JEE Main 2023 April 6 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) 
 
 
  
06/04/2023 
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) 
 
 
  
06/04/2023 
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 straight lines l
1
 and l
2
 pass through the origin 
and trisect the line segment of the line L : 9x + 5y = 
45 between the axes. If m
1
 and m
2
 are the slopes 
of the lines l
1
 and l
2
, then the point of intersection of 
the line y = (m
1
 + m
2
)x with L lies on 
 (1) y – 2x = 5 
 (2) 6x + y = 10 
 (3) y – x = 5 
 (4) 6x – y = 15 
Answer (3) 
Sol.  
 L : 9x + 5y = 45 
 ? 1
59
xy
+= 
 ? 
10
,3
3
C
??
?
??
??
 
  
5
,6
3
D
??
?
??
??
 
 ? 
12
9 6 3 18
,
10 5 5
mm
?
= = = 
 ? 
9 36 9
10 10 2
y x x
??
= + =
??
??
 …(i) 
 So, intersection point with L 
  
45 10
7 45 ,
77
y y x = ? = = 
 ? Option (3) is correct. 
2. If the ratio of the fifth term from the beginning to the 
fifth term from the end in the expansion of 
4
4
1
2
3
n
??
+
??
??
 is 6 : 1 , then the third term from the 
beginning is: 
 (1) 30 2 (2) 30 3 
 (3) 60 2 (4) 60 3 
Answer (4) 
Sol. Given expansion 
4
1
2
43
n
??
+
??
??
 
 (5
th
 term from beginning)  
 
( )
4
4
4
54
4
1
2
3
n
n
TC
-
??
=?
??
??
 
 (5
th
 term from end) 
 
( )
4
4
4
54
4
1
2
3
n
n
TC
-
??
?
=
??
??
 
 Now, 
88
5
44
5
6
2 3 6
1
nn
T
T
--
= ? ? =
?
 
 ? 
8
2
( 6) ( 6)
n-
= 
 ? n = 10 
 ? 
( )
2
8
10 4
32
4
1 45 4
2
3 3
TC
?? ?
==
??
??
 
    = 60 3 
3. The mean and variance of a set of 15 numbers are 
12 and 14 respectively. The mean and variance of 
another set of 15 numbers are 14 and ?
2
 
respectively. If the variance of all the 30 numbers in 
the two sets is 13, then ?
2
 is equal to 
 (1) 10 
 (2) 11 
 (3) 9 
 (4) 12 
Answer (1) 
Page 3


   
 
  
Answers & Solutions 
for 
JEE (Main)-2023 (Online) Phase-2 
(Mathematics, Physics and Chemistry) 
 
 
  
06/04/2023 
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 straight lines l
1
 and l
2
 pass through the origin 
and trisect the line segment of the line L : 9x + 5y = 
45 between the axes. If m
1
 and m
2
 are the slopes 
of the lines l
1
 and l
2
, then the point of intersection of 
the line y = (m
1
 + m
2
)x with L lies on 
 (1) y – 2x = 5 
 (2) 6x + y = 10 
 (3) y – x = 5 
 (4) 6x – y = 15 
Answer (3) 
Sol.  
 L : 9x + 5y = 45 
 ? 1
59
xy
+= 
 ? 
10
,3
3
C
??
?
??
??
 
  
5
,6
3
D
??
?
??
??
 
 ? 
12
9 6 3 18
,
10 5 5
mm
?
= = = 
 ? 
9 36 9
10 10 2
y x x
??
= + =
??
??
 …(i) 
 So, intersection point with L 
  
45 10
7 45 ,
77
y y x = ? = = 
 ? Option (3) is correct. 
2. If the ratio of the fifth term from the beginning to the 
fifth term from the end in the expansion of 
4
4
1
2
3
n
??
+
??
??
 is 6 : 1 , then the third term from the 
beginning is: 
 (1) 30 2 (2) 30 3 
 (3) 60 2 (4) 60 3 
Answer (4) 
Sol. Given expansion 
4
1
2
43
n
??
+
??
??
 
 (5
th
 term from beginning)  
 
( )
4
4
4
54
4
1
2
3
n
n
TC
-
??
=?
??
??
 
 (5
th
 term from end) 
 
( )
4
4
4
54
4
1
2
3
n
n
TC
-
??
?
=
??
??
 
