JEE Main 2023 April 10 Shift 2 Paper & Solutions | Mock Tests for JEE Main and Advanced 2025 PDF Download

 Page 1


   
 
 
Answers & Solutions 
for 
JEE (Main)-2023 (Online) Phase-2 
(Mathematics, Physics and Chemistry) 
 
 
  
10/04/2023 
Evening 
Time : 3 hrs. M.M. : 300 
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 Mathematics, Physics and Chemistry 
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. 
Page 2


   
 
 
Answers & Solutions 
for 
JEE (Main)-2023 (Online) Phase-2 
(Mathematics, Physics and Chemistry) 
 
 
  
10/04/2023 
Evening 
Time : 3 hrs. M.M. : 300 
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 Mathematics, Physics and Chemistry 
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. 
 
   
  
MATHEMATICS 
SECTION - A 
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. 
Choose the correct answer: 
1. Let ? be the mean and ? be the standard deviation 
of the distribution 
x
i
 0 1 2 3 4 5 
f
i
 k + 2 2k 
k
2 
– 1 k
2 
– 1 k
2 
+ 1 
k – 3 
 where 62.
i
f =
?
 If [x] denotes the greatest integer 
 ? x, then [?
2
 + ?
2
] is equal to 
 (1) 9 (2) 8 
 (3) 7 (4) 6 
Answer (2) 
Sol. 
x
i
 0 1 2 3 4 5 
 f
i
 k + 2 2k 
k
2 
– 1 k
2 
– 1 k
2 
+ 1 
k – 3 
 
2 2 2 2
2 2 – 2 3 – 3 4 4 5 – 15 9 7 – 16
62 62
k k k k k k k + + + + + +
? = =
 
 and 62
i
f =
?
? 3k
2
 + 4k – 2 = 62 
    ? 3k
2
 + 4k – 64 = 0 
    ? (3k
 
+ 16) (k – 4) = 0 
    4 k ?= 
 ? 
156
62
?= 
 
2 2 2 2
2
1 (2 ) 4( – 1) 9( – 1) 16( 1) 25( – 3) 156
–
62 62
k k k k k ? + + + + + ??
?=
??
??
 
  
2
2
29 27 – 72
–
62
kk +
=? 
  
500
62
= 
 ? 
22
500
62
? + ? = ? [?
2
 + ?
2
] = 8 
2. Let the image of the point P(1, 2, 6) in the plane 
passing through the points A(1, 2, 0) and B(1, 4, 1) 
C(0, 5, 1) be Q(?, ?, ?). Then (?
2
 + ?
2
 + ?
2
) is equal 
to 
 (1) 65 (2) 62 
 (3) 76 (4) 70 
Answer (1) 
Sol. Plane passes through the points 
 A(1, 2, 0), B(1, 4, 1) and C(0, 5, 1) 
 Normal vector 
ˆˆ
ˆ ˆ ˆ
0 2 1 (–1) – (1) 2
–1 3 1
i j k
i j k = = + 
   = <–1, –1, 2> 
 ? –1(x – 1) –1 (y – 2) + 2z = 0 
 ? x + y – 2z = 3 …(i) 
 Now image of P(1, 2, 6) in (i) 
 
– 1 – 2 – 6 –2(–9 – 3)
4
1 1 –2 6
x y z
= = = = 
 ? x = 5, y = 6, z = –2 
 ? ? = 5, ? = 6, ? = –2 
 ? ?
2
 + ?
2
 + ?
2
 = 65 
3. Let the number (22)
2022
 + (2022)
22
 leave the 
remainder ? when divided by 3 and ? when divided 
by 7. Then (?
2
 + ?
2
) is equal to 
 (1) 20 (2) 13 
 (3) 5 (4) 10 
Answer (3) 
Sol. For ? : 
2022 22
divisible by 3
(21 1) (2022) ++ 
 = 3K
1
 + 1 
 For ? : (21 + 1)
2022
 + (2023 – 1)
22
 
 = 7? + 1 + 7? + 1 
 = 7K2 + 2 
 So, ? = 1, ? = 2 
 ?
2
 + ?
2
 = 5 
Page 3


   
 
 
Answers & Solutions 
for 
JEE (Main)-2023 (Online) Phase-2 
(Mathematics, Physics and Chemistry) 
 
 
  
10/04/2023 
Evening 
Time : 3 hrs. M.M. : 300 
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 Mathematics, Physics and Chemistry 
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. 
 
