JEE Main 2023 April 6 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) 
 
 
  
06/04/2023 
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) 
 
 
  
06/04/2023 
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. 
11 1 1 1 1
35 2 2 2 2 1
lim 2 2 2 2 ..... 2 2
n
n
+
??
?? ? ?? ???
??
? ?? ???
- - -
??
? ?? ???
??
?? ? ?? ? ??
 is 
equal to 
 (1) 1 (2) 0 
 (3) 2 (4) 
1
2
 
Answer (2) 
Sol. 
11 1 1 1 1
35 2 2 2 2 1
lim 2 2 2 2 ..... 2 2
n
n
+
??
?? ? ?? ???
??
? ?? ???
- - -
??
? ?? ???
??
?? ? ?? ? ??
 
 Since 
1 1
3 2
2 2 1 -? 
  
1 1
5 2
2 2 1 -? 
 -------------------- 
 
11
2 2 1
2 2 1
n+
-? 
n ??
 
 ? 
11 1 1 1 1
35 2 2 2 2 1
lim 2 2 2 2 ..... 2 2
n
n
+
??
?? ? ?? ???
??
? ?? ???
- - -
??
? ?? ???
??
?? ? ?? ? ??
 
 = 0  
2. If gcd (m, n) = 1 and 1
2
 – 2
2
 + 3
2
 – 4
2
 + ….. + (2021)
2
 
– (2022)
2
 + (2023)
2
 = 1012m
2
n then m
2
 – n
2
 is equal 
to 
 (1) 240 (2) 200 
 (3) 220 (4) 180 
Answer (1) 
Sol. 1
2
 – 2
2
 + 3
2
 – 4
2
 + ….. + (2021)
2
 – (2022)
2
 + (2023)
2
 
 = 
2
1011 times
3 7 11 (2023) - - - ?+ 
 = 
2
1011
[6 (1010)4] (2023)
2
-
++ 
 = 2023(1012) 
 ? 2023 = 17
2
 × 7 
 ? m = 17, n = 7 
 ? m
2
 – n
2
 = 289 – 49 
  = 240 
3. In a group of 100 persons 75 speak English and 40 
speak Hindi. Each person speaks at least one of the 
two languages. If the number of persons who speak 
only English is ? and the number of persons who 
speak only Hindi is ?, then the eccentricity of the 
ellipse 25(?
2
x
2
 + ?
2
y
2
) = ?
2
?
2
 is 
 (1) 
119
12
 (2) 
117
12
 
 (3) 
3 15
12
 (4) 
129
12
 
Answer (1) 
Sol.  
 Now ? = 100 – 75 = 25 
 ? ? = 75 – [40 – 25] 
  = 60 
 Now, ellipse 
22
22
25 1
(60) 25
xy
??
+= ??
??
??
 
 ? 
22
1
36 4 25
xy
+=
?
 
 ? 
25 119
1
36 4 12
e = - =
?
 
4. Let the vectors , , a b c represent three 
coterminous edges of a parallelopiped of volume V. 
Then the volume of the parallelopiped, whose 
coterminous edges are represented by , + a b c 
and  + 2 + 3 a b c is equal to 
 (1) 2V (2) 6V 
 (3) V (4) 3V 
Answer (3) 
Page 3


   
 
  
Answers & Solutions 
for 
JEE (Main)-2023 (Online) Phase-2 
(Mathematics, Physics and Chemistry) 
 
 
  
06/04/2023 
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. 
11 1 1 1 1
35 2 2 2 2 1
lim 2 2 2 2 ..... 2 2
n
n
+
??
?? ? ?? ???
??
? ?? ???
- - -
??
? ?? ???
??
?? ? ?? ? ??
 is 
equal to 
 (1) 1 (2) 0 
 (3) 2 (4) 
1
2
 
Answer (2) 
Sol. 
11 1 1 1 1
35 2 2 2 2 1
lim 2 2 2 2 ..... 2 2
n
n
+
??
?? ? ?? ???
??
? ?? ???
- - -
??
? ?? ???
??
?? ? ?? ? ??
 
 Since 
1 1
3 2
2 2 1 -? 
  
1 1
5 2
2 2 1 -? 
 -------------------- 
 
11
2 2 1
2 2 1
n+
-? 
n ??
 
 ? 
11 1 1 1 1
35 2 2 2 2 1
lim 2 2 2 2 ..... 2 2
n
n
+
??
?? ? ?? ???
??
? ?? ???
- - -
??
? ?? ???
??
?? ? ?? ? ??
 