 Now, 
88
5
44
5
6
2 3 6
1
nn
T
T
--
= ? ? =
?
 
 ? 
8
2
( 6) ( 6)
n-
= 
 ? n = 10 
 ? 
( )
2
8
10 4
32
4
1 45 4
2
3 3
TC
?? ?
==
??
??
 
    = 60 3 
3. The mean and variance of a set of 15 numbers are 
12 and 14 respectively. The mean and variance of 
another set of 15 numbers are 14 and ?
2
 
respectively. If the variance of all the 30 numbers in 
the two sets is 13, then ?
2
 is equal to 
 (1) 10 
 (2) 11 
 (3) 9 
 (4) 12 
Answer (1) 
 
   
   
Sol. 15 12
i
x=?
?
 and 
2
2
12 14
15
i
x
-=
?
 
 And 15 14
i
y=?
?
 and 
2
22
14
15
i
y
- = ?
?
 
 Now, 
2
2
(14 144) 15 ( 196) 15
13 13
30
+ ? + ? + ?
=- 
 ? 
2
10 ?= 
4. A pair of dice is thrown 5 times. For each throw, a 
total of 5 is considered a success. If the probability 
of at least 4 successes is 
11
,
3
k
 then k is equal to 
 (1) 82 (2) 75 
 (3) 164 (4) 123 
Answer (4) 
Sol. P(success) = 
41
36 9
= 
 P(failure) = 
8
9
 
 ? Required probability = 
45
55
45
1 8 1
9 9 9
CC
? ? ? ?
+
? ? ? ?
? ? ? ?
 
   = 
5 5 5
8 1 41
5
9 9 9
? + = 
   = 
11
123
3
 
 ? k = 123 
5. Let the position vectors of the points A, B, C and D 
be 
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
5 5 2 , 2 3 , 2 4 i j k i j k i j k + + ? + + - + ? + and 
ˆ ˆ ˆ
5 6 . i j k - + + Let the set S = {? ? : the points A, 
B, C and D are coplanar}. The ( )
2
2
S ??
?+
?
 is equal 
to 
 (1) 25 (2) 
37
2
 
 (3) 13 (4) 41 
Answer (4) 
Sol. A(5, 5, 2?) 
 B(1, 2, 3) 
 C(–2, ?, 4) 
 D(–1, 5, 6) 
 (–4,–3,3 2 ) AB -? 
 (–7, 5,4 2 ) AC ? - - ? 
 (–6,0,6 2 ) AD -? 
 ? [ ] 0 AB AC AD = 
 
4 3 3 2
7 5 4 2 0
6 0 6 2
- - - ?
- ? - - ? =
- - ?
 
 ? –4(? – 3) (? – 2) = 0 
 ? ? = 3, ? = 2 
  
2 2 2
( 2) 5 4
s ??
? + = +
?
 
   = 41 
6. Let 
( )
( )
22
2
sec tan
( ) .
tan 1
x x x
I x dx
xx
+
=
+
?
 If I(0) = 0, then 
4
I
? ??
??
??
 is equal to 
 (1) 
( )
( )
2
2
4
log
16 4 4
e
?+
?
+
?+
 
 (2) 
( )
( )
2
2
4
log –
16 4 4
e
?+
?
?+
 
 (3) 
( )
( )
2
2
4
log –
32 4 4
e
?+
?
?+
 
 (4) 
( )
( )
2
2
4
log
32 4 4
e
?+
?
+
?+
 
Answer (3) 
Sol. 
( )
2
2
2
sec tan
tan 1
x x x
x dx
xx
??
+
??
??
+
??
?
 
 
( )
2
2
tan 1 tan 1
xx
dx
x x x x
=+
++
?
 
 2
tan 1
x
I dx
xx
=
+
?
 
 
cos
2
sin cos
xx
dx
x x x
=
+
?
 
 Let xsinx + cosx = t 
 (x cosx + sinx – sinx) dx = dt 
 2 2log
dt
tc
t
= = +
?
 