   
  
MATHEMATICS 
SECTION - A 
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. 
Choose the correct answer: 
1. Let ? be the mean and ? be the standard deviation 
of the distribution 
x
i
 0 1 2 3 4 5 
f
i
 k + 2 2k 
k
2 
– 1 k
2 
– 1 k
2 
+ 1 
k – 3 
 where 62.
i
f =
?
 If [x] denotes the greatest integer 
 ? x, then [?
2
 + ?
2
] is equal to 
 (1) 9 (2) 8 
 (3) 7 (4) 6 
Answer (2) 
Sol. 
x
i
 0 1 2 3 4 5 
 f
i
 k + 2 2k 
k
2 
– 1 k
2 
– 1 k
2 
+ 1 
k – 3 
 
2 2 2 2
2 2 – 2 3 – 3 4 4 5 – 15 9 7 – 16
62 62
k k k k k k k + + + + + +
? = =
 
 and 62
i
f =
?
? 3k
2
 + 4k – 2 = 62 
    ? 3k
2
 + 4k – 64 = 0 
    ? (3k
 
+ 16) (k – 4) = 0 
    4 k ?= 
 ? 
156
62
?= 
 
2 2 2 2
2
1 (2 ) 4( – 1) 9( – 1) 16( 1) 25( – 3) 156
–
62 62
k k k k k ? + + + + + ??
?=
??
??
 
  
2
2
29 27 – 72
–
62
kk +
=? 
  
500
62
= 
 ? 
22
500
62
? + ? = ? [?
2
 + ?
2
] = 8 
2. Let the image of the point P(1, 2, 6) in the plane 
passing through the points A(1, 2, 0) and B(1, 4, 1) 
C(0, 5, 1) be Q(?, ?, ?). Then (?
2
 + ?
2
 + ?
2
) is equal 
to 
 (1) 65 (2) 62 
 (3) 76 (4) 70 
Answer (1) 
Sol. Plane passes through the points 
 A(1, 2, 0), B(1, 4, 1) and C(0, 5, 1) 
 Normal vector 
ˆˆ
ˆ ˆ ˆ
0 2 1 (–1) – (1) 2
–1 3 1
i j k
i j k = = + 
   = <–1, –1, 2> 
 ? –1(x – 1) –1 (y – 2) + 2z = 0 
 ? x + y – 2z = 3 …(i) 
 Now image of P(1, 2, 6) in (i) 
 
– 1 – 2 – 6 –2(–9 – 3)
4
1 1 –2 6
x y z
= = = = 
 ? x = 5, y = 6, z = –2 
 ? ? = 5, ? = 6, ? = –2 
 ? ?
2
 + ?
2
 + ?
2
 = 65 
3. Let the number (22)
2022
 + (2022)
22
 leave the 
remainder ? when divided by 3 and ? when divided 
by 7. Then (?
2
 + ?
2
) is equal to 
 (1) 20 (2) 13 
 (3) 5 (4) 10 
Answer (3) 
Sol. For ? : 
2022 22
divisible by 3
(21 1) (2022) ++ 
 = 3K
1
 + 1 
 For ? : (21 + 1)
2022
 + (2023 – 1)
22
 
 = 7? + 1 + 7? + 1 
 = 7K2 + 2 
 So, ? = 1, ? = 2 
 ?
2
 + ?
2
 = 5 
 
   
  
4. Eight persons are to be transported from city A to 
city B in three cars of different makes. If each car 
can accommodate at most three persons, then the 
number of ways, in which they can be transported, 
is 
 (1) 1120 (2) 3360 
 (3) 1680 (4) 560 
Answer (2) 
Sol. Total ways = 
8
C
3
 × 
5
C
3
 × 
2
C
2
 × 3! 
   = 3360 
5. Let 
ˆ ˆ ˆ ˆ ˆ ˆ
2 7 , 3 5 a i j k b i k = + - = + and 
ˆ ˆ ˆ
– 2 . c i j k =+ 
Let d be a vector which is perpendicular to both 
 and ab , and  · 12. cd = Then 
( ) ( )
ˆ ˆ ˆ
· i j k c d - + - ? 
is equal to 
 (1) 24 (2) 44 
 (3) 42 (4) 48 
Answer (2) 
Sol. () d a b = ? ? 
 
