JEE Advanced 2022 · Paper 1

Let α be a positive real number. Let f: ℝ → ℝ and g: (α, ∞) → ℝ be the functions defined by f(x) = sin(πx/12) and g(x) = 2 logₑ(√x − √α)/logₑ(e√x − e√α). Then the value of limₓ→α⁺ f(g(x)) is __________.

In a study about a pandemic, data of 900 persons was collected. It was found that 190 persons had symptom of fever, 220 persons had symptom of cough, 220 persons had symptom of breathing problem, 330 persons had symptom of fever or cough or both, 350 persons had symptom of cough or breathing problem or both, 340 persons had symptom of fever or breathing problem or both, 30 persons had all three symptoms (fever, cough and breathing problem). If a person is chosen randomly from these 900 persons, then the probability that the person has at most one symptom is _____________.

Let z be a complex number with non-zero imaginary part. If (2 + 3z + 4z²)/(2 − 3z + 4z²) is a real number, then the value of |z|² is _____________.

Let z̅ denote the complex conjugate of a complex number z and let i = √−1. In the set of complex numbers, the number of distinct roots of the equation z̅ − z² = i(z̅ + z²) is _____________.

Let l₁, l₂, …, l₁₀₀ be consecutive terms of an arithmetic progression with common difference d₁, and let w₁, w₂, …, w₁₀₀ be consecutive terms of another arithmetic progression with common difference d₂, where d₁d₂ = 10. For each i = 1, 2, …, 100, let Rᵢ be a rectangle with length lᵢ, width wᵢ and area Aᵢ. If A₅₁ − A₅₀ = 1000, then the value of A₁₀₀ − A₉₀ is ____________.

The number of 4-digit integers in the closed interval [2022, 4482] formed by using the digits 0, 2, 3, 4, 6, 7 is ____________.

Let ABC be the triangle with AB = 1, AC = 3 and ∠BAC = π/2. If a circle of radius r > 0 touches the sides AB, AC and also touches internally the circumcircle of the triangle ABC, then the value of r is _____________.