202260 marks18 questions
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Source: JEE (Advanced) 2022, Paper 2, Maths, published by JEE (Advanced). The official PDF
If y(x) is the solution of the differential equation x dy − (y² − 4y) dx = 0 for x > 0, y(1) = 2, and the slope of the curve y = y(x) is never zero, then the value of 10 y(√2) is _____________.
Let β be a real number. Consider the matrix A = (β 0 1; 2 1 −2; 3 1 −2). If A⁷ − (β − 1)A⁶ − βA⁵ is a singular matrix, then the value of 9β is ___________.
Consider the functions f, g : ℝ → ℝ defined by f(x) = x² + 5/12 and g(x) = {2(1 − 4|x|/3), |x| ≤ 3/4; 0, |x| > 3/4}. If α is the area of the region {(x, y) ∈ ℝ × ℝ : |x| ≤ 3/4, 0 ≤ y ≤ min{f(x), g(x)}}, then the value of 9α is _____________.
Let G be a circle of radius R > 0. Let G₁, G₂, …, Gₙ be n circles of equal radius r > 0. Suppose each of the n circles G₁, G₂, …, Gₙ touches the circle G externally. Also, for i = 1, 2, …, n − 1, the circle Gᵢ touches Gᵢ₊₁ externally, and Gₙ touches G₁ externally. Then, which of the following statements is/are TRUE ?
Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls are distinct. In how many different ways can 10 balls be chosen from these 4 boxes so that from each box at least one red ball and one blue ball are chosen ?