**Example Problems on Properties of Modulus of Complex Numbers :**

Here we are going to see some example problems to understand properties of modulus of complex numbers.

**Question 1 :**

Find the modulus of the following complex numbers

(i) 2/(3 + 4i)

**Solution :**

We have to take modulus of both numerator and denominator separately.

|(2/(3+4i))| = |2|/|(3 + 4i)|

= 2 / √(3^{2} + 4^{2})

= 2 / √(9 + 16)

= 2 / √25

= 2/5

(ii) (2 - i)/(1 + i) + (1 - 2i)/(1 - i)

**Solution :**

**(2 - i)/(1 + i) + (1 - 2i)/(1 - i)**

** = |(2 - i)|/|(1 + i)| + |****(1 - 2i)|/|(1 - i)|**

**|(2 - i)|**** = **√(2

**|1 + i|**** = **√(1

**|1 - 2i|**** = **√(1

**|1 - i|**** = **√(1

= (√5/√2) + (√5/√2)

= 2√5/√2

= √2√5

= √10

(iii) (1 - i)^{10}

**Solution :**

To solve this problem, we may use the property

|z^{n}| = |z|^{n}

(1 - i)^{10 }= {(1 - i)^{2}}^{5}

= (1^{2} + i^{2} - 2i)^{5}

= (1 - 1 - 2i)^{5}

= (- 2i)^{5}

= -32i^{5}

= |-32i|

= √(-32)^{2}

= 32

(iv) 2i(3− 4i)(4 − 3i)

**Solution :**

|2i(3− 4i)(4 − 3i)| = |2i| |3 - 4i||4 - 3i|

= √2^{2 } √3^{2 }+ (-4)^{2 }√4^{2 }+ (-3)^{2 }

= √4 √25 √25

= 2 (5)(5)

= 50

**Question 2 :**

For any two complex numbers z_{1} and z_{2} , such that |z_{1}| = |z_{2}| = 1 and z_{1} z_{2} ≠ -1, then show that z_{1} + z_{2}/(1 + z_{1} z_{2}) is a real number.

**Solution :**

Let z_{1} = 1 and z_{2 } = i

|z_{1}| = √1^{2 }+ 0^{2 } = 1

|z_{2}| = √0^{ }+ 1^{2 } = 1

z_{1 }+ z_{2 }= 1 + i

z_{1 } z_{1 }= i

By applying the values of z_{1 }+ z_{2 }and z_{1 } z_{2 }in the given statement, we get

z_{1} + z_{2}/(1 + z_{1} z_{2}) = (1 + i)/(1 + i) = 1

1 is real. Hence it is proved.

**Question 3 :**

Which one of the points 10 − 8i , 11 + 6i is closest to 1 + i

**Solution :**

Let the given points as A(10 - 8i), B (11 + 6i) and C (1 + i).

To find which point is more closer, we have to find the distance between the points AC and BC.

AC = √(1-10) = √9 = √(81 + 81) AC = √162 |
BC = √(1-11) = √10 = √(100 + 25) BC = √125 |

√162 > √125

Hence the point B is closer to C.

After having gone through the stuff given above, we hope that the students would have understood, "Example Problems on Properties of Modulus of Complex Numbers".

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