JEE Main 2024 January 27 Shift 2 Paper & Solutions | JEE Main & Advanced Mock Test Series 2025 PDF Download

 Page 1


FINAL JEE –MAIN EXAMINATION – JANUARY, 2024 
(Held On Saturday 27
th
 January, 2024)                TIME : 3 : 00 PM  to  6 : 00 PM 
MATHEMATICS TEST PAPER WITH SOLUTION 
 
 
 
 
 
SECTION-A 
1. Considering only the principal values of inverse 
trigonometric functions, the number of positive 
real values of x satisfying 
1 1
tan (x) tan (2x)
4
?M?M
?? ?K?] 
is :
 
 (1) More than 2 
 (2) 1 
 (3) 2 
 (4) 0 
Ans. (2) 
Sol. 
1 1
tan x tan 2x
4
?M?M
?? ?K?] ; x > 0 
 ?? 1 1
tan 2x tan x
4
?M?M
?? ?]?M 
 Taking tan both sides 
 ?? 1 x
2x
1 x
?M ?] ?K 
 
2
2x 3x 1 0 ?? ?K ?M ?] 
 
3 9 8 3 17
x
8 8
?M ?? ?K ?M ?? ?]?] 
 Only possible 
3 17
x
8
?M?K
?] 
2. Consider the function f :(0,2) R ?? defined by 
x 2
f (x)
2 x
?]?K and the function g(x) defined by 
min{f (t)}, 0 t x and 0 x 1
g(x) 3
x, 1 x 2
2
?\ ?? ?\ ?? ?? ?? ?] ?? ?K ?\ ?\ ?? ?? . Then 
 (1) g is continuous but not differentiable at x = 1 
 (2) g is not continuous for all x (0,2) ?? 
 (3) g is neither continuous nor differentiable at x = 1 
 (4) g is continuous and differentiable for all x (0,2) ?? 
Ans. (1) 
Sol. f :(0,2) R ?? ; 
x 2
f (x)
2 x
?]?K 
 
1 2
f (x)
2 x
?R ?? ?]?M 
 f (x) ?| is decreasing in domain.  
 
2
2
x
f(x)
 
 
x 2
0 x 1
2 x
g(x)
3
x 1 x 2
2
?? ?K ?\ ?? ?? ?] ?? ?? ?K ?\ ?\ ?? 
 
1 2 O
g(x)
  
3. Let the image of the point (1, 0, 7) in the line 
x y 1 z 2
1 2 3
?M?M
?]?] be the point ( ?? , ?? , ?? ). Then 
which one of the following points lies on the line 
passing through ( ?? , ?? , ?? ) and making angles 
2
3
?? 
and 
3
4
?? with y-axis and z-axis respectively and an 
acute angle with x-axis ? 
 (1) 
?H ?I 1, 2,1 2 ?M?K 
 (2) 
?H ?I 1,2,1 2 ?M 
 (3) 
?H ?I 3,4,3 2 2 ?M 
 (4) 
?H ?I 3, 4,3 2 2 ?M?K 
Ans. (3) 
Page 2


FINAL JEE –MAIN EXAMINATION – JANUARY, 2024 
(Held On Saturday 27
th
 January, 2024)                TIME : 3 : 00 PM  to  6 : 00 PM 
MATHEMATICS TEST PAPER WITH SOLUTION 
 
 
 
 
 
SECTION-A 
1. Considering only the principal values of inverse 
trigonometric functions, the number of positive 
real values of x satisfying 
1 1
tan (x) tan (2x)
4
?M?M
?? ?K?] 
is :
 
 (1) More than 2 
 (2) 1 
 (3) 2 
 (4) 0 
Ans. (2) 
Sol. 
1 1
tan x tan 2x
4
?M?M
?? ?K?] ; x > 0 
 ?? 1 1
tan 2x tan x
4
?M?M
?? ?]?M 
 Taking tan both sides 
 ?? 1 x
2x
1 x
?M ?] ?K 
 
2
2x 3x 1 0 ?? ?K ?M ?] 
 
3 9 8 3 17
x
8 8
?M ?? ?K ?M ?? ?]?] 
 Only possible 
3 17
x
8
?M?K
?] 
2. Consider the function f :(0,2) R ?? defined by 
x 2
f (x)
2 x
?]?K and the function g(x) defined by 
min{f (t)}, 0 t x and 0 x 1
g(x) 3
x, 1 x 2
2
?\ ?? ?\ ?? ?? ?? ?] ?? ?K ?\ ?\ ?? ?? . Then 
 (1) g is continuous but not differentiable at x = 1 
 (2) g is not continuous for all x (0,2) ?? 
 (3) g is neither continuous nor differentiable at x = 1 
 (4) g is continuous and differentiable for all x (0,2) ?? 
Ans. (1) 
Sol. f :(0,2) R ?? ; 
x 2
f (x)
2 x
?]?K 
 
1 2
f (x)
2 x
?R ?? ?]?M 
 f (x) ?| is decreasing in domain.  
 
2
2
x
f(x)
 
 
x 2
0 x 1
2 x
g(x)
3
x 1 x 2
2
?? ?K ?\ ?? ?? ?] ?? ?? ?K ?\ ?\ ?? 
 
1 2 O
g(x)
  
3. Let the image of the point (1, 0, 7) in the line 
x y 1 z 2
1 2 3
?M?M
?]?] be the point ( ?? , ?? , ?? ). Then 
which one of the following points lies on the line 
passing through ( ?? , ?? , ?? ) and making angles 
2
3
?? 
and 
3
4
?? with y-axis and z-axis respectively and an 
acute angle with x-axis ? 
 (1) 
?H ?I 1, 2,1 2 ?M?K 
 (2) 
?H ?I 1,2,1 2 ?M 
 (3) 
?H ?I 3,4,3 2 2 ?M 
 (4) 
?H ?I 3, 4,3 2 2 ?M?K 
Ans. (3) 
 
 
 
Sol. 
1
x y 1 z 2
L
1 2 3
?M?M
?] ?] ?] ?] ?? 
  
 M( ,1 2 ,2 3 ) ?? ?K ?? ?K ?? 
 
