Sec: SR.

IIT_*CO-SC(MODEL-A,B&C) GTM-12(N) Date: 10-01-24

Time: 3 HRS JEE-MAIN Max. Marks: 300
KEY SHEET
PHYSICS
1 2 2 1 3 4 4 3 5 1
6 4 7 2 8 1 9 4 10 3
11 3 12 1 13 3 14 1 15 1
16 3 17 2 18 4 19 4 20 1
21 8 22 0 23 25 24 13 25 1
26 180 27 1 28 5 29 1 30 0

CHEMISTRY
31 4 32 2 33 3 34 1 35 1
36 3 37 2 38 1 39 4 40 1
41 2 42 1 43 2 44 3 45 3
46 4 47 3 48 3 49 4 50 1
51 2 52 138 53 7 54 9 55 158
56 6 57 7 58 144 59 26 60 12

MATHEMATICS
61 B 62 B 63 A 64 A 65 B
66 B 67 A 68 A 69 B 70 C
71 C 72 D 73 D 74 D 75 B
76 D 77 A 78 A 79 B 80 D
81 5 82 7 83 0 84 2 85 2
86 6 87 5 88 3 89 4 90 0
Narayana IIT Academy 10-01-24_SR.IIT_*CO-SC(MODEL-A,B&C)_JEE-MAIN_GTM-12(N)_KEY&SOL
SOLUTIONS
PHYSICS
1. v cos  300   v0 cos 600
v0
v
3
v0
1
e 2 3 
v0 3 3
2
2. P  2 Pl  Pm
3. Conceptual
4. Applying conservation of energy
2
1 1 1  5v  9 MR 2
Mgh  Mv02  I 02  M  0   I 
2 2 2  4  16
3R
Mx 2  I  x 
4
n3
5. For conservative forces, dU   dW
U f  U i  Wi  f
Or Wi  f  U i  U f  q Vi  V f 
5  105  2  106  2a  0.1  0
2
 
Or a  1.25  10 V / m  1250 V / m 2
3 2

6. Magnetic moment M  NIA

3
  3
2
M  I0 2a  I0a2
4 2
7.

Mass of coal

P 1
8. S av   0 cE02
4 R 2
2
P
 E0 
2 R 2 0 c
3

2  3.14  100  8.85  1012  3  108
=1.34 V/m

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4
9. Mass of the sphere is given by M   y 3 
3
4 
G   y3  m
Gravitational force F   
3
2
y
F
a .
m
a0 g
10.  tan   a0 
g 3
11. v cos 370  20cos 530
3
v cos 37 0  20   v  15m / s
5
m
12.  
l r 2
 m 2r l
  
 m r l
0.003 2  0.005 0.06 4
     4%
0.3 0.5 6 100
D   2D 
13.  n  1 t  
d d
 
 n  1 t  5892 A
2
3
14.  2v0   e  v0  3e
2
Here, v0  gR
kT
15.    p  constant
2 D 2 p
16. Conceptual
17. Since ML = Pt
Pt
L
M
18. When one e is removed from neutral helium atom, it
For one e species we know
13.6Z 2
En  eV/atom
n2
For helium ion, Z=2 and for first orbit n=1
13.6 2
 E1   2  54.4eV
1
2

 Energy required to removed this e =+54.4eV

 total energy required =54.4+24.6=79eV
103
19. n
200  106  1.6  10 19
20. use wheatstone bridge condition
21. Wave velocity as a function of distance (x) from top is

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T
v  g 9  x

2h
g
h
dx  2h 
   dt  time takenby stone  
0 g 9  x  0  g 
 h  8m
22. At steady state 10 resistor will be short circuited
u2
23.  100
g
  450
u 2 sin 2 
 25m
2g
24. Heat is extracted from the source in path DA and AB is
3  PV  5  2 PV  13
Q  R  0 0   R  0 0   PV 0 0
2  R  2  R  2
  
25. B  B1  B2
  
B  0  B1   B2
0 I 0 0 I1
 0
2 d 2 R
I R
I1  0
d
1
s1   60 18 
26. 2
s2  40  18
27. Conceptual
20   3
t

28. I1  1  e 5104
   1.5 A
10   2
4

20V
L  5mH

I1

6
5

5
20V I2

C  0.1mF

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t
20 103
I2  e  1.0 A
10
From superposition I  I1  I 2  2.5 A
dl
29.  dR  k  l
R1  k 2 l

R2  k 2  2 l 
R1  R2
30. No current flow through capacitor, it remains un charged

CHEMISTRY
31. Hybradisation
7.8
32. S  10 4 mol / l  10 5 mol / l
78
K SP  4 S 3  4  10 15
log10 K SP  log 4  15  14.4
33. Friedal Crafts alkylation, rearrangement
34.
35. Conceptual
36. H  H
C   A
0.06 0.001
Ecell   log  0.06 log 2  102  0.06  0.3  2   0.06  2.3  138V
1 0.2

37. A  I 2 , B  IO3
38. Conceptual
39. Ncert Points
40. Conceptual
41. BCD are correct
42. H  120  350  380  610 Kj / mol
43. Conceptual
44. Conceptual
45. Sulphur
46. Conceptual
47. Conceptual
48. Conceptual
49. T f  K f mi  5  0.4  3  6
50. Conceptual
51. 2 r  n , 2  0.53  2  
OH
COOH

52.
53. Conceptual
54. Conceptual
55.

