[k.M&c
y?kq rjkRed iz' u
4.
fl) dhft, okLrfod la[;kvksa R ij ifjHkkf"kr Qyu f(x) = 3 – 4x ,dSdh vkPNknd Qyu gSA
5.
;fn 2 A B 2
6.
;fn A 2 1 gks rks vkO;wg A dk izfryke vkO;wg Kkr djksA
7.
fuEufyf[kr lehdj.k fudk; dks gy dhft,A
3 1
1 5
B
rFkk
gks rks AB Kkr dhft,A
4
2
0
1
2
2x 5 y 1
3x 2 y 7
8.
kx 1 x 5
f ( x)
}kjk ifjHkkf"kr Qyu x 5 ij lrr gks rks k dk eku Kkr djksA
3x 5 x 5
9.
x sin x dk x ds lkis{k vodyu dhft,A
d2y
gks rks
Kkr djksA
dx 2
10.
;fn x a sin rFkk y a 1 cos
11.
F(x) = 2x2 - 3x gks rks og vUrjky Kkr djks ftlesa Qyu fujUrj o/kZeku vkSj fujUrj gzkleku gks\
(87)
'ks [kkokVh fe'ku&
l= %
3 x 2 dx
12.
13.
izFke prqFkkZa'k esa oØ y x 2 rFkk x 1 ,oa x 2 ds chp dk {ks=Qy Kkr djksA
14.
vody lehdj.k
15.
,d ikls dks nks ckj mNkyk tkrk gS vkSj izdV gqbZ la[;kvksa dk ;ksx 6 ik;k tkrk gSA la[;k 4 ds U;wure ,d ckj izdV gksus
dk eku Kkr djksA
2 3x
3 3
dy 1 y 2
dk O;kid gy Kkr djksA
dx 1 x 2
dh izkf;drk Kkr djksA
[k.M&l
nh?kZm Ùkjh; iz ' u
16.
e
x 1 sin
x
1 cos x dx dk eku Kkr djksA
2
vFkok
2
sin x
1 cos
0
17.
2
x
dx dk eku Kkr djksA
vody lehdj.k x 2 y 2 dx 2 xydy 0 dk O;kid gy Kkr djksA
vFkok
vody lehdj.k x y dy x y dx 0 dk fof'k"V gy Kkr djks ;fn x 1 rFkk y 1
18.
fcUnqvksa (-1, 0, 2) vkSj (3, 4, 6) ls gksdj tkus okyh js[kkvksa ds dkrhZ; o lfn'k lehdj.k Kkr djksA
vFkok
,d f=Hkqt dh Hkqtkvksa dh fnd~ dkslkbu Kkr djksA ;fn f=Hkqt ds 'kh"kZ (3, 5, -4), (-1, 1, 2) rFkk (-5, -5, -2) gSA
19.
,d FkSys esa 4 yky vkSj 4 dkyh xsan gS rFkk vU; FkSys esa 2 yky vkSj 6 dkyh xsan gSA nksuksa FkSyks esa ls ,d dk p;u ;kn`PN;k
fd;k tkrk gS vkSj mlesa ls ,d xsan fudkyh tkrh gS tks yky jax dh gS rks bl xsan ds nwljs FkSys ls fudyus dh izkf;drk Kkr
djksA
vFkok
,d e'khu leqfpr <ax ls LFkkfir dh tkrh gS rks 90 izfr'kr Lohdk;Z oLrq mRikfnr djrh gSA vkSj leqfpr <ax ls LFkkfir ugha
gksus ij ek= 40 izfr'kr Lohdk;Z oLrq mRikfnr djrh gSA iwoZ vuqHko crkrk gS fd e'khu 80 izfr'kr leqfpr gSA ;fn fuf'pr LFkkiu
ds ckn e'khu 2 Lohdk;Z oLrqvksa dk mRiknu djrh gS rks e'khu ds leqfpr <ax ls LFkkfir gksus dh izkf;drk Kkr djksA
(88)
'ks [kkokVh fe'ku&
l= %
fuca/kkRed iz ' u
20.