ˆ ˆ ˆ
PM ( 1)i (1 2 )j (3 5)k ?] ?? ?M ?K ?K ?? ?K ?? ?M 
 PM is perpendicular to line L
1
 
 PM.b 0 ?]     (
ˆ ˆ ˆ
b i 2j 3k ?] ?K ?K ) 
 1 4 2 9 15 0 ?? ?? ?M ?K ?? ?K ?K ?? ?M ?] 
 14 14 1 ?? ?] ?? ?? ?] 
 M (1,3,5) ?|?] 
 Q 2M P ?]?M [M is midpoint of P & Q ] 
 
ˆ ˆ ˆ ˆ ˆ
Q 2i 6j 10k i 7k ?] ?K ?K ?M ?M 
 
ˆ ˆ ˆ
Q i 6j 3k ?] ?K ?K 
 ( , , ) (1,6,3) ?| ?? ?? ?? ?] 
 Required line having direction cosine (l, m, n) 
 
2 2 2
1 ?K ?K ?] l m n 
 
2 2
2
1 1
1
2 2
???? ????
?? ?K ?M ?K ?M ?] ???? ????
???? ????
l 
 
2
1
4
?] l 
 
1
2
?|?] l [Line make acute angle with x-axis] 
 Equation of line passing through (1, 6, 3) will be 
 
1 1 1
ˆ ˆ ˆ ˆ ˆ ˆ
r (i 6j 3k) i j k
2 2 2
????
?] ?K ?K ?K ?? ?M ?M????
????
 
 Option (3) satisfying for ?? = 4 
4. Let R be the interior region between the lines 
3x y 1 0 ?M ?K ?] and x 2y 5 0 ?K ?M ?] containing the 
origin. The set of all values of a, for which the 
points (a
2
, a + 1) lie in R, is : 
 (1) 
1
( 3, 1) ,1
3
????
?M ?M ?? ?M ????
????
 
 (2) 
1
( 3,0) ,1
3
????
?M??
????
????
 
 (3) 
2
( 3,0) ,1
3
????
?M??
????
????
 
 (4) 
1
( 3, 1) ,1
3
????
?M ?M ?? ????
????
 
Ans. (2) 
Sol. P(a
2
, a + 1) 
 L
1
 = 3x – y + 1 = 0 
 Origin and P lies same side w.r.t. L
1
 
 ?? L
1
(0) . L
1
(P) > 0 
 ?| 3(a
2
) – (a + 1) + 1 > 0 
 
y
O
(0,0)
L : x+2y–5=0
2
x
L : 3x–y+1=0
1
 
 ?? 3a
2
 – a > 0 
 
1
a ( ,0) ,
3
????
?? ?M ?? ?? ?? ????
????
…………….(1) 
 Let L
2
 : x + 2y – 5 = 0 
 Origin and P lies same side w.r.t. L
2
 
 ?? 2 2
L (0).L (P) 0 ?^ 
 
2
a 2(a 1) 5 0 ?? ?K ?K ?M ?\ 
 ?? 2
a 2a 3 0 ?K ?M ?\ 
 ?? (a 3)(a 1) 0 ?K ?M ?\ 
 ?| a ( 3,1) ???M …………….(2) 
 Intersection of (1) and (2) 
 
1
a ( 3,0) ,1
3
????
?? ?M ?? ????
????
 
Page 3


FINAL JEE –MAIN EXAMINATION – JANUARY, 2024 
(Held On Saturday 27
th
 January, 2024)                TIME : 3 : 00 PM  to  6 : 00 PM 
MATHEMATICS TEST PAPER WITH SOLUTION 
 
 
 
 
 
SECTION-A 
1. Considering only the principal values of inverse 
trigonometric functions, the number of positive 
real values of x satisfying 
1 1
tan (x) tan (2x)
4
?M?M
?? ?K?] 
is :
 
 (1) More than 2 
 (2) 1 
 (3) 2 
 (4) 0 
Ans. (2) 
Sol. 
1 1
tan x tan 2x
4
?M?M
?? ?K?] ; x > 0 
 ?? 1 1
tan 2x tan x
4
?M?M
?? ?]?M 
 Taking tan both sides 
 ?? 1 x
2x
1 x
?M ?] ?K 
 
2
2x 3x 1 0 ?? ?K ?M ?] 
 
3 9 8 3 17
x
8 8
?M ?? ?K ?M ?? ?]?] 
 Only possible 
3 17
x
8
?M?K
?] 
2. Consider the function f :(0,2) R ?? defined by 
x 2
f (x)
2 x
?]?K and the function g(x) defined by 
min{f (t)}, 0 t x and 0 x 1
g(x) 3
x, 1 x 2
2
?\ ?? ?\ ?? ?? ?? ?] ?? ?K ?\ ?\ ?? ?? . Then 
 (1) g is continuous but not differentiable at x = 1 
 (2) g is not continuous for all x (0,2) ?? 
 (3) g is neither continuous nor differentiable at x = 1 
 (4) g is continuous and differentiable for all x (0,2) ?? 
Ans. (1) 
Sol. f :(0,2) R ?? ; 
x 2
f (x)
2 x
?]?K 
 
1 2
f (x)
2 x
?R ?? ?]?M 
 f (x) ?| is decreasing in domain.  
 
2
2
x
f(x)
 
 
x 2
0 x 1
2 x
g(x)
3
x 1 x 2
2
?? ?K ?\ ?? ?? ?] ?? ?? ?K ?\ ?\ ?? 
 
1 2 O
g(x)
  
3. Let the image of the point (1, 0, 7) in the line 
x y 1 z 2
1 2 3
?M?M
?]?] be the point ( ?? , ?? , ?? ). Then 
which one of the following points lies on the line 
passing through ( ?? , ?? , ?? ) and making angles 
2
3
?? 
and 
3
4
?? with y-axis and z-axis respectively and an 
acute angle with x-axis ? 
 (1) 
?H ?I 1, 2,1 2 ?M?K 
 (2) 
?H ?I 1,2,1 2 ?M 
 (3) 
?H ?I 3,4,3 2 2 ?M 
 (4) 
?H ?I 3, 4,3 2 2 ?M?K 
Ans. (3) 
 
 
 
Sol. 
1
x y 1 z 2
L
1 2 3
?M?M
?] ?] ?] ?] ?? 
  