 = 0  
2. If gcd (m, n) = 1 and 1
2
 – 2
2
 + 3
2
 – 4
2
 + ….. + (2021)
2
 
– (2022)
2
 + (2023)
2
 = 1012m
2
n then m
2
 – n
2
 is equal 
to 
 (1) 240 (2) 200 
 (3) 220 (4) 180 
Answer (1) 
Sol. 1
2
 – 2
2
 + 3
2
 – 4
2
 + ….. + (2021)
2
 – (2022)
2
 + (2023)
2
 
 = 
2
1011 times
3 7 11 (2023) - - - ?+ 
 = 
2
1011
[6 (1010)4] (2023)
2
-
++ 
 = 2023(1012) 
 ? 2023 = 17
2
 × 7 
 ? m = 17, n = 7 
 ? m
2
 – n
2
 = 289 – 49 
  = 240 
3. In a group of 100 persons 75 speak English and 40 
speak Hindi. Each person speaks at least one of the 
two languages. If the number of persons who speak 
only English is ? and the number of persons who 
speak only Hindi is ?, then the eccentricity of the 
ellipse 25(?
2
x
2
 + ?
2
y
2
) = ?
2
?
2
 is 
 (1) 
119
12
 (2) 
117
12
 
 (3) 
3 15
12
 (4) 
129
12
 
Answer (1) 
Sol.  
 Now ? = 100 – 75 = 25 
 ? ? = 75 – [40 – 25] 
  = 60 
 Now, ellipse 
22
22
25 1
(60) 25
xy
??
+= ??
??
??
 
 ? 
22
1
36 4 25
xy
+=
?
 
 ? 
25 119
1
36 4 12
e = - =
?
 
4. Let the vectors , , a b c represent three 
coterminous edges of a parallelopiped of volume V. 
Then the volume of the parallelopiped, whose 
coterminous edges are represented by , + a b c 
and  + 2 + 3 a b c is equal to 
 (1) 2V (2) 6V 
 (3) V (4) 3V 
Answer (3) 
 
   
   
Sol. [ , ,  2 3 ] a b c a b c + + + 
 = 
1 0 0
0 1 1 [ ]
1 2 3
a b c 
 = [] a b c 
 = V 
5. If the solution curve f(x, y) = 0 of the differential 
equation ( ) 1 log log ,  > 0,
y
ee
dx
x x x e x
dy
+ - = 
passes through the points (1, 0) and (?, 2), then a
a
 
is equal to 
 (1) 
2
2e
e (2) 
2
e
e 
 (3) 
2
2e
e (4) 
2
2e
e 
Answer (1) 
Sol. ( ) 1 ln ln
y
dx
x x x e
dy
+ - = 
 Put x ln x = t 
 (1 + ln x) dx = dt 
 ? 
y
dt
te
dy
-= 
 I.F
dy
e
-
?
= 
 = e
–y
 
 
y y y
t e e e dy c
--
? = ? +
?
 
 t × e
–y
 = y + c 
 x ln x = y e
y
 + c e
y
 
 Put x = 1, y = 0 
 ? c = 0 
 Put x = a, y = 2 
 a ln a = 2e
2
 
 ? 
2
2 ae
ae = 
6. Let f(x) be a function satisfying ( ) ( )
2
, + ? - = ? f x f x
. x ?? Then ( )
0
sin f x x dx
?
?
 is equal to 
 (1) 
2
4
?
 (2) 2?
2
 
 (3) ?
2
 (4) 
2
2
?
 
Answer (3) 
Sol. 
( )
0
sin I f x x dx
?
=
?
 
 
( )
0
sin I f x x dx
?
= ? -
?
 
 
( ) ( ) ( )
0
2 sin I x f x f x dx
?
= + ? -
?
 
 
2
0
2 sin I x dx
?
=?
?
 
 
2
2
0
2 2 sin I x dx
?
=?
?
 
 
2
I =? 
7. If the tangents at the points P and Q on the circle 
22
25 + - + = x y x y meet at the point 
9
, 2 ,
4
??
??
??
R 
then the area of the triangle PQR is 
 (1) 
5
4
 (2) 
13
8
 
 (3) 
5
8
 (4) 
13
4
 
Answer (3) 
Sol.  
 ( )
2
2
1
99
2 2 2 5
44
LS
??
= = + - ? + -
??
??
 
