If $M=\left(\begin{array}{rr}\frac{5}{2} & \frac{3}{2} \\ -\frac{3}{2} & -\frac{1}{2}\end{array}\right)$, then which of the

following matrices is equal to $M^{2022} ?$

Let $$p, q, r$$ be nonzero real numbers that are, respectively, the $$10^{\text {th }}, 100^{\text {th }}$$ and $$1000^{\text {th }}$$ terms of a harmonic progression. Consider the system of linear equations

$$$ \begin{gathered} x+y+z=1 \\ 10 x+100 y+1000 z=0 \\ q r x+p r y+p q z=0 \end{gathered} $$$

List-I	List-II
(I) If $$\frac{q}{r}=10$$, then the system of linear equations has	(P) $$x=0, \quad y=\frac{10}{9}, z=-\frac{1}{9}$$ as a solution
(II) If $$\frac{p}{r} \neq 100$$, then the system of linear equations has	(Q) $$x=\frac{10}{9}, y=-\frac{1}{9}, z=0$$ as a solution
(III) If $$\frac{p}{q} \neq 10$$, then the system of linear equations has	(R) infinitely many solutions
(IV) If $$\frac{p}{q}=10$$, then the system of linear equations has	(S) no solution
	(T) at least one solution

The correct option is:

Let $$M = \left[ {\matrix{ {{{\sin }^4}\theta } \cr {1 + {{\cos }^2}\theta } \cr } \matrix{ { - 1 - {{\sin }^2}\theta } \cr {{{\cos }^4}\theta } \cr } } \right] = \alpha I + \beta {M^{ - 1}}$$,

where $$\alpha $$ = $$\alpha $$($$\theta $$) and $$\beta $$ = $$\beta $$($$\theta $$) are real numbers, and I is the 2 $$ \times $$ 2 identity matrix. If $$\alpha $$^* is the minimum of the set {$$\alpha $$($$\theta $$) : $$\theta $$ $$ \in $$ [0, 2$$\pi $$)} and {$$\beta $$($$\theta $$) : $$\theta $$ $$ \in $$ [0, 2$$\pi $$)}, then the value of $$\alpha $$^* + $$\beta $$^* is

How many 3 $$ \times $$ 3 matrices M with entries from {0, 1, 2} are there, for which the sum of the diagonal entries of M^TM is 5?