JEE Advanced 2016 · Paper 1

Let −π/6 < θ < −π/12. Suppose α₁ and β₁ are the roots of the equation x² − 2x sec θ + 1 = 0 and α₂ and β₂ are the roots of the equation x² + 2x tan θ − 1 = 0. If α₁ > β₁ and α₂ > β₂, then α₁ + β₂ equals

1. A2(sec θ − tan θ)
2. B2 sec θ
3. C−2 tan θ
4. D0

A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the selection of a captain (from among these 4 members) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is

1. A380
2. B320
3. C260
4. D95

Let S = {x ∈ (−π, π) : x ≠ 0, ±π/2}. The sum of all distinct solutions of the equation √3 sec x + cosec x + 2(tan x − cot x) = 0 in the set S is equal to

1. A−7π/9
2. B−2π/9
3. C0
4. D5π/9

A computer producing factory has only two plants T₁ and T₂. Plant T₁ produces 20% and plant T₂ produces 80% of the total computers produced. 7% of computers produced in the factory turn out to be defective. It is known that P (computer turns out to be defective given that it is produced in plant T₁) = 10P (computer turns out to be defective given that it is produced in plant T₂), where P(E) denotes the probability of an event E. A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is produced in plant T₂ is

1. A36/73
2. B47/79
3. C78/93
4. D75/83

The least value of α ∈ ℝ for which 4αx² + 1/x ≥ 1, for all x > 0, is

1. A1/64
2. B1/32
3. C1/27
4. D1/25

Consider a pyramid OPQRS located in the first octant (x ≥ 0, y ≥ 0, z ≥ 0) with O as origin, and OP and OR along the x-axis and the y-axis, respectively. The base OPQR of the pyramid is a square with OP = 3. The point S is directly above the mid-point T of diagonal OQ such that TS = 3. Then

1. Athe acute angle between OQ and OS is π/3
2. Bthe equation of the plane containing the triangle OQS is x − y = 0
3. Cthe length of the perpendicular from P to the plane containing the triangle OQS is 3/√2
4. Dthe perpendicular distance from O to the straight line containing RS is √(15/2)

Let f : (0, ∞) → ℝ be a differentiable function such that f′(x) = 2 − f(x)/x for all x ∈ (0, ∞) and f(1) ≠ 1. Then

1. Alimₓ→₀₊ f′(1/x) = 1
2. Blimₓ→₀₊ x f(1/x) = 2
3. Climₓ→₀₊ x² f′(x) = 0
4. D|f(x)| ≤ 2 for all x ∈ (0, 2)

Let P = [[3, −1, −2], [2, 0, α], [3, −5, 0]], where α ∈ ℝ. Suppose Q = [qᵢⱼ] is a matrix such that PQ = kI, where k ∈ ℝ, k ≠ 0 and I is the identity matrix of order 3. If q₂₃ = −k/8 and det(Q) = k²/2, then

1. Aα = 0, k = 8
2. B4α − k + 8 = 0
3. Cdet(P adj(Q)) = 2⁹
4. Ddet(Q adj(P)) = 2¹³