x
2
1 log x 2 1 2 log x
x
4
dx
Kkr dhft,A
vFkok
3
sin x cos x dx
sin 2 x
dk eku Kkr djksA
6
21.
fuEufyf[kr js[kkvksa r iˆ 2 ˆj 4kˆ 2iˆ 3 ˆj 6kˆ rFkk r 3iˆ 3 ˆj 5kˆ 6iˆ 9 ˆj 18kˆ ds chp U;wure nwjh Kkr
djksA
vFkok
fl) dhft, js[kk,a
22.
x 3 y 1 z 5
x 1 y 2 z 5
rFkk
lgruh; gSA
3
1
5
1
2
5
fuEu vojks/kksa ds vUrxZr z = -x + 2y dk vf/kdlehdj.k Kkr djksA
x y 5
x 2y 6
x3
y0
vFkok
fuEu vojks/kksa ds vUrxZr z 200 x 500 y dk U;wure eku Kkr djksA
x 2 y 10
3 x 4 y 24
x0
y0
(89)
'ks [kkokVh fe'ku&
l= %
1.
cgqfodYih; iz'u &
(i)
;fn f : R R, f ( x) sin x rFkk g : R R, g ( x) x 2 rc ( fog )( x)
(1) sin x 2
(ii)
sin 1 2 x cos 1
(1)
(iii)
(2) sin 2 x
1
2
(2) 2
1
logb a
log a b
1
(2) f=Hkqtkdkj vkO;wg
(2) 0
(4) None of these
(3) fod.kZ vkO;wg
(4) buesa ls dksbZ ugha
(3) log a b
(4) log b a
(3) log e (1 x )
(4)
(3) 0
(4) vfLrRo ugha gS
(3) log | log x | C
(4)
d2y
dk eku gksxk&
dx 2
1
1 x
(2) 1 log e x
1
x
| x 2 | 1 dk U;wure eku gS&
(1) 1
(vii)
1
4
dk eku gksxk&
; y x log e x rks
(1)
(vi)
(3)
;fn ,d vkO;wg lefer ,oa fo"ke&lefer gks rks og vkO;wg gksxk&
(1) 1
(v)
(4) sin 2 x 2
1
gks rks x dk eku gS&
2 2
(1) 'kwU; vkO;wg
(iv)
(3) sin x x 2
(2) -1
dx
x log
e
x dk eku gksxkA
(1) | log x | C
(2)
1
C
2
1
C
x2
(viii) izFke prqFkkZa'k esa o`r x 2 y 2 4 ls f?kjs {ks= dk {ks=Qy gksxk&
(1)
(ix)
(2)
(3)
(4)
ijoy; x 2 4 y rFkk blds ukfHkyEc }kjk ifjc) {ks=Qy gS&
(1)
5
3
(2)
2
3
(3)
(90)
8
3
(4)
4
3
'ks [kkokVh fe'ku&
l= %
4
(x)
d2y
dy
x cos x dh ?kkr gksxh&
vody lehdj.k
2
dx
dx
(1) 1
(xi)
(2) 2
(3) 4
(4) vifjHkkf"kr
(3) 0
(4) 0
(3) -1, 0, 0
(4) 0, 0, 1
;fn nks lfn'kksa a vkSj b ds chp dk dks.k gS rks a.b 0 gksxkA ;fn
(1) 0
(2) 0
2
2
3
(xii) fuEu esa ls dkSulk lewg ,d js[kk dh fnd~dksT;k,a ugha gSA
(1) 1, 1, 1
(xiii) js[kk,a
(2) 0, 0, -1
x5 y 2 z
x
y
z
yEcor gks rks P dk eku gksxk&
rFkk
7
P
1
1 2 3
(1) 1
(2) 2
(3) 4
(4) 5
1
3
(xiv) ;fn A vkSj B nks Lora= ?kVuk,a gS tgka P( A) , P( A B) rFkk P( B) x rc x dk eku gksxk&
2
5
(1)
1
10
(2)
2
10
(3)
3
10
(4)
2
5
(xv) ;fn nks fu"i{k iklksa dh ,d tksM+h dks ,d ckj QSadk tkrk gS rks nksuksa iklksa ij vadksa dk ;ksx 5 gksus dh izk;drk gS&
(1)
5
36
(2)
1
12
(3)
1
18
(4)
1
9
2.
