24/01/2023
Morning
Corporate Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 | Ph.: 011-47623456
Answers & Solutions
for
Time : 3 hrs.
M.M. : 300
JEE (Main)-2023 (Online) Phase-1
(Physics, Chemistry and Mathematics)
IMPORTANT INSTRUCTIONS:
(1)
The test is of 3 hours duration.
(2)
The Test Booklet consists of 90 questions. The maximum marks are 300.
(3)
There are three parts in the question paper consisting of Physics, Chemistry and Mathematics
having 30 questions in each part of equal weightage. Each part (subject) has two sections.
(i)
Section-A: This section contains 20 multiple choice questions which have only one correct
answer. Each question carries 4 marks for correct answer and –1 mark for wrong answer.
(ii)
Section-B: This section contains 10 questions. In Section-B, attempt any five questions out of
10. The answer to each of the questions is a numerical value. Each question carries 4 marks for
correct answer and –1 mark for wrong answer. For Section-B, the answer should be rounded off
to the nearest integer.
-1-
JEE (Main)-2023 : Phase-1 (24-01-2023)-Morning
PHYSICS
3.
SECTION - A
Consider the following radioactive decay process
Multiple Choice Questions: This section contains 20
multiple choice questions. Each question has 4 choices
(1), (2), (3) and (4), out of which ONLY ONE is correct.
The mass number and the atomic number of A6 are
given by:
Choose the correct answer:
1. A travelling wave is described by the equation
(1) 210 and 80
(2) 210 and 82
(3) 210 and 84
(4) 211 and 80
Answer (1)
y(x, t) = [0.05 sin (8x – 4t)] m
Sol.
The velocity of the wave is : [all the quantities are in
SI unit]
(1) 8 ms–1
(2) 0.5 ms–1
(3) 4 ms–1
(4) 2 ms–1
2.
Coefficient of t
Coefficient of x
(3) 3 2 : 2
(4) 1: 2
(1) 8 N
(2) 19.6 N
(3) 9.8 N
(4) 4.9 N
Sol. Wearth = 18 N
mgearth = 18
A circular loop of radius r is carrying current I A. The
ratio of magnetic field at the center of circular loop
and at a distance r from the center of the loop on its
axis is:
(2) 1: 3 2
The weight of a body at the surface of earth is
18 N. The weight of the body at an altitude of
3200 km above the earth’s surface is (given, radius
of earth Re = 6400 km):
Answer (1)
4
= 0.5 ms−1
8
(1) 2 2 : 1
γ
Atomic number = 80
4.
Sol.  y(x, t) = [0.05sin(8x – 4t)] m
=
+
γ
α 210
β

→ 214
→ 81 A4 
→ 210
→ 210
83 A3 
80 A5 
80 A6
Mass number = 210
Answer (2)
Speed of wave =
β−
α 214
218
→ 82 A1 → 214
84 A 
83 A2
 R 
Also mg h = mg earth 

R +h
2
6400


= 18 

 6400 + 3200 
= 18 ×
5.
Answer (1)
2
4
= 8
9
Given below are two statements : one is labelled as
Assertion A and the other is labelled as Reason R.
Assertion A : Photodiodes are preferably operated
in reverse bias condition for light intensity
measurement.
Sol.
Reason R : The current in the forward bias is more
than the current in the reverse bias for a p – n
junction diode.
BP =
1
=
BP
2
µ0I
In the light of the above statements, choose the
correct answer from the options given below:
µ0Ir 2
µ0I
=
2
2 3/2
2(r + r )
25/2 r
(1) Both A and R are true and R is the correct
explanation of A
2r
BP
1
∴ =
BP
2
(2) A is false but R is true
(3) A is true but R is false
µ0I
2 2
2r
=
µ0I
1
(4) Both A and R are true but R is NOT the correct
explanation of A
25/2 r
Answer (4)
-2-
JEE (Main)-2023 : Phase-1 (24-01-2023)-Morning
Sol. Photodiodes are preferably operated in reverse
bias condition for light intensity measurement
because it increases the width of depletion layer,
therefore both are correct but not the correct
explanation.
6. Given below are two statements :
Statement I : If the Brewster's angle for the light
propagating from air to glass is θB, then the
Brewster's angle for the light propagating from
π
glass to air is − θB
2
Statement II : The Brewster's angle for the light
propagating from glass to air is tan–1 (µg) where µg
is the refractive index of glass.
In the light of the above statements, choose the
correct answer from the options given below:
(1) Both statement I and Statement II are true
(2) Both statement I and statement II are false
(3) Statement I is true but statement II is false
(4) Statement I is false but statement Il is true
Answer (3)
Sol. Case I :
i + r = 90° as transmitted is ⊥ to reflected.
tan i=
7.
µa
µ
π
⇒ i= tan−1 a =
− θB
µg
µg 2
A modulating signal is a square wave, as shown in
the figure.
If the carrier wave is given as c(t) = 2 sin (8πt) volts,
the modulation index is:
(1)
1
2
(2)
1
4
(3) 1
(4)
1
3
Answer (1)
Sol. Am = Amplitude of modulating wave = 1 volt
Ac = Amplitude of carrier wave = 2 volt
Modulation index =
8.
Transmitted is ⊥ to reflected.
i + r = 90°
Snell’s law
µasini = µgsinr
µg
tan i =
µa
 µg 
i = tan−1 
 = θB
 µa 
Case II :
As shown in the figure, a network of resistors is
connected to a battery of 24V with an internal
resistance of 3Ω. The currents through the resistors
R4 and R5 are I4 and I5 respectively. The values of
I4 and I5 are:
=
(1) I4
=
(2) I4
8
2
=
A and I5
A
5
5
24
6
=
A and I5
A
5
5
=
(3) I4
2
8
=
A and I5
A
5
5
=
(4) I4
6
24
=
A and I5
A
5
5
Answer (3)
-3-
Am 1 1
= =
Ac
2 2
JEE (Main)-2023 : Phase-1 (24-01-2023)-Morning
Sol. R=
eq
µ0i1i 2
F1 µ0i1
=
=
i2
Sol.
2πr
2πr
l
R4R5
R1R2
+ R3 +
+ R6 + r
R1 + R2
R5 + R4
r 
If i1 and i 2 both are doubled, if rf =  
2
= (1 + 2 + 4 + 2 + 3)
= 12 Ω
=
I
24
= 2A
12
 R5
I4 = 
 R4 + R5

5
2
× 2=
A
I =
25
5

Ff µ0 (2i1)(2i 2 ) 8µ0 i1i 2
=
=
r
l
2πr
2π  
2
Ff = 8F1
9.
10
cm is
π
placed perpendicular to a uniform magnetic field of
0.5 T. The magnetic field is decreased to zero in 0.5
s at a steady rate. The induced emf in the circular
loop at 0.25 s is:
11. A conducting circular loop of radius
R4I
20
8
I5=
=
× I=
A
R4 + R5 25
5
Given below are two statements :
Statement I : The temperature of a gas is –73°C.
When the gas is heated to 527°C, the root mean
square speed of the molecules is doubled.
Statement II : The product of pressure and volume
of an ideal gas will be equal to translational kinetic
energy of the molecules.
(1) emf = 1 mV
(2) emf = 5 mV
(3) emf = 100 mV
(4) emf = 10 mV
Answer (4)
Sol.
In the light of the above statements, choose the
correct answer from the options given below:
(1) Both statement I and statement II are true
(2) Statement I is false but statement Il is true
(3) Both statement I and statement II are false
(4) Statement I is true but statement II is false
Answer (4)
=
Sol. Ti 200K
v rms ∝ T
dφ
εind =
–
dt
Tf = 800K
Vi
=
Vf
Ti
=
Tf
200
=
800
1 1
=
4 2
 dB 
=
A
 dt 
Vf = 2Vi
 0.5   1 
= 
π
 0.5   10 π 
3

