JEE Main 2024 · 1 Feb 2024 Shift 2

If the domain of the function f(x) = √(x² − 25)/(4 − x²) + log₁₀ (x² + 2x − 15) is (−∞, α) ∪ [β, ∞), then α² + β³ is equal to :

1. A125
2. B140
3. C150
4. D175

Consider the relations R₁ and R₂ defined as aR₁b ⇔ a² + b² = 1 for all a, b ∈ R and (a, b)R₂(c, d) ⇔ a + d = b + c for all (a, b), (c, d) ∈ N × N. Then

1. AOnly R₁ is an equivalence relation
2. BOnly R₂ is an equivalence relation
3. CR₁ and R₂ both are equivalence relations
4. DNeither R₁ nor R₂ is an equivalence relation

Let α and β be the roots of the equation px² + qx − r = 0, where p ≠ 0. If p, q and r are the consecutive terms of a non-constant G.P. and 1/α + 1/β = 3/4, then the value of (α − β)² is :

1. A9
2. B80/9
3. C20/3
4. D8

Let the system of equations x + 2y + 3z = 5, 2x + 3y + z = 9, 4x + 3y + λz = μ have infinite number of solutions. Then λ + 2μ is equal to :

1. A17
2. B22
3. C15
4. D28

Let Sₙ denote the sum of the first n terms of an arithmetic progression. If S₁₀ = 390 and the ratio of the tenth and the fifth terms is 15 : 7, then S₁₅ − S₅ is equal to :

1. A690
2. B800
3. C890
4. D790

Let f(x) = |2x² + 5|x| − 3|, x ∈ R. If m and n denote the number of points where f is not continuous and not differentiable respectively, then m + n is equal to :

1. A0
2. B2
3. C3
4. D5

Let f(x) = {x − 1, x is even, 2x, x is odd, x ∈ N. If for some a ∈ N, f(f(f(a))) = 21, then limₓ→ₐ⁻ (|x|³/a − [x/a]), where [t] denotes the greatest integer less than or equal to t, is equal to :

1. A121
2. B144
3. C169
4. D225