JEE Main 2025 · 7 Apr 2025 Shift 1

If for θ ∈ [−π/3, 0], the points (x, y) = (3 tan(θ + π/3), 2 tan(θ + π/6)) lie on xy + αx + βy + γ = 0, then α² + β² + γ² is equal to

1. A75
2. B80
3. C96
4. D72

Among the statements (S1) : The set {z ∈ C − {−i} : |z| = 1 and (z − i)/(z + i) is purely real} contains exactly two elements, and (S2) : The set {z ∈ C − {−1} : |z| = 1 and (z − 1)/(z + 1) is purely imaginary} contains infinitely many elements.

1. Aboth are correct
2. Bboth are incorrect
3. Conly (S1) is correct
4. Donly (S2) is correct

Let the set of all values of p ∈ R, for which both the roots of the equation x² − (p + 2)x + (2p + 9) = 0 are negative real numbers, be the interval (α, β]. Then β − 2α is equal to

1. A20
2. B9
3. C5
4. D0

Let the system of equations: 2x + 3y + 5z = 9, 7x + 3y − 2z = 8, 12x + 3y − (4 + λ)z = 16 − μ, have infinitely many solutions. Then the radius of the circle centred at (λ, μ) and touching the line 4x = 3y is

1. A7
2. B21/5
3. C7/5
4. D17/5

Let A be a 3 × 3 matrix such that |adj (adj (adj A))| = 81. If S = {n ∈ Z : (|adj (adj A)|)^((n−1)²/2) = |A|^(3n²−5n−4)}, then ∑ₙ∈S |A^(n²+n)| is equal to

1. A732
2. B750
3. C820
4. D866

Let x₁, x₂, x₃, x₄ be in a geometric progression. If 2, 7, 9, 5 are subtracted respectively from x₁, x₂, x₃, x₄, then the resulting numbers are in an arithmetic progression. Then the value of 1/24 (x₁ x₂ x₃ x₄) is:

1. A216
2. B72
3. C36
4. D18

The remainder when ((64)⁶⁴)⁶⁴ is divided by 7 is equal to

1. A1
2. B3
3. C4
4. D6

From a group of 7 batsmen and 6 bowlers, 10 players are to be chosen for a team, which should include atleast 4 batsmen and atleast 4 bowlers. One batsmen and one bowler who are captain and vice-captain respectively of the team should be included. Then the total number of ways such a selection can be made, is

1. A135
2. B145
3. C155
4. D165