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. Four
B. Two
C. One
D. Three

Answer

B

Explanation

`x+3y+7z=0, " " -x+4y+7z=0`
`(sin 3 theta)x +(cos 2 theta)y+2z=0`
Since the given set of equation are homogeneous, it will have a non-trivial solution if D = 0
`|(1,3,7),(-1,4,7),(sin3theta,cos2theta,2)|=0`
`R_(1) to R_(1) +R_(2)`
`7|(0,1,2),(-1,4,7),(sin3theta,cos2theta,2)|=0`
`therefore -1(-2-7sin3theta)+2(-cos2theta-4sin3theta)=0`
`2-2cos2theta=sin3theta`
`4 sin^(2)theta=3sin theta-4sin^(3)theta therefore 4sin^(2)+4sin theta-3=0`
`rArr (2 sin theta+3)(2sin theta-1)=0 therefore sin theta=(1)/(2)" " theta " have 2 values in "(0,pi)`
Since `sin theta ne 0" in " (0,pi)`