ˆ ˆ ˆ
2 7 1
3 0 5
i j k
ab ? = - 
 = 
ˆ ˆ ˆ
(35) 13 21 i j k -- 
 
ˆ ˆ ˆ
(35 13 21 ) d i j k = ? - - 
 (35 13 42) 12 cd ? = ? + - = 
 ? ? = 2 
 
ˆ ˆ ˆ
2 1 1 2
35 13 21
i j k
cd ? = ? -
--
 
 = 
ˆ ˆ ˆ
2(47 91 22 ) i j k ++ 
 
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
( ) ( ) 2( ) (47 91 22 ) 44 i j k c d i j k i j k - + - ? ? = - + - + + = 
6. Let f be a continuous function satisfying 
( )
( )
2
23
0
4
, 0.
3
t
f x x dx t t + = ? ?
?
 Then 
2
4
??
?
??
??
??
f is 
equal to 
 (1) 
2
2
1
16
??
?
?-
??
??
??
 (2) 
3
1
16
??
?
-? +
??
??
??
 
 (3) 
3
1
16
??
?
?-
??
??
??
 (4) 
2
2
1
16
??
?
-? +
??
??
??
 
Answer (3) 
Sol. 
( ) ( )
2 4 2
24 f t t t t += 
 
( )
24
2 f t t t += 
 
( )
24
2 f x x x = - + 
 Let x
2
 = u 
 ( )
2
2 f u u u = - + 
 ( )
2
2 f x x x = - + 
 
24
2
2
42
4
f
??
? ? ?
= - + ?
??
??
??
 
   
4
16
-?
= + ? 
   
3
1
16
??
?
= ? -
??
??
??
 
7. For , , , , ? ? ? ? ? if 
22
log
xx
e
xe
x
ex
??
? ? ? ?
?? +
? ? ? ?
??
? ? ? ?
??
?
11
,
??
? ? ? ?
= - +
? ? ? ?
??
? ? ? ?
xx
xe
dx C
ex
 where 
0
1
!
?
=
=
?
n
e
n
 and 
C is constant of integration, then ? + 2? + 3? – 4? is 
equal to 
 (1) 1 (2) 4 
 (3) –4 (4) –8 
Answer (2) 
Sol. 
2
Let 
x
x
t
e
??
=
??
??
 
 ( ) 2 ln 1 ln x x t -= 
 ( )
11
2 ln 1 2 x x dx dt
xt
??
- + ? =
??
??
 
 
1
ln 
2
x dx dt
t
= 
 
2
1 1 1 1
1
22
I t dt dt
tt
t
? ? ? ?
= + ? = +
? ? ? ?
? ? ? ?
??
 
  
11
2
tC
t
??
= - +
??
??
 
  
22
11
22
xx
xe
C
ex
? ? ? ?
= - +
? ? ? ?
? ? ? ?
 
 ? ? = 2, ? = 2, ? = 2, ? = 2 
 ? 2 3 4 2 4 6 8 4 ? + ? + ? - ? = + + - = 
Page 4


   
 
 
Answers & Solutions 
for 
JEE (Main)-2023 (Online) Phase-2 
(Mathematics, Physics and Chemistry) 
 
 
  
10/04/2023 
Evening 
Time : 3 hrs. M.M. : 300 
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 Mathematics, Physics and Chemistry 
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. 
 
   
  
MATHEMATICS 
SECTION - A 
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. 
Choose the correct answer: 
1. Let ? be the mean and ? be the standard deviation 
of the distribution 
x
i
 0 1 2 3 4 5 
f
i
 k + 2 2k 
k
2 
– 1 k
2 
– 1 k
2 
+ 1 
k – 3 
 where 62.
i
f =
?
 If [x] denotes the greatest integer 
 ? x, then [?
2
 + ?
2
] is equal to 
 (1) 9 (2) 8 
 (3) 7 (4) 6 
Answer (2) 
Sol. 
x
i
 0 1 2 3 4 5 
 f
i
 k + 2 2k 
k
2 
– 1 k
2 
– 1 k
2 
+ 1 
k – 3 
 
2 2 2 2
2 2 – 2 3 – 3 4 4 5 – 15 9 7 – 16
62 62
k k k k k k k + + + + + +
? = =
 
 and 62
i
f =
?
? 3k
2
 + 4k – 2 = 62 
    ? 3k
2
 + 4k – 64 = 0 
    ? (3k
 
+ 16) (k – 4) = 0 
    4 k ?= 
 ? 
156
62
?= 
 
2 2 2 2
2
1 (2 ) 4( – 1) 9( – 1) 16( 1) 25( – 3) 156
–
62 62
k k k k k ? + + + + + ??
?=
??
??
 
  
2
2
29 27 – 72
–
62
kk +
=? 
  
