JEE Main 2024 January 27 Shift 2 Paper & Solutions | Mock Tests for JEE Main and Advanced 2025 PDF Download

 Page 1


  
            
  
 
 
 
 
 
 
 
 
SECTION-A 
1. Considering only the principal values of inverse 
trigonometric functions, the number of positive 
real values of x satisfying 
11
tan (x) tan (2x)
4
??
?
?? 
is :
 
 (1) More than 2 
 (2) 1 
 (3) 2 
 (4) 0 
Ans. (2) 
Sol. 
11
tan x tan 2x
4
??
?
?? ; x > 0 
 ?
11
tan 2x tan x
4
??
?
?? 
 Taking tan both sides 
 ?
1x
2x
1x
?
?
?
 
 
2
2x 3x 1 0 ? ? ? ? 
 
3 9 8 3 17
x
88
? ? ? ? ?
?? 
 Only possible 
3 17
x
8
??
? 
2. Consider the function f :(0,2) R ? defined by 
x2
f (x)
2x
?? 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
? ? ? ? ?
?
?
?
? ? ?
?
?
. 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 ? ; 
x2
f (x)
2x
?? 
 
12
f (x)
2x
?
? ?? 
 f (x) ? is decreasing in domain.  
 
2
2
x
f(x)
 
 
x2
0 x 1
2x
g(x)
3
x 1 x 2
2
?
? ? ?
?
?
?
?
? ? ?
?
 
 
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
??
?? 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) 
? ?
1, 2,1 2 ?? 
 (2) 
? ?
1,2,1 2 ? 
 (3) 
? ?
3,4,3 2 2 ? 
 (4) 
? ?
3, 4,3 2 2 ?? 
Ans. (3) 
Page 2


  
            
  
 
 
 
 
 
 
 
 
SECTION-A 
1. Considering only the principal values of inverse 
trigonometric functions, the number of positive 
real values of x satisfying 
11
tan (x) tan (2x)
4
??
?
?? 
is :
 
 (1) More than 2 
 (2) 1 
 (3) 2 
 (4) 0 
Ans. (2) 
Sol. 
11
tan x tan 2x
4
??
?
?? ; x > 0 
 ?
11
tan 2x tan x
4
??
?
?? 
 Taking tan both sides 
 ?
1x
2x
1x
?
?
?
 
 
2
2x 3x 1 0 ? ? ? ? 
 
3 9 8 3 17
x
88
? ? ? ? ?
?? 
 Only possible 
3 17
x
8
??
? 
2. Consider the function f :(0,2) R ? defined by 
x2
f (x)
2x
?? 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
? ? ? ? ?
?
?
?
? ? ?
?
?
. 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 ? ; 
x2
f (x)
2x
?? 
 
12
f (x)
2x
?
? ?? 
 f (x) ? is decreasing in domain.  
 
2
2
x
f(x)
 
 
x2
0 x 1
2x
g(x)
3
x 1 x 2
2
?
? ? ?
?
?
?
?
? ? ?
?
 
 
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
??
?? 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) 
? ?
1, 2,1 2 ?? 
 (2) 
? ?
1,2,1 2 ? 
 (3) 
? ?
3,4,3 2 2 ? 
 (4) 
? ?
3, 4,3 2 2 ?? 
Ans. (3) 
 
 
 
 
 
Sol. 
1
x y 1 z 2
L
1 2 3
??
? ? ? ? ? 
  
 M( ,1 2 ,2 3 ) ? ? ? ? ? 
 
ˆ ˆ ˆ
PM ( 1)i (1 2 )j (3 5)k ? ? ? ? ? ? ? ? ? 
 PM is perpendicular to line L
1
 
 PM.b 0 ?     (
ˆ ˆ ˆ
b i 2j 3k ? ? ? ) 
 1 4 2 9 15 0 ? ? ? ? ? ? ? ? ? ? 
 14 14 1 ? ? ? ? ? 
 M (1,3,5) ?? 
 Q 2M P ?? [M is midpoint of P & Q ] 
 
ˆ ˆ ˆ ˆ ˆ
Q 2i 6j 10k i 7k ? ? ? ? ? 
 