 M( ,1 2 ,2 3 ) ?? ?K ?? ?K ?? 
 
ˆ ˆ ˆ
PM ( 1)i (1 2 )j (3 5)k ?] ?? ?M ?K ?K ?? ?K ?? ?M 
 PM is perpendicular to line L
1
 
 PM.b 0 ?]     (
ˆ ˆ ˆ
b i 2j 3k ?] ?K ?K ) 
 1 4 2 9 15 0 ?? ?? ?M ?K ?? ?K ?K ?? ?M ?] 
 14 14 1 ?? ?] ?? ?? ?] 
 M (1,3,5) ?|?] 
 Q 2M P ?]?M [M is midpoint of P & Q ] 
 
ˆ ˆ ˆ ˆ ˆ
Q 2i 6j 10k i 7k ?] ?K ?K ?M ?M 
 
ˆ ˆ ˆ
Q i 6j 3k ?] ?K ?K 
 ( , , ) (1,6,3) ?| ?? ?? ?? ?] 
 Required line having direction cosine (l, m, n) 
 
2 2 2
1 ?K ?K ?] l m n 
 
2 2
2
1 1
1
2 2
???? ????
?? ?K ?M ?K ?M ?] ???? ????
???? ????
l 
 
2
1
4
?] l 
 
1
2
?|?] l [Line make acute angle with x-axis] 
 Equation of line passing through (1, 6, 3) will be 
 
1 1 1
ˆ ˆ ˆ ˆ ˆ ˆ
r (i 6j 3k) i j k
2 2 2
????
?] ?K ?K ?K ?? ?M ?M????
????
 
 Option (3) satisfying for ?? = 4 
4. Let R be the interior region between the lines 
3x y 1 0 ?M ?K ?] and x 2y 5 0 ?K ?M ?] containing the 
origin. The set of all values of a, for which the 
points (a
2
, a + 1) lie in R, is : 
 (1) 
1
( 3, 1) ,1
3
????
?M ?M ?? ?M ????
????
 
 (2) 
1
( 3,0) ,1
3
????
?M??
????
????
 
 (3) 
2
( 3,0) ,1
3
????
?M??
????
????
 
 (4) 
1
( 3, 1) ,1
3
????
?M ?M ?? ????
????
 
Ans. (2) 
Sol. P(a
2
, a + 1) 
 L
1
 = 3x – y + 1 = 0 
 Origin and P lies same side w.r.t. L
1
 
 ?? L
1
(0) . L
1
(P) > 0 
 ?| 3(a
2
) – (a + 1) + 1 > 0 
 
y
O
(0,0)
L : x+2y–5=0
2
x
L : 3x–y+1=0
1
 
 ?? 3a
2
 – a > 0 
 
1
a ( ,0) ,
3
????
?? ?M ?? ?? ?? ????
????
…………….(1) 
 Let L
2
 : x + 2y – 5 = 0 
 Origin and P lies same side w.r.t. L
2
 
 ?? 2 2
L (0).L (P) 0 ?^ 
 
2
a 2(a 1) 5 0 ?? ?K ?K ?M ?\ 
 ?? 2
a 2a 3 0 ?K ?M ?\ 
 ?? (a 3)(a 1) 0 ?K ?M ?\ 
 ?| a ( 3,1) ???M …………….(2) 
 Intersection of (1) and (2) 
 
1
a ( 3,0) ,1
3
????
?? ?M ?? ????
????
 
 
 
 
 
5. The 20
th
 term from the end of the progression 
1 1 3 1
20,19 ,18 ,17 ,...., 129
4 2 4 4
?M is :- 
 (1) –118 
 (2) –110 
 (3) –115 
 (4) –100 
Ans. (3) 
Sol. 
1 1 3 1
20,19 ,18 ,17 ,......, 129
4 2 4 4
?M 
 This is A.P. with common difference  
 
1
1 3
d 1
4 4
?] ?M ?K ?] ?M 
 
1 1
129 ,..............,19 ,20
4 4
?M 
 This is also A.P. 
1
a 129
4
?]?M and 
3
d
4
?] 
 Required term =  
 
1 3
129 (20 1)
4 4
????
?M ?K ?M ????
????
 
 
1 3
129 15 115
4 4
?] ?M ?M ?K ?M ?] ?M 
6. Let 
1
f : R R
2
?M ????
?M??
????
????
 and 
5
g : R R
2
?M????
?M??
????
????
 be 
defined as 
2x 3
f (x)
2x 1
?K ?] ?K and 
| x | 1
g(x)
2x 5
?K ?] ?K . Then 
the domain of the function fog is : 
 (1) 
5
R
2
????
?M?M
????
????
 
 (2) R 
 (3) 
7
R
4
????
?M?M
????
????
 
 (4) 
5 7
R ,
2 4
????
?M ?M ?M????
????
 
Ans. (1) 
Sol. 
2x 3 1
f (x) ;x
2x 1 2
?K ?] ?? ?M ?K 
 
| x | 1 5
g(x) ,x
2x 5 2
?K ?] ?? ?M ?K 
 Domain of f(g(x)) 
 
2g(x) 3
f (g(x))
2g(x) 1
?K ?] ?K 
 
5
x
2
???M and 
| x | 1 1
2x 5 2
?K ???M
?K 
 
5
x R
2
????
?? ?M ?M ????
????
 and x R ?? 
 ?| Domain will be 
5
R
2
????
?M?M
????
????
 
7. For 0 < a < 1, the value of the integral 
2
0
dx
1 2a cos x a
?? ?M?K
?? is : 
 (1) 
2
2
a
?? ???K
 
 (2) 
2
2
a
?? ???M
 
 (3) 
2
1 a
?? ?M 
 (4) 
2
1 a
?? ?K 
Ans. (3) 
Sol. 
2
0
dx
I ; 0 a 1
1 2a cos x a
?? ?] ?\ ?\ ?M?K
?? 
 
2
0
dx
I
1 2a cos x a
?? ?] ?K?K
??   
 
/2
2
2 2 2 2
0
2(1 a )
2I 2 dx
(1 a ) 4a cos x
?? ?K ?] ?K?M
?? 
 
