Question

Let, α and β be real. Find the set of all values of β for which the system of equation βx + sin α*y + cosα*z = 0, x + cosα * y + sinα * z = 0 , -x + sinα*y – cosα * z = 0 has a non-trivial solution. For β = 1 what are all values of α?

(a) 2α = 2nπ ± π/2 + π/2

(b) 2α = 2nπ ± π/2 + π/4

(c) 2α = 2nπ ± π/4 + π/4

(d) 2α = 2nπ ± π/4 + π/2

This question was posed to me in exam.

The question is from Application of Determinants topic in portion Determinants of Mathematics – Class 12

Answer 1

Right answer is (c) 2α = 2nπ ± π/4 + π/4

The explanation is: The given system have non-trivial solution if \(\begin{vmatrix}\beta & sin \alpha & cos \alpha \\1 & cos \alpha & sin \alpha \\ -1 & sin \alpha & -cos \alpha \end {vmatrix}\) = 0

On opening the determinant we get β = sin 2α + cos 2 α

Therefore, -√2 ≤ β ≤ √2

Now, for β = 1,

sin 2α + cos 2 α = 1

=> (1/√2)sin 2α + (1/√2) cos 2α = (1/√2)

Or, cos(2α – π/4) = 1/√2 = cos(2nπ ± π/4)

=> 2α = 2nπ ± π/4 + π/4