If Δ=1sinθ1−sinθ1sinθ−1−sinθ1 then find the range of Δ.

Δ=1[1+sin2θ]−sinθ[−sinθ+sinθ]+1[sin2θ+1]=2[1+sin2θ]=0≤sin2θ≤1=1≤1+sin2θ≤22≤2(1+sin2θ)≤4

If Δ=1sinθ1−sinθ1sinθ−1−sinθ1 then find the range of Δ.

Δ=1[1+sin2θ]−sinθ[−sinθ+sinθ]+1[sin2θ+1]=2[1+sin2θ]=0≤sin2θ≤1=1≤1+sin2θ≤22≤2(1+sin2θ)≤4

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If $Δ = |\begin{matrix} 1 & \sin θ & 1 \\ - \sin θ & 1 & \sin θ \\ - 1 & - \sin θ & 1 \end{matrix}|$ then find the range of $Δ$ .

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$[1, 2]$

$[2, 4]$

$[- 2, 2]$

$[4, 2]$

answer is B.

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Detailed Solution

$Δ = 1 [1 + \sin^{2} θ] - \sin θ [- \sin θ + \sin θ] + 1 [\sin^{2} θ + 1]$

$= 2 [1 + \sin^{2} θ]$

$= 0 \leq \sin^{2} θ \leq 1$

$= 1 \leq 1 + \sin^{2} θ \leq 2$

$2 \leq 2 (1 + \sin^{2} θ) \leq 4$

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If Δ=1sinθ1−sinθ1sinθ−1−sinθ1 then find the range of Δ.

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If $Δ = |\begin{matrix} 1 & \sin θ & 1 \\ - \sin θ & 1 & \sin θ \\ - 1 & - \sin θ & 1 \end{matrix}|$ then find the range of $Δ$ .