2025 (22 Jan Shift 1) JEE Main Previous Year Paper
JEE Main 2025 January MathonGo
Q1. Let a 1, a2 , a3 , … be a G.P. of increasing positive terms. If a 1 a5 = 28 and a 2 + a 4 = 29 , then a is equal to:
6
(1) 628 (2) 812
(3) 526 (4) 784
Q2. Let x = x(y) be the solution of the differential equation y 2
dx + (x −
1
y
)dy = 0 . If x(1) = 1, then x ( 1
2
) is :
(1) 1
2
+ e (2) 3 + e
(3) 3 − e (4) 3
2
+ e
Q3. Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black
balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is m
n
,
where gcd(m, n) = 1, then m + n is equal to :
(1) 4 (2) 14
(3) 13 (4) 11
2
Q4. The product of all solutions of the equation e 5(log e x) +3 8
= x ,x > 0 , is :
(1) e 8/5
(2) e 6/5
(3) e 2
(4) e
Q5. Let the triangle PQR be the image of the triangle with vertices (1, 3), (3, 1) and (2, 4) in the line x + 2y = 2. If
the centroid of △PQR is the point (α, β), then 15(α − β) is equal to :
(1) 19 (2) 24
(3) 21 (4) 22
Q6. Let for f (x) = 7 tan 8
x + 7 tan
6
x − 3 tan
4
x − 3 tan
2
x, I1 = ∫
0
π/4
f (x)dx and I 2 = ∫
π/4
0
xf (x)dx. Then
7I 1 + 12I 2 is equal to :
(1) 2 (2) 1
(3) 2π (4) π
Q7. Let the parabola y = x 2
+ px − 3 , meet the coordinate axes at the points P, Q and R . If the circle C with centre
at (−1, −1) passes through the points P , Q and R, then the area of △P QR is :
(1) 7 (2) 4
(3) 6 (4) 5
Q8. Let L 1
:
x−1
2
=
y−2
3
=
z−3
4
and L 2
:
x−2
3
=
y−4
4
=
z−5
5
be two lines. Then which of the following points lies
on the line of the shortest distance between L and L ? 1 2
(1) ( 14
3
, −3,
22
3
) (2) (− 5
3
, −7, 1)
(3) (2, 3, 1
3
) (4) ( 8
3
, −1,
1
3
)
Q9. Let f (x) be a real differentiable function such that f (0) = 1 and f (x + y) = f (x)f ′ ′
(y) + f (x)f (y) for all
x, y ∈ R . Then ∑ 100
n=1
log
e
f (n) is equal to :
(1) 2525 (2) 5220
(3) 2384 (4) 2406
Q10. From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number
of ways, in which the middle letter is ' M ', is :
, 2025 (22 Jan Shift 1) JEE Main Previous Year Paper
JEE Main 2025 January MathonGo
(1) 5148 (2) 6084
(3) 4356 (4) 14950
Q11. Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum
values of 16 ((sec is :
2 2
−1 −1
x) + (cosec x) )
(1) 24π 2
(2) 22π 2
(3) 31π 2
(4) 18π 2
Q12. Let f : R → R be a twice differentiable function such that f (x + y) = f (x)f (y) for all x, y ∈ R. If
′
f (0) = 4a and f satisfies f ′′ ′
(x) − 3af (x) − f (x) = 0 , a > 0, then the area of the region
R = {(x, y) ∣ 0 ≤ y ≤ f (ax), 0 ≤ x ≤ 2} is:
(1) e 2
− 1 (2) e 2
+ 1
(3) e 4
+ 1 (4) e 4
− 1
Q13. The area of the region, inside the circle (x − 2√3) 2
+ y
2
= 12 and outside the parabola y 2
= 2√ 3x is :
(1) 3π + 8 (2) 6π − 16
(3) 3π − 8 (4) 6π − 8
Q14. Let the foci of a hyperbola be (1, 14) and (1, −12). If it passes through the point (1, 6), then the length of its
latus-rectum is :
(1) 24
5
(2) 25
6
(3) 144
5
(4) 288
5
Q15. If ∑ n
r=1
Tr =
(2n−1)(2n+1)(2n+3)(2n+5)
64
, then lim n→∞ ∑
n
r=1
(
1
Tr
) is equal to :
(1) 0 (2) 2
3
(3) 1 (4) 1
3
Q16. A coin is tossed three times. Let X denote the number of times a tail follows a head. If μ and σ denote the 2
mean and variance of X, then the value of 64 (μ + σ 2
) is :
(1) 51 (2) 64
(3) 32 (4) 48
Q17. The number of non-empty equivalence relations on the set {1, 2, 3} is :
(1) 6 (2) 5
(3) 7 (4) 4
Q18. A circle C of radius 2 lies in the second quadrant and touches both the coordinate axes. Let r be the radius of a
circle that has centre at the point (2, 5) and intersects the circle C at exactly two points. If the set of all
possible values of r is the interval (α, β), then 3β − 2α is equal to :
(1) 10 (2) 15
(3) 12 (4) 14
Q19. Let A = {1, 2, 3, … , 10} and B = { m
n
: m, n ∈ A, m < n and gcd(m, n) = 1}. Then n(B) is equal to :