fjDr LFkkuksa dh iwfrZ djks&
(i)
7
cos 1 cos
dk eq[; eku -------------------------- gSA
6
(ii)
1
sin sin 1
dk eku -------------------- gksxkA
2
4
(iii)
3 tan 1 x cot 1 x gks rks x --------------- gksxkA
(iv)
;fn y cos x rks
----------------------- gksxkA
dx
(v)
fdlh mRikn dh x bdkb;ksa ds foØ; ls izkIr dqy vk; #i;s esa R( x ) 4 x 2 49 x 7 gS x 6 rks lhekUr vk; ----------- gksxhA
(vi)
vody lehdj.k
(vii)
iˆ j .kˆ iˆ. ˆj
dy
dy y
x 2 dk lekdyu xq.kkad ¼ I .F . ½ --------------------- gksxk
dx x
dk eku ----------------------- gksxkA
(91)
'ks [kkokVh fe'ku&
3.
(i)
l= %
vfry?kq rjkRed iz 'u&
x
x 1
x 1
x
dk eku Kkr djksA
2 3
6 0
5
4
ds vo;o aij dk lg[k.M Aij gS rks a32 A32 dk eku Kkr djksA
(ii)
;fn lkjf.kd
(iii)
o`r ds {ks=Qy esa ifjorZu dh nj bldh f=T;k r ds lkis{k Kkr djks tcfd r 7cm gSA
(iv)
n'kkbZ, dh F ( x) ex , R esa ,d fujUrj o/kZeku Qyu gSA
(v)
(vi)
a
1
5
4
1 sin x
dx dk eku Kkr djksA
cos 2 x
3 log a x
dx dk eku Kkr djksA
dy y
dk O;kid gy Kkr djksA
dx x
(viii) lfn'kksa a 2iˆ ˆj 2 kˆ vkSj b iˆ ˆj kˆ ds ;ksxQy ds vuqfn'k ek=d lfn'k Kkr djksA
(ix) ;fn nks lfn'k a vkSj b bl izdkj gS fd | a | 2, | b | 3 vkSj a.b 4 rks | a b | Kkr djksA
(vii)
(x)
;fn xiˆ 2 ˆj ( z 1)kˆ rFkk 2iˆ ( y 2) ˆj kˆ leku lfn'k gS rks x, y, z ds eku gSA
[k.M&c
y?kq rjkRed iz' u
4.
tkap dhft, fd D;k leqPp; {1,2,3,4,5,6} esa R {(a, b) : b a 1} }kjk ifjHkkf"kr laca/k R ,d rqY;rk laca/k gSA
5.
5 2
3 6
X rFkk Y Kkr dhft, ;fn X Y 0 9 rFkk X Y 0 1 gSA
6.
;fn A 6 7 gks rks fl) djks A A1 ,d fo"ke lefer vkO;wg gSA
7.
lkjf.kd dh lgk;rk ls fcUnq 3,1 vkSj 9,3 ls tkus okyh js[kk dk lehdj.k Kkr djksA
8.
