JEE Main 2024 · 30 Jan 2024 Shift 2

If the domain of the function f(x) = logₑ((2x + 3)/(4x² + x − 3)) + cos⁻¹((2x − 1)/(x + 2)) is (α, β], then the value of 5β − 4α is equal to

1. A9
2. B10
3. C11
4. D12

If z is a complex number, then the number of common roots of the equations z¹⁹⁸⁵ + z¹⁰⁰ + 1 = 0 and z³ + 2z² + 2z + 1 = 0, is equal to

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

Let R = ⎛x 0 0; 0 y 0; 0 0 z⎞ be a non-zero 3×3 matrix, where x sin θ = y sin(θ + 2π/3) = z sin(θ + 4π/3) ≠ 0, θ ∈ (0, 2π). For a square matrix M, let trace (M) denote the sum of all the diagonal entries of M. Then, among the statements: (I) Trace (R) = 0 (II) If trace (adj(adj(R)) = 0, then R has exactly one non-zero entry.

1. AOnly (I) is true
2. BOnly (II) is true
3. CBoth (I) and (II) are true
4. DNeither (I) nor (II) is true

Consider the system of linear equations x + y + z = 5, x + 2y + λ²z = 9, x + 3y + λz = μ, where λ, μ ∈ R. Then, which of the following statement is NOT correct?

1. ASystem is consistent if λ ≠ 1 and μ = 13
2. BSystem is inconsistent if λ = 1 and μ ≠ 13
3. CSystem has unique solution if λ ≠ 1 and μ ≠ 13
4. DSystem has infinite number of solutions if λ = 1 and μ = 13

Let f : R − {0} → R be a function satisfying f(x/y) = f(x)/f(y) for all x, y, f(y) ≠ 0. If f′(1) = 2024, then

1. Axf′(x) − 2024 f(x) = 0
2. Bxf′(x) + 2024 f(x) = 0
3. Cxf′(x) − 2023 f(x) = 0
4. Dxf′(x) + f(x) = 2024

Let a and b be real constants such that the function f defined by f(x) = {x² + 3x + a, x ≤ 1; bx + 2, x > 1} be differentiable on R. Then, the value of ∫₋₂² f(x) dx equals

1. A15/6
2. B19/6
3. C17
4. D21

Let f(x) = (x + 3)² (x − 2)³, x ∈ [−4, 4]. If M and m are the maximum and minimum values of f, respectively in [−4, 4], then the value of M − m is

1. A600
2. B392
3. C608
4. D108