JEE Main 2024 Jan 29 (Shift 2) Question Paper
2 1 2 1 2 0
1. Let A = 6 2 11 and P = 5 0 2. The sum of the prime factors of |P −1 AP −
3 3 2 7 1 5
2I| is equal to:
(1) 26
(2) 27
(3) 66
(4) 23
2. Number of ways of arranging 8 identical books into 4 identical shelves where any
number of shelves may remain empty is equal to:
(1) 18
(2) 16
(3) 12
(4) 15
3. Let P (3, 2, 3), Q(4, 6, 2), and R(7, 3, 2) be the vertices of △P QR. Then, the angle
∠QP R is:
1
, π
(1) 6
(2) cos−1 7
18
(3) cos−1 1
18
π
(4) 3
24 194
4. If the mean and variance of five observations are 5 and 25 respectively, and the
mean of the first four observations is 72 , then the variance of the first four observations is
equal to:
4
(1) 5
77
(2) 12
5
(3) 4
105
(4) 4
2
5. The function f (x) = 2x + 3(x) 3 , x ∈ R, has
(1) exactly one point of local minima and no point of local maxima
(2) exactly one point of local maxima and no point of local minima
(3) exactly one point of local maxima and exactly one point of local minima
(4) exactly two points of local maxima and exactly one point of local minima
6. Let r and θ respectively be the modulus and amplitude of the complex number
z = 2 − i 2 tan 5π
8 . Then, (r, θ) is equal to:
(1) 2 sec 3π 3π
8 , 8
(2) 2 sec 3π 5π
8 , 8
(3) 2 sec 5π 3π
8 , 8
(4) 2 sec 11π 11π
8 , 8
2
, 7. The sum of the solutions x ∈ R of the equation
3 cos 2x + cos3 2x
= x3 − x2 + 6
cos6 x − sin6 x
is:
(1) 0
(2) 1
(3) -1
(4) 3
−→ −−→ −→
8. Let OA = ⃗a, OB = 12⃗a + 4⃗b and OC = ⃗b, where O is the origin. If S is the parallelo-
gram with adjacent sides OA and OC , then
area of OABC
area of the quadrilateral OABC is equal to
area of S
is equal to:
(1) 6
(2) 10
(3) 7
(4) 8
9. If loge a, loge b, loge c are in an A.P. and loge a − loge 2b, loge 2b − loge 3c, loge 3c − loge a
are also in an A.P., then a : b : c is equal to:
(1) 9 : 6 : 4
(2) 16 : 4 : 1
(3) 25 : 10 : 4
(4) 6 : 3 : 2
3
, 10. If
sin3 x + cos3 x
Z
cos θ tan x − sin θ cos θ − sin θ cot x
3
dx = A 2
+ B + C,
sin x cos3 x · sin(x − θ) sin θ cos2 θ
where C is the integration constant, then AB is equal to:
(1) 4 csc(2θ)
(2) 4 sec θ
(3) 2 sec θ
(4) 8 csc(2θ)
11. The distance of the point (2, 3) from the line 2x − 3y + 28 = 0, measured parallel to
√
the line 3x − y + 1 = 0, is equal to
√
(1) 4 2
√
(2) 6 3
√
(3) 3 + 4 2
√
(4) 4 + 6 3
y α
12. If sin x = loge |x| + 2 is the solution of the differential equation
y dy y
x cos = y cos +x
x dx x
and y(1) = π3 , then α2 is equal to:
(1) 3
(2) 12
(3) 4
(4) 9
4
2 1 2 1 2 0
1. Let A = 6 2 11 and P = 5 0 2. The sum of the prime factors of |P −1 AP −
3 3 2 7 1 5
2I| is equal to:
(1) 26
(2) 27
(3) 66
(4) 23
2. Number of ways of arranging 8 identical books into 4 identical shelves where any
number of shelves may remain empty is equal to:
(1) 18
(2) 16
(3) 12
(4) 15
3. Let P (3, 2, 3), Q(4, 6, 2), and R(7, 3, 2) be the vertices of △P QR. Then, the angle
∠QP R is:
1
, π
(1) 6
(2) cos−1 7
18
(3) cos−1 1
18
π
(4) 3
24 194
4. If the mean and variance of five observations are 5 and 25 respectively, and the
mean of the first four observations is 72 , then the variance of the first four observations is
equal to:
4
(1) 5
77
(2) 12
5
(3) 4
105
(4) 4
2
5. The function f (x) = 2x + 3(x) 3 , x ∈ R, has
(1) exactly one point of local minima and no point of local maxima
(2) exactly one point of local maxima and no point of local minima
(3) exactly one point of local maxima and exactly one point of local minima
(4) exactly two points of local maxima and exactly one point of local minima
6. Let r and θ respectively be the modulus and amplitude of the complex number
z = 2 − i 2 tan 5π
8 . Then, (r, θ) is equal to:
(1) 2 sec 3π 3π
8 , 8
(2) 2 sec 3π 5π
8 , 8
(3) 2 sec 5π 3π
8 , 8
(4) 2 sec 11π 11π
8 , 8
2
, 7. The sum of the solutions x ∈ R of the equation
3 cos 2x + cos3 2x
= x3 − x2 + 6
cos6 x − sin6 x
is:
(1) 0
(2) 1
(3) -1
(4) 3
−→ −−→ −→
8. Let OA = ⃗a, OB = 12⃗a + 4⃗b and OC = ⃗b, where O is the origin. If S is the parallelo-
gram with adjacent sides OA and OC , then
area of OABC
area of the quadrilateral OABC is equal to
area of S
is equal to:
(1) 6
(2) 10
(3) 7
(4) 8
9. If loge a, loge b, loge c are in an A.P. and loge a − loge 2b, loge 2b − loge 3c, loge 3c − loge a
are also in an A.P., then a : b : c is equal to:
(1) 9 : 6 : 4
(2) 16 : 4 : 1
(3) 25 : 10 : 4
(4) 6 : 3 : 2
3
, 10. If
sin3 x + cos3 x
Z
cos θ tan x − sin θ cos θ − sin θ cot x
3
dx = A 2
+ B + C,
sin x cos3 x · sin(x − θ) sin θ cos2 θ
where C is the integration constant, then AB is equal to:
(1) 4 csc(2θ)
(2) 4 sec θ
(3) 2 sec θ
(4) 8 csc(2θ)
11. The distance of the point (2, 3) from the line 2x − 3y + 28 = 0, measured parallel to
√
the line 3x − y + 1 = 0, is equal to
√
(1) 4 2
√
(2) 6 3
√
(3) 3 + 4 2
√
(4) 4 + 6 3
y α
12. If sin x = loge |x| + 2 is the solution of the differential equation
y dy y
x cos = y cos +x
x dx x
and y(1) = π3 , then α2 is equal to:
(1) 3
(2) 12
(3) 4
(4) 9
4
