JEE Advanced 2020 · Paper 1

Suppose a, b denote the distinct real roots of the quadratic polynomial x² + 20x − 2020 and suppose c, d denote the distinct complex roots of the quadratic polynomial x² − 20x + 2020. Then the value of ac(a − c) + ad(a − d) + bc(b − c) + bd(b − d) is

1. A0
2. B8000
3. C8080
4. D16000

If the function f: ℝ ⟶ ℝ is defined by f(x) = |x|(x − sin x), then which of the following statements is TRUE?

1. Af is one-one, but NOT onto
2. Bf is onto, but NOT one-one
3. Cf is BOTH one-one and onto
4. Df is NEITHER one-one NOR onto

Let the functions f: ℝ ⟶ ℝ and g: ℝ ⟶ ℝ be defined by f(x) = eˣ⁻¹ − e⁻|ˣ⁻¹| and g(x) = 1/2 (eˣ⁻¹ + e¹⁻ˣ). Then the area of the region in the first quadrant bounded by the curves y = f(x), y = g(x) and x = 0 is

1. A(2 − √3) + 1/2 (e − e⁻¹)
2. B(2 + √3) + 1/2 (e − e⁻¹)
3. C(2 − √3) + 1/2 (e + e⁻¹)
4. D(2 + √3) + 1/2 (e + e⁻¹)

Let a, b and λ be positive real numbers. Suppose P is an end point of the latus rectum of the parabola y² = 4λx, and suppose the ellipse x²/a² + y²/b² = 1 passes through the point P. If the tangents to the parabola and the ellipse at the point P are perpendicular to each other, then the eccentricity of the ellipse is

1. A1/√2
2. B1/2
3. C1/3
4. D2/5

Let the function f: ℝ → ℝ be defined by f(x) = x³ − x² + (x − 1) sin x and let g: ℝ → ℝ be an arbitrary function. Let fg: ℝ → ℝ be the product function defined by (fg)(x) = f(x)g(x). Then which of the following statements is/are TRUE?

1. AIf g is continuous at x = 1, then fg is differentiable at x = 1
2. BIf fg is differentiable at x = 1, then g is continuous at x = 1
3. CIf g is differentiable at x = 1, then fg is differentiable at x = 1
4. DIf fg is differentiable at x = 1, then g is differentiable at x = 1

Let M be a 3 × 3 invertible matrix with real entries and let I denote the 3 × 3 identity matrix. If M⁻¹ = adj (adj M), then which of the following statements is/are ALWAYS TRUE?

1. AM = I
2. Bdet M = 1
3. CM² = I
4. D(adj M)² = I

Let S be the set of all complex numbers z satisfying |z² + z + 1| = 1. Then which of the following statements is/are TRUE?

1. A|z + 1/2| ≤ 1/2 for all z ∈ S
2. B|z| ≤ 2 for all z ∈ S
3. C|z + 1/2| ≥ 1/2 for all z ∈ S
4. DThe set S has exactly four elements

Let x, y and z be positive real numbers. Suppose x, y and z are the lengths of the sides of a triangle opposite to its angles X, Y and Z, respectively. If tan(X/2) + tan(Z/2) = 2y/(x + y + z), then which of the following statements is/are TRUE?

1. A2Y = X + Z
2. BY = X + Z
3. Ctan(X/2) = x/(y+z)
4. Dx² + z² − y² = xz