Question

Show that if the determinant ∆ = `|(3, -2, sin3theta),(-7, 8, cos2theta),(-11, 14, 2)|` = 0, then sinθ = 0 or `1/2`.

Answer 1

Applying R₂ → R₂ + 4R₁ and R₃ → R₃ + 7R₁, we get

`|(3, -2, sin3theta),(5, 0, cos2theta + 4sin3theta = 0),(10, 0, 2 + 7sin3theta)|` = 0

or 2 [5(2 + 7 sin³θ) – 10(cos²θ + 4sin³θ)] = 0

or 2 + 7sin³θ – 2cos²θ – 8sin³θ = 0

or 2 – 2cos²θ – sin³θ = 0

sinθ (4sin²θ + 4sinθ – 3) = 0

or sinθ = 0 or (2sinθ – 1) = 0 or (2sinθ + 3) = 0

or sinθ = 0 or sinθ = `1/2` .....(why?)