Consider the line $L: 5y + (2k-4)x - 10k = 0$ on the coordinate plane (where $k$ is a real number), and the rectangle $OABC$ with vertices at $O(0,0)$, $A(10,0)$, $B(10,6)$, $C(0,6)$. Let $L$ intersect the line $OC$ and the line $AB$ at points $D$ and $E$ respectively. Select the correct options.
(1) When $k = 4$, the line $L$ passes through point $A$
(2) If the line $L$ passes through point $C$, then the slope of $L$ is $-\frac{5}{2}$
(3) If point $D$ is on the line segment $\overline{OC}$, then $0 \leq k \leq 3$
(4) If $k = \frac{1}{2}$, then the line segment $\overline{DE}$ is inside the rectangle $OABC$ (including the boundary)
(5) If the line segment $\overline{DE}$ is inside the rectangle $OABC$ (including the boundary), then the slope of $L$ could be $\frac{3}{10}$

In the Cartesian coordinate plane, the perpendicular drawn from point $A ( 1,0 )$ to the line $\mathbf { y } + \mathbf { 2 x } - \mathbf { 1 } = \mathbf { 0 }$ intersects the Y-axis at which point?
A) $\frac { - 1 } { 2 }$
B) $\frac { - 1 } { 3 }$
C) $\frac { - 1 } { 4 }$
D) $\frac { - 1 } { 5 }$
E) $\frac { - 1 } { 6 }$

In the rectangular coordinate plane, the line $y = \frac { x } { 7 }$ intersects the lines $x = 2$ and $x = 9$ at points $P$ and $R$ respectively.
Accordingly, what is the length $| \mathrm { PR } |$ in units?
A) $5 \sqrt { 2 }$
B) $6 \sqrt { 2 }$
C) $4 \sqrt { 10 }$
D) 8
E) 9

The square shown in the figure in the Cartesian coordinate plane is divided into two regions of equal area by a line with slope $\frac { - 1 } { 4 }$.
If this line intersects the x-axis at point $(a, 0)$, what is a?
A) 12 B) 14 C) 16 D) 18 E) 20

Emre marks a point on the x-axis of the Cartesian coordinate plane in a mathematics class activity. Then, by decreasing the x-coordinate of this marked point by 1 unit and increasing the y-coordinate by 3 units, he obtains a second point, and when he applies the same operation to the second point, he obtains a third point on the y-axis.
What is the sum of the coordinates of the fourth point that Emre will obtain by applying the same operation to the third point?
A) 4
B) 5
C) 6
D) 7
E) 8

In the rectangular coordinate plane, it is known that a line $d$ passes through point $A(-4, 1)$ and is perpendicular to the line $2x - y = 5$. If the point where line $d$ intersects the x-axis is $(a, 0)$ and the point where it intersects the y-axis is $(0, b)$, what is the sum $a + b$?
A) -3
B) -1
C) 0
D) 1
E) 3

In the rectangular coordinate plane, points A and B lie on the line $y = x + 2$, and the distance between them is 3 units.
Given that the coordinates of the midpoint of segment [AB] are $( -1, 1 )$, in which regions of the analytic plane are points A and B located?
A) Both in region II
B) Both in region III
C) One in region I, the other in region II
D) One in region I, the other in region III
E) One in region II, the other in region III

In the rectangular coordinate plane, when point $A$ is translated 15 units in the negative direction along the $x$-axis, the resulting point lies on the line $d: 4x - 3y + 24 = 0$.
Accordingly, if point $A$ is translated how many units in the positive direction along the $y$-axis, the resulting point will lie on line $d$?
A) 9 B) 12 C) 16 D) 20 E) 25