There are three boxes containing white balls and black balls.

Box-1 contains 2 black and 1 white balls.

Box-2 contains 1 black and 2 white balls.

Box-3 contains 3 black and 3 white balls.

In a random experiment, one of these boxes is selected, where the probability of choosing Box-1 is $\frac{1}{2}$, Box-2 is $\frac{1}{6}$, and Box-3 is $\frac{1}{3}$. A ball is drawn at random from the selected box. Given that the ball drawn is white, the probability that it is drawn from Box-2 is ____________. (Round off to two decimal places)

$$\mathop {\lim }\limits_{t \to + \infty } \sqrt{t^2+t}-t= $$

(Round

off to one decimal place)

Let $Y=Z^2, Z=\frac{X-\mu}{\sigma}$, where $X$ is a normal random variable with mean $\mu$ and variance $\sigma^2$. The variance of $Y$ is

Let $A \in \mathbb{R}^{n \times n}$ be such that $A^3=A$. Which one of the following statements is ALWAYS correct?