Chap 1 The Number System

Let z z1 z2 show that z is a 12 th of unity 33 use de

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Unformatted text preview: 2i by taking the following steps: a) Rewrite the equation z 2 = 5 + 12i in real variables x and y , where z = x + iy . b) By considering the real and imaginary parts of your result in (a), solve for x and y . 22. Consider the equation z 3 − 3z 2 + 4z − 2 = 0. a) Verify that 1 + i is a solution of the equation. b) Find also the other solutions. 23. You are given that z = 1 is a solution of the cubic equation z 3 − 5z 2 + 9z − 5 = 0. Find the other two solutions. 24. You are given that z = 2 is a solution of the cubic equation z 3 − 6z 2 + 13z − 10 = 0. Find the other two solutions. 25. You are given that z = −1 is a solution of the equation z 3 + 3z 2 + 6z + 4 = 0. Use this to find the other two solutions. Then indicate the positions of the three solutions in the Argand diagram. 26. Suppose that a non-zero complex number z has modulus r and argument θ. Write down the modulus and argument of each of the following: a) z b) z 3 c) z −1 d) −z e) z z 27. Express each of the√ follow...
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