This exercise is a multiple choice questionnaire. No justification is required. For each question, only one of the four propositions is correct. Each correct answer is worth 1 point. An incorrect answer or no answer does not deduct any points.
Let $z _ { 1 } = \sqrt { 6 } \mathrm { e } ^ { \mathrm { i } \frac { \pi } { 4 } }$ and $z _ { 2 } = \sqrt { 2 } \mathrm { e } ^ { - \mathrm { i } \frac { \pi } { 3 } }$. The exponential form of $\mathrm { i } \frac { z _ { 1 } } { z _ { 2 } }$ is: a. $\sqrt { 3 } \mathrm { e } ^ { \mathrm { i } \frac { 19 \pi } { 12 } }$ b. $\sqrt { 12 } \mathrm { e } ^ { - \mathrm { i } \frac { \pi } { 12 } }$ c. $\sqrt { 3 } \mathrm { e } ^ { \mathrm { i } \frac { 7 \pi } { 12 } }$ d. $\sqrt { 3 } \mathrm { e } ^ { \mathrm { i } \frac { 13 \pi } { 12 } }$
The equation $- z = \bar { z }$, with unknown complex number $z$, admits: a. one solution b. two solutions c. infinitely many solutions whose image points in the complex plane are located on a line. d. infinitely many solutions whose image points in the complex plane are located on a circle.
In a coordinate system of space, consider the three points $A ( 1 ; 2 ; 3 ) , B ( - 1 ; 5 ; 4 )$ and $C ( - 1 ; 0 ; 4 )$. The line parallel to the line $( A B )$ passing through point $C$ has the parametric representation: a. $\left\{ \begin{array} { l } x = - 2 t - 1 \\ y = 3 t \\ z = t + 4 \end{array} , t \in \mathbb { R } \right.$ b. $\left\{ \begin{array} { l } x = - 1 \\ y = 7 t \\ z = 7 t + 4 \end{array} , t \in \mathbb { R } \right.$ c. $\left\{ \begin{array} { l } x = - 1 - 2 t \\ y = 5 + 3 t \\ z = 4 + t \end{array} , t \in \mathbb { R } \right.$ d. $\left\{ \begin{array} { l } x = 2 t \\ y = - 3 t \\ z = - t \end{array} , t \in \mathbb { R } \right.$
In an orthonormal coordinate system of space, consider the plane $\mathscr { P }$ passing through point $D ( - 1 ; 2 ; 3 )$ and with normal vector $\vec { n } ( 3 ; - 5 ; 1 )$, and the line $\Delta$ with parametric representation $\left\{ \begin{array} { l } x = t - 7 \\ y = t + 3 \\ z = 2 t + 5 \end{array} , t \in \mathbb { R } \right.$. a. The line $\Delta$ is perpendicular to the plane $\mathscr { P }$. b. The line $\Delta$ is parallel to the plane $\mathscr { P }$ and has no common point with the plane $\mathscr { P }$. c. The line $\Delta$ and the plane $\mathscr { P }$ are secant. d. The line $\Delta$ is contained in the plane $\mathscr { P }$.
This exercise is a multiple choice questionnaire. No justification is required. For each question, only one of the four propositions is correct. Each correct answer is worth 1 point. An incorrect answer or no answer does not deduct any points.
\begin{enumerate}
\item Let $z _ { 1 } = \sqrt { 6 } \mathrm { e } ^ { \mathrm { i } \frac { \pi } { 4 } }$ and $z _ { 2 } = \sqrt { 2 } \mathrm { e } ^ { - \mathrm { i } \frac { \pi } { 3 } }$. The exponential form of $\mathrm { i } \frac { z _ { 1 } } { z _ { 2 } }$ is:\\
a. $\sqrt { 3 } \mathrm { e } ^ { \mathrm { i } \frac { 19 \pi } { 12 } }$\\
b. $\sqrt { 12 } \mathrm { e } ^ { - \mathrm { i } \frac { \pi } { 12 } }$\\
c. $\sqrt { 3 } \mathrm { e } ^ { \mathrm { i } \frac { 7 \pi } { 12 } }$\\
d. $\sqrt { 3 } \mathrm { e } ^ { \mathrm { i } \frac { 13 \pi } { 12 } }$
\item The equation $- z = \bar { z }$, with unknown complex number $z$, admits:\\
a. one solution\\
b. two solutions\\
c. infinitely many solutions whose image points in the complex plane are located on a line.\\
d. infinitely many solutions whose image points in the complex plane are located on a circle.
\item In a coordinate system of space, consider the three points $A ( 1 ; 2 ; 3 ) , B ( - 1 ; 5 ; 4 )$ and $C ( - 1 ; 0 ; 4 )$. The line parallel to the line $( A B )$ passing through point $C$ has the parametric representation:\\
a. $\left\{ \begin{array} { l } x = - 2 t - 1 \\ y = 3 t \\ z = t + 4 \end{array} , t \in \mathbb { R } \right.$\\
b. $\left\{ \begin{array} { l } x = - 1 \\ y = 7 t \\ z = 7 t + 4 \end{array} , t \in \mathbb { R } \right.$\\
c. $\left\{ \begin{array} { l } x = - 1 - 2 t \\ y = 5 + 3 t \\ z = 4 + t \end{array} , t \in \mathbb { R } \right.$\\
d. $\left\{ \begin{array} { l } x = 2 t \\ y = - 3 t \\ z = - t \end{array} , t \in \mathbb { R } \right.$
\item In an orthonormal coordinate system of space, consider the plane $\mathscr { P }$ passing through point $D ( - 1 ; 2 ; 3 )$ and with normal vector $\vec { n } ( 3 ; - 5 ; 1 )$, and the line $\Delta$ with parametric representation $\left\{ \begin{array} { l } x = t - 7 \\ y = t + 3 \\ z = 2 t + 5 \end{array} , t \in \mathbb { R } \right.$.\\
a. The line $\Delta$ is perpendicular to the plane $\mathscr { P }$.\\
b. The line $\Delta$ is parallel to the plane $\mathscr { P }$ and has no common point with the plane $\mathscr { P }$.\\
c. The line $\Delta$ and the plane $\mathscr { P }$ are secant.\\
d. The line $\Delta$ is contained in the plane $\mathscr { P }$.
\end{enumerate}