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# Exercise 2.5: Modulus of a Complex Number

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EXERCISE 2.5

1. Find the modulus of the following complex numbers  2. For any two complex numbers z1 and z2 , such that |z1| = |z2| = 1 and z1z2 ã  -1, then show that [z1  + z2] / [1+ z1z2] is a real number. 3. Which one of the points10 - 8i, 11 + 6is closest to1+ . 4. If | |= 3 , show that 7 ãÊ | + 6 ã8| ãÊ13 . 5. If  |z| = 1, show that 2 ãÊ |z2 ã 3| ãÊ 4. 6. If = 2 , show that the greatest and least value of | z | are ã3 +1 and ã3 - 1 respectively. 7.  If z1, z2, and z3 are three complex numbers such that |z1| = 1, |z2| = 2, |z3| = 3 and |z1 + z2 + z3| = 1 , show that |9z1z2 + 4z1z3 + z2z3| = 6. 8. If the area of the triangle formed by the vertices ziz , and iz is 50 square units, find the value of |z|. 9. Show that the equation z3 + 2 = 0 has five solutions. 10. Find the square roots of (i) 4 + 3i  (ii) -6 + 8(iii) -5 -12.  