Translational K.E. =  PV 
2

= 1×
10. Two long straight wires P and Q carrying equal
current 10A each were kept parallel to each other
at 5 cm distance. Magnitude of magnetic force
experienced by 10 cm length of wire P is F1. If
distance between wires is halved and currents on
them are doubled, force F2 on 10 cm length of wire
P will be:
(1) 10F1
(3)
F1
10
1
V
100
= 0.01 V
= 10 mV
12. 1 g of a liquid is converted to vapour at 3 × 105 Pa
pressure. If 10% of the heat supplied is used for
increasing the volume by 1600 cm3 during this
phase change, then the increase in internal energy
in the process will be:
(2) 8F1
(4)
2
F1
8
Answer (2)
(1) 4800 J
(2) 432000 J
(3) 4.32 × 108 J
(4) 4320 J
Answer (4)
-4-
JEE (Main)-2023 : Phase-1 (24-01-2023)-Morning

Sol. ∆Q = ∆U + ∆W

15. In E and K represent electric field and propagation
10 ∆W = ∆W + ∆W
∆U = 9 ∆W
vectors of the EM waves in vacuum, then magnetic
field vector is given by:
1 

= 9 × (3 × 105 ) 1600 × 6 
10 

(1)
(ω – angular frequency):
= 4320 J
(3) K × E
13. If two charges q1 and q2 are separated with distance
‘d’ and placed in a medium of dielectric constant K.
What will be the equivalent distance between
charges in air for the same electrostatic force?
(1) 2d k
(2) d k
(3) 1.5d k
(4) k d
(2) ω ( E × K )
(4) ω ( K × E )
Answer (1)

Sol. E ⇒ electric field

K ⇒ propagation vector
 1  
=
B
(K × E )
ω
Answer (2)
16. As per given figure, a weightless pulley P is
attached on a double inclined frictionless surfaces.
The tension in the string (massless) will be (if g =
10 m/s2)
Sol.
dielectric constant = K
F
=
medium
1(
K ×E)
ω
qq
1
× 1 22
4π(Kε0 ) d
 Fair = Fmedium
q1q2
1 q1q2
1
=
2
4πε0 (dair )
4π(Kε0 ) d 2
∴
dair = K d
(1)
14. Given below are two statements:
(4
3 + 1) N
(3) 4 ( 3 + 1) N
Statement I: An elevator can go up or down with
uniform speed when its weight is balanced with the
tension of its cable.
(2)
(4
3 − 1) N
(4) 4 ( 3 − 1) N
Answer (3)
Sol.
Statement II: Force exerted by the floor of an
elevator on the foot of a person standing on it is
more than his/her weight when the elevator goes
down with increasing speed.
In the light of the above statements, choose the
correct answer from the options given below:
(1) Statement I is true but Statement II is false
(2) Both Statement I and Statement II are true
Let’s consider T tension in string and a acceleration
of blocks.
(3) Statement I is false but Statement II is true
FBD of 4kg
(4) Both statement I and Statement II are false
Answer (1)
Sol. Statement I is correct, because lift is moving with
zero acceleration.
Statement II is incorrect as force exerted will be
less than the weight.
4gsinθ – T = 4a
-5-
JEE (Main)-2023 : Phase-1 (24-01-2023)-Morning
4g 3
−T =
4a
2
Choose the correct answer from the options given
below:
…(1)
(1) A, B, D only
FBD of 1kg
(2) A, C, D only
(3) B only
(4) B, C only
Answer (3)
Sol. (A) From Einstein’s equation
T – gsin30° = a
T −
g
=
a
2
Kmax = eVs = hν – φ
…(2)
Form the stopping potential (Vs) depends on φ
& ν.
From (1) and (2)
=
T 4 (1 + 3 ) N
(B) Saturation current is proportional to intensity,
i.e., number of incident photons.
17. Match List I with List II
(C) Kmax only depends on nature of photon and φ.
LIST I
LIST II
A.
Planck's constant (h)
I.
[M1L2T–2]
B.
Stopping potential (Vs)
II.
[M1L1T–1]
C.
Work function (φ)
III.
[M1L2T–1]
D.
Momentum (p)
IV.
[M1L2T–3A–1]
(D) Einstein used particle behaviour of photon to
explain photon electric effect.
Only B is correct.
19. The maximum vertical height to which a man can
throw a ball is 136 m. The maximum horizontal
distance upto which he can throw the same ball is:
(1) 272 m
Choose the correct answer from the options given
below:
(2) 68 m
(3) 192 m
(1) A-III, B-IV, C-I, D-II
(4) 136 m
(2) A-I, B-III, C-IV, D-II
Answer (1)
(3) A-II, B-IV, C-III, D-I
Sol. For vertical throw,
(4) A-III, B-I, C-II, D-IV
h=
Answer (1)
Sol. [h] = [ML2T–1] → A-III
=
ν
[Vs] = [ML2T–3A–1] → B-IV
v2
2g
2gh
=
2g × 136
[φ] = [ML2T–2] → C-I
For max range, θ = 45°
[p] = [MLT–1] → D-II
Rmax =
18. From the photoelectric effect experiment, following
observations are made. Identify which of these are
correct.
ν2
g
…(1)
…(2)
From (1) and (2)
=
Rmax
A. The stopping potential depends only on the
work function of the metal.
ν 2 2g × 136
=
g
g
= 272 m
B. The saturation current increases as the
intensity of incident light increases.
20. A 100 m long wire having cross-sectional area
6.25 × 10–4 m2 and Young's modulus is 1010 Nm–2
is subjected to a load of 250 N, then the elongation
in the wire will be:
C. The maximum kinetic energy of a photo
electron depends on the intensity of the
incident light.
D. Photoelectric effect can be explained using
wave theory of light.
(1) 6.25 × 10–6 m
(2) 6.25 × 10–3 m
(3) 4 × 10–3 m
(4) 4 × 10–4 m
Answer (3)
-6-
JEE (Main)-2023 : Phase-1 (24-01-2023)-Morning
Sol. Both the springs are in parallel so net spring
constant is Knet = K1 + K2 = 40 N/m
Fl
Sol. ∆l =
AY
So T = 2π
m
Knet
2
40
= 2π
F= 250 N
π
=
l = 100 m
5
x=5
A = 6.25 × 10–4 m2
23. In the circuit shown in the figure, the ratio of the
quality factor and the band width is _______ s.
Y = 1010
∴ ∆l = 4 × 10–3 m
SECTION - B
Numerical Value Type Questions: This section
contains 10 questions. In Section B, attempt any five
questions out of 10. The answer to each question is a
NUMERICAL VALUE. For each question, enter the
correct numerical value (in decimal notation,
truncated/rounded-off to the second decimal place; e.g.
06.25, 07.00, –00.33, –00.30, 30.27, –27.30) using the
mouse and the on-screen virtual numeric keypad in the
place designated to enter the answer.
Answer (10)
Sol. Bandwidth ∆ω =
R
L
Quality factor Q =
21. A hole is drilled in a metal sheet. At 27°C, the
diameter of hole is 5 cm. When the sheet is heated
to 177°C, the change in the diameter of hole is d ×
10–3 cm. The value of d will be ______ if coefficient
of linear expansion of the metal is 1.6 × 10–5/°C.
1 L
R C
1 L
Q
R
C
So
=
R
∆ω
L
3
Answer (12)
=
Sol. ∆D = Dα∆t
L2
R2 C
= 5 × 1.6 × 10–5 (177 – 27)
3
= 0.012 cm
32
=
= 12 × 10–3 cm
10
so, d = 12
=
22.
2
1
−6 2
(27 × 10 )
3 3
100(3 3 × 10−3 )
= 10
A block of a mass 2 kg is attached with two identical
springs of spring constant 20 N/m each. The block
is placed on a frictionless surface and the ends of
the springs are attached to rigid supports (see
figure). When the mass is displaced from its
equilibrium position, it executes a simple harmonic
π
motion. The time period of oscillation is
in SI
x
unit. The value of x is ______.
24. As shown in the figure, a combination of a thin
plano concave lens and a thin plano convex lens is
used to image an object placed at infinity. The
radius of curvature of both the lenses is 30 cm and
refraction index of the material for both the lenses
is 1.75. Both the lenses are placed at distance of 40
cm from each other. Due to the combination, the
image of the object is formed at distance x = ____
cm, from concave lens.
Answer (5)
-7-
JEE (Main)-2023 : Phase-1 (24-01-2023)-Morning
26. A stream of a positively charged particles having