   
5
4
= 
 
3
3
2 2 2 2
55
24
Area
55
24
RL
RL
??
?
??
??
==
+
? ? ? ?
+
? ? ? ?
? ? ? ?
 
   
25
5
8
4 1 8
==
+
 
Page 4


   
 
  
Answers & Solutions 
for 
JEE (Main)-2023 (Online) Phase-2 
(Mathematics, Physics and Chemistry) 
 
 
  
06/04/2023 
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. 
11 1 1 1 1
35 2 2 2 2 1
lim 2 2 2 2 ..... 2 2
n
n
+
??
?? ? ?? ???
??
? ?? ???
- - -
??
? ?? ???
??
?? ? ?? ? ??
 is 
equal to 
 (1) 1 (2) 0 
 (3) 2 (4) 
1
2
 
Answer (2) 
Sol. 
11 1 1 1 1
35 2 2 2 2 1
lim 2 2 2 2 ..... 2 2
n
n
+
??
?? ? ?? ???
??
? ?? ???
- - -
??
? ?? ???
??
?? ? ?? ? ??
 
 Since 
1 1
3 2
2 2 1 -? 
  
1 1
5 2
2 2 1 -? 
 -------------------- 
 
11
2 2 1
2 2 1
n+
-? 
n ??
 
 ? 
11 1 1 1 1
35 2 2 2 2 1
lim 2 2 2 2 ..... 2 2
n
n
+
??
?? ? ?? ???
??
? ?? ???
- - -
??
? ?? ???
??
?? ? ?? ? ??
 
 = 0  
2. If gcd (m, n) = 1 and 1
2
 – 2
2
 + 3
2
 – 4
2
 + ….. + (2021)
2
 
– (2022)
2
 + (2023)
2
 = 1012m
2
n then m
2
 – n
2
 is equal 
to 
 (1) 240 (2) 200 
 (3) 220 (4) 180 
Answer (1) 
Sol. 1
2
 – 2
2
 + 3
2
 – 4
2
 + ….. + (2021)
2
 – (2022)
2
 + (2023)
2
 
 = 
2
1011 times
3 7 11 (2023) - - - ?+ 
 = 
2
1011
[6 (1010)4] (2023)
2
-
++ 
 = 2023(1012) 
 ? 2023 = 17
2
 × 7 
 ? m = 17, n = 7 
 ? m
2
 – n
2
 = 289 – 49 
  = 240 
3. In a group of 100 persons 75 speak English and 40 
speak Hindi. Each person speaks at least one of the 
two languages. If the number of persons who speak 
only English is ? and the number of persons who 
speak only Hindi is ?, then the eccentricity of the 
ellipse 25(?
2
x
2
 + ?
2
y
2
) = ?
2
?
2
 is 
 (1) 
119
12
 (2) 
117
12
 
 (3) 
3 15
12
 (4) 
129
12
 
Answer (1) 
Sol.  
 Now ? = 100 – 75 = 25 
 ? ? = 75 – [40 – 25] 
  = 60 
 Now, ellipse 
22
22
25 1
(60) 25
xy
??
+= ??
??
??
 
 ? 
22
1
36 4 25
xy
+=
?
 
 ? 
25 119
1
36 4 12
e = - =
?
 
4. Let the vectors , , a b c represent three 
coterminous edges of a parallelopiped of volume V. 
Then the volume of the parallelopiped, whose 
coterminous edges are represented by , + a b c 
and  + 2 + 3 a b c is equal to 
 (1) 2V (2) 6V 
 (3) V (4) 3V 
Answer (3) 
 
   
   
Sol. [ , ,  2 3 ] a b c a b c + + + 
 = 
1 0 0
0 1 1 [ ]
1 2 3
a b c 
 = [] a b c 
 = V 
5. If the solution curve f(x, y) = 0 of the differential 
equation ( ) 1 log log ,  > 0,
y
ee
dx
x x x e x
dy
+ - = 
passes through the points (1, 0) and (?, 2), then a
a
 
is equal to 
 (1) 
2
2e
e (2) 
2
e
e 
 (3) 
2
2e
e (4) 
2
2e
e 
Answer (1) 
Sol. ( ) 1 ln ln
y
dx
x x x e
dy
+ - = 
 Put x ln x = t 
 (1 + ln x) dx = dt 
 ? 
y
dt
te
dy
-= 
 I.F
dy
e
-
?
= 
 = e
–y
 
 
y y y
t e e e dy c
--
? = ? +
?
 
