JEE Main 2023 April 13 Shift 1 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) 
 
 
  
13/04/2023 
Morning 
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) 
 
 
  
13/04/2023 
Morning 
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. The number of symmetric matrices of order 3, with 
all the entries from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} 
is 
 (1) 6
10 
 (2) 10
6
 
 (3) 9
10 
 (4) 10
9 
Answer (2) 
Sol. A = {0, 1, 2, 3, …8, 9} 
 
* * *
0 **
00 *
M
??
??
=
??
??
??
 
 ? Total symmetric matrix = 10
6
 
 
2. Let 
13
1 2 3 , 2
4
B
? ??
??
= ? ?
??
?? ??
??
be the adjoint of a matrix A 
and |A| = 2. Then ? ? 2 2 B
? ??
??
? - ? ? - ?
??
?? ?
??
 is equal to 
 (1) 0 (2) 16 
 (3) –16 (4) 32 
Answer (3) 
Sol. 
3 1
3 1 2 , 2
4
B
??
?
??
= ? ?
??
??
??
??
 
 And adj (A) = B, |A| = 2 
 ? |adj (A)| = |B| 
 ? 2
2
 = (8 – 3?) – 3(4 – 3?) + ?(–?) 
 ? ?
2
 – 6? + 8 = 0 
 ? (? – 4) (? – 2) = 0 
  ? = 4, 2 but ? > 2 so 4 ?= 
 Now 
 ? ? ? ?
3 14 4
3 2 2 4 8 4 1 2 8
444 4
B
?? ? ? ? ?
?
?? ? ? ? ?
? - ? ? - ? = - -
?? ? ? ? ?
?? ? ? ? ?
?
? ? ? ? ??
 
  ? ?
4
12 12 8 8
4
??
??
=-
??
??
??
 
  ? ? 48 96 32 16 - + = - 
3. Among  
 ( ) ( )
2
1
1 : lim 2 4 6 ... 2 1
n
Sn
n
??
+ + + + = 
 ( )
( )
15 15 15 15
16
11
2 : lim 1 2 3 ...
16 n
Sn
n
??
+ + + + = 
 (1) Both (S1) and (S2) are true 
 (2) Only (S1) is true 
 (3) Both (S1) and (S2) are false 
 (4) Only (S2) is true 
Answer (1) 
Sol. 
? ?
1
2
1
: lim 2 4 6 ... 2
n
Sn
n
??
+ + + + 
 
( )
2
1
lim 2 1
2
n
nn
n
??
+
= 
 
( )
15 15 15 15
2
16
1
: lim 1 2 3 ...
n
Sn
n
??
+ + + + 
 ? 
1
1
1
lim
1
n
k
r
k
n
r
k
n
=
+
??
=
+
?
 
 Hence k = 15 
 ? 
15
1
16
1
lim
16
n
r
n
r
n
=
??
=
?
 
 ? Both S1 and S2 are correct 
Page 3


   
 
  
Answers & Solutions 
for 
JEE (Main)-2023 (Online) Phase-2 
(Mathematics, Physics and Chemistry) 
 
 
  
13/04/2023 
Morning 
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. The number of symmetric matrices of order 3, with 
all the entries from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} 
is 
 (1) 6
10 
 (2) 10
6
 
 (3) 9
10 
 (4) 10
9 
Answer (2) 
Sol. A = {0, 1, 2, 3, …8, 9} 
 
* * *
0 **
00 *
M
??
??
=
??
??
??
 
 ? Total symmetric matrix = 10
6
 
 
2. Let 
13
1 2 3 , 2
4
B
? ??
??
= ? ?
??
?? ??
??
be the adjoint of a matrix A 
and |A| = 2. Then ? ? 2 2 B
? ??
??
? - ? ? - ?
??
?? ?
??
 is equal to 
 (1) 0 (2) 16 
 (3) –16 (4) 32 
Answer (3) 
Sol. 
3 1
3 1 2 , 2
4
B
??
?
??
= ? ?
??
??
??
??
 
