aieee (pcm) 2009
TRANSCRIPT
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Physics AIEEE Solution (26 - 04 - 09)
e See- See - O e e e e e ee e e
e be ebe e w ee
1. Statement 1 : e le P e e w e by ele el
e le eee e e P
Statement 2 : e e w e by e e bje l le l ze
Sol. See- See- b e e b See- e e el See-
2.
e be l b eey e le b e le M; e
ee le e e:
A B C+ + C A B + + D E F+ + F D E , + + wee e eey elee? I w e e?
Sol. A B C+ + le B e
F D E + + le B e
3. j w e e ee le e e V ee e
e e I e e be w by :
Sol.
4. e l w belw e we w P e e
we
O
Set # A
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Sol.
5. I ee e lee e ely e ele le ee le
e e e w e llw e e w e ?
Sol.2
x Asin tT
=
A2 2 tv cosT T
=
2 2 2
2 2
2 2
4 A 4 tv cosT T
=
2
2
4 2 ta Asin
T T
=
4 2 2
2 2 2
2 2
16 cos 4 t4 v A
T T
=
4
2 2
4
16 A 2 ta sin
T T
=
4
2 2 2 2
2
16 2 ta T A .sin
T T
=
S42 2 2 2 2
2
16a T 4 v A const.T
+ = =
2aT const. 4
x= =
2 24 A 2 t 4 A 2 t
aT 2 v sin cosT T T T
+ = +
24 A 2 t 2 tcos sin
T T T
=
2aT Acos T.cos t
A sin t
=
Note : Both 1 and 2 are correct, but option 1 is a function of time period and option 2 is independent of
unit of time.
6. I eee w e e bje e e e e e le
e e ee je e le e e bje bewee e bje e e
e e e e e e le le e e le e w e
le e le 5 w e - ee e eeel e P e
e P wll be:
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Sol.1 1 1
v u f =
1 1 1
u v f=
1 f v
u fv=
2fv f (v f f ) f u f
v f v f v f
+= = =
2fu f
v f+ =
7. le l w eely b zl e I
l ee I ee e e :
Sol. h sin (1 sin )2 2 2
= + = +
2
21 m. mg. h2 3
=
2 2
6g
8. Le 4Q
P(r) r R
=
be e e ey b l ee l e P
e e ee e
e ee e ee e e ele el :
Sol. 4Q
P(r) .r R
=
r 2
40
Qrq 4 r dr.
R=
4 4
4 4
4Q r Qr q .
R 4 R= =
4 2
4 2 4
0 0
1 Qr Qr E .4 R r 4 R = =
9. e e e = = ye le el lle Ie e
wll be be e :
Sol. e lwe eey eee b ee
10 . Oe ee 8 / e ey e / W e eey
e e el ?
Sol.4
2 3
N kgP 8 10 , 4
m m= =
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5KE PV
2= PV =
PV =m
RTM
PM m
RT V= =
45 1KE 8 102 4
= 4KE 5 10 J=
e See- See - O e e e e e ee e e
e be ebe e w ee
11. Statement 1 : e eee eeee ee lly e =
+ e ee
we e 5 we eee ee 7 7 le = 5 /
Statement 2 : =
+ l ly we e e e eee ll =
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19 . der rbber b reey rm e = m rz e pe me e dr
ebe d e w e pe y e
Te e ey me d e e me w be:
Sol.1
vg ve is downward direction.
t
= = +
1 1 1 1t 2t t t= =
(d)
20 . re ped e e ppe rer re re ped e e er w
e I e e elel e ze e / el:
Sol.
2
2 2
Qq 2 Q0
d 2d+ =
Qq 2
2
=
Q2 2
q=
21 . l ell b y e e e e e e e ey-e e
eee l e le e b e be ebe by w e llw?
Sol.d kA dT
dt dx
=
dT ( dx)
I we e e -e e ell be
22 . e l yll ee e 2
.3
e by l y e e
e e e w e e
e e le w e l y ze l e wll e :
Sol2
sin sin3
=
2sin(90 ) 1 1
3 =
3cos cos30
2 = =
30 = P
2 1sin sin 30
3 3 = =
1 1sin
3
=
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23 . ee we el le e eee + ey ee e be ebe be e e e wll be:
Sol.A B C
f 1 f f 1 +
Se & w be pe e e d & w be pe e e
& w be pe we eS w be pe e e der mm we d re pe mey
ee pe e eery e be reey =
Note : in some books it is written 2 but is not corresponding to absolute maxima.
