JEE Advanced 2021 · Paper 1

Consider a triangle ∆ whose two sides lie on the x-axis and the line x + y + 1 = 0. If the orthocenter of ∆ is (1, 1), then the equation of the circle passing through the vertices of the triangle ∆ is

1. Ax² + y² − 3x + y = 0
2. Bx² + y² + x + 3y = 0
3. Cx² + y² + 2y − 1 = 0
4. Dx² + y² + x + y = 0

Consider three sets E₁ = {1, 2, 3}, F₁ = {1, 3, 4} and G₁ = {2, 3, 4, 5}. Two elements are chosen at random, without replacement, from the set E₁, and let S₁ denote the set of these chosen elements. Let E₂ = E₁ − S₁ and F₂ = F₁ ∪ S₁. Now two elements are chosen at random, without replacement, from the set F₂ and let S₂ denote the set of these chosen elements. Let G₂ = G₁ ∪ S₂. Finally, two elements are chosen at random, without replacement, from the set G₂ and let S₃ denote the set of these chosen elements. Let E₃ = E₂ ∪ S₃. Given that E₁ = E₃, let p be the conditional probability of the event S₁ = {1, 2}. Then the value of p is

1. A1/5
2. B3/5
3. C1/2
4. D2/5

Let θ₁, θ₂, …, θ₁₀ be positive valued angles (in radian) such that θ₁ + θ₂ + ⋯ + θ₁₀ = 2π. Define the complex numbers z₁ = eⁱᶿ¹, zₖ = zₖ₋₁ eⁱᶿᵏ for k = 2, 3, …, 10, where i = √−1. Consider the statements P and Q given below: P : |z₂ − z₁| + |z₃ − z₂| + ⋯ + |z₁₀ − z₉| + |z₁ − z₁₀| ≤ 2π; Q : |z₂² − z₁²| + |z₃² − z₂²| + ⋯ + |z₁₀² − z₉²| + |z₁² − z₁₀²| ≤ 4π. Then,

1. AP is TRUE and Q is FALSE
2. BQ is TRUE and P is FALSE
3. Cboth P and Q are TRUE
4. Dboth P and Q are FALSE

For any 3 × 3 matrix M, let |M| denote the determinant of M. Let E = [1 2 3; 2 3 4; 8 13 18], P = [1 0 0; 0 0 1; 0 1 0] and F = [1 3 2; 8 18 13; 2 4 3]. If Q is a nonsingular matrix of order 3 × 3, then which of the following statements is (are) TRUE ?

1. AF = PEP and P² = [1 0 0; 0 1 0; 0 0 1]
2. B|EQ + PFQ⁻¹| = |EQ| + |PFQ⁻¹|
3. C|(EF)³| > |EF|²
4. DSum of the diagonal entries of P⁻¹EP + F is equal to the sum of diagonal entries of E + P⁻¹FP

Let f: ℝ → ℝ be defined by f(x) = (x² − 3x − 6)/(x² + 2x + 4). Then which of the following statements is (are) TRUE ?

1. Af is decreasing in the interval (−2, −1)
2. Bf is increasing in the interval (1, 2)
3. Cf is onto
4. DRange of f is [−3/2, 2]

For any 3 × 3 matrix M, let |M| denote the determinant of M. Let I be the 3 × 3 identity matrix. Let E and F be two 3 × 3 matrices such that (I − EF) is invertible. If G = (I − EF)⁻¹, then which of the following statements is (are) TRUE ?

1. A|FE| = |I − FE||FGE|
2. B(I − FE)(I + FGE) = I
3. CEFG = GEF
4. D(I − FE)(I − FGE) = I

For any complex number w = c + id, let arg(w) ∈ (−π, π], where i = √−1. Let α and β be real numbers such that for all complex numbers z = x + iy satisfying arg((z + α)/(z + β)) = π/4, the ordered pair (x, y) lies on the circle x² + y² + 5x − 3y + 4 = 0. Then which of the following statements is (are) TRUE ?

1. Aα = −1
2. Bαβ = 4
3. Cαβ = −4
4. Dβ = 4