500
62
= 
 ? 
22
500
62
? + ? = ? [?
2
 + ?
2
] = 8 
2. Let the image of the point P(1, 2, 6) in the plane 
passing through the points A(1, 2, 0) and B(1, 4, 1) 
C(0, 5, 1) be Q(?, ?, ?). Then (?
2
 + ?
2
 + ?
2
) is equal 
to 
 (1) 65 (2) 62 
 (3) 76 (4) 70 
Answer (1) 
Sol. Plane passes through the points 
 A(1, 2, 0), B(1, 4, 1) and C(0, 5, 1) 
 Normal vector 
ˆˆ
ˆ ˆ ˆ
0 2 1 (–1) – (1) 2
–1 3 1
i j k
i j k = = + 
   = <–1, –1, 2> 
 ? –1(x – 1) –1 (y – 2) + 2z = 0 
 ? x + y – 2z = 3 …(i) 
 Now image of P(1, 2, 6) in (i) 
 
– 1 – 2 – 6 –2(–9 – 3)
4
1 1 –2 6
x y z
= = = = 
 ? x = 5, y = 6, z = –2 
 ? ? = 5, ? = 6, ? = –2 
 ? ?
2
 + ?
2
 + ?
2
 = 65 
3. Let the number (22)
2022
 + (2022)
22
 leave the 
remainder ? when divided by 3 and ? when divided 
by 7. Then (?
2
 + ?
2
) is equal to 
 (1) 20 (2) 13 
 (3) 5 (4) 10 
Answer (3) 
Sol. For ? : 
2022 22
divisible by 3
(21 1) (2022) ++ 
 = 3K
1
 + 1 
 For ? : (21 + 1)
2022
 + (2023 – 1)
22
 
 = 7? + 1 + 7? + 1 
 = 7K2 + 2 
 So, ? = 1, ? = 2 
 ?
2
 + ?
2
 = 5 
 
   
  
4. Eight persons are to be transported from city A to 
city B in three cars of different makes. If each car 
can accommodate at most three persons, then the 
number of ways, in which they can be transported, 
is 
 (1) 1120 (2) 3360 
 (3) 1680 (4) 560 
Answer (2) 
Sol. Total ways = 
8
C
3
 × 
5
C
3
 × 
2
C
2
 × 3! 
   = 3360 
5. Let 
ˆ ˆ ˆ ˆ ˆ ˆ
2 7 , 3 5 a i j k b i k = + - = + and 
ˆ ˆ ˆ
– 2 . c i j k =+ 
Let d be a vector which is perpendicular to both 
 and ab , and  · 12. cd = Then 
( ) ( )
ˆ ˆ ˆ
· i j k c d - + - ? 
is equal to 
 (1) 24 (2) 44 
 (3) 42 (4) 48 
Answer (2) 
Sol. () d a b = ? ? 
 
ˆ ˆ ˆ
2 7 1
3 0 5
i j k
ab ? = - 
 = 
ˆ ˆ ˆ
(35) 13 21 i j k -- 
 
ˆ ˆ ˆ
(35 13 21 ) d i j k = ? - - 
 (35 13 42) 12 cd ? = ? + - = 
 ? ? = 2 
 
ˆ ˆ ˆ
2 1 1 2
35 13 21
i j k
cd ? = ? -
--
 
 = 
ˆ ˆ ˆ
2(47 91 22 ) i j k ++ 
 
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
( ) ( ) 2( ) (47 91 22 ) 44 i j k c d i j k i j k - + - ? ? = - + - + + = 
6. Let f be a continuous function satisfying 
( )
( )
2
23
0
4
, 0.
3
t
f x x dx t t + = ? ?
?
 Then 
2
4
??
?
??
??
??
f is 
equal to 
 (1) 
2
2
1
16
??
?
?-
??
??
??
 (2) 
3
1
16
??
?
-? +
??
??
??
 
 (3) 
3
1
16
??
?
?-
??
??
??
 (4) 
2
2
1
16
??
?
-? +
??
??
??
 
Answer (3) 
Sol. 
( ) ( )
2 4 2
24 f t t t t += 
 
( )
24
2 f t t t += 
 
( )
24
2 f x x x = - + 
 Let x
2
 = u 
 ( )
2
2 f u u u = - + 
 ( )
2
2 f x x x = - + 
 
24
2
2
42
4
f
??
? ? ?
= - + ?
??
??
??
 
   
4
16
-?
= + ? 
   
3
1
16
??
?
= ? -
??
??
??
 