 t × e
–y
 = y + c 
 x ln x = y e
y
 + c e
y
 
 Put x = 1, y = 0 
 ? c = 0 
 Put x = a, y = 2 
 a ln a = 2e
2
 
 ? 
2
2 ae
ae = 
6. Let f(x) be a function satisfying ( ) ( )
2
, + ? - = ? f x f x
. x ?? Then ( )
0
sin f x x dx
?
?
 is equal to 
 (1) 
2
4
?
 (2) 2?
2
 
 (3) ?
2
 (4) 
2
2
?
 
Answer (3) 
Sol. 
( )
0
sin I f x x dx
?
=
?
 
 
( )
0
sin I f x x dx
?
= ? -
?
 
 
( ) ( ) ( )
0
2 sin I x f x f x dx
?
= + ? -
?
 
 
2
0
2 sin I x dx
?
=?
?
 
 
2
2
0
2 2 sin I x dx
?
=?
?
 
 
2
I =? 
7. If the tangents at the points P and Q on the circle 
22
25 + - + = x y x y meet at the point 
9
, 2 ,
4
??
??
??
R 
then the area of the triangle PQR is 
 (1) 
5
4
 (2) 
13
8
 
 (3) 
5
8
 (4) 
13
4
 
Answer (3) 
Sol.  
 ( )
2
2
1
99
2 2 2 5
44
LS
??
= = + - ? + -
??
??
 
   
5
4
= 
 
3
3
2 2 2 2
55
24
Area
55
24
RL
RL
??
?
??
??
==
+
? ? ? ?
+
? ? ? ?
? ? ? ?
 
   
25
5
8
4 1 8
==
+
 
 
   
   
8. The area bounded by the curves 12 = - + - y x x 
and y = 3 is equal to 
 (1) 4 (2) 6 
 (3) 3 (4) 5 
Answer (1) 
Sol.  
 Area  = ? ?
1
1 3 2
2
+? 
  = 4 
9. If the coefficients of x
7
 in 
11
2
1
2
??
+
??
??
ax
bx
 and x
–7
 in 
11
2
1
3
??
-
??
??
ax
bx
 are equal, then 
 (1) 729ab = 32 (2) 32ab = 729 
 (3) 64ab = 243 (4) 243ab = 64 
Answer (1) 
Sol. Coefficient of x
7
 in 
11
2
1
2
ax
bx
??
+
??
??
 
 
( )
11
11 2
1
1
2
r
r
rr
T C ax
bx
-
+
??
=
??
??
 
  ( )
11
11 22 3
1
2
r
r
r
r
C a x
b
-
-
??
=
??
??
 
  22 3 7 5 rr - = ? = 
 Coefficient of x
–7
 in 
11
2
1
3
ax
bx
??
-
??
??
 
 ( )
11
11
1
2
1
3
r
r
rr
T C ax
bx
-
+
??
=-
??
??
  
  
11 11 11 3
1
3
r
rr
r
C a x
b
--
??
=-
??
??
 
  11 3 7 6 rr - = - ? = 
 ? ( )
56
6
11 11 5
56
11
 
23
C a C a
bb
? ? ? ?
=-
? ? ? ?
? ? ? ?
 
 ? 
6
3 32 ab = 
 ? 729 ab = 32 
10. Let the sets A and B denote the domain and range 
respectively of the function 
( )
1
, fx
xx
=
- ??
??
 where 
x ??
??
 denotes the smallest integer greater than or 
equal to x. Then among the statements  
 
( ) ( )
( ) ( )
S1 : 1 ,  and 
S2 : 1 , 
AB
AB
? = ? -
? = ?
  
 (1) Only (S2) is true 
 (2) Only (S1) is true  
 (3) Neither (S1) nor (S2) is true 
 (4) Both (S1) and (S2) are true 
Answer (3) 
Sol. 
( )
? ?
1
fx
xx
=
-
 
  
? ?
1
x
=
-
 
 ? Domain = ? 
11. Let P be a square matrix such that P
2
 = I – P. For 
? ? ? ? ? , , , , if 
??
+ = ? – 29 P P I P and 
??
=? – – 13 , P P I P then ? + ? + ? – ? is equal to 
 (1) 18 (2) 40 
 (3) 22 (4) 24 
Answer (4) 
Sol. P
2
 = I – P 
 
