Let `lambda` and `alpha` be real. Then the numbers of intergral values `lambda` for which the system of linear equations
`lambdax +(sin alpha) y+ (cos alpha) z=0`
`x + (cos alpha) y+ (sin alpha) z=0`
`-x+(sin alpha) y -(cos alpha) z=0` has non-trivial solutions is

Options

A. 0
B. 1
C. 2
D. 3

Answer

D

Explanation

The given system has non=trivial solution if
`|{:(lambda,,sin alpha,,cos alpha),(1,,cos alpha,,sin alpha),(-1,,sin alpha,,-cos alpha):}|=0`
By expanding the determinant along first column we get
`lambda =sin 2alpha +cos 2alpha`
We know that
`-sqrt(2) le sin 2alpha + cos 2 alpha le sqrt(2)`
`:. -sqrt(2) le lambda le sqrt(2)`
hence integral values of `lambda` are -1 ,0 and 1