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8/9/2019 Eng Model Test 1
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This paper contains 80 questions from Mathematics, 40 questions each from
Physics and Chemistry.
Each of the 160 questions carries one mark. No negative marking for wrong
answers.
Maximum trying allowed is 3 hours (180 minuites)
Use Ball - Point Pen while entering the Hall - Ticket Number and filling in Part - A
of the First page.
Use H.B. Pencil only to darken the circle on OMR Answer Sheet.
Over - writing or blackening of more than one circle will not count for marks.
If you wish to change your answer, erase the already darkened circle completely
and then darken the appropriate circle.
Candidates are prohibited from carrying any sheet of paper to the Examination
Hall except the Hall - Ticket.
Do any rough / scratch work on the Test Paper itself.
Calculators, watches with calculators, pagers & cellular phones will not be allowed
into the Examination Hall.
Candidates have to write suitable answers on the Answer Sheet only.
Candidates have to return the Answer Sheets and the Question Papers at the time
of leaving the Examination Hall.
Candidates can leave the Examination Hall only in the last half - an hour before
the close of the Test.
Total number of pages of the Test is 48.
The questions of this paper were set by Senior Faculty Members of
Sri Chaitanya EAMCET Coaching Centre, Vijayawada.
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θ 2 2x y
12 2a b
− =
2 2x y1
2 2a b− =
θ
( )2 4 2a e Sec 1θ + ( )2 4 2a e Sec 1θ − ( )2 4 2b e Sec 1θ + ( )2 4 2b e Sec 1θ − −
−
a b
1
3
2
3
1
3
2
3
3 1 x − ≤
3 1 x −
5x y 1− =
5x y 1− =
2 0± = x 5y 5 0+ ± = x 5y 5 2 0+ ± =
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( )2 2x y 4x 2y 4 2 3x 4y 10 0+ − − − + λ + + =
2 7 7 4 7 8 7
14
14
→ → →
2
π
→ → →
2
π
1−
( )1 3 2
216 9
Tanh x
x
− −
−
( )1 3 2
216 9
Tanh x
x
− −
−
1
, 13
1
, 13
−
[ ]2, 3− 3, 13 2, 13 2, 13 −
∈ 2 2
2 4 3
x x a
x x a
+ +
+ +
∈ 2
22 4 3
x x a
x x a+ ++ +
( )0, 2a ∈ ( )0,1a ∈ [ ]1,1a ∈ − [ ]0, 2a ∈
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( )( )2 2 A B A B A B− = − +
( )( )2 2 A B A B A B− = − +
2/3 2/3
1 x yCot
x y
+− +
xV +yV =x y
1
Sin2V
6
2
Sin2V
3
3
Sin2V
2
Sin2V
λ λ
λ λ
, ,α β γ 3 3 1 0 x x− + =
1 1 1, ,α β γ
βγ γα αβ − − −
33 1 0 x x− + = , ,α β γ
1 1 1, ,α β γ
βγ γα αβ − − −
3 3 8 0 x x− + = 3 6 8 0 x x− + = 3 9 8 0 x x− + = 3 12 8 0 x x− + =
4 3 24 2 4 2 0 x x x x+ + − − =
4 3 24 2 4 2 0 x x x x+ + − − =
4 24 1 0 y y+ + = 4 224 65 55 0 y y y− + − =
4 24 1 0 y y− + = 4 24 1 0 y y+ − =
1
5 x π ≤ ≤
4 3
3 4 or
4 3
3 4or
ABC ∆
ABC ∆
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6 6 2 1Sin Cos KCosθ θ θ + + =
1 2 22
Tan θ 1 2
4C o s 24 2Cot θ 23 Sin
65 25
65 25
Sin Sin
Cos Cos° + ° =° + ° 3
70 40
70 40
Sin Cos
Cos Sin
° + °=
° + °
3 3
33 27
33 27
Cos Cos
Cos Cos
3
4
( )1 1 21 12
Tan x x Sin x x π − −+ + + + =
θ θ 2 2Cosh y Cos x+ =
1−
13 2 2 3 3 3 2−
ABC ∆ 1 1 1: :1 2 3r r r
=
ABC ∆
r
1R
r
1R
r
R
R1
r
5 5.8 5.8.11
....
