In space, with respect to an orthonormal coordinate system, we consider the points $\mathrm { A } ( 1 ; - 1 ; - 1 )$, $\mathrm { B } ( 1 ; 1 ; 1 ) , \mathrm { C } ( 0 ; 3 ; 1 )$ and the plane $\mathscr { P }$ with equation $2 x + y - z + 5 = 0$.
Let $\mathscr { D } _ { 1 }$ be the line with direction vector $\vec { u } ( 2 ; - 1 ; 1 )$ passing through A.
A parametric representation of the line $\mathscr { D } _ { 1 }$ is : a. $\left\{ \begin{array} { l } x = 2 + t \\ y = - 1 - t \\ z = 1 - t \end{array} \quad ( t \in \mathbb { R } ) \right.$ b. $\left\{ \begin{array} { l } x = - 1 + 2 t \\ y = 1 - t \\ z = 1 + t \end{array} \quad ( t \in \mathbb { R } ) \right.$ c. $\left\{ \begin{array} { l } x = 5 + 4 t \\ y = - 3 - 2 t \\ z = 1 + 2 t \end{array} \quad ( t \in \mathbb { R } ) \right.$ d. $\left\{ \begin{array} { l } x = 4 - 2 t \\ y = - 2 + t \\ z = 3 - 4 t \end{array} \quad ( t \in \mathbb { R } ) \right.$

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, multiple answers, or the absence of an answer to a question earns neither points nor deducts points. The four questions are independent.
Space is referred to an orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath}, \vec{k})$.

1. Consider the points $A(1; 0; 3)$ and $B(4; 1; 0)$.
   A parametric representation of the line (AB) is: a. $\left\{ \begin{aligned} x & = 3 + t \\ y & = 1 \\ z & = -3 + 3t \end{aligned} \right.$ with $t \in \mathbb{R}$ b. $\left\{ \begin{array}{l} x = 1 + 4t \\ y = 3 \\ z = 3 \end{array} \right.$ with $t \in \mathbb{R}$ c. $\left\{ \begin{aligned} x & = 1 + 3t \\ y & = t \\ z & = 3 - 3t \end{aligned} \right.$ with $t \in \mathbb{R}$ d. $\left\{ \begin{aligned} x & = 4 + t \\ y & = 1 \\ z & = 3 - 3t \end{aligned} \right.$ with $t \in \mathbb{R}$
   
2. Consider the line (d) with parametric representation $\left\{ \begin{aligned} x & = 3 + 4t \\ y & = 6t \\ z & = 4 - 2t \end{aligned} \right.$ with $t \in \mathbb{R}$
   Among the following points, which one belongs to the line (d)? a. $M(7; 6; 6)$ b. $N(3; 6; 4)$ c. $P(4; 6; -2)$ d. $R(-3; -9; 7)$
   
3. Consider the line $(d')$ with parametric representation $\left\{ \begin{aligned} x & = -2 + 3k \\ y & = -1 - 2k \\ z & = 1 + k \end{aligned} \right.$ with $k \in \mathbb{R}$
   The lines $(d)$ and $(d')$ are: a. secant b. non-coplanar c. parallel d. coincident
   
4. Consider the plane $(P)$ passing through the point $I(2; 1; 0)$ and perpendicular to the line (d).
   An equation of the plane $(P)$ is: a. $2x + 3y - z - 7 = 0$ b. $-x + y - 4z + 1 = 0$ c. $4x + 6y - 2z + 9 = 0$ d. $2x + y + 1 = 0$

16. The parametric equation of line $l$ is $\left\{ \begin{array} { l } x = 1 + 2 t \\ y = 2 - t \end{array} ( t \in \mathbb{R} ) \right.$. Then a direction vector $\overrightarrow { Therefore $\frac { ( 1 - \cos \theta ) ^ { 2 } } { 4 } + \frac { ( 1 + \sin \theta ) ^ { 2 } } { 4 } < 1$ , simplifying gives $\sin \theta - \cos \theta < \frac { 1 } { 2 } , \sin \left( \theta - \frac { \pi } { 4 } \right) < \frac { \sqrt { 2 } } { 4 }$ , Also $0 < \theta < \pi$ , that is $- \frac { \pi } { 4 } < \theta - \frac { \pi } { 4 } < \frac { 3 \pi } { 4 }$ , so $- \frac { \pi } { 4 } < \theta - \frac { \pi } { 4 } < \arcsin \frac { \sqrt { 2 } } { 4 }$ , Thus the range of values for $\theta$ is $\left( 0 , \frac { \pi } { 4 } + \arcsin \frac { \sqrt { 2 } } { 4 } \right)$ .

Let $L _ { 1 }$ and $L _ { 2 }$ denote the lines $$\begin{aligned} & \vec { r } = \hat { i } + \lambda ( - \hat { i } + 2 \hat { j } + 2 \hat { k } ) , \lambda \in \mathbb { R } \text { and } \\ & \vec { r } = \mu ( 2 \hat { i } - \hat { j } + 2 \hat { k } ) , \mu \in \mathbb { R } \end{aligned}$$ respectively. If $L _ { 3 }$ is a line which is perpendicular to both $L _ { 1 }$ and $L _ { 2 }$ and cuts both of them, then which of the following options describe(s) $L _ { 3 }$?
(A) $\vec { r } = \frac { 2 } { 9 } ( 4 \hat { i } + \hat { j } + \hat { k } ) + t ( 2 \hat { i } + 2 \hat { j } - \hat { k } ) , t \in \mathbb { R }$
(B) $\vec { r } = \frac { 2 } { 9 } ( 2 \hat { i } - \hat { j } + 2 \hat { k } ) + t ( 2 \hat { i } + 2 \hat { j } - \hat { k } ) , t \in \mathbb { R }$
(C) $\vec { r } = \frac { 1 } { 3 } ( 2 \hat { i } + \hat { k } ) + t ( 2 \hat { i } + 2 \hat { j } - \hat { k } ) , t \in \mathbb { R }$
(D) $\vec { r } = t ( 2 \hat { i } + 2 \hat { j } - \hat { k } ) , t \in \mathbb { R }$

