This exercise is a multiple choice questionnaire. For each question, only one of the four proposed answers is correct. No justification is required. A wrong answer, a multiple answer, or the absence of an answer to a question neither awards nor deducts points. The five questions are independent.
We consider the function $f$ defined on the interval $]1; +\infty[$ by $$f(x) = 0{,}05 - \frac{\ln x}{x-1}$$ The limit of the function $f$ at $+\infty$ is equal to: a. $+\infty$ b. 0.05 c. $-\infty$ d. 0
We consider a function $h$ continuous on the interval $[-2;4]$ such that: $$h(-1) = 0, \quad h(1) = 4, \quad h(3) = -1.$$ We can affirm that: a. the function $h$ is increasing on the interval $[-1; 1]$. b. the function $h$ is positive on the interval $[-1; 1]$. c. there exists at least one real number $a$ in the interval $[1; 3]$ such that $h(a) = 1$. d. the equation $h(x) = 1$ has exactly two solutions in the interval $[-2; 4]$.
We consider two sequences $(u_{n})$ and $(v_{n})$ with strictly positive terms such that $\lim_{n \rightarrow +\infty} u_{n} = +\infty$ and $(v_{n})$ converges to 0. We can affirm that: a. the sequence $\left(\dfrac{1}{v_{n}}\right)$ converges. b. the sequence $\left(\dfrac{v_{n}}{u_{n}}\right)$ converges. c. the sequence $(u_{n})$ is increasing. d. $\lim_{n \rightarrow +\infty} \left(-u_{n}\right)^{n} = -\infty$
To participate in a game, a player must pay $4\,€$. They then roll a fair six-sided die:
if they get 1, they win $12\,€$;
if they get an even number, they win $3\,€$;
otherwise, they win nothing.
On average, the player: a. wins $3.50\,€$ b. loses $3\,€$. c. loses $1.50\,€$ d. loses $0.50\,€$.
We consider the random variable $X$ following the binomial distribution $\mathscr{B}(3; p)$. We know that $P(X = 0) = \dfrac{1}{125}$. We can affirm that: a. $p = \dfrac{1}{5}$ b. $P(X = 1) = \dfrac{124}{125}$ c. $p = \dfrac{4}{5}$ d. $P(X = 1) = \dfrac{4}{5}$
This exercise is a multiple choice questionnaire. For each question, only one of the four proposed answers is correct. No justification is required. A wrong answer, a multiple answer, or the absence of an answer to a question neither awards nor deducts points. The five questions are independent.
\begin{enumerate}
\item We consider the function $f$ defined on the interval $]1; +\infty[$ by
$$f(x) = 0{,}05 - \frac{\ln x}{x-1}$$
The limit of the function $f$ at $+\infty$ is equal to:\\
a. $+\infty$\\
b. 0.05\\
c. $-\infty$\\
d. 0
\item We consider a function $h$ continuous on the interval $[-2;4]$ such that:
$$h(-1) = 0, \quad h(1) = 4, \quad h(3) = -1.$$
We can affirm that:\\
a. the function $h$ is increasing on the interval $[-1; 1]$.\\
b. the function $h$ is positive on the interval $[-1; 1]$.\\
c. there exists at least one real number $a$ in the interval $[1; 3]$ such that $h(a) = 1$.\\
d. the equation $h(x) = 1$ has exactly two solutions in the interval $[-2; 4]$.
\item We consider two sequences $(u_{n})$ and $(v_{n})$ with strictly positive terms such that $\lim_{n \rightarrow +\infty} u_{n} = +\infty$ and $(v_{n})$ converges to 0.\\
We can affirm that:\\
a. the sequence $\left(\dfrac{1}{v_{n}}\right)$ converges.\\
b. the sequence $\left(\dfrac{v_{n}}{u_{n}}\right)$ converges.\\
c. the sequence $(u_{n})$ is increasing.\\
d. $\lim_{n \rightarrow +\infty} \left(-u_{n}\right)^{n} = -\infty$
\item To participate in a game, a player must pay $4\,€$. They then roll a fair six-sided die:
\begin{itemize}
\item if they get 1, they win $12\,€$;
\item if they get an even number, they win $3\,€$;
\item otherwise, they win nothing.
\end{itemize}
On average, the player:\\
a. wins $3.50\,€$\\
b. loses $3\,€$.\\
c. loses $1.50\,€$\\
d. loses $0.50\,€$.
\item We consider the random variable $X$ following the binomial distribution $\mathscr{B}(3; p)$. We know that $P(X = 0) = \dfrac{1}{125}$. We can affirm that:\\
a. $p = \dfrac{1}{5}$\\
b. $P(X = 1) = \dfrac{124}{125}$\\
c. $p = \dfrac{4}{5}$\\
d. $P(X = 1) = \dfrac{4}{5}$
\end{enumerate}