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(A) $A$. Given $A = \begin{bmatrix} \cos^2 37^{\circ} & \cos^2 53^{\circ} & -1 \ \sin^2 76^{\circ} & -1 & \sin^2 14^{\circ} \ -1 & \cos^2 28^{\circ}

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$A$. If $A = \begin{bmatrix} \cos^2 37^{\circ} & \cos^2 53^{\circ} & \cot 135^{\circ} \ \sin^2 76^{\circ} & \sin 270^{\circ} & \sin^2 14^{\circ} \ \cos 180^{\circ} & \cos^2 28^{\circ} & \cos^2 62^{\circ} \end{bmatrix}$,then $3 - |A| = $

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(A) $A$. Given $A = \begin{bmatrix} \cos^2 37^{\circ} & \cos^2 53^{\circ} & -1 \ \sin^2 76^{\circ} & -1 & \sin^2 14^{\circ} \ -1 & \cos^2 28^{\circ} & \cos^2 62^{\circ} \end{bmatrix}$. Using properties of determinants,$|A| = 0$. Thus,$3 - |A| = 3 - 0 = 3$. Matches $(iii)$.$B$. The period of $\cos(6x-4)$ is $\frac{\pi}{3}$,$\sec(3-4x)$ is $\frac{\pi}{2}$,$\cot(5x+3)$ is $\frac{\pi}{5}$,and $\sin(3x+4)$ is $\frac{2\pi}{3}$. The period of the expression is $LCM(\frac{\pi}{3}, \frac{\pi}{2}, \frac{\pi}{5}, \frac{2\pi}{3}) = 2\pi$. Given $\frac{2k\pi}{5} = 2\pi$,so $k = 5$. Matches $(v)$.$C$. $y = \frac{1}{2}(\cos x + \sin x)^2 + (\sin x - \cos x)^2 = \frac{3}{2} - \frac{\sin 2x}{2}$. Max value is $\frac{3}{2} + \frac{1}{2} = 2$. Matches $(ii)$.$D$. If $x+y+z=0$,then $\sin 2x + \sin 2y + \sin 2z = -4\sin x \sin y \sin z$. The expression becomes $\frac{-4\sin x \sin y \sin z}{-\sin x \sin y \sin z} = 4$. Matches $(iv)$.

Match the items of List-$I$ with those of List-$II$. The correct match is:

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