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

Suppose $ABCD$ is a quadrilateral such that $\angle BAC = 50^\circ, \angle CAD = 60^\circ, \angle CBD = 30^\circ$ and $\angle BDC = 25^\circ$. If $E$ is the point of intersection of $AC$ and $BD$, then the value of $\angle AEB$ is
(A) $75^\circ$
(B) $85^\circ$
(C) $95^\circ$
(D) $110^\circ$

Let $P = ( - 1,0 ) , Q = ( 0,0 )$ and $R = ( 3,3 \sqrt { 3 } )$ be three points. The equation of the bisector of the angle PQR
(1) $\sqrt { 3 } x + y = 0$
(2) $x + \frac { \sqrt { 3 } } { 2 } y = 0$
(3) $\frac { \sqrt { 3 } } { 2 } x + y = 0$
(4) $x + \sqrt { 3 } y = 0$

If one of the lines of $m y ^ { 2 } + \left( 1 - m ^ { 2 } \right) x y - m x ^ { 2 } = 0$ is a bisector of the angle between the lines $x y = 0$, then $m$ is
(1) $- 1 / 2$
(2) - 2
(3) 1
(4) 2

Consider the straight lines $$\begin{aligned} & L _ { 1 } : x - y = 1 \\ & L _ { 2 } : x + y = 1 \\ & L _ { 3 } : 2 x + 2 y = 5 \\ & L _ { 4 } : 2 x - 2 y = 7 \end{aligned}$$ The correct statement is
(1) $L _ { 1 } \left\| L _ { 4 } , L _ { 2 } \right\| L _ { 3 } , L _ { 1 }$ intersect $L _ { 4 }$.
(2) $L _ { 1 } \perp L _ { 2 } , L _ { 1 } \| L _ { 3 } , L _ { 1 }$ intersect $L _ { 2 }$.
(3) $L _ { 1 } \perp L _ { 2 } , L _ { 2 } \| L _ { 3 } , L _ { 1 }$ intersect $L _ { 4 }$.
(4) $L _ { 1 } \perp L _ { 2 } , L _ { 1 } \perp L _ { 3 } , L _ { 2 }$ intersect $L _ { 4 }$.

Let $\theta _ { 1 }$ be the angle between two lines $2 x + 3 y + c _ { 1 } = 0$ and $- x + 5 y + c _ { 2 } = 0$ and $\theta _ { 2 }$ be the angle between two lines $2 x + 3 y + c _ { 1 } = 0$ and $- x + 5 y + c _ { 3 } = 0$, where $c _ { 1 } , c _ { 2 } , c _ { 3 }$ are any real numbers: Statement-1: If $c _ { 2 }$ and $c _ { 3 }$ are proportional, then $\theta _ { 1 } = \theta _ { 2 }$. Statement-2: $\theta _ { 1 } = \theta _ { 2 }$ for all $c _ { 2 }$ and $c _ { 3 }$.
(1) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation of Statement-1.
(2) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation of Statement-1.
(3) Statement-1 is false; Statement-2 is true.
(4) Statement-1 is true; Statement-2 is false.

Suppose that the points $( h , k )$, $( 1,2 )$ and $( - 3,4 )$ lie on the line $L _ { 1 }$. If a line $L _ { 2 }$ passing through the points $( h , k )$ and $( 4,3 )$ is perpendicular to $L _ { 1 }$, then $\frac { k } { h }$ equals:
(1) $- \frac { 1 } { 7 }$
(2) 3
(3) 0
(4) $\frac { 1 } { 3 }$

Slope of a line passing through $P ( 2,3 )$ and intersecting the line $x + y = 7$ at a distance of 4 units from $P$, is
(1) $\frac { \sqrt { 7 } - 1 } { \sqrt { 7 } + 1 }$
(2) $\frac { 1 - \sqrt { 7 } } { 1 + \sqrt { 7 } }$
(3) $\frac { \sqrt { 5 } - 1 } { \sqrt { 5 } + 1 }$
(4) $\frac { 1 - \sqrt { 5 } } { 1 + \sqrt { 5 } }$