q
C
and velocity v 0= 3 × 107 iˆ m / s is
= 2 × 1011
m
kg
deflected by an electric field 1.8 ˆj kV/m. The electric
field exists in a region of 10 cm along x direction.
Due to the electric field, the deflection of the charge
particles in the y direction is ____ mm.
Answer (120)
Sol.
1
fconcave
Answer (2)
1 
0.75
1
=
(1.75 − 1)  −
=
−

30
 ∞ +30 
Sol.
fconcave = – 40 cm
1
 1 1  0.75
= (1.75 − 1) 
− =
fconvex
30
 30 ∞ 
fconvex = 40 cm
Fy =
Let the first image is formed at v1 so
qE y
m
ay = 2 × 1011 × 1800
1 1
1
1
− =
=
−
v1 ∞ fconcave
40
= 36 × 1013 m/s2
for second image
∴
1
1
1
−
=
x − 40 −80 40
⇒ x = 120 cm
25. A spherical body of mass 2 kg starting from rest
acquires a kinetic energy of 10000 J at the end of
5th second. The force acted on the body is _____
N.
Sol. Let the force be F so acceleration a =
F
m
1
1

× 36 × 1013 ×  × 10 −8 
2
3

=
2 × 10–3 m
=
2 mm
2
If the resistivity of the material is 2.4 × 10–8 Ωm. The
value of n is ______.
1 2 Ft 2
=
at
2
2m
Answer (2)
Sol.
F 2t 2
So work done W = F.S =
2m
From work energy Theorem
∆KE = W
F=
1 2
at
2
27. A hollow cylindrical conductor has length of 3.14 m,
while its inner and outer diameters are 4 mm and
8 mm respectively. The resistance of the conductor
is n × 10–3 Ω.
Answer (40)
=
W
y=
⇒y=
So displacement
=
S
10 × 10−2
0.1
1

=
=  × 10−8  sec.
v0
3 × 107  3

=
Time =
⇒ v1 = −40 cm
F 2t 2
= 10000
2m
10000 × 2 × 2
52
F = 40 N
-8-
JEE (Main)-2023 : Phase-1 (24-01-2023)-Morning
R
=
ρl 2.4 × 10−8 × 3.14
=
A
π(16 − 4) × 10−6
=
2.4
× 10−2
12
=
24
× 10−3
12
2 × 10−3 Ω
=
Answer (110)
value of n is 2.
Sol.
28. Assume that protons and neutrons have equal
masses. Mass of a nucleon is 1.6 × 10–27 kg and
radius of nucleus is 1.5 × 10–15 A1/3 m. The
approximate ratio of the nuclear density and water
density is n × 1013. The value of n is _____.
Answer (11)
Sol. Radius = 1.5 × 10−15 A1/3
Volume =
=
I Icm + Md 2
4π 3
r
3
Mass of nucleus = (1.6 × 10–27) A kg
Density of nucleus =
1.6 × 10
−27
=
K
×A
1

4
× π ×  1.5 × 10−15 A 3 


3


3
K=
30. Vectors
32
× 1017
9π
aiˆ + bjˆ + kˆ
and
2iˆ – 3 ˆj + 4kˆ
are
perpendicular to each other when 3a + 2b = 7, the
x
. The value of x is _______.
ratio of a to b is
2
32
× 1017
Density of nucleus 9π
=
Density of water
1000
Answer (1)
Sol. ( aiˆ + bjˆ + kˆ ) ⋅ ( 2iˆ – 3 jˆ + 4kˆ ) =
0
2a – 3b + 4 = 0
320
× 1013
9π
(2a – 3b = –4 ×) 7
also, (3a + 2b = 7 ×) 4
= 11.32 × 1013
adding (1) and (2)
value of n = 11
14a + 12a – 21b + 8b = 0
29. Solid sphere A is rotating about an axis PQ. If the
26a – 13b = 0
radius of the sphere is 5 cm then its radius of
gyration about PQ will be
2 2
× 5 + 102 cm
5
Value of x = 110 .
Density of water = 1000 kg/m3
=
2 2
R + d2
5
K = 110 cm
1.6 × 3 × 8 × 1018
=
4π × 27
=
2
MR 2 + Md 2
5
⇒ =
MK 2
a 13 1
= =
b 26 2
x cm. The value of x
is ______.
Value of x = 1 .
-9-
… (1)
… (2)
JEE (Main)-2023 : Phase-1 (24-01-2023)-Morning
CHEMISTRY
SECTION – A
(1) Both statement I and Statement II are true
Multiple Choice Questions: This section contains 20
multiple choice questions. Each question has 4 choices
(1), (2), (3) and (4), out of which ONLY ONE is correct.
(2) Both statement I and Statement II are false
(3) Statement I is true but Statement II is false
(4) Statement I is false but Statement II is true
Answer (3)
Choose the correct answer :
31. Given below are two statements:
Statement I : Noradrenaline is a neurotransmitter.
Sol. For colloidal particles value of colligative properties
is less as compared to true solutions at same
concentration as number of particles are less.
Statement II : Low level of noradrenaline is not the
cause of depression in human.
In the light of the above statements, choose the
correct answer from the options given below.
(1) Statement I is correct but Statement II is
incorrect
(2) Statement I is incorrect but Statement II is
correct
But fixed layer and diffused layer have opposite
charges.
34. In the depression of freezing point experiment
(3) Both statement I and Statement II are incorrect
A. Vapour pressure of the solution is less than that
of pure solvent
(4) Both statement I and Statement II are correct
B. Vapour pressure of the solution is more than
that of pure solvent
Answer (1)
Sol. •
•
Noradrenaline is a neurotransmitter.
C. Only solute molecules solidify at the freezing
point
Low level of noradrenaline is a cause for
depression in human.
D. Only solvent molecules solidify at the freezing
point
32. Reaction of BeO with ammonia and hydrogen
fluoride gives A which on thermal decomposition
gives BeF2 and NH4F. What is ‘A’?
(1) H3NBeF3
(3) (NH4)Be2F5
Choose the most appropriate answer from the
options given below:
(2) (NH4)BeF3
(1) A and D only
(2) B and C only
(4) (NH4)2BeF4
(3) A only
(4) A and C only
Answer (4)
Answer (1)
Sol.
Sol. A and D are correct
as (vp)solution < (vp)solvent