ˆ ˆ ˆ
Q i 6j 3k ? ? ? 
 ( , , ) (1,6,3) ? ? ? ? ? 
 Required line having direction cosine (l, m, n) 
 
2 2 2
1 ? ? ? l m n 
 
2 2
2
11
1
2 2
?? ??
? ? ? ? ? ?
?? ??
?? ??
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
22 2
??
? ? ? ? ? ? ?
??
??
 
 Option (3) satisfying for ? = 4 
4. Let R be the interior region between the lines 
3x y 1 0 ? ? ? and x 2y 5 0 ? ? ? 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
??
? ? ? ?
??
??
 
 (2) 
1
( 3,0) ,1
3
??
??
??
??
 
 (3) 
2
( 3,0) ,1
3
??
??
??
??
 
 (4) 
1
( 3, 1) ,1
3
??
? ? ?
??
??
 
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
??
? ? ? ? ?
??
??
…………….(1) 
 Let L
2
 : x + 2y – 5 = 0 
 Origin and P lies same side w.r.t. L
2
 
 ?
22
L (0).L (P) 0 ? 
 
2
a 2(a 1) 5 0 ? ? ? ? ? 
 ?
2
a 2a 3 0 ? ? ? 
 ? (a 3)(a 1) 0 ? ? ? 
 ? a ( 3,1) ?? …………….(2) 
 Intersection of (1) and (2) 
 
1
a ( 3,0) ,1
3
??
? ? ?
??
??
 
Page 3


  
            
  
 
 
 
 
 
 
 
 
SECTION-A 
1. Considering only the principal values of inverse 
trigonometric functions, the number of positive 
real values of x satisfying 
11
tan (x) tan (2x)
4
??
?
?? 
is :
 
 (1) More than 2 
 (2) 1 
 (3) 2 
 (4) 0 
Ans. (2) 
Sol. 
11
tan x tan 2x
4
??
?
?? ; x > 0 
 ?
11
tan 2x tan x
4
??
?
?? 
 Taking tan both sides 
 ?
1x
2x
1x
?
?
?
 
 
2
2x 3x 1 0 ? ? ? ? 
 
3 9 8 3 17
x
88
? ? ? ? ?
?? 
 Only possible 
3 17
x
8
??
? 
2. Consider the function f :(0,2) R ? defined by 
x2
f (x)
2x
?? 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
? ? ? ? ?
?
?
?
? ? ?
?
?
. 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 ? ; 
x2
f (x)
2x
?? 
 
12
f (x)
2x
?
? ?? 
 f (x) ? is decreasing in domain.  
 
2
2
x
f(x)
 
 
x2
0 x 1
2x
g(x)
3
x 1 x 2
2
?
? ? ?
?
?
?
?
? ? ?
?
 
 
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
??
?? 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) 
? ?
1, 2,1 2 ?? 
 (2) 
? ?
1,2,1 2 ? 
 (3) 
? ?
3,4,3 2 2 ? 
 (4) 
? ?
3, 4,3 2 2 ?? 
Ans. (3) 
 
 
 
 
 
Sol. 
1
x y 1 z 2
L
1 2 3
??
? ? ? ? ? 
  
 M( ,1 2 ,2 3 ) ? ? ? ? ? 
 
ˆ ˆ ˆ
PM ( 1)i (1 2 )j (3 5)k ? ? ? ? ? ? ? ? ? 
 PM is perpendicular to line L
1
 
 PM.b 0 ?     (
ˆ ˆ ˆ
b i 2j 3k ? ? ? ) 
 1 4 2 9 15 0 ? ? ? ? ? ? ? ? ? ? 
 14 14 1 ? ? ? ? ? 
 M (1,3,5) ?? 
 Q 2M P ?? [M is midpoint of P & Q ] 
 
ˆ ˆ ˆ ˆ ˆ
Q 2i 6j 10k i 7k ? ? ? ? ? 
 
ˆ ˆ ˆ
Q i 6j 3k ? ? ? 
 ( , , ) (1,6,3) ? ? ? ? ? 
 Required line having direction cosine (l, m, n) 
 
2 2 2
1 ? ? ? l m n 
 
2 2
2
11
1
2 2
?? ??
? ? ? ? ? ?
?? ??
?? ??
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
22 2
??
? ? ? ? ? ? ?
??
??
 
