Maths

Let A + 2B = [, [6, -3, 3], [-5, 3, 1]] and 2A - B = [[2, -1, 5], [2, -1, 6],]. If Tr(A) denotes the sum of all diagonal elements of the matrix A, then Tr(A) – Tr(B) has value equal to:

If the equation a|z|² + α'z + αz' + d = 0 represents a circle where a, d are real constants, then which of the following condition is correct?

If the function f(x) = (cos(sin x) - cos x) / x⁴ is continuous at each point in its domain and f(0) = 1/k, then k is .

If [.] represents the greatest integer function, then the value of ∫[from 1 to 5] ([x²] - cos x) dx is.

If the equation of the plane passing through the line of intersection of the planes 2x - 7y + 4z - 3 = 0, 3x – 5y + 4z + 11 = 0 and the point (-2, 1, 3) is ax + by + cz - 7 = 0, then the value of 2a + b + c - 7 is.

The maximum value of z in the following equation z = 6xy + y², where 3x + 4y ≤ 100 and 4x + 3y ≤ 75 for x ≥ 0 and y ≥ 0 is.

If a = αi + βj + 3k, b = -βi - αj - k and c = i - 2j - k such that a ⋅ b = 1 and b ⋅ c = -3, then (1/3)((a * b) ⋅ c) is equal to.

The minimum distance between any two points P₁ and P₂ while considering point P₁ on one circle and point P₂ on the other circle for the given circles' equations x² + y² - 10x - 10y + 41 = 0, x² + y² - 24x - 10y + 160 = 0 is.

If A = [, [0, -1]], then the value of det(A⁴) + det(A¹⁰ - (Adj(2A))¹⁰) is equal to.

Let there be three independent events E₁, E₂ and E₃. The probability that only E₁ occurs is α, only E₂ occurs is β and only E₃ occurs is γ. Let 'p' denote the probability of none of events occurs that satisfies the equations (α - 2β)p = αβ and (β - 3γ)p = 2βγ. All the given probabilities are assumed to lie in the interval (0, 1). Then, (Probability of occurrence of E₁) / (Probability of occurrence of E₃) is equal to.