42
( )( ) – 2 2 3 P I P I P I P P I P = - - = + = - 
 
62
2 5 3 2 5 3( ) 5 8 P I P P I P IP I P = - + = - + = - …(i) 
 
82
5 13 8 13 21 P I P P I P = - + = - …(ii) 
 (ii) + (i) 
 
86
18 29 P P I P + = - 
 (ii) – (i) 
 
86
– 8 13 P P I P =- 
 ? = 8, ? = 6, ? = 18, ? = 8 
 8 + 6 + 18 + 8 = 24 
Page 5


   
 
  
Answers & Solutions 
for 
JEE (Main)-2023 (Online) Phase-2 
(Mathematics, Physics and Chemistry) 
 
 
  
06/04/2023 
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. 
11 1 1 1 1
35 2 2 2 2 1
lim 2 2 2 2 ..... 2 2
n
n
+
??
?? ? ?? ???
??
? ?? ???
- - -
??
? ?? ???
??
?? ? ?? ? ??
 is 
equal to 
 (1) 1 (2) 0 
 (3) 2 (4) 
1
2
 
Answer (2) 
Sol. 
11 1 1 1 1
35 2 2 2 2 1
lim 2 2 2 2 ..... 2 2
n
n
+
??
?? ? ?? ???
??
? ?? ???
- - -
??
? ?? ???
??
?? ? ?? ? ??
 
 Since 
1 1
3 2
2 2 1 -? 
  
1 1
5 2
2 2 1 -? 
 -------------------- 
 
11
2 2 1
2 2 1
n+
-? 
n ??
 
 ? 
11 1 1 1 1
35 2 2 2 2 1
lim 2 2 2 2 ..... 2 2
n
n
+
??
?? ? ?? ???
??
? ?? ???
- - -
??
? ?? ???
??
?? ? ?? ? ??
 
 = 0  
2. If gcd (m, n) = 1 and 1
2
 – 2
2
 + 3
2
 – 4
2
 + ….. + (2021)
2
 
– (2022)
2
 + (2023)
2
 = 1012m
2
n then m
2
 – n
2
 is equal 
to 
 (1) 240 (2) 200 
 (3) 220 (4) 180 
Answer (1) 
Sol. 1
2
 – 2
2
 + 3
2
 – 4
2
 + ….. + (2021)
2
 – (2022)
2
 + (2023)
2
 
 = 
2
1011 times
3 7 11 (2023) - - - ?+ 
 = 
2
1011
[6 (1010)4] (2023)
2
-
++ 
 = 2023(1012) 
 ? 2023 = 17
2
 × 7 
 ? m = 17, n = 7 
 ? m
2
 – n
2
 = 289 – 49 
  = 240 
3. In a group of 100 persons 75 speak English and 40 
speak Hindi. Each person speaks at least one of the 
two languages. If the number of persons who speak 
only English is ? and the number of persons who 
speak only Hindi is ?, then the eccentricity of the 
ellipse 25(?
2
x
2
 + ?
2
y
2
) = ?
2
?
2
 is 
 (1) 
119
12
 (2) 
117
12
 
 (3) 
3 15
12
 (4) 
129
12
 
Answer (1) 
Sol.  
 Now ? = 100 – 75 = 25 
 ? ? = 75 – [40 – 25] 
  = 60 
 Now, ellipse 
22
22
25 1
(60) 25
xy
??
+= ??
??
??
 
 ? 
22
1
36 4 25
xy
+=
?
 
 ? 
25 119
1
36 4 12
e = - =
?
 
4. Let the vectors , , a b c represent three 
coterminous edges of a parallelopiped of volume V. 
Then the volume of the parallelopiped, whose 
coterminous edges are represented by , + a b c 
and  + 2 + 3 a b c is equal to 
 (1) 2V (2) 6V 
 (3) V (4) 3V 
Answer (3) 
 
   
   
Sol. [ , ,  2 3 ] a b c a b c + + + 
 = 
1 0 0
0 1 1 [ ]
1 2 3
a b c 
 = [] a b c 
 = V 
5. If the solution curve f(x, y) = 0 of the differential 
equation ( ) 1 log log ,  > 0,
y
ee
dx
x x x e x
dy
+ - = 
passes through the points (1, 0) and (?, 2), then a
a
 
is equal to 
 (1) 
2
2e
e (2) 
2
e
e 
 (3) 
2
2e
e (4) 
2
2e
e 
Answer (1) 
Sol. ( ) 1 ln ln
y
dx
x x x e
dy
+ - = 
 Put x ln x = t 
 (1 + ln x) dx = dt 
 ? 
y
dt
te
dy
-= 
 I.F
dy
e
-
?
= 
 = e
–y
 
 
y y y
t e e e dy c
--
? = ? +
?
 