24 . e e w e ele e y bee /9 wee = e ele e y e
e e e e e e e :
Sol
2g R h g
9g ' R g / 9
+ = = =
h
1 3R
+ = h 2R=
25 . w we e e e e el e e e le Hwee we -el e
we -el e I e le we ee by ly e w e eee e we by e e ?
Sol.
=
=
= 1
x=
= ?2
x=
FLy
A=
S 2F
y A
2
21 2 2
2 12 2
1 2 1
F F AF F 3 F
A A A
= = =
2F 9F=
26 . I eee e le e ee be ee e 9 e le ely
e w e e ee le I e lle e le l--eee =5
e e le e e :
Sol. e le = 5
Le =s (1/2)
n 30
=
11'
60
=
= e e
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27 .
e L = H e ee
= = e ee bey e V w e e e el ee e bey elble He w S
le = e el L e :
Sol. = e L = H
=
= =
2R 2=
Rt
L0
v v e
=
0v 12 volt=
2R tLv 12e
=
32 105t400v 12e 12e v
= =
Directions: e be 8 9 e be e llw w le el e
e e e yle w e P-
28 . e be el e w e e :
Sol. PV =
B A B AW P(V V ) nR(T T )= =
[ ]2 R 500 300=
W 400R=
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29 . e w e e :
Sol. I Iel Pe
1
e
2
PW nRTlog
P=
2e
1
PnRTlogP
= =
e2R 300log 2=
600R 0.6932 415.9200R= =
DAW 414R =
30 . e e w e e e yle :
Sol. W
=
W
=
W
= C
BC e
B
PW nRT log
P=
2
e e
1 102 R log 2R log
2 100
= = =
= 9 5 = 9
W
=
l W = 7
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CHEMISTRY AIEEE SOLUTION (26 - 04 - 09)
31 . Kw e ey l L e + e w e llw
ee e ?
ee e le ze e L III e b eely e
b e ze L III eee eel w e be
L III e eelly lle
L III ye e ly b e
Sol. (c) e eee e ele L III e eelly le
32 . l w e w el ee HSO
w e w y ell w
e e l w:
HOH b HHO H
O H
H
OOH
Sol. (d) y ell e e ee
H SO2 42 5 3 3 2 5 2C H OH CH COOH CH COOC H H O + +
33 . e e b 3 3(CH )C, CCl H CH H5 2CH e e ee bly :
H
5
2CH > 3CCl > H C > H CH b H CH > 3CCl > H5 2CH > H C
3CCl > H5 2CH > H CH > H C H C > H CH >H5 2CH > 3CCl
Sol. (c) I3CCl
26 5C H C H
ee ee
I3CCl e e elle bewee b l ee e ee l well l e elee-
e
I6 5 2C H CH ee e elle bewee b b b le eleee
e le
Hee3CCl e ble 6 5 2C H CH
3 2 3 3(CH ) CH (CH ) CH
stability
>
I e le +I ee b
34 . e lee eb eel e :
ee b -eyl ee -bee -eyl--bee
Sol. (c)
a
bC C
a
b
b
ye lele w w ee bly be b w eel e
35 . I w e llw ee e eee ly e ey we ?
O
< SO
< SO
< PbO
: e we
b H < Hl < H < HI : e e
H
< PH
< H
< SbH
: e b e
< < O < : e z ely
Set # A
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Sol. (c) I ye e ly H SbH
by eee e ee ze el le
36 . e j be e el w ye b e :
bez b lylleye lyl l
Sol. (c) Klbe e
OH
+ NaOH2 + CO2
OH
COOH
Salicylic acid
heating
underpressure
37 . W e llw ee e e y ?
I bee e Wl e
b Me ely leble e e be ely
Ue ee el l lel lye be e
ly adsorption( H ) lw e
Sol. (a)
38 . W e llw e w e KOH e eleye ?
