—•No notes, personal aids or calculators are permitted.•Answer all questions in the space provided. If you require more space to write your answer, you may continueon the back of the page. There is a blank page at the end of the exam for rough work.•Explain your work!Little or no points will be given for a correct answer with no explanation of how you got it.Good luck!Problem 1. (10 points)LetAbe a5×5matrix with eigenvalues±3i,1,1,1.(a) Suppose that the eigenvalueλ= 1has defect1. Does the equationx′=Axhave (nonzero) solutions of one ofthe following forms?(v1t+v2)etparenleftBigv1t22+v2t+v3parenrightBigetparenleftBigv1t36+v2t22+v3t+v4parenrightBiget(v1t+v2)sin(3t)v1etcos(3t)Circle those that are solutions (for appropriate choices of the coefficientsv1,v2,v3,v4).(b) Now, consider the differential equationx′=Ax+(3t2,0,cos(t),0,-1)T. Write down a particular solutionxpwith undetermined coefficients.Solution.