(1) 36 (2) 31
(3) 37 (4) 29
JEE Main 2025 January MathonGo
Q1. Let a 1, a2 , a3 , … be a G.P. of increasing positive terms. If a 1 a5 = 28 and a 2 + a 4 = 29 , then a is equal to:
6
(1) 628 (2) 812
(3) 526 (4) 784
Q2. Let x = x(y) be the solution of the differential equation y 2
dx + (x −
1
y
)dy = 0 . If x(1) = 1, then x ( 1
2
) is :
(1) 1
2
+ e (2) 3 + e
(3) 3 − e (4) 3
2
+ e
Q3. Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black
balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is m
n
,
where gcd(m, n) = 1, then m + n is equal to :
(1) 4 (2) 14
(3) 13 (4) 11
2
Q4. The product of all solutions of the equation e 5(log e x) +3 8
= x ,x > 0 , is :
(1) e 8/5
(2) e 6/5
(3) e 2
(4) e
Q5. Let the triangle PQR be the image of the triangle with vertices (1, 3), (3, 1) and (2, 4) in the line x + 2y = 2. If
the centroid of △PQR is the point (α, β), then 15(α − β) is equal to :
(1) 19 (2) 24
(3) 21 (4) 22
Q6. Let for f (x) = 7 tan 8
x + 7 tan
6
x − 3 tan
4
x − 3 tan
2
x, I1 = ∫
0
π/4
f (x)dx and I 2 = ∫
π/4
0
xf (x)dx. Then
7I 1 + 12I 2 is equal to :
(1) 2 (2) 1
(3) 2π (4) π
Q7. Let the parabola y = x 2
+ px − 3 , meet the coordinate axes at the points P, Q and R . If the circle C with centre
at (−1, −1) passes through the points P , Q and R, then the area of △P QR is :
(1) 7 (2) 4
(3) 6 (4) 5
Q8. Let L 1
:
x−1
2
=
y−2
3
=
z−3
4
and L 2
:
x−2
3
=
y−4
4
=
z−5
5
be two lines. Then which of the following points lies
on the line of the shortest distance between L and L ? 1 2
(1) ( 14
3
, −3,
22
3
) (2) (− 5
3
, −7, 1)
(3) (2, 3, 1
3
) (4) ( 8
3
, −1,
1
3
)
Q9. Let f (x) be a real differentiable function such that f (0) = 1 and f (x + y) = f (x)f ′ ′
(y) + f (x)f (y) for all
x, y ∈ R . Then ∑ 100
n=1
log
e
f (n) is equal to :
(1) 2525 (2) 5220
(3) 2384 (4) 2406
Q10. From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number
of ways, in which the middle letter is ' M ', is :
, 2025 (22 Jan Shift 1) JEE Main Previous Year Paper
JEE Main 2025 January MathonGo
(1) 5148 (2) 6084
(3) 4356 (4) 14950
Q11. Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum
values of 16 ((sec is :
2 2
−1 −1
x) + (cosec x) )
(1) 24π 2
(2) 22π 2
(3) 31π 2
(4) 18π 2
Q12. Let f : R → R be a twice differentiable function such that f (x + y) = f (x)f (y) for all x, y ∈ R. If
′
f (0) = 4a and f satisfies f ′′ ′
(x) − 3af (x) − f (x) = 0 , a > 0, then the area of the region
R = {(x, y) ∣ 0 ≤ y ≤ f (ax), 0 ≤ x ≤ 2} is:
(1) e 2
− 1 (2) e 2
+ 1
(3) e 4
+ 1 (4) e 4
− 1
Q13. The area of the region, inside the circle (x − 2√3) 2
+ y
2
= 12 and outside the parabola y 2
= 2√ 3x is :
(1) 3π + 8 (2) 6π − 16
(3) 3π − 8 (4) 6π − 8
Q14. Let the foci of a hyperbola be (1, 14) and (1, −12). If it passes through the point (1, 6), then the length of its
latus-rectum is :
(1) 24
5
(2) 25
6
(3) 144
5
(4) 288
5
Q15. If ∑ n
r=1
Tr =
(2n−1)(2n+1)(2n+3)(2n+5)
64
, then lim n→∞ ∑
n
r=1
(
1
Tr
) is equal to :
(1) 0 (2) 2
3
(3) 1 (4) 1
3
Q16. A coin is tossed three times. Let X denote the number of times a tail follows a head. If μ and σ denote the 2
mean and variance of X, then the value of 64 (μ + σ 2
) is :
(1) 51 (2) 64
(3) 32 (4) 48
Q17. The number of non-empty equivalence relations on the set {1, 2, 3} is :
(1) 6 (2) 5
(3) 7 (4) 4
Q18. A circle C of radius 2 lies in the second quadrant and touches both the coordinate axes. Let r be the radius of a
circle that has centre at the point (2, 5) and intersects the circle C at exactly two points. If the set of all
possible values of r is the interval (α, β), then 3β − 2α is equal to :
(1) 10 (2) 15
(3) 12 (4) 14
Q19. Let A = {1, 2, 3, … , 10} and B = { m
n
: m, n ∈ A, m < n and gcd(m, n) = 1}. Then n(B) is equal to :
(1) 36 (2) 31
(3) 37 (4) 29
