Let $A$ be the fixed point $(0,4)$ and $B$ be a moving point $(2t, 0)$. Let $M$ be the mid-point of $A B$ and let the perpendicular bisector of $A B$ meet the $y$-axis at $R$. The locus of the mid-point $P$ of $M R$ is
(A) $y + x ^ { 2 } = 2$
(B) $x ^ { 2 } + ( y - 2 ) ^ { 2 } = 1 / 4$
(C) $( y - 2 ) ^ { 2 } - x ^ { 2 } = 1 / 4$
(D) none of the above

Let $L$ be the point $(t, 2)$ and $M$ be a point on the $y$-axis such that $LM$ has slope $-t$. Then the locus of the midpoint of $LM$, as $t$ varies over all real values, is
(A) $y = 2 + 2x^2$
(B) $y = 1 + x^2$
(C) $y = 2 - 2x^2$
(D) $y = 1 - x^2$

Let $L$ be the point $(t, 2)$ and $M$ be a point on the $y$-axis such that $L M$ has slope $-t$. Then the locus of the midpoint of $L M$, as $t$ varies over all real values, is
(A) $y = 2 + 2 x ^ { 2 }$
(B) $y = 1 + x ^ { 2 }$
(C) $y = 2 - 2 x ^ { 2 }$
(D) $y = 1 - x ^ { 2 }$

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 unit square has its corners chopped off to form a regular polygon with eight sides. What is the area of this polygon?
(A) $2(\sqrt{3} - \sqrt{2})$
(B) $2\sqrt{2} - 2$
(C) $\frac{\sqrt{2}}{2}$
(D) $\frac{7}{9}$.

Consider a triangle $ABC$. The sides $AB$ and $AC$ are extended to points $D$ and $E$, respectively, such that $AD = 3AB$ and $AE = 3AC$. Then one diagonal of $BDEC$ divides the other diagonal in the ratio
(A) $1 : 3$
(B) $1 : \sqrt{3}$
(C) $1 : 2$
(D) $1 : \sqrt{2}$.

A moving line intersects the lines $x + y = 0$ and $x - y = 0$ at the points $A$ and $B$ such that the area of the triangle with vertices $(0,0)$, $A$ and $B$ has a constant area $C$. The locus of the midpoint of $AB$ is given by the equation
(A) $\left(x^2 + y^2\right)^2 = C^2$
(B) $\left(x^2 - y^2\right)^2 = C^2$
(C) $(x + y)^2 = C^2$
(D) $(x - y)^2 = C^2$.

Two persons, both of height $h$, are standing at a distance of $h$ from each other. The shadow of one person cast by a vertical lamp-post placed between the two persons is double the length of the shadow of the other. If the sum of the lengths of the shadows is $h$, then the height of the lamp post is
(A) $\frac{\sqrt{3}}{2}h$
(B) $2h$
(C) $\left(\frac{1 + \sqrt{2}}{2}\right)h$
(D) $\left(\frac{\sqrt{3} + 1}{2\sqrt{2}}\right)h$.

The sides of a regular hexagon $A B C D E F$ are extended by doubling them (for example, $B A$ extends to $B A ^ { \prime }$ with $B A ^ { \prime } = 2 B A$) to form a bigger regular hexagon $A ^ { \prime } B ^ { \prime } C ^ { \prime } D ^ { \prime } E ^ { \prime } F ^ { \prime }$. Then, the ratio of the areas of the bigger to the smaller hexagon is:
(A) 2
(B) 3
(C) $2 \sqrt { 3 }$
(D) $\pi$.

Between 12 noon and 1 PM, there are two instants when the hour hand and the minute hand of a clock are at right angles. The difference in minutes between these two instants is:
(A) $32 \frac { 8 } { 11 }$
(B) $30 \frac { 8 } { 11 }$
(C) $32 \frac { 5 } { 11 }$
(D) $30 \frac { 5 } { 11 }$.

Let $A$ and $B$ be variable points on $x$-axis and $y$-axis respectively such that the line segment $AB$ is in the first quadrant and of a fixed length $2d$. Let $C$ be the mid-point of $AB$ and $P$ be a point such that
(a) $P$ and the origin are on the opposite sides of $AB$ and,
(b) $PC$ is a line segment of length $d$ which is perpendicular to $AB$.
Find the locus of $P$.

Let $A , B , C$ be finite subsets of the plane such that $A \cap B , B \cap C$ and $C \cap A$ are all empty. Let $S = A \cup B \cup C$. Assume that no three points of $S$ are collinear and also assume that each of $A , B$ and $C$ has at least 3 points. Which of the following statements is always true?
(A) There exists a triangle having a vertex from each of $A , B , C$ that does not contain any point of $S$ in its interior.
(B) Any triangle having a vertex from each of $A , B , C$ must contain a point of $S$ in its interior.
(C) There exists a triangle having a vertex from each of $A , B , C$ that contains all the remaining points of $S$ in its interior.
(D) There exist 2 triangles, both having a vertex from each of $A , B , C$ such that the two triangles do not intersect.

Consider the parabola $C: y^2 = 4x$ and the straight line $L: y = x + 2$. Let $P$ be a variable point on $L$. Draw the two tangents from $P$ to $C$ and let $Q_1$ and $Q_2$ denote the two points of contact on $C$. Let $Q$ be the mid-point of the line segment joining $Q_1$ and $Q_2$. Find the locus of $Q$ as $P$ moves along $L$.

