JEE Advanced 2019 · Paper 1

Let S be the set of all complex numbers z satisfying |z − 2 + i| ≥ √5. If the complex number z₀ is such that 1/|z₀ − 1| is the maximum of the set {(1/|z − 1|) : z ∈ S}, then the principal argument of (4 − z₀ − z̄₀)/(z₀ − z̄₀ + 2i) is

1. A−π/2
2. Bπ/4
3. Cπ/2
4. D3π/4

Let M = [sin⁴θ −1 − sin²θ; 1 + cos²θ cos⁴θ] = αI + βM⁻¹, where α = α(θ) and β = β(θ) are real numbers, and I is the 2 × 2 identity matrix. If α* is the minimum of the set {α(θ): θ ∈ [0, 2π]} and β* is the minimum of the set {β(θ): θ ∈ [0, 2π]}, then the value of α* + β* is

1. A−37/16
2. B−31/16
3. C−29/16
4. D−17/16

A line y = mx + 1 intersects the circle (x − 3)² + (y + 2)² = 25 at the points P and Q. If the midpoint of the line segment PQ has x-coordinate −3/5, then which one of the following options is correct?

1. A−3 ≤ m < −1
2. B2 ≤ m < 4
3. C4 ≤ m < 6
4. D6 ≤ m < 8

The area of the region {(x, y) : xy ≤ 8, 1 ≤ y ≤ x²} is

1. A16 logₑ 2 − 14/3
2. B8 logₑ 2 − 14/3
3. C16 logₑ 2 − 6
4. D8 logₑ 2 − 7/3

Let α and β be the roots of x² − x − 1 = 0, with α > β. For all positive integers n, define aₙ = (αⁿ − βⁿ)/(α − β), n ≥ 1, b₁ = 1 and bₙ = aₙ₋₁ + aₙ₊₁, n ≥ 2. Then which of the following options is/are correct?

1. Aa₁ + a₂ + a₃ + ⋯ + aₙ = aₙ₊₂ − 1 for all n ≥ 1
2. B∑ₙ₌₁^∞ aₙ/10ⁿ = 10/89
3. Cbₙ = αⁿ + βⁿ for all n ≥ 1
4. D∑ₙ₌₁^∞ bₙ/10ⁿ = 8/89

Let M = [0 1 a; 1 2 3; 3 b 1] and adj M = [−1 1 −1; 8 −6 2; −5 3 −1], where a and b are real numbers. Which of the following options is/are correct?

1. Aa + b = 3
2. B(adj M)⁻¹ + adj M⁻¹ = −M
3. Cdet(adj M²) = 81
4. DIf M[α β γ]ᵀ = [1 2 3]ᵀ, then α − β + γ = 3

There are three bags B₁, B₂ and B₃. The bag B₁ contains 5 red and 5 green balls, B₂ contains 3 red and 5 green balls, and B₃ contains 5 red and 3 green balls. Bags B₁, B₂ and B₃ have probabilities 3/10, 3/10 and 4/10 respectively of being chosen. A bag is selected at random and a ball is chosen at random from the bag. Then which of the following options is/are correct?

1. AProbability that the chosen ball is green, given that the selected bag is B₃, equals 3/8
2. BProbability that the chosen ball is green equals 39/80
3. CProbability that the selected bag is B₃, given that the chosen ball is green, equals 5/13
4. DProbability that the selected bag is B₃ and the chosen ball is green equals 3/10

In a non-right-angled triangle ΔPQR, let p, q, r denote the lengths of the sides opposite to the angles at P, Q, R respectively. The median from R meets the side PQ at S, the perpendicular from P meets the side QR at E, and RS and PE intersect at O. If p = √3, q = 1, and the radius of the circumcircle of the ΔPQR equals 1, then which of the following options is/are correct?

1. ALength of RS = √7/2
2. BArea of ΔSOE = √3/12
3. CLength of OE = 1/6
4. DRadius of incircle of ΔPQR = (√3/2)(2 − √3)