The number of possible value of `theta` lies in `(0,pi)`, such that system of equation `x+3y+7z=0`, `-x+4y+7z=0`, `xsin3theta+ycos2theta+2z=0` has non trivial solution is/are equal to (a) 2 (b) 3 (c) 5 (d) 4

Options

A. one
B. two
C. three
D. none of these

Answer

D

Explanation

The given system of equations has a non-trivial solution
`{:abs((1,3,7),(-1,4,7),(sin3theta,cos2theta,2))=0:}`
`rArr{:abs((0,7,14),(-1,4,7),(sin3theta,cos2theta,2))=0:}` [Applying `R_1toR_1+R_2`]
`rArr -7(-2-7sin3theta)+14(-cos2theta-4sin3theta)=0`
`rArr 2+7sin3theta-2cos2theta-8sin3theta=0`
`rArr 2cos2theta +sin3theta=2`
`rArr 2(1-2sin^2theta)+3sintheta-4sin^3theta=0`
`rArr 4sin^3theta+4sin^2theta-3sintheta=0`
`rArr sintheta(4sin^2theta+4sintheta-3)=0`
`rArr sintheta (2sintheta-1)(2sintheta+3)=0`
`rArr sintheta =0or, sintheta=1/2`
`rArr theta=0,pi,pi/6,(5pi)/6`