9.18 9.18.27 9.18.27.36
+ + + ∞ =
1 9 1132 4 18
− 33 18 32
12
−
33 9 11
12
−
310 5
6
−
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α α 23 .....α + 1nnα − =
α α 23 .....α + 1nnα − =
1
n
α −
1
n
α
−−
( )2
1
n
α
−
−
( )
21
n
α −
2 0 z pz q+ + =
2 0 z pz q+ + =
2 p q= 2q p= 2 3q p= 2 3 p q=
4 4
3 3Cos iSin
π π
+
3n1 a
2
( )1 n−
( )132
n
n
−
1
32 n
( )2, 1,1− ABC ∆
( )2, 1,1− ABC ∆
1
2
3
2
1
4
,OA a OC c= =
BC
,OA a OC c= =
BC
( )1r s a sc= − + ( )( )1r s a c sc= − + + r c ta= + ( )r s a c= +
2 3 2F i j k = − +
2 3 2F i j k = − +
4− 2−
2 , 2i j k i j k + + + + i j k + +
i j k + + 2 , 2i j k i j k + + + +
2
i k +
2
i k −
2
j k −
2
j k +
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, ,a b c ( ) 1
3a b c b c a× × = θ
b c θ
, ,a b c ( ) 1
3a b c b c a× × =
b c θ
θ
2 2
3
2
3
5
6
5
4
5
2 6 3
5
6
1
72
1
52
1
4
1
13
1
26
π
π
( )1
3
k a
k
+
0k ≥
( )1
3
k a
k
+ 0k ≥
0.3
e−
0.5
e−
2
e−
10 10
5100
Lt Tan x Sin x
k x x
−=
+→
2− 1−
( )1 22
Lt Sinx
Sinx x Cos x
π − =→
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1 1 1
.....1.2 2.3 3.4
+ + +
1 1 1
.....
1.3 3.5 5.7
+ + +
1 1 1
.....1.4 4.7 7.10
+ + +
1 1 1
.....1.5 5.9 9.13
1/21 1 1
1 2
xTan x
x
+ −− −
dy
dx=
2
41
x
x+
2
41
x
x− 21
x
x− 2 1
x
x −
3
5 ( )2 xe xe
0
dy
dx x
= =
3
5 ( )2 xe xe
0
dy
dx x
= =
( )2
4Sinx Cosx Cot x
π − −
− −
2 3
A5 6
1
,2
∞
1
,2
− ∞
1
,2
∞
1
,2
−∞ −
2 2x y
125 9
+ =
2 2x y1
25 9+ =
3 5 , 5 5 5 3 , 3 3 5 2 , 3 2
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( )2x ax 2a 3 0− + − =
( )2x ax 2a 3 0− + − =
a 1= −
2 2
x y 2y 0+ − =
2 2x y 2y 0+ − =
2 2x y y 0 2 2x y y 0 2 2x y 3y 0 2 2x y 3y 0
( )21 Sin 2x
d 4x1 Cos4x
+=
+∫
− + − −
( )
x 2dx
2x 3x 3 x 1
+=
+ + +∫
( )
1 x1Tan C3 3 x 1
− + +
( )
2 x1Tan C3 3 x 1
− + +
( )
2 x1Tan Cx 13
− + +
2 x1Tan C3 x 1
− + +
1Tan x
dx4x
−=∫
1 1 11Tan x
3 2 33x 6x
x
C21 x
++
x
C21 x
−+
+
2xC
21 x
++
x
C22 1 x
++
/2Cos3x 1
dx2Cosx 1
0
π+−∫
xsinx
dx21 Cos x0
π=
+∫
2π 2
4
π
2
2
π
2
8
π
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2x 4y= x 4y 2= −
2x 4y= x 4y 2= −
2 2Lt 1 2 1....
