1. (a) Find the six complex solutions of the equation -240 in modulus-principal

Question

1. (a) Find the six complex solutions of the equation -240 in modulus-principal argument form and plot them on an Argand diagram. (b) Show that (1 + cos2+ isin 2)os* 1 for n a positive integer (c) Let n be a positive integer and : a complex number with 1-1 and za, -1. Show that is a real number. 2. Use Gaussian elimination to salve 3. Let 1 0 0 0 -2 9 (a) Use the Gauss Jordan method to find S- (b) If A-SDS- show that A-72-17-8 -72 18 8 (c) Use the definition, (1). of A to show that A"-A when n is an odd, positive number. [ (d) Find A" when n is an even, positive number.

Explanation / Answer

(1) This question is simple application of Rouche's Theorem which says that

" Let f(z) and g(z) are two analytic function inside and on a simple closed curve C and let

|g(z)| <|f(z)| on C . Then f+g and f have the same number of zeros inside C."

Now Consider Curve C1; |z|=2 in our given case and f(z)=z6 and g(z)=-2z3 +4 . Then we will apply

Rouche's theorem as |g(z)|=|-2z3+4| <2|z|3+4=20 < f(z)= |z|6=26 . Note that in this chosen curve f has all its roots and so f+g also has its all root in the chosen curve.

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