H
Ol b H
H
l H
l H
l H
Hl
Sol. (d) HH
Cl
Cl
+ KOH H
H
OH
OH+ Kl
H
H
OH
OH 2H O 3CH CHO
39 . I ele vg w peed f / w uy f 5% ey w w e
p f e ele be led = kg f ele e
= 9 kg
5 b 5 9 d 8
Sol. (c) 5600 5 10 = = = /e
h
4
34
31 2
6.6 10x
9.1 10 3 10 4 3.14
= 9
40 . I fuel ell el ued fuel d yge g ued dze Te e
3 2 2 2CH OH( ) 3 /2O (g) CO (g) 2H O( )+ + 98 K dd Gbb eege f f f O
O d O
g e 7 d 9 kJ l epevely If dd elpy f bu f
el 7 kJ l effey f e fuel ell wll be :
8 % b 87 % 9% d 97 %
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Sol. (d)| G |
100H
= Tedy effey
G = { } { }394.4 2(237.2) 166.2
= 7 kJ
702.6100 %726 = = 97 % = 97 %
41 . Tw lud X d Y f del lu K vpu peue f e lu g l f X d l
f Y 55 g e e epeue f l f Y fue dded lu vpu peue f e
lu ee by g Vpu peue g f X d Y e pue e wll be epevely :
d b d d d 5 d
Sol. (c)A A B BP P X P X= +
AP = f g
BP
= f g
42 . Te lf lfe ped f f de el e 9 ue Te e eued f e ple f 99%
f e el e wll be lg = :
ue b ue ue d ue
Sol. (c)0.693 2.303 100
log6.93 t 1
=
=
43 . Gve: 0 03 2Fe FeFe Fe
E 0.36V,E 0.439V+ += = Te vlue f dd elede pel f e ge
3 2(aq)Fe e Fe (aq)
+ ++ wll be
7 V b 85 V 77 V d 7 V
Sol. (c) Fe+ e + Fe = V
Fe Fe+ + e = + 9
G
= G
+ G
F
=
F
+ F
= [ ] + [ 9] = 77 V
44 . O e b f e fllwg eel d : (aq)( fG H 0)+ =
O l + + O ; = 57 kJ
g +1
2O
g
O l ; = 8 kJ
Te vlue f elpy f f f O 5 :
88 kJ b 888 kJ + 888 kJ d 5 kJ
Sol. (b) = 888 kJ
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45 . ppe ylle f w u ell leg f p W e du f ppe ?
8 p b 7 p 57 p d 8 p
Sol. (b)a
r2 2
= = 7 p
46 . W f e fllwg pl e ?
3 3[CO(NH ) Cl]
+ b 23 2[CO(en)(NH ) ]
+
32 4[CO(H O) (en)
+ d 32 3 2[CO(en) (NH ) ]
+
Sol. (c) f f ple wll be plly ve wle e
47 . Sld NO
gdully dlved M N
O
lu w e f + wll
pepe beg f ? Kp
f O
= 5 9 :
5 M b 5 5 M 8 8 M d 8 7 M
Sol. (b) Q > kp
5 9 = [+]
[+] = 5 5
48 . W e f e fllwg e f Xe pud notfeble ?
Xe O
+ F XeF+
O
b Xe F
+ O Xe + Xe O
+ F + 5 O
Xe F
+ O Xe + F + O
d Xe F
+ RbF Rb [XeF7]
Sol. (a)
49 . Ug MO ey ped w f e fllwg pee e e bd leg ?
22O + b 2O+ 2O d22O
Sol. (a) Species Bond order
O
+
O
+ 5
O 5
O
d de f O+ lge bd leg le
50 . I e w e elee w f e fllwg e e ?
I dd e l d e e ze d e l w by ee elee pleeb I e ge d e e el w b e d f plee
I e ge d e f e f fve elee S M ll e d d ele e ued
f bdg
d Oe e 5 fgu eeeded e edey vlve ll e d ele bdg deee
Sol. (b) Lw d e e b bu ge d e elee e d ue d ge
d e ey f ple
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51 . lule e wveleg ee ed w p vg
M f p = 7 7 kg d = J :
b 5 d
Sol. (b)
34
27 3
h 6.63 10m
mv 1.67 10 10
= =
=
52 . by lud lu peped by g -epe d el W e f e fllwg ee e
egdg e bevu f e lu ?
