AP EAMCET202420 May 2024Evening ShiftMathematicsComplex NumberActual
If z=x+i y, x^2+y^2=1 and z₁=z e^ i then z₁^ 2 n -1 z₁^ 2 n +1 =
Options
- A-i n ( + ⁻¹ ( y x ) )
- Bi (n ( + ⁻¹ y x ) )
- Ci n ( + ⁻¹ x y )
- Di (n ( + ⁻¹ y x ) )
Correct answer
D. i (n ( + ⁻¹ y x ) )
Step-by-step solution
z=x+i y=e^ i aligned & z₁=z e^ i =e^ i( + ) & z₁^ 2 n -1 z₁^ 2 n +1 = (2 n( + )-1+i (2 n( + )) 1+ (2 n( + )+i (2 n( + )) & = 2 i n( + ) (n( + )-2 ^2(n( + )) . 2 ^2(n( + )+2 i n( + ) n( + ) & =i n( + )=i n ( + ⁻¹ y x ) aligned