 Option (3) satisfying for ? = 4 
4. Let R be the interior region between the lines 
3x y 1 0 ? ? ? and x 2y 5 0 ? ? ? 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
??
? ? ? ?
??
??
 
 (2) 
1
( 3,0) ,1
3
??
??
??
??
 
 (3) 
2
( 3,0) ,1
3
??
??
??
??
 
 (4) 
1
( 3, 1) ,1
3
??
? ? ?
??
??
 
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
??
? ? ? ? ?
??
??
…………….(1) 
 Let L
2
 : x + 2y – 5 = 0 
 Origin and P lies same side w.r.t. L
2
 
 ?
22
L (0).L (P) 0 ? 
 
2
a 2(a 1) 5 0 ? ? ? ? ? 
 ?
2
a 2a 3 0 ? ? ? 
 ? (a 3)(a 1) 0 ? ? ? 
 ? a ( 3,1) ?? …………….(2) 
 Intersection of (1) and (2) 
 
1
a ( 3,0) ,1
3
??
? ? ?
??
??
 
 
 
 
 
 
 
5. The 20
th
 term from the end of the progression 
1 1 3 1
20,19 ,18 ,17 ,...., 129
4 2 4 4
? 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
? 
 This is A.P. with common difference  
 
1
13
d1
44
? ? ? ? ? 
 
11
129 ,..............,19 ,20
44
? 
 This is also A.P. 
1
a 129
4
?? and 
3
d
4
? 
 Required term =  
 
13
129 (20 1)
44
??
? ? ?
??
??
 
 
13
129 15 115
44
? ? ? ? ? ? ? 
6. Let 
1
f : R R
2
? ??
??
??
??
 and 
5
g : R R
2
? ??
??
??
??
 be 
defined as 
2x 3
f (x)
2x 1
?
?
?
 and 
| x | 1
g(x)
2x 5
?
?
?
. Then 
the domain of the function fog is : 
 (1) 
5
R
2
??
??
??
??
 
 (2) R 
 (3) 
7
R
4
??
??
??
??
 
 (4) 
57
R,
24
??
? ? ?
??
??
 
Ans. (1) 
Sol. 
2x 3 1
f (x) ;x
2x 1 2
?
? ? ?
?
 
 
| x | 1 5
g(x) ,x
2x 5 2
?
? ? ?
?
 
 Domain of f(g(x)) 
 
2g(x) 3
f (g(x))
2g(x) 1
?
?
?
 
 
5
x
2
?? and 
| x | 1 1
2x 5 2
?
??
?
 
 
5
xR
2
??
? ? ?
??
??
 and xR ? 
 ? Domain will be 
5
R
2
??
??
??
??
 
7. For 0 < a < 1, the value of the integral 
2
0
dx
1 2a cos x a
?
??
?
 is : 
 (1) 
2
2
a
?
??
 
 (2) 
2
2
a
?
??
 
 (3) 
2
1a
?
?
 
 (4) 
2
1a
?
?
 
Ans. (3) 
Sol. 
2
0
dx
I ; 0 a 1
1 2a cos x a
?
? ? ?
??
?
 
 
2
0
dx
I
1 2a cos x a
?
?
??
?
   
 
/2
2
2 2 2 2
0
2(1 a )
2I 2 dx
(1 a ) 4a cos x
?
?
?
??
?
 
 
/2
22
2 2 2 2
0
2(1 a ).sec x
I dx
(1 a ) .sec x 4a
?
?
??
??
?
 
 
/2
22
2 2 2 2 2
0
2.(1 a ).sec x
I dx
(1 a ) .tan x (1 a )
?
?
??
? ? ?
?
 
Page 4


  
            
  
 
 
 
 
 
 
 
 
SECTION-A 
1. Considering only the principal values of inverse 
trigonometric functions, the number of positive 
real values of x satisfying 
11
tan (x) tan (2x)
4
??
?
?? 
is :
 
 (1) More than 2 
 (2) 1 
 (3) 2 
 (4) 0 
Ans. (2) 
Sol. 
11
tan x tan 2x
4
??
?
?? ; x > 0 
 ?
11
tan 2x tan x
4
??
?
?? 
 Taking tan both sides 
 ?
1x
2x
1x
?
?
?
 