 = 2log |xsinx + cosx| + c 
Page 4


   
 
  
Answers & Solutions 
for 
JEE (Main)-2023 (Online) Phase-2 
(Mathematics, Physics and Chemistry) 
 
 
  
06/04/2023 
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 straight lines l
1
 and l
2
 pass through the origin 
and trisect the line segment of the line L : 9x + 5y = 
45 between the axes. If m
1
 and m
2
 are the slopes 
of the lines l
1
 and l
2
, then the point of intersection of 
the line y = (m
1
 + m
2
)x with L lies on 
 (1) y – 2x = 5 
 (2) 6x + y = 10 
 (3) y – x = 5 
 (4) 6x – y = 15 
Answer (3) 
Sol.  
 L : 9x + 5y = 45 
 ? 1
59
xy
+= 
 ? 
10
,3
3
C
??
?
??
??
 
  
5
,6
3
D
??
?
??
??
 
 ? 
12
9 6 3 18
,
10 5 5
mm
?
= = = 
 ? 
9 36 9
10 10 2
y x x
??
= + =
??
??
 …(i) 
 So, intersection point with L 
  
45 10
7 45 ,
77
y y x = ? = = 
 ? Option (3) is correct. 
2. If the ratio of the fifth term from the beginning to the 
fifth term from the end in the expansion of 
4
4
1
2
3
n
??
+
??
??
 is 6 : 1 , then the third term from the 
beginning is: 
 (1) 30 2 (2) 30 3 
 (3) 60 2 (4) 60 3 
Answer (4) 
Sol. Given expansion 
4
1
2
43
n
??
+
??
??
 
 (5
th
 term from beginning)  
 
( )
4
4
4
54
4
1
2
3
n
n
TC
-
??
=?
??
??
 
 (5
th
 term from end) 
 
( )
4
4
4
54
4
1
2
3
n
n
TC
-
??
?
=
??
??
 
 Now, 
88
5
44
5
6
2 3 6
1
nn
T
T
--
= ? ? =
?
 
 ? 
8
2
( 6) ( 6)
n-
= 
 ? n = 10 
 ? 
( )
2
8
10 4
32
4
1 45 4
2
3 3
TC
?? ?
==
??
??
 
    = 60 3 
3. The mean and variance of a set of 15 numbers are 
12 and 14 respectively. The mean and variance of 
another set of 15 numbers are 14 and ?
2
 
respectively. If the variance of all the 30 numbers in 
the two sets is 13, then ?
2
 is equal to 
 (1) 10 
 (2) 11 
 (3) 9 
 (4) 12 
Answer (1) 
 
   
   
Sol. 15 12
i
x=?
?
 and 
2
2
12 14
15
i
x
-=
?
 
 And 15 14
i
y=?
?
 and 
2
22
14
15
i
y
- = ?
?
 
 Now, 
2
2
(14 144) 15 ( 196) 15
13 13
30
+ ? + ? + ?
=- 
 ? 
2
10 ?= 
4. A pair of dice is thrown 5 times. For each throw, a 
total of 5 is considered a success. If the probability 
of at least 4 successes is 
11
,
3
k
 then k is equal to 
 (1) 82 (2) 75 
 (3) 164 (4) 123 
Answer (4) 
Sol. P(success) = 
41
36 9
= 
 P(failure) = 
8
9
 
 ? Required probability = 
45
55
45
1 8 1
9 9 9
CC
? ? ? ?
+
? ? ? ?
? ? ? ?
 
   = 
5 5 5
8 1 41
5
9 9 9
? + = 
   = 
11
123
3
 
 ? k = 123 
5. Let the position vectors of the points A, B, C and D 
be 
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
5 5 2 , 2 3 , 2 4 i j k i j k i j k + + ? + + - + ? + and 
ˆ ˆ ˆ
5 6 . i j k - + + Let the set S = {? ? : the points A, 
B, C and D are coplanar}. The ( )
2
2
S ??
?+
?
 is equal 
to 
 (1) 25 (2) 
37
2
 
 (3) 13 (4) 41 
Answer (4) 
Sol. A(5, 5, 2?) 
 B(1, 2, 3) 
 C(–2, ?, 4) 
 D(–1, 5, 6) 
 (–4,–3,3 2 ) AB -? 
 (–7, 5,4 2 ) AC ? - - ? 
 (–6,0,6 2 ) AD -? 
 ? [ ] 0 AB AC AD = 
 
4 3 3 2
7 5 4 2 0
6 0 6 2
- - - ?
- ? - - ? =
- - ?
 
