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Evaluation of a Determinant

If cosec α 1...

Question

If $∣ ∣ ∣ ∣ \begin{matrix} cosec α & 1 & 0 1 & 2 cosec α & 1 0 & 1 & 2 cosec α \end{matrix} ∣ ∣ ∣ ∣ = λ {cosec}^{3} α - μ cosec α,$
then the value of $λ + μ$ is

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Solution

$L H S = ∣ ∣ ∣ ∣ \begin{matrix} cosec α & 1 & 0 1 & 2 cosec α & 1 0 & 1 & 2 cosec α \end{matrix} ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ \begin{matrix} cosec α & 1 & 0 0 & 2 cosec α - sin α & 1 0 & 1 & 2 cosec α \end{matrix} ∣ ∣ ∣ ∣ [R_{2} \to R_{2} - sin α R_{1}] = 4 {cosec}^{3} α - 3 cosec α \Rightarrow λ + μ = 7$

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Evaluation of a Determinant

Standard XII Mathematics

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