/2
2 2
2 2 2 2
0
2(1 a ).sec x
I dx
(1 a ) .sec x 4a
?? ?K ???]
?K?M
?? 
 
/2
2 2
2 2 2 2 2
0
2.(1 a ).sec x
I dx
(1 a ) .tan x (1 a )
?? ?K ???]
?K ?K ?M ?? 
Page 4


FINAL JEE –MAIN EXAMINATION – JANUARY, 2024 
(Held On Saturday 27
th
 January, 2024)                TIME : 3 : 00 PM  to  6 : 00 PM 
MATHEMATICS TEST PAPER WITH SOLUTION 
 
 
 
 
 
SECTION-A 
1. Considering only the principal values of inverse 
trigonometric functions, the number of positive 
real values of x satisfying 
1 1
tan (x) tan (2x)
4
?M?M
?? ?K?] 
is :
 
 (1) More than 2 
 (2) 1 
 (3) 2 
 (4) 0 
Ans. (2) 
Sol. 
1 1
tan x tan 2x
4
?M?M
?? ?K?] ; x > 0 
 ?? 1 1
tan 2x tan x
4
?M?M
?? ?]?M 
 Taking tan both sides 
 ?? 1 x
2x
1 x
?M ?] ?K 
 
2
2x 3x 1 0 ?? ?K ?M ?] 
 
3 9 8 3 17
x
8 8
?M ?? ?K ?M ?? ?]?] 
 Only possible 
3 17
x
8
?M?K
?] 
2. Consider the function f :(0,2) R ?? defined by 
x 2
f (x)
2 x
?]?K and the function g(x) defined by 
min{f (t)}, 0 t x and 0 x 1
g(x) 3
x, 1 x 2
2
?\ ?? ?\ ?? ?? ?? ?] ?? ?K ?\ ?\ ?? ?? . Then 
 (1) g is continuous but not differentiable at x = 1 
 (2) g is not continuous for all x (0,2) ?? 
 (3) g is neither continuous nor differentiable at x = 1 
 (4) g is continuous and differentiable for all x (0,2) ?? 
Ans. (1) 
Sol. f :(0,2) R ?? ; 
x 2
f (x)
2 x
?]?K 
 
1 2
f (x)
2 x
?R ?? ?]?M 
 f (x) ?| is decreasing in domain.  
 
2
2
x
f(x)
 
 
x 2
0 x 1
2 x
g(x)
3
x 1 x 2
2
?? ?K ?\ ?? ?? ?] ?? ?? ?K ?\ ?\ ?? 
 
1 2 O
g(x)
  
3. Let the image of the point (1, 0, 7) in the line 
x y 1 z 2
1 2 3
?M?M
?]?] be the point ( ?? , ?? , ?? ). Then 
which one of the following points lies on the line 
passing through ( ?? , ?? , ?? ) and making angles 
2
3
?? 
and 
3
4
?? with y-axis and z-axis respectively and an 
acute angle with x-axis ? 
 (1) 
?H ?I 1, 2,1 2 ?M?K 
 (2) 
?H ?I 1,2,1 2 ?M 
 (3) 
?H ?I 3,4,3 2 2 ?M 
 (4) 
?H ?I 3, 4,3 2 2 ?M?K 
Ans. (3) 
 
 
 
Sol. 
1
x y 1 z 2
L
1 2 3
?M?M
?] ?] ?] ?] ?? 
  
 M( ,1 2 ,2 3 ) ?? ?K ?? ?K ?? 
 
ˆ ˆ ˆ
PM ( 1)i (1 2 )j (3 5)k ?] ?? ?M ?K ?K ?? ?K ?? ?M 
 PM is perpendicular to line L
1
 
 PM.b 0 ?]     (
ˆ ˆ ˆ
b i 2j 3k ?] ?K ?K ) 
 1 4 2 9 15 0 ?? ?? ?M ?K ?? ?K ?K ?? ?M ?] 
 14 14 1 ?? ?] ?? ?? ?] 
 M (1,3,5) ?|?] 
 Q 2M P ?]?M [M is midpoint of P & Q ] 
 
ˆ ˆ ˆ ˆ ˆ
Q 2i 6j 10k i 7k ?] ?K ?K ?M ?M 
 
ˆ ˆ ˆ
Q i 6j 3k ?] ?K ?K 
 ( , , ) (1,6,3) ?| ?? ?? ?? ?] 
 Required line having direction cosine (l, m, n) 
 
2 2 2
1 ?K ?K ?] l m n 
 
2 2
2
1 1
1
2 2
???? ????
?? ?K ?M ?K ?M ?] ???? ????
???? ????
l 
 
2
1
4
?] l 
 
1
2
?|?] l [Line make acute angle with x-axis] 
 Equation of line passing through (1, 6, 3) will be 
 
1 1 1
ˆ ˆ ˆ ˆ ˆ ˆ
r (i 6j 3k) i j k
2 2 2
????
?] ?K ?K ?K ?? ?M ?M????
????
 
 Option (3) satisfying for ?? = 4 
4. Let R be the interior region between the lines 
3x y 1 0 ?M ?K ?] and x 2y 5 0 ?K ?M ?] containing the 
origin. The set of all values of a, for which the 
points (a
2
, a + 1) lie in R, is : 
 (1) 
1
( 3, 1) ,1
3
????
?M ?M ?? ?M ????
????
 
 (2) 
1
( 3,0) ,1
3
????
?M??
????
????
 
 (3) 
2
( 3,0) ,1
3
????
?M??
????
????
 
 (4) 
1
( 3, 1) ,1
3
????
?M ?M ?? ????
????
 
Ans. (2) 
Sol. P(a
2
, a + 1) 
 L
1
 = 3x – y + 1 = 0 
 Origin and P lies same side w.r.t. L
1
 
 ?? L
1
(0) . L
1
(P) > 0 
 ?| 3(a
2
) – (a + 1) + 1 > 0 
 
y
O
(0,0)
L : x+2y–5=0
2
x
L : 3x–y+1=0
1
 
 ?? 3a
2
 – a > 0 
 
1
a ( ,0) ,
3
????
?? ?M ?? ?? ?? ????
????
…………….(1) 
 Let L
2
 : x + 2y – 5 = 0 
 Origin and P lies same side w.r.t. L
2
 
 ?? 2 2
L (0).L (P) 0 ?^ 
 
2
a 2(a 1) 5 0 ?? ?K ?K ?M ?\ 
 ?? 2
a 2a 3 0 ?K ?M ?\ 
 ?? (a 3)(a 1) 0 ?K ?M ?\ 
 ?| a ( 3,1) ???M …………….(2) 
 Intersection of (1) and (2) 
 
1
a ( 3,0) ,1
3
????
?? ?M ?? ????
????
 