3 3 3 3n 2nn 1 n 2
+ + + =
→ ∞ + +
1
log22
1
log23
1
log33
1
log32
2 x 3 x 9
7
9
7
( ) 1
3 2 x −−
8
81
( ) 1
3 2 x −−
8
81
dydx
= 3x y 2 x ye x e+ ++
31x y x
e e e C3
−− + = 31x y x
e e e C3
−+ + =
31x y x
e e e C3
− −+ + = 31x y x
e e e C3
+ − =
( )dy 22xy e x 1dx
− = +
x 1e + ( )x 1− +
1
x 1+ α α α Cos− α ABC∆
ABC∆
2 2
x y 2x 4y 1 0+ − − + = ( )2 23 x y 2x 4y 1 0+ − − + =
2 2x y 2x 4y 3 0+ − − + = 2 2x y 2x 4y 3 0+ + + + =
α 1, 3 2,0 α
α
1, 3
2,0
α
6
π
4
π
3
π
2
π
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8 4.4 4P P
2 2 2 3 2 4 2
( ) ( )2
x 7y 4 2 x 7y 42 0+ + + − =
( ) ( )2
x 7y 4 2 x 7y 42 0+ + + − =
2 2
− −
1
172
1
662
7
2
17
2
5
3
x xe e
xe
+
5
3
x xe e
xe
+
2n
n
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λ − λ 2 2x y 4x 8y 5+ − − = 2 2x y 4x 8y 5+ − − = − λ λ
( )35,15− [ ]15, 35−
9 9
,8 2
9 9,
8 2
( )0,1 p
q∈ ∈
m n−
∈ ( )0,1 p
q∈
m n−
1−
θ 2π θ +
2 2 2 2 2 2b x a y a b+ =
θ 2π θ + 2 2 2 2 2 2b x a y a b+ =
2 2
2 2
1
2
x y
a b
2 2
2 2 2
x y
a b
2 2
2 2 4
x y
a b
2 2
2 2 8
x y
a b
( ) ( ) ( )
4 2
2 21 11 1 1
x C D E Ax B
x x x x x
+= + + + +
− +− + − −
(4 3cos ) 8r
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α
α
− −
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θ
1
3 θ
θ
1
3 θ
µP
mg
µmg
P µmgP
P
µmg
12
5
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1
2
1
2
1
2
14
3
2
3
2
2a t
bx
−
2a tbx−
2 3M LT− 2MT− 3 1ML T− 3LT−
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ˆ ˆ ˆ0.5i 0.8j ck + + ˆ ˆ ˆ0.5i 0.8j ck + +
0.89 0.11
200
m / s3
100
m / s3
50
m / s3
400
m / s3
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2
2
10 10 20 10 2 10 10
β α β α
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3000A
3000A
1 C− µ
1 C− µ
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Ω Ω Ω Ω
0λ 0 / 2λ
0 / 5λ 0λ 0 / 2λ 0 / 5λ
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2 CH CHC 3CH Cl ( )3 2CH CHCl 6 5 2C H CH Cl
22 2 3
Z n H OC H C H O X Y
+= + → →
,2504 2
Cu C CH O
°+ → ,400
34 2
M oO C CH O
°+ →
,1000
2 34 3
A l O C C H N H
°+ → 23
Z n H OC H C H O
+≡ + →
4 3 3
ClO CH CH COO OH 4 3 3OH ClO CH COO CH − − − −> > >
3 3 4CH COO CH OH ClO− − − −> > > 3 3 4CH OH CH COO ClO
− − − −> > >
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127.
2SiO
2I
3 3 2 NaOH CH COOH CH COONa H O+ → + 2 NaOH HCl NaCl H O+ → + 3 22 2 3KClO KCl O→ + 4 2 2 22 NH NO N H O→ +
. / 2 4 / 3 2 402 60
dil H SO HgSOCH MgI Ag I KOH A B C D E
H O C → → → →∆
133. ( ) ( ) ( )2 33
2 .2
s g gS O SO X K Cal+ → + ( ) ( ) ( )2 31
.2 2
g g gSO O SO Y K Cal+ → +
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( ) ( )3 z y xCo NH Cl Cl
( ) ( )3 z y xCo NH Cl Cl
( )
2 72 22r ,/ ,2 4 300
Ca OH K C O H H O H Cu
C C H X A B C D
∆°
++ → → → → →
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− → 2O
−
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α
α
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t
x
m
t
x
m
t
x
m
t
x
m
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20Sn, HCl NaNO , HCl z
6 5 2 6 60 5 CC H NO X Y C H .
