In space referred to an orthonormal coordinate system, consider the lines $\mathscr{D}_1$ and $\mathscr{D}_2$ which have the following parametric representations respectively: $$\left\{ \begin{array}{l} x = 1 + 2t \\ y = 2 - 3t \\ z = 4t \end{array} \right., t \in \mathbb{R} \quad \text{and} \quad \left\{ \begin{array}{l} x = -5t' + 3 \\ y = 2t' \\ z = t' + 4 \end{array} \right., t' \in \mathbb{R}$$
Statement 3: The lines $\mathscr{D}_1$ and $\mathscr{D}_2$ are secant.
Indicate whether this statement is true or false, justifying your answer.

A passage of an aerial acrobatics show in a duo is modelled as follows:

• we place ourselves in an orthonormal coordinate system $(O ; \vec { \imath } , \vec { \jmath } , \vec { k })$, where one unit represents one metre;
• plane no. 1 must travel from point O to point $A(0 ; 200 ; 0)$ along a straight trajectory, at the constant speed of $200 \mathrm {~m/s}$;
• plane no. 2 must travel from point $B(-33 ; 75 ; 44)$ to point $C(87 ; 75 ; -116)$ also along a straight trajectory, and at the constant speed of $200 \mathrm {~m/s}$;
• at the same instant, plane no. 1 is at point O and plane no. 2 is at point B.

1. Justify that plane no. 2 will take the same time to travel segment $[BC]$ as plane no. 1 to travel segment $[OA]$.
2. Show that the trajectories of the two planes intersect.
3. Is there a risk of collision between the two planes during this passage?

In coordinate space, when the line passing through two points $\mathrm { A } ( 5,5 , a ) , \mathrm { B } ( 0,0,3 )$ is perpendicular to the line $x = 4 - y = z - 1$, what is the value of $a$? [3 points]
(1) 3
(2) 5
(3) 7
(4) 9
(5) 11

8. As shown in the figure, point $N$ is the center of square $A B C D$ , $\triangle E C D$ is an equilateral triangle, plane $E C D \perp$ plane $A B C D$ , and $M$ is the midpoint of segment $E D$ . Then [Figure]
A. $B M = E N$ , and lines $B M$ and $E N$ are intersecting lines
B. $B M \neq E N$ , and lines $B M$ and $E N$ are intersecting lines
C. $B M = E N$ , and lines $B M$ and $E N$ are skew lines
D. $B M \neq E N$ , and lines $B M$ and $E N$ are skew lines

8. As shown in the figure, point $N$ is the center of square $A B C D$ , $\triangle E C D$ is an equilateral triangle, plane $E C D \perp$ plane $A B C D$ , and $M$ is the midpoint of segment $E D$ . Then [Figure]
A. $B M = E N$ , and lines $B M$ and $E N$ are intersecting lines
B. $B M \neq E N$ , and lines $B M$ and $E N$ are intersecting lines
C. $B M = E N$ , and lines $B M$ and $E N$ are skew lines
D. $B M \neq E N$ , and lines $B M$ and $E N$ are skew lines

Two lines $L _ { 1 } : x = 5 , \frac { y } { 3 - \alpha } = \frac { z } { - 2 }$ and $L _ { 2 } : x = \alpha , \frac { y } { - 1 } = \frac { z } { 2 - \alpha }$ are coplanar. Then $\alpha$ can take value(s)
(A) 1
(B) 2
(C) 3
(D) 4

Let $L$ be the line of intersection of the planes $2 x + 3 y + z = 1$ and $x + 3 y + 2 z = 2$. If $L$ makes an angles $\alpha$ with the positive $x$-axis, then $\cos \alpha$ equals
(1) $\frac { 1 } { \sqrt { 3 } }$
(2) $\frac { 1 } { 2 }$
(3) 1
(4) $\frac { 1 } { \sqrt { 2 } }$

If a line makes an angle of $\frac { \pi } { 4 }$ with the positive directions of each of $x$-axis and $y$-axis, then the angle that the line makes with the positive direction of the $z$-axis is
(1) $\frac { \pi } { 6 }$
(2) $\frac { \pi } { 3 }$
(3) $\frac { \pi } { 4 }$
(4) $\frac { \pi } { 2 }$

