Question 156
Um estudante realizou um experimento e obteve os seguintes dados:

$x$	$y$
1	3
2	5
3	7
4	9

A função que melhor representa a relação entre $x$ e $y$ é
(A) $y = x + 2$ (B) $y = 2x + 1$ (C) $y = 3x$ (D) $y = x^2 + 2$ (E) $y = 2x^2 - 1$

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) $y = x + 7$

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) $y = x + 3$

The equation of the line passing through the points $(0, 3)$ and $(2, 7)$ is:
(A) $y = x + 3$
(B) $y = 2x + 3$
(C) $y = 3x + 1$
(D) $y = 2x + 1$
(E) $y = x + 5$

In a digital game, there are three characters: one hero and two villains. The programming is done in such a way that the hero will always be attacked by the villain closest to him. One way to ``confuse'' the villains is to move the hero along trajectories that keep him equidistant from the villains, creating uncertainty between them, and thus preventing him from being attacked.
For the programming of one of the stages of this game, the programmer considered, in the Cartesian plane, the square STUV as the region of movement of the characters, where V and $T$ represent the fixed positions of the villains, and $S$, the initial position of the hero, as shown in the figure.
What is the equation of the trajectory along which the hero can move without being attacked?
(A) $y = -3x + 20$
(B) $y = -3x + 16$
(C) $y = -3x - 20$
(D) $y = 3x + 16$
(E) $y = 3x - 16$

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]

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. $\left\{ \begin{array} { l } x = 1 + 4 t \\ y = 1 + 3 t \end{array} \right.$

Draw a graph of $\Delta\left(0, \vec{e}_1\right)$ and $\Delta\left(2, \frac{\vec{e}_1 + \vec{e}_2}{\sqrt{2}}\right)$.

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 $\mathbb{R}$.

The equation $x ^ { 3 } y + x y ^ { 3 } + x y = 0$ represents
(a) a circle
(b) a circle and a pair of straight lines
(c) a rectangular hyperbola
(d) a pair of straight lines.

The equation $x ^ { 3 } y + x y ^ { 3 } + x y = 0$ represents
(a) a circle
(b) a circle and a pair of straight lines
(c) a rectangular hyperbola
(d) a pair of straight lines.

The equation $x^3 y + xy^3 + xy = 0$ represents
(A) a circle
(B) a circle and a pair of straight lines
(C) a rectangular hyperbola
(D) a pair of straight lines

The equation $x ^ { 3 } y + x y ^ { 3 } + x y = 0$ represents
(A) a circle
(B) a circle and a pair of straight lines
(C) a rectangular hyperbola
(D) a pair of straight lines

A circle $C$ of radius 1 is inscribed in an equilateral triangle $P Q R$. The points of contact of $C$ with the sides $P Q , Q R , R P$ are $D , E , F$, respectively. The line $P Q$ is given by the equation $\sqrt { 3 } x + y - 6 = 0$ and the point $D$ is $\left( \frac { 3 \sqrt { 3 } } { 2 } , \frac { 3 } { 2 } \right)$. Further, it is given that the origin and the centre of $C$ are on the same side of the line $P Q$.
Equations of the sides $Q R , R P$ are
(A) $y = \frac { 2 } { \sqrt { 3 } } x + 1 , y = - \frac { 2 } { \sqrt { 3 } } x - 1$
(B) $y = \frac { 1 } { \sqrt { 3 } } x , y = 0$
(C) $y = \frac { \sqrt { 3 } } { 2 } x + 1 , y = - \frac { \sqrt { 3 } } { 2 } x - 1$
(D) $y = \sqrt { 3 } x , y = 0$

Let $P \left( x _ { 1 } , y _ { 1 } \right)$ and $Q \left( x _ { 2 } , y _ { 2 } \right)$ be two distinct points on the ellipse
$$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$$
such that $y _ { 1 } > 0$, and $y _ { 2 } > 0$. Let $C$ denote the circle $x ^ { 2 } + y ^ { 2 } = 9$, and $M$ be the point $( 3,0 )$.
Suppose the line $x = x _ { 1 }$ intersects $C$ at $R$, and the line $x = x _ { 2 }$ intersects C at $S$, such that the $y$-coordinates of $R$ and $S$ are positive. Let $\angle R O M = \frac { \pi } { 6 }$ and $\angle S O M = \frac { \pi } { 3 }$, where $O$ denotes the origin $( 0,0 )$. Let $| X Y |$ denote the length of the line segment $X Y$.
Then which of the following statements is (are) TRUE?