2 2 7H , K Cr O 3 2H PO 4OH , KMnO
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KEY SHEET
MATHEMATICS
1) 1 2) 2 3) 4 4) 3 5) 4 6) 3 7) 4 8) 1 9) 3 10) 4
11) 1 12) 1 13) 2 14) 4 15) 2 16) 4 17) 1 18) 2 19) 4 20) 3
21) 3 22) 2 23) 4 24) 3 25) 3 26) 4 27) 4 28) 1 29) 4 30) 2
31) 1 32) 2 33) 4 34) 2 35) 2 36) 3 37) 4 38) 3 39) 2 40) 2
41) 4 42) 4 43) 2 44) 3 45) 2 46) 1 47) 3 48) 2 49) 4 50) 2
51) 1 52) 3 53) 3 54) 4 55) 2 56) 3 57) 2 58) 1 59) 3 60) 2
61) 4 62) 2 63) 3 64) 2 65) 4 66) 2 67) 3 68) 4 69) 2 70) 1
71) 3 72) 2 73) 4 74) 3 75) 2 76) 2 77) 2 78) 3 79) 2 80) 3
PHYSICS
81) 1 82) 1 83) 1 84) 3 85) 2 86) 2 87) 4 88) 4 89) 1 90) 1
91) 3 92) 1 93) 2 94) 2 95) 4 96) 2 97) 1 98) 1 99) 4 100) 4
101) 3 102) 2 103) 1 104) 1 105) 4 106) 4 107) 2 108) 3 109) 2 110) 4
111) 1 112) 3 113) 3 114) 1 115) 1 116) 1 117) 2 118) 2 119) 4 120) 1
CHEMISTRY
121) 1 122) 2 123) 4 124) 2 125) 2 126) 4 127) 4 128) 2 129) 2 130) 1
131) 2 132) 1 133) 1 134) 2 135) 3 136) 4 137) 2 138) 2 139) 1 140) 2
141) 2 142) 1 143) 4 144) 4 145) 3 146) 2 147) 3 148) 4 149) 1 150) 3
151) 3 152) 2 153) 2 154) 3 155) 2 156) 3 157) 1 158) 4 159) 2 160) 2
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Page No : 28
HINTS & SOLUTIONS
A
BCD6030
40 x
h h
tan3040 x
h
tan60 x
40 x h
3
3x h
3x 40 x x 20
2 2ax by a b
sec tan
2 2
2a b sec
x ae seca
2G ae sec ,0
1 2 2G.A G ae sec a (ae sec a)
2 2 2a (e sec 1)
k k : m : n : : k
2 2
: m : n k : k : k
1 2 1 2 1 2
2 2 2 2 2 21 1 1 2 2 2
a a b b c c 1cos
2a b c a b c
120
bc ac
cb a
1
1
a
b
bc
ca
1
1
6 2 3
3 .... ....2 2 3
x x x x x
3 1 213 3
x
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Page No : 29
x 3
3 x 27
30 x 1 26
2 2c k 5
2 ab 2 5
k 2 5 x 5y 2 5 0
4 6 ( 2 8 )C ,
2 2
C 2 3 , 1 4
4 17
5
1
2 3 7 11 4 7C ,
5 5
22 3 7 11 4 7
C ,5 5
1 2
10C C 7 2 7
5
2 2ax by 1
21 2a b y yy 0......................(2)
21 2ax b y yy x 0.....................(3)
22 1 1 xyy xy yy 0
2S 4 r log S log 4 2log r
S r 100 0 2 100
S r
0.03 100
2. 23
k 1
k 214 1 14 k 1 28 k 26,30
24 9 1
ho(gof ) h g h(1) 02 4
2 2 16 4 416 9x 0 x 0 x ,
9 3 3
3x 2 1 1 3x 2 1 1 3x 3 1 x 1
3
1
x ,13
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Page No : 30
