For each of the following questions, only one of the four statements is correct. Indicate on your answer sheet the question number and copy the letter corresponding to the correct statement. One point is awarded if the letter corresponds to the correct statement, 0 otherwise. Throughout the exercise, we work in an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$ in space. The four questions are independent. No justification is required.
Consider the plane $P$ with Cartesian equation $3x + 2y + 9z - 5 = 0$ and the line $d$ with parametric representation: $\left\{\begin{array}{l} x = 4t + 3 \\ y = -t + 2 \\ z = -t + 9 \end{array}, t \in \mathbb{R}\right.$. Statement A: the intersection of plane $P$ and line $d$ is reduced to the point with coordinates $(3;2;9)$. Statement B: plane $P$ and line $d$ are orthogonal. Statement C: plane $P$ and line $d$ are parallel. Statement D: the intersection of plane $P$ and line $d$ is reduced to the point with coordinates $(-353; 91; 98)$.
Consider the cube ABCDEFGH and the points I, J and K defined by the vector equalities: $$\overrightarrow{\mathrm{AI}} = \frac{3}{4}\overrightarrow{\mathrm{AB}}, \quad \overrightarrow{\mathrm{DJ}} = \frac{1}{4}\overrightarrow{\mathrm{DC}}, \quad \overrightarrow{\mathrm{HK}} = \frac{3}{4}\overrightarrow{\mathrm{HG}}$$ Statement A: the cross-section of cube ABCDEFGH by plane (IJK) is a triangle. Statement B: the cross-section of cube ABCDEFGH by plane (IJK) is a quadrilateral. Statement C: the cross-section of cube ABCDEFGH by plane (IJK) is a pentagon. Statement D: the cross-section of cube ABCDEFGH by plane (IJK) is a hexagon.
Consider the line $d$ with parametric representation $\left\{\begin{aligned} x &= t + 2 \\ y &= 2 \\ z &= 5t - 6 \end{aligned}\right.$, with $t \in \mathbb{R}$, and the point $\mathrm{A}(-2; 1; 0)$. Let $M$ be a variable point on line $d$. Statement A: the smallest length $AM$ is equal to $\sqrt{53}$. Statement B: the smallest length $AM$ is equal to $\sqrt{27}$. Statement C: the smallest length $AM$ is attained when point $M$ has coordinates $(-2; 1; 0)$. Statement D: the smallest length $AM$ is attained when point $M$ has coordinates $(2; 2; -6)$.
Consider the plane $P$ with Cartesian equation $x + 2y - 3z + 1 = 0$ and the plane $P'$ with Cartesian equation $2x - y + 2 = 0$. Statement A: planes $P$ and $P'$ are parallel. Statement B: the intersection of planes $P$ and $P'$ is a line passing through points $\mathrm{A}(5; 12; 10)$ and $\mathrm{B}(3; 1; 2)$. Statement C: the intersection of planes $P$ and $P'$ is a line passing through point $\mathrm{C}(2; 6; 5)$ and having a direction vector $\vec{u}(1; 2; 2)$. Statement D: the intersection of planes $P$ and $P'$ is a line passing through point $\mathrm{D}(-1; 0; 0)$ and having a direction vector $\vec{v}(3; 6; 5)$.
For each of the following questions, only one of the four statements is correct. Indicate on your answer sheet the question number and copy the letter corresponding to the correct statement. One point is awarded if the letter corresponds to the correct statement, 0 otherwise.
Throughout the exercise, we work in an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$ in space. The four questions are independent. No justification is required.
\begin{enumerate}
\item Consider the plane $P$ with Cartesian equation $3x + 2y + 9z - 5 = 0$ and the line $d$ with parametric representation: $\left\{\begin{array}{l} x = 4t + 3 \\ y = -t + 2 \\ z = -t + 9 \end{array}, t \in \mathbb{R}\right.$.\\
Statement A: the intersection of plane $P$ and line $d$ is reduced to the point with coordinates $(3;2;9)$.\\
Statement B: plane $P$ and line $d$ are orthogonal.\\
Statement C: plane $P$ and line $d$ are parallel.\\
Statement D: the intersection of plane $P$ and line $d$ is reduced to the point with coordinates $(-353; 91; 98)$.
\item Consider the cube ABCDEFGH and the points I, J and K defined by the vector equalities:
$$\overrightarrow{\mathrm{AI}} = \frac{3}{4}\overrightarrow{\mathrm{AB}}, \quad \overrightarrow{\mathrm{DJ}} = \frac{1}{4}\overrightarrow{\mathrm{DC}}, \quad \overrightarrow{\mathrm{HK}} = \frac{3}{4}\overrightarrow{\mathrm{HG}}$$
Statement A: the cross-section of cube ABCDEFGH by plane (IJK) is a triangle.\\
Statement B: the cross-section of cube ABCDEFGH by plane (IJK) is a quadrilateral.\\
Statement C: the cross-section of cube ABCDEFGH by plane (IJK) is a pentagon.\\
Statement D: the cross-section of cube ABCDEFGH by plane (IJK) is a hexagon.
\item Consider the line $d$ with parametric representation $\left\{\begin{aligned} x &= t + 2 \\ y &= 2 \\ z &= 5t - 6 \end{aligned}\right.$, with $t \in \mathbb{R}$, and the point $\mathrm{A}(-2; 1; 0)$. Let $M$ be a variable point on line $d$.\\
Statement A: the smallest length $AM$ is equal to $\sqrt{53}$.\\
Statement B: the smallest length $AM$ is equal to $\sqrt{27}$.\\
Statement C: the smallest length $AM$ is attained when point $M$ has coordinates $(-2; 1; 0)$.\\
Statement D: the smallest length $AM$ is attained when point $M$ has coordinates $(2; 2; -6)$.
\item Consider the plane $P$ with Cartesian equation $x + 2y - 3z + 1 = 0$ and the plane $P'$ with Cartesian equation $2x - y + 2 = 0$.\\
Statement A: planes $P$ and $P'$ are parallel.\\
Statement B: the intersection of planes $P$ and $P'$ is a line passing through points $\mathrm{A}(5; 12; 10)$ and $\mathrm{B}(3; 1; 2)$.\\
Statement C: the intersection of planes $P$ and $P'$ is a line passing through point $\mathrm{C}(2; 6; 5)$ and having a direction vector $\vec{u}(1; 2; 2)$.\\
Statement D: the intersection of planes $P$ and $P'$ is a line passing through point $\mathrm{D}(-1; 0; 0)$ and having a direction vector $\vec{v}(3; 6; 5)$.
\end{enumerate}