# EXAMPLE PROBLEMS ON PROPERTIES OF MODULUS OF COMPLEX NUMBERS

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.

## Example Problems on Properties of Modulus of Complex Numbers - Questions

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 / √(32 + 42)

=  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)|  = √(22 + 12)  =  √5

|1 + i|  = √(12 + 12)  =  √2

|1 - 2i|  = √(12 + 22)  =  √5

|1 - i|  = √(12 + 12)  =  √2

=  (√5/√2) + (√5/√2)

=  2√5/√2

=  √2√5

=  √10

(iii)  (1 - i)10

Solution :

To solve this problem, we may use the property

|zn|  =  |z|n

(1 - i)10 =  {(1 - i)2}5

=  (12 + i2 - 2i)5

=  (1 - 1 - 2i)5

=  (- 2i)5

=  -32i5

=  |-32i|

=   √(-32)2

=   32

(iv) 2i(3− 4i)(4 − 3i)

Solution :

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

=  √2 √32 + (-4)√4+ (-3)

=  √4 √25 √25

=  2 (5)(5)

=  50

Question 2 :

For any two complex numbers z1 and z2 , such that |z1| = |z2|  =  1 and z1 z2 ≠ -1, then show that z1 + z2/(1 + z1 z2) is a real number.

Solution :

Let z1  =  1 and z =  i

|z1|  =  √1+ 0 =  1

|z2|  =  √0 + 1 =  1

z1 z2  =  1 + i

z z1  =   i

By applying the  values of zz2 and z zin the given statement, we get

z1 + z2/(1 + z1 z2)    =  (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)2 + (1+8)2   =  √92 + 92  =  √(81 + 81)AC  =  √162 BC  =  √(1-11)2 + (1-6)2   =  √102 + (-5)2  =  √(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".

Apart from the stuff given in this section "Example Problems on Properties of Modulus of Complex Numbers"if you need any other stuff in math, please use our google custom search here.

WORD PROBLEMS

HCF and LCM  word problems

Word problems on simple equations

Word problems on linear equations

Word problems on quadratic equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6