Let $L _ { 1 }$ and $L _ { 2 }$ be the following straight lines.
$$L _ { 1 } : \frac { x - 1 } { 1 } = \frac { y } { - 1 } = \frac { z - 1 } { 3 } \text { and } L _ { 2 } : \frac { x - 1 } { - 3 } = \frac { y } { - 1 } = \frac { z - 1 } { 1 } .$$
Suppose the straight line
$$L : \frac { x - \alpha } { l } = \frac { y - 1 } { m } = \frac { z - \gamma } { - 2 }$$
lies in the plane containing $L _ { 1 }$ and $L _ { 2 }$, and passes through the point of intersection of $L _ { 1 }$ and $L _ { 2 }$. If the line $L$ bisects the acute angle between the lines $L _ { 1 }$ and $L _ { 2 }$, then which of the following statements is/are TRUE?
(A) $\alpha - \gamma = 3$
(B) $l + m = 2$
(C) $\alpha - \gamma = 1$
(D) $l + m = 0$

The equation of the line through the point $( 0,1,2 )$ and perpendicular to the line $\frac { x - 1 } { 2 } = \frac { y + 1 } { 3 } = \frac { z - 1 } { - 2 }$ is :
(1) $\frac { x } { 3 } = \frac { y - 1 } { - 4 } = \frac { z - 2 } { 3 }$
(2) $\frac { x } { 3 } = \frac { y - 1 } { 4 } = \frac { z - 2 } { 3 }$
(3) $\frac { x } { - 3 } = \frac { y - 1 } { 4 } = \frac { z - 2 } { 3 }$
(4) $\frac { x } { 3 } = \frac { y - 1 } { 4 } = \frac { z - 2 } { - 3 }$

In coordinate space, $O$ is the origin, and point $P$ is in the first octant with $\overline{OP} = 1$. The line $OP$ makes an angle of $45^\circ$ with the $x$-axis, and the distance from point $P$ to the $y$-axis is $\frac{\sqrt{6}}{3}$. Select the $z$-coordinate of point $P$.
(1) $\frac{1}{2}$
(2) $\frac{\sqrt{2}}{4}$
(3) $\frac{\sqrt{3}}{3}$
(4) $\frac{\sqrt{6}}{6}$
(5) $\frac{\sqrt{3}}{6}$

Problem 4

For the real numbers $\theta$ and $\alpha$ within the regions $0 \leq \theta < 2 \pi$ and $0 \leq \alpha \leq \pi$, consider the line L that passes through two points: point $\mathrm { P } ( \cos \theta$, $\sin \theta , 1 )$ and point $\mathrm { Q } ( \cos ( \theta + \alpha ) , \sin ( \theta + \alpha ) , - 1 )$ in a three-dimensional Cartesian coordinate system $x y z$. I. Represent the line L as a linear function of a parameter $t$. Here, the point on the line L at $t = 0$ should represent the point Q and the point at $t = 1$ should represent the point P. II. Find the surface S swept by the line L as an equation of $x , y$ and $z$ when $\theta$ varies in the region $0 \leq \theta < 2 \pi$. Let C be the intersection lines of the surface S with the plane $y = 0$. Find the equation of C in terms of $x$ and $z$, and sketch the shape of C.
Next, examine the Gaussian curvature of the surface S. Generally, when the position vector $r$ of a point R on a curved surface is represented using parameters $u$ and $v$ by
$$\boldsymbol { r } ( u , v ) = ( x ( u , v ) , y ( u , v ) , z ( u , v ) ) ,$$
the Gaussian curvature $K$ is represented as the following equation:
$$K = \frac { \left( \boldsymbol { r } _ { u u } \cdot \boldsymbol{e} \right) \left( \boldsymbol { r } _ { v v } \cdot \boldsymbol { e } \right) - \left( \boldsymbol { r } _ { u v } \cdot \boldsymbol { e } \right) ^ { 2 } } { \left( \boldsymbol { r } _ { u } \cdot \boldsymbol { r } _ { u } \right) \left( \boldsymbol { r } _ { v } \cdot \boldsymbol { r } _ { v } \right) - \left( \boldsymbol { r } _ { u } \cdot \boldsymbol { r } _ { v } \right) ^ { 2 } } ,$$
where $\boldsymbol { r } _ { u }$ and $\boldsymbol { r } _ { v }$ are first-order partial differentials of $\boldsymbol { r } ( u , v )$ with respect to the parameters $u$ and $v$, and $\boldsymbol { r } _ { u u } , \boldsymbol { r } _ { u v }$ and $\boldsymbol { r } _ { v v }$ are second-order partial differentials of $\boldsymbol { r } ( u , v )$ with respect to the parameters $u$ and $v$. $( \boldsymbol { a } \cdot \boldsymbol { b } )$ represents the inner product of two three-dimensional vectors $a$ and $b$, and $e$ is the unit vector of the normal direction at the point R. III. Let the point W be the intersection of the surface S and the $x$ axis in the region $x > 0$. Calculate the Gaussian curvature of S at the point W for $\alpha$ within the region $0 \leq \alpha < \pi$. IV. For $\alpha$ within the region $0 \leq \alpha < \pi$, prove that the Gaussian curvature is less than or equal to 0 at arbitrary points on the surface $S$.