Question 158
A figura mostra um retângulo $ABCD$ com $AB = 8$ cm e $BC = 6$ cm. O ponto $E$ é o ponto médio de $AB$.
[Figure]
O comprimento do segmento $CE$, em cm, é
(A) 5 (B) 7 (C) 8 (D) 10 (E) 12

A distância entre os pontos $A = (1, 2)$ e $B = (4, 6)$ no plano cartesiano é
(A) 3 (B) 4 (C) 5 (D) 6 (E) 7

A equação da reta que passa pelos pontos $(0, 3)$ e $(2, 7)$ é
(A) $y = x + 3$ (B) $y = 2x + 3$ (C) $y = 3x + 1$ (D) $y = 2x - 3$ (E) $y = x + 7$

O coeficiente angular da reta $3x - 2y + 6 = 0$ é
(A) $-3$ (B) $-\dfrac{3}{2}$ (C) $\dfrac{3}{2}$ (D) $2$ (E) $3$

QUESTION 154
The equation of a line passing through the points $(1, 2)$ and $(3, 6)$ is
(A) $y = x + 1$
(B) $y = 2x$
(C) $y = 2x + 1$
(D) $y = 3x - 1$
(E) $y = x + 3$

QUESTION 158
The distance between the points $A(1, 2)$ and $B(4, 6)$ is
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

QUESTION 179
The midpoint of the segment with endpoints $A(2, 4)$ and $B(6, 8)$ is
(A) $(3, 5)$
(B) $(4, 6)$
(C) $(5, 7)$
(D) $(6, 8)$
(E) $(8, 12)$

The distance between the points $A = (1, 2)$ and $B = (4, 6)$ is:
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

As shown in the figure, there are two points $\mathrm { A } ( - 1,0 )$ and $\mathrm { P } ( t , t + 1 )$ on the line $y = x + 1$. Let Q be the point where the line passing through P and perpendicular to the line $y = x + 1$ meets the $y$-axis. What is the value of $\lim _ { t \rightarrow \infty } \frac { \overline { \mathrm { AQ } } ^ { 2 } } { \overline { \mathrm { AP } } ^ { 2 } }$? [3 points]
(1) 1
(2) $\frac { 3 } { 2 }$
(3) 2
(4) $\frac { 5 } { 2 }$
(5) 3

On the coordinate plane, a line passes through the point $( 4,1 )$ and is perpendicular to the vector $\vec { n } = ( 1,2 )$. Let the coordinates of the points where this line meets the $x$-axis and $y$-axis be $( a , 0 ) , ( 0 , b )$ respectively. Find the value of $a + b$. [3 points]

In the coordinate plane, consider two lines $$\frac { x + 1 } { 2 } = y - 3 , \quad x - 2 = \frac { y - 5 } { 3 }$$ If $\theta$ is the acute angle between these lines, what is the value of $\cos \theta$? [3 points]
(1) $\frac { 1 } { 2 }$
(2) $\frac { \sqrt { 5 } } { 4 }$
(3) $\frac { \sqrt { 6 } } { 4 }$
(4) $\frac { \sqrt { 7 } } { 4 }$
(5) $\frac { \sqrt { 2 } } { 2 }$

11. Let the area of the closed region bounded by the lines $l _ { 2 } : n x + y - n = 0$, $l _ { 3 } : x + n y - n = 0$ ($n \in \mathbb{N} ^ { * } , n \geq 2$), the $x$-axis, and the $y$-axis be denoted by $S _ { n }$. Then $\lim _ { n \rightarrow \infty } S _ { n } =$ $\_\_\_\_$ $1$.
Analysis: $B \left( \frac { n } { n + 1 } , \frac { n } { n + 1 } \right)$, so $OB \perp AC$, $S _ { n } = \frac { 1 } { 2 } \times \sqrt { 2 } \times \frac { n } { n + 1 } \sqrt { 2 } = \frac { n } { n + 1 }$, so $\lim _ { n \rightarrow \infty } S _ { n } = 1$ [Figure]

5. If line $l$ passes through point $(3,4)$ and $(1,2)$ is a normal vector to it, then the equation of line $l$ is $\_\_\_\_$

17. (This problem is worth 14 points) There are two mutually perpendicular straight-line highways on the periphery of a mountainous area. To further improve the traffic situation in the mountainous area, a plan is made to build a straight-line highway connecting the two highways and the boundary of the mountainous area. Let the two mutually perpendicular highways be $l _ { 1 } , l _ { 2 }$, the boundary curve of the mountainous area be C, and the planned highway be l. As shown in the figure, $\mathrm { M } , \mathrm { N }$ are two endpoints of C. It is measured that the distances from point M to $l _ { 1 } , l _ { 2 }$ are 5 kilometers and 40 kilometers respectively, and the distances from point N to $l _ { 1 } , l _ { 2 }$ are 20 kilometers and 2.5 kilometers respectively. Taking the lines where $l _ { 1 } , l _ { 2 }$ are located as the $\mathrm { x } , \mathrm { y }$ axes respectively, establish a rectangular coordinate system xOy. Assume that the curve C conforms to the function model $y = \frac { a } { x ^ { 2 } + b }$ (where $\mathrm { a } , \mathrm { b }$ are constants). (I) Find the values of $\mathrm { a } , \mathrm { b }$; (II) Let the highway l be tangent to curve C at point P, and the x-coordinate of P is t.
(1) Write out the function expression $f ( t )$ for the length of highway l and its domain;
(2) When t takes what value is the length of highway l shortest? Find the shortest length.