BeO  NH3  HF (NH4 )2 BeF4 
 BeF2  NH4F
and only solvent particles undergoes solidification.
(A)
Compound A is (NH4)2BeF4
33. Statement I : For colloidal particles, the values of
colligative properties are of small order as
compared to values shown by true solutions at
same concentration.
35. ‘A’ and ‘B’ formed in the following set of reactions
are:
Statement II : For colloidal particles, the potential
difference between the fixed layer and the diffused
layer of same charges is called the electrokinetic
potential or zeta potential.
In the light of the above statements, choose the
correct answer from the options given below
- 10 -
JEE (Main)-2023 : Phase-1 (24-01-2023)-Morning
(1)
(2)
(1)
(2)
(3)
(4)
Answer (1)
Sol.
(3)
(4)

38. It is observed that characteristic X-ray spectra of
elements show regularity. When frequency to the
power “n” i.e. n of X-rays emitted is plotted against
atomic number “Z”, following graph is obtained.
Answer (1)
Sol.
36. The primary and secondary valencies of cobalt
(1) 1
(2) 2
(3) 3
(4)
1
2
respectively in [Co(NH3)5CI]Cl2 are :
Answer (4)
(1) 3 and 5
(2) 2 and 6
(3) 2 and 8
(4) 3 and 6
 1
1
Sol. h = E = 13.6 × Z2  2  2 
 n1 n2 
Answer (4)
   Z2 
Sol. [Co(NH3)5CI]Cl2
 (  )1/ 2  Z
Oxidation no. = primary valencies = 3
Co-ordination no. = secondary valencies = 6
37. ‘R’ formed in the following sequence of reactions is
1

n  2 


39. The magnetic moment of a transition metal
compound has been calculated to be 3.87 B.M. The
metal ion is
(1) Ti2+
(2) Mn2+
(3) Cr2+
(4) V2+
Answer (4)
- 11 -
JEE (Main)-2023 : Phase-1 (24-01-2023)-Morning
Sol.  = 3.87
Answer (1)
n=3
HO
2
Sol. CH3 – Cl 
 CH3OH
V23 = 4s23d3
 slow process  .... r1 
HO
 I being a Good 
 leaving group 


NaI
 I being a Good 
 Nucleophile 


2
CH3 – Cl 
CH3 – I 
CH3 – OH ...(r2 )
–
–
V2+ = 4s03d3 (n = 3)
40. Which of the following is true about freons?
r2 >> r1
(1) These are chemicals causing skin cancer
(2) These are radicals of chlorine and chlorine
monoxide
(3) All radicals are called freons
as I– is a good nucleophile as well as good leaving
group.
43. Match List I with List II
LIST I
(4) These are chlorofluorocarbon compounds
Answer (4)
Sol. Freons are chlorofluorocarbons
41. Increasing order of stability of the resonance
structures is:
LIST II
A.
Chlorophyll
I.
Na2CO3
B.
Soda ash
II.
CaSO4
C.
Dentistry,
Ornamental work
III.
Mg2+
D.
Used in
washing
IV.
Ca(OH)2
white
Choose the correct answer from the options given
below:
(1) A-II, B-I, C-III, D-IV
(2) A-III, B-IV, C-I, D-II
(3) A-II, B-III, C-IV, D-I
(4) A-III, B-I, C-II, D-IV
Answer (4)
Choose the correct answer from the options given
below:
Sol. Chlorophyll contains Mg2+ Ions (A – III)
Soda ash is Na2Co3
(B – I)
(1) C, D, A, B
(2) D, C, B, A
Dentistry; ornamental work – CaSO4 (C – II)
(3) D, C, A, B
(4) C, D, B, A
Used in white washing – Ca(OH)2
Answer (Not given in the options)
44. In the following given reaction, ‘A’ is
Sol. Correct stabilising order is
C<A<B<D
(This question should be given bonus)
42. Assertion A: Hydrolysis of an alkyl chloride is a
slow reaction but in the presence of NaI, the rate of
the hydrolysis increases.
Reason R: I– is a good nucleophile as well as a
good leaving group.
In the light of the above statements, choose the
correct answer from the options given below.
(1)
(2)
(3)
(4)
(1) Both A and R are true and R is the correct
explanation of A
(2) A is false but R is true
(3) A is true but R is false
(4) Both A and R are true but R is NOT the correct
explanation of A
- 12 -
(D – IV)
JEE (Main)-2023 : Phase-1 (24-01-2023)-Morning
Answer (4)
Sol.
(1)
(2)
(3)
45. Match List I with List II
LIST I
LIST II
A.
Reverberatory
furnace
I.
Pig Iron
B.
Electrolytic cell
II.
Aluminum
C.
Blast furnace
III.
Silicon
D.
Zone
furnace
IV.
Copper
refining
(4)
Answer (1)
Sol.
Choose the correct answer from the options given
below:
(1) A-I, B-IV, C-II, D-III
(2) A-I, B-III, C-II, D-IV
(3) A-IV, B-II, C-I, D-III
(4) A-III, B-IV, C-I, D-II
Answer (3)
Sol. (A) Reverberatory furnance is used for extraction
of copper.
(B) Electrolytic cell is used for obtaining highly
reactive metals like aluminium.
(C) Blast furnace is used for extraction of Iron.
(D) Zone refining furnace is used for silicon.
46. Compound (X) undergoes following sequence of
reactions to give the Lactone (Y).
47. Which of the phosphorus oxoacid can create silver
mirror from AgNO3 solution?
(1) H4P2O5
(2) (HPO3)n
(3) H4P2O7
(4) H4P2O6
Compound (X) is
Answer (1)
- 13 -
JEE (Main)-2023 : Phase-1 (24-01-2023)-Morning
Sol. H4P2O5 can act as a reducing agent due to (P – H)
bond.
50. Order of covalent bond :
A. KF > KI; LiF > KF
B. KF < KI; LiF > KF
C. SnCl4 > SnCl2; CuCl > NaCl
D. LiF > KF; CuCl < NaCl
48. Decreasing order of the hydrogen bonding in
following forms of water is correctly represented by
E. KF < KI; CuCl > NaCl
Choose the correct answer from the options given
below:
A. Liquid water
B. Ice
C. Impure water
Choose the correct answer from the options given
below:
(1) A > B > C
(1) B, C only
(2) C, E only
(3) B, C, E only
(4) A, B only
Answer (3)
Sol. B is correct
C is correct
(2) B > A > C
E is correct
(4) C > B > A
Sol. Extent of hydrogen bonding :
Ice > liquid water > impure water

In ice, 4 molecules of H2O are connected to
H2O molecule.

