Decomposers are

$A$ mass $M$ hangs on a massless rod of length $l$ which rotates at a constant angular frequency $\omega$. The mass $M$ moves with steady speed in a circular path of constant radius $r$. The angular momentum of $M$ about point $A$ is $L_A$,which lies in the positive $z$-direction,and the angular momentum of $M$ about point $B$ is $L_B$. Which of the following statements is correct for this system?

Suppose that $E_1$ and $E_2$ are two events of a random experiment such that $P(E_1) = \frac{1}{4}$, $P(E_2 / E_1) = \frac{1}{2}$ and $P(E_1 / E_2) = \frac{1}{4}$. Observe the lists given below. The correct matching of List-$I$ from List-$II$ is:

List-$I$	List-$II$
$(A)$ $P(E_2)$	$(i)$ $1/4$
$(B)$ $P(E_1 \cup E_2)$	$(ii)$ $5/8$
$(C)$ $P(\bar{E}_1 / \bar{E}_2)$	$(iii)$ $1/8$
$(D)$ $P(E_1 / \bar{E}_2)$	$(iv)$ $1/2$
	$(v)$ $3/8$
	$(vi)$ $3/4$

Four particles,each of mass $M$,are placed at the corners of a square of side $L$. The radius of gyration of the system about an axis perpendicular to the square and passing through its centre is:

The equation of the circle with radius $5$ and touching the coordinate axes in the third quadrant is

If $a > 0$ and $\lim _{x \rightarrow a} \frac{a^x - x^a}{x^x - a^a} = -1$,then $a$ is equal to