 
 
 
 
5. The 20
th
 term from the end of the progression 
1 1 3 1
20,19 ,18 ,17 ,...., 129
4 2 4 4
?M is :- 
 (1) –118 
 (2) –110 
 (3) –115 
 (4) –100 
Ans. (3) 
Sol. 
1 1 3 1
20,19 ,18 ,17 ,......, 129
4 2 4 4
?M 
 This is A.P. with common difference  
 
1
1 3
d 1
4 4
?] ?M ?K ?] ?M 
 
1 1
129 ,..............,19 ,20
4 4
?M 
 This is also A.P. 
1
a 129
4
?]?M and 
3
d
4
?] 
 Required term =  
 
1 3
129 (20 1)
4 4
????
?M ?K ?M ????
????
 
 
1 3
129 15 115
4 4
?] ?M ?M ?K ?M ?] ?M 
6. Let 
1
f : R R
2
?M ????
?M??
????
????
 and 
5
g : R R
2
?M????
?M??
????
????
 be 
defined as 
2x 3
f (x)
2x 1
?K ?] ?K and 
| x | 1
g(x)
2x 5
?K ?] ?K . Then 
the domain of the function fog is : 
 (1) 
5
R
2
????
?M?M
????
????
 
 (2) R 
 (3) 
7
R
4
????
?M?M
????
????
 
 (4) 
5 7
R ,
2 4
????
?M ?M ?M????
????
 
Ans. (1) 
Sol. 
2x 3 1
f (x) ;x
2x 1 2
?K ?] ?? ?M ?K 
 
| x | 1 5
g(x) ,x
2x 5 2
?K ?] ?? ?M ?K 
 Domain of f(g(x)) 
 
2g(x) 3
f (g(x))
2g(x) 1
?K ?] ?K 
 
5
x
2
???M and 
| x | 1 1
2x 5 2
?K ???M
?K 
 
5
x R
2
????
?? ?M ?M ????
????
 and x R ?? 
 ?| Domain will be 
5
R
2
????
?M?M
????
????
 
7. For 0 < a < 1, the value of the integral 
2
0
dx
1 2a cos x a
?? ?M?K
?? is : 
 (1) 
2
2
a
?? ???K
 
 (2) 
2
2
a
?? ???M
 
 (3) 
2
1 a
?? ?M 
 (4) 
2
1 a
?? ?K 
Ans. (3) 
Sol. 
2
0
dx
I ; 0 a 1
1 2a cos x a
?? ?] ?\ ?\ ?M?K
?? 
 
2
0
dx
I
1 2a cos x a
?? ?] ?K?K
??   
 
/2
2
2 2 2 2
0
2(1 a )
2I 2 dx
(1 a ) 4a cos x
?? ?K ?] ?K?M
?? 
 
/2
2 2
2 2 2 2
0
2(1 a ).sec x
I dx
(1 a ) .sec x 4a
?? ?K ???]
?K?M
?? 
 
/2
2 2
2 2 2 2 2
0
2.(1 a ).sec x
I dx
(1 a ) .tan x (1 a )
?? ?K ???]
?K ?K ?M ?? 
 
 
 
2
/2
2
2
2
0
2
2
2.sec x
.dx
1 a
I
1 a
tan x
1 a
?? ?K ???]
???? ?M ?K????
?K????
?? 
 
2
2
I 0
(1 a ) 2
??????
?? ?] ?M????
?M ????
 
 
2
I
1 a
?? ?] ?M  
8. Let 
x
g(x) 3f f (3 x)
3
????
?] ?K ?M ????
????
 and f (x) 0 ???? ?^ for all 
x (0,3) ?? . If g is decreasing in (0, ?? ) and 
increasing in ( ?? , 3), then 8 ?? is 
 (1) 24 
 (2) 0 
 (3) 18 
 (4) 20 
Ans. (3) 
Sol. 
x
g(x) 3f f (3 x)
3
????
?] ?K ?M ????
????
 and f (x) 0 ???? ?^ ?B x ?? (0, 3) 
 f (x) ?? ?? is increasing function 
 
1 x
g (x) 3 .f f (3 x)
3 3
????
?? ?? ?? ?] ?? ?M ?M ????
????
 
 
x
f f (3 x)
3
????
???? ?] ?M ?M ????
????
 
 If g is decreasing in (0, ?? ) 
 g (x) 0 ?? ?\ 
 
x
f f (3 x) 0
3
????
???? ?M ?M ?\ ????
????
 
 
x
f f (3 x)
3
????
???? ?\?M
????
????
 
 
x
3 x
3
?? ?\ ?M  
 
9
x
4
???\ 
 Therefore 
9
4
???] 
 Then 
9
8 8 18
4
?? ?] ?? ?]  
9. If 
e
2
x 0
3 sin x cos x log (1 x)
1
lim
3tan x 3
?? ?K ?? ?K ?? ?K ?M ?] , then 
2 ?? – ?? is equal to : 
 (1) 2 
 (2) 7 
 (3) 5 
 (4) 1 
Ans. (3) 
Sol. 
e
2
x 0
3 sin x cos x log (1 x)
1
lim
3tan x 3
?? ?K ?? ?K ?? ?K ?M ?] 
 
3 2 4 2 3
2
x 0
x x x x x
3 x .... 1 .... x ...
3! 2! 4! 2 3 1
lim
3tan x 3
?? ?? ?? ?? ?? ?? ?? ?K ?? ?M ?K ?K ?? ?M ?K ?K ?M ?M ?M???? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ???]
 