(A)	The equation of the line joining $P$ and $Q$ is $2 x + 3 y = 3 ( 1 + \sqrt { 3 } )$
(B)	The equation of the line joining $P$ and $Q$ is $2 x + y = 3 ( 1 + \sqrt { 3 } )$
(C)	If $N _ { 2 } = \left( x _ { 2 } , 0 \right)$, then $3 \left| N _ { 2 } Q \right| = 2 \left| N _ { 2 } S \right|$
(D)	If $N _ { 1 } = \left( x _ { 1 } , 0 \right)$, then $9 \left| N _ { 1 } P \right| = 4 \left| N _ { 1 } R \right|$

Let $P S$ be the median of the triangle with vertices $P ( 2,2 ) , Q ( 6 , - 1 )$ and $R ( 7,3 )$. The equation of the line passing through $( 1 , - 1 )$ and parallel to $P S$ is
(1) $4 x + 7 y + 3 = 0$
(2) $2 x - 9 y - 11 = 0$
(3) $4 x - 7 y - 11 = 0$
(4) $2 x + 9 y + 7 = 0$

If a straight line passing through the point $P ( - 3,4 )$ is such that its intercepted portion between the coordinate axes is bisected at $P$, then its equation is :
(1) $4 x + 3 y = 0$
(2) $4 x - 3 y + 24 = 0$
(3) $3 x - 4 y + 25 = 0$
(4) $x - y + 7 = 0$

If the system of linear equations $$\begin{aligned} & x - 2 y + k z = 1 \\ & 2 x + y + z = 2 \\ & 3 x - y - k z = 3 \end{aligned}$$ has a solution $( x , y , z )$, $z \neq 0$, then $( x , y )$ lies on the straight line whose equation is:
(1) $4 x - 3 y - 4 = 0$
(2) $3 x - 4 y - 4 = 0$
(3) $3 x - 4 y - 1 = 0$
(4) $4 x - 3 y - 1 = 0$

If the perpendicular bisector of the line segment joining the points $P ( 1,4 )$ and $Q ( k , 3 )$ has $y$-intercept equal to $-4$, then a value of $k$ is:
(1) $-2$
(2) $-4$
(3) $\sqrt { 14 }$
(4) $\sqrt { 15 }$

A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate axes is $\frac { 1 } { 4 }$. Three stones $A , B$ and $C$ are placed at the points $1,1,2,2$ and $4,4$ respectively. Then which of these stones is / are on the path of the man?
(1) $C$ only
(2) All the three
(3) $B$ only
(4) $A$ only

The number of integral values of $m$ so that the abscissa of point of intersection of lines $3 x + 4 y = 9$ and $y = m x + 1$ is also an integer, is:
(1) 1
(2) 2
(3) 3
(4) 0

A line, with the slope greater than one, passes through the point $A(4,3)$ and intersects the line $x - y - 2 = 0$ at the point $B$. If the length of the line segment $AB$ is $\frac { \sqrt { 29 } } { 3 }$, then $B$ also lies on the line
(1) $2x + y = 9$
(2) $3x - 2y = 7$
(3) $x + 2y = 6$
(4) $2x - 3y = 3$

A line passing through the point $A ( 9,0 )$ makes an angle of $30 ^ { \circ }$ with the positive direction of $x$-axis. If this line is rotated about $A$ through an angle of $15 ^ { \circ }$ in the clockwise direction, then its equation in the new position is
(1) $\frac { y } { \sqrt { 3 } - 2 } + x = 9$
(2) $\frac { x } { \sqrt { 3 } - 2 } + y = 9$
(3) $\frac { x } { \sqrt { 3 } + 2 } + y = 9$
(4) $\frac { y } { \sqrt { 3 } + 2 } + x = 9$

The equations of two sides AB and AC of a triangle ABC are $4 x + y = 14$ and $3 x - 2 y = 5$, respectively. The point $\left( 2 , - \frac { 4 } { 3 } \right)$ divides the third side BC internally in the ratio $2 : 1$. the equation of the side BC is
(1) $x + 3 y + 2 = 0$
(2) $x - 6 y - 10 = 0$
(3) $x - 3 y - 6 = 0$
(4) $x + 6 y + 6 = 0$