# PRACTICE QUESTIONS BASED ON PROPERTIES OF COMPLEX NUMBERS

Practice Questions Based on Properties of Complex Numbers :

Here we are going to see some example problems based on practice questions based on properties of complex numbers.

## Practice Questions Based on Properties of Complex Numbers - Questions

Question 1 :

If z1  =  1 - 3i, z2  =  -4i and z3  =  5 show that

(i)  (z1 + z2) + z3  =  z1 + (z2 + z3)

Solution :

L.H.S :

z1 + z2  =  1 - 3i - 4i  =  1 - 7i

(z1 + z2) + z3  =  (1 - 7i) + 5

=  (1 + 5) - 7i

=  6 - 7i  ---(1)

R.H.S :

z2 + z3  =  - 4i + 5

z1 + (z2 + z3)  =  (1 - 3i) + (-4i + 5)

=  (1 + 5) + i(-3 - 4)

=  6 - 7i  ---(2)

(1)  = (2)

(ii)  (z1 z2) z=  z1 (z2 z3)

Solution :

L.H.S :

z1 z=  (1 - 3i)(-4i)

=  -4i + 12i2

=  -4i + 12(-1)

=  -4i - 12

=  -4i + 12(-1)

z1 z2  =  -4i - 12

(z1 z2) z=  (-4i - 12) (5)  = -20i - 60  ----(1)

R.H.S :

z2 z =  -4i (5)  =  -20i

z1 (z2 z3)  =  (1 - 3i)(-20i)

=  -20i + 60i2

=  -20i + 60(-1)

= -20i - 60  -----(2)

(1)  =  (2)

Hence proved.

Let us look into the next problem on "practice questions based on properties of complex numbers."

Question 2 :

If z1 = 3, z2 = -7i and z3  =  5 + 4i

(i)  z1 (z2 + z3)  =  z1 z2 +  z1 z3

Solution :

L.H.S :

z2 + z=  -7i + (5 + 4i)

=  -7i + 5 + 4i

z2 + z3   =  -3i + 5

z1 (z2 + z3)  =  3 (-3i + 5)

z1 (z2 + z3)  =  15 - 9i  -----(1)

R.H.S :

z1 z=  3(-7i)  =  -21i

z1 z3  =  3(5 + 4i)  =  15 + 12i

z1 z2 z1 z= -21i + 15 + 12i

=  -9i + 15

=  15 - 9i ------(2)

(1)  =  (2)

(ii)  (z1 + z2)z3  =  z1 z3 +  z2 z3

Solution :

L.H.S :

z1 + z2  =  3 - 7i

(z1 + z2)z3  =  (3 - 7i) (5 + 4i)

=  15 + 12i - 35i - 28i2

=  15 - 23i - 28(-1)

=  15 - 23i + 28

(z1 + z2)z3  =  43 - 23i  ----(1)

z1 z3  =  3(5 + 4i)  =  15 + 12i

z2 z3  = (-7i) (5 + 4i)

=  -35i - 28i2

z2 z3  =  -35i - 28(-1)

=  -35i + 28

z1 z3 + z2 z  =  15 + 12i - 35i + 28

=  43 - 23i  ---(2)

(1)  =  (2)

Hence proved.

Question 3 :

If z1 = 2 + 5i, z2 = -3 - 4i and z3 = 1 + i , find the additive and multiplicative inverse of z1, z2 and z3.

Solution :

z1 = 2 + 5i

z1 = 2 + 5i

Multiplicative inverse of (z1) :

z1 = 2 + 5i

1/z1 = 1/(2 + 5i)

= (1/(2 + 5i))((2 - 5i)/(2 - 5i))

=  (2 - 5i)/(4 - 25(-1))

=  (2 - 5i)/29

Hence multiplicative inverse of 2 + 5i is (2 - 5i)/29.

z2 = -3 - 4i

z2 = -3 - 4i

Multiplicative inverse of (z1) :

z2 = -3-4i

1/z2 = -1/(3 + 4i)

= (-1/(3 + 4i))((3 - 4i)/(3 - 4i))

=  -(3 - 4i)/(9 - 16(-1))

=  -(3 - 4i)/(9 + 16)

=  -(3 - 4i)/25

Hence multiplicative inverse of -3-4i is -(3 - 4i)/25.

z3 = 1 + i

z3 = 1 + i

Additive inverse  = -1 - i

Multiplicative inverse of (z3) :

z3 = -1 - i

1/z2 = -1/(1 + i)

= (-1/(1 + i))((1 - i)/(1 - i))

=  (-1 + i)/(1 - (-1))

=  (-1 + i)/2

Hence multiplicative inverse of  1 + i is (-1 + i)/2 After having gone through the stuff given above, we hope that the students would have understood, "Practice Questions Based on Properties of Complex Numbers".

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