find all the possible triplets `(a_(1), a_(2), a_(3))` such that `a_(1)+a_(2) cos (2x)+a_(3) sin^(2) (x)=0` for all real x.

Options

A. 0
B. 1
C. 3
D. infinite

Answer

D

Explanation

We have, ltBrgt `a_(1) + a_(2) "cos" 2x + a_(3) "sin"^(2) x = 1 "for all" x`
`rArr a_(1) + a_(2) (2 "cos"^(2)x -1) + a_(3) (1-"cos"^(2) x) = 1 "for all" x`
`rArr (a_(1)-a_(2) + a_(3)-1) + (2a_(2)-a_(3)) "cos"^(2)x = 0 "for all"x`
`rArr a_(1) - a_(2) + a_(3) - 1 = 0 "and " 2a_(2) -a_(3) = 0`
`rArr a_(1) = 1-a_(2), a_(3) = 2a_(2), "where "a_(2) in R`
Thus, there are infinitely many triplets `(a_(1), a_(2), a_(3)).