if a=cos 0 + in sin 0,=cos 20 -I sin 2 0, c= cos 0 + I sin 3 0 and if `|{:(a,,b,,c),(b,,c,,a),(c,,a,,b):}| =0 ` thenA. `=0 2 k pi, k in Z`B. ` 0= (2k +1) pi k in Z`C. `0= (4k +1) pi k in Z`D. none of these

if a=cos 0 + in sin 0,=cos 20 -I sin 2 0, c= cos 0 + I sin 3 0 and if

1.	if a=cos 0 + in sin 0,=cos 20 -I sin 2 0, c= cos 0 + I sin 3 0 and if `\|{:(a,,b,,c),(b,,c,,a),(c,,a,,b):}\| =0 ` thenA. `=0 2 k pi, k in Z`B. ` 0= (2k +1) pi k in Z`C. `0= (4k +1) pi k in Z`D. none of these
Answer» Correct Answer - A `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`