JEE Main 2024 · 8 Apr 2024 Shift 1

Let [t] be the greatest integer less than or equal to t. Let A be the set of all prime factors of 2310 and f : A → Z be the function f(x) = [log₂(x² + [x³/5])]. The number of one-to-one functions from A to the range of f is

1. A20
2. B24
3. C120
4. D25

The sum of all the solutions of the equation (8)²ˣ − 16·(8)ˣ + 48 = 0 is :

1. Alog₈(4)
2. Blog₈(6)
3. C1 + log₆(8)
4. D1 + log₈(6)

Let A = [[2, a, 0], [1, 3, 1], [0, 5, b]]. If A³ = 4A² − A − 21I, where I is the identity matrix of order 3×3, then 2a + 3b is equal to

1. A−9
2. B−13
3. C−12
4. D−10

The equations of two sides AB and AC of a triangle ABC are 4x + y = 14 and 3x − 2y = 5, respectively. The point (2, −4/3) divides the third side BC internally in the ratio 2:1. The equation of the side BC is

1. Ax + 3y + 2 = 0
2. Bx − 3y − 6 = 0
3. Cx + 6y + 6 = 0
4. Dx − 6y − 10 = 0

If the set R = {(a, b) : a + 5b = 42, a, b ∈ N} has m elements and ∑ₙ₌₁ᵐ (1 − iⁿ!) = x + iy, where i = √−1, then the value of m + x + y is

1. A12
2. B8
3. C5
4. D4

For the function f(x) = (cos x) − x + 1, x ∈ R, between the following two statements (S1) f(x) = 0 for only one value of x in [0, π]. (S2) f(x) is decreasing in [0, π/2] and increasing in [π/2, π].

1. ABoth (S1) and (S2) are correct.
2. BOnly (S1) is correct.
3. COnly (S2) is correct.
4. DBoth (S1) and (S2) are incorrect.

The number of critical points of the function f(x) = (x − 2)²⁄³ (2x + 1) is

1. A0
2. B1
3. C2
4. D3

Let f(x) = 4 cos³ x + 3√3 cos² x − 10. The number of points of local maxima of f in interval (0, 2π) is

1. A1
2. B2
3. C3
4. D4