If m and l are the greatest and least values of the determinant 1+sin22xcos22xsin3xsin22x1+cos22xsin3xsin22xcos22x1+sin3x respectively, then

1+sin22xcos22xsin3xsin22x1+cos22xsin3xsin22xcos22x1+sin3x=1+sin22xcos22xsin3x−110−101 · R2→R2−R1,R3→R3−R1=1+sin22x+cos22x+sin3x=2+sin3xm=3l=1

If m and l are the greatest and least values of the determinant 1+sin22xcos22xsin3xsin22x1+cos22xsin3xsin22xcos22x1+sin3x respectively, then

1+sin22xcos22xsin3xsin22x1+cos22xsin3xsin22xcos22x1+sin3x=1+sin22xcos22xsin3x−110−101 · R2→R2−R1,R3→R3−R1=1+sin22x+cos22x+sin3x=2+sin3xm=3l=1

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$If m and l are the greatest and least values of the determinant$ $|\begin{matrix} 1 + \sin^{2} 2 x & \cos^{2} 2 x & \sin 3 x \\ \sin^{2} 2 x & 1 + \cos^{2} 2 x & \sin 3 x \\ \sin^{2} 2 x & \cos^{2} 2 x & 1 + \sin 3 x \end{matrix}| respectively, then$

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$m = 2 l$

$m = 3 l$

$m - l, 3 l, 2 l + m are the lengths of the sides of a right angled triangle$

$m^{4} - l^{4} = 80$

answer is B, C.

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Detailed Solution

$\begin{matrix} |\begin{matrix} 1 + \sin^{2} 2 x & \cos^{2} 2 x & \sin 3 x \\ \sin^{2} 2 x & 1 + \cos^{2} 2 x & \sin 3 x \\ \sin^{2} 2 x & \cos^{2} 2 x & 1 + \sin 3 x \end{matrix}| = |\begin{matrix} 1 + \sin^{2} 2 x & \cos^{2} 2 x & \sin 3 x \\ - 1 & 1 & 0 \\ - 1 & 0 & 1 \end{matrix}| \cdot R_{2} \to R_{2} - R_{1}, R_{3} \to R_{3} - R_{1} \\ = 1 + \sin^{2} 2 x + \cos^{2} 2 x + \sin 3 x \\ = 2 + \sin 3 x \\ m = 3 \\ l = 1 \end{matrix}$