 
2
2
2 2
x 0
1
(3 ) ( 1)x x ....
x 1 2 2
lim
3x tan x 3
?? ?? ????
?K ?? ?K ?? ?M ?K ?M ?M ?K????
????
?? ?? ?] 
 3 0, 1 0 ?? ?? ?K ?] ?? ?M ?] and 
1
1
2 2
3 3
?? ?M?M
?] 
 3, 1 ?? ?? ?] ?M ?? ?] 
 2 2 3 5 ?? ?? ?M ?? ?] ?K ?] 
10. If ?? , ?? are the roots of the equation, 
2
x x 1 0 ?M ?M ?] 
and 
n n
n
S 2023 2024 ?] ?? ?K ?? , then 
 (1) 
12 11 10
2S S S ?]?K 
 (2) 
12 11 10
S S S ?]?K 
 (3) 
11 12 10
2S S S ?]?K 
 (4) 
11 10 12
S S S ?]?K 
Ans. (2) 
Sol. 
2
x x 1 0 ?M ?M ?] 
 
n n
n
S 2023 2024 ?] ?? ?K ?? 
 
n 1 n 1 n 2 n 2
n 1 n 2
S S 2023 2024 2023 2024
?M ?M ?M ?M ?M?M
?K ?] ?? ?K ?? ?K ?? ?K ?? 
 
n 2 n 2
2023 [1 ] 2024 [1 ]
?M?M
?] ?? ?K ?? ?K ?? ?K ?? 
 
n 2 2 n 2 2
2023 [ ] 2024 [ ]
?M?M
?] ?? ?? ?K ?? ?? 
 
n n
2023 2024 ?] ?? ?K ?? 
 
n 1 n 2 n
S S S
?M?M
?K?] 
 Put n = 12 
 
11 10 12
S S S ?K?] 
Page 5


FINAL JEE –MAIN EXAMINATION – JANUARY, 2024 
(Held On Saturday 27
th
 January, 2024)                TIME : 3 : 00 PM  to  6 : 00 PM 
MATHEMATICS TEST PAPER WITH SOLUTION 
 
 
 
 
 
SECTION-A 
1. Considering only the principal values of inverse 
trigonometric functions, the number of positive 
real values of x satisfying 
1 1
tan (x) tan (2x)
4
?M?M
?? ?K?] 
is :
 
 (1) More than 2 
 (2) 1 
 (3) 2 
 (4) 0 
Ans. (2) 
Sol. 
1 1
tan x tan 2x
4
?M?M
?? ?K?] ; x > 0 
 ?? 1 1
tan 2x tan x
4
?M?M
?? ?]?M 
 Taking tan both sides 
 ?? 1 x
2x
1 x
?M ?] ?K 
 
2
2x 3x 1 0 ?? ?K ?M ?] 
 
3 9 8 3 17
x
8 8
?M ?? ?K ?M ?? ?]?] 
 Only possible 
3 17
x
8
?M?K
?] 
2. Consider the function f :(0,2) R ?? defined by 
x 2
f (x)
2 x
?]?K and the function g(x) defined by 
min{f (t)}, 0 t x and 0 x 1
g(x) 3
x, 1 x 2
2
?\ ?? ?\ ?? ?? ?? ?] ?? ?K ?\ ?\ ?? ?? . Then 
 (1) g is continuous but not differentiable at x = 1 
 (2) g is not continuous for all x (0,2) ?? 
 (3) g is neither continuous nor differentiable at x = 1 
 (4) g is continuous and differentiable for all x (0,2) ?? 
Ans. (1) 
Sol. f :(0,2) R ?? ; 
x 2
f (x)
2 x
?]?K 
 
1 2
f (x)
2 x
?R ?? ?]?M 
 f (x) ?| is decreasing in domain.  
 
2
2
x
f(x)
 
 
x 2
0 x 1
2 x
g(x)
3
x 1 x 2
2
?? ?K ?\ ?? ?? ?] ?? ?? ?K ?\ ?\ ?? 
 
1 2 O
g(x)
  
3. Let the image of the point (1, 0, 7) in the line 
x y 1 z 2
1 2 3
?M?M
?]?] be the point ( ?? , ?? , ?? ). Then 
which one of the following points lies on the line 
passing through ( ?? , ?? , ?? ) and making angles 
2
3
?? 
and 
3
4
?? with y-axis and z-axis respectively and an 
acute angle with x-axis ? 
 (1) 
?H ?I 1, 2,1 2 ?M?K 
 (2) 
?H ?I 1,2,1 2 ?M 
 (3) 
?H ?I 3,4,3 2 2 ?M 
 (4) 
?H ?I 3, 4,3 2 2 ?M?K 
Ans. (3) 
 
 
 
Sol. 
1
x y 1 z 2
L
1 2 3
?M?M
?] ?] ?] ?] ?? 
  
 M( ,1 2 ,2 3 ) ?? ?K ?? ?K ?? 
 
ˆ ˆ ˆ
PM ( 1)i (1 2 )j (3 5)k ?] ?? ?M ?K ?K ?? ?K ?? ?M 
 PM is perpendicular to line L
1
 
 PM.b 0 ?]     (
ˆ ˆ ˆ
b i 2j 3k ?] ?K ?K ) 
 1 4 2 9 15 0 ?? ?? ?M ?K ?? ?K ?K ?? ?M ?] 
 14 14 1 ?? ?] ?? ?? ?] 
 M (1,3,5) ?|?] 
 Q 2M P ?]?M [M is midpoint of P & Q ] 
 
ˆ ˆ ˆ ˆ ˆ
Q 2i 6j 10k i 7k ?] ?K ?K ?M ?M 
 
ˆ ˆ ˆ
Q i 6j 3k ?] ?K ?K 
 ( , , ) (1,6,3) ?| ?? ?? ?? ?] 
 Required line having direction cosine (l, m, n) 
 
2 2 2
1 ?K ?K ?] l m n 
 
2 2
2
1 1
1
2 2
???? ????
?? ?K ?M ?K ?M ?] ???? ????
???? ????
l 
 
2
1
4
?] l 
 
1
2
?|?] l [Line make acute angle with x-axis] 
 Equation of line passing through (1, 6, 3) will be 
 
1 1 1
ˆ ˆ ˆ ˆ ˆ ˆ
r (i 6j 3k) i j k
2 2 2
????
?] ?K ?K ?K ?? ?M ?M????
????
 