2 2 2 2a b , a b
2 22 3 13
2
2
x 2x 9
y x 4x 3a
2(y 1)x (4y 2)x (3ay a) 0
x R
y R 216(1 a ) 4(4 3a )(1 a ) 0
2a a 0
(A B)(A B) A.A A.B B.A B.B
2 2 2 2B A AB BA B
.B B.A
2/3 2/3 x y cotv
x y
2 1n 1
3 3
2 2 1 x.( cos ec v)vx y( cos ec v)vy .cot v 3
21 cosv x vx y vy .sin v 3 sinv
1 sinv.cosv 3
1
2sinvcosv 6
sin2v
6
1 1 1 x 6
1 2 y 0
1 2 3 z 10
1 1 11 2 0
1 2 3
1(6 2 ) 1(3 ) 0 0
3
0
3
1
1 2 32 2 2, ,
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Page No : 31
1 2 3 2 0
1 2 3 2 2
8 88
( ) ( 1)
1 2 2 3 3 11 1 14
312
1
3 21 2 3 1 2 2 3 3 1 1 2 3 x x x 0
3 x 0 12x 8 0 3 x 12x 8 0
n r 1 4 2 1 3f f f
4 3 2 x 4x 2x 4x 2 0 3 24x 12x 4x 4 0
212x 24x 4 0
1 4 2 -4 -2
0 -1 -3 1 3-1
1
2
0 -1
-3
1-1
3 -1
-2
-3
3
1
0 -1-1
-1
0
21
0 -1-1
1 0
-4
4 2 y 4y 1 0
2
2 2
2tan x/2 tan x /2 1
1 tan x /2 1 tan x /2 5
2 2 212 tan xh 1 tan xh (1 tan xh) 5 tan xh 10 tan xh 5 1 tan xh5
2 26 tan xh 10 tan xh 4 0 3 tan xh 5 tan xh 1 0 tan xh 2, 1/3
22tan xh 4 3tan x or 1 tan xh 5 4
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Page No : 32
B C 180 tan A tanB tanC 3 tan A tanB 3
tanA 1,tanB 2,tanC 3
36 6 2 2 2 3 2sin cos k cos 1 (sin ) (cos ) k cos 1
2 2 2 2 2 2 21 3sin cos k cos 1 3sin cos k cos k 3sin
sin65 sin25 2sin75cos20
1cos65 cos25 2cos 45cos20
sin70 cos40 sin70 sin50 2sin60.cos10
3cos70 cos40 cos70 cos50 2cos60.cos20
3 32 2 2cos 33 cos 27 1cos 33 cos33cos27 cos 27 1 cos 33 sin27 (2cos33cos27)
cos33 cos27 2
1 1 1 31 cos 60.cos 6 cos 60 cos 63 1 .
2 2 2 4
32 1 3 3 1 2 21 1 1 1 1 1 1 1
9 10
1 1
C . C C . C C . C C C 2 9 2 6 19
C . C 9.10 90
1 1 2tan x(x 1) sin x x 1
2
1 1 2
22
1cos sin x x 1
2 x x 1
2
2 2
1 x x 1
(x x) 1
2(a 1)(a 1) 1
3 2a a a 1 0 2a(a a 1) 0
1 3
a 2
sin xs inh y cos , cos x cosh y sin
2 2 2 2 2 2cos sin sin x sin hy cos x cos hy
2 2 2 2 21 sin x sin hy cos hy sin x cos hy
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Page No : 33
2 21 sin x cos hy
2 21 1 cos x cos hy
2 2cos x cos hy 2
24 576 368k 2
12 2 3
1 2 3
1 1 1 s a s b s c: : : : 6 :12 :18 : 1: 2 : 3
r r r
B C
cos A cos B cos C 1 4 sin sin sin2 2 2
4R A B C r 1 sin sin sin 1
R 2 2 2 R
2 33.5 1 3.5.8 1
3S .................2! 9 3! 9
3/ 21 2
1 3. 3S 19 9
3/2 3/21 7 9
1 3S3 9 7
4 9 93S .
