JEE Main 2023 · 15 Apr 2023 Shift 1

If the domain of the function f(x) = logₑ(4x² + 11x + 6) + sin⁻¹(4x + 3) + cos⁻¹((10x + 6)/3) is (α, β], then 36|α + β| is equal to

1. A45
2. B54
3. C63
4. D72

The number of real roots of the equation x|x| − 5|x + 2| + 6 = 0, is

1. A6
2. B3
3. C5
4. D4

If the set {Re((z − z̄ + z z̄)/(2 − 3z + 5z̄)) : z ∈ C, Re(z) = 3} is equal to the interval (α, β], then 24(β − α) is equal to

1. A27
2. B30
3. C36
4. D42

Let the determinant of a square matrix A of order m be m − n, where m and n satisfy 4m + n = 22 and 17m + 4n = 93. If det(n adj(adj(mA))) = 3ᵃ5ᵇ6ᶜ, then a + b + c is equal to

1. A84
2. B96
3. C101
4. D109

Let the system of linear equations −x + 2y − 9z = 7, −x + 3y + 7z = 9, −2x + y + 5z = 8, −3x + y + 13z = λ has a unique solution x = α, y = β, z = γ. Then the distance of the point (α, β, γ) from the plane 2x − 2y + z = λ is

1. A7
2. B9
3. C11
4. D13

The total number of three-digit numbers, divisible by 3, which can be formed using the digits 1, 3, 5, 8, if repetition of digits is allowed, is

1. A18
2. B20
3. C21
4. D22

Let (a + bx + cx²)¹⁰ = ∑ᵢ₌₀²⁰ pᵢxⁱ, a, b, c ∈ N. If p₁ = 20 and p₂ = 210, then 2(a + b + c) is equal to

1. A6
2. B8
3. C12
4. D15

Let A₁ and A₂ be two arithmetic means and G₁, G₂, G₃ be three geometric means of two distinct positive numbers. Then G₁⁴ + G₂⁴ + G₃⁴ + G₁²G₃² is equal to

1. A2(A₁ + A₂)G₁G₃
2. B(A₁ + A₂)G₁²G₃²
3. C(A₁ + A₂)²G₁G₃
4. D2(A₁ + A₂)G₁²G₃²