 And adj (A) = B, |A| = 2 
 ? |adj (A)| = |B| 
 ? 2
2
 = (8 – 3?) – 3(4 – 3?) + ?(–?) 
 ? ?
2
 – 6? + 8 = 0 
 ? (? – 4) (? – 2) = 0 
  ? = 4, 2 but ? > 2 so 4 ?= 
 Now 
 ? ? ? ?
3 14 4
3 2 2 4 8 4 1 2 8
444 4
B
?? ? ? ? ?
?
?? ? ? ? ?
? - ? ? - ? = - -
?? ? ? ? ?
?? ? ? ? ?
?
? ? ? ? ??
 
  ? ?
4
12 12 8 8
4
??
??
=-
??
??
??
 
  ? ? 48 96 32 16 - + = - 
3. Among  
 ( ) ( )
2
1
1 : lim 2 4 6 ... 2 1
n
Sn
n
??
+ + + + = 
 ( )
( )
15 15 15 15
16
11
2 : lim 1 2 3 ...
16 n
Sn
n
??
+ + + + = 
 (1) Both (S1) and (S2) are true 
 (2) Only (S1) is true 
 (3) Both (S1) and (S2) are false 
 (4) Only (S2) is true 
Answer (1) 
Sol. 
? ?
1
2
1
: lim 2 4 6 ... 2
n
Sn
n
??
+ + + + 
 
( )
2
1
lim 2 1
2
n
nn
n
??
+
= 
 
( )
15 15 15 15
2
16
1
: lim 1 2 3 ...
n
Sn
n
??
+ + + + 
 ? 
1
1
1
lim
1
n
k
r
k
n
r
k
n
=
+
??
=
+
?
 
 Hence k = 15 
 ? 
15
1
16
1
lim
16
n
r
n
r
n
=
??
=
?
 
 ? Both S1 and S2 are correct 
 
   
   
4. Fractional part of the number 
2022
4
15
 is equal to 
 (1) 
8
15
 (2) 
4
15
 
 (3) 
14
15
 (4) 
1
15
 
Answer (4) 
Sol. 
2022
4
15
 
 4 4 (mod 15) ? 
 
2
4 1(mod 15) ? 
 
2022
4 1(mod 15) ? 
 ? Fractional part of 
2022
41
15 15
= 
5. 
0
1
max 2sin cos sin3
3 x
x x x x
? ??
??
- + =
??
??
 
 (1) 
2 3 3
6
? + -
 (2) ? 
 (3) 0 (4) 
5 2 3 3
6
? + +
 
Answer (4) 
Sol. ( ) 1 2cos2 cos3 f x x x ? = - + 
 ( ) 4sin2 3sin3 f x x x ??=- 
 ( ) 0 fx ? = 
 
( )
23
1 2 2cos 1 4cos 3cos 0 x x x ? - - + - = 
  
( )( )
( ) 2cos 3 2cos 3 cos 1 0 x x x + - - = 
  
– 3 3
cos , , 1
22
x = 
  
5
, , 0
66
x
??
= 
 
5
2 3 3 0
6
f
? ??
?? = - - ?
??
??
 
 2 3 3 0
6
f
? ??
?? = - ?
??
??
 
 (0) 0 f ?? = 
 So 
5
6
x
?
= is local maxima point 
 Maximum value of 
5 5 3 1
()
6 6 2 3
f x f
?? ??
= = + +
??
??
 
   
5 2 3 3
6
? + +
= 
6. The negation of the statement ( ) ( ( ) A B C ??
( )) A B A ? ? ? is 
 (1) equivalent to ~ C 
 (2) equivalent to ~ BC ? 
 (3) a fallacy 
 (4) equivalent to ~ A 
Answer (4) 
Sol. ( ) ( ) ( ) A B C A B A ? ? ? ? ? 
 ( ) ( ) ( ) ( )
~~ A B C A B A ? ? ? ? ? 
 ( ) ( ) ( ) ~ A B C A B A ? ? ? ? ? 
 = A 
 ? Negation of statement = ~A 
7. For , x ? two real valued functions () fx and () gx
are such that, ( ) 1 g x x =+ and 
( ) 3 fog x x x = + - . Then (0) f is equal to 
 (1) 1 (2) 5 
 (3) 0 (4) –3 
Answer (2) 
Sol. ( ) ( )
3 f g x x x = + - 
 