Te lu fed del lu
b Te lu -del wg +ve dev f Rul Lw
Te lu -del wg Ve dev f Rul Lw
d -epe w + ve dev wle el w ve dev f Rul Lw
Sol. (b)
53 . Te ube f eee pble f pud f e leul ful =OMe
b d
Sol. (c)
OH
CH3
CH CH CH Me
: = f duble bd Geel
: = f l b pl
Tl ee e :
54 . Te IUP e f epee :
eylbue b -deylppe eylppe d -deylbue
Sol. (b) CH3 C
CH3
CH3
CH32 1
deyl ppe
55 . Te e epeeg e e de f du :
L+ > e+ > N+ > Mg+ b N+ > L+ > Mg+ > e+
L+ > N+ > Mg+ > e+ d Mg+ > e+ > L+ > N+
Sol. (b)
56 . Te w ful gup pee ypl byde e :
O d OO b O d OO > = O d O d O d O
Sol. (c,d)
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57 . Te bd d eegy f F F
kJ l wee f F F
55 kJ l Te e
e f ge F bd d eegy ped f F :
lle ze f - ped f -
b ge bd bewee d F F
ped bewee d F F
gf p - p e bewee d F F
wee ee pbly f u e
bewee d F F
d lwe degee f p-p e bewee d F F
bewee d F F
Sol. (d)
58 . I zz e gve belwOH
222Ph CHO Ph CH OH Ph CO
+i i
e lwe ep :
e k f : OH
e byl gup
b e fe f ydde e byl gup
e b f p f e byl gup
d e dep f P O
Sol. (b) Re ulepl dd d ee e fe f ydde uleple wll be e e deeg
ep
59 . W f e fllwg p epee lkge e ?
[u N
] [P l
] d [P N
] [ul
]
b [Pd P P
NS
] d [Pd P P
SN
]
[O N
5NO
]SO
d [ON
5SO
] NO
d [P l
N
]
d [P
N
] l
Sol. (b) I f e N e d e wle ed ulpe e d e
60 . u-N ye ubbe plye f :
2 2
Cl |H C CH C CH= = d = =
b = =
d
5
=
= N d
= =
d = N d
2 2
3
H C CH C CH |CH
= =
Sol. (c)
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Mathematics AIEEE Solution (26 - 04 - 09)
Directions : Que ube 5 e e - Re ype ue f ee ue
w ee :
Statement - 1 (Assertion ) d Statement - 2 (Reason).
f ee ue l fu leve e ly e f w e e we
Yu ve ele e e e
31. Statement - 1 : ~ p ~ euvle p
Statement - 2 : ~ p ~ ulgy
See - ue See - ue; See - e epl f See -
b See - ue See - fle
See - fle See - ue
d See - ue See - ue ;
See - e epl f See -
Sol. ~ p ~
p
1 2
(1) (2) (3) (4) (5) (6)
~ ~ ( )p q q p q s p q s p q
T T F F T T
T F T T F F
F T F T F F
F F T F T T
F lu 5 &
euvle ee ee - ue
f lu 5 ~ p ~ ulgy ee ee - fle
See - ue See - fle
32 . Le be
Statement - 1 : dj dj = ;
Statement - 2 : |dj | = | |
See - ue See - ue; See - e epl f See -
b See - ue See - fle
See - fle See - ue
d See - ue See - ue ;
See - e epl f See -
Sol. We kw
dj = | | I | dj | = | | |dj | = | |
Nw pu = dj dj dj dj = | dj | I dj dj = | | I
dj dj =(| | )
| |
A AA
A I
= djdj =
Set # B
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33 . Le f = +
Statement - 1 : Te e { : f = f } = { }
Statement - 2 : f bje
See - ue See - ue; See - e epl f See -
b See - ue See - fle
See - fle See - ued See - ue See - ue ;
See - e epl f See -
Sol. f = +
See -: Te e { : f = f} = { }
we kw lu e f f = f e lu e f f =
Tu lvg f = = d =
S lu e { }
e fu e-e e bjeve
See - ue See - ue
34. Statement - 1 : Te ve f f eve ul ube
2 1
4
n
Statement - 2 : Te u f f ul ube ( 1)
2
n n +d e u f ue f f ul ube
( 1)(2 1)
6
n n n+ +
See - ue See - ue; See - e epl f See -
b See - ue See - fle
See - fle See - ue
d See - ue See - ue ;
See - e epl f See -
Sol. Me = +
S D =
2
1
(2 ( 1))n
r
r n
n
=
+ SD =
2 1
3
n
V = SD =1
3
n ee ee - fle
35 . Le f = | | d g =
Statement - 1 : gf dffeeble = d devve uu p
