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