`a_(1), a_(2),a_(3) in R - {0} and a_(1)+ a_(2)cos2x+ a_(3)sin^(2)x=0` " for all " x in R then (a) vectors `veca=a_(1) hati+ a_(2) hatj + a_(3) hatk and vecb = 4hati + 2hatj + hatk` are perpendicular to each other (b)vectors `veca=a_(1) hati+ a_(2) hatj + a_(3) hatk and vecb = hati + hatj + 2hatk` are parallel to each each other (c)if vector `veca=a_(1) hati+ a_(2) hatj + a_(3) hatk` is of length `sqrt6` units, then on of the ordered trippplet `(a_(1), a_(2),a_(3)) = (1, -1,-2)` (d)if `2a_(1) + 3 a_(2) + 6 a_(3) = 26 , " then " |veca hati + a_(2) hatj + a_(3) hatk | is 2sqrt6`

Options

A. vectors `veca=a_(1) hati+ a_(2) hatj + a_(3) hatk and vecb = 4hati + 2hatj + hatk` are perpendicular to each other
B. vectors `veca=a_(1) hati+ a_(2) hatj + a_(3) hatk and vecb = hati + hatj + 2hatk` are parallel to each each other
C. if vector `veca=a_(1) hati+ a_(2) hatj + a_(3) hatk` is of length `sqrt6` units, then on of the ordered trippplet `(a_(1), a_(2),a_(3)) = (1, -1,-2)`
D. if `2a_(1) + 3 a_(2) + 6 a_(3) + 6a_(3) = 26 , " then " |veca hati + a_(2) hatj + a_(3) hatk | is 2sqrt6`

Answer

a,b,c,d

Explanation

`a_(1)+a_(2)cos2x+a_(3)sin^(2)x=0AAx inR`
`or (a_(1)+ a_(2)) + sin ^(2)x (a_(3)- 2a_(2)=0`
`Rightarrow a_(1)+a_(2)=0 and a_(3)-2a_(2)=0`
` a_(1)/(-1)=a_(2)/1=a_(3)/2 = lambda(ne0)`
`Rightarrow a_(1) =lamda, a_(2)-lamda, a_(3)= 2lambda`