( )
13 f x x x + = + - 
 Put 1 xt += 
  1 xt =- 
  x = (t – 1)
2
 
 f(t) = (t – 1)
2
 + 3 – (t – 1) 
 f(0) = 1 + 3 + 1 = 5 
Page 4


   
 
  
Answers & Solutions 
for 
JEE (Main)-2023 (Online) Phase-2 
(Mathematics, Physics and Chemistry) 
 
 
  
13/04/2023 
Morning 
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. The number of symmetric matrices of order 3, with 
all the entries from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} 
is 
 (1) 6
10 
 (2) 10
6
 
 (3) 9
10 
 (4) 10
9 
Answer (2) 
Sol. A = {0, 1, 2, 3, …8, 9} 
 
* * *
0 **
00 *
M
??
??
=
??
??
??
 
 ? Total symmetric matrix = 10
6
 
 
2. Let 
13
1 2 3 , 2
4
B
? ??
??
= ? ?
??
?? ??
??
be the adjoint of a matrix A 
and |A| = 2. Then ? ? 2 2 B
? ??
??
? - ? ? - ?
??
?? ?
??
 is equal to 
 (1) 0 (2) 16 
 (3) –16 (4) 32 
Answer (3) 
Sol. 
3 1
3 1 2 , 2
4
B
??
?
??
= ? ?
??
??
??
??
 
 And adj (A) = B, |A| = 2 
 ? |adj (A)| = |B| 
 ? 2
2
 = (8 – 3?) – 3(4 – 3?) + ?(–?) 
 ? ?
2
 – 6? + 8 = 0 
 ? (? – 4) (? – 2) = 0 
  ? = 4, 2 but ? > 2 so 4 ?= 
 Now 
 ? ? ? ?
3 14 4
3 2 2 4 8 4 1 2 8
444 4
B
?? ? ? ? ?
?
?? ? ? ? ?
? - ? ? - ? = - -
?? ? ? ? ?
?? ? ? ? ?
?
? ? ? ? ??
 
  ? ?
4
12 12 8 8
4
??
??
=-
??
??
??
 
  ? ? 48 96 32 16 - + = - 
3. Among  
 ( ) ( )
2
1
1 : lim 2 4 6 ... 2 1
n
Sn
n
??
+ + + + = 
 ( )
( )
15 15 15 15
16
11
2 : lim 1 2 3 ...
16 n
Sn
n
??
+ + + + = 
 (1) Both (S1) and (S2) are true 
 (2) Only (S1) is true 
 (3) Both (S1) and (S2) are false 
 (4) Only (S2) is true 
Answer (1) 
Sol. 
? ?
1
2
1
: lim 2 4 6 ... 2
n
Sn
n
??
+ + + + 
 
( )
2
1
lim 2 1
2
n
nn
n
??
+
= 
 
( )
15 15 15 15
2
16
1
: lim 1 2 3 ...
n
Sn
n
??
+ + + + 
 ? 
1
1
1
lim
1
n
k
r
k
n
r
k
n
=
+
??
=
+
?
 
 Hence k = 15 
 ? 
15
1
16
1
lim
16
n
r
n
r
n
=
??
=
?
 
 ? Both S1 and S2 are correct 
 
   
   
4. Fractional part of the number 
2022
4
15
 is equal to 
 (1) 
8
15
 (2) 
4
15
 
 (3) 
14
15
 (4) 
1
15
 
Answer (4) 
Sol. 
2022
4
15
 
 4 4 (mod 15) ? 
 
2
4 1(mod 15) ? 
 
2022
4 1(mod 15) ? 
 ? Fractional part of 
2022
41
15 15
= 
5. 
0
1
max 2sin cos sin3
3 x
x x x x
? ??
??
- + =
??
??
 