Impure water will have less hydrogen bonding.
49. An ammoniacal metal salt solution gives a brilliant
red precipitate on addition of dimethylglyoxime. The
metal ion is
(2)
Cu2+
(3) Fe2+
(4) Co2+
Answer (1)
2
KF < KI; CuCl > NaCl
SECTION - B
Answer (2)
(1)
SnCl4  SnCl2 ; CuCl  NaCl
4
(3) A = B > C
Ni2+
KF < KI; LiF > KF
Numerical Value Type Questions: This section
contains 10 questions. In Section B, attempt any five
questions out of 10. The answer to each question is a
NUMERICAL VALUE. For each question, enter the
correct numerical value (in decimal notation,
truncated/rounded-off to the second decimal place; e.g.
06.25, 07.00, –00.33, –00.30, 30.27, –27.30) using the
mouse and the on-screen virtual numeric keypad in the
place designated to enter the answer.
51. The d-electronic configuration of [CoCl4]2– in
tetrahedral crystal field is em t 2n . Sum of “m” and
“number of unpaired electrons” is _______.
Answer (7)
Sol. [CoCl4]2–
Co2+ = 4s03d7
Sol. Ni2+ forms cherry red ppt with dmg
t2g
m=4
eg
=
eg
4t
2g
3
Number of unpaired electrons = 3
52. The dissociation constant of acetic acid is x × 10 –5.
When 25 mL of 0.2 M CH3COONa solution is mixed
with 25 mL of 0.02 M CH3COOH solution, the pH of
the resultant solution is found to be equal to 5. The
value of x is _______.
Answer (10)
- 14 -
JEE (Main)-2023 : Phase-1 (24-01-2023)-Morning
 25  0.2 
Sol. pH = pKa + log 

 25  0.02 
(C)
5 = pKa + log10
pKa = 4  Ka = 10–4 = 10 × 10–5
x = 10
(D) ln k  ln A 
53. Uracil is a base present in RNA with the following
structure. % of N in uracil is ________.
Ea
RT
Slope of ln k vs
1
 E 
is  a 
T
 R 
55. At 298 K, a 1 litre solution containing 10 mmol of
Cr2O72– and 100 mmol of Cr3+ shows a pH of 3.0.
Given : Cr2O72–  Cr3+; E° = 1.330V
Given :
Molar mass
N = 14 g mol–1
and
O = 16 g mol–1
2.303 RT
= 0.059 V
F
The potential for the half cell reaction is x × 10–3 V.
C = 12 g mol–1
The value of x is _______.
H = 1 g mol–1
Answer (25)
Answer (917)
Sol. Uracil is C4H4N2O2
Sol. 6e  14H  Cr2O72  2Cr 3  7H2O
10–3 m
14  2
 100
% by mass of N =
112
Ecell  1.330 –
= 25%
54. The number of correct statement/s from the
following is __________.
 1.330 –
A. Larger the activation energy, smaller is the
value of the rate constant.
D. A plot of In k vs
equal to –
0.059
 42 
6
= 0.917 = 917 ×10–3
x = 917
56. 5 g of NaOH was dissolved in deionized water to
prepare a 450 mL stock solution. What volume (in
mL) of this solution would be required to prepare
500 mL of 0.1 M solution?________
1
is a straight line with slope
T
Given : Molar Mass of Na, O and H is 23, 16 and
Ea
.
R
1 g mol–1 respectively
Answer (180)
Answer (3)
Sol. (A) k  Ae
0.059
102
log
6
(102 )(10–42 )
= 1.330 – 0.413
B. The higher is the activation energy, higher is
the value of the temperature coefficient.
C. At lower temperatures, increase in temperature
causes more change in the value of k than at
higher temperature.
10–1m
10 –2 m
E
 a
RT
 Ea 
(B) ln k  ln A 
k 
Ea
RT
Sol. Molarity of solution =
5 (1000)
(40) (450)
 M  V  500  .1
5
1000

 V  500  .1
450 450
1 dk E a
·

k dT RT 2

E a  temp. coefficient 
V  180 mL
- 15 -
JEE (Main)-2023 : Phase-1 (24-01-2023)-Morning
57. For independent processes at 300 K
B : Fe0.93O
Process
H/kJ mol–1
S/J K–1
A
–25
–80
B
–22
40
C
25
–50
D
22
3
Fex2 Fe(93
 x)O100
2x + 3(93 – x) = 200
x = 79
20
% of Fe3+ =
14
 100  15%
93
= Fe0.93O is metal deficient compound
Answer (2)
A, B, C, D are correct.
C is non  spon taneous as ( H  0, S  0)
59. If wavelength of the first line of the Paschen series
D is Non - spontaneous
of hydrogen atom is 720 nm, then the wavelength
of the second line of this series is______ nm.
For D,
(Nearest integer)
G  H – TS
Answer (492)
= 22,000 – (300) (20)
Sol.
= (22,000 – 6000) > 0
Non-spontaneous as (G > 0)
The
equivalent
weight
of
Fe0.93O
1
1 1 
 R –

720
 9 16 
R
58. When Fe0.93O is heated in presence of oxygen, it
converts to Fe2O3. The number of correct
statement/s from the following is______
A.
79
 100  85%
93
% of Fe2+ =
The number of non-spontaneous processes from
the following is______
Sol.
Fe93O100
9  16
720  7
1
9  16  1 1 


–
 ' 720  7  9 25 
is
Molecular weight
0.79
' = 492.18 nm
B. The number of moles of Fe2+ and Fe3+ in 1 mole
of Fe0.93O is 0.79 and 0.14 respectively.
 '  492 nm (nearest integer)
60. Number of moles of AgCl formed in the following
reaction is_____
C. Fe0.93O is metal deficient with lattice comprising
of cubic closed packed arrangement of O2–
ions.
D. The % composition of Fe2+ and Fe3+ in Fe0.93O
is 85% and 15% respectively.
Answer (4)
Sol. A : Fe0.93O  Fe2 O3
200 