 t × e
–y
 = y + c 
 x ln x = y e
y
 + c e
y
 
 Put x = 1, y = 0 
 ? c = 0 
 Put x = a, y = 2 
 a ln a = 2e
2
 
 ? 
2
2 ae
ae = 
6. Let f(x) be a function satisfying ( ) ( )
2
, + ? - = ? f x f x
. x ?? Then ( )
0
sin f x x dx
?
?
 is equal to 
 (1) 
2
4
?
 (2) 2?
2
 
 (3) ?
2
 (4) 
2
2
?
 
Answer (3) 
Sol. 
( )
0
sin I f x x dx
?
=
?
 
 
( )
0
sin I f x x dx
?
= ? -
?
 
 
( ) ( ) ( )
0
2 sin I x f x f x dx
?
= + ? -
?
 
 
2
0
2 sin I x dx
?
=?
?
 
 
2
2
0
2 2 sin I x dx
?
=?
?
 
 
2
I =? 
7. If the tangents at the points P and Q on the circle 
22
25 + - + = x y x y meet at the point 
9
, 2 ,
4
??
??
??
R 
then the area of the triangle PQR is 
 (1) 
5
4
 (2) 
13
8
 
 (3) 
5
8
 (4) 
13
4
 
Answer (3) 
Sol.  
 ( )
2
2
1
99
2 2 2 5
44
LS
??
= = + - ? + -
??
??
 
   
5
4
= 
 
3
3
2 2 2 2
55
24
Area
55
24
RL
RL
??
?
??
??
==
+
? ? ? ?
+
? ? ? ?
? ? ? ?
 
   
25
5
8
4 1 8
==
+
 
 
   
   
8. The area bounded by the curves 12 = - + - y x x 
and y = 3 is equal to 
 (1) 4 (2) 6 
 (3) 3 (4) 5 
Answer (1) 
Sol.  
 Area  = ? ?
1
1 3 2
2
+? 
  = 4 
9. If the coefficients of x
7
 in 
11
2
1
2
??
+
??
??
ax
bx
 and x
–7
 in 
11
2
1
3
??
-
??
??
ax
bx
 are equal, then 
 (1) 729ab = 32 (2) 32ab = 729 
 (3) 64ab = 243 (4) 243ab = 64 
Answer (1) 
Sol. Coefficient of x
7
 in 
11
2
1
2
ax
bx
??
+
??
??
 
 
( )
11
11 2
1
1
2
r
r
rr
T C ax
bx
-
+
??
=
??
??
 
  ( )
11
11 22 3
1
2
r
r
r
r
C a x
b
-
-
??
=
??
??
 
  22 3 7 5 rr - = ? = 
 Coefficient of x
–7
 in 
11
2
1
3
ax
bx
??
-
??
??
 
 ( )
11
11
1
2
1
3
r
r
rr
T C ax
bx
-
+
??
=-
??
??
  
  
11 11 11 3
1
3
r
rr
r
C a x
b
--
??
=-
??
??
 
  11 3 7 6 rr - = - ? = 
 ? ( )
56
6
11 11 5
56
11
 
23
C a C a
bb
? ? ? ?
=-
? ? ? ?
? ? ? ?
 
 ? 
6
3 32 ab = 
 ? 729 ab = 32 
10. Let the sets A and B denote the domain and range 
respectively of the function 
( )
1
, fx
xx
=
- ??
??
 where 
x ??
??
 denotes the smallest integer greater than or 
equal to x. Then among the statements  
 
( ) ( )
( ) ( )
S1 : 1 ,  and 
S2 : 1 , 
AB
AB
? = ? -
? = ?
  