 Option (3) satisfying for ?? = 4 
4. Let R be the interior region between the lines 
3x y 1 0 ?M ?K ?] and x 2y 5 0 ?K ?M ?] containing the 
origin. The set of all values of a, for which the 
points (a
2
, a + 1) lie in R, is : 
 (1) 
1
( 3, 1) ,1
3
????
?M ?M ?? ?M ????
????
 
 (2) 
1
( 3,0) ,1
3
????
?M??
????
????
 
 (3) 
2
( 3,0) ,1
3
????
?M??
????
????
 
 (4) 
1
( 3, 1) ,1
3
????
?M ?M ?? ????
????
 
Ans. (2) 
Sol. P(a
2
, a + 1) 
 L
1
 = 3x – y + 1 = 0 
 Origin and P lies same side w.r.t. L
1
 
 ?? L
1
(0) . L
1
(P) > 0 
 ?| 3(a
2
) – (a + 1) + 1 > 0 
 
y
O
(0,0)
L : x+2y–5=0
2
x
L : 3x–y+1=0
1
 
 ?? 3a
2
 – a > 0 
 
1
a ( ,0) ,
3
????
?? ?M ?? ?? ?? ????
????
…………….(1) 
 Let L
2
 : x + 2y – 5 = 0 
 Origin and P lies same side w.r.t. L
2
 
 ?? 2 2
L (0).L (P) 0 ?^ 
 
2
a 2(a 1) 5 0 ?? ?K ?K ?M ?\ 
 ?? 2
a 2a 3 0 ?K ?M ?\ 
 ?? (a 3)(a 1) 0 ?K ?M ?\ 
 ?| a ( 3,1) ???M …………….(2) 
 Intersection of (1) and (2) 
 
1
a ( 3,0) ,1
3
????
?? ?M ?? ????
????
 
 
 
 
 
5. The 20
th
 term from the end of the progression 
1 1 3 1
20,19 ,18 ,17 ,...., 129
4 2 4 4
?M is :- 
 (1) –118 
 (2) –110 
 (3) –115 
 (4) –100 
Ans. (3) 
Sol. 
1 1 3 1
20,19 ,18 ,17 ,......, 129
4 2 4 4
?M 
 This is A.P. with common difference  
 
1
1 3
d 1
4 4
?] ?M ?K ?] ?M 
 
1 1
129 ,..............,19 ,20
4 4
?M 
 This is also A.P. 
1
a 129
4
?]?M and 
3
d
4
?] 
 Required term =  
 
1 3
129 (20 1)
4 4
????
?M ?K ?M ????
????
 
 
1 3
129 15 115
4 4
?] ?M ?M ?K ?M ?] ?M 
6. Let 
1
f : R R
2
?M ????
?M??
????
????
 and 
5
g : R R
2
?M????
?M??
????
????
 be 
defined as 
2x 3
f (x)
2x 1
?K ?] ?K and 
| x | 1
g(x)
2x 5
?K ?] ?K . Then 
the domain of the function fog is : 
 (1) 
5
R
2
????
?M?M
????
????
 
 (2) R 
 (3) 
7
R
4
????
?M?M
????
????
 
 (4) 
5 7
R ,
2 4
????
?M ?M ?M????
????
 
Ans. (1) 
Sol. 
2x 3 1
f (x) ;x
2x 1 2
?K ?] ?? ?M ?K 
 
| x | 1 5
g(x) ,x
2x 5 2
?K ?] ?? ?M ?K 
 Domain of f(g(x)) 
 
2g(x) 3
f (g(x))
2g(x) 1
?K ?] ?K 
 
5
x
2
???M and 
| x | 1 1
2x 5 2
?K ???M
?K 
 
5
x R
2
????
?? ?M ?M ????
????
 and x R ?? 
 ?| Domain will be 
5
R
2
????
?M?M
????
????
 
7. For 0 < a < 1, the value of the integral 
2
0
dx
1 2a cos x a
?? ?M?K
?? is : 
 (1) 
2
2
a
?? ???K
 
 (2) 
2
2
a
?? ???M
 
 (3) 
2
1 a
?? ?M 
 (4) 
2
1 a
?? ?K 
Ans. (3) 
Sol. 
2
0
dx
I ; 0 a 1
1 2a cos x a
?? ?] ?\ ?\ ?M?K
?? 
 
2
0
dx
I
1 2a cos x a
?? ?] ?K?K
??   
 
/2
2
2 2 2 2
0
2(1 a )
2I 2 dx
(1 a ) 4a cos x
?? ?K ?] ?K?M
?? 
 
/2
2 2
2 2 2 2
0
2(1 a ).sec x
I dx
(1 a ) .sec x 4a
?? ?K ???]
?K?M
?? 
 
/2
2 2
2 2 2 2 2
0
2.(1 a ).sec x
I dx
(1 a ) .tan x (1 a )
?? ?K ???]
?K ?K ?M ?? 
 
 
 
2
/2
2
2
2
0
2
2
2.sec x
.dx
1 a
I
1 a
tan x
1 a
?? ?K ???]
???? ?M ?K????
?K????
?? 
 
2
2
I 0
(1 a ) 2
??????
?? ?] ?M????
?M ????
 
 
2
I
1 a
?? ?] ?M  
8. Let 
x
g(x) 3f f (3 x)
3
????
?] ?K ?M ????
????
 and f (x) 0 ???? ?^ for all 
x (0,3) ?? . If g is decreasing in (0, ?? ) and 
increasing in ( ?? , 3), then 8 ?? is 
 (1) 24 
 (2) 0 
 (3) 18 
 (4) 20 
Ans. (3) 
Sol. 
x
g(x) 3f f (3 x)
3
????
?] ?K ?M ????
????
 and f (x) 0 ???? ?^ ?B x ?? (0, 3) 
 f (x) ?? ?? is increasing function 
 
1 x
g (x) 3 .f f (3 x)
3 3
????
?? ?? ?? ?] ?? ?M ?M ????
????
 
 
x
f f (3 x)
3
????
???? ?] ?M ?M ????
????
 
