JEE Main 2023 · 6 Apr 2023 Shift 2

1. Aonly (S1) is true
2. Bonly (S2) is true
3. Cboth (S1) and (S2) are true
4. Dneither (S1) nor (S2) is true

In a group of 100 persons 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. If the number of persons, who speak only English is α and the number of persons who speak only Hindi is β, then the eccentricity of the ellipse 25(β²x² + α²y²) = α²β² is

1. A√117/12
2. B√119/12
3. C√129/12
4. D3√15/12

Let P be a square matrix such that P² = I − P. For α, β, γ, δ ∈ N, if Pᵅ + Pᵝ = γI − 29P and Pᵅ − Pᵝ = δI − 13P, then α + β + γ − δ is equal to

1. A40
2. B24
3. C22
4. D18

For the system of equations x + y + z = 6, x + 2y + αz = 10, x + 3y + 5z = β, which one of the following is NOT true?

1. ASystem has a unique solution for α = −3, β = 14.
2. BSystem has no solution for α = 3, β = 24.
3. CSystem has infinitely many solutions for α = 3, β = 14.
4. DSystem has a unique solution for α = 3, β ≠ 14.

Among the statements: (S1) : 2023²⁰²² − 1999²⁰²² is divisible by 8 (S2) : 13(13)ⁿ − 11n − 13 is divisible by 144 for infinitely many n ∈ N

1. Aonly (S1) is correct
2. Bonly (S2) is correct
3. Cboth (S1) and (S2) are incorrect
4. Dboth (S1) and (S2) are correct

If the coefficients of x⁷ in (ax² + 1/(2bx))¹¹ and x⁻⁷ in (ax − 1/(3bx²))¹¹ are equal, then

1. A32ab = 729
2. B64ab = 243
3. C243ab = 64
4. D729ab = 32

All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is

1. A576
2. B578
3. C580
4. D582