The sides of a regular hexagon $A B C D E F$ is extended by doubling them to form a bigger hexagon $A ^ { \prime } B ^ { \prime } C ^ { \prime } D ^ { \prime } E ^ { \prime } F ^ { \prime }$ as in the figure below. Then the ratio of the areas of the bigger to the smaller hexagon is:
(A) $\sqrt { 3 }$
(B) 3
(C) $2 \sqrt { 3 }$
(D) 4

A straight road has walls on both sides of height 8 feet and 4 feet respectively. Two ladders are placed from the top of one wall to the foot of the other as in the figure below. What is the height (in feet) of the maximum clearance $x$ below the ladders?
(A) 3
(B) $2 \sqrt { 2 }$
(C) $\frac { 8 } { 3 }$
(D) $2 \sqrt { 3 }$

In the figure below, $A B C D$ is a square and $\triangle C E F$ is a triangle with given sides inscribed as in the figure. Find the length $B E$.
(A) $\frac { 13 } { \sqrt { 17 } }$
(B) $\frac { 14 } { \sqrt { 17 } }$
(C) $\frac { 15 } { \sqrt { 17 } }$
(D) $\frac { 16 } { \sqrt { 17 } }$

Two ships are approaching a port along straight routes at constant velocities. Initially, the two ships and the port formed an equilateral triangle. After the second ship travelled 80 km, the triangle became right-angled. When the first ship reaches the port, the second ship was still 120 km from the port. Find the initial distance of the ships from the port.
(A) 240 km
(B) 300 km
(C) 360 km
(D) 180 km

In the following diagram, four triangles and their sides are given. Areas of three of them are also given. Find the area $x$ of the remaining triangle. The four triangles have areas 4, 5, $x$, and 13 respectively.
(A) 12
(B) 13
(C) 14
(D) 15

In a triangle $A B C$, consider points $D$ and $E$ on $A C$ and $A B$, respectively, and assume that they do not coincide with any of the vertices $A , B , C$. If the segments $B D$ and $C E$ intersect at $F$, consider the areas $w , x , y , z$ of the quadrilateral $A E F D$ and the triangles $B E F , B F C , C D F$, respectively.
(a) Prove that $y ^ { 2 } > x z$.
(b) Determine $w$ in terms of $x , y , z$.

The straight line $O A$ lies in the second quadrant of the $(x, y)$-plane and makes an angle $\theta$ with the negative half of the $x$-axis, where $0 < \theta < \frac { \pi } { 2 }$. The line segment $C D$ of length 1 slides on the $(x, y)$-plane in such a way that $C$ is always on $O A$ and $D$ on the positive side of the $x$-axis. The locus of the mid-point of $C D$ is
(A) $x ^ { 2 } + 4 x y \cot \theta + y ^ { 2 } \left( 1 + 4 \cot ^ { 2 } \theta \right) = \frac { 1 } { 4 }$.
(B) $x ^ { 2 } + y ^ { 2 } = \frac { 1 } { 4 } + \cot ^ { 2 } \theta$.
(C) $x ^ { 2 } + 4 x y \cot \theta + y ^ { 2 } = \frac { 1 } { 4 }$.
(D) $x ^ { 2 } + y ^ { 2 } \left( 1 + 4 \cot ^ { 2 } \theta \right) = \frac { 1 } { 4 }$.

Let $K$ be the set of all points $(x , y)$ such that $| x | + | y | \leq 1$. Given a point $A$ in the plane, let $F _ { A }$ be the point in $K$ which is closest to $A$. Then the points $A$ for which $F _ { A } = ( 1,0 )$ are
(A) all points $A = ( x , y )$ with $x \geq 1$.
(B) all points $A = ( x , y )$ with $x \geq y + 1$ and $x \geq 1 - y$.
(C) all points $A = ( x , y )$ with $x \geq 1$ and $y = 0$.
(D) all points $A = ( x , y )$ with $x \geq 0$ and $y = 0$.

45. Let $O ( 0,0 ) , P ( 3,4 ) , Q ( 6,0 )$ be the vertices of the triangle $O P Q$. The point $R$ inside the triangle $O P Q$ is such that the triangles $O P R , P Q R , O Q R$ are of equal area. The coordinates of $R$ are
(A) $\left( \frac { 4 } { 3 } , 3 \right)$
(B) $\left( 3 , \frac { 2 } { 3 } \right)$
(C) $\left( 3 , \frac { 4 } { 3 } \right)$
(D) $\left( \frac { 4 } { 3 } , \frac { 2 } { 3 } \right)$
Answer ◯ [Figure] [Figure] [Figure]
(A)
(B)
(C)
(D)

Let $O(0,0)$, $P(3,4)$, $Q(6,0)$ be the vertices of the triangle $OPQ$. The point $R$ inside the triangle $OPQ$ is such that the triangles $OPR$, $PQR$, $OQR$ are of equal area. The coordinates of $R$ are
(A) $\left(\frac{4}{3}, 3\right)$
(B) $(3, \frac{2}{3})$
(C) $(3, \frac{4}{3})$
(D) $\left(\frac{4}{3}, \frac{2}{3}\right)$

Consider three points $P = ( - \sin ( \beta - \alpha ) , - \cos \beta ) , Q = ( \cos ( \beta - \alpha ) , \sin \beta )$ and $R = ( \cos ( \beta - \alpha + \theta ) , \sin ( \beta - \theta ) )$, where $0 < \alpha , \beta , \theta < \frac { \pi } { 4 }$. Then,
(A) $P$ lies on the line segment $R Q$
(B) $Q$ lies on the line segment $P R$
(C) $R$ lies on the line segment $Q P$
(D) $P , Q , R$ are non-collinear