Question

Question 20 If [(1+i) / (1-i)]^m = 1, then find the least positive integral value of m.

Class X1 - Maths -Complex Numbers and Quadratic Equations Page 113

Answer 1

{(1 + i)/(1 - i)}^m = 1

multiply (1 + i) numberator as well as denominator .

{(1 + i)(1 + i)/(1 - i)(1 + i)}^m = 1

{(1 + i)²/(1² - (i)²)}^m = 1

{(1 + i² +2i)/2 }^m = 1

{( 2i)/2}^m = 1

{i}^m = 1

we know, i^4n = 1 where , n is an integer.

so, m = 4n where n is an integers
e.g m = 4 { because least positive integer 1 }
hence, m = 4 ( answer )

Answer 2

{(1 + i)/(1 - i)}^m = 1

multiply (1 + i) numberator as well as denominator .

{(1 + i)(1 + i)/(1 - i)(1 + i)}^m = 1

{(1 + i)²/(1² - (i)²)}^m = 1

{(1 + i² +2i)/2 }^m = 1

{( 2i)/2}^m = 1

{i}^m = 1

we know, i^4n = 1 where , n is an integer.

so, m = 4n where n is an integers

e.g m = 4 { because least positive integer 1 }

hence, m = 4 ( answer )