 (1) 
2 3 3
6
? + -
 (2) ? 
 (3) 0 (4) 
5 2 3 3
6
? + +
 
Answer (4) 
Sol. ( ) 1 2cos2 cos3 f x x x ? = - + 
 ( ) 4sin2 3sin3 f x x x ??=- 
 ( ) 0 fx ? = 
 
( )
23
1 2 2cos 1 4cos 3cos 0 x x x ? - - + - = 
  
( )( )
( ) 2cos 3 2cos 3 cos 1 0 x x x + - - = 
  
– 3 3
cos , , 1
22
x = 
  
5
, , 0
66
x
??
= 
 
5
2 3 3 0
6
f
? ??
?? = - - ?
??
??
 
 2 3 3 0
6
f
? ??
?? = - ?
??
??
 
 (0) 0 f ?? = 
 So 
5
6
x
?
= is local maxima point 
 Maximum value of 
5 5 3 1
()
6 6 2 3
f x f
?? ??
= = + +
??
??
 
   
5 2 3 3
6
? + +
= 
6. The negation of the statement ( ) ( ( ) A B C ??
( )) A B A ? ? ? is 
 (1) equivalent to ~ C 
 (2) equivalent to ~ BC ? 
 (3) a fallacy 
 (4) equivalent to ~ A 
Answer (4) 
Sol. ( ) ( ) ( ) A B C A B A ? ? ? ? ? 
 ( ) ( ) ( ) ( )
~~ A B C A B A ? ? ? ? ? 
 ( ) ( ) ( ) ~ A B C A B A ? ? ? ? ? 
 = A 
 ? Negation of statement = ~A 
7. For , x ? two real valued functions () fx and () gx
are such that, ( ) 1 g x x =+ and 
( ) 3 fog x x x = + - . Then (0) f is equal to 
 (1) 1 (2) 5 
 (3) 0 (4) –3 
Answer (2) 
Sol. ( ) ( )
3 f g x x x = + - 
 
( )
13 f x x x + = + - 
 Put 1 xt += 
  1 xt =- 
  x = (t – 1)
2
 
 f(t) = (t – 1)
2
 + 3 – (t – 1) 
 f(0) = 1 + 3 + 1 = 5 
 
   
   
8. Let 
1 2 3 10
, , ...., s s s s respectively be the sum of 12 
terms of 10 A.Ps whose first terms are 1, 2, 3,....,10 
and the common differences are 1, 3, 5,...,19 
respectively. Then 
10
1
i
i
s
=
?
is equal to 
 (1) 7220 
 (2) 7360 
 (3) 7260 
 (4) 7380 
Answer (3) 
Sol. 1 10 r ?? 
 ( )( ) ( )
12
2 12 1 2 1 6 2 22 11
2
i
S r r r r ?? = + - - = + -
??
 
  = 6(24r – 11) 
 ( ) ? ?
10 10
11
10
6 24 11 6 24 11 240 11
2
r
rr
Sr
==
= - = ? - + -
??
 
  = 30 (242) = 7260 
9. Let the equation of plane passing through the line 
of intersection of the planes 22 x y az + + = and 
3 x y z - + = be 5 11 6 1 x y bz a - + = - . For c ? , 
if the distance of this plane from the point ( , , ) a c c -
is 
2
,
a
then 
ab
c
+
is equal to 
 (1) 2 (2) 4 
 (3) – 4 (4) – 2 
Answer (3) 
Sol. Let the equation of plane passing through the 
intersection of two given planes: 
 x + 2y + az – 2 + ?(x – y + z – 3) = 0 
 ? x(? + 1) + y(2 – ?) + z(a + ?) –2 –3? = 0 
 This is same as 5x –11y + bz = 6a – 1 
 
1 2 2 3
5 11 6 1
a
ba
? + - ? + ? + ?
= = =
--
 
 ? –11? – 11 = 10 – 5? 
 ? 
7
6 21
2
-
? = - ? ? = 
 
7 21
22
2 2 3
22
11 6 1 11 6 1 aa
+-
- ? + ?
= ? =
- - - -
 
 ? 
( )
1 17
6 1 17 3
2 2 6 1
aa
a
-
- = ? - = ? =
-
 
  
7
3
21
2
11 2
a
bb
-
- ? + ?
= ? - =
-
 
 ? 
1
1
22
b
b - = - ? = 
 ? point (a, – c, c) ? (3, – c, c) 
  Distance 
22
3 a
== 
  Plane: 5x – 11y + z = 17 
  
