JEE Main 2024 · 27 Jan 2024 Shift 1

Let S = {1, 2, 3, . . . , 10}. Suppose M is the set of all the subsets of S, then the relation R = {(A, B) : A ∩ B ≠ ϕ; A, B ∈ M} is :

1. Asymmetric and transitive only
2. Bsymmetric and reflexive only
3. Csymmetric only
4. Dreflexive only

The function f : N - {1} → N; defined by f(n) = the highest prime factor of n, is :

1. Aone-one only
2. Bonto only
3. Cboth one-one and onto
4. Dneither one-one nor onto

If S = {z ∈ C : |z − i| = |z + i| = |z − 1|}, then, n(S) is :

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

Consider the matrix f(x) = [cos x −sin x 0; sin x cos x 0; 0 0 1]. Given below are two statements : Statement I : f(−x) is the inverse of the matrix f(x). Statement II : f(x) f(y) = f(x + y). In the light of the above statements, choose the correct answer from the options given below

1. ABoth Statement I and Statement II are true
2. BBoth Statement I and Statement II are false
3. CStatement I is true but Statement II is false
4. DStatement I is false but Statement II is true

ⁿ⁻¹Cᵣ = (k² − 8) ⁿCᵣ₊₁ if and only if :

1. A2√2 < k ≤ 3
2. B2√3 < k ≤ 3√2
3. C2√2 < k < 2√3
4. D2√3 < k < 3√3

If A denotes the sum of all the coefficients in the expansion of (1 − 3x + 10x²)ⁿ and B denotes the sum of all the coefficients in the expansion of (1 + x²)ⁿ, then :

1. AA = 3B
2. BA = B³
3. CB = A³
4. D3A = B

The number of common terms in the progressions 4, 9, 14, 19, . . . . . . , up to 25ᵗʰ term and 3, 6, 9, 12, . . . . . . , up to 37ᵗʰ term is :

1. A5
2. B7
3. C8
4. D9

Consider the function f(x) = { a(7x − 12 − x²)/(b|x² − 7x + 12|), x < 3; sin(x − 3)/2^(x − [x]), x > 3; b, x = 3 }, where [x] denotes the greatest integer less than or equal to x. If S denotes the set of all ordered pairs (a, b) such that f(x) is continuous at x = 3, then the number of elements in S is :

1. A4
2. BInfinitely many
3. C1
4. D2