2 x 3 ; fn x 2
F ds vlkarR; ds fcUnqvksa dks Kkr dhft, ;fn F ( x ) 2 x 3 ; fn x 2 gSA
9.
d2y
y 0 gSA
;fn y A sin x B cos x gS rks fl) djks fd
dx 2
10.
sin x x sin 1
11.
varjky Kkr djks ftuesa F ( x) 4 x 3 6 x 2 72 x 30 }kjk iznr Qyu F fujarj o/kZeku rFkk fujarj gzkleku gSA
12.
x cos x dx
1 5
x dk x ds lkis{k vodyu djksA
Kkr djksA
(92)
'ks [kkokVh fe'ku&
l= %
x2 y2
1 ls f?kjs {ks= dk {ks=Qy Kkr djksA
16 9
13.
nh?kZo`r
14.
vody lehdj.k e x tan ydx 1 e x sec 2 ydy 0 dk O;kid gy Kkr djksA
15.
52 irksa dh vPNh rjg QsaVh xbZ xM~Mh esa ls ,d ds ckn ,d rhu iRrs fcuk izfrLFkkiu fd, fudkys tkrs gS igys nks irks dk
ckn'kkg vkSj rhljs dk bDdk gksus dh izk;drk Kkr djksA
[k.M&l
nh?kZm Ùkjh; iz ' u
2
16.
sin cos 5 d dk eku Kkr djksA
0
vFkok
sin 3/ 2 x dx
0 sin3/2 x cos3/2 x dk eku Kkr djksA
2
17.
vody lehdj.k
dy
y cot x 4 x cos ecx dk fof'k"V gy Kkr djks ;fn y 0 rFkk x
gksA
dx
2
vFkok
x/ y
x/ y
vody lehdj.k 1 e dx e 1 y dy 0 dk O;kid gy Kkr djksA
18.
fn, x, js[kk ;qXe
x
r 3iˆ 2 ˆj 4kˆ iˆ 2 ˆj 2kˆ
r 5iˆ 2 ˆj 3iˆ 2 ˆj 6kˆ ds e/; dks.k Kkr djksA
vFkok
A(1,2,3) B (4,5,7)C (4,3,6) vkSj D (2,9,2) pkj fcUnq gks rks AB vkSj CD js[kkvksa ds chp dks.k Kkr djksA
19.
nks FkSys I vkSj II fn, gS] Fksys I esa 3 yky vkSj 4 dkyh xsan gS tcfd FkSys II esa 5 yky vksj 6 dkyh xsan gSA fdlh ,d FkSys esa
ls ;kn`PN;k ,d xsan fudkyh tkrh gS tks fd yky jax dh gS bl ckr dh D;k izk;drk gS fd ;g xsan FkSys II ls fudkyh xbZ gSA
vFkok
52 rk'k dh xM~Mh ls ,d irk [kks tkrk gS 'ks"k irks ls nks irs fudkys tkrs gS tks bZaV ds gS [kks x, irs ds bZVa ds gksus dh izk;drk
D;k gS\
(93)
'ks [kkokVh fe'ku&
l= %
fuca/kkRed iz ' u
20.
log(1 cos x ) dx
dk eku Kkr djksA
0
vFkok
tan
21.
js[kkvksa
1
1 x
dk eku Kkr djksA
1 x
x 1 y 1 z 1
x 3 y 5 z 7
rFkk
ds e/; nwjh Kkr djksA
7
6
1
1
2
1
vFkok
,d js[kk ,d ?ku ds fod.kksZa ds lkFk , , , dks.k cukrh gS rks fl) djksA
cos 2 cos 2 cos 2 cos 2
22.
4
3
fuEu vojks/kksa ds vUrxZr z 5 x 3 y dk U;wurehdj.k dhft,A
3x 3 y 15,5 x 2 y 10, x 0, y 0
vFkok
vkys[kh; fof/k dks fuEu leL;k dks gy dhft,A
fuEu O;ojks/kksa ds vUrxZr
x 3 y 60
x y 10
x y
x 0, y 0
z 3 x 9 y dk U;wure vkSj vf/kdre eku Kkr dhft,A
'ks[kkokVh fe'ku 100 dh d{kk 10 ,oa 12 ds
fofHkUu fo"k;ksa dh uohure cqdysV MkmuyksM
djus gsrq Vsyhxzke QR CODE LdSu djsa
(94)