Statement - 2 : gf we dffeeble =
See - ue See - ue; See - e epl f See -
b See - ue See - fle
See - fle See - ue
d See - ue See - ue ;
See - e epl f See -
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Sol. f = | | g =
le = gf = gf = g | | = | |
= gf =
2
2
sin( ) 0
sin( ) 0
x x
x x
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P ee =
1 0 1
P(1)P(1)
[ e ee wll be w e]
Ge P < P
el [ ] P b P le P
( P
38 . e e e bewee e le y = e e = y
:
2 3
8b
3 2
5
3
4
3 2
8
Sol. Le y y
be e e bl = y
Ge le y = +
eee e w 21 1( , ) 1
11
2y y
dy
dx y
= =
1
1
2y = 21
p y , y
y = x +1
0x
-e 1 1
,4 2
p
e le
1 11
3 3 24 2
82 4 2
+= =
39 . Le e le2 1 2
3 5 2
x y z += =
le e le + y z + = e , el :
7 b 5 5 5 5 7
Sol. Se e le2 1 2
3 5 2
x y z += =
le e le + y + =
S 5 = =
+ + + = 5 + = = 7
40 . ee el ee e el y e be elee e
w el e y lwy e le e e be ee :
le 5 b le 75 b le 75 b le
le le 5
Sol. wy ele el le =6
4
6.515
2.1c = =
wy ele y y = 31 3c =
wy e el Se y be le = 4 24=
l ee = 5 = 8 le
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41 . I bl b1
,4
B n p =
e bbly le e e ee el 9
10 e
ee :
10 10
1
log 4 log 3+ b 10 10
9
log 4 log 3 10 10
4
log 4 log 3 10 10
1
log 4 log 3
Sol. = = =0
0
3 11
4 4
n
nc
=3
14
n
e 9
10
3 91
4 10
n
1 3
10 4
n
3/ 4
1log
10
3/ 4log 10n ;
10
10
log 103
log4
n ;10 10
1log 4 log 3
n
42 . e le + y + = + + + y + = e eel le :
ely e le b ely w le e w le le
Sol. I ly ble we b le e llel
S + = 2 2
2
( 1)
( 1)
p
p
++
+ + + = + + = =
43 . I e ee e = = e :
= b = = =
Sol. by e =
44 . el le = + 5 + e
Rb e-e b e-e R
ee e-e R e-e b R
Sol. = + 5 + ; = + 5 >
Hee e-e R
45 . e eel e w eee e ly e 21
c xy c e= wee
e by
y = yy b yy = y yy = y y = y
Sol. 21
c x y c e=
2
1 2
c xy c c e = 2y c y =
2 y c y =
yy y
y
=
yy = y
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46 . Le b be b + I
1 1
1 1
1 1
+
+
+
a a a
b b b
c c c+
( ) ( ) ( )2 1
1 1 1
1 1 1
1 1 1+ +
+ +
+
n n n
a b c
a b c
a b c= e e le
:
y ee ee b y ee y ee ze
Sol.( )
1 1 1 1 1
1 1 1 1 1 1 0
1 1
+ + +
+ + + =
+
n
a a a a b c
b b b a b c
c c c a b c
by w l
( )
1 1 1 1
1 1 1 1 1 0
1 1 1 1
+ +
+ + + =
+ +
n
a a a a a a
b b b b b b
c c c c c c
ly ble we ee
47 . e ee le we ( )2 128 62
+nn e by 9 :
b 7 8
Sol. 2 2 18 62 +n n
( ) ( )2 1
64 62+
n n
( ) ( )2 1
63 1 63 1+
+ n n
( ) ( )2 1
1 63 1 63+
+ + n n
2
1 21 63 63 ... 63 + + + +
n n n n
nC C C +2 1 2 1 2 2 1 3 2 1 2 1
1 2 3 2 11 63 63 63 ... 63+ + + + ++ + +
n n n n n
nC C C C
( ) ( )2 1 2 11 1 2 22 63 63 ....+ + + + + + n n n nC C C C
O by 9 ee 7
48 . Le y be l ee by 2 2 cot 1 0 =x x x x y e ( )' 1y el :
b l l
Sol. = 2 2 cot 1 0 =x x x x y
= y = y =2
[ ] ( )2
2
.2 1 log 2cot 1 log
2. cos
+ +
=
x x
x
x x y x xdy
dx x ec y
=
[ ]21 2 1 log1 2 02
+ =
dy
dx
12
2= =
dy
dx
49 . I e e e 2 0+ + =bx cx a be y e ll el le e ee
2 2 23 6 2+ +b x bcx c :
le b b ee b le b ee b
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Sol. we e e e b + + = be y e
b < < bS le ee
/ =
2 2 2 2
2
36 4 3 2
4 3
b c b c
b
=
2 2
221212 =
c b cb
e le ee ee b
50 . e y e ee 2 3 42 6 10 14
1 ....3 3 3 3
+ + + + + :
b
Sol. (a) Le S = 2 3 42 6 10 14
....3 3 3 3
+ + + +
2 3 4
2 3 4
2 6 10....