nF   3 –
 .93
93 

Answer (2)
Sol.
= .79
eq. wt 
mw
.79
Circled Cl will get precipitated
- 16 -
JEE (Main)-2023 : Phase-1 (24-01-2023)-Morning
MATHEMATICS
v v w  =  v 2 + u  v
SECTION - A
Multiple Choice Questions: This section contains 20
Substituting values we have
multiple choice questions. Each question has 4 choices
6 + 3 = 0   = −
(1), (2), (3) and (4), out of which ONLY ONE is correct.
1
2
 Equation (i) becomes
Choose the correct answer :
61. Let N denote the number that turns up when a fair
die is rolled. If the probability that the system of
equations
v w = u −
0 = u w −
2x + Ny + 2z = 2
3x + 3y + Nz = 3
 u w =
(1) 18
(2) 20
(3) 21
(4) 19
...(ii)
Taking dot with w of (ii) we get
x+y+z=1
k
has unique solution is , then the sum of value of
6
k and all possible values of N is
v
2
v w
2
2
=1
2
(as v  w = 2 given)
63. The distance of the point (–1, 9, –16) from the plane
2x + 3y – z = 5 measured parallel to the line
x +4 2−y z −3
is
=
=
3
4
12
Answer (2)
(1) 20 2
(2) 13 2
Sol. For unique solution   0
(3) 26
(4) 31
1
1
1
Answer (3)
i.e. 2 N 2  0
3 3 N
Sol.
x +4 y −2 z−3
=
=
3
−4
12
 (N2 – 6) – (2N – 6) + (6 – 3N)  0
 N2 – 5N + 6  0
 N  2 and N  3
 Probability of not getting 2 or 3 in a throw of dice =
As given
2
3
2 k
=
k =4
3 6
Equation of line AP
 Required value = 1 + 4 + 5 + 6 + 4 = 20
u = iˆ − ˆj − 2kˆ, v = 2iˆ + ˆj − kˆ, v  w = 2
62. Let
and
v  w = u + v . Then u  w is equal to
(1) 2
(2) 1
3
2
(4) −
(3)
Point A(3 – 1, –4 + 9, 12 – 16) lies on 2x + 3y –
z = 5,  6 – 2 – 12 + 27 – 12 + 16 = 5   = 2
2
3
 Point A(5, 1, 8)
 AP2 = 62 + 82 + 242 = 4(9 + 16 + 144) = 4 × 169
Answer (2)
Sol. Given v  w = v + u
x + 1 y − 9 z + 16
=
=
3
−4
12
AP = 26
...(i)
Taking dot with v we get
Option (3) is correct.
- 17 -
JEE (Main)-2023 : Phase-1 (24-01-2023)-Morning
 A2(B – I) – I(B – I) = I
64. Let y = y(x) be the solution of the differential
 1
equation x3 dy + (xy – 1)dx = 0, x > 0, y   = 3 − e.
2
 (A2 – I)(B – I) = I
 A2 – I is the inverse matrix of B – I and vice
versa.
Then y(1) is equal to
(1) e
(2) 2 – e
So, (B – I)(A2 – I) = I
(3) 3
(4) 1
 BA2 – B – A2 + I = I
Answer (4)
 A2 + B = BA2
Sol. x3dy + xy dx – dx = 0

dy 1 − xy
=
dx
x3

dy
y
1
+
=
dx x 2 x 3

dx
−
1
x
I.F. = e

ye
x2
=e
=




1
−
ye x
ye
ye
−
1
x
−
1
x
 Option (1) is correct.
1
x
−
1
x
66. Let PQR be a triangle. The points A, B and C are
on the sides QR, RP and PQ respectively such that
Area ( PQR )
QA RB PC 1
=
=
= . Then
is equal
AR BP CQ 2
Area ( ABC )
to
dx
x3
For RHS put −
So, by (i) and (ii)
A2B = BA2
−
e
...(ii)
5
2
(2) 4
(3) 3
(4) 2
(1)
1
dx
=t 
= dt
x
x2
Answer (3)
= −  tet dt
Sol.
= − tet − et  + c


=
e
−
x
1
x
+e
−
1
x
+c
...(i)
 1
 y  = 3−e
2
 (3 – e)e–2 = 2e–2 + e–2 + c
1
 c=−
e
By PSY formula
ABC ( PC  QA  RB ) + (CQ  AR  BP )
=
PQR
PQ  QR  RP
...(ii)
For y(1) put x = 1, c = –e–1 in equation (i) we get
=
ye–1 = e–1 + e–1 – e–1
 y=1
65. If A and B are two non-zero n × n matrics such that
A2 + B = A2B, then
(1)
A2B
(3)
A2
=
BA2
= I or B = I
(2)
A2B
8 +1
1
=
333 3
2
2
67. For three positive integers p, q, r , x pq = y qr = z p r
and r = pq + 1 such that 3, 3 logy x, 3logz y, 7 logx
=I
z are in A.P with common difference
(4) AB = I
q is equal to
Answer (1)
(1) 12
(2) 2
 A2 + B – I = A2B – I
(3) –6
(4) 6
 A2B – A2 – B + I = I
Answer (2)
Sol. Given : A2 + B = A2B
...(i)
- 18 -
1
. Then r–p–
2
JEE (Main)-2023 : Phase-1 (24-01-2023)-Morning
Sol.
2
2
Sol. x pq = y qr = z p r
7
9
3logy x = ,3logz y = 4,7logx z =
2
2
7

7
y6

4
9
x = y 6 , y = z 3 , z = x 14
pq 2
3 2
p r
= y qr = y 4
7 2
3
pq = qr = p2r
6
4
 7pq = 6r, 4q = 3p2
r = pq + 1
r =
6r
+1  r = 7
7
pq = 6
Required area =
 3 p2 
p
=6
 4 


2
2
68. The relation
R = ( a, b ) : gcd ( a, b ) = 1,2a  b, a, b 

is
= 9 square units
(1) Reflexive but not symmetric
70.
(2) Neither symmetric nor transitive
(3) Symmetric but not transitive
1
1
 1
2
2
2

lim 1sin t + 2 sin t + ... + n sin t
t →0 

(1) n2
(4) Transitive but not reflexive
Answer (3)





sin2 t
is equal to
(2) n2 + n
n ( n + 1)
(3)
Sol. gcd (a, a) = a so (a, a) so  R  not reflexive
(4) n
2
Answer (4)
If gcd (a, b) = 1  gcd (b, a) = 1


cosec 2 t
 1
2
2

+ ... + 1
Sol. l = lim n   cosec t +  

t →0   n 
n


 (b, a)  R  Symmetric
If gcd (a, b) = 1 and gcd (b, c) = 1
 gcd (a, c) = 1
l=n
 R is not transitive
cosec 2t
r 
 lim  
t →0  n 
69. The area enclosed by the curves y2 + 4x = 4 and y
– 2x = 2 is
Answer (4)
4
y −2
 dy
2 
1 
y 3 
= 8 y − y 2 −

4 
3 
−4
r–p–q=7–5=2
25
(3)
3

−
8 − 2y − y 2
dy
−4
4
=
p = 2, q = 3
23
(1)
3
 4 − y2
2
−4 
22
(2)
3
= 0,  1  r  n
22
71. The value of

