Q. If `z=-2 + 2sqrt3i` then `z^(2n)+2^(2n) z^n+2^(4n)` may be equal to A) `2^(2n)` B) 0 C) `3.2^(4n)` D) none of these

justaguide | Certified Educator

The complex number `z = -2 + 2*sqrt 3*i`

`z = 2*2*e^(-i*60) = 4*e^(-i*60)`

`z^(2n) = 4^(2n)*(e^(-i*120))^n`

= `4^(2n)*(-0.5 + i*sqrt 3/2)^n`

= `2^(2n)*(-2 + 2*sqrt 3*i)^n`

= `2^(2n)*z^n`

`z^(2n) + 2^(2n)*z^n + 2^(4*n)`

= `2^(2n)*z^n + 2^(2n)*z^n + 2^(4*n)`

= `2*2^(2n)*z^n + 2^(4n)`

= `2^(2n+1)*z^n + 2^(4n)`

The correct answer is D) None of these

aruv | Student

because

`Z=-2(1-sqrt(3)i)`

`Z=-4(1/2-sqrt(3)/2i) `

`Z=(-4)(cos(pi/3)-isin(pi/3))`

`Z^(2n)=(-4)^(2n)(cos((2npi)/3)-isin((2npi)/3))`

`2^(2n)Z^n=2^(2n)(-4)^n(cos((npi)/3)-isin((npi)/3))`

`Z^(4n)=(-4)^(4n)(cos((4npi)/3)-isin((4npi)/3))`

`Z^(2n)+2^(2n)Z^n+Z^(4n)`

`=(-4)^n{(-4)^n(cos((2npi)/3)-isin((2npi)/3))`

`+2^(2n)(cos((npi)/3)-isin((npi)/3))`

`+(-4)^(3n)(cos((4npi)/3)-isin((4npi)/3))}`

which is not equal to either A or B or C .

Thus only option with us part D.