3 7 7
2 n 1S 1 2 3 ........n
2 nS 0 2 ..................n
n n
2
( 1) n.S
(1 ) 1
n 1
2 n 1 n(1 )S 1 ....... n
n nS 0
1 1
nn1(1 )S n
1
2z pz q 0
1 2 1 2z z p, z z q
2 2 21 2 1 2z z z z p 3q
21 2 1 2(z z ) 3z z
4 4 1 3
a cos i sin cos 60 i sin 60 i3 3 2 2
3n nnn
3n 3n 3n 3n1 3 1 a 1 1 1 ( 1)1 a i 1 a (cos x i sin x) cos i sin2 2 2 2 2 2 2
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Page No : 34
B
A
D C
5 5 3 1D , ,2
2 2
BD AB 1
DC AC 1
1 1 6 3 AD 1
4 4 4 2
A
BC
D a
c
a at b
r c ta
F 2i 3 j 4k
a AB ( 2 2 2)
c x i y j zk
B C 0
i j k . x i y j zk 0
k 2
1
(a b) c b c a 3
1(a.c).b (b.c)a b c a
3
1 1 b c cos b c cos
3 3
1 8 2 2sin 1
9 9 3
6 6 42 1n(E) C . C .5
6 6 4 5
2 1
6 4
C . C .5 5
6 2 6
21
54
C 2 1
C 52 26
2 2 2tan x sec x 2cos x 1 sin x 2cos x 1 sin x 2 2sin x 2sin x sin x 1 0
1sin x ,sin x 1
2
[0 2 ]
3, ,
6 6 2
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Page No : 35
k (k 1)a
p(x k)3
2
2 3a 1 ........ 1
3 3
2
1a 1 15
22
a 13
23 9 4
a 1 a 1 a 2 4 9
250 1
P500 2
12 12
0 1 01e . e .1P(x 0) e
0! 0!
10 10
10 k x 0
tan x sin x Lt 5
x
10 10
10 k x 0
sin x 1 cos x Lt . 5
x x
10
k x 10
1 cos x
Lt 1. 5 x
10 9
2 x 0
1 cos x 10 cos x sin x Lt 5 5 5(1).1 5 5 5
x 20
sin x
2 x
2
1 (sin x)Lt
cos x
sin x
x 2
(sin x) cos x 1 log(sinx)Lt
2cos x.sin x
sinx 1
x 2
(sin x) 1Lt 1 log(sin x)2 2
1 1
A 10.11 110
1
B15.17
1
C19.22
1
D21.25
2 2dy 1 1 x 1 x 1 x 1
dx 4 1 x (1 x) 2(1 x )
2 21 1
2(1 x ) 2(1 x )
2 2 2
4 4
1 x 1 x x
2(1 x ) 1 x
2x x y f e e
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Page No : 36
2x x x 1 2x 2x dy f e e e .f (e ).e .2dx
1
x 0
dy f 1 .1 1.f (1).2
dx
3 60 .25 5
2 cot x 1(sinx cos x)
cot x 1
2 cos x sin x (cos x sinx)cos x sin x
2 2cos x sin x cos 2x
502 cos 2x
2 3 A 5 6
I 0
(2 )(6 ) 15 0
2 8 3 0
2 8A 3I 0
(1,1)
dy 1
dx
dy 0
dx
b x
2
1
x 2
1
x , 2
2 2 x y
125 9
4ab.sin .cos 2absin2
22
4
2a cos , 2bsin
1 12(5). , 2(3).