 ? –4(? – 3) (? – 2) = 0 
 ? ? = 3, ? = 2 
  
2 2 2
( 2) 5 4
s ??
? + = +
?
 
   = 41 
6. Let 
( )
( )
22
2
sec tan
( ) .
tan 1
x x x
I x dx
xx
+
=
+
?
 If I(0) = 0, then 
4
I
? ??
??
??
 is equal to 
 (1) 
( )
( )
2
2
4
log
16 4 4
e
?+
?
+
?+
 
 (2) 
( )
( )
2
2
4
log –
16 4 4
e
?+
?
?+
 
 (3) 
( )
( )
2
2
4
log –
32 4 4
e
?+
?
?+
 
 (4) 
( )
( )
2
2
4
log
32 4 4
e
?+
?
+
?+
 
Answer (3) 
Sol. 
( )
2
2
2
sec tan
tan 1
x x x
x dx
xx
??
+
??
??
+
??
?
 
 
( )
2
2
tan 1 tan 1
xx
dx
x x x x
=+
++
?
 
 2
tan 1
x
I dx
xx
=
+
?
 
 
cos
2
sin cos
xx
dx
x x x
=
+
?
 
 Let xsinx + cosx = t 
 (x cosx + sinx – sinx) dx = dt 
 2 2log
dt
tc
t
= = +
?
 
 = 2log |xsinx + cosx| + c 
 
   
   
 ? 
22
2
( sec tan )
( tan 1)
+
+
?
x x x x
dx
x
 
 
2
tan 1
-
=
+
x
xx
 + 2 log |xsinx + cosx| + c 
 I(0) = 0 
 ? c = 0 
 
2
1 4
I 2log 1
44
2
11
4
? ??
??
?? ? ? ? ? ??
= - + +
? ? ? ?
?
? ? ? ?
?+
 
 = 
( )
( )
2
2
e
4
log
32 4 4
?+
?
-
?+
 
7. Let 
ˆ ˆ ˆ ˆ ˆ ˆ
2 3 4 , 2 2 a i j k b i j k = + + = - - and 
ˆ ˆ ˆ
43 c i j k = - + + . If d is a vector perpendicular to both 
b and , c and · 18, ad = then 
2
ad ? is equal to 
 (1) 640 (2) 680 
 (3) 720 (4) 760 
Answer (3) 
Sol. 
ˆ ˆ ˆ
22 ? = - + b c i j k 
 ? 
( )
ˆ ˆ ˆ
22 = ? - + d i j k 
 · 18 = ad 
 ? ? = 2 
 ? 
( )
2 22
2
·· ? = - a d a d a d 
 = 720 
8. Let 
11
5 ( ) 4 3, 0. f x f x
xx
??
+ = + ?
??
??
 Then 
2
1
18 ( ) f x dx
?
 
is equal to 
 (1) 5 loge 2 + 3 (2) 10 loge 2 + 6 
 (3) 10 loge 2 – 6 (4) 5 loge 2 – 3 
Answer (3) 
Sol. 
11
5 ( ) 4 3
??
+ = +
??
??
f x f
xx
 …(i) 
 Replace 
1
? x
x
 
 
1
5 4 ( ) 3
??
+ = +
??
??
f f x x
x
 …(ii)  
 Solving (i) and (ii) we get 
 
5
9 ( ) 4 3 = - + f x x
x
 
 
22
11
15
( ) 4 3
9
??
= - +
??
??
??
f x dx x dx
x
 
 
2
2
1
1
5log 2 3
9
??
= - +
??
x x x 
 
? ?
1
5log 3
9
=- x 
 ? 
2
1
18 ( ) 10log 2 6 =-
? e
f x dx 
9. Statement ( ) ( ) P Q R Q ? ? ? is logically 
equivalent to   
 (1) ( ) ( ) P R Q R ? ? ? 
 (2) ( ) P R Q ?? 
 (3) ( ) ( ) P R Q R ? ? ? 
 (4) ( ) P R Q ?? 
Answer (4) 
Sol. ( ) ( ) ? ? ? P Q R Q 
 ? ( ) ( ) P Q R Q ? ? ? ? ?  
 ? () Q P R ? ? ? ? 
 ? () Q P R ? ? ? 
 ? () P R Q ? ? ? 
 ? () P R Q ?? 
10. If the system of equations 
 x + y + az = b 
 2x + 5y + 2z = 6 
 x + 2y + 3z = 3 
 has infinitely many solutions, then 2a + 3b is equal 
to 
 (1) 25 (2) 20 
 (3) 23 (4) 28 
Answer (3) 
Sol. x + y + az = b …(i) 
 2x + 5y + 2z = 6 …(ii) 
 x + 2y + 3z = 3 …(iii) 
 3 (iii) – (ii) 
 x + y + 7z = 3 
 x + y + az = b 
 ? a = 7, b = 3 ( ? solutions are infinite) 
 ? 2a + 3b 
  = 14 + 9 = 23 
Page 5