If the lines $\frac{x-1}{2} = \frac{y+1}{3} = \frac{z-1}{4}$ and $\frac{x-3}{1} = \frac{y-k}{2} = \frac{z}{1}$ intersect, then $k$ is equal to
(1) $\frac{2}{9}$
(2) $\frac{9}{2}$
(3) 0
(4) $-1$

If the lines $\frac{x-2}{1} = \frac{y-3}{1} = \frac{z-4}{-k}$ and $\frac{x-1}{k} = \frac{y-4}{2} = \frac{z-5}{1}$ are coplanar, then $k$ can be
(1) $-1$ or $-3$
(2) $-1$ or $3$
(3) $1$ or $-1$
(4) $0$ or $-3$

If two lines $L _ { 1 }$ and $L _ { 2 }$ in space, are defined by
$$\begin{gathered} L _ { 1 } = \{ x = \sqrt { \lambda } y + ( \sqrt { \lambda } - 1 ) , \\ z = ( \sqrt { \lambda } - 1 ) y + \sqrt { \lambda } \} \text { and } \\ L _ { 2 } = \{ x = \sqrt { \mu } y + ( 1 - \sqrt { \mu } ) , \\ z = ( 1 - \sqrt { \mu } ) y + \sqrt { \mu } \} \end{gathered}$$
then $L _ { 1 }$ is perpendicular to $L _ { 2 }$, for all nonnegative reals $\lambda$ and $\mu$, such that :
(1) $\sqrt { \lambda } + \sqrt { \mu } = 1$
(2) $\lambda \neq \mu$
(3) $\lambda + \mu = 0$
(4) $\lambda = \mu$

If the lines $\frac{x-2}{1} = \frac{y-3}{1} = \frac{z-4}{-k}$ and $\frac{x-1}{k} = \frac{y-4}{2} = \frac{z-5}{1}$ are coplanar, then $k$ can have
(1) exactly two values.
(2) exactly three values.
(3) any value.
(4) exactly one value.

The lines $\vec { r } = ( \hat { i } - \hat { j } ) + l ( 2 \hat { i } + \widehat { k } )$ and $\vec { r } = ( 2 \hat { i } - \hat { j } ) + m ( \hat { i } + \hat { j } - \widehat { k } )$
(1) Do not intersect for any values of $l$ and $m$
(2) Intersect for all values of $l$ and $m$
(3) Intersect when $l = 2$ and $m = \frac { 1 } { 2 }$
(4) Intersect when $l = 1$ and $m = 2$

The angle between the straight lines, whose direction cosines $l , m , n$ are given by the equations $2 l + 2 m - n = 0$ and $m n + n l + \operatorname { lm } = 0$, is: (1) $\frac { \pi } { 3 }$ (2) $\frac { \pi } { 2 }$ (3) $\cos ^ { - 1 } \left( \frac { 8 } { 9 } \right)$ (4) $\pi - \cos ^ { - 1 } \left( \frac { 4 } { 9 } \right)$

If the lines $\vec { r } = ( \hat { i } - \hat { j } + \widehat { k } ) + \lambda ( 3 \hat { j } - \widehat { k } )$ and $\vec { r } = ( \alpha \hat { i } - \hat { j } ) + \mu ( 2 \hat { i } - 3 \widehat { k } )$ are co-planar, then the distance of the plane containing these two lines from the point $( \alpha , 0,0 )$ is
(1) $\frac { 2 } { 9 }$
(2) $\frac { 2 } { 11 }$
(3) $\frac { 4 } { 11 }$
(4) 2

If the two lines $l _ { 1 } : \frac { x - 2 } { 3 } = \frac { y + 1 } { - 2 } , z = 2$ and $l _ { 2 } : \frac { x - 1 } { 1 } = \frac { 2 y + 3 } { \alpha } = \frac { z + 5 } { 2 }$ are perpendicular, then an angle between the lines $l _ { 2 }$ and $l _ { 3 } : \frac { 1 - x } { 3 } = \frac { 2 y - 1 } { - 4 } = \frac { z } { 4 }$ is
(1) $\cos ^ { - 1 } \left( \frac { 29 } { 4 } \right)$
(2) $\sec ^ { - 1 } \left( \frac { 29 } { 4 } \right)$
(3) $\cos ^ { - 1 } \left( \frac { 2 } { 29 } \right)$
(4) $\cos ^ { - 1 } \left( \frac { 2 } { \sqrt { 29 } } \right)$