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 ray of light is incident along the line $x = 2y$ and hits a mirror. The angle of incidence equals the angle of reflection. Find the equation of the reflected ray passing through the point $(2, 1)$.
(A) $4x - 3y = 5$ (B) $3x - 4y = 2$ (C) $x - y = 1$ (D) $2x - y = 3$

Let $L$ be the line $y = 2 x$, in the two dimensional plane. Statement 1: The image of the point $( 0,1 )$ in $L$ is the point $\left( \frac { 4 } { 5 } , \frac { 3 } { 5 } \right)$. Statement 2: The points $( 0,1 )$ and $\left( \frac { 4 } { 5 } , \frac { 3 } { 5 } \right)$ lie on opposite sides of the line $L$ and are at equal distance from it.
(1) Statement 1 is true, Statement 2 is false.
(2) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.
(3) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.
(4) Statement 1 is false, Statement 2 is true.

A light ray emerging from the point source placed at $\mathrm { P } ( 1,3 )$ is reflected at a point Q in the axis of $x$. If the reflected ray passes through the point $R ( 6,7 )$, then the abscissa of $Q$ is:
(1) 1
(2) 3
(3) $\frac { 7 } { 2 }$
(4) $\frac { 5 } { 2 }$

If the image of point $\mathrm { P } ( 2,3 )$ in a line L is $\mathrm { Q } ( 4,5 )$, then the image of point $\mathrm { R } ( 0,0 )$ in the same line is:
(1) $( 2,2 )$
(2) $( 4,5 )$
(3) $( 3,4 )$
(4) $( 7,7 )$

A ray of light along $x + \sqrt{3}y = \sqrt{3}$ gets reflected upon reaching $X$-axis, the equation of the reflected ray is
(1) $y = \sqrt{3}x - \sqrt{3}$
(2) $\sqrt{3}y = x - 1$
(3) $y = x + \sqrt{3}$
(4) $\sqrt{3}y = x - \sqrt{3}$

A ray of light is incident along a line which meets another line $7 x - y + 1 = 0$ at the point $( 0,1 )$. The ray is then reflected from this point along the line $y + 2 x = 1$. Then the equation of the line of incidence of the ray of light is :
(1) $41 x - 25 y + 25 = 0$
(2) $41 x + 25 y - 25 = 0$
(3) $41 x - 38 y + 38 = 0$
(4) $41 x + 38 y - 38 = 0$

A ray of light coming from the point $( 2,2 \sqrt { 3 } )$ is incident at an angle $30 ^ { \circ }$ on the line $x = 1$ at the point $A$. The ray gets reflected on the line $x = 1$ and meets $x$-axis at the point $B$. Then, the line $A B$ passes through the point
(1) $\left( 3 , - \frac { 1 } { \sqrt { 3 } } \right)$
(2) $\left( 4 , - \frac { \sqrt { 3 } } { 2 } \right)$
(3) $( 3 , - \sqrt { 3 } )$
(4) $( 4 , - \sqrt { 3 } )$

Let $L$ denote the line in the $xy$-plane with $x$ and $y$ intercepts as 3 and 1 respectively. Then the image of the point $(-1,-4)$ in the line is:
(1) $\left(\frac{11}{5},\frac{28}{5}\right)$
(2) $\left(\frac{29}{5},\frac{8}{5}\right)$
(3) $\left(\frac{8}{5},\frac{29}{5}\right)$
(4) $\left(\frac{29}{5},\frac{11}{5}\right)$

A light ray emits from the origin making an angle $30 ^ { \circ }$ with the positive $x$-axis. After getting reflected by the line $\mathrm { x } + \mathrm { y } = 1$, if this ray intersects x-axis at Q , then the abscissa of Q is
(1) $\frac { 2 } { ( \sqrt { 3 } - 1 ) }$
(2) $\frac { 2 } { 3 + \sqrt { 3 } }$
(3) $\frac { 2 } { 3 - \sqrt { 3 } }$
(4) $\frac { \sqrt { 3 } } { 2 ( \sqrt { 3 } + 1 ) }$

Let a ray of light passing through the point $( 3,10 )$ reflects on the line $2 x + y = 6$ and the reflected ray passes through the point $( 7,2 )$. If the equation of the incident ray is $a x + b y + 1 = 0$, then $a ^ { 2 } + b ^ { 2 } + 3 a b$ is equal to $\_\_\_\_$

Let the triangle PQR be the image of the triangle with vertices $( 1,3 ) , ( 3,1 )$ and $( 2,4 )$ in the line $x + 2 y = 2$. If the centroid of $\triangle \mathrm { PQR }$ is the point $( \alpha , \beta )$, then $15 ( \alpha - \beta )$ is equal to:
(1) 19
(2) 24
(3) 21
(4) 22

Let $ABC$ be a triangle formed by the lines $7x - 6y + 3 = 0$, $x + 2y - 31 = 0$ and $9x - 2y - 19 = 0$. Let the point $(h, k)$ be the image of the centroid of $\triangle ABC$ in the line $3x + 6y - 53 = 0$. Then $h^2 + k^2 + hk$ is equal to:
(1) 47
(2) 37
(3) 36
(4) 40

In the rectangular coordinate plane, the symmetric point of $( 4,4 )$ with respect to a line passing through $( 1,0 )$ is $( a , 0 )$. Accordingly, what is the product of the values that $a$ can take?
A) $-24$
B) $-16$
C) $-8$
D) $16$
E) $32$

In a rectangular coordinate plane, point $A(a, b)$; its reflection with respect to point $B(3, 0)$ is point $C$, and its reflection with respect to the $y$-axis is point $D$.
Given that the equation of the line passing through points $C$ and $D$ is $y = -x - 1$, what is the sum $a + b$?
A) 7 B) 13 C) 15 D) 19 E) 24