22
Cr 23Cr is
r =0
(4) 9
(1)
44C
(3)
44C
Answer (4)
- 19 -
22
23
(2)
45C
24
(4)
45C
23
sin2 t
JEE (Main)-2023 : Phase-1 (24-01-2023)-Morning
Sol. (1 + x )22 =
73. The equation x2 – 4x + [x] + 3 = x[x], where [x]
denotes the greatest integer function, has
22
C0 + 22C1x + 22C2 x 2 + ......
+ 22C21x 21 + 22C22 x 22
(1) Exactly two solutions in (–, )
...(i)
(2) No solution
( x + 1)23 = 23C0 x 23 + 23C1x 22 + 23C2 x 21 + ......
+
23
2
C21x +
23
C22 x +
(3) A unique solution in (–, 1)
23
C23 …(ii)
Multiplying (i) & (ii) and comparing coefficients of x23
on both sides
(4) A unique solution in (–, )
Answer (4)
Sol. x2 – 4x + [x] + 3 = x[x]
(x – 1)(x – 3) = (x – 1)[x]
45
C23 = 22C0 · 23C0 + 22C1· 23C1 + 22C2 · 23C2 + ......
+
22
C22 ·
(x – 1)(x – 3 – [x]) = 0
23
x = 1 or x – 3 – [x] = 0
C22
{x} = 3
22

x=
22
Cr · 23Cr = 45C23
 a unique solution in (–, )
r =0
72. Let a tangent to the curve y2 = 24x meet the curve
xy = 2 at the points A and B. Then the mid points of
such line segments AB lie on a parabola with the
74. The distance of a point (7, –3, –4) from the plane
passing through the points (2, –3, 1), (–1, 1, –2) and
(3, –4, 2) is
(1) 4 2
(1) Directrix 4x = 3
(2) 4
3
(2) Length of latus rectum
2
(3) 5 2
(4) 5
(3) Length of latus rectum 2
Answer (3)
(4) Directrix 4x = –3
Sol. A(2, –3, 1), B(–1, 1, –2), C(3, –4, 2)
Answer (1)
AB = −3iˆ + 4 ˆj − 3kˆ
Sol. y2 = 24x, xy = 2
iˆ
Let the equation of tangent to y2 = 24x is ty = x + 6t2
ty = x + 6t2 meet the curve xy = 2 at points A and B.
Let mid-point of AB is P(h, k).
ty =
2
+ 6t 2
y
t·
n = −3
1
 h=
–3t2,
−1
1
Distance of point (7, –3, –4) from the plane x – z –
1 = 0 is 5 2
equivalent to
2
(1) ((~ P )  Q )  (~ Q )
(2) (~ Q )  P
 y2 = –3x
(3) (~ P )  Q
 Length of L.R. = 3
Equation of directrix is x =
−3 = iˆ − kˆ
75. The compound statement
(~ (P  Q ))  ((~ P )  Q )  ((~ P )  (~ Q )) is
k = 3t
 h  k
 
= 
 –3   3 
4
Equation of plane is x – z – 1 = 0
x1 + x2 = −6t 2
 Mid-point P is (–3t2, 3t)
kˆ
Let equation of plane is x – z +  = 0 passes through
point A(2, –3, 1)   = –1
2
= x + 6t 2
x
ty 2 − 6t 2 y − 2 = 0 x 2 + 6t 2 x − 2t = 0
y1 + y 2 = 6t
jˆ
AC = iˆ − jˆ + kˆ
3
4
(4) ((~ P )  Q )  ((Q )  P )
Answer (4)
- 20 -
JEE (Main)-2023 : Phase-1 (24-01-2023)-Morning
Sol. (~ (P  Q ))  ((~ P )  Q )  ((~ P )  (~ Q ))
(1) f is continuous but not differentiable
(~ P  ~ Q )  (~ P  Q )  (~ P  ~ Q )
(2) f and f' both are continuous
 (~ P  ~ Q  ~ P )  (~ P  ~ Q  Q )  (~ P  ~ Q )
(3) f' is continuous but not differentiable
(4) f is continuous but f' is not continuous
 (~ P  ~ Q )  (T )  (~ P  ~ Q )
Answer (4)
 (~ P  ~ Q )  (~ P  ~ Q )
1
 2
 x sin , x  0
Sol. f ( x ) = 
x
 0
, x =0
 ~ (~ P  ~ Q )  (~ P  ~ Q )
 (P  Q )  (~ P  ~ Q )  (~ P  Q )  (• Q  P )
lim f ( x ) = lim f ( x ) = f (0) = 0
76. Let  be the sample space and A   be an event.
x →0+
Given below are two statements:
x →0 –
 f is continuous at x = 0
(S1) : If P(A) = 0, then A = 
(S2) : If P(A) = 1, then A = 
Now, R.H.D = lim
h→0
Then
(1) Both (S1) and (S2) are false
at x = 0
(2) Only (S1) is true
and LHD
(3) Only (S2) is true
= lim
(4) Both (S1) and (S2) are true
77. Let  be a root of the equation (a – c)x2 + (b – a) x
+ (c – b) = 0 where a, b, c are distinct real numbers

 1

1 1 is singular. Then,
b c 

(1) 12
(2) 6
(3) 9
(4) 3
Sol.
=
3 ( a − c )( b − a )( c − b )
( a − c )( b − a )( c − b )
=
h
1
h =0
 1
−h 2 sin  
h =0
=
−h

 1
 1
2 x sin   − cos   , x  0
f (x) = 
x
x

0,
x =0

'
lim f ' ( x ) is oscillatory
x →0+
 f is continuous but f’ is not at x = 0
79.
Answer (4)
( a − c )3 + ( b − a )3 + ( c − a )3
( a − c )( c − b )( b − a )
−h
h
h2 sin
 RHD = LHD  f is differentiable at x = 0
Sol. Both statements are correct
( a − c )2
( b − a )2
( c − b )2
+
+
is
( b − a )( c − b ) ( a − c )( c − b ) ( a − c )( b − a )
f ( −h ) − f ( 0 )
h →0
Answer (4)
2

such that the matrix  1
a

the value of
f (h ) − f (0)
 8+4 3 
 1+ 3 
 is equal to:
tan−1 
+ sec −1 

3+ 3 
 6+3 3 




(1)

3
(2)

6
(3)

2
(4)

4
Answer (1)
 1+ 3 
 1  
Sol. tan−1 
= tan−1 
=
 3 + 3 
 3 6


=3
and sec
 2
 1
 x sin   , x  0
78. Let f ( x ) = 
x

0,
x =0


Then at x = 0
- 21 -
(
(
)
)
 4 2+ 3 


−1  2 

 = sec  3  = 6


 3 2+ 3 


−1 
 8+4 3    
 1+ 3 
= + =
tan−1 
+ sec −1 
 3 + 3 
 6+3 3  6 6 3




JEE (Main)-2023 : Phase-1 (24-01-2023)-Morning
80. Let p, q 
(1 −
3i
)
and
200

= 2199 ( p + iq ) , i = −1 , then p + q + q2

(2) x2 – 4x – 1 = 0
(3) x2 + 4x – 1 = 0
(4) x2 + 4x + 1 = 0
(
Sol. Given 1 − 3i
)
199
=2

r 2023Cr x r
( 2023 )  x 2022 (1 + x )2021 + (1 + x )2022 
=
( p + iq ) …(1)

