if `a=cos theta + i sin theta` , `b=cos 2 theta- i sin 2 theta`, `c= cos 3theta + i sin 3theta` and if `|{:(a,,b,,c),(b,,c,,a),(c,,a,,b):}| =0 ` then

Options

A. `theta = 2 k pi, k in Z`
B. `theta= (2k +1) pi , k in Z`
C. `theta= (4k +1) pi , k in Z`
D. none of these

Answer

A

Explanation

`Delta =|{:(a,,b,,c),(b,,c,,a),(c,,a,,b):}|`
`=-(a+b+c)(a^(2)+b^(2)+c^(2)-ab-bc-ca)`
`=(1)/(2) (a+b+c)[(a-b)^(2)+(b-c)^(2)+(c-a)^(2)]=0`
`rArr a+b+c=0" or " a=b=c `
`" if " a+b+c =0 , ` we have
`cos0 + cos 2 0 + cos 3 0 =0` and s sin 0 -sin 2 0 + sin 3 0 =0
`" or " cos 2 0 (2 cos 0 + 1) =0 " and " sin 2 0 (1 -2 cos 0 ) =0 (1)`
which is not possible as cos 2 0 = 0 given sin 2 0 `ne 0` cos 0 `ne 1//2`
And cos 0 = `1//2` gives sin 2 0 `ne 0 cos 0 ne 1//2` . Therefore , Eq.
(1) does not hold simultaneously .Therefore ,
`a+b+c ne 0`
`rArr a=b=c`
`:. e^(i0) =e^(-2i0) =e^(3i0)`
which is satisfied only by by `e^(i 0) =1, i.e., cos 0 =1 sin 0 =0`
`so 0 = 2k pi , k in Z`