   
 
  
Answers & Solutions 
for 
JEE (Main)-2023 (Online) Phase-2 
(Mathematics, Physics and Chemistry) 
 
 
  
06/04/2023 
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 straight lines l
1
 and l
2
 pass through the origin 
and trisect the line segment of the line L : 9x + 5y = 
45 between the axes. If m
1
 and m
2
 are the slopes 
of the lines l
1
 and l
2
, then the point of intersection of 
the line y = (m
1
 + m
2
)x with L lies on 
 (1) y – 2x = 5 
 (2) 6x + y = 10 
 (3) y – x = 5 
 (4) 6x – y = 15 
Answer (3) 
Sol.  
 L : 9x + 5y = 45 
 ? 1
59
xy
+= 
 ? 
10
,3
3
C
??
?
??
??
 
  
5
,6
3
D
??
?
??
??
 
 ? 
12
9 6 3 18
,
10 5 5
mm
?
= = = 
 ? 
9 36 9
10 10 2
y x x
??
= + =
??
??
 …(i) 
 So, intersection point with L 
  
45 10
7 45 ,
77
y y x = ? = = 
 ? Option (3) is correct. 
2. If the ratio of the fifth term from the beginning to the 
fifth term from the end in the expansion of 
4
4
1
2
3
n
??
+
??
??
 is 6 : 1 , then the third term from the 
beginning is: 
 (1) 30 2 (2) 30 3 
 (3) 60 2 (4) 60 3 
Answer (4) 
Sol. Given expansion 
4
1
2
43
n
??
+
??
??
 
 (5
th
 term from beginning)  
 
( )
4
4
4
54
4
1
2
3
n
n
TC
-
??
=?
??
??
 
 (5
th
 term from end) 
 
( )
4
4
4
54
4
1
2
3
n
n
TC
-
??
?
=
??
??
 
 Now, 
88
5
44
5
6
2 3 6
1
nn
T
T
--
= ? ? =
?
 
 ? 
8
2
( 6) ( 6)
n-
= 
 ? n = 10 
 ? 
( )
2
8
10 4
32
4
1 45 4
2
3 3
TC
?? ?
==
??
??
 
    = 60 3 
3. The mean and variance of a set of 15 numbers are 
12 and 14 respectively. The mean and variance of 
another set of 15 numbers are 14 and ?
2
 
respectively. If the variance of all the 30 numbers in 
the two sets is 13, then ?
2
 is equal to 
 (1) 10 
 (2) 11 
 (3) 9 
 (4) 12 
Answer (1) 
 
   
   
Sol. 15 12
i
x=?
?
 and 
2
2
12 14
15
i
x
-=
?
 
 And 15 14
i
y=?
?
 and 
2
22
14
15
i
y
- = ?
?
 
 Now, 
2
2
(14 144) 15 ( 196) 15
13 13
30
+ ? + ? + ?
=- 
 ? 
2
10 ?= 
4. A pair of dice is thrown 5 times. For each throw, a 
total of 5 is considered a success. If the probability 
of at least 4 successes is 
11
,
3
k
 then k is equal to 
 (1) 82 (2) 75 
 (3) 164 (4) 123 
Answer (4) 
Sol. P(success) = 
41
36 9
= 
 P(failure) = 
8
9
 
 ? Required probability = 
45
55
45
1 8 1
9 9 9
CC
? ? ? ?
+
? ? ? ?
? ? ? ?
 
