Let $A\left(t\right)=\left[a_{ij}\right]$ is a matrix of order  $3\times 3$ given by $a_{ij}=\{\begin{array}{cc}2\text{cos }t& if i=j\\ 1& if |i-j|=1, \text{\hspace{0.17em} \hspace{0.17em}\hspace{0.17em}then} \\ 0& \text{otherwise} \end{array}$

	Column-I		Column-II
P)	$The number of t in interval [-2\pi ,4\pi ]$  such that  $\left|\text{A}\right(t\left)\right|=4$ is equal to	(1)	0
Q)	$\left|\text{A}\left(\frac{\pi }{17}\right)\right|\left|\text{A}\left(\frac{4\pi }{17}\right)\right|$  is equal to	(2)	1
R)	The maximum value of $\left|\text{A}\right(t\left)\right|+\left|\text{A}\right(2t\left)\right|,\forall t\in \text{R}$ is equal to	(3)	4
S)	$\underset{0}{\overset{\pi }{\int }}\left|\text{A}\right(\text{t})\parallel \text{A}(4\text{t}\left)\right| dt$ is equal to	(4)	6
		(5)	8

$A\left(t\right)=\left[\begin{array}{ccc}2\text{cos}t& 1& 0\\ 1& 2\text{cos}t& 1\\ 0& 1& 2\text{cos}t\end{array}\right]$

$$\begin{gathered} \left|A\right(t\left)\right|=2\text{cost}(4{\text{cos}}^{2}t-1)-2\text{cos}t=8{\text{cos}}^{3}t-4\text{cos}t \\ \left|A\right(t\left)\right|=4\text{cos}t\text{\hspace{0.17em}cos}2t \end{gathered}$$

(A)  $\left|A\right(t\left)\right|=4\Rightarrow t=2n\pi ,n\in I\Rightarrow t=-2\pi ,0,2\pi ,4\pi$
(B)   $\left|\text{A}\left(\frac{\pi }{17}\right)\right|\left|\text{A}\left(\frac{4\pi }{17}\right)\right|=16\text{cos}\frac{\pi }{17}\text{cos}\frac{2\pi }{17}\text{cos}\frac{4\pi }{17}\text{cos}\frac{8\pi }{17}=\frac{\text{sin}\frac{16\pi }{16\pi }}{\text{sin}\frac{\pi }{17}}=1$
(C)  $\left|\text{A}\right(\text{t}\left)\right|+\left|\text{A}\right(2\text{t}\left)\right|=4\text{cos}t\text{\hspace{0.17em}\hspace{0.17em}cos}2\text{t}+4\text{cos}2\text{t\hspace{0.17em}\hspace{0.17em}cos}4\text{t}\le 8$
(D)  $\underset{0}{\overset{\pi }{ }}16\text{\hspace{0.17em}cos }t\text{\hspace{0.17em}cos}2t\text{\hspace{0.17em}\hspace{0.17em}cos}4t\text{\hspace{0.17em}\hspace{0.17em}cos}8tdt=\underset{0}{\overset{\pi }{ }}\frac{\text{sin}16tdt}{\text{sin}t}$

$=\underset{0}{\overset{\pi /2}{ }}(\frac{\text{sin}16t}{\text{sin}t}+\frac{\text{sin}(16\pi -16t)}{\text{sin}(\pi -t)})dt=0$