 (1) Only (S2) is true 
 (2) Only (S1) is true  
 (3) Neither (S1) nor (S2) is true 
 (4) Both (S1) and (S2) are true 
Answer (3) 
Sol. 
( )
? ?
1
fx
xx
=
-
 
  
? ?
1
x
=
-
 
 ? Domain = ? 
11. Let P be a square matrix such that P
2
 = I – P. For 
? ? ? ? ? , , , , if 
??
+ = ? – 29 P P I P and 
??
=? – – 13 , P P I P then ? + ? + ? – ? is equal to 
 (1) 18 (2) 40 
 (3) 22 (4) 24 
Answer (4) 
Sol. P
2
 = I – P 
 
42
( )( ) – 2 2 3 P I P I P I P P I P = - - = + = - 
 
62
2 5 3 2 5 3( ) 5 8 P I P P I P IP I P = - + = - + = - …(i) 
 
82
5 13 8 13 21 P I P P I P = - + = - …(ii) 
 (ii) + (i) 
 
86
18 29 P P I P + = - 
 (ii) – (i) 
 
86
– 8 13 P P I P =- 
 ? = 8, ? = 6, ? = 18, ? = 8 
 8 + 6 + 18 + 8 = 24 
 
   
   
12. Among the statements 
 (S1) : ( ) ( ) ? ? ? (~ ) p q p q is a tautology 
 (S2) : ( ) ( ) ? ? ? (~ ) q p p q is a contradiction 
 (1) Neither (S1) and (S2) is True 
 (2) Both (S1) and (S2) are True 
 (3) Only (S2) is True 
 (4) Only (S1) is True 
Answer (1) 
Sol. S-I : ( ) ( ) ~ p q p q ? ? ? 
 ? ( ) ( ) p q p q ? ? ? ? ? 
 ? ( ) () p p q q ? ? ? ? ? 
  = ( ) pq ?? (not a tautology) 
 S-2 : ( ) ( ) q p p q ? ? ? ? 
 ? ( ) ( ) q p p q ? ? ? ? ? ? 
 ? ( ) ( ) q p p q ? ? ? ? ? 
  = pq ?? (not a contradiction) 
13. All the letters of the word PUBLIC are written in all 
possible orders and these words are written as in a 
dictionary with serial numbers. Then the serial 
number of the word PUBLIC is 
 (1) 576 (2) 578 
 (3) 580 (4) 582 
Answer (4) 
Sol. 
5 6 1 4 3 2
P U B L I C
4 4 0 2 1 0
5! 4! 3! 2! 1! 0!
 
 Rank = (1 × 1! + 2 × 2! + 4 × 4! + 4 × 5!) + 1 
 = (1 + 4 + 96 + 480) + 1 
 = 582 
14. Three dice are rolled. If the probability of getting 
different numbers on the three dice is ,
p
q
 where p 
and q are co-prime, then q – p is equal to 
 (1) 2 (2) 1 
 (3) 3 (4) 4 
Answer (4) 
Sol. If numbers are different on all three dice then 
number of ways 
  = 6 × 5 × 4 = 120 
 
3
120 120 5
()
216 9
6
p
PE
q
= = = = 
 Now, q – p = 9 – 5 = 4  
15. Among the statements :  
 (S1)  :  2023
2022
 – 1999
2022
 is divisible by 8. 
 (S2) : 13(13)
n
 – 11n – 13 is divisible by 144 for 
infinitely many n? 
 (1) Only (S2) is correct 
 (2) Only (S1) is correct 
 (3) Both (S1) and (S2) are correct 
 (4) Both (S1) and (S2) are incorrect 
Answer (2) 
Sol. (S1) : (2023)
2022
 – (1999)
2022
 is divisible by 8 
 We know that (x – y) divides (x
n
 – y
n
) n ?? 
 ? (2023 – 1999) divides (2023)
2022
 – (1999)
2022 
 
? 24 divides 
2022 2002
(2023) – (1999) 
 ? 8 will divide 
2022 2002
(2023) – (1999) 
 ? (S1) is correct. 
 (S2) : 13(13) – 11 13
n
n - is divisible by 144 for 
n? . 
 13(1 12) – 11 13
n
n +- 
 ( )
2
0 1 2
13 12 12 ... 12 – 11 13
n n n n n
n
C C C C n + + + + - 
 
2
12 13 11 12 nn ? - + ? 
 145 144 n+? is not divisible by 144. 
 ? (S2) is incorrect. 
16. Let the line L pass through the point (0, 1, 2), 
intersect the line 
1 2 3
2 3 4
x y z - - -
== and be 
parallel to the plane 2x + y – 3z = 4. Then the 
distance of the point P(1, –9, 2) from the line L is 
 (1) 74 (2) 69 
 (3) 54 (4) 9 
Answer (1)