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) $y - 2x = \Lambda$

136- In rectangle $ABCD$, point $(5,3)$ is vertex $B$ and the lengths of sides $C$ and $D$ are $5/8$ and $3$ respectively. If vertex $D$ is reflected over the $x$-axis, the distance from the image of vertex $C$ to the line $BD$ from the origin of coordinates is how much?
\[ \text{(1)}\ 2/5 \qquad \text{(2)}\ \sqrt{6/5} \qquad \text{(3)}\ \sqrt{6} \qquad \text{(4)}\ 2 \]

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

Q83. 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 $\_\_\_\_$

Q65. A ray of light coming from the point $P ( 1,2 )$ gets reflected from the point $Q$ on the $x$-axis and then passes through the point $R ( 4,3 )$. If the point $S ( h , k )$ is such that PQRS is a parallelogram, then $h k ^ { 2 }$ is equal to :
(1) 70
(2) 80
(3) 60
(4) 90

The image of the parabola $\mathrm { x } ^ { 2 } = 4 \mathrm { y }$ in the line $\mathrm { x } - \mathrm { y } = 1$ is
(A) $( y - 1 ) ^ { 2 } = 4 ( x + 1 )$
(B) $( y + 1 ) ^ { 2 } = 4 ( x + 1 )$
(C) $( y + 1 ) ^ { 2 } = 4 ( x - 1 )$
(D) $( y - 1 ) ^ { 2 } = 4 ( x - 1 )$

4. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \end{array} \right\}$ ONLY.

Mathematics \& Computer Science and Computer Science applicants should turn to page 14.
Let $P$ and $Q$ be the points with co-ordinates $( 7,1 )$ and $( 11,2 )$.
(i) The mirror image of the point $P$ in the $x$-axis is the point $R$ with co-ordinates $( 7 , - 1 )$. Mark the points $P , Q$ and $R$ on the grid provided opposite.
(ii) Consider paths from $P$ to $Q$ each of which consists of two straight line segments $P X$ and $X Q$ where $X$ is a point on the $x$-axis. Find the length of the shortest such parth, giving clear reasoning for your answer. (You may refer to the diagram to help your explanation, if you wish.)
(iii) Sketch in the line $\ell$ with equation $y = x$. Find the co-ordinates of $S$, the mirror image in the line $\ell$ of the point $Q$, and mark in the point $S$.
(iv) Consider paths from $P$ to $Q$ each of which consists of three straight line segments $P Y , Y Z$ and $Z Q$, where $Y$ is on the $x$-axis and $Z$ is on the line $\ell$. Find the shortest such path, giving clear reasoning for your answer. [Figure]

4. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \end{array} \right\}$ ONLY.

Mathematics \& Computer Science and Computer Science applicants should turn to page 14.
Let $P$ and $Q$ be the points with co-ordinates $( 7,1 )$ and $( 11,2 )$.
(i) The mirror image of the point $P$ in the $x$-axis is the point $R$ with co-ordinates $( 7 , - 1 )$. Mark the points $P , Q$ and $R$ on the grid provided opposite.
(ii) Consider paths from $P$ to $Q$ each of which consists of two straight line segments $P X$ and $X Q$ where $X$ is a point on the $x$-axis. Find the length of the shortest such parth, giving clear reasoning for your answer. (You may refer to the diagram to help your explanation, if you wish.)
(iii) Sketch in the line $\ell$ with equation $y = x$. Find the co-ordinates of $S$, the mirror image in the line $\ell$ of the point $Q$, and mark in the point $S$.
(iv) Consider paths from $P$ to $Q$ each of which consists of three straight line segments $P Y , Y Z$ and $Z Q$, where $Y$ is on the $x$-axis and $Z$ is on the line $\ell$. Find the shortest such path, giving clear reasoning for your answer. [Figure]

3. Let $P$ and $Q$ be the points with co-ordinates $( 7,1 )$ and $( 11,2 )$.
(a) The mirror image of the point $P$ in the $x$-axis is the point $R$ with coordinates $( 7 , - 1 )$. Mark the points $P , Q$ and $R$ on the grid provided.
(b) Consider paths from $P$ to $Q$ each of which consists of two straight line segments $P X$ and $X Q$ where $X$ is a point on the $x$-axis. Find the length of the shortest such path, giving clear reasoning for your answer. (You may refer to the diagram to help your explanation, if you wish.)
(c) Sketch in the line $\ell$ with equation $y = x$. Find the co-ordinates of $S$, the mirror image in the line $\ell$ of the point $Q$, and mark in the point $S$.
(d) Consider paths from $P$ to $Q$ each of which consists of three straight line segments $P Y , Y Z$ and $Z Q$, where $Y$ is on the $x$-axis and $Z$ is on the line $\ell$. Find the length of the shortest such path, giving clear reasoning for your answer. [Figure]

The curve with equation
$$x = y ^ { 2 } - 6 y + 11$$
is rotated $90 ^ { \circ }$ clockwise about the point $P$ to give the curve $C$. $P$ has $x$-coordinate - 2 and $y$-coordinate 3 . What is the equation of $C$ ?
A $y = - x ^ { 2 } - 4 x - 3$ B $y = - x ^ { 2 } - 4 x - 5$ C $y = - x ^ { 2 } - 6 x - 7$ D $y = - x ^ { 2 } - 6 x - 11$ E $y = x ^ { 2 } - 4 x + 5$ F $y = x ^ { 2 } + 4 x + 3$ G $y = x ^ { 2 } - 6 x + 11$ H $y = x ^ { 2 } + 6 x + 7$

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