 
2
2x 3x 1 0 ? ? ? ? 
 
3 9 8 3 17
x
88
? ? ? ? ?
?? 
 Only possible 
3 17
x
8
??
? 
2. Consider the function f :(0,2) R ? defined by 
x2
f (x)
2x
?? 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
? ? ? ? ?
?
?
?
? ? ?
?
?
. 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 ? ; 
x2
f (x)
2x
?? 
 
12
f (x)
2x
?
? ?? 
 f (x) ? is decreasing in domain.  
 
2
2
x
f(x)
 
 
x2
0 x 1
2x
g(x)
3
x 1 x 2
2
?
? ? ?
?
?
?
?
? ? ?
?
 
 
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
??
?? 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) 
? ?
1, 2,1 2 ?? 
 (2) 
? ?
1,2,1 2 ? 
 (3) 
? ?
3,4,3 2 2 ? 
 (4) 
? ?
3, 4,3 2 2 ?? 
Ans. (3) 
 
 
 
 
 
Sol. 
1
x y 1 z 2
L
1 2 3
??
? ? ? ? ? 
  
 M( ,1 2 ,2 3 ) ? ? ? ? ? 
 
ˆ ˆ ˆ
PM ( 1)i (1 2 )j (3 5)k ? ? ? ? ? ? ? ? ? 
 PM is perpendicular to line L
1
 
 PM.b 0 ?     (
ˆ ˆ ˆ
b i 2j 3k ? ? ? ) 
 1 4 2 9 15 0 ? ? ? ? ? ? ? ? ? ? 
 14 14 1 ? ? ? ? ? 
 M (1,3,5) ?? 
 Q 2M P ?? [M is midpoint of P & Q ] 
 
ˆ ˆ ˆ ˆ ˆ
Q 2i 6j 10k i 7k ? ? ? ? ? 
 
ˆ ˆ ˆ
Q i 6j 3k ? ? ? 
 ( , , ) (1,6,3) ? ? ? ? ? 
 Required line having direction cosine (l, m, n) 
 
2 2 2
1 ? ? ? l m n 
 
2 2
2
11
1
2 2
?? ??
? ? ? ? ? ?
?? ??
?? ??
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
22 2
??
? ? ? ? ? ? ?
??
??
 
 Option (3) satisfying for ? = 4 
4. Let R be the interior region between the lines 
3x y 1 0 ? ? ? and x 2y 5 0 ? ? ? 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
??
? ? ? ?
??
??
 
 (2) 
1
( 3,0) ,1
3
??
??
??
??
 
 (3) 
2
( 3,0) ,1
3
??
??
??
??
 
 (4) 
1
( 3, 1) ,1
3
??
? ? ?
??
??
 
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
??
? ? ? ? ?
??
??
…………….(1) 
 Let L
2
 : x + 2y – 5 = 0 
 Origin and P lies same side w.r.t. L
2
 
 ?
22
L (0).L (P) 0 ? 
 
2
a 2(a 1) 5 0 ? ? ? ? ? 
 ?
2
a 2a 3 0 ? ? ? 
 ? (a 3)(a 1) 0 ? ? ? 
 ? a ( 3,1) ?? …………….(2) 
 Intersection of (1) and (2) 
 
1
a ( 3,0) ,1
3
??
? ? ?
??
??
 
 
 
 
 
 
 
5. The 20
th
 term from the end of the progression 
1 1 3 1
20,19 ,18 ,17 ,...., 129
4 2 4 4
? 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
? 
 This is A.P. with common difference  
 
1
13
d1
44
? ? ? ? ? 
 
11
129 ,..............,19 ,20
44
? 
 This is also A.P. 
1
a 129
4
?? and 
3
d
4
? 
 Required term =  
 
13
129 (20 1)
44
??
? ? ?
??
??
 
 
13
129 15 115
44
? ? ? ? ? ? ? 
6. Let 
1
f : R R
2
? ??
??
??
??
 and 
5
g : R R
2
? ??
??
??
??
 be 
defined as 
2x 3
f (x)
2x 1
?
?
?
 and 
| x | 1
g(x)
2x 5
?
?
?
. Then 
the domain of the function fog is : 
 (1) 
5
R
2
??
??
??
??
 
