JEE Main 2024 · 1 Feb 2024 Shift 1

Let S = {z ∈ C : |z − 1| = 1 and (√2 − 1)(z + z̄) − i(z − z̄) = 2√2}. Let z₁, z₂ ∈ S be such that |z₁| = max_{z ∈ S} |z| and |z₂| = min_{z ∈ S} |z|. Then |√2 z₁ − z₂|² equals :

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

Let S = {x ∈ R : (√3 + √2)ˣ + (√3 − √2)ˣ = 10}. Then the number of elements in S is :

1. A0
2. B1
3. C2
4. D4

If A = [√2 1; −1 √2], B = [1 0; 1 1], C = ABAᵀ and X = AᵀC²A, then det X is equal to :

1. A27
2. B729
3. C891
4. D243

If n is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then n is equal to :

1. A51
2. B47
3. C53
4. D43

If the system of equations 2x + 3y − z = 5, x + αy + 3z = −4, 3x − y + βz = 7 has infinitely many solutions, then 13 αβ is equal to __________.

1. A1210
2. B1120
3. C1220
4. D1110

Let 3, a, b, c be in A.P. and 3, a − 1, b + 1, c + 9 be in G.P. Then, the arithmetic mean of a, b and c is :

1. A11
2. B−1
3. C13
4. D−4

If 5f(x) + 4f(1/x) = x² − 2, ∀ x ≠ 0 and y = 9x²f(x), then y is strictly increasing in :

1. A(−1/√5, 0) ∪ (0, 1/√5)
2. B(−∞, 1/√5) ∪ (0, 1/√5)
3. C(−1/√5, 0) ∪ (1/√5, ∞)
4. D(0, 1/√5) ∪ (1/√5, ∞)

Let f : R → R be defined as: f(x) = (a − b cos 2x)/x², x < 0; x² + cx + 2, 0 ≤ x ≤ 1; 2x + 1, x > 1. If f is continuous everywhere in R and m is the number of points where f is NOT differential then m + a + b + c equals :

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