 If g is decreasing in (0, ?? ) 
 g (x) 0 ?? ?\ 
 
x
f f (3 x) 0
3
????
???? ?M ?M ?\ ????
????
 
 
x
f f (3 x)
3
????
???? ?\?M
????
????
 
 
x
3 x
3
?? ?\ ?M  
 
9
x
4
???\ 
 Therefore 
9
4
???] 
 Then 
9
8 8 18
4
?? ?] ?? ?]  
9. If 
e
2
x 0
3 sin x cos x log (1 x)
1
lim
3tan x 3
?? ?K ?? ?K ?? ?K ?M ?] , then 
2 ?? – ?? is equal to : 
 (1) 2 
 (2) 7 
 (3) 5 
 (4) 1 
Ans. (3) 
Sol. 
e
2
x 0
3 sin x cos x log (1 x)
1
lim
3tan x 3
?? ?K ?? ?K ?? ?K ?M ?] 
 
3 2 4 2 3
2
x 0
x x x x x
3 x .... 1 .... x ...
3! 2! 4! 2 3 1
lim
3tan x 3
?? ?? ?? ?? ?? ?? ?? ?K ?? ?M ?K ?K ?? ?M ?K ?K ?M ?M ?M???? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ???]
 
 
2
2
2 2
x 0
1
(3 ) ( 1)x x ....
x 1 2 2
lim
3x tan x 3
?? ?? ????
?K ?? ?K ?? ?M ?K ?M ?M ?K????
????
?? ?? ?] 
 3 0, 1 0 ?? ?? ?K ?] ?? ?M ?] and 
1
1
2 2
3 3
?? ?M?M
?] 
 3, 1 ?? ?? ?] ?M ?? ?] 
 2 2 3 5 ?? ?? ?M ?? ?] ?K ?] 
10. If ?? , ?? are the roots of the equation, 
2
x x 1 0 ?M ?M ?] 
and 
n n
n
S 2023 2024 ?] ?? ?K ?? , then 
 (1) 
12 11 10
2S S S ?]?K 
 (2) 
12 11 10
S S S ?]?K 
 (3) 
11 12 10
2S S S ?]?K 
 (4) 
11 10 12
S S S ?]?K 
Ans. (2) 
Sol. 
2
x x 1 0 ?M ?M ?] 
 
n n
n
S 2023 2024 ?] ?? ?K ?? 
 
n 1 n 1 n 2 n 2
n 1 n 2
S S 2023 2024 2023 2024
?M ?M ?M ?M ?M?M
?K ?] ?? ?K ?? ?K ?? ?K ?? 
 
n 2 n 2
2023 [1 ] 2024 [1 ]
?M?M
?] ?? ?K ?? ?K ?? ?K ?? 
 
n 2 2 n 2 2
2023 [ ] 2024 [ ]
?M?M
?] ?? ?? ?K ?? ?? 
 
n n
2023 2024 ?] ?? ?K ?? 
 
n 1 n 2 n
S S S
?M?M
?K?] 
 Put n = 12 
 
11 10 12
S S S ?K?] 
 
 
 
11. Let A and B be two finite sets with m and n 
elements respectively. The total number of subsets 
of the set A is 56 more than the total number of 
subsets of B. Then the distance of the point P(m, n) 
from the point Q(–2, –3) is 
 (1) 10 
 (2) 6 
 (3) 4 
 (4) 8 
Ans. (1) 
Sol. 
m n
2 2 56 ?M?] 
 
n m n 3
2 (2 1) 2 7
?M ?M ?] ?? 
 
n 3
2 2 ?] and 
m n
2 1 7
?M ?M?] 
 
m n
n 3 and 2 8
?M ?? ?] ?] 
 n 3 and m n 3 ?? ?] ?M ?] 
 n 3 and m 6 ?? ?] ?] 
 P(6,3) and Q(–2, –3) 
 
2 2
PQ 8 6 100 10 ?] ?K ?] ?] 
 Hence option (1) is correct 
 
 
12. The values of ?? , for which  
 
3 3
1
2 2
1 1
1 0
3 3
2 3 3 1 0
???K
?? ?K ?] ?? ?K ?? ?K , lie in the interval 
 (1) (–2, 1) 
 (2) (–3, 0) 
 (3) 
3 3
,
2 2
????
?M????
????
 
 (4) (0, 3) 
Ans. (2) 
Sol. 
3 3
1
2 2
1 1
1 0
3 3
2 3 3 1 0
???K
?? ?K ?] ?? ?K ?? ?K 
 
7 7
(2 3) (3 1) 0
6 6
???M ?? ?? ?? ?? ?? ?? ?K ?M ?? ?K ?] ?? ?? ?? ?? ?? ?? ?? ?? 
 
7 7
(2 3). (3 1). 0
6 6
?? ?? ?? ?K ?K ?? ?K ?] 
 
2
2 3 3 1 0 ?? ?? ?K ?? ?K ?? ?K ?] 
 
2
2 6 1 0 ?? ?? ?K ?? ?K ?] 
 
3 7 3 7
,
2 2
?M ?K ?M ?M ?? ?? ?] 
 Hence option (2) is correct. 
13. An urn contains 6 white and 9 black balls. Two 
successive draws of 4 balls are made without 
replacement. The probability, that the first draw 
gives all white balls and the second draw gives all 
black balls, is : 
 (1) 
5
256
 (2) 
5
715
 
 (3) 
3
715
  (4) 
3
256
 
Ans. (3) 
Sol. 
6 9
4 4
15 11
4 4
C C
3
C C 715
???] 
 Hence option (3) is correct. 
14. The integral 
8 2
12 6 1 3
3
(x x )dx
1
(x 3x 1) tan x
x
?M ?M ????
?K ?K ?K????
????
?? is 
equal to :  
 (1) 
1/3
1 3
3 e
1
log tan x C
x
?M????
????
?K?K
????
????
????
????
 
 (2) 
1/2
1 3
3 e
1
log tan x C
x
?M????
????
?K?K
????
????
????
????
 
 (3) 
1 3
3 e
1
log tan x C
x
?M????
????
?K?K
????
????
????
????
 
 (4) 
3
1 3
3 e
1
log tan x C
x
?M????
????
?K?K
????
????
????
????
 
Ans. (1)