15 11 17 2
147 3
cc + + -
= 
 ? |12c – 2| = 14 
 ? c = – 1, 
4
3
 
  c = – 1 ( ) c ? 
 ? 
31
4
1
ab
c
++
= = -
-
 
10. Let PQ be a focal chord of the parabola 
2
36 yx =
of length 100, making an acute angle with the 
positive x-axis. Let the ordinate of P be positive and 
M be the point on the line segment PQ such that 
PM : MQ = 3 : 1. Then which of the following points 
does NOT lie on the line passing through M and 
perpendicular to the line PQ?  
 (1) ( 6, 45) - (2) (6, 29) 
 (3) (3, 33) (4) ( 3, 43) - 
Answer (4) 
Sol.  
Page 5


   
 
  
Answers & Solutions 
for 
JEE (Main)-2023 (Online) Phase-2 
(Mathematics, Physics and Chemistry) 
 
 
  
13/04/2023 
Morning 
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. The number of symmetric matrices of order 3, with 
all the entries from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} 
is 
 (1) 6
10 
 (2) 10
6
 
 (3) 9
10 
 (4) 10
9 
Answer (2) 
Sol. A = {0, 1, 2, 3, …8, 9} 
 
* * *
0 **
00 *
M
??
??
=
??
??
??
 
 ? Total symmetric matrix = 10
6
 
 
2. Let 
13
1 2 3 , 2
4
B
? ??
??
= ? ?
??
?? ??
??
be the adjoint of a matrix A 
and |A| = 2. Then ? ? 2 2 B
? ??
??
? - ? ? - ?
??
?? ?
??
 is equal to 
 (1) 0 (2) 16 
 (3) –16 (4) 32 
Answer (3) 
Sol. 
3 1
3 1 2 , 2
4
B
??
?
??
= ? ?
??
??
??
??
 
 And adj (A) = B, |A| = 2 
 ? |adj (A)| = |B| 
 ? 2
2
 = (8 – 3?) – 3(4 – 3?) + ?(–?) 
 ? ?
2
 – 6? + 8 = 0 
 ? (? – 4) (? – 2) = 0 
  ? = 4, 2 but ? > 2 so 4 ?= 
 Now 
 ? ? ? ?
3 14 4
3 2 2 4 8 4 1 2 8
444 4
B
?? ? ? ? ?
?
?? ? ? ? ?
? - ? ? - ? = - -
?? ? ? ? ?
?? ? ? ? ?
?
? ? ? ? ??
 
  ? ?
4
12 12 8 8
4
??
??
=-
??
??
??
 
  ? ? 48 96 32 16 - + = - 
3. Among  
 ( ) ( )
2
1
1 : lim 2 4 6 ... 2 1
n
Sn
n
??
+ + + + = 
 ( )
( )
15 15 15 15
16
11
2 : lim 1 2 3 ...
16 n
Sn
n
??
+ + + + = 
 (1) Both (S1) and (S2) are true 
 (2) Only (S1) is true 
 (3) Both (S1) and (S2) are false 
 (4) Only (S2) is true 
Answer (1) 
Sol. 
? ?
1
2
1
: lim 2 4 6 ... 2
n
Sn
n
??
+ + + + 
 
( )
2
1
lim 2 1
2
n
nn
n
??
+
= 
 
( )
15 15 15 15
2
16
1
: lim 1 2 3 ...
n
Sn
n
??
+ + + + 
 ? 
1
1
1
lim
1
n
k
r
k
n
r
k
n
=
+
??
=
+
?
 
 Hence k = 15 
 ? 
15
1
16
1
lim
16
n
r
n
r
n
=
??
=
?
 
