For each of the four following statements, indicate whether it is true or false, and justify the answer. An unjustified answer is not taken into account. An absence of response is not penalized.

In questions 1 and 2, the space is equipped with an orthonormal coordinate system, and we consider the planes $\mathscr{P}_{1}$ and $\mathscr{P}_{2}$ with equations $x + y + z - 5 = 0$ and $7x - 2y + z - 2 = 0$ respectively.

\begin{enumerate}
  \item Statement 1: the planes $\mathscr{P}_{1}$ and $\mathscr{P}_{2}$ are perpendicular.
  \item Statement 2: the planes $\mathscr{P}_{1}$ and $\mathscr{P}_{2}$ intersect along the line with parametric representation:
$$\left\{ \begin{aligned}
x & = t \\
y & = 2t + 1, \quad t \in \mathbb{R} \\
z & = -3t + 4
\end{aligned} \right.$$
  \item A video game player always adopts the same strategy. Out of the first 312 games played, he wins 223. The games played are treated as a random sample of size 312 from the set of all games.\\
It is desired to estimate the proportion of games that the player will win in the next games he plays, while maintaining the same strategy.\\
Statement 3: at the 95\% confidence level, the proportion of games won should belong to the interval $[0.658; 0.771]$.
  \item Consider the following algorithm:
\end{enumerate}

\begin{center}
\begin{tabular}{|l|l|}
\hline
VARIABLES & \begin{tabular}{l}
$a, b$ are two real numbers such that $a < b$ \\
$x$ is a real number \\
$f$ is a function defined on the interval $[a; b]$ \\
\end{tabular} \\
\hline
PROCESSING & \begin{tabular}{l}
Read $a$ and $b$ \\
While $b - a > 0.3$ \\
$x$ takes the value $\frac{a + b}{2}$ \\
If $f(x)f(a) > 0$, then $a$ takes the value $x$ otherwise $b$ takes the value $x$ \\
End If \\
End While \\
Display $\frac{a + b}{2}$ \\
\end{tabular} \\
\hline
\end{tabular}
\end{center}

Statement 4: if we enter $a = 1, b = 2$ and $f(x) = x^{2} - 3$, then the algorithm displays as output the number 1.6875.