7. For , , , , ? ? ? ? ? if 
22
log
xx
e
xe
x
ex
??
? ? ? ?
?? +
? ? ? ?
??
? ? ? ?
??
?
11
,
??
? ? ? ?
= - +
? ? ? ?
??
? ? ? ?
xx
xe
dx C
ex
 where 
0
1
!
?
=
=
?
n
e
n
 and 
C is constant of integration, then ? + 2? + 3? – 4? is 
equal to 
 (1) 1 (2) 4 
 (3) –4 (4) –8 
Answer (2) 
Sol. 
2
Let 
x
x
t
e
??
=
??
??
 
 ( ) 2 ln 1 ln x x t -= 
 ( )
11
2 ln 1 2 x x dx dt
xt
??
- + ? =
??
??
 
 
1
ln 
2
x dx dt
t
= 
 
2
1 1 1 1
1
22
I t dt dt
tt
t
? ? ? ?
= + ? = +
? ? ? ?
? ? ? ?
??
 
  
11
2
tC
t
??
= - +
??
??
 
  
22
11
22
xx
xe
C
ex
? ? ? ?
= - +
? ? ? ?
? ? ? ?
 
 ? ? = 2, ? = 2, ? = 2, ? = 2 
 ? 2 3 4 2 4 6 8 4 ? + ? + ? - ? = + + - = 
 
   
  
8. Let ( ) ( ) ( ) ( ) ( ) 1 and 0, 0, 1 . = + - ?? ? ? g x f x f x f x x 
If g is decreasing in the interval (0, ?) and 
increasing in the interval (?, 1), then tan
–1
(2?) + 
11
11
tan tan
--
?+ ? ? ? ?
+
? ? ? ?
??
? ? ? ?
 is equal to  
 (1) ? (2) 
5
4
?
 
 (3) 
3
4
?
 (4) 
3
2
?
 
Answer (1) 
Sol. ( ) ( ) ( ) 1 g x f x f x = + - 
 ( ) ( ) ( ) 1 g x f x f x ? = ? - ? - 
 ( ) ( ) ( ) 10 g x f x f x ?? = ?? + ?? - ? 
 g?(x) is increasing 
 g?(0) < g?(1) 
 (0) (1) (1) (0) f f f f ? - ? ? ? - ? 
 ? (0) (1) ff ? ? ? 
  g?(x) = 0 
 ? ( ) (1 ) f x f x ? = ? - 
  x = 1 – x 
  
1
2
x = 
 
1
( ) is positive for 0, 
2
g x x
??
??
??
??
 
 
1
( ) is negative for , 1
2
g x x
??
??
??
??
 
 ? 
1
2
?= 
 ( )
1 1 1
11
tan 2 tan tan
- - -
?+ ? ? ? ?
? + +
? ? ? ?
??
? ? ? ?
 
 ( ) ( ) ( )
1 1 1
tan 1 tan 2 tan 3
- - -
= + + 
 = ? 
9. The statement ( ) ( )
~~?? ??
??
p p q is equivalent to  
 (1) ( ) ~ ? pq (2) ( ) ( ) ~ ?? p q p 
 (3) ( ) ( )
~ ?? p q q (4) ( ) ~ ? pq 
Answer (4) 
Sol. ( ) ( )
~~?? ??
??
p p q 
 ( ) = ~ ~ ~ p p q ?? 
 ( ) = ~ ~ p p q ?? 
 ( ) ( ) = ~ ~ ~ p p p q ? ? ? 
 = ~ ~ pq ? 
 ( ) = ~ pq ? 
10. Let a circle of radius 4 be concentric to the ellipse 
15x
2
 + 19y
2
 = 285. Then the common tangents are 
inclined to the minor axis of the ellipse at the angle 
 (1) 
3
?
 (2) 
4
?
 
 (3) 
6
?
 (4) 
12
?
 
Answer (1) 
Sol.  
 Let the equation of tangent of ellipse be 
 
2
19 15 y mx m = ? + 
 If it is tangent to circle x
2
 + y
2
 = 16 
 
2
2
19 15
4
1
m
m
+
=
+
 
 ? 
22
19 15 16 16 mm + = + 
 ? 
2
31 m = 
  
1
3
m =? 
 ? Angle made by tangent with minor axis i.e., with 
y axis is 
3
?
 
Page 5


   
 
 
Answers & Solutions 
for 
JEE (Main)-2023 (Online) Phase-2 
(Mathematics, Physics and Chemistry) 
 
 
  
10/04/2023 
Evening 
Time : 3 hrs. M.M. : 300 
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 Mathematics, Physics and Chemistry 
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. 
 