 ? Both S1 and S2 are correct 
 
   
   
4. Fractional part of the number 
2022
4
15
 is equal to 
 (1) 
8
15
 (2) 
4
15
 
 (3) 
14
15
 (4) 
1
15
 
Answer (4) 
Sol. 
2022
4
15
 
 4 4 (mod 15) ? 
 
2
4 1(mod 15) ? 
 
2022
4 1(mod 15) ? 
 ? Fractional part of 
2022
41
15 15
= 
5. 
0
1
max 2sin cos sin3
3 x
x x x x
? ??
??
- + =
??
??
 
 (1) 
2 3 3
6
? + -
 (2) ? 
 (3) 0 (4) 
5 2 3 3
6
? + +
 
Answer (4) 
Sol. ( ) 1 2cos2 cos3 f x x x ? = - + 
 ( ) 4sin2 3sin3 f x x x ??=- 
 ( ) 0 fx ? = 
 
( )
23
1 2 2cos 1 4cos 3cos 0 x x x ? - - + - = 
  
( )( )
( ) 2cos 3 2cos 3 cos 1 0 x x x + - - = 
  
– 3 3
cos , , 1
22
x = 
  
5
, , 0
66
x
??
= 
 
5
2 3 3 0
6
f
? ??
?? = - - ?
??
??
 
 2 3 3 0
6
f
? ??
?? = - ?
??
??
 
 (0) 0 f ?? = 
 So 
5
6
x
?
= is local maxima point 
 Maximum value of 
5 5 3 1
()
6 6 2 3
f x f
?? ??
= = + +
??
??
 
   
5 2 3 3
6
? + +
= 
6. The negation of the statement ( ) ( ( ) A B C ??
( )) A B A ? ? ? is 
 (1) equivalent to ~ C 
 (2) equivalent to ~ BC ? 
 (3) a fallacy 
 (4) equivalent to ~ A 
Answer (4) 
Sol. ( ) ( ) ( ) A B C A B A ? ? ? ? ? 
 ( ) ( ) ( ) ( )
~~ A B C A B A ? ? ? ? ? 
 ( ) ( ) ( ) ~ A B C A B A ? ? ? ? ? 
 = A 
 ? Negation of statement = ~A 
7. For , x ? two real valued functions () fx and () gx
are such that, ( ) 1 g x x =+ and 
( ) 3 fog x x x = + - . Then (0) f is equal to 
 (1) 1 (2) 5 
 (3) 0 (4) –3 
Answer (2) 
Sol. ( ) ( )
3 f g x x x = + - 
 
( )
13 f x x x + = + - 
 Put 1 xt += 
  1 xt =- 
  x = (t – 1)
2
 
 f(t) = (t – 1)
2
 + 3 – (t – 1) 
 f(0) = 1 + 3 + 1 = 5 
 
   
   
8. Let 
1 2 3 10
, , ...., s s s s respectively be the sum of 12 
terms of 10 A.Ps whose first terms are 1, 2, 3,....,10 
and the common differences are 1, 3, 5,...,19 
respectively. Then 
10
1
i
i
s
=
?
is equal to 
 (1) 7220 
 (2) 7360 
 (3) 7260 
 (4) 7380 
Answer (3) 
Sol. 1 10 r ?? 
 ( )( ) ( )
12
2 12 1 2 1 6 2 22 11
2
i
S r r r r ?? = + - - = + -
??
 
  = 6(24r – 11) 
 ( ) ? ?
10 10
11
10
6 24 11 6 24 11 240 11
2
r
rr
Sr
==
= - = ? - + -
??
 
