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M135F07A9 - MATH 135 Assignment#9 Fall 2007 Due Wednesday...

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MATH 135 Fall 2007 Assignment #9 Due: Wednesday 28 November 2007, 8:20 a.m. N.B. This is the final regularly scheduled Assignment. There will be an Assignment 10 created and solutions posted. While you should not hand Assignment 10 in, working through the problems will be helpful in learning the material from Chapter 9. Hand-In Problems 1. (a) Convert - 3 7 - 21 i to polar form. (b) Convert (12 , - 31 π/ 6) to standard form. 2. Determine the modulus and argument of (1 - i ) 42 . 3. Express ( 2 - 6 i ) 32 in standard form. 4. Determine all z C such that z 8 = 81 i . Plot your solutions in the complex plane. 5. Determine all z C such that iz 3 + 1 + 3 i = 0. Plot your solutions in the complex plane. 6. (a) Prove directly that if z = r cis( θ ) and w = s cis( φ ), then z w = r s cis( θ - φ ). (b) Using this result, evaluate 1 - i 3 + i . Express your answer in polar form. 7. Determine all z C such that z 9 +8 iz 6 + z 3 +8 i = 0. Plot your solutions in the complex plane. 8. If a, b, c, d R with c + id = ( a + ib ) n , show that c 2 + d 2 = ( a 2 + b 2 ) n . 9. Suppose n P , with n 3. Suppose that z C has z = z n - 1 and z = 0. (a) Determine | z | . (b) Determine all possible values for z (in terms of
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