5 
 5 
L.H.S = 2200 cos   + i sin 
3
 3 

2023

r =0
r 22023Cr x r −1
Put x = 1
200
 2023 2022  22021 + 22022  =


1000
1000 

= 2200 cos
+ i sin
3
3 


2023

r =0
 1
3 
= 2200  − −
i
 2 2 
2023

r =0
r 2 2023Cr
r 2 2023Cr = 2023  22022 (1012 )
  = 1012
82. Let C be the largest circle centred at (2, 0) and
So, by (1)

2023
( 2023 ) x (1 + x )2022 = 
r =0
Answer (1)
200
Cr rx r −1
2023
r =0
and p – q + q2 are roots of the equation.
(1) x2 – 4x + 1 = 0
2023
( 2023 )(1 + x )2022 = 
x2 y 2
+
= 1.
36 16
P = – 1, q = − 3
inscribed in the ellipse
p + q + q 2 = −1 − 3 + 3 = 2 − 3 = 
If (1, ) lies on C, then 102 is equal to _________
Answer (118.00)
and p − q + q 2 = −1 + 3 + 3 = 2 + 3 = 
Sol.
 quadratic equation whose roots are  and 
x2 y 2
+
=1
36 16
r2 = (x – 2)2 + y2
x2 – 4x + 1 = 0
Solving simultaneously
Option (1) is correct.
–5x2 + 36x + (9r2 – 180) = 0
SECTION - B
D=0
Numerical Value Type Questions: This section
r2 =
contains 10 questions. In Section B, attempt any five
128
10
questions out of 10. The answer to each question is a
Distance between (1, ) and (2, 0) should be r
NUMERICAL VALUE. For each question, enter the
1 + 2 =
correct
numerical
value
(in
decimal
notation,
2 =
truncated/rounded-off to the second decimal place; e.g.
06.25, 07.00, –00.33, –00.30, 30.27, –27.30) using the
mouse and the on-screen virtual numeric keypad in the
place designated to enter the answer.
2023
81. Suppose

r =0
r 2 2023Cr = 2023    22022. Then
the value of  is ___________
118
10
= 118.00
83. The number of 9 digit numbers, that can be formed
using all the digits of the number 123412341 so that
the even digits occupy only even places, is
___________
Answer (60)
Sol. Given number 123412341

Answer (1012)
2023
Sol. 
128
10
(1 + x )2023 = 
r =0

Total number =
2023
Cr x r
- 22 -


4!
5!

= 6  10 = 60
2!2! 3!2!
JEE (Main)-2023 : Phase-1 (24-01-2023)-Morning
84. Let a tangent to the curve 9x2 + 16y2 = 144 intersect
the coordinate axes at the points A and B. Then, the
minimum length of the line segment AB is
____________
1
2
3
0
1
2
=  ( x – 1)( x – 2)dx –  ( x – 1)( x – 2)dx +  ( x – 1)( x – 2)dx
1
2
 x3 3x2

 x3 3x2

=
–
+ 2x  – 
–
+ 2x 
2
2
 3
 0  3
1
Answer (07)
Sol. Given curve : 9x2 + 16y2 = 144
3
 x3 3x 2

+
–
+ 2x 
2
 3
 2
x2 y 2
+
=1
16 9

Let P(4cos, 3sin) be any point on it.
1 3
  8
 1 3

=  – + 2 –  – 6 + 4  –  – + 2  
3
2
3
3
2

 
 

Now tangent at P
x cos  y sin 
+
=1
4
3

27
 8

+  9 –
+ 6 –  – 6 + 4
2
3





 A  (4sec, 0) B  (0, 3cosec)
=
AB = 16 sec 2  + 9cosec 2 
= 16 + 9 + 16 tan2  + 9 cot 2 
11
6
12 I =
ABmin = 25 + 2  12
11
 12
6
= 22
85. A boy needs to select five courses from 12 available
courses, out of which 5 courses are language
courses. If he can choose at most two language
courses, then the number of ways he can choose
five courses is _____________
87. The 4th term of GP is 500 and its common ratio is
1
, m  . Let Sn denote the sum of the first n terms
m
1
of this GP. If S6 > S5 + 1 and S7  S6 + , then the
2
number of possible values of m is
Answer (546)
Answer (12)
Sol. Case 1 If no language course is selected.
Sol. T4 = 500
=7
= 7 C5 
ar3 = 500  a =
Case 2 If one language course is selected.
C4  5 C1 
S6 > S5 + 1
Case 3 If two language course is selected.
a(1– r 6 ) a(1– r 5 )
–
1
1– r
1– r
7
C3  5C2 
Total = 7C5 + 7C4  5C1 + 7C3  5C2
ar5 > 1
= 21 + 175 + 350
Now, r =
= 546
2
1
500
and a =
m
r3
 m 2  500
3
x
r3
Now,
7
86. The value of 12
500
– 3 x + 2 dx is _______.

0
Answer (22)
(
m  0  m  0, 10 5
S7  S6 +
3
Sol. 12 x 2 – 3 x + 2 dx
)
1
2
a(1– r 6 ) a(1– r 6 ) 1

+
1– r
1– r
2
0
3
Let I =  ( x – 2)( x – 1)dx
ar 6 
0
- 23 -
1
2
...(i)
JEE (Main)-2023 : Phase-1 (24-01-2023)-Morning

r =
1
m
3

x
2
and
let
the
equation
E
be
– 2 x +  – 3 = 0 . Then the largest element in
the set S = {x + : x is an integer solution of E} is
________.
1
1000
 m  (10,  )
Answer (05)
...(ii)
Sol. D  0  4 – 4  – 3  0
Possible values of m is {11, 12, .........22}
–3 1
 mN
–1   – 3  1
Total 12 values
24
88. The shortest distance between the lines
x – 2 y +1 z – 6
x – 6 1– y z + 8
and
is
=
=
=
=
3
3
2
2
2
0
equal to
x =
2 4– 4 –3
2
= 1  1–  – 3
Answer (14)
xlargest = 1 + 1 = 2, when  = 3
Sol. a1 = 2iˆ – jˆ + 6kˆ
Largest element of S = 2 + 3 = 5
a2 = 6iˆ + ˆj – 8kˆ

2
a = 3iˆ + 2 jˆ + 2kˆ
90. The value of
8
(cos x )2023
dx

 (sin x )2023 + (cos x )2023
is
0
b = 3iˆ – 2 jˆ + 0kˆ
S.D =

89. Let
1
500
and a =
m
r5
________.
a2 – a1

b 
a
Answer (02)
ab
4
2
a2 – a1 a b  = 3 2


3 –2
Sol. I =
–14

2
8
(cos x )2023
dx
  (sin x )2023 + (cos x )2023
2
0
I=
= 4× (4) – 2 (–6) – 14 (–12)

2
8
(sin x )2023
dx
  (sin x )2023 + (cos x )2023
jˆ
kˆ
ab = 3
2
2 = 4iˆ + 6 jˆ – 12kˆ
...(ii)
0
= 16 + 12 + 168 = 196
iˆ
...(i)
0



3 –2 0
(i) + (ii)  2I =
a  b = 16 + 36 + 144 = 196 = 14

2
8
8 
1dx =  = 4


 2
0
196
S.D =
= 14
14
I=2
❑ ❑
- 24 -
❑
b


f
(
x
)
dx
=
f
(
a
+
b
–
x
)
dx



a
a
b