 (2) R 
 (3) 
7
R
4
??
??
??
??
 
 (4) 
57
R,
24
??
? ? ?
??
??
 
Ans. (1) 
Sol. 
2x 3 1
f (x) ;x
2x 1 2
?
? ? ?
?
 
 
| x | 1 5
g(x) ,x
2x 5 2
?
? ? ?
?
 
 Domain of f(g(x)) 
 
2g(x) 3
f (g(x))
2g(x) 1
?
?
?
 
 
5
x
2
?? and 
| x | 1 1
2x 5 2
?
??
?
 
 
5
xR
2
??
? ? ?
??
??
 and xR ? 
 ? Domain will be 
5
R
2
??
??
??
??
 
7. For 0 < a < 1, the value of the integral 
2
0
dx
1 2a cos x a
?
??
?
 is : 
 (1) 
2
2
a
?
??
 
 (2) 
2
2
a
?
??
 
 (3) 
2
1a
?
?
 
 (4) 
2
1a
?
?
 
Ans. (3) 
Sol. 
2
0
dx
I ; 0 a 1
1 2a cos x a
?
? ? ?
??
?
 
 
2
0
dx
I
1 2a cos x a
?
?
??
?
   
 
/2
2
2 2 2 2
0
2(1 a )
2I 2 dx
(1 a ) 4a cos x
?
?
?
??
?
 
 
/2
22
2 2 2 2
0
2(1 a ).sec x
I dx
(1 a ) .sec x 4a
?
?
??
??
?
 
 
/2
22
2 2 2 2 2
0
2.(1 a ).sec x
I dx
(1 a ) .tan x (1 a )
?
?
??
? ? ?
?
 
 
 
 
 
 
2
/2
2
2
2
0
2
2
2.sec x
.dx
1a
I
1a
tan x
1a
?
?
??
?? ?
?
??
?
??
?
 
 
2
2
I0
(1 a ) 2
? ??
? ? ?
??
?
??
 
 
2
I
1a
?
?
?
  
8. Let 
x
g(x) 3f f (3 x)
3
??
? ? ?
??
??
 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
??
? ? ?
??
??
 and f (x) 0 ?? ? ? x ? (0, 3) 
 f (x) ? ? is increasing function 
 
1x
g (x) 3 .f f (3 x)
33
??
? ? ? ? ? ? ?
??
??
 
 
x
f f (3 x)
3
??
?? ? ? ?
??
??
 
 If g is decreasing in (0, ?) 
 g (x) 0 ? ? 
 
x
f f (3 x) 0
3
??
?? ? ? ?
??
??
 
 
x
f f (3 x)
3
??
?? ??
??
??
 
 
x
3x
3
? ? ?  
 
9
x
4
?? 
 Therefore 
9
4
?? 
 Then 
9
8 8 18
4
? ? ? ?  
9. If 
e
2
x0
3 sin x cos x log (1 x)
1
lim
3tan x 3
?
? ? ? ? ? ?
? , then 
2 ?– ? is equal to : 
 (1) 2 
 (2) 7 
 (3) 5 
 (4) 1 
Ans. (3) 
Sol. 
e
2
x0
3 sin x cos x log (1 x)
1
lim
3tan x 3
?
? ? ? ? ? ?
? 
 
3 2 4 2 3
2
x0
x x x x x
3 x .... 1 .... x ...
3! 2! 4! 2 3 1
lim
3tan x 3
?
? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
?? ? ? ? ?
? ? ? ? ? ?
??
 
 
2
2
22
x0
1
(3 ) ( 1)x x ....
x1 22
lim
3x tan x 3
?
? ??
? ? ? ? ? ? ? ? ?
??
??
? ? ?
 
