MTH632 Practice Questions Set 2 by VUBuddy

By Pin2 MTH632

Modules ( 5 – 7 )

Q1. Verify commutative law with respect to multiplication for complex numbers z1 = 2 + i, z2 = –3 + 9i.

Solution: The commutative law is z1 + z2 = z2 +z1 Consider z1 + z2 = ( 2 + i ) + ( –3 + 9i ) = 2 – 3 + i +9i = – 1 + 10 i Now z2 + z1 = ( –3 + 9i ) +( 2 + i ) = – 3 + 2 + 9i +i = – 1 + 10 i Hence z1 + z2 = z2 +z1

Q2. Verify the associative law with respect to multiplication for complex numbers z1 = 2 + i, z2 = –3 + 9i and z3 = – 8 + i.

Solution: The associative law is ( z1 z2 ) z3= z1( z2 z3 ) Consider z1 z2 = ( 2 + i ) ( –3 + 9i ) = 2 (– 3) +2(9i)+ i(–3) + (i )(9i) = – 6 + 18 i –3i+9i2 = – 6 + 18 i –3i+9(–1) = – 6 –9+ 18 i –3i = – 12 + 15 i Now ( z1 z2 ) z3 = (– 12 + 15 i) (– 8 + i) = – 12 (– 8) +(– 12 (i))+ 15i(–8) + (15i )(i) = 96 – 12 i –120 i + 15i 2 = 96 – 12 i –120 i + 15(–1) = 81 – 132 i Similarly, you can find z1( z2 z3 ).

Q3. Find the multiplicative inverse z -1 of the complex numbers z = – 5 + i.

Solution: z-1 = 1 z = – 5 + 1 = -5 (– 5)2 + 12 -i (– 5)2 + 12 by formula

-5 25 +1 -i 25+1 = -5 26 -i 1

Q4. Write down the Distributive law of multiplication over addition. Solution:

The Distributive law is z1 ( z2 + z3 )= z1z2 +z1 z3

Q5. Let z1 = – 8 + i, z2 = –3 + 8i, then calculate

z z .

Solution: z1 z2 = (x1+iy1)( x2 x2 2+y2 2 -i y2 x2 2+y2 2 )

x1x 2+y1y2 x2 2+y2 2 , y1x2-x1y x2 2+y2 2 )

(-8)(-3)+(1)(8) (-3)2+(8)2 , (1)(-3)-(-8)(8) (-3)2+(8)2 )

(-3)2+(8)2 , (-3)2+(8)2 )

Q6. Find the modulus | z | of the complex numbers z = – 7 + 5i. Solution:

| z | = | – 7 + 5i | = 72+52 = 49+25 =