This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. To answer, indicate on your paper the question number and the letter of the chosen answer. No justification is required.
A wrong answer, no answer, or multiple answers, neither earn nor lose points.
- We consider the function $f$ defined on $\mathbb{R}$ by: $f(x) = (x + 1)e^x$.
An antiderivative $F$ of $f$ on $\mathbb{R}$ is defined by: a. $F(x) = 1 + xe^x$ b. $F(x) = (1 + x)e^x$ c. $F(x) = (2 + x)e^x$ d. $F(x) = \left(\frac{x^2}{2} + x\right)e^x$.
- We consider the lines $(d_1)$ and $(d_2)$ whose parametric representations are respectively: $$\left(d_1\right) \left\{\begin{array}{l}
x = 2 + r \\
y = 1 + r \\
z = -r
\end{array} \quad (r \in \mathbb{R}) \quad \text{and} \quad (d_2) \left\{\begin{array}{rl}
x & = 1 - s \\
y & = -1 + s \\
z & = 2 - s
\end{array} \quad (s \in \mathbb{R})\right.\right.$$ The lines $(d_1)$ and $(d_2)$ are: a. secant. b. strictly parallel. c. coincident. d. non-coplanar.
- We consider the plane $(P)$ whose Cartesian equation is: $$2x - y + z - 1 = 0$$ We consider the line $(\Delta)$ whose parametric representation is: $$\left\{\begin{array}{l}
x = 2 + u \\
y = 4 + u \quad (u \in \mathbb{R}) \\
z = 1 - u
\end{array}\right.$$ The line $(\Delta)$ is: a. secant and non-orthogonal to the plane $(P)$. b. included in the plane $(P)$. c. strictly parallel to the plane $(P)$. d. orthogonal to the plane $(P)$.
- We consider the plane $(P_1)$ whose Cartesian equation is $x - 2y + z + 1 = 0$, as well as the plane $(P_2)$ whose Cartesian equation is $2x + y + z - 6 = 0$. The planes $(P_1)$ and $(P_2)$ are: a. secant and perpendicular. b. coincident. c. secant and non-perpendicular. d. strictly parallel.
- We consider the points $E(1; 2; 1)$, $F(2; 4; 3)$ and $G(-2; 2; 5)$.
We can affirm that the measure $\alpha$ of the angle $\widehat{FEG}$ satisfies: a. $\alpha = 90°$ b. $\alpha > 90°$ c. $\alpha = 0°$ d. $\alpha \approx 71°$