3 3 3 3
2 2 4 4 4........
3 3 3 3 3
= + + +
= + + + +
S
S
2 3 4
2 2 4 4 4......
3 3 3 3 3
= + + + + S
4
2 2 913 3
13
= +
S
2
3S=
2 4 3
3 9 2+
2 4
3 3=S
S =4 3
23 2 = Hee 2 3
2 6 101 .....
3 3 3+ + + + = S + =
51 . e je e e ee e e eeely e e e e e
e :
6 3 2
, ,5 5 5
b
6 3 2, ,
7 7 7
6 3 2, ,
7 7 7
Sol. lely6 3 2
, ,7 7 7
52 . Le A B ee e ee
A : cos cos cos 0+ + =
B : sin sin sin 0+ + =
I ( ) ( ) ( )3
cos cos cos2
+ + = e :
A le B e b b A B e e
b A B e le A e B le
Sol. Le = + b = + = + + b + =
( ) ( ) ( )cos sin cos sin cos sin + + + + +i i i =
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cos cos cos + + = = sin sin sin+ +
( )2
cos cos cos+ + + ( )2
sin sin sin+ + =
+ ( ) ( ) ( )( )cos cos cos + + =
( ) ( ) ( )3
cos cos cos2
+ + =
53 . Oe e elee 5 e be 9 e e bbly e e
e elee e 8 e e ee ze el :
1
7b
5
14
1
50
1
14
Sol. S : { 5 7 8 9 }
S =
: {8}
=
( )( )n E 1
P E = =n(S) 14
54 . ee e e e - el e le e e e
y e e e e e e el 1
3 e e ee
e le e :
5
,04
b5
,02
5
,03
Sol. lely le le ee e ee wll be ee l
I P y
1
3=
PA
PB
2 29 = PA PB
( ) ( )2 22 29 1 1 + = + + x y x y
2 2 5 10 02
+ + = x y x 5/
55 . I e e e e be + + + e e 55 e e el :
b
Sol. ( )101 100 / 2 101101
+ = dx = + 5
S e = | 5| + |9| + |8| + + || + | | + | | + + | 5|
= 5 5
Me e = 55 =50 51
101
d
=
56 . e elle 2 24 4+ =x y be ele le w e e e w be
e elle e e e e e e elle :
2 212 16+ =x y b 2 24 48 48+ =x y 2 24 64 48+ =x y 2 216 16+ =x y
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Sol.
2 2
14 1
+ =x y
( )2,1
( )4,0O
ee ele
le e ee elle
wll y e e2 2
21
16+ =
x y
b
2
4 11
16+ =
b
2 4
3=b
e elle2 23
116 4
+ =x y
2 212 16+ =x y
57 . I4
2 =Zz
e e le | Z | el :
5 1+ b 2 2+ 3 1+
Sol.4
2 =zz
4 4| |
| | z z
z z | z |
4
| |
22 4 0 z z
( ) ( )1 5 1 5 0 + + z z
1 5,1 5 + z
max1 5= +z
58 . I P e e ee e le + y + + 7y + 5 = + y + + y =
e ee le P :
ll ee e le b ll ee w le
ely e le ll le
Sol. 2 2x 3 7 2 5 0+ + + + = y x y p
2 2 2x 2 2 0+ + + = y x y p
ee le
( )2 2 2 2 2x 3 7 2 5 2 2 0+ + + + + + + + =y x y p x y x y p e
+ + +7 + 5 + + + + =
2
2 7
6
+=
p
p
ee ll le ee = 6, 6
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59 . I , , u v w e -l e e el be e e ely
[ ]3 u p v p w [ ]
pv w qu [ ]2
w qv qu = l :
ely w le b e w b ll le
ll le ely e le
Sol. we e [ ] 0 u v w
[ ] [ ] [ ]2 2
3 2 0 + =
p u v w pq u v w q u v w
[ ]( )2 23 2 0 + = u v w p pq q
ly ble we 2 23 2 0 + = p pq q ly ble we
= =
60 . [ ]0
cot x dx
wee [ . ] ee e ee ee el :
b 2
2
Sol. I = [ ]0
cot x dx
I = [ ]0
cot x dx
by ( ) ( )0 0
= a a
f x dx f a x dx
[] = []
Hee I = [ ]( )0
cot 1 x dx
I = [ ]0 0
cot 1 x dx dx
I = 2
= I