202066 marks18 questions
14 of its 18 questions are here. The rest have a figure or are still being checked, so for now they're only in the official PDF.
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Source: JEE (Advanced) 2020, Paper 2, Maths, published by JEE (Advanced). The official PDF
For a complex number z, let Re(z) denote the real part of z. Let S be the set of all complex numbers z satisfying z⁴ − |z|⁴ = 4iz², where i = √−1. Then the minimum possible value of |z₁ − z₂|², where z₁, z₂ ∈ S with Re(z₁) > 0 and Re(z₂) < 0, is _____
The probability that a missile hits a target successfully is 0.75. In order to destroy the target completely, at least three successful hits are required. Then the minimum number of missiles that have to be fired so that the probability of completely destroying the target is NOT less than 0.95, is _____
Let O be the centre of the circle x² + y² = r², where r > √5/2. Suppose PQ is a chord of this circle and the equation of the line passing through P and Q is 2x + 4y = 5. If the centre of the circumcircle of the triangle OPQ lies on the line x + 2y = 4, then the value of r is _____
The trace of a square matrix is defined to be the sum of its diagonal entries. If A is a 2 × 2 matrix such that the trace of A is 3 and the trace of A³ is −18, then the value of the determinant of A is _____
Let the functions f: (−1, 1) → ℝ and g: (−1, 1) → (−1, 1) be defined by f(x) = |2x − 1| + |2x + 1| and g(x) = x − [x], where [x] denotes the greatest integer less than or equal to x. Let f ∘ g: (−1, 1) → ℝ be the composite function defined by (f ∘ g)(x) = f(g(x)). Suppose c is the number of points in the interval (−1, 1) at which f ∘ g is NOT continuous, and suppose d is the number of points in the interval (−1, 1) at which f ∘ g is NOT differentiable. Then the value of c + d is _____
Let b be a nonzero real number. Suppose f: ℝ → ℝ is a differentiable function such that f(0) = 1. If the derivative f′ of f satisfies the equation f′(x) = f(x)/(b² + x²) for all x ∈ ℝ, then which of the following statements is/are TRUE?
Let a and b be positive real numbers such that a > 1 and b < a. Let P be a point in the first quadrant that lies on the hyperbola x²/a² − y²/b² = 1. Suppose the tangent to the hyperbola at P passes through the point (1, 0), and suppose the normal to the hyperbola at P cuts off equal intercepts on the coordinate axes. Let ∆ denote the area of the triangle formed by the tangent at P, the normal at P and the x-axis. If e denotes the eccentricity of the hyperbola, then which of the following statements is/are TRUE?
Let f: ℝ → ℝ and g: ℝ → ℝ be functions satisfying f(x + y) = f(x) + f(y) + f(x)f(y) and f(x) = xg(x) for all x, y ∈ ℝ. If limₓ→₀ g(x) = 1, then which of the following statements is/are TRUE?