 3 0, 1 0 ? ? ? ? ? ? ? and 
1
1
22
33
?
??
? 
 3, 1 ? ? ? ? ? ? 
 2 2 3 5 ? ? ? ? ? ? ? 
10. If ?, ? are the roots of the equation, 
2
x x 1 0 ? ? ? 
and 
nn
n
S 2023 2024 ? ? ? ? , then 
 (1) 
12 11 10
2S S S ?? 
 (2) 
12 11 10
S S S ?? 
 (3) 
11 12 10
2S S S ?? 
 (4) 
11 10 12
S S S ?? 
Ans. (2) 
Sol. 
2
x x 1 0 ? ? ? 
 
nn
n
S 2023 2024 ? ? ? ? 
 
n 1 n 1 n 2 n 2
n 1 n 2
S S 2023 2024 2023 2024
? ? ? ?
??
? ? ? ? ? ? ? ? ? 
 
n 2 n 2
2023 [1 ] 2024 [1 ]
??
? ? ? ? ? ? ? ? 
 
n 2 2 n 2 2
2023 [ ] 2024 [ ]
??
? ? ? ? ? ? 
 
nn
2023 2024 ? ? ? ? 
 
n 1 n 2 n
S S S
??
?? 
 Put n = 12 
 
11 10 12
S S S ?? 
Page 5


  
            
  
 
 
 
 
 
 
 
 
SECTION-A 
1. Considering only the principal values of inverse 
trigonometric functions, the number of positive 
real values of x satisfying 
11
tan (x) tan (2x)
4
??
?
?? 
is :
 
 (1) More than 2 
 (2) 1 
 (3) 2 
 (4) 0 
Ans. (2) 
Sol. 
11
tan x tan 2x
4
??
?
?? ; x > 0 
 ?
11
tan 2x tan x
4
??
?
?? 
 Taking tan both sides 
 ?
1x
2x
1x
?
?
?
 
 
2
2x 3x 1 0 ? ? ? ? 
 
3 9 8 3 17
x
88
? ? ? ? ?
?? 
 Only possible 
3 17
x
8
??
? 
2. Consider the function f :(0,2) R ? defined by 
x2
f (x)
2x
?? 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
? ? ? ? ?
?
?
?
? ? ?
?
?
. 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 ? ; 
x2
f (x)
2x
?? 
 
12
f (x)
2x
?
? ?? 
 f (x) ? is decreasing in domain.  
 
2
2
x
f(x)
 
 
x2
0 x 1
2x
g(x)
3
x 1 x 2
2
?
? ? ?
?
?
?
?
? ? ?
?
 
 
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
??
?? 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) 
? ?
1, 2,1 2 ?? 
 (2) 
? ?
1,2,1 2 ? 
 (3) 
? ?
3,4,3 2 2 ? 
 (4) 
? ?
3, 4,3 2 2 ?? 
Ans. (3) 
 
 
 
 
 
Sol. 
1
x y 1 z 2
L
1 2 3
??
? ? ? ? ? 
  
 M( ,1 2 ,2 3 ) ? ? ? ? ? 
 
ˆ ˆ ˆ
PM ( 1)i (1 2 )j (3 5)k ? ? ? ? ? ? ? ? ? 
 PM is perpendicular to line L
1
 
 PM.b 0 ?     (
ˆ ˆ ˆ
b i 2j 3k ? ? ? ) 
 1 4 2 9 15 0 ? ? ? ? ? ? ? ? ? ? 
 14 14 1 ? ? ? ? ? 
 M (1,3,5) ?? 
 Q 2M P ?? [M is midpoint of P & Q ] 
 
ˆ ˆ ˆ ˆ ˆ
Q 2i 6j 10k i 7k ? ? ? ? ? 
 
ˆ ˆ ˆ
Q i 6j 3k ? ? ? 
 ( , , ) (1,6,3) ? ? ? ? ? 
 Required line having direction cosine (l, m, n) 
 
2 2 2
1 ? ? ? l m n 
 
2 2
2
11
1
2 2
?? ??
? ? ? ? ? ?
?? ??
?? ??
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
22 2
??
? ? ? ? ? ? ?
??
??
 
 Option (3) satisfying for ? = 4 
4. Let R be the interior region between the lines 
3x y 1 0 ? ? ? and x 2y 5 0 ? ? ? 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
??
? ? ? ?
??
??
 
 (2) 
1
( 3,0) ,1
3
??
??
??
??
 
 (3) 
2
( 3,0) ,1
3
??
??
??
??
 
 (4) 
1
( 3, 1) ,1
3
??
? ? ?
??
??
 
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
??
? ? ? ? ?
??
??
…………….(1) 
 Let L
2
 : x + 2y – 5 = 0 
 Origin and P lies same side w.r.t. L
2
 
 ?
22
L (0).L (P) 0 ? 
 
