Determine each integrating factor to make the Differential Equation exact then solve each equation. (2xy³ + 1)dx + (3x²y²-y-¹)dy = 0

(1 5 -4 Given the matrix A = 1-3 4 2 2 4 (a) Find the determinant of A. [4 marks] (b) Calculate the inverse of A. [8 marks] (c) Hence find the general solution of the linear simultaneous equations x+5y -4z -w = 3 x-3y +4z +w = 3 2x + 2y +4z = 10 [6 marks] (d) Find the eigenvalues and eigenvectors of A. [12 marks]

Apply de'Moivre's theorem or otherwise to solve for Zo and C from these expressions given below: ZO-Z/Y and C=Z*Y Where: .Z is a complex number. • Y is also a complex number. • Re (Z0) >0 and Re (C) >0 Find Z0 and C when: Z=1+5j,Y=1-3j

Problem 6: Inputs: ● ● ● a = 600 mm b = 400 mm c = 200 mm F = 100 N The line of action of the force vector passes through points B and C. B X Z (a) Use the cross product to compute the moment about point A. For this part, you must use B in your vector formulation. F That is, compute MA = B x F. Express your answer as a vector and convert "mm" to "m". IB a A IC с (b) Now use a different position vector, r, in the cross product. This time use rc to to compute the moment about point A. For this part, you must use rc in your vector formulation. Compute MA = rcx F. Express your answer as a vector and convert "mm" to "m". c) Do your answers agree? Yes or no. F y