   
  
MATHEMATICS 
SECTION - A 
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. 
Choose the correct answer: 
1. Let ? be the mean and ? be the standard deviation 
of the distribution 
x
i
 0 1 2 3 4 5 
f
i
 k + 2 2k 
k
2 
– 1 k
2 
– 1 k
2 
+ 1 
k – 3 
 where 62.
i
f =
?
 If [x] denotes the greatest integer 
 ? x, then [?
2
 + ?
2
] is equal to 
 (1) 9 (2) 8 
 (3) 7 (4) 6 
Answer (2) 
Sol. 
x
i
 0 1 2 3 4 5 
 f
i
 k + 2 2k 
k
2 
– 1 k
2 
– 1 k
2 
+ 1 
k – 3 
 
2 2 2 2
2 2 – 2 3 – 3 4 4 5 – 15 9 7 – 16
62 62
k k k k k k k + + + + + +
? = =
 
 and 62
i
f =
?
? 3k
2
 + 4k – 2 = 62 
    ? 3k
2
 + 4k – 64 = 0 
    ? (3k
 
+ 16) (k – 4) = 0 
    4 k ?= 
 ? 
156
62
?= 
 
2 2 2 2
2
1 (2 ) 4( – 1) 9( – 1) 16( 1) 25( – 3) 156
–
62 62
k k k k k ? + + + + + ??
?=
??
??
 
  
2
2
29 27 – 72
–
62
kk +
=? 
  
500
62
= 
 ? 
22
500
62
? + ? = ? [?
2
 + ?
2
] = 8 
2. Let the image of the point P(1, 2, 6) in the plane 
passing through the points A(1, 2, 0) and B(1, 4, 1) 
C(0, 5, 1) be Q(?, ?, ?). Then (?
2
 + ?
2
 + ?
2
) is equal 
to 
 (1) 65 (2) 62 
 (3) 76 (4) 70 
Answer (1) 
Sol. Plane passes through the points 
 A(1, 2, 0), B(1, 4, 1) and C(0, 5, 1) 
 Normal vector 
ˆˆ
ˆ ˆ ˆ
0 2 1 (–1) – (1) 2
–1 3 1
i j k
i j k = = + 
   = <–1, –1, 2> 
 ? –1(x – 1) –1 (y – 2) + 2z = 0 
 ? x + y – 2z = 3 …(i) 
 Now image of P(1, 2, 6) in (i) 
 
– 1 – 2 – 6 –2(–9 – 3)
4
1 1 –2 6
x y z
= = = = 
 ? x = 5, y = 6, z = –2 
 ? ? = 5, ? = 6, ? = –2 
 ? ?
2
 + ?
2
 + ?
2
 = 65 
3. Let the number (22)
2022
 + (2022)
22
 leave the 
remainder ? when divided by 3 and ? when divided 
by 7. Then (?
2
 + ?
2
) is equal to 
 (1) 20 (2) 13 
 (3) 5 (4) 10 
Answer (3) 
Sol. For ? : 
2022 22
divisible by 3
(21 1) (2022) ++ 
 = 3K
1
 + 1 
 For ? : (21 + 1)
2022
 + (2023 – 1)
22
 
 = 7? + 1 + 7? + 1 
 = 7K2 + 2 
 So, ? = 1, ? = 2 
 ?
2
 + ?
2
 = 5 
 
   
  
4. Eight persons are to be transported from city A to 
city B in three cars of different makes. If each car 
can accommodate at most three persons, then the 
number of ways, in which they can be transported, 
is 
 (1) 1120 (2) 3360 
 (3) 1680 (4) 560 
Answer (2) 
Sol. Total ways = 
8
C
3
 × 
5
C
3
 × 
2
C
2
 × 3! 
   = 3360 
5. Let 
ˆ ˆ ˆ ˆ ˆ ˆ
2 7 , 3 5 a i j k b i k = + - = + and 
ˆ ˆ ˆ
– 2 . c i j k =+ 
Let d be a vector which is perpendicular to both 
 and ab , and  · 12. cd = Then 
( ) ( )
ˆ ˆ ˆ
· i j k c d - + - ? 
is equal to 
 (1) 24 (2) 44 
 (3) 42 (4) 48 
Answer (2) 
Sol. () d a b = ? ? 
 
ˆ ˆ ˆ
2 7 1
3 0 5
i j k
ab ? = - 
 = 
ˆ ˆ ˆ
(35) 13 21 i j k -- 
 
ˆ ˆ ˆ
(35 13 21 ) d i j k = ? - - 
 (35 13 42) 12 cd ? = ? + - = 
 ? ? = 2 
 
ˆ ˆ ˆ
2 1 1 2
35 13 21
i j k
cd ? = ? -
--
 
 = 
ˆ ˆ ˆ
2(47 91 22 ) i j k ++ 
 
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
( ) ( ) 2( ) (47 91 22 ) 44 i j k c d i j k i j k - + - ? ? = - + - + + = 
6. Let f be a continuous function satisfying 
( )
( )
2
23
0
4
, 0.
3
t
f x x dx t t + = ? ?
?
 Then 
2
4
??
?
??
??
??
f is 
equal to 
 (1) 
2
2
1
16
??
?
?-
??
??
??
 (2) 
3
1
16
??
?
-? +
??
??
??
 