2
a 2(a 1) 5 0 ? ? ? ? ? 
 ?
2
a 2a 3 0 ? ? ? 
 ? (a 3)(a 1) 0 ? ? ? 
 ? a ( 3,1) ?? …………….(2) 
 Intersection of (1) and (2) 
 
1
a ( 3,0) ,1
3
??
? ? ?
??
??
 
 
 
 
 
 
 
5. The 20
th
 term from the end of the progression 
1 1 3 1
20,19 ,18 ,17 ,...., 129
4 2 4 4
? 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
? 
 This is A.P. with common difference  
 
1
13
d1
44
? ? ? ? ? 
 
11
129 ,..............,19 ,20
44
? 
 This is also A.P. 
1
a 129
4
?? and 
3
d
4
? 
 Required term =  
 
13
129 (20 1)
44
??
? ? ?
??
??
 
 
13
129 15 115
44
? ? ? ? ? ? ? 
6. Let 
1
f : R R
2
? ??
??
??
??
 and 
5
g : R R
2
? ??
??
??
??
 be 
defined as 
2x 3
f (x)
2x 1
?
?
?
 and 
| x | 1
g(x)
2x 5
?
?
?
. Then 
the domain of the function fog is : 
 (1) 
5
R
2
??
??
??
??
 
 (2) R 
 (3) 
7
R
4
??
??
??
??
 
 (4) 
57
R,
24
??
? ? ?
??
??
 
Ans. (1) 
Sol. 
2x 3 1
f (x) ;x
2x 1 2
?
? ? ?
?
 
 
| x | 1 5
g(x) ,x
2x 5 2
?
? ? ?
?
 
 Domain of f(g(x)) 
 
2g(x) 3
f (g(x))
2g(x) 1
?
?
?
 
 
5
x
2
?? and 
| x | 1 1
2x 5 2
?
??
?
 
 
5
xR
2
??
? ? ?
??
??
 and xR ? 
 ? Domain will be 
5
R
2
??
??
??
??
 
7. For 0 < a < 1, the value of the integral 
2
0
dx
1 2a cos x a
?
??
?
 is : 
 (1) 
2
2
a
?
??
 
 (2) 
2
2
a
?
??
 
 (3) 
2
1a
?
?
 
 (4) 
2
1a
?
?
 
Ans. (3) 
Sol. 
2
0
dx
I ; 0 a 1
1 2a cos x a
?
? ? ?
??
?
 
 
2
0
dx
I
1 2a cos x a
?
?
??
?
   
 
/2
2
2 2 2 2
0
2(1 a )
2I 2 dx
(1 a ) 4a cos x
?
?
?
??
?
 
 
/2
22
2 2 2 2
0
2(1 a ).sec x
I dx
(1 a ) .sec x 4a
?
?
??
??
?
 
 
/2
22
2 2 2 2 2
0
2.(1 a ).sec x
I dx
(1 a ) .tan x (1 a )
?
?
??
? ? ?
?
 
 
 
 
 
 
2
/2
2
2
2
0
2
2
2.sec x
.dx
1a
I
1a
tan x
1a
?
?
??
?? ?
?
??
?
??
?
 
 
2
2
I0
(1 a ) 2
? ??
? ? ?
??
?
??
 
 
2
I
1a
?
?
?
  
8. Let 
x
g(x) 3f f (3 x)
3
??
? ? ?
??
??
 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
??
? ? ?
??
??
 and f (x) 0 ?? ? ? x ? (0, 3) 
 f (x) ? ? is increasing function 
 
1x
g (x) 3 .f f (3 x)
33
??
? ? ? ? ? ? ?
??
??
 
 
x
f f (3 x)
3
??
?? ? ? ?
??
??
 
 If g is decreasing in (0, ?) 
 g (x) 0 ? ? 
 
x
f f (3 x) 0
3
??
?? ? ? ?
??
??
 
 
x
f f (3 x)
3
??
?? ??
??
??
 
 
x
3x
3
? ? ?  
 
