CMI-CE4S - CC: AD [2011-12] 242: College Mathematics I -...

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CC: AD [ 2011-12 ] 242: College Mathematics I — Class Exercises [ Sol ] Chapter 4: Complex Numbers ª 4.1 Definitions and Operations on Complex Numbers ª 4.2 De Moivre’s Theorem and its Applications
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Ch4: Complex Numbers [ CE4S-1 ] q 4.1 Definitions and Operations on Complex Numbers Class Ex. 1 Given that z = - 3 + 4 i and zw = - 14 + 2 i , find (a) w in the form of p + iq where p and q are real, [ 2 ] (b) | z | and arg z , principal argument in radians to 2 decimal places, [ 2 ] (c) the values of the real constants m and n such that mz + nzw = - 10 - 20 i . [ 4 ] Solution (a) w = - 14 + 2 i - 3 + 4 i = ( - 14 + 2 i )( - 3 - 4 i ) ( - 3 + 4 i )( - 3 - 4 i ) = ( 42 + 8 )+ i ( - 6 + 56 ) 9 + 16 = 2 + 2 i . (b) | z | = p 3 2 + 4 2 = 5 and arg z = tan - 1 4 - 3 = π - 0.927 = 2.21 (to 2 d.p.). (c) mz + nzw = m ( - 3 + 4 i )+ n ( - 14 + 2 i ) = - 3 m - 14 n +( 4 m + 2 n ) i Equating real and imaginary parts, we have ¤ 3 m + 14 n = 10 4 m + 2 n = - 20 . Solving to obtain m = - 6 and n = 2. AD: College Mathematics I [ 2011-12 ] Louis Lau 1
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4.1 Definitions and Operations on Complex Numbers [ CE4S-1 ] Class Ex. 2 The complex number z is given by z = ( 1 + 3 i )( p + qi ) , where p and q are real and p > 0. Given that arg z = π 4 , (a) write an equation connecting p and q . [ 2 ] Given also that | z | = 10 p 2, (b) find the values of p and q . [ 4 ] (c) Write down the values of arg z and arg 1 z . [ 2 ] Solution (a) z = ( 1 + 3 i )( p + qi ) = ( p - 3 q )+( 3 p + q ) i tan ( arg z ) = tan π 4 3 p + q p - 3 q = 1 3 p + q = p - 3 q p + 2 q = 0 (b) | z | 2 = ( p - 3 q ) 2 +( 3 p + q ) 2 = ( 10 p 2 ) 2 p 2 - 6 pq + 9 q 2 + 9 p 2 + 6 pq + q 2 = 200 10 p 2 + 10 q 2 = 200 p 2 + q 2 = 20 ··· ( * ) From (a), p = - 2 q ( > 0 ) , substituting into ( * ) , we have ( - 2 q ) 2 + q 2 = 20 5 q 2 = 20 q = - p 4 = - 2 ( q < 0 ) When q = - 2, p = 4. (c) arg z = arg 1 z = - arg z = - π 4 . AD: College Mathematics I [ 2011-12 ] Louis Lau 2
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4.1 Definitions and Operations on Complex Numbers [ CE4S-1 ] Class Ex. 3 If z = 2 [ cos ( - π 3 ) + i sin ( - π 3 )] , express z , z 2 and 1 z in the form x + y i where x and y are real. State the modulus and principal argument in radians of each of these complex numbers. Represent the three numbers in an Argand diagram.
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CMI-CE4S - CC: AD [2011-12] 242: College Mathematics I -...

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