The parametric equation of line $l$ is $\left\{ \begin{array} { l } x = 1 + 3 t , \\ y = 2 + 4 t \end{array} \right.$ ($t$ is the parameter). The distance from point $(1,0)$ to line $l$ is (A) $\frac { 1 } { 5 }$ (B) $\frac { 2 } { 5 }$ (C) $\frac { 4 } { 5 }$ (D) $\frac { 6 } { 5 }$

16. A line $l$ with slope $k ( k < 0 )$ passes through point $F ( 0,1 )$ and intersects the curve $y = \frac { 1 } { 4 } x ^ { 2 } ( x \geq 0 )$ and the line $y = - 1$ at points $A$ and $B$ respectively. If $| F B | = 6 | F A |$, then $k = $ \_\_\_\_.
III. Solution Questions: This section contains 6 questions, totaling 70 points. Show your working, proofs, or calculation steps. Questions 17-21 are required questions that all candidates must answer. Questions 22 and 23 are optional questions; candidates should answer according to the requirements. (I) Required Questions: Total 60 points.

The maximum distance from the point $( 0 , - 1 )$ to the line $y = k ( x + 1 )$ is
A. 1
B. $\sqrt { 2 }$
C. $\sqrt { 3 }$
D. 2

Given that the equation of line $l$ is $3 x - 4 y + 1 = 0$, which of the following is a parametric equation of $l$? ( )
A. $\left\{ \begin{array} { l } x = 4 + 3 t \\ y = 3 - 4 t \end{array} \right.$
B. $\left\{ \begin{array} { l } x = 4 + 3 t \\ y = 3 + 4 t \end{array} \right.$
C. $\left\{ \begin{array} { l } x = 1 - 4 t \\ y = 1 + 3 t \end{array} \right.$
D. $\left\{ \begin{array} { l } x = 1 + 4 t \\ y = 1 + 3 t \end{array} \right.$

Determine the coordinates of $D$ and give the coordinates of the midpoint $M$ of the line segment $[ A C ]$.

Determine a Cartesian equation of $\Delta\left(q, \vec{u}_\theta\right)$.

Show that a parametrization of $\Delta\left(q, \vec{u}_\theta\right)$ is given by $\left\{ \begin{array}{l} x(t) = q\cos\theta - t\sin\theta \\ y(t) = q\sin\theta + t\cos\theta \end{array} \right.$ when $t$ ranges over $\mathbb{R}$.

Under what condition are the lines $\Delta(q, \vec{u})$ and $\Delta(r, \vec{v})$ identical?

Exercise VII

Consider the points $A$ and $B$ with respective coordinates in an orthonormal coordinate system: $$A ( 2 ; 0 ) \text { and } \mathrm { B } ( 0 ; - 4 ) \text {. }$$ VII-A- An equation of the line $( A B )$ is $2 x - y - 4 = 0$. VII-B- An equation of the perpendicular bisector of segment $[ A B ]$ is $x + 2 y + 3 = 0$. VII-C- An equation of the circle with diameter $[ A B ]$ is $x ^ { 2 } + y ^ { 2 } - 2 x - 4 y = 0$. VII-D- The point with coordinates $( - 1 ; - 1 )$ belongs to the circle with diameter $[ A B ]$. VII-E- The line with equation $2 x - y + 1 = 0$ is tangent to the circle with diameter $[ A B ]$.
For each statement, indicate whether it is TRUE or FALSE.

131- The reflection of line $\Delta$ across the line $y = -x$ is line $\Delta'$. The equation of line $\Delta$ is $2y + x = 6$. With respect to line $x = -x$, the equation of line $\Delta'$ is which of the following?
(1) $y + 2x = -6$ (2) $y + 2x = 2$ (3) $y + 2x = -2$ (4) $y - 2x = \Lambda$

109. The distance of point $A$ from the line $x + y = a$ is $a$. The two points $B(-3, 2)$ and $C(-1, 4)$ are on this line, and $5$ is the distance. What is the value of $a$?
(1) $2$ (2) $\dfrac{1}{2}$ (3) $-\dfrac{1}{2}$ (4) $-2$