The equation of one of the straight lines which passes through the point $(1, 3)$ and makes an angle $\tan ^ { - 1 } ( \sqrt { 2 } )$ with the straight line, $y + 1 = 3 \sqrt { 2 } x$ is
(1) $4 \sqrt { 2 } x + 5 y - ( 15 + 4 \sqrt { 2 } ) = 0$
(2) $5 \sqrt { 2 } x + 4 y - ( 15 + 4 \sqrt { 2 } ) = 0$
(3) $4 \sqrt { 2 } x + 5 y - 4 \sqrt { 2 } = 0$
(4) $4 \sqrt { 2 } x - 5 y - ( 5 + 4 \sqrt { 2 } ) = 0$

Let the equation of the pair of lines, $y = p x$ and $y = q x$, can be written as $( y - p x ) ( y - q x ) = 0$. Then the equation of the pair of the angle bisectors of the lines $x ^ { 2 } - 4 x y - 5 y ^ { 2 } = 0$ is:
(1) $x ^ { 2 } - 3 x y + y ^ { 2 } = 0$
(2) $x ^ { 2 } + 4 x y - y ^ { 2 } = 0$
(3) $x ^ { 2 } + 3 x y - y ^ { 2 } = 0$
(4) $x ^ { 2 } - 3 x y - y ^ { 2 } = 0$

The distance between the two points $A$ and $A ^ { \prime }$ which lie on $y = 2$ such that both the line segments $A B$ and $A ^ { \prime } B$ (where $B$ is the point $( 2,3 )$ ) subtend angle $\frac { \pi } { 4 }$ at the origin, is equal to
(1) 10
(2) $\frac { 48 } { 5 }$
(3) $\frac { 52 } { 5 }$
(4) 3

The portion of the line $4 x + 5 y = 20$ in the first quadrant is trisected by the lines $\mathrm { L } _ { 1 }$ and $\mathrm { L } _ { 2 }$ passing through the origin. The tangent of an angle between the lines $L _ { 1 }$ and $L _ { 2 }$ is:
(1) $\frac { 8 } { 5 }$
(2) $\frac { 25 } { 41 }$
(3) $\frac { 2 } { 5 }$
(4) $\frac { 30 } { 41 }$

In a $\triangle \mathrm { ABC }$, suppose $\mathrm { y } = \mathrm { x }$ is the equation of the bisector of the angle $B$ and the equation of the side $A C$ is $2 x - y = 2$. If $2 A B = B C$ and the point $A$ and $B$ are respectively $( 4,6 )$ and $( \alpha , \beta )$, then $\alpha + 2 \beta$ is equal to
(1) - 4
(2) 42
(3) 2
(4) - 1

If the line segment joining the points $( 5,2 )$ and $( 2 , a )$ subtends an angle $\frac { \pi } { 4 }$ at the origin, then the absolute value of the product of all possible values of $a$ is : (1) 6 (2) 8 (3) 2 (4) - 4

Two equal sides of an isosceles triangle are along $- x + 2 y = 4$ and $x + y = 4$. If m is the slope of its third side, then the sum, of all possible distinct values of $m$, is :
(1) $- 2 \sqrt { 10 }$
(2) 12
(3) 6
(4) $-6$

Two lines $L _ { 1 } , L _ { 2 }$ on the coordinate plane both have positive slopes, and the angle bisector of one of the angles formed by $L _ { 1 } , L _ { 2 }$ has slope $\frac { 11 } { 9 }$ . Another line $L$ passes through the point $( 2 , \frac { 1 } { 3 } )$ and forms a bounded region with $L _ { 1 } , L _ { 2 }$ that is an equilateral triangle. Which of the following options is the equation of $L$?
(1) $11 x - 9 y = 19$
(2) $9 x + 11 y = 25$
(3) $11 x + 9 y = 25$
(4) $27 x - 33 y = 43$
(5) $27 x + 33 y = 65$

In the right coordinate plane shown in the figure, lines d and e are perpendicular to each other.
Accordingly, what is the abscissa of the point where line d intersects the x-axis?
A) $\frac { 9 } { 2 }$
B) $\frac { 11 } { 2 }$
C) $\frac { 13 } { 3 }$
D) $\frac { 14 } { 3 }$
E) $\frac { 25 } { 6 }$

In the rectangular coordinate plane, the line $2x + y = 12$ and a line d intersect at point $\mathrm{A}(4,4)$. These two lines divide every circle centered at point $\mathrm{A}(4,4)$ into four equal areas.
Accordingly, which of the following is the equation of line d?
A) $-2x + y = -4$ B) $x - 3y = -8$ C) $3x + y = 16$ D) $x + 2y = 12$ E) $x - 2y = -4$