2 2
5 2, 3 2
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Page No : 37
a
2a 3
3 3 3f ( ) 3 ( )
3a 3a(2a 3)
3 2
f a 6a 9a
2df 3a 12a 9da
23a 12a 9 0 29 4a 3 0
a 1
1 10
x 2x 2
1 10
y 2y 2
( , )
2 2 2 0
2 21 1 14x 4y 4y 0
2 2 x y y 0
2
2
1 sin 2x d(4x)
1 cos 2x 1
2
2
1 sin 2x d(4x)
2cos 2x
2 1sec 2xd(4x) ...d(4x)2
1dx dt
4
2
2 2 2
(t 1)tdt 2
(t 1) 3(t 1) 3t
2
4 2
(t 1)dt 2
t t 1
2 2
1 1dt dt
t t 1 t t 1
2 22 2
dt dt
1 3 1 3t t 2 2 2 2
1 12 2t 1 2t 1tan tan c3 3 3
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Page No : 38
12 x
tan c3 3(x 1)
1 4tan x x dx
4 1 31
2
x 1 x tan x dx
4 1 1 x 3
1
3 2 3
1 1 1tan x dx
3x 3 (1 x )x
1
3 2 3 2
1 1 A B C dx etan x dx
3x 3 x x x x 1
1
3 2 2
1 1 1 x tan x log c
3x 6x 3 1 x
2
0
cos 3x 1 sin x dx
2 cos x 1 sin x
22
0
3x cos sin x
2 dx sin2x sin x
22
0
3x 2cos sin x
2 dx 3x x
2 cos sin2 2
2
0
(cos 2x cos 2x)dx
2
20
0
sin2x sin x 1
2
20
x sin x I dx
1 cos x
2
0
sinx 2I dx
1 cos x
1
2
1
dt 2I 1 t
11
12I tan t
2
2I I2 4
2 2
1
x 2 x dx
4 4
2
2
1
1(x 2 x )dx
4
22 3
1
1 x x 2x
4 2 3
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Page No : 39
1 8 1 1 1 10 72 4 2
4 3 2 3 4 3 6
1 27 9
4 6 8
2
n
3 x a r 1
r
1 nLt
n r
1 n
1 2
3
0
x dx
1 x
1 2
3
0
1 3x dx
3 1 x
3 1
0
1log(1 x )
3
1 1log 2 log1 log 2
3 3
9 2 7 9 3 6
2 3C k .3 C k 3
99 9 3
2 3 92
C3 9 3 1 7C .3 C .k
k C 3 3
1 2 3
1 2 1 2x 2x 2x 3 1 x 13 3 3 3 3
1 8 8
.3 27 81
3 y x 2 x dy e e x e
dx
3 x 2 x dy e x e dx dx
3 x y x 1e e e c3
2x dy y e (1 x)
dx 1 x
dy py Q
dx
e1
dx pdx log (1 x)1 x 1
e e e1 x
1 1
cos sin 1 sin cos 2 x , y
3 3
1(3x 1) cos sin .................(1)
1(3y 2) sin cos .................(2)
2 2
1 13(x y ) 2x 4y 12 0
1 2cos 0(sin )...............(1)
3 2sin 0(cos )...............(2)
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Page No : 40
(2)tan 3
(1) 3
1 1ax by cPQ r
a cos b sin
1 1(x , y ) ( 3,5) tan 1 45
3 5 6 4PQ 2 2
1 1 2
2 2
2 2 x 3xy 2y (x y)........(1)(x 2y)..........(2)
x +
2 y x-y+22=0
x+y=0
2g ac g 2 2d 2
a(a b) a 1 c 42
8 42d 2 2
1(1 49)
7 1
, ,12 2
2 2
27 11 2 (3 1)2 2
25 25 664
4 4 2
2x 2x e e
r d
6 16 59 16
(10 ) 5
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Page No : 41
15, 35
( 35, 15)
dy dw
2y 18dx dt
d y d n
2d x d t
dx dx (2y)2 18
dt dt
9 9 y , x
2 8
9 9(x, y) ,
8 2
1 1 1 1 2 2 3 3 4, , , , , , , ,
2 3 . 1 4 . 1 5 3 5 4 5 5
cos sin 1 x y
a b
sin cos 1 x y
a b
2 2
2 2 2
x y
a b
4 2
2 2 2
x 2 2x 1 x 1
(x 1) (x 1) (x 1) (x 1)
3 3 3 3D 2E 1 2 1 1
2 4 2 2
8
4 3cosr
3 214
coxr
3
4 e
21mgh mv
2
21 mv
2
A
v
B