9
x
4
?? 
 Therefore 
9
4
?? 
 Then 
9
8 8 18
4
? ? ? ?  
9. If 
e
2
x0
3 sin x cos x log (1 x)
1
lim
3tan x 3
?
? ? ? ? ? ?
? , then 
2 ?– ? is equal to : 
 (1) 2 
 (2) 7 
 (3) 5 
 (4) 1 
Ans. (3) 
Sol. 
e
2
x0
3 sin x cos x log (1 x)
1
lim
3tan x 3
?
? ? ? ? ? ?
? 
 
3 2 4 2 3
2
x0
x x x x x
3 x .... 1 .... x ...
3! 2! 4! 2 3 1
lim
3tan x 3
?
? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
?? ? ? ? ?
? ? ? ? ? ?
??
 
 
2
2
22
x0
1
(3 ) ( 1)x x ....
x1 22
lim
3x tan x 3
?
? ??
? ? ? ? ? ? ? ? ?
??
??
? ? ?
 
 3 0, 1 0 ? ? ? ? ? ? ? and 
1
1
22
33
?
??
? 
 3, 1 ? ? ? ? ? ? 
 2 2 3 5 ? ? ? ? ? ? ? 
10. If ?, ? are the roots of the equation, 
2
x x 1 0 ? ? ? 
and 
nn
n
S 2023 2024 ? ? ? ? , then 
 (1) 
12 11 10
2S S S ?? 
 (2) 
12 11 10
S S S ?? 
 (3) 
11 12 10
2S S S ?? 
 (4) 
11 10 12
S S S ?? 
Ans. (2) 
Sol. 
2
x x 1 0 ? ? ? 
 
nn
n
S 2023 2024 ? ? ? ? 
 
n 1 n 1 n 2 n 2
n 1 n 2
S S 2023 2024 2023 2024
? ? ? ?
??
? ? ? ? ? ? ? ? ? 
 
n 2 n 2
2023 [1 ] 2024 [1 ]
??
? ? ? ? ? ? ? ? 
 
n 2 2 n 2 2
2023 [ ] 2024 [ ]
??
? ? ? ? ? ? 
 
nn
2023 2024 ? ? ? ? 
 
n 1 n 2 n
S S S
??
?? 
 Put n = 12 
 
11 10 12
S S S ?? 
 
 
 
 
 
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. 
mn
2 2 56 ?? 
 
n m n 3
2 (2 1) 2 7
?
? ? ? 
 
n3
22 ? and 
mn
2 1 7
?
?? 
 
mn
n 3 and 2 8
?
? ? ? 
 n 3 and m n 3 ? ? ? ? 
 n 3 and m 6 ? ? ? 
 P(6,3) and Q(–2, –3) 
 
22
PQ 8 6 100 10 ? ? ? ? 
 Hence option (1) is correct 
 
 
12. The values of ?, for which  
 
33
1
22
11
10
33
2 3 3 1 0
??
? ? ?
? ? ? ?
, lie in the interval 
 (1) (–2, 1) 
 (2) (–3, 0) 
 (3) 
33
,
22
??
?
??
??
 
 (4) (0, 3) 
Ans. (2) 
Sol. 
33
1
22
11
10
33
2 3 3 1 0
??
? ? ?
? ? ? ?
 
 
77
(2 3) (3 1) 0
66
?? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ?
? ? ? ?
 
 
77
(2 3). (3 1). 0
66
?
? ? ? ? ? ? ? 
 
2
2 3 3 1 0 ? ? ? ? ? ? ? ? 
 
2
2 6 1 0 ? ? ? ? ? ? 
 
3 7 3 7
,
22
? ? ? ?
? ? ? 
 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. 
69
44
15 11
44
CC
3
C C 715
?? 
 Hence option (3) is correct. 
14. The integral 
82
12 6 1 3
3
(x x )dx
1
(x 3x 1) tan x
x
?
?
??
? ? ?
??
??
?
 is 
equal to :  
 (1) 
1/3
13
3 e
1
log tan x C
x
?
??
??
??
??
??
??
??
 
 (2) 
1/2
13
3 e
1
log tan x C
x
?
??
??
??
??
??
??
??
 
 (3) 
13
3 e
1
log tan x C
x
?
??
??
??
??
??
??
??
 
 (4) 
3
13
3 e
1
log tan x C
x
?
??
??
??
??
??
??
??
 
Ans. (1)