   = 
5 5 5
8 1 41
5
9 9 9
? + = 
   = 
11
123
3
 
 ? k = 123 
5. Let the position vectors of the points A, B, C and D 
be 
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
5 5 2 , 2 3 , 2 4 i j k i j k i j k + + ? + + - + ? + and 
ˆ ˆ ˆ
5 6 . i j k - + + Let the set S = {? ? : the points A, 
B, C and D are coplanar}. The ( )
2
2
S ??
?+
?
 is equal 
to 
 (1) 25 (2) 
37
2
 
 (3) 13 (4) 41 
Answer (4) 
Sol. A(5, 5, 2?) 
 B(1, 2, 3) 
 C(–2, ?, 4) 
 D(–1, 5, 6) 
 (–4,–3,3 2 ) AB -? 
 (–7, 5,4 2 ) AC ? - - ? 
 (–6,0,6 2 ) AD -? 
 ? [ ] 0 AB AC AD = 
 
4 3 3 2
7 5 4 2 0
6 0 6 2
- - - ?
- ? - - ? =
- - ?
 
 ? –4(? – 3) (? – 2) = 0 
 ? ? = 3, ? = 2 
  
2 2 2
( 2) 5 4
s ??
? + = +
?
 
   = 41 
6. Let 
( )
( )
22
2
sec tan
( ) .
tan 1
x x x
I x dx
xx
+
=
+
?
 If I(0) = 0, then 
4
I
? ??
??
??
 is equal to 
 (1) 
( )
( )
2
2
4
log
16 4 4
e
?+
?
+
?+
 
 (2) 
( )
( )
2
2
4
log –
16 4 4
e
?+
?
?+
 
 (3) 
( )
( )
2
2
4
log –
32 4 4
e
?+
?
?+
 
 (4) 
( )
( )
2
2
4
log
32 4 4
e
?+
?
+
?+
 
Answer (3) 
Sol. 
( )
2
2
2
sec tan
tan 1
x x x
x dx
xx
??
+
??
??
+
??
?
 
 
( )
2
2
tan 1 tan 1
xx
dx
x x x x
=+
++
?
 
 2
tan 1
x
I dx
xx
=
+
?
 
 
cos
2
sin cos
xx
dx
x x x
=
+
?
 
 Let xsinx + cosx = t 
 (x cosx + sinx – sinx) dx = dt 
 2 2log
dt
tc
t
= = +
?
 
 = 2log |xsinx + cosx| + c 
 
   
   
 ? 
22
2
( sec tan )
( tan 1)
+
+
?
x x x x
dx
x
 
 
2
tan 1
-
=
+
x
xx
 + 2 log |xsinx + cosx| + c 
 I(0) = 0 
 ? c = 0 
 
2
1 4
I 2log 1
44
2
11
4
? ??
??
?? ? ? ? ? ??
= - + +
? ? ? ?
?
? ? ? ?
?+
 
 = 
( )
( )
2
2
e
4
log
32 4 4
?+
?
-
?+
 
7. Let 
ˆ ˆ ˆ ˆ ˆ ˆ
2 3 4 , 2 2 a i j k b i j k = + + = - - and 
ˆ ˆ ˆ
43 c i j k = - + + . If d is a vector perpendicular to both 
b and , c and · 18, ad = then 
2
ad ? is equal to 
 (1) 640 (2) 680 
 (3) 720 (4) 760 
Answer (3) 
Sol. 
ˆ ˆ ˆ
22 ? = - + b c i j k 
 ? 
( )
ˆ ˆ ˆ
22 = ? - + d i j k 
 · 18 = ad 
 ? ? = 2 
 ? 
( )
2 22
2
·· ? = - a d a d a d 
 = 720 
8. Let 
11
5 ( ) 4 3, 0. f x f x
xx
??
+ = + ?
??
??
 Then 
2
1
18 ( ) f x dx
?
 
is equal to 
 (1) 5 loge 2 + 3 (2) 10 loge 2 + 6 
 (3) 10 loge 2 – 6 (4) 5 loge 2 – 3 
Answer (3) 
Sol. 
11
5 ( ) 4 3
??
+ = +
??
??
f x f
xx
 …(i) 
 Replace 
1
? x
x
 
