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Study Notes IX-XII - Notes & Assignments XI

Extra question complex numbers

Example 1: If 4x + i (3x - y) = 3 + i (-6), where x and y are real numbers, then find the values of x and y.        (x= and y= )

Based on example first

(i) (3x - 7) + 2iy = -5y + (5+x) i              (x = -1, y = 2)
(ii) (1 - i) x + (1 + i) y = 1 - 3i                 (x = 2, y = -1)
(iii) (x + iy) (2 - 3i) = 4 + i                       (x = , y = )
(iv) + = i                                (x = -4, y = 6)
(v) (1 + i) y2 + (6 + i) = (2 + i) x             (x= 5, y = 2 / x=5 , y = -2)
(vi) 2x + 3yi = 2 + 9i                               (x = 1, y = 3)
(vii) x + 4yi = ix + y + 3                          (x = 4, y = 1)
(viii) (3y - 2) + (5 - 4x) i = 0                   (x = , y = )
(ix) 3x + 2iy - ix + 5y = 7 + 5i                 (x = -1, y = 2)

Question based on power of i

(i) i ¹³⁵                                               (-i)
(ii) i ¹⁹                                                (-i)
(iii) i ⁹⁹⁹                                              (i)
(iv)                              (-4)
(v)                               (2i)
(vi) i ⁿ + i ⁿ⁺¹ + i ⁿ⁺² + i ⁿ⁺³                  (0)
(vii) 2i ³ + 6i ³ - 3i ¹⁶ + 6i ¹⁹ - 4i ²⁵      (1 + 4i)
(viii)                              (0)
(ix) (i ⁷⁷ + i ⁷⁰ + i ⁸⁷ + i ⁴¹⁴) ³             (-8)

(x) i ³⁰ + i ⁴⁰ + i ⁶⁰                              (1)
(xi) i ⁴⁹ + i ⁶⁸ + i ⁸⁹ + i ¹¹⁰                   (2i)
(xii) i ³⁰ + i ⁸⁰ + i ¹²⁰                           (1)
(xiii)       (-1)
(xiv) 1 + i ² + i ⁴ + i ⁶ + i ⁸ + -----+ i ²⁰ (1)

Find the modulus and the arguments of each of the complex numbers in Exercise 1 to 2.

1. Z = -1 -i                                   (2, )
2. Z = - + i                                  (2, )

Find the modulus and argument of the following complex numbers and hence express each of them in the polar form:

(i) 1 + i                          
(ii) + i                     
(iii) 1 - i                         
(iv)                       
(v)                        
(vi)                     
(vii)                 
(viii)                   

Convert each of the complex numbers given in Exercise 3 to 8 in the polar form:-

(i) 1 - i                       
(ii) -1 + i                    
(iii) -1 -i                     
(iv) -3                          3
(v) + i                    2
(vi) i                             cos

Represent the complex number z = 1 + i in the polar form.
                                                     Z = 2

Convert the complex number into polar form.
                             8

Convert the complex number z = in the polar form.
                                                       

Convert the following in the polar form:
(i) ,                          
(ii)                                

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