ECONOMICS STATISTICS

# ECONOMICS STATISTICS - COURSE:BASIC MATHEMATICS TRIMESTER I...

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COURSE:BASIC MATHEMATICS TRIMESTER I: CAT Course Code & Title: XET 101: BASIC MATHEMATICS 1

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QUESTION TWO a) i) State Demoivres theorem and use to express (1-2i) 4 in the form x+iy Demoivres theorem states that the modulus of the product of two complex numbers is the product of their moduli and the argument of the product of two complex numbers is the sum of their products . i.e, Let Z be any complex number in the complex plane (with real and imaginary parts and polar coordinates in the Cartesian plane), r the modulus i.e. distance from the origin and Z ≠ 0 with argument Ɵ being a real number and n is a positive integer power of Z. 1. z 1 z 2 = r 1 r 2 = z 1 z 2 2. Arg (z 1 z 2 )= Ɵ 1 + Ɵ 2 = arg (z 1 )+ arg (z 2 ) Suppose: 1. z = r ( cos +i sin ) Ɵ Ɵ then z in its n th power is 2. z n =r n (cos (n ) +i sin (n )) Ɵ Ɵ Solution: (1-2i) 4 in x+iy form using the theorem 1. Let z= (1-2i) (this is the part where z=r ( cos +i sin ) Ɵ Ɵ then z = r=√ 1 2 +(-2 )2 =√1+4=√5= 2.2 Arg (z)=tan -1 =y/x=-2/1=- 63.43 0 Therefore z=2.2 (cos -63.43 0 + isin-63.43 0 ) 2. z 4 =(1-2i) 4 = (the 2 nd

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