  = 30 (242) = 7260 
9. Let the equation of plane passing through the line 
of intersection of the planes 22 x y az + + = and 
3 x y z - + = be 5 11 6 1 x y bz a - + = - . For c ? , 
if the distance of this plane from the point ( , , ) a c c -
is 
2
,
a
then 
ab
c
+
is equal to 
 (1) 2 (2) 4 
 (3) – 4 (4) – 2 
Answer (3) 
Sol. Let the equation of plane passing through the 
intersection of two given planes: 
 x + 2y + az – 2 + ?(x – y + z – 3) = 0 
 ? x(? + 1) + y(2 – ?) + z(a + ?) –2 –3? = 0 
 This is same as 5x –11y + bz = 6a – 1 
 
1 2 2 3
5 11 6 1
a
ba
? + - ? + ? + ?
= = =
--
 
 ? –11? – 11 = 10 – 5? 
 ? 
7
6 21
2
-
? = - ? ? = 
 
7 21
22
2 2 3
22
11 6 1 11 6 1 aa
+-
- ? + ?
= ? =
- - - -
 
 ? 
( )
1 17
6 1 17 3
2 2 6 1
aa
a
-
- = ? - = ? =
-
 
  
7
3
21
2
11 2
a
bb
-
- ? + ?
= ? - =
-
 
 ? 
1
1
22
b
b - = - ? = 
 ? point (a, – c, c) ? (3, – c, c) 
  Distance 
22
3 a
== 
  Plane: 5x – 11y + z = 17 
  
15 11 17 2
147 3
cc + + -
= 
 ? |12c – 2| = 14 
 ? c = – 1, 
4
3
 
  c = – 1 ( ) c ? 
 ? 
31
4
1
ab
c
++
= = -
-
 
10. Let PQ be a focal chord of the parabola 
2
36 yx =
of length 100, making an acute angle with the 
positive x-axis. Let the ordinate of P be positive and 
M be the point on the line segment PQ such that 
PM : MQ = 3 : 1. Then which of the following points 
does NOT lie on the line passing through M and 
perpendicular to the line PQ?  
 (1) ( 6, 45) - (2) (6, 29) 
 (3) (3, 33) (4) ( 3, 43) - 
Answer (4) 
Sol.  
 
   
   
 Length of focal chord at ( )
2
1
100 t a t
t
??
= + =
??
??
 
 Where a = 9 
  
1 10
3
t
t
+ = ? 
 ? 
11
3, , 3,
33
t
-
=- 
 Since ordinate of P is +ve 
 ? t = 3 or 
1
3
 
 
   (t = 3 as slope of PQ is positive) 
 ? M(21, 9) 
 Required line : ( )
80
9 21
60
yx
-
- = - 
 ? 4x + 3y = 111 
 (–3, 43) does not lie on line 
 ? Option (4) is correct. 
11. Let 
ˆ ˆ ˆ ˆ ˆ ˆ
4 2 , 3 2 7 a i j k b i j k and 
ˆ ˆ ˆ
24 c i j k . If a vector d satisfies 
d b c b and 24 da , then 
2
d is equal to 
 (1) 323 
 (2) 423 
 (3) 313 
 (4) 413 
Answer (4) 
Sol.  d b c b ? = ? 
 ? 
( )
0 d c b - ? = 
 ? d c b - = ? 
( )
as d c c a b ? ? = 
 ? d c b = + ? 
  24 ad ?= (given) 
  24 a c b a ? + ? ? = 
 ? 6 + ?(3 – 8 + 14) = 24 
 ? 9? = 18 
 ? ? = 2 
 ? 2 d c b =+ 
   
ˆ
8 5 18 i j k = - + 
  
2
64 25 324 d = + + 
   = 413 
 ? Option (4) is correct. 
12. A coin is biased so that the head is 3 times as likely 
to occur as tail. This coin is tossed until a head or 
three tails occur. If X denotes the number of tosses 
of the coin, then the mean of X is  
 (1) 
37
16
 
 (2) 
15
16
 
 (3) 
21
16
 
 (4) 
81
64
 
Answer (3) 
Sol.  P(H) = 3P(T) 
 ? ( ) ( )
31
,
44
P H P T ==  
 Since coin is tossed till either 1H or 3T occurs. So, 
process will end in the maximum of 3 throws. 
 Xi  1 2 3 
 P(Xi) 
3
4
 
13
44
? 
1 1 1 1 1 3
4 4 4 4 4 4
? ? + ? ? 
 Mean 
3 3 1
1 2 3
4 16 16
= ? + ? + ? 
   
21
16
= 
 Option (3) is correct.