 
1
5 4 ( ) 3
??
+ = +
??
??
f f x x
x
 …(ii)  
 Solving (i) and (ii) we get 
 
5
9 ( ) 4 3 = - + f x x
x
 
 
22
11
15
( ) 4 3
9
??
= - +
??
??
??
f x dx x dx
x
 
 
2
2
1
1
5log 2 3
9
??
= - +
??
x x x 
 
? ?
1
5log 3
9
=- x 
 ? 
2
1
18 ( ) 10log 2 6 =-
? e
f x dx 
9. Statement ( ) ( ) P Q R Q ? ? ? is logically 
equivalent to   
 (1) ( ) ( ) P R Q R ? ? ? 
 (2) ( ) P R Q ?? 
 (3) ( ) ( ) P R Q R ? ? ? 
 (4) ( ) P R Q ?? 
Answer (4) 
Sol. ( ) ( ) ? ? ? P Q R Q 
 ? ( ) ( ) P Q R Q ? ? ? ? ?  
 ? () Q P R ? ? ? ? 
 ? () Q P R ? ? ? 
 ? () P R Q ? ? ? 
 ? () P R Q ?? 
10. If the system of equations 
 x + y + az = b 
 2x + 5y + 2z = 6 
 x + 2y + 3z = 3 
 has infinitely many solutions, then 2a + 3b is equal 
to 
 (1) 25 (2) 20 
 (3) 23 (4) 28 
Answer (3) 
Sol. x + y + az = b …(i) 
 2x + 5y + 2z = 6 …(ii) 
 x + 2y + 3z = 3 …(iii) 
 3 (iii) – (ii) 
 x + y + 7z = 3 
 x + y + az = b 
 ? a = 7, b = 3 ( ? solutions are infinite) 
 ? 2a + 3b 
  = 14 + 9 = 23 
 
   
   
11. From the top A of a vertical wall AB of height 30 m, 
the angles of depression of the top P and bottom Q 
of a vertical tower PQ are 15° and 60° respectively, 
B and Q are on the same horizontal level. If C is a 
point on AB such that CB = PQ, then the area (in 
m
2
) of the quadrilateral BCPQ is equal to 
 (1) 
( )
300 3 1 - (2) 
( )
300 3 1 + 
 (3) 
( )
600 3 1 - (4) 
( )
200 3 1 - 
Answer (3) 
Sol.  
 
30
3;
x
= 
 
30
23
h
x
-
=- 
  ? 30 20 3 30 h - = - 
 ? 10 3 x = 
   ? 60 20 3 h=- 
 ? area = hx 
 = (60 20 3)10 3 - 
 = 200 3(3 3) - 
 = 600( 3 1) - 
 ? (3) is correct. 
12. One vertex of a rectangular parallelopiped is at the 
origin O and the lengths of its edges along x, y and 
z axes are 3, 4 and 5 units respectively. Let P be 
the vertex (3, 4, 5). Then the shortest distance 
between the diagonal OP and an edge parallel to z 
axis, not passing through O or P is 
 (1) 
12
5
 (2) 12 5 
 (3) 
12
55
 (4) 
12
5
 
Answer (4) 
Sol.  
 Line OP : 
3 4 5
x y z
== 
 Line AB : 
3
0 0 1
x y z -
== 
 
12
ˆ ˆ ˆ
3 4 5
0 0 1
i j k
nn ?= 
   = 
ˆ ˆ ˆ
(4) (3) (0) i j k -+ 
   = 
ˆˆ
43 ij - 
 Distance = 
2 1 1 2
12
( ) ( )
||
a a n n
nn
- ? ?
?
 
   = 
ˆ ˆ ˆ
(3 ) (4 3 )
5
i i j ?-
 
   = 
12
5
 
 Option (4) is correct. 
13. If 
2n
C3 : 
n
C3 = 10 : 1, then the ratio (n
2
 + 3n) : (n
2
 – 
3n + 4) is 
 (1) 35 : 16 (2) 27 : 11 
 (3) 65 : 37 (4) 2 : 1 
Answer (4) 
Sol. 
2n
C3 : 
n
C3 = 10 : 1 
 
(2 )! 3! ( 3)!
10
3! (2 3)! !
nn
nn
-
?=
-
 
 ? 4(2n – 1) = 10n – 20 
 ? n = 8 
 Now 
2
2
( 3 ) 64 24 88
2
64 24 4 44
( 3 4)
nn
nn
++
= = =
-+
-+
 
P
Q
h
C
B
h
A
x
15°
60°
30 m
z
y
x
D
P
C
O
5
3
4
B
A