Let Z1 and Z2 be roots of the equation z + pz + q = 0, where the coefficients p and q may be complex numbers. Let A and B represent z₁ and z2 in the complex plane, respectvely. If ZAOB = 0#0 and OA = OB, where O is the origin, prove that p² = 4q cos² (8/2). 2

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A: Given that A is 3×3 matrix and det(A)= 4

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A: We will solve both these questions one by one .

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A: Total borrowed Php 10,000 Total payout Php 15,000 Total time 20 month

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Q: 54. Properties of integrals Suppose f(x) > 0 on [0, 2], f(x) ≤ 0 on [2, 5], ₁² f(x) dx c = 6, and ſ₂…

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Q: [. I. 7. In x 12 x² dx x² sin(4x) dx I cos² cos¹ x dx 4x+8 (x + 2)² +5 dx

A: We will use properties of integrals. In 1st problem we use substitution method .

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A: The given problem is to solve the given boundary value problem with given boundary conditions.

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A: A is an invertible nxn matrix. Since A is invertible, A is full rank matrix.

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Q: Find the divergence of the vector field F. F(x, y, z)= In (3x²+ 6y2)i + 18xyj +In(6y²+72²)k F(x, y,…

A: Given function F(x,y,z) = ln(3x2 +6y2)i + 18xy j + ln(6y2+7z2)k    We need to find the divergence of…

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A: Use the vector knowledge to solve. A vector is resultant of its y component and y component…

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A: Given, Cost function C(q) = 80+15q Price function p = 63-2q

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A: The objective was to prove for all z∈ℂ, |z|=|z¯|.

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Q: IV. Determine the non-trivial solutions of the system of homogeneous equation, if it exists: x₁ +…

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Q: Use the definition of a Taylor series to find the first four nonzero terms of the series for f(x)…

A: We will use taylor series expansion  f(x) = fa+f'a1!x-a+f''a2!x-a2+f'''a3!x-a3+fiva4!x-a4+........

Q: Use the definition of the Laplace Transform and the properties of the Dirac Delta generalized…

A: As per guidelines ,I will give solution for first three problems only.

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Q: For the system of differential equations do the following. n' = n(1-2m), m' = m(2 - 2n - m), n, m≥ 0

A: Given:  n'=n1-2m, m'=2-2n-m , n, m≥0 To find: All equilibria n^ , m^.

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A: The given problem is to solve the given matrix differential equation initial value problem with…

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A: We have to evaluate the line integral using the green’s theorem ∮(6x+3y3)dx+(3xy2+5y)dy

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Q: 9. For the linear operator LA R³ R³ defined by L₁(x) = Ax, for x column vectors in : R³, and 0 1 0 0…

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A: To find formula for m,  we have to rearrange the terms.

Q: 4) F Consider the following weighted graph: 1 11 A E 15 8 8 9 w/ 3 7 10 B D 8 12 4 Use the…

A: Given : Weighted-Graph   To find : Minimum cost Hamiltonian count for the given weighted-graph.

Q: Compute the determinant 2 1 1 1 1 1 3 1 1 1 1 1 1 1 1 4 1 1 151 1 1 1 6

A: To compute the determinant of the following matrix,