JEE Main 2024 · 31 Jan 2024 Shift 2

If the function f : (−∞, −1] → (a, b] defined by f(x) = eˣ³⁻³ˣ⁺¹ is one-one and onto, then the distance of the point P(2b + 4, a + 2) from the line x + e⁻³y = 4 is :

1. A4√(1 + e⁶)
2. B3√(1 + e⁶)
3. C2√(1 + e⁶)
4. D√(1 + e⁶)

The number of solutions, of the equation eˢⁱⁿ ˣ − 2e⁻ˢⁱⁿ ˣ = 2, is :

1. A0
2. B1
3. C2
4. Dmore than 2

Let z₁ and z₂ be two complex numbers such that z₁ + z₂ = 5 and z₁³ + z₂³ = 20 + 15i. Then, |z₁⁴ + z₂⁴| equals -

1. A30√3
2. B15√15
3. C25√3
4. D75

Let A be a 3 × 3 real matrix such that A(1, 0, 1)ᵀ = 2(1, 0, 1)ᵀ, A(−1, 0, 1)ᵀ = 4(−1, 0, 1)ᵀ, A(0, 1, 0)ᵀ = 2(0, 1, 0)ᵀ. Then, the system (A − 3I)(x, y, z)ᵀ = (1, 2, 3)ᵀ has

1. Ano solution
2. Binfinitely many solutions
3. Cunique solution
4. Dexactly two solutions

The number of ways in which 21 identical apples can be distributed among three children such that each child gets at least 2 apples, is

1. A130
2. B136
3. C142
4. D406

If for some m, n; ⁶Cₘ + 2(⁶Cₘ₊₁) + ⁶Cₘ₊₂ > ⁸C₃ and ⁿ⁻¹P₃ : ⁿP₄ = 1 : 8, then ⁿPₘ₊₁ + ⁿ⁺¹Cₘ is equal to

1. A372
2. B376
3. C380
4. D384

Let 2nd, 8th and 44th terms of a non-constant A. P. be respectively the 1st, 2nd and 3rd terms of a G. P. If the first term of the A. P. is 1, then the sum of its first 20 terms is equal to -

1. A960
2. B970
3. C980
4. D990

Consider the function f : (0, ∞) → ℝ defined by f(x) = e⁻|logₑ x|. If m and n be respectively the number of points at which f is not continuous and f is not differentiable, then m + n is

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