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OH OH

Br Br

S= =331 T= =173

Br Br
56. Conceptual
57. Conceptual
58. x  6, y  0, z  4, p  2
0.693 2.303  0.3010
59. t1/ 2    0.3010
k 2.303
t75%  2t1/2  0.6020
2.303 100
t99%   log 2
2.303 1
 2  0.6020  2.6020
60. DU=12

MATHS
 8 
61. 61   cis  
 11 
1
Re     2   3   4   5   
2
1          ....    0
2 3 4 10

 2 Re     2  ....   5   1
62. Use the theory of combined mean and combined variance formulae
 1 1 1 1 1
63.  1  3    3  3    3  3       1
 2  2 3  3 4 
64. x 3  6x 2  3px  2p  0
x 4  x 2  x3  2
  
 0 
2 4 4
1 x
x2 x3
65. f  x   4e 2  1  x  
2 3
 7 1 1
g '    
 6  f ' 1 5

66. g  t    2 cot 1  3 t 
2
 
g   t    2 tan 1  3 t     2 cot 1  3 x 
2 2
 odd
1
g '  t   2. 2t
.3 t   log 3
1 3
 decreasing

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dy  1  cos x  
67.     sin x , f    1
dx  y  2
dy  sin x.y
 
dx 1  cos x
1 x
 dy   tan dx
y 2
x
log y  2 log sec    c
2
0   log 2  c
x
 y  2 cos 2  
2
y  0  2
x y
68.  1
a b
1 4
 1
a b
b
Sab  b
b4
4
 1 b 
b4
ds 4
 1
 b  4
2
db
b  4  2, 2
b  6, 2
b6;a 3
A9
69.  a  b  .c  1
c  a b
1
a  2, b  3, c 
3
1
   2  3  sin  
3
1
 sin  
3 2
  a  b  .c    2  3  sin 2    1
6sin   3 2
1
sin  
2


4

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x 1 y z  2
70. Any point on the line   is B  t  1, t , 2t  2  , t  R
1 1 2
Also AB is perpendicular true the line where A is (1,2,-4)
That is 1 t    t  2   2   2t  2   0
 t  1
 B=  0,1, 4 
 AB  2
71. Standard problem
x 1 x2  2 x2  x
72. ax 2  bx 6  cx 5  ....  h  x 2  x x  1 x2  1
x2  2 x2  x x  1
1 2 0
 h  0 1 1  1  2  2   5
2 0 1
73. sin 4 x   k  2  sin 2 x   k  3  0
k  2  k  3 1
 sin 2 x   k  3  sin 2 x  1  0
 0  k 3 1
 3  k  2 
74. 112012  232014  32012  1
1 1
I dx
0 1
1  x2  x 
x
1
x 
 dx 
0 1 x  x 1 x
2 2 4
2
75. A) Area  2  ydy
0

B) Standard formula
 1
C) Area  ab  ab .
4 2
2
D) Area   1dx
1

77. S  1, 2,3,........,50

Ways  250  225
 225  225  1
78. 1, 2,3,........,50
6m  9n multiple of 5
6m  6
9n  9 / 1
ways  50  25
63
79. P E  1
127

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tan 1  x  sin x cos x
80. Lim
x 0 x 1  x 
x  0 , n  0  x  x
x  0 , n  1  x  x  1
81. V  ai  bj  ck ; a, b, c  1,1
ways 8 C3  6 4 C3  8  7  6  4  25
82. 0  a, b, c, d  
2 cos a  6cos b  7 cos c  9 cos d  0
2 cos a  6sin b  7 cos c  9sin d  0
 4  81  36cos  a  d   36  4  84 cos  b  c 
cos  a  d  21 7
 
cos  b  c  9 3
a3 a7 a13
83. a4 a6 a12 
a5 a4 a11
16
 1  x  x 2    a r x16  r
8

r 0
So,   0
n  2  2r  1 
84. Sn   tan 1  
 4  r 2  r 2  2r  1 
r 1
 
n 
2
 tan 1  
2 
n
   lim   cot Sn 1  cot Sn 
n 
x2

  cot S1  cot Sn 
I    x 2  1   x  1 e x  dx
2
85.
x 2
 1 e x  t
  x  1 e x dx  dx
2

1

 x 2  1 e x 
2
I c
2
1
 f  x   c
2

2
2A  f  0   1  1  2
86. Basic type
A  95 
87. P     
 B   25  9  5 
 1 2 
 23  1
88. P E   
 1 1  2  2
 2 3

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89. ways  D 4  D5
x  y
2
90. min
 x  2y   2
 x  y   2 1 y  y 
2 2

y
1 y  
2
3y y
 1, 1 
2 2
2
y , y2
3
y  2, x  2
2 2
y , x
3 3

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