20 m
250 1mgh mv 100 2
21 mv mgh2
v 2gh 2 10 20 20 m/s
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Page No : 42
2
i10 1 R
32iR 10 1 10
10 1 R
R 19989
1B A 1 2 T 10
1S A 1 2 T 20
B S1 2 T 10 A 1 2 T 20
B S T 10 T 20
B S&
1 1 2 2P V P V
33atm w atm
4 4P P r P 2r
3 3
atm w atmP P 8P
w atmP 7P
h 70 m
45 60 1050 T
100 M S
40
3
1 300 10150 C
2 10 10
0 T 150 C
1 2 2
1 1 2
1 2 1 2
m m 2m v u u
m m m m
2 1m m 12
1m10
2m
2 21 2
2 2
m 2m v u u
m m
1 2u 2u 12 20 8 m/s
22 21/3 1/3 1/ 3 1
cos 2 1/3 1/3 2
21cm
1cm
3
1/ 3
1/31/3
0120
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Page No : 43
3
power P velocity
force mg
O M.I of ABaboutO 6
2
2M 21 M 31 612
A B
C
DE
F
2L,M
2L,M
2L,M
2L,M
2L,M
2L,M
O
22 2ML 3ML 6 20 ML
3
2 3R
2 3
2 2
1 1
T r
T r
22 T 64
25
2 T 40
1
2
2 2 2 21 1m x m A2 4
22 Ax
2
Ax
2
1
4
2 2 2 2 21 1m (A x ) m A
2 8
22 3Ax4
3Ax
2
ext e F
ext F mg
ext F 3mg /2
22
3mge 32 e ee mg 2
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Page No : 44
21 2
2
LT ML T
L
c .11
2 21 gh2Mgh M 4v v v 22 3
P
A
B
h
h H
1h
2 1 4gh1
mv 2 32h 2h/3
Mg g
2h 5h
h3 3
2u sin2R
g
100 110
10
2 04 sin 60 200
10 u m/sg 3
02T
P hdgR
P V C C R
V Q nC dT
V 60 3 C 5
V C 4
PC 4 R 4 2 6 cal/mol/k
PQ nC dt 5 6 10 300 cal
KA t
Q2
v
n2L
340L 1m2 425
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Page No : 45
2
11 2
sinrsin45 1 12; sin r
sin r 2 sin i 2
2
1sini
2
20
3/22 2
nir B
2 r x
ray e e
e e
e & e
2E 2 0.51 2.1Mev
E 0.54Mev
50 50
0.981 1 50 51
1 2
2
1 2
4i ia
i i
1 2
2
1 2
4i i 3
4i i
1 22
1
2
4 i /i 3 4x 3 x 34 4i x 1
1i
max
min
i3
i
1
1
1 A
1
1 .72
4 .54
1 5.33
d sin n
sin 1
d n
0
0
d 9000An 3
3000A
C
XT
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Page No : 46
2PK.E.
2m
1K.E. m
v
t 102
10 2 10 2 1t
v 20 2
2 20 1/ 22usin 1t
Eq g 210m
045
Eq 10 40
m
mE 30 30 N/C
q
2 1 26 i 10 i i 0 1
D
B
C
EF 1 0 v 2
6 v 1
1 0
2i
1i 1 1 210 2i 10 i i 0
1i 1.56A
2e at bt
2
2
a 50 b 508 a 5b 8
15 2a 200b 15a 100 b 100
a
850 b
0a 850
425 C2b 2
3L 5 10
R
90
3L 0.5 10R 90
R 9010 R 10
R
2
L 5 10 H 50 mH
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Page No : 47
0
hc hc
0
2
0
hc hc
0 0 0
c hc hc
/2
2e
0
1 hcme v
2
2e
0
2hc v
me
0
5
2max e
0 0 0
5hc hc 4hc 1K.E mev
2
21 2e e v v 4
1 6e e v 2v 2 10 m/s
2 33
S O SO ; H 2xK.Cal i2
2 2 3
1SO O SO ; H y .K.Cal ii
2
2 2S O SO ; H 2x y
H S
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1 1 1 1 x y z2s 2p 2p 2p
0
2.1A
RadialProbability
Distance
2mg of MgCl1000 100
95
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