The values of `theta`, `lambda` for which the following equations `sinthetax-costhetay+(lambda+1)z=0` , `costhetax+sinthetay-lambdaz=0` , `lambdax+(lambda+1)y+costhetaz=0`
have non trivial solution, is

Options

A. `theta=npi`, `lambda in R-{0}`
B. `theta=2npi`, `lambda` is any rational number
C. `theta=(2n+1)pi`, `lambda in R^(+)`, `n in I`
D. `theta=(2n+1)(pi)/(2)`, `lambda in R`, `n in I`

Answer

D

Explanation

`(d)` For an trivial solution `|{:(sintheta,-costheta,lambda+1),(costheta,sintheta,-lambda),(lambda,lambda+1,costheta):}|=0`
`impliessin^(2)thetacostheta+lambda^(2)costheta+(lambda+1)^(2)costheta-sinthetalambda(lambda+1)+cos^(3)theta+sinthetalambda(lambda+1)=0`
`implies(sin^(2)theta+cos^(2)theta)costheta+lambda^(2)costheta+(lambda+1)^(2)costheta=0`
`impliescostheta+lambda^(2)costheta+(lambda+1)^(2)costheta=0`
`implies2costheta(lambda^(2)+lambda+1)=0`
`impliescostheta=0`
`impliestheta=(2n+1)(pi)/(2)`, `lambda in R`, `n in I`