 (3) 
3
1
16
??
?
?-
??
??
??
 (4) 
2
2
1
16
??
?
-? +
??
??
??
 
Answer (3) 
Sol. 
( ) ( )
2 4 2
24 f t t t t += 
 
( )
24
2 f t t t += 
 
( )
24
2 f x x x = - + 
 Let x
2
 = u 
 ( )
2
2 f u u u = - + 
 ( )
2
2 f x x x = - + 
 
24
2
2
42
4
f
??
? ? ?
= - + ?
??
??
??
 
   
4
16
-?
= + ? 
   
3
1
16
??
?
= ? -
??
??
??
 
7. For , , , , ? ? ? ? ? if 
22
log
xx
e
xe
x
ex
??
? ? ? ?
?? +
? ? ? ?
??
? ? ? ?
??
?
11
,
??
? ? ? ?
= - +
? ? ? ?
??
? ? ? ?
xx
xe
dx C
ex
 where 
0
1
!
?
=
=
?
n
e
n
 and 
C is constant of integration, then ? + 2? + 3? – 4? is 
equal to 
 (1) 1 (2) 4 
 (3) –4 (4) –8 
Answer (2) 
Sol. 
2
Let 
x
x
t
e
??
=
??
??
 
 ( ) 2 ln 1 ln x x t -= 
 ( )
11
2 ln 1 2 x x dx dt
xt
??
- + ? =
??
??
 
 
1
ln 
2
x dx dt
t
= 
 
2
1 1 1 1
1
22
I t dt dt
tt
t
? ? ? ?
= + ? = +
? ? ? ?
? ? ? ?
??
 
  
11
2
tC
t
??
= - +
??
??
 
  
22
11
22
xx
xe
C
ex
? ? ? ?
= - +
? ? ? ?
? ? ? ?
 
 ? ? = 2, ? = 2, ? = 2, ? = 2 
 ? 2 3 4 2 4 6 8 4 ? + ? + ? - ? = + + - = 
 
   
  
8. Let ( ) ( ) ( ) ( ) ( ) 1 and 0, 0, 1 . = + - ?? ? ? g x f x f x f x x 
If g is decreasing in the interval (0, ?) and 
increasing in the interval (?, 1), then tan
–1
(2?) + 
11
11
tan tan
--
?+ ? ? ? ?
+
? ? ? ?
??
? ? ? ?
 is equal to  
 (1) ? (2) 
5
4
?
 
 (3) 
3
4
?
 (4) 
3
2
?
 
Answer (1) 
Sol. ( ) ( ) ( ) 1 g x f x f x = + - 
 ( ) ( ) ( ) 1 g x f x f x ? = ? - ? - 
 ( ) ( ) ( ) 10 g x f x f x ?? = ?? + ?? - ? 
 g?(x) is increasing 
 g?(0) < g?(1) 
 (0) (1) (1) (0) f f f f ? - ? ? ? - ? 
 ? (0) (1) ff ? ? ? 
  g?(x) = 0 
 ? ( ) (1 ) f x f x ? = ? - 
  x = 1 – x 
  
1
2
x = 
 
1
( ) is positive for 0, 
2
g x x
??
??
??
??
 
 
1
( ) is negative for , 1
2
g x x
??
??
??
??
 
 ? 
1
2
?= 
 ( )
1 1 1
11
tan 2 tan tan
- - -
?+ ? ? ? ?
? + +
? ? ? ?
??
? ? ? ?
 
 ( ) ( ) ( )
1 1 1
tan 1 tan 2 tan 3
- - -
= + + 
 = ? 
9. The statement ( ) ( )
~~?? ??
??
p p q is equivalent to  
 (1) ( ) ~ ? pq (2) ( ) ( ) ~ ?? p q p 
 (3) ( ) ( )
~ ?? p q q (4) ( ) ~ ? pq 
Answer (4) 
Sol. ( ) ( )
~~?? ??
??
p p q 
 ( ) = ~ ~ ~ p p q ?? 
 ( ) = ~ ~ p p q ?? 
 ( ) ( ) = ~ ~ ~ p p p q ? ? ? 
 = ~ ~ pq ? 
 ( ) = ~ pq ? 
10. Let a circle of radius 4 be concentric to the ellipse 
15x
2
 + 19y
2
 = 285. Then the common tangents are 
inclined to the minor axis of the ellipse at the angle 
 (1) 
3
?
 (2) 
4
?
 
 (3) 
6
?
 (4) 
12
?
 
Answer (1) 
Sol.  
 Let the equation of tangent of ellipse be 
 
2
19 15 y mx m = ? + 
 If it is tangent to circle x
2
 + y
2
 = 16 
 
2
2
19 15
4
1
m
m
+
=
+
 
 ? 
22
19 15 16 16 mm + = + 
 ? 
2
31 m = 
  
1
3
m =? 
 ? Angle made by tangent with minor axis i.e., with 
y axis is 
3
?
 
 
   
  
11. Let 
22
1–tan tan
, : 9 9 10
22
xx
Sx
?? ?? ??
= ? - + =
?? ??
????
 and 
?
??
?=
??
??
?
2
tan ,
3
xs
x
 then ( ) ?-
2 1
14
6
 is equal to 
 (1) 16 (2) 8 
 (3) 64 (4) 32 
Answer (4) 
Sol. 
22
1 tan tan
9 9 10
xx -
+= 
 Let 
2
tan
9
x
y = 
 
9
10 y
y
+= 
 ? 
2
10 9 0 yy - + = 
 ? y = 1, y = 9 
  
22
tan tan
9 1 OR 9 9
xx
== 
 ? tan x = 0 OR tan x = ±1 
  x = 0          ,
44
x
? -?
= 
 ? 
2 2 2 2
tan tan 0 tan tan
3 12 12
xs
x
?
??
? = = + +
?
 
  = 
2
2(2 3) - 
  = 2(7 4 3) - 
  = 140 – 8 3 
 ? 
2
11
( 14) 64 3 32
66
? - = ? ? = 
12. Let A = {2, 3, 4} and B = {8, 9, 12}. Then the number 
of elements in the relation  
 = ? ? ?
1 1 2 2 1
{(( , ),( , )) ( , ) : R a b a b A B A B a divides 
b
2
 and a
2
 divides b
1
} is 
 (1) 36 (2) 24 
 (3) 18 (4) 12 
Answer (1) 
Sol. ? ? ? ? 2, 3, 4 , 8, 9, 12 AB == 
 
12
, a A b B ?? 
 a
1
 divides b
2
 
 ? ?
12
( , ) (2, 4),(2, 12), (3, 9), (3, 12), (4, 8),(4, 12) ab ? 
 
21
, a A b B ?? 
 a
2
 divides b
1 
 
…. same as above… 
 ? Number of relations = 6 × 6 = 36 
13. Let a die be rolled n times. Let the probability of 
getting odd numbers seven times be equal to the 
probability of getting odd numbers nine times. If the 
probability of getting even numbers twice is 
15
,
2
k
 
then k is equal to 
 (1) 60 (2) 15 
 (3) 90 (4) 30 
Answer (1) 
Sol. P(getting odd 7 time) = P(getting odd 9 times) 
 
7 7 9 9
79
1 1 1 1
2 2 2 2
nn
nn
CC
--
? ? ? ? ? ? ? ?
=
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
 
 
79
nn
CC = 
 ? n = 16 
 P(2 times even) = 
14 2
16
2
11
22
C
? ? ? ?
? ? ? ?
? ? ? ?
 
 = 
15 15 15
15 4 60
2 2 2
k ?
== 
 k = 60 
14. Let 
? ?
23
: is a real number
42
zi
S z x iy
zi
-
= = +
+
. 
Then which of the following is NOT correct? 
 (1) + + ?
22
1
–
4
y x y 
 (2) 
??
=-
??
??
1
( , ) 0,
2
xy 
 (3) x = 0 
 (4) 
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
11
– , – – ,
22
y 
Answer (2) 
Sol.  z x iy =+ and 
23
42
zi
zi
-
+
 is real number 
 ? 
2 (2 3)
4 (4 2)
xyi
x y i
+-
++
 is real number. 
 ? 
( )( )
22
2 (2 3) 4 (4 2)
(4 ) (4 2)
x y i x y i
xy
+ - - +
++
 is real number 
 ? 2 (4 2) (2 3)4 0 x y y x - + + - = 
 ? x = 0 
 Here x = 0 and y ? but 
1
–
2
y ? is not acceptable 
in this case  
 Denominator will be zero. 
 Hence, option (2) is correct.