For $x \in \mathbb{R}$, the floor function is denoted by $f(x)=\lfloor x\rfloor$ and defined as follows $\lfloor x\rfloor=k, k \leq x

where $k$ is an integer. Let $Y=\lfloor X\rfloor$, where $X$ is an exponentially distributed random variable with mean $\frac{1}{\ln 10}$, where In denotes natural logarithm. For any positive integer $l$, one can write the probability of the event $Y=l$ as follows

$$ P(Y=l)=q^l(1-q) $$

The value of $q$ is

A random experiment consists of throwing 100 fair dice, each die having six faces numbered 1 to 6 . An event $A$ represents the set of all outcomes where at least one of the dice shows a 1 . Then, $\mathrm{P}(\mathrm{A})=$

Consider the function

$$ f(\mathrm{x})=\frac{x^3}{3}+\frac{7}{2} x^2+10 x+\frac{133}{2}, x \in[-8,0] . $$

Which of the following statements is/are correct?

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a twice-differentiable function and suppose its second derivative

satisfies $f^{\prime \prime}(